__________ 

An Empirical Comparison of Methods for Temporal

Disaggregation at the National Accounts 

Baoline Chen 

Office of Directors 

Bureau of Economic Analysis

1441 L St., NW, Washington, DC 20230

Email: Baoline.chen@bea.gov 

August 30, 2007 

Abstract 

This study evaluates five mathematical and five statistical 

methods for temporal disaggregation in an attempt to select the

most suitable method(s) for routine compilation of sub-annual

estimates through temporal distribution and interpolation in the

national accounts at BEA. The evaluation is conducted using 60

series of annual data from the National Economic Accounts, and

the final sub-annual estimates are evaluated according to 

specific criteria to ensure high quality final estimates that are

in compliance with operational policy at the national accounts.

The study covers the cases of temporal disaggregation when 1)

both annual and sub-annual information is available; 2) only

annual data are available; 3) sub-annual estimates have both

temporal and contemporaneous constraints; and 4) annual data 

contain negative values. The estimation results show that the 

modified Denton proportional first difference method outperforms

the other methods, though the Casey-Trager growth preservation

model is a close competitor in certain cases. Lagrange 

polynomial interpolation procedure is inferior to all other

methods. 

I would like to express our great appreciation to Pierre Cholette from

Statistics Canada, David Findley and Brian Monsell from the Census Bureau,

Roberto Barcellan from Eurostats, and Nils Maehle from the MIF for providing us

the software programs and related documentations used in the estimation 

experiment and for their helpful discussions on the subject. I am also very 

grateful to Stephen Morris and Aaron Elrod for the tremendous amount of 

estimation work and tedious statistical analysis they have helped with on the

project. Thanks go to Brent Moulton, Marshall Reinsdorf and Ana Aizcorbe for

their helpful comments. 
1 

1. Introduction 

National accounts face the task of deriving a large number 

of monthly or quarterly estimates using mostly annual data and 

some less reliable or less comprehensive quarterly or monthly 

information from proxy variables known as indicators. The annual 

data are usually detailed and of high precision but not very 

timely; they provide the most reliable information on the overall 

level and long-term movements in the series. The quarterly or 

monthly indicators are less detailed and of lower precision but 

they are timely; they provide the only explicit information about 

the short-term movements in the series. Statistically speaking, 

the process of deriving high frequency data from low frequency 

data and, if it is available, related high frequency information 

can be described as temporal disaggregation. To simplify the 

exposition, we follow the terminology used in the literature to 

refer to the low frequency data as annual data and the high 

frequency data as sub-annual data (Cholette and Dagum, 2006). 

Typically the annual sums of the sub-annual indicator 

series are not consistent with the annual values, which are 

regarded as temporal aggregation constraints or benchmarks, and 

quite often the annual and sub-annual series exhibit inconsistent 

annual movements. The primary objective of temporal 

disaggregation is to obtain sub-annual estimates that preserve as 

much as possible the short-term movements in the indicator series 

under the restriction provided by the annual data which exhibit 

long term movements of the series. Temporal disaggregation 

combines the relative strength of the less frequent annual data 

and the more frequent sub-annual information. 

Temporal disaggregation also covers the cases where subannual information is not available. In addition, it covers the 

cases where a set of sub-annual series are linked by some 

accounting relationship regarded as a contemporaneous aggregation 

constraint. In such cases, temporal disaggregation should 
2 

produce sub-annual estimates that are consistent with both 

temporal and contemporaneous aggregation constraints. 

Temporal disaggregation is closely related to but different 

from benchmarking. They are related in that benchmarking or 

temporal disaggregation arises because of the need to remove 

discrepancies between annual benchmarks and corresponding sums of 

the sub-annual values. They are different because the 

benchmarking problem arises when time series data for the same 

target variable are measured at different frequencies with 

different level of accuracy, whereas temporal disaggregation 

deals with the problem where the sub-annual data are not for the 

same target variable as the annual data. 

There are two facets of the temporal disaggregation 

problem: temporal distribution and interpolation. Temporal 

distribution is needed when annual data are either the sum or 

averages of sub-annual data. In general distribution deals with 

all flow and all index variables. Interpolation, on the other 

hand, deals with estimation of missing values of a stock 

variable, which at points in time have been systematically 

skipped by the observation process. In the estimation of 

quarterly variables, interpolation is used for all stock 

variables whose annual values equal those of the fourth (or the 

first) quarter of the same year. An example is population at the 

end of a year which equals the population at the end of the 

fourth quarter of the same year. Sometimes sub-annual estimates 

need to be made before the relevant annual and/or quarterly 

information is available. In such cases temporal distribution 

and interpolation are combined with extrapolation to produce subannual estimates. 

There is a wide variety of methods for temporal 

distribution, interpolation and extrapolation. Because 

benchmarking and temporal disaggregation deal with similar 

statistical problems, the methods used for temporal 

disaggregation are very similar to those used for benchmarking. 

To a large extent, the choice of methods depends on two factors: 
 

3 

basic information available for estimation, and preference or 

operational criteria suitable for either a mathematical approach. 

There are three general cases regarding the availability of 

information: 1) information is available on both an annual and 

sub-annual basis; 2) information is only available on an annual 

basis; and 3) annual and/or sub-annual information is not yet 

available at the time of estimation. 

If both annual and sub-annual information is available, the 

choices are pure numerical procedures, mathematical benchmarking 

methods, or statistical methods. Examples of pure numerical 

procedures are linear interpolation or distribution procedures, 

and Lagrange interpolation procedure. The most widely used 

mathematical benchmarking methods are the Denton adjustment 

method (Denton, 1971) and its variants (Helfand et al, 1977; 

Cholette, 1979) and the Causey-Trager method (Causey and Trager, 

1981; Bozik and Otto, 1988). The most commonly used statistical 

methods are the Chow-Lin regression methods (1971) and their 

extensions (Bournay and Laroque, 1979; Fernandez, 1981; 

Litterman, 1983); time-series ARIMA models (Hillmer and Trabelsi, 

1987); and generalized regression-based methods (Cholette and 

Dagum, 2006). More recently, time series models such as 

benchmarking based on signal extraction (Durbin and Quenneville, 

1997; Chen, Cholette and Dagum, 1997) and the structural time 

series models in state space representation for interpolation and 

temporal distribution (Harvey and Pierse, 1984; Gudmundson, 1999; 

Proietti, 1999; and Aadland, 2000) have also been developed. 

If, on the other hand, only annual data are available, 

choices are mathematical, numerical smoothing methods, and time 

series ARIMA models. Commonly used smoothing methods are those 

developed by Boot, Feibes, and Lisman (1970), and by Denton 

(1971). A commonly used numerical smoothing procedure is the 

cubic spline. The ARIMA models developed to generate sub-annual 

estimates in the absence of sub-annual information are those by 

Stram and Wei (1986) and Wei and Stram (1990). 
4 

When annual and/or sub-annual information for the current 

year is not yet available at the time of estimation, 

extrapolation is necessary. If only sub-annual indicator values 

are available, they can be used to extrapolate the annual 

aggregate, which can then be used to distribute or interpolate 

the sub-annual estimates. The idea is that the indicator and the 

annual series have the same time profile and, thus, they have the 

same growth rate. However, if no information on the indicator is 

yet available, the sub-annual estimates for the current periods 

can only be derived from the extrapolation of previous sub-annual 

estimates or from interpolation of the extrapolated annual data. 

If the available information is not sufficiently reliable and/or 

not complete, it is then necessary to consider some classical 

extrapolation methods based on previously available sub-annual 

information or based on the methods that use related series. 

Like most government statistical agencies, deriving subannual estimates using only annual data and incomplete sub-annual 

information is a routine practice in the national accounts at 

BEA. The national accounts at BEA have experimented with a 

variety of techniques over the years. Up to the 1970s, the 

Bassie adjustment procedure (Bassie, 1958) was the major method 

used at BEA. Bassie was the first to develop a method for 

constructing sub-annual series whose short-term movements would 

closely reflect those of a related (indicator) series while 

maintaining consistency with annual aggregates. The Bassie 

procedure tends to smooth the series, and hence, it can seriously 

disturb the period-to-period rates of change in sub-annual series 

that exhibit strong short-term variation. Because the Bassie 

method operates on only two consecutive years of data, using the 

Bassie method often results in a step problem if data of several 

years are adjusted simultaneously. Moreover, the Bassie method 

does not support extrapolation. 

Because of the unsatisfactory results from the Bassie 

procedure, the Minimum Constrained Variance Internal Moving 

Average (MCVIM) procedure was introduced to the national accounts 
5 

during the 1980s, and both Bassie and MCVIM were used for 

interpolation and temporal distribution. MCVIM method is based 

on the idea of deriving sub-annual estimates by minimizing the 

variance in the period-to-period changes in the estimated subannual series. However, in the 1990s, the Bassie and MCVIM 

procedures were largely replaced by a purely numerical procedure 

known as the Lagrange polynomial interpolation procedure when BEA 

transferred to the AREMOS time-series database. 

Polynomial interpolation is a method that takes a 

collection of n points of annual data and constructs a polynomial 

function of degree n-1 that passes these annual values. Using 

the Lagrange polynomial interpolation procedure assumes that the 

polynomial function that passes the annual values is a Lagrange 

polynomial; the n points of annual values are called Lagrange 

data. In general, Lagrange polynomial interpolation can be 

considered if only the approximation of level is needed. 

Lagrange polynomial interpolation procedure has some serious 

known drawbacks. First of all, Lagrange polynomial interpolation 

is a global method, and hence, it requires some information about 

the function globally. In practice, the approximated values 

based on a certain assumed degree of polynomial could sharply 

disagree with the actual values of the function due to lack of 

information about the function globally. Secondly, the 

approximation error may be unbounded as the degree of polynomial 

increases. Thirdly, computation of the Lagrange is costly, 

because it requires a large number of evaluations. The Lagrange 

polynomial interpolation formula uses 3n additions, 2n divisions 

and 2n multiplications even when done most efficiently. 

Lagrange polynomial interpolation has not proven 

satisfactory in the national accounts. The major problems 

encountered are inconsistency in the short-term movements shown 

in the indicator and final estimated sub-annual series, the sharp 

increase or decrease at the end of the interpolated series, and 

the jumpy links between the newly interpolated span of a series 

and the previously benchmarked span of the series. An added 
 

 

6 

difficulty arises in practice because Lagrange polynomial 

interpolation in AREMOS uses five annual data points, but under 

the current revision policy at the national accounts, only data 

of the most recent three years can be used in estimation. Thus, 

it requires a forward extrapolation of two annual data points 

before Lagrange polynomial interpolation can be performed. 

Although there is a recommended extrapolation procedure, analysts 

often have to make judgmental decisions on the extrapolation 

procedure to be used when encountering various data problems. 

Sub-annual estimates generated from the Lagrange polynomial 

interpolation procedure in AREMOS are directly affected by the 

choices of extrapolation procedures. 

Because of the difficulties experienced using Lagrange 

polynomial interpolation, alternative methods such as linear 

interpolation, Bassie and MCVIM methods continue to be used in 

various parts of the national accounts. To improve the quality 

of sub-annual estimates in the national accounts, there is strong 

interest in finding an appropriate standardized method for 

temporal disaggregation at the national accounts. Various 

attempts have been made in recent years by researchers at BEA in 

a search for a better method for temporal disaggregation (Klick, 

2000; Loebach 2002). 

The objective of this study is to evaluate a number of 

existing mathematical and statistical methods for temporal 

disaggregation based on certain specific criteria and to 

recommend the most suitable method(s) for practice in the 

national accounts. The evaluation is conducted using 60 annual 

data series from the National Economic Accounts. The study will 

cover the cases where 1) both annual and indicator series are 

available; 2) only annual data are available; 3) annual series 

contain both positive and negative values; and 4) annual series 

have both temporal and contemporaneous constraints. To evaluate 

different methods for temporal disaggregation, three software 

packages were used: ECOTRIM from Eurostats, BENCH from the 

Statistics of Canada, and BMARK from the U.S. Census Bureau. 
7 

Each software package offers some unique features and has 

advantages over the others in certain aspects of computation. 

The plan for the rest of the paper is as follows. Section 

2 presents various mathematical and statistical methods for 

temporal disaggregation. Section 3 discusses the five criteria 

for evaluation. Section 4 presents and discusses the estimation 

results. Section 5 evaluates the methods and software package 

used in the experiment and suggests method(s) and software for 

temporal disaggregation at the national accounts. Section 6 

concludes the paper. 

2. Methods for Temporal Disaggregation 

There is a variety of mathematical and statistical methods 

developed for temporal disaggregation. The distinction between a 

mathematical and a statistical method is that a mathematical 

model treats the process of an unknown sub-annual series as 

deterministic and treats the annual constraints as binding, 

whereas a statistical model treats the process of an unknown subannual series as stochastic and allows the annual constraints to 

be either binding or not binding. The choice of a particular 

method depends to a large extend on the information available for 

estimation and subjective preference or operational criteria. 

Because the objective of this study is to search for the 

most suitable method(s) for interpolation and temporal 

distribution for routine compilation of sub-annual estimates in 

the national accounts, we focus on three mathematical methods: 

the Denton adjustment method (1971) and its variants; the CauseyTrager growth preservation method (Causey and Trager, 1981); and 

the smoothing method (Boot, Feibes and Lisman, 1970). We shall 

also review five extensions of the Chow-Lin regression method 

(Chow and Lin, 1971), which include an AR(1) model estimated by 

applying maximum likelihood (ML) and generalized least squares 

(GLS), the Fernandez random walk model (1981), and the Litterman 

random walk-Markov model (1998) estimated by applying ML and GLS. 
 

 

8 

In this section we shall briefly describe the mathematical 

and statistical methods being reviewed in this study and discuss 

the advantages and disadvantages of these methods. 

2.1 Mathematical Methods for Temporal Disaggregation 

We shall start with the original Denton adjustment method 

(Denton, 1971) and its variants, and then describe the CauseyTrager method (Causey and Trager, 1981). These methods are 

suitable for interpolation or distribution when both annual and 

sub-annual indicator values are available. We shall then discuss 

the Boot, Feibes and Lisman smoothing method for distribution and 

interpolation when only annual values are available. 

To formalize our discussion, we first define some notation. 

Let zt and xt denote the sub-annual indicator series and subannual estimates from distribution or interpolation for subannual period t = 1, 2 ..., T, where T is the total number of subannual periods in the sample. Let ym denote the annual value of 

year m for m = 1, 2 ..., M, where M is the total number of annual 

values in the sample. Let kmt denote the coverage fraction of 

sub-annual estimate of period t, e.g. for quarterly or monthly 

series of index variables, kmt = 1/4 or kmt = 1/12; for flow 

variables, kmt = 1. Also let t1m and t2m denote the first and last 

periods covered by the m-th annual value, e.g. if m = 2, for a 

quarterly series, t = t1m = 5 and t = t2m = 8 respectively. Let ∆

denote backward first difference operator, e.g., ∆xt = xt – xt-1, 

∆zt = zt – zt-1, ∆xt - ∆zt = ∆(xt – zt) = (xt – zt) – (xt-1 – zt-1). 

Let ∆2

 denote the backward second difference operator, e.g., ∆2xt 

- ∆2

zt = ∆2(xt – zt) = ∆(xt – zt) – ∆(xt-1 – zt-1). 

2.1.1 The original Denton adjustment method and its variants 

The original Denton adjustment method is based on the 

principle of movement preservation. According to this principle, 
 

9 

the sub-annual series xt should preserve the movement in the 

indicator series, because the movement in the indicator series is 

the only information available. Formally, we specify a penalty 

function, P(x, z), where x = (x1 ..., xT)’ and z = (z1 ..., zT)’ are 

Tx1 column vectors of final sub-annual and indicator values. The 

mathematical problem of the original Denton adjustment model is 

to choose final sub-annual estimates, x, so as to minimize the 

penalty function P(x, z), subject to the temporal aggregation 

constraints, 

t 

ym = ∑

2m

kmtx t , t=t1m 

for m = 1, 2 ..., M, in the cases of distribution of index or flow 

variables. In the cases of interpolation of stock variables, the 

constraint becomes 

ym = x , or ym = , t1 m x t2 m 

for m = 1, 2 ..., M, with either the first or the last sub-annual 

estimate equals the annual value for year m. 

Let y = (y1 ..., yM)’ be a column vector of M annual values; 

let B be a TxM matrix that maps sub-annual estimates to annual 

constraints; and let A be a TxT weighting matrix. Then, the 

original Denton model can be specified in the matrix form as 

(1) Minx P(x, z) = (x − z)' A(x − z), 

Subject to 

(2) y = B’x. 

The solution to the final sub-annual estimate is 
10 

(3) x = z + C(y – B’z), 

where y – B’z measures the annual discrepancy and C represents 

the distribution rule determining how the annual discrepancy is 

distributed to each sub-annual period. If A is ITxT, then C will 

be the inverse of the number of sub-annual periods in a year. 

The annual discrepancy would then be distributed evenly to each 

sub-annual period. Apparently, that is not a good choice of 

distribution rule. 

Denton proposed several variants of the original movement 

preservation model based on the first or higher order difference 

of the final sub-annual estimates and the indicator series. The 

most widely used are the additive and proportional first and 

second difference variants. 

1) The additive first difference variant preserves the sample 

period-to-period change in the level of the final sub-annual 

estimates and the indicator values, (xt – zt), under the annual 

constraint. As a result, xt tends to be parallel to zt. The 

objective in this case is achieved by minimizing the sum of 

squares of ∆(xt – zt), so the penalty function is P(x, z) = 

T 2 ∑[∆(x − z )] . t t 

t=1 

2) The proportional first difference variant preserves the 

proportional period-to-period change in the final sub-annual 

estimates and the indicator series, xt/zt. As a result, xt tends 

to have the same period-to-period growth rate as zt. The 

objective here is achieved by minimizing the sum of squares of 

∆(xt/zt) = (xt/zt) – (xt-1/zt-1), and the penalty function in this 

T x x t t−1 2 case is P(x, z) = ∑( − ) . t=1 zt zt−1 
 

 

11 

3) The additive second difference variant preserves the sample 

period-to-period change in ∆(xt – zt) as linear as possible. In 

this case, the objective is to minimize the sum of squares of 

T 2 2 ∆2

(xt – zt), and the penalty function is P(x, z) = ∑[∆ (x − z )] . t t 

t=1 

4) The proportional second difference variant preserves the 

sample period-to-period change in ∆(xt/zt) as linear as possible. 

The objective in this case is to minimize the sum of squares of 

x t T x x t t−1 2 ∆2

( ), and the penalty function is P(x, z) = ∑ [∆( − )] . z t=1 z z t t t−1 

Denton imposed the initial conditions that no adjustments 

are to be made in the indicator values outside of the sample. 

Thus, in the first difference variants, the initial condition 

boils down to x0 = z0, and in the second difference variants, the 

initial conditions boil down to xo = zo and x-1 = z-1. Such 

initial conditions result in a major short-coming of the original 

Denton method, because it induces a transient movement at the 

beginning of the series. It forces the final sub-annual 

estimates to equal the original series at time zero and results 

in the minimization of the first change (x1 – z1). Such transient 

movement defeats the principle of movement preservation. Helfand 

et al. (1977) solved this problem by modifying the penalty 

T 2 function to P(x, z) = ∑[∆(x t t − z )] for the additive first

t= 2 

T x 2 difference variant, and P(x, z) = ∑(∆ t) for the proportional

t= 2 zt 

first difference variant. The modified first difference variants 

precisely omit this first term to solve this short-coming of the 

original Denton method. The penalty functions for the second 

difference variants are modified accordingly. 
12 

2.1.2 The Causey-Trager growth preservation model 

The growth preservation model is first developed by Causey 

and Trager (1981) and later revised by Trager (1982). They 

propose that instead of preserving the proportional period-toperiod change in final sub-annual estimates and indicator series 

xt/zt, the objective should be to preserve the period-to-period 

percentage change in the indicator series. As a result, the 

period-to-period percentage change in xt tends to be very close 

to that in zt. This objective is achieved by minimizing the sum 

of squares of (xt/xt-1 – zt/zt-1). Formally, the mathematical 

problem of the Causey-Trager model is 

T x z t t 2 (4) Minx P(x, z) = ∑[ − ] , t= 2 xt−1 zt−1 

subject to (2), the same temporal constraints as those specified 

in the original Denton model. 

The Causey-Trager model is non-linear in the final subannual estimates xt, and thus, it does not have a closed form 

solution. The Causey-Trager model is solved iteratively using 

the solution to the Denton proportional first difference model as 

the starting values. Causey (1981) provided a numerical 

algorithm using steepest descent to obtain the final sub-annual 

estimates xt, t = 1, 2 ..., T. This numerical algorithm was later 

revised by Trager (1982). Because the Causey-Trager model is 

non-linear and is solved iteratively, the computational cost is 

higher compared to the Denton proportional first difference 

model, which has a closed form solution. 

2.1.3. The Boot, Feibes and Lisman smoothing method (BFL) 

Smoothing methods are suitable for univariate temporal 

disaggregation when only annual data are available. The basic 
 

13 

assumption of smoothing methods is that the unknown sub-annual 

trend can be conveniently described by a mathematical function of 

time. Because no sub-annual information is available, the subannual path is given a priori or chosen within a larger class, 

such as that the necessary condition of satisfying temporal 

aggregation constraints and the desirable condition of smoothness 

are both met. One such smoothing method is the BFL method. 

There are two versions of the BFL method, the first difference 

model and the second difference model. 

The objective of the first difference model is to minimize 

period-to-period change in the level of final sub-annual 

estimates ∆xt subject to the annual constraints. Formally, the 

problem is to 

T

(5) Minx P(x) = ∑(xt − xt−1)

2 , t= 2 

subject to (2). The constraints are the same as those specified 

in the original Denton model. 

The objective of the second difference model is to keep the 

period-to-period change in ∆xt as linear as possible. This 

objective is achieved by minimizing the sum of squares of ∆2

xt = 

(∆xt - ∆xt-1) subject to annual constraints. Formally, the second 

difference model is 

(6) Minx P(x) = ∑

T

[∆(x − x )]2 t t−1 , t= 2 

subject to (2). 

Like other smoothing methods, the BFL method does not allow 

extrapolation of sub-annual estimates, because it is designed 

only to give a sub-annual breakdown of the available annual data. 

Thus, to produce the current year estimates, a forecast value of 
14 

the current year annual value is needed. All sub-annual periods 

of the current year are then estimated at the same time. 

2.2 Regression Methods 

There have been quite a number of statistical methods 

developed for temporal disaggregation. These methods are 

designed to improve upon the mathematical methods discussed 

above, which do not take into consideration certain behaviors of 

economic time series data. Some examples of the statistical 

benchmarking methods are optimal regression models (Chow and Lin, 

1971; Bournay and Laroque; 1979; Fernandez, 1983; Litterman, 

1983), dynamic regression models (De Fonzo, 2002), unobserved 

component models or structural time series models using Kalman 

filter to solve for optimal estimates of the missing sub-annual 

observations (Gudmundson, 1999; Hotta and Vasconcellos, 1999; 

Proietti, 1999; Gomez, 2000), and Cholette-Dagum regression-based 

benchmarking methods (Cholette and Dagum, 1994, 2006). 

For our purpose, we will limit our discussion on the more 

widely used Chow-Lin regression model and its variants, which are 

used in our experiment. We shall also briefly discuss the 

Cholette-Dagum regression-based method, because, in some respect, 

it can be considered a generalization of the Denton benchmarking 

approach. The structural time series models formulated in state 

space representation for interpolation and temporal distribution 

are not yet fully corroborated by empirical applications, and, 

therefore, will not be discussed here. 

2.2.1 The Chow-Lin regression method 

Chow-Lin (1971) developed a static multivariate regression 

based method for temporal disaggregation. They argue that subannual series to be estimated could be related to multiple 

series, and the relationship between the sub-annual series and 

the observed related sub-annual series is 
15 

(7) xt = ztβ + ut, 

where x is a Tx1 vector, z is a Txp matrix of p related series, β

is px1 vector of coefficients, ut is a Tx1 vector of random 

variables with mean zero and TxT covariance matrix V. They 

further assume that there is no serial correlation in the 

residuals of the sub-annual estimates. 

In matrix form, the relationship between annual and subannual series can be expressed as 

(8) y = B’x = B’zβ + B’u. 

Chow-Lin derives the solutions by means of the minimum 

variance linear unbiased estimator approach. The estimated 

coefficient β

variable and annual sums of the related sub-annual series as the 

independent variables. The estimated coefficients are 

)

 is the GLS estimator with y being the dependent 

(9) )

β = [z’B(B’VB)-1B’z]-1z’B(B’VB)-1y, 

) ) )

and the linear unbiased estimator of x is 

x = zβ + VB(B’VB)-1 (10) [y – B’zβ ]. 

) The first term in (10) applies β

sub-annual series of the explanatory variables. The second term 

is an estimate of the Tx1 vector u of residuals obtained by 

to the observed related 

) distributing the annual residuals y – B’zβ

VB(B’VB)-1. This implies that if the sub-annual residuals are 

serially uncorrelated, each with variance σ2

, then V = σ2ITxT, and 

 with the TxM matrix 
 

16 

then annual discrepancies are distributed in exactly the same 

fashion as Denton’s basic model with A = ITxT. 

The assumption of no serial correlation in the residuals of 

sub-annual estimates is generally not supported by empirical 

evidence. Chow-Lin proposes a method to estimate the covariance 

matrix V under the assumption that the errors follow a firstorder autoregressive AR(1) process. There are two static 

variants of the Chow-Lin approach intended to correct the serial 

correlation in the sub-annual estimates. One is the random walk 

model developed by Fernandez (1981), and the other is the random 

walk-Markov model developed by Litterman (1983). 

2.2.2 The Random walk model 

Fernandez argues that economic time series data are often 

composed of a trend and a cyclical component, and he proposes to 

transform the series to eliminate the trend before estimation. 

He sets up the relationship between the annual and the sub-annual 

) series as that in equation (8) and derives β and the linear 

unbiased estimator of x as those in equations (9) and (10). 

However, Fernandez argues that the sub-annual residuals follow 

the process 

(11) ut = ut-1 + εt, 

where εt ∼ N(0, VTxT), is a vector of random variables with mean 

zero and covariance matrix VTxT. Based on this specification, the 

relationship between x and z is 

(12) xt – xt-1 = ztβ - zt-1β + ut – ut-1, 

which can be expressed as 

(13) Dxt = Dztβ + Dut, 
17 

where D is the first difference operator. 

The relationship between the annual and sub-annual series 

also needs to be transformed accordingly, because the sum of Dxt 

is not equal to y. Fernandez shows such relationship in the 

matrix form as 

)

(14) ∆y = QDx = QDzβ + QDu, 

where ∆ is similar to D but the dimension of ∆y is MxM, and Q is 

MxT matrix. This specification holds if the final sub-annual 

estimates x in year 0 are constant, an assumption considered 

reasonable for large sample size. 

Given the sub-annual residual process specified by 

Fernandez and setting QD = B’, the solutions are 

x = zβ + (D’D)-1B(B’(D’D)-1 B)-1[y – B’zβ ) ) (15) ], 

(16) β

) = [z’B(B’(D’D)-1B)-1B’z]-1z’B(B’(D’D)-1B)-1y. 

If A = D’D, these solutions are identical to those derived 

from the first difference regression model. Fernandez concludes 

that 1) before estimating the sub-annual series through 

interpolation or distribution, the behavior of the series should 

be studied. If the series is non-stationary and serially 

correlated, then the first difference data should be used to 

transform the data in order to obtain stationary and uncorrelated 

series; 2) if the first difference is not enough, other 

transformation is needed to convert residuals to serially 

uncorrelated and stationary variables; and 3) given proper 

transformation, the degree of serial correlation can be tested by 

generalized least square estimation. 
18 

2.2.3 The Random walk-Markov model 

Litterman (1983) argues that the relationship between 

short-run movements in x and zβ is fairly stable in most cases, 

but the levels of x and zβ may vary over time. He points out 

that Chow-Lin’s specification of the covariance matrix, V = 

Inxnσ2

, is not adequate if the sub-annual residuals exhibit serial 

correlation, because this specification would lead to step 

discontinuity of the sub-annual estimates between the annual 

periods as it allocates each annual residual among all sub-annual 

estimates. He also argues that Chow-Lin’s treatment of serial 

correlation is only adequate if the error process is stationary. 

Litterman argues that Fernandez’ random walk assumption for 

the sub-annual residual term could remove all serial correlation 

in the annual residuals when the model is correct. However, in 

some cases, Fernandez’ specification does not remove all serial 

correlation. Litterman proposes the following generalization of 

the Fernandez method, 

(17) xt = ztβ + ut, 

(18) ut = ut-1 + εt, 

(19) εt = αεt-1 + et, 

where et ∼ N(0, V), is a vector of random variables with mean 

zero and covariance matrix V, and the implicit initial condition 

is that u0 = 0. In fact, given the specification of his model, 

Litterman’s model is considered an ARIMA (1, 1, 0) model. 

Under this assumption of the sub-annual residual process, 

Litterman transforms the annual residual vector into E = HDu and 

derived the covariance matrix V as 
 

19 

(20) V = (D’H’HD)-1σ2, 

where H is an TxT matrix with 1 in the diagonal elements and -α

in the entries below the diagonal elements. The solutions of x 

and β are respectively 

(21) xˆ = zβ

ˆ + (D’H’HD)-1B’[B(D’H’HD)-1B’]-1[Y – B’zˆβ], 

(22) β

ˆ = [z’B(B’(D’H’HD)-1B’)-1B’z]-1z’B(B’(D’H’HD)-1B’)-1Y. 

Litterman suggests two steps to estimate β and derives the 

linear unbiased estimator of x. The first step is to follow the 

estimator derived by Fernandez and to generate the annual 

residuals, Uˆ = B uˆ. The second step is to estimate α by forming 

the first-order autocorrelation coefficient of the first 

difference of the annual residuals and solving for α. Therefore, 

Litterman’s method also uses first difference data rather than 

the level data. 

2.2.4 AR(1) model 

Apart from the random walk models, there are also attempts 

to model the errors as an AR(1) process. Bournay and Laroque 

(1979) propose that the sub-annual errors follow an AR(1) process 

(11) ut = ρut-1 + εt, 

where εt ∼ N(0, σ2

) is white noise, and |ρ| < 1. The value of the 

coefficient ρ represents the strength of movement preservation of 

the distribution or interpolation model. Bournay and Laroque 

suggested that ρ = .999, which represents very strong movement 

preservation. The AR(1) model can be estimated by applying ML or 
 

 

20 

GLS. Cholette and Dagum (2006) suggest that ρ be set to .9 for 

monthly series and to .93

 for quarterly series. 

Cholette and Dagum developed Regression-based benchmarking 

method (1994), which consists of two basic models, the additive 

model and the multiplicative model. 

The Cholette-Dagum additive benchmarking model is 

formulated as follows, 

H 

(12) zt = ∑ rthβh + xt + et, t = 1, ..., T,

h=1 

subject to 

t2m

(13) ym = ∑ jmtxt + εt, m = 1, ..., M,

t=t1m 

λ where E(et) = 0, E(etet-1) = σλ

t t σ −1 ωt, λ measures degree of 

heteroscedasticity, E(εm) = 0 and E(εmet) = 0. 

The first term in (12) specifies deterministic time 

effects. If H = 1 and rth = -1, this term captures the average 

level discrepancy between the annual and the sub-annual data. In 

some cases, a second regressor is used to capture a deterministic 

trend in the discrepancy. In some cases, rth may be absent, which 

implies H = 0. The Cholette-Dagum model allows the annual 

constraint to be not binding. This is the case if εt is nonzero. 

The above additive model can be modified into proportional model. 

One can see from (12) and (13) that by setting the parameters to 

certain default values, the Cholette-Dagum additive model can be 

modified to approximate the Denton additive and proportional 

models. 

The Cholette-Dagum multiplicative benchmarking model is 

formulated as follows, 

H 

(14) lnzt = ∑ rthlnβh + lnxt + lnet, t = 1, ..., T,

h=1 
21 

subject to (13), where E(εm) = 0 and E(εmεm) = σε

2

m . 

The multiplicative model requires that both annual and sub-annual 

indicator observations to be positive in order to avoid negative final 

estimates of sub-annual values. Typically, the deterministic regressor 

is a constant, i.e. H = 1 and rHt = -1, which captures the average 

proportional level difference between the annual and the sub-annual 

indicator data. If this is the case, the first term in (13) is a 

weighted average of the proportional annual discrepancies. Similar as 

in the additive model, in some cases, a second regressor is used to 

capture a deterministic trend in the proportional discrepancy, or the 

regressor may be absent, in which case H = 0. By setting the 

parameters in the multiplicative model to certain default values, the 

multiplicative model can be modified to approximate the Causey-Trager 

growth preservation model. 

3. Test Criteria and Estimation Strategy 

Our objective is to select a method or methods most 

suitable for routine temporal distribution and interpolation in 

the national accounts. The most suitable method(s) should 

generate final sub-annual estimates that best satisfy certain 

pre-specified criteria and should be easy to implement given the 

operational criteria set by the national accounts. 

3.1 Test Criteria 

Five basic criteria should be used to evaluate the final 

sub-annual estimates generated using different methods. 

1) Temporal aggregation constraint must be satisfied. This means 

that for each annual period, the sub-annual estimates must 

aggregate or average to the annual benchmarks. The temporal 

discrepancy can be measured with respect to the indicator series 

or to the estimated final sub-annual series. Temporal 
 

 

22 

discrepancies with respect to indicator series show how much the 

indicator series need to be adjusted so that the temporal 

constraints can be satisfied, and could be large because the 

indicator and the annual benchmark in general do not directly 

measure the same target variable. Temporal discrepancies with 

respect to the final sub-annual estimates measure how well the 

temporal aggregation constraints are satisfied, and should be 

null if the annual benchmarks are binding. 

Temporal discrepancies can be measured algebraically (in 

level) or proportionally. For year m = 1, 2 ..., M, algebraic 

temporal discrepancy with respect to final sub-annual estimates 

is computed as 

t A D = ym

2m 

∑

=

(3.1) x - k mtxt , for indicator and flow variables, t t1m 

or 

A D = y x m – xt1m (or ym – xt2m), for stock variables.

Correspondingly, proportional temporal discrepancy with respect 

to final sub-annual estimates is computed as 

∑

=

t P D = ym/ 2m

(3.2) x kmtxt , for indicator and flow variables, t t1m 

or 

P D = y x m/xt1m (or ym/xt2m), for stock variables.

Algebraic and proportional discrepancies with respect to the 

A P indicator series D and D can be written out simply by z z 

replacing xt with zt in (3.1) and (3.2). 

Two statistics of temporal discrepancies are empirically 

useful. The means of discrepancies measure the level or 

proportional difference between the annual benchmarks and the 

indicator series, or between the annual benchmarks and the 

estimated sub-annual series, of all annual periods in the sample. 
23 

The standard deviation of discrepancies measures the dispersion 

of discrepancies of all sample periods. A large standard 

deviation may imply erratic discrepancies over sample periods, 

suggesting a contradiction between the annual and indicator 

variables, and it may also imply that in the process of 

satisfying the annual benchmarks, temporal distribution or 

interpolation distorts the movements of the indicator series. 

Erratic discrepancies may suggest low reliability of the 

indicator series. 

2) Short-term movements in the indicator series should be 

preserved as much as possible. Short-term movement preservation 

can be measured in terms of level, proportion, and growth rates. 

Different methods are designed to achieve different objectives of 

short-term movement preservation. For example, the Denton 

additive first difference method is designed to preserve periodto-period movements in the indicator series. Thus, the objective 

is to minimize period-to-period change between the sub-annual and 

indicator series. The resulting sub-annual estimates tend to be 

parallel to the indicator series. The Denton proportional first 

difference method is designed to preserve proportional period-toperiod movements in the final sub-annual estimates and the 

indicator series. Therefore, the objective is to minimize 

period-to-period change in the ratio of the final sub-annual 

estimates to the indicator series. Final sub-annual series 

estimated using this method tends to have the same period-toperiod percentage changes as the indicator series. The CauseyTrager method is designed to preserve period-to-period growth 

rate in the indicator series. The resulting sub-annual estimates 

and the indicator series tend to have the same growth rates. 

Two statistics can be used to measure short-term movement 

preservation: 1) the average absolute change in period-to-period 

differences between the final sub-annual estimates and the 

indicator series of all sub-annual periods in the sample, cL

; and 

2) the average absolute change in period-to-period growth rates 
24 

between the final sub-annual estimates and the indicator series 

of all periods in the sample, cP . These two statistics are 

computed as follows: 

T 

(3.3) cL

 = ∑|(x − x ) − (z − z )| /(T − 1), t t−1 t t−1 

t= 2 

T 

(3.4) cP = ∑| [(x / x )/(z / z )] − 1.0 | /(T − 1). t t−1 t t−1 

t= 2 

The first statistic cL

 measures changes in the period-toperiod differences, and thus, it is more relevant to additive 

benchmarking. The second statistic cP

 measures changes in the 

period-to-period growth rates, and thus, it is more relevant to 

proportional or growth rate preserving benchmarking. 

3) Final sub-annual estimates should not exhibit drastic 

distortions at the breaks between years. By distortion we mean 

that the movements in the sub-annual estimates are inconsistent 

with the movements in the indicator series, unless such 

inconsistent movements are observed in the annual values. Some 

benchmarking methods may generate large percentage changes in the 

sub-annual periods at the breaks between years. In a comparative 

study of benchmarking methods Hood (2002) shows that the average 

absolute percentage change between the estimated sub-annual 

series and the indicator is larger during the months from 

November to February than that during the months from March to 

October. For some benchmarking methods, the distortion can be 

quite large. 

To evaluate estimates obtained using different methods, for 

monthly series we compute the average absolute change in periodto-period growth rate from November to February, and from March 

to October, of all years in the sample. We denote the first 

grouped average as CB and the second grouped average as CM, where 
 

25 

B stands for periods at breaks and M stands for periods in the 

middle of a year. For quarterly series we compute CB as the 

average absolute period-to-period change in growth rate from the 

fourth quarter to the following first quarter, and CM as the 

average from the second to the third quarter, of all sample 

years. We compare the two grouped averages computed using final 

sub-annual series estimated by each method. A good benchmarking 

method should generate the least distortions at the breaks 

between years when compared with other methods. 

4. Final sub-annual estimates should not exhibit step 

discontinuity or drastic distortions at the beginning and ending 

periods of the sample. To evaluate final estimates from 

different methods, we compare the absolute change in period-toperiod growth rate between the final estimates and the indicator 

values for the second and the last periods of the sample. Note 

that the second period is when the first period-to-period growth 

can be computed. 

A related issue is how smoothly the final estimates 

interpolated using revised annual and indicator data link to the 

previously benchmarked series. The national accounts revise the 

annual and the subannual values of the previous three calendar 

years during annual revision in each July. Under the current 

revision policy only the final sub-annual estimates of the most 

recent three calendar years are updated during annual revision. 

Final estimates for the periods prior to the three years being 

revised are considered previously benchmarked and not to be 

revised. 

There are two alternative ways to comply with the current 

revision policy. The first alternative is to incorporate linking 

as an initial condition in the optimization problem. That is to 

require the optimal solution to satisfy the condition that the 

newly revised final estimates be linked to the unrevised estimate 

of the last sub-annual period prior to the three years being 

revised. The second alternative is to simply replace the 
 

26 

previous three years of annual and indicator data with the 

revised data in a sample of many years, and re-interpolate the 

whole sample. Then use the newly interpolated final estimates of 

the most recent three years to replace the estimates obtained 

prior to the annual revision. The rationale for the second 

alternative is that re-interpolating the whole sample rather than 

just the sample of the previous three years may lead to more 

gradual transition from the span of the sample that is previously 

benchmarked to the span of the newly revised sample. To find out 

which alternative provides smoother linking, we compute the 

absolute percentage change in the linking period using the final 

estimates generated using each method and denote it as CL. 

5. Contemporaneous constraint, if present, should be satisfied. 

Some series have both temporal and contemporaneous constraints. 

Two examples are the 16 quarterly series of government taxes on 

production and imports and the 15 quarterly series of industry 

transfer payments to government. Each tax and transfer payment 

series has temporal constraints to be satisfied. For each 

quarter, the quarterly total of the taxes serves as 

contemporaneous constraint for the 16 series on taxes, and the 

quarterly total of the transfer payments serve as contemporaneous 

constraint for the 15 series on transfer payments. 

Satisfying contemporaneous constraint is a reconciliation 

issue rather than benchmarking issue. Ideally, the software for 

benchmarking should also be able to handle reconciliation. 

Unfortunately, most benchmarking programs are designed for 

benchmarking only and do not provide the option for 

reconciliation. To evaluate different methods, we compute the 

contemporaneous discrepancy as the percentage difference between 

the sum of the final sub-annual estimates and the contemporaneous 

aggregate for each sub-annual period, and compare the 

contemporaneous discrepancies of the final sub-annual estimates 

interpolated using different methods. 
27 

3.2 Methods for Evaluation and Software Used for Estimation 

The methods selected for evaluation are five mathematical 

methods and five regression methods discussed in Section 2. The 

five mathematical methods are: the modified Denton additive first 

difference and proportional first difference methods; the CauseyTrager growth preservation method, and the first and second 

difference Boot-Feibes-Lisman smoothing methods. The five 

regression methods are: an AR(1) model by Bournay and Laroque 

(1975) estimated by applying ML and GLS; the random walk model by 

Fernandez (1981); and the Random walk-Markov model by Litterman 

(1983) estimated by applying ML and GLS. 

We use three software programs for estimation: 1) a FORTRAN 

program BMARK developed by the Statistical Methodology Research 

Division at the Census Bureau; 2) a FORTRAN program BENCH 

developed by the Statistical Research Division at Statistics 

Canada; and 3) ECOTRIM program for Windows based on Visual Basic 

and C++ languages developed by Eurostat. The BMARK program is 

designed for univariate benchmarking and it supplies procedures 

based on four mathematical benchmarking methods, two for 

benchmarking seasonally unadjusted series and two for 

benchmarking seasonally adjusted series. The two options 

relevant for interpolation and distribution for the national 

accounts are the modified Denton proportional first difference 

method and the iterative, non-linear Causey-Trager growth 

preservation method. In the BMARK program these methods are 

referred to as RATIO and TREND models. We shall refer to these 

methods as RATIO and TREND in the following discussion. 

The BENCH program is designed for univariate benchmarking 

and for temporal disaggregation. It is developed for a 

generalization of the Denton methods based on GLS regression 

techniques. It provides options for specifying binding or nonbinding benchmarks, benchmarks for particular years only, and 

sub-annual benchmarks. It allows for incorporating particular 

information about the error generating process. For instance, 
 

28 

the autocorrelation of the errors may be modeled by assuming that 

the errors follow a stationary ARMA process, and the reliability 

of each annual and sub-annual observation may be characterized by 

their variances. Although the program is designed for 

benchmarking using regression based methods, the program can be 

used to approximate the modified Denton additive and proportional 

first difference methods by assigning a set of parameters to the 

default values. 

The ECOTRIM program is developed for Windows by Eurostat. 

It supplies procedures based on temporal disaggregation of low 

frequency series using mathematical and statistical methods. It 

allows for univariate and multivariate temporal disaggregation of 

time series. For univariate series with indicator or related 

series, ECOTRIM provides the options of five regression methods 

listed above. For univariate series with no indicators, ECOTRIM 

provides the options of first and second difference smoothing 

methods by Denton and by Boot-Feibes-Lisman. For temporal 

disaggregation of multivariate series with respect to both 

temporal and contemporaneous constraints, ECOTRIM provides 

procedures using the Fernandez random walk model, the Chow-Lin 

white noise model, the Denton adjustment methods and the Rossi 

regression model (1982). Moreover, ECOTRIM provides both 

interactive and batch mode for temporal disaggregation. 

We use all three software programs to generate final subannual estimates using the selected methods and evaluate the 

final estimates according to the five criteria discussed above. 

4. Estimation Results 

When compiling sub-annual estimates by temporal 

distribution and interpolation, the national accounts encounter 

the following cases: 1) both annual and sub-annual indicator data 

are available; 2) only annual data are available; 3) both 

temporal and contemporaneous constraints are presents; and 4) 
29 

annual data contain negative values. In order to have a proper 

understanding of how each method works, we choose series so that 

all these cases are included in our experiments. 

We have selected 60 series for temporal distribution and 

interpolation, 15 from the National Income and Wealth division 

(NIWD) and 45 from the Government division (GOVD). Table A1 in 

the Appendix lists the annual series and their indicator series, 

if available, included in the estimation experiment. Data used 

in estimation were obtained after the 2005 annual revision. For 

the 15 series from NIWD, some are quarterly variables and some 

are monthly variables. Indicator series are available for 14 out 

of the 15 series. Of the 45 series from the GOVD, 16 are 

government taxes on production and imports, 15 are transfer 

payments to the federal government, and 14 are series from the 

Federal and the State and Local Government branches. The series 

on taxes and transfer payments are quarterly variables, and they 

have both temporal and contemporaneous constraints. The 

contemporaneous constraints for taxes and transfer payments are, 

respectively, quarterly total of the taxes and quarterly total of 

the transfer payments. No indicator series are available for the 

14 series from the Federal and State and Local Governments 

branches, and some of these series have multiple negative annual 

values in the sample. 

Because choices of methods for temporal distribution or 

interpolation largely depend on the basic information available 

for estimation, we separate the 60 series into two categories: 1) 

annual series with sub-annual indicators; and 2) annual series 

without sub-annual indicators. We shall discuss the results in 

each category according to the criteria discussed in Section 3. 

4.1 Temporal disaggregation of annual series with indicators 

Of the 60 series included in the experiment, 45 series fall 

into this category, 14 of which are from NIWD, and the remaining 

31 are from the GOVD, which have both temporal and 
30 

contemporaneous constraints. The sample sizes are between 8 to 

12 years. For some series we have pre- and post-revised annual 

and indicator values from 2002 to 2004. 

The indicator series selected for distribution and 

interpolation are not the same target variables measured by the 

annual data. They are intended to provide information on the 

short-term movements in the target variables. Thus, indicators 

selected should be closely correlated with the target variables. 

To see if the indicators are closely correlated with the target 

variables, we computed the correlation coefficient ρ between each 

pair of annual series and annual aggregates of sub-annual 

indicator series. Table 1 shows that all but one ρ values are in 

the range of .8130 and .9999, and 39 of the 45 pairs have a ρ

value greater than .9, an indication of strong correlation 

between the annual and the corresponding indicator series. (Most 

tables are included at the end of the report.) 

Table 1 is here 

We estimated these series using the following methods: the 

modified Denton additive first difference (DAFD) and proportional 

first difference (DPFD) methods; the Casey-Trager growth rate 

preservation method (TREND); the Fernandez random-walk model 

(RAWK); the Litterman random-walk Markov model estimated by 

applying ML (RAWKM MAX) and GLS (RAWKM MIN) and the Bournay and 

Laroque AR(1) model also estimated by applying maximum likelihood 

(AR(1) MAX) and GLS (AR(1) MIN). To compare these methods with 

Lagrange polynomial interpolation, we also included the final 

estimates from Lagrange polynomial interpolation procedure (LPI). 

We used the options of the modified Denton additive and 

proportional first difference methods from BENCH program; the 

options of the modified Denton proportional first difference 

method and the Casey-Trager method from BMARK program; and the 

options for the five regression based methods from ECOTRIM 
31 

program. For the regression models, the parameter to be 

estimated is the one in the autocorrelation process of the 

errors. ECOTRIM program allows the options of having the user 

choose the parameter’s value or having the parameter be optimally 

estimated. By choosing the parameter’s value, the user decides 

how strong the short-term movement preservation should be. In 

this experiment, we choose the option of having the parameter be 

optimally determined in the estimation. Both BENCH and BMARK 

programs provide the option of the modified Denton proportional 

first difference method, and we used both to compare the results 

from the two software programs. We continue to refer to the 

Denton proportional first difference option from BMARK as RATIO. 

To simplify the exposition, we shall use the abbreviations of the 

methods in the following discussion of the results. 

The final sub-annual estimates from the mathematical and 

regression methods are quite close in level. See Figures 1-1 to 

1-10 for details. (All figures are included at the end of the 

report.) However, the final estimates from LPI procedure 

sometimes display a significant jump or dip at the beginning 

and/or ending periods of the sample. Four such examples are 

provided in Figures 1-1 to 1-4. In Figure 1-1, the big dip at 

the end of the final estimates from LPI exhibits contradictory 

movement as seen in the indicator series. In Figure 1-2 and 1-3, 

such contradiction can be seen both at the beginning and at the 

end of the final estimates from LPI. In Figure 1-4, the final 

estimates from LPI are flat for all periods in 2004, while the 

indicator values increase mildly and the annual value for 2004 

increases sharply. In this case, the balance between the shortterm movements in the indicator series and the long-term 

movements in the annual series is lost in the final estimates 

from LPI. In some cases, the final estimates from LPI display 

movements inconsistent with those seen in the annual and 

indicator series for some periods in the sample. For example, in 

Figure 1-5 the movements in the LPI estimates from the beginning 

of 2001 to the end of 2002 do not match the movements seen in the 
32 

indicator series. In Figure 1-6 the LPI estimates display a 

zigzagged pattern that is in sharp contrast to the smooth 

movements in the indicator series. Reasons for such behaviors in 

the LPI final estimates are discussed in the introduction. 

Final estimates from the modified Denton additive first 

difference method (DAFD) may also display a pattern that is not 

observed in the indicator series. Two such examples are shown in 

Figure 1-7 and 1-8. In Figure 1-7a and 1-7b, the indicator 

series is quite volatile, especially in the later periods in the 

sample. However, these movements are quite moderate compared 

with the progressively sharper zigzagged pattern seen in the 

final estimates from DAFD. This zigzagged pattern may be caused 

by the mechanism of the DAFD model to minimize period-to-period 

difference between the indicator series and estimated final subannual series. Figure 1-8a and 1-8b show a similar example. 

These examples suggest that the DAFD method may not be the proper 

choice for distribution and interpolation if there are frequent 

rises and falls in the indicator series, because by keeping the 

indicator values and final estimates parallel, some volatile 

pattern is generated. Next we shall compare the final estimates 

according to the 5 test criteria. Final estimates generated by 

the 5 regression methods are quite close in level. Figure 1-9 

and 1-10 are two examples. 

Temporal aggregation constraint 

As discussed in Section 3 proportional annual discrepancy 

P with respect to indicator series D z measures the annual values 

relative to the annual aggregates of the indicator values. The 

P more different is D z from one, the more adjustments in the 

indicator series are needed to satisfy annual constraints. In 

P most cases, the computed Dz is very different from one, 

indicating that the annual values and the annual aggregates of 

the indicator values of these series are quite different in level 
 

 

 

 

33 

(See Table 2-1 and 2-2). This is because the indicator and the 

annual data do not directly measure the same target variables. 

The computed proportional annual discrepancy with respect to 

P final sub-annual estimates D is equal to one for all final x 

estimates, except for 6 final sub-annual series from LPI. In 

P these 6 cases, D is significantly different from one, indicating x 

that the temporal constraints are not satisfied. For all other 

methods evaluated in this study, temporal aggregation constraints 

are satisfied. 

Table 2-1 and 2-2 are here 

Short-term movement preservation 

Recall that the two statistics which measure the short-term 

preservation are the average absolute change in period-to-period 

difference CL

x and the average absolute change in period-toperiod growth rate between the final sub-annual estimates and the 

indicator series CP

x . These two statistics can be interpreted as 

correction in level and in percentage or as adjustment to the 

indicator series in order to satisfy temporal constraints. 

Because in the national accounts the emphasis is placed on 

achieving smooth period-to-period percentage changes, we focus on 

P the comparison of C x value from final estimates interpolated 

using each method. 

P Table 3-1 contains the C values of the 14 NIWD series from x

P each method. The final estimates from LPI often have larger Cx

values when compared with the final estimates from other methods. 

One can also see from Figure 2-1 to 2-4 that LPI final estimates 

have larger dispersions in the period-to-period difference in 

growth rates. In some cases, C of the final estimates from LPI P

x
 

34 

is greater by a factor of at least 10 compared with C values of P

x

the final estimates from the other methods. 

Table 3-1 is here 

P For each of the 14 series, the C values of the final x

estimates from DAFD, DPFD, RATIO and TREND are quite close, 

P except for the two cases where C values of the final estimates x

P from DAFD are unreasonably large. These large C values x

correspond to the cases where the DAFD final estimates exhibit a 

P sharp zigzagged pattern. For some series, the C values of the x 

final estimates from the five regression based methods are more 

varied. See Figure 3-1 to 3-4 for details. 

P No single method produces the minimum C in all 14 cases. x 

To have some idea about which method on average better preserves 

P the short-term movements, we computed the mean of the 14 Cx 

values for each method. From Table 3-1 we can see that the 

P smallest mean of the 14 C values is from the final estimates of x

DPFD. Thus, for the 14 series from NIWD, the modified Denton 

proportional first difference (DPFD) option from BENCH program is 

on average the best in preserving the short-term movements in the 

P indicator series. A comparison of the C values from the x

P mathematical methods with the C values from the regression x

P methods shows that the means of C values from the regression x 

based methods are in general significantly larger. 

For the 16 series of taxes on production and imports and 

the 15 series of transfer payments to the federal government, we 

only compare the final sub-annual estimates from the DPFD, RATIO, 

TREND and LPI methods1

. From Table 3-2 one can see that for each 

1 Because of the erratic behaviors of the final estimates from DAFD, we excluded DAFD from

experiment after estimating the 14 series from NIWD. We did not include the estimates 
 

 

 

35 

P tax or transfer payment series, the C values from DPFD, RATIO x 

and TREND are only slightly different. The final estimates of 

P taxes from DPFD has the smallest mean of the 16 C x values, and 

the final estimates of transfer payments from TREND has the 

P smallest mean of the 15 C values. We should point out that the x 

P differences between the means of C values are only marginal. x 

Thus, we could say that for the series on taxes and transfer 

payments, the DPFD option from BENCH and the TREND option from 

BMAK preserve short-term movements equally well. 

Table 3-2 is here 

Minimum distortion at breaks between years 

Final estimates from all methods included in the experiment 

exhibit some degree of distortion at breaks between years. Table 

4-1a and 4-1b contain the CB and CM values for the final subannual estimates of the 14 series from NIWD estimated using each 

method. It is clear that for each series of final estimates, CB 

is greater than CM. The pattern of distortion at breaks between 

years can also be observed easily from Figures 2-1 to 2-10. In 

almost all cases, the final estimates from LPI generate larger 

distortions at breaks between years. See Figures 2-1 to 2-8 for 

examples. In a few cases, the final estimates from DAFD also 

display much larger distortions than the final estimates from the 

other methods. Figures 2-9 and 2-10 give two such examples. The 

unusually large distortions occur when the final estimates from 

DAFD exhibit zigzag patterns as shown in Figure 1-9 and 1-10. 

Table 4-1a and 4-1b are here 

from regression models because ECOTRIM program is currently experiencing problems in the 

two-stage estimation of multivariate series with contemporaneous constraints. 
 

36 

A comparison of the CB values in Table 4-1a shows that DPFD 

method has out-performed the other methods for the 14 series from 

NIWD, and DPFD has also produced the smallest mean of all 14 CB 

values. A comparison of the CB values from the regression 

methods show that the AR(1) methods generate smaller distortions 

than the other regression methods. However, when compared with 

the CB values from the mathematical methods, the CB values from 

the regression methods are in general larger, indicating that the 

final estimates from these regression methods generate larger 

distortions at breaks between years. 

Table 4-2a and 4-2b contain the CB and CM values for the 

final estimates of taxes and transfer payments. From Table 4-2a 

we can see that for each series, the CB values from DPFD, RATIO 

and TREND are very close. The minimum mean of the 16 CB values 

for tax series and the minimum of the 15 CB values for the 

transfer payment series are both from DPFD, though the means of 

CB values from different methods are only marginally different. 

Thus, we can say that the modified Denton proportional first 

difference method (DPFD from BENCH and RATIO from BMAK) and the 

Casey-Trager method (TREND) generate similar distortions at 

breaks between years. 

Table 4-2a and 4-2b are here 

Minimum distortion in the beginning and ending periods 

A good method for temporal disaggregation should not 

generate final estimates that impose large distortion at the 

beginning and ending periods of the sample. Such distortion is 

measured by inconsistent period-to-period growth rate between the 

final estimates and the indicator series. There are two aspects 

we need to consider when examining distortions at the beginning 

and ending periods of final estimates. First, we look at the 

absolute change in period-to-period growth rates between the 
 

37 

final estimates and the indicator values in the second and the 

last sub-annual periods, and we denote them as C2 and CT. 

Table 5-1a and 5-1b contain the values of C2 and CT of the 

14 NIWD series from the final sub-annual estimates of all methods 

under evaluation. One can see that for almost all series, C2 and 

CT values from LPI estimate are much larger than those from the 

final estimates of the other methods. This clearly shows that 

Lagrange polynomial interpolation method has a tendency to 

generate distortions at the beginning and ending periods of the 

sample. Such distortions can also be observed in Figures 2-1 to 

2-8. In two cases, C2 and CT from DAFD have unreasonably large 

values, which correspond to the cases where the final estimates 

exhibit sharp zigzagged patterns. 

Table 5-1a and 5-1b are here 

A careful comparison of the C2 and CT values show that there 

is not a single method that produces the minimum C2 and CT for all 

14 series. We computed the means of the 14 C2 and CT values and 

found that the minimum means of C2 and CT are both computed from 

DPFD final estimates, though the differences between the means of 

C2 and CT of DPFD, RATIO and TREND methods are fairly small. 

However, for most series, the C2 and CT values from the regression 

methods are much larger, and so are the means of the C2 and CT 

values. These results suggest that on average, DPFD method 

generates the least distortion at the beginning and ending subannual periods of the sample. 

Table 5-2a and 5-2b are here 

Table 5-2a and 5-2b show the C2 and CT values of the final 

sub-annual estimates of the 16 tax and 15 transfer payment series 

from LPI, DPFD, RATIO and TREND methods. Again, for almost all 

series, the C2 and CT values from LPI are much larger than those 
38 

from the other methods. However, the results for the other 

methods are mixed. For the final estimates of both taxes and 

transfer payments the minimum mean of C2 is from the DPFD method, 

whereas the minimum CT is from the TREND method for both taxes 

and transfer payments. One should note that the means for C2 and 

CT from different methods differ very marginally. 

Next we shall look at how smoothly the revised estimates 

link to the previously benchmarked series. We have both prerevised and revised data from 2002 to 2004 for 9 of the 14 series 

from NIWD, and we experimented with both alternatives for linking 

using DPFD, RATIO and TREND methods. We decided not to use the 

regression methods for the linking test, because the sample size 

of 3 years is too small for any reliable statistical results. 

Tables 6-1 and 6-2 compare the absolute change in periodto-period growth rate between the newly revised final sub-annual 

estimate and the indicator value in the first period of the 

revised sample. Table 6-1 shows the results if the revised final 

estimates are linked to the previously benchmarked estimates 

obtained using the same method. Table 6-2 shows the results if 

the revised final estimates are linked to the previously 

benchmarked estimates from LPI. One would expect less smooth 

transition if the revised estimates are linked to the previously 

benchmarked estimates obtained using a different method, 

especially a method with serious known problems. 

Table 6-1 and 6-2 are here 

The left panels in both Table 6-1 and 6-2 show the results 

using the first alternative for linking; and the right panels 

show the results using the second alternative for linking. We 

observe the following from these tables: 1) the absolute 

difference in growth rates shown in the left panels are smaller 

in most cases than those shown in the corresponding columns in 

the right panels, indicating that setting linking as an initial 
39 

condition in the optimization problem tend to lead to smoother 

transition; 2) a comparison of the means of the 9 series in Table 

6-1 and 6-2 show that DPFD method combined with the first 

alternative for linking generates the smoothest transition; and 

3) transition is less smooth if the revised final estimates are 

linked to the previously benchmarked estimates obtained using 

LPI. 

Contemporaneous constraint 

The 16 series on taxes have a contemporaneous constraint, 

and so do the 15 series on transfer payments. Ideally, a 

benchmarking program should provide an option for temporal 

disaggregation when contemporaneous constraint is present. 

Unfortunately, BENCH and BMARK programs do not provide such an 

option. ECOTRIM provides an option of a two-stage method for 

disaggregation of multivariate series. In the first stage, 

temporal constraint of each series is satisfied, and the 

estimates from the first stage are used as preliminary estimates 

in the second stage when the contemporaneous constraint is 

included in estimation. The options for the second stage 

temporal distribution and interpolation are the modified Denton 

proportional first and second difference methods and a regression 

method developed by Rossi (1982). 

We chose to use the option of the modified Denton 

proportional first difference for multivariate series in ECOTRIM 

for the second stage estimation, using the estimates obtained 

from LPI, DPFD, RATIO, and TREND as the preliminary estimates in 

the second stage. Table 7-1 and 7-2 show how well the first 

stage estimates from DPFD, RATIO, TREND, and LPI satisfy the 

contemporaneous constraint, and how well the second stage 

estimates satisfy the contemporaneous constraint. We computed 

the contemporaneous discrepancy measured by the level difference 

and by the percentage difference between the sum of the final 
 

40 

estimates of the 16 taxes (15 transfer payments) and the total 

taxes (total transfer payments) for each quarter in the sample. 

Table 7-1 and 7-2 are here 

From Table 7-1 and 7-2 one can see that for both taxes and 

transfer payments the contemporaneous discrepancies from the 

first stage estimation are very small for estimates from DPFD, 

RATIO and TREND, whereas the contemporaneous discrepancies for 

the estimates obtained using LPI is much larger. One can also 

see that the contemporaneous constraints for both taxes and 

transfer payments are perfectly satisfied in the second stage 

estimation. For each component series of taxes and transfer 

payments, the estimates obtained using DPFD, RATIO and TREND in 

the first stage estimation are only adjusted mildly in the second 

stage estimation, whereas some component series estimated using 

LPI in the first stage are adjusted by much bigger margins. The 

results suggest that the modified Denton proportional first 

difference for multivariate series in ECOTRIM is a good candidate 

for the second stage temporal disggregation when contemporaneous 

constraint is present. 

4.2 Temporal Disaggregation of Annual Series without Indicators 

Of the 60 series included in the experiment, 15 series have 

no sub-annual information available, 14 of which are from GOVD 

and 1 is from NIWD. The annual data are from 1993 to 2004. 

out of the 15 series have negative values in the annual series. 

The methods available for disaggregation without sub-annual 

information are the smoothing methods developed by Boot, Feibes 

and Lisman (BFL) (1970), ARIMA models by Stram and Wei (1986, 

1990), and numerical procedures such as LPI and cubic spline 

interpolation. We focus on the BFL and LPI methods. We did not 

include the ARIMA models in our experiment, because the annual 

5 
 

41 

series are not sufficiently long and because such methods require 

estimation of unobserved covariance matrix. ECOTRIM program 

provides the options of the first difference and second 

difference BFL models (BFLFD, BFLSD); BENCH program provides an 

equivalent version of the first difference BFL model by setting 

the indicator value to 1 for all sub-annual periods in the DAFD 

method; BMARK program also provides a similar option in the RATIO 

method. ECOTRIM and BENCH programs are able to disaggregate the 

annual series that have negative values, but BMARK program is not 

able to handle such cases. This is probably due to programming 

details. Nevertheless, this is a limitation of BMARK program. 

When sub-annual information is not available, some test 

criteria need to be modified. Instead of examining short-term 

movement preservation measured by the average absolute change in 

period-to-period growth rate between the final and the indicator 

series, we compare the smoothness of final sub-annual estimates. 

Instead of examining distortions of sub-annual movements at the 

beginning and/or ending periods, we compare the smoothness of the 

final estimates at the beginning and ending periods. 

Temporal aggregation constraint 

Temporal aggregation constraints are satisfied by the final 

sub-annual estimates obtained using LPI, DAFD, BFLFD, BFLSD, and 

RATIO methods. The final estimates from FAFD, BFLFD and RATIO 

are fairly close in level. From Figures 4-1 to 4-5 one can see 

that the final estimates obtained from LPI may display sharp 

increase or decrease at the beginning or the ending period of the 

series, and they may also exhibit sharp turns in the middle 

portion of the series. Similarly, as shown in Figure 4-5 and 4-

6, the final estimates from BFLSD method may also show large 

increase or decrease at the beginning or ending periods of the 

final sub-annual series. 
42 

Smoothness of the final estimates 

We compare smoothness of the final estimates from different 

methods using the average absolute period-to-period growth rates 

of the final sub-annual estimates. For the 5 series which have 

negative values in the annual series, the estimated final subannual series also has negative values. Consequently, period-toperiod growth rate cannot be computed. Thus, for these series, 

we compare period-to-period change in level as a proxy for 

smoothness. When comparing the final estimates obtained using 

different methods, the smoothest series should have the least 

volatile changes between periods. 

Table 8-1a shows the average absolute period-to-period 

growth rates of the final sub-annual estimates of the 10 nonnegative series estimated using LPI, DAFD, BFLFD, BFLSD and RATIO 

methods. DAFD and RATIO are BFLFD equivalent from BENCH and 

BMARK. One can see that no single method has generated the 

minimum average absolute period-to-period growth rate in all 10 

cases. A comparison of the means of the average absolute periodto-period growth rates of the 10 series shows that DAFD (BENCH 

version of BFLFD) generates the smoothest final estimates among 

the five methods, though the means from DAFD, BFLFD, BFLSD and 

RATIO are quite close. Figure 4-7 to 4-10 show that period-toperiod growth rate of final estimates from LPI and BFLSD can be 

quite volatile. 

Table 8-1a is here 

Table 8-1b shows the average absolute period-to-period 

change in level of the 5 series which have negative values in the 

annual data. No results from RATIO because BMARK program cannot 

disaggregate series that have negative values. A comparison of 

the means of the average absolute period-to-period level change 

of the 5 series shows that DAFD generates smoothest final 

estimates. Figure 4-11 to 4-12 are examples to show that the 
 

43 

final estimates from LPI or BFLSD may have much larger period-toperiod change than the final estimates from BFLFD. 

Table 8-1b is here 

Smoothness in the beginning and ending periods 

A good smoothing method should not produce sharp jump or 

dip in the beginning and/or ending periods of the final subannual series, if such movement is not observed in the annual 

data. We evaluate the smoothness of all final sub-annual 

estimates in the beginning and ending periods by comparing the 

period-to-period growth rate in the second and the last period of 

the series. The left panel of Table 8-2a shows the absolute 

percentage change of the 10 non-negative series in the second 

period. It is obvious that the final estimates from LPI and 

BFLSD may have much larger percentage change in the second 

period. In fact, this is the case for 9 out 10 series of LPI 

estimates and 6 out of 10 series of BFLSD estimates. 

Table 8-2a is here 

The right panel of Table 8-2a shows the absolute period-toperiod growth rates of the 10 series in the last period of the 

sample. Similarly, the final estimates from LPI and BFLSD 

exhibit much larger percentage change compared with other 

estimates. Four examples are shown in Figure 4-7 to 4-10. A 

comparison of the growth rates from all series and from all 

methods shows that on average the final estimates from DAFD have 

the smallest period-to-period growth rate in the second and the 

last periods of the sample. 

Table 8-2b shows the absolute period-to-period level change 

in the beginning and ending periods of the 5 series that have 

negative values. Again, the final estimates from LPI and BFLSD 
44 

exhibit much larger period-to-period change in the second and 

last periods. Figure 4-11 and 4-12 show two such examples. 

Moreover, on average, the final estimates from DAFD method 

exhibit the smallest period-to-period level change in the 

beginning and ending periods of the series. 

Table 8-2b is here 

In sum, for diaggregation of annual series that have no 

sub-annual information available, BFLFD method seems to perform 

the best. Out of three versions of BFLFD model, the one provided 

in BENCH program performs slightly better than the other 

versions. BMARK program fail to handle disaggregation of annual 

series that have negative values. 

4.3 Comparative Summary of the Estimation Results 

In this section we shall provide a comparative summary of 

the results from the estimation of 60 series using a variety of 

methods. Table 9-1 compares the 4 mathematical methods and 5 

regression methods used to estimate the 14 series from NIWD. The 

P 6 statistics used for the comparative evaluation are: 1) D x , the

mean of proportional temporal discrepancy with respect to the 

final estimates; 2) CP , the mean of the average absolute change 

in period-to-period growth rate between the final estimates and 

the indicator series; 3) CB , the mean of the average absolute 

change in period-to-period growth rate between the final 

estimates and the indictor series during all periods at breaks 

between years in the sample; 4) C2 , the mean of the absolute 

change in growth rate between the final estimates and the 

indicator values in the second period of the sample; 5) CT , the 

mean of the absolute change in growth rate between the final 

estimates and the indicator values in the last period of the 

sample; and 6) CL , the mean of the absolute change in growth rate 
 

 

45 

between the final estimates and the indicator values in the 

linking period. These statistics are taken from the tables used 

in earlier discussion. 

Recall that CP 

x measures short-term movement preservation, CB

is the statistic to measure the distortion at the breaks between 

years, C2 and CT measure the distortions at the beginning and 

ending periods of the sample, and CL measures how well the newly 

revised final estimates link to the previously benchmarked 

series. These 5 statistics measure the difference in various 

aspects of short-term movements between the final estimates and 

the indicator series. Thus, the smaller the number the better 

the results. 

Table 9-1 is here 

It is clear from Table 9-1 that the modified Denton 

proportional first difference method (DPFD) from BENCH program 

outperforms the other methods, though the differences between 

DPFD, RATIO, and TREND are fairly small. The results from DAFD 

are contaminated by the two cases in which the final estimates 

exhibit very volatile patterns. The results also show that the 

mean statistics are in general larger for regression methods, and 

LPI is inferior to all the other methods. 

Table 9-2 contains the comparative results for the 4 

mathematical methods used to estimate the 30 series from GOVD. 

Because these series are not used to test linking of the newly 

revised data due to lack of pre-revision data, and because they 

C have contemporaneous constraints, CL is replaced with Dx the mean 

of the contemporaneous discrepancy. The left panel shows the 

comparative statistics for the taxes and the right panel shows 

the comparative statistics for the transfer payments. 

Table 9-2 is here 
46 

There is no clear winner in all 5 test criteria. However, 

the modified Denton proportional first difference model (DPFD) 

from BENCH and the Causey-Trager model (TREND) from BMARK are 

close competitors. Again, differences between DPFD, RATIO and 

TREND are only marginal. LPI is inferior to the others methods. 

Table 9-3 compares the mathematical methods used to 

estimate the 15 annual series that have no sub-annual 

information. For the 10 series that have no negative values, the 

statistics used to evaluate the smoothness of the final estimates 

are: 1) gt , the mean of the average period-to-period growth rate 

of the sample; 2) g2 and gT , the mean of the period-to-period 

percentage change in the second and the last period. For the 5 

series with negative values, means of the corresponding periodto-period level changes, ∆xt , ∆x2 and ∆xT , are used instead. 

Table 9-3 is here 

The comparative results show that for series with or 

without negative values in the annual data, the first difference 

BFL approximation provided by BENCH (DAFD) outperforms the other 

methods. The BMARK approximation of the first difference BFL 

model cannot handle the series with negative annual values. The 

LPI and BFLSD methods often generate large percentage change in 

the beginning and ending periods of the final estimates. 

5. Evaluation of the Methods for Temporal Disaggregation in NEA 

We have experimented with 6 mathematical methods, 5 

regression methods and 3 software programs for temporal 

disaggregation in our search for the most suitable method(s) for 

routine compilation of sub-annual estimates through temporal 

distribution and interpolation in the national accounts. We now 

discuss how each method has performed. 
 

 

47 

It is very clear from our results that Lagrange polynomial 

interpolation procedure is inferior to all other methods when 

evaluated according to the 5 test criteria, and thus, is not an 

appropriate choice for temporal distribution and interpolation in 

the national accounts. 

Among the mathematical methods for temporal disaggregation 

of annual series that have sub-annual indicators, the modified 

Denton proportional first difference method (DPFD) from BENCH 

program outperforms the other methods, though the Causey-Trager 

growth preservation method (TREND) is a close competitor in the 

temporal distribution of the tax and transfer payment series. 

BMARK version of the modified Denton proportional first 

difference model (RATIO) slightly underperforms DPFD. Cautions 

should be taken when considering the modified Denton additive 

first difference method (DAFD), because it may generate final 

estimates that exhibit erratic patterns. 

For temporal disaggregation of annual series with no subannual indicator, the BENCH version of the first difference FBL 

smoothing method (DAFD) outperforms the ECOTRIM version of the 

same method (BFLFD) and the second difference smoothing method 

(BFLSD). TREND cannot be used for temporal disaggregation of 

annual series without sub-annual indicators. BMARK version of 

the first difference BFL smoothing model (RATIO) cannot handle 

annual series with negative values. 

We should evaluate the regression methods used in the 

experiment with caution, because these methods are not 

extensively tested. We chose the option of having parameter ρ in 

the autoregressive error process be optimally estimated, and the 

outcome of this choice may be more smoothed final estimates but 

somewhat weaker preservation of the short-term movement in the 

indicator series. There is a trade-off between achieving 

smoothness of the final estimates and strong short-term movement 

preservation. Some studies suggest that setting ρ to a certain 

value helps achieve proper short-term movement preservation. 
48 

More tests with different parameter values should be conducted to 

reach more conclusive evaluation. 

We should also point out that the current revision policy 

in the national accounts limit the advantages that regression 

methods can offer. Most of the series used in the experiment are 

clearly serially correlated. Using regression methods to correct 

serial correlation should improve the final estimates. However, 

because only the most recent 3 years of data can be used in 

estimation, it would be very difficult for any regression method 

to produce reliable results. 

6. Conclusion 

We have experimented with 5 mathematical and 5 regression 

methods for temporal distribution and interpolation using 3 

software programs designed for benchmarking and temporal 

disaggregation. Because we have used a large number of series 

that allow us to study a variety of cases routinely encountered 

in temporal distribution and interpolation at the national 

accounts, we have acquired a good understanding of how each 

method works in each case. Such knowledge provides us with a 

solid basis for selecting the most suitable method(s) for routine 

compilation at the national accounts. 

This study has also helped familiarize us with the frontier 

research in this area. New research on temporal disaggregation 

and new software programs developed for that purpose continue to 

emerge, which should further benefit our research in the future. 

One can see from the estimation results that the mathematical 

methods are easier to use, but they do not correct any serial 

correlation in the time-series data. Thus, in our future 

research, we would like to further examine regression-based 

methods which would help correct serial correlation in the timeseries data. The best method for temporal disaggregation at the 

national accounts should produces sub-annual estimates that 
 

49 

satisfy the criteria used in our evaluation and are also free of 

serial correlation. Toward that goal we shall continue our 

research. 

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Figure 1-1: Annual, Indicator and Interpolated Series 

Monetary Interests (MIRPERSON 1995-2004) 

4 00000 

4 50000 

5 00000 

5 50000 

6 00000 

6 50000 

7 00000 

7 50000 

8 00000 

J an-9 5 Apr-95 

Jul-95

O ct-95J an-9 6 Apr-96 J ul-96 O ct-96 J an-9 7 Apr-97 J ul-97O ct-97J an-9 8

Apr-98 J ul-98

O ct-98 J an-9 9 Apr-99 J ul-99 O ct-99 Jan-0 0 Apr-00 

Jul-00

O ct-00J an-0 1 Apr-01 J ul-01 O ct-01 J an-0 2 Apr-02 J ul-02 O ct-02J an-0 3

Apr-03 

Jul-03

O ct-03 Jan-0 4 Apr-04 J ul-04 O ct-04 

tim e 

Q u a r t e r l y E s t i m a t e s 

I N DI CA T O R * A n n ual LP I DP F D RA T I O TR E N D 

Figure 1-2: Annual, Indicator and Interpolated Series 

Housing owner occupied closing cost (XOTHOTHCCO 1995-2004) 

0 

50 00 

10 00 0 

15 00 0 

20 00 0 

25 00 0 

30 00 0 

35 00 0 

40 00 0 

Jan-9 5 Apr-95

Jul-95

Oct-95Jan-9 6 Apr-96 Jul-96O ct-96 Jan-9 7 Apr-97 Jul-97Oct-97 Jan-9 8

Apr-98

Jul-98 Oct-98 

Jan-9 9 Apr-99 Jul-99Oct-99 Jan-0 0 Apr-00 Jul-00O ct-00 Jan-0 1 Apr-01

Jul-01

Oct-01 Jan-0 2

Apr-02

Jul-02

Oct-02 Jan-0 3 Apr-03 Jul-03 Oct-03Jan-0 4 Apr-04 Jul-04O ct-04 

time 

Q u a rt e rly E s t im a t e s 

I N D I CA T O R* A n nu al LP I DP F D RA T I O TR E N D 
 

53 

Figure 1-3: Annual, Indicator and Interpolated Series 

Crop Output (GFOCXLOSS 1994-2004 (revised data)) 

75 00 0 

80 00 0 

85 00 0 

90 00 0 

95 00 0 

10 00 00 

10 50 00 

J an-9 4

Apr-94 J ul-94 

O ct-94 Jan-9 5

Apr-95 J ul-95 O ct-95 

Jan-9 6

Apr-96

Jul-96 O ct-96 

Jan-9 7

Apr-97

Jul-97 O ct-97 

Jan-9 8 

Apr-98

Jul-98

O ct-98 Jan-9 9 

Apr-99 J ul-99

O ct-99 Jan-0 0 

Apr-00 J ul-00 

O ct-00 Jan-0 1 

Apr-01 J ul-01 

O ct-01 Jan-0 2 

Apr-02 J ul-02 

O ct-02 Jan-0 3 

Apr-03 J ul-03 

O ct-03 Jan-0 4 

Apr-04 J ul-04 

O ct-04 

time 

Q u a r t e r l y E s t i m a t e s 

I N DI CA T O R* A n nu al LP I DP F D DA F D RA T I O TR E N D 

Figure 1-4: Annual, Indicator and Interpolated Series 

Owner-Occupied Mobile Homes (EOMHP 1995-2004 revised data) 

20 000 

24 000 

28 000 

32 000 

36 000 

40 000 

Jan-9 5

Apr-95

Jul-95 

Oct-95 

Jan-9 6

Apr-96

Jul-96 

Oct-96 

Jan-9 7

Apr-97

Jul-97

O ct-97 

Jan-9 8

Apr-98

Jul-98 

Oct-98 

Jan-9 9

Apr-99

Jul-99

O ct-99 

Jan-0 0

Apr-00

Jul-00 

Oct-00 

Jan-0 1 Apr-01

Jul-01 O ct-01 

Jan-0 2

Apr-02

Jul-02 

Oct-02 

Jan-0 3

Apr-03

Jul-03

O ct-03 

Jan-0 4 

Apr-04 Jul-04 

Oct-04 

time 

M o n t h ly E s t im a t e s 

I N D I CA T O R* A n n ual LP I DP F D DA F D RA T I O TR E N D 
 

54 

Figure 1-5: Annual, Indicator and Interpolated Series 

Tennant-Occupied Mobile Homes (ETMHP 1995-2004 revised data) 

470 0 

490 0 

510 0 

530 0 

550 0 

570 0 

590 0 

610 0 

630 0 

Jan-9 5 

Apr-95 Jul-95 O ct-95 Jan-9 6Apr-96

Jul-96 

Oct-96 Jan-9 7

Apr-97 Jul-97

O ct-97 Jan-9 8 Apr-98 Jul-98O ct-98 Jan-9 9 

Apr-99 Jul-99

O ct-99 Jan-0 0

Apr-00

Jul-00 O ct-00 Jan-0 1

Apr-01

Jul-01

O ct-01 Jan-0 2

Apr-02

Jul-02

O ct-02 Jan-0 3 Apr-03 Jul-03O ct-03 Jan-0 4 

Apr-04 Jul-04

O ct-04 

time 

M o n t h ly E s t im a t e s 

I N DI CA T O R * A nnua l LP I DP F D DA F D RA T I O TR E N D 

Figure 1-6: Annual, Indicator and Interpolated Series 

Livestock Output (GFOL$ 1994-2004 revised data) 

80000 

85000 

90000 

95000 

100000 

105000 

110000 

115000 

Jan-9 4

Apr-94

Jul-94 

Oct-94

Jan-9 5

Apr-95

Jul-95 

Oct-95

Jan-9 6

Apr-96

Jul-96 

Oct-96

Jan-9 7

Apr-97

Jul-97 

Oct-97

Jan-9 8

Apr-98

Jul-98 

Oct-98

Jan-9 9

Apr-99 Jul-99 

Oct-99 

Jan-0 0 Apr-00 Jul-00 

Oct-00 

Jan-0 1

Apr-01 Jul-01 

Oct-01 

Jan-0 2

Apr-02

Jul-02 O ct-02 Jan-0 3

Apr-03

Jul-03 O ct-03 

time 

Q u a rt e rly E s t im a t e s 

I NDI CA T O R* A nnual LP I DP F D DA F D RA T I O TR E N D 
55 

Figure 1-7a: Annual, Indicator and Interpolated Series 

Gasoline Consumption (E1GAS1 1993-2004) 

0 

50 000 

100 000 

150 000 

200 000 

250 000 

300 000 

350 000 

400 000 

450 000 

Jan-9 3

Apr-93 Jul-93 

O ct-93 Jan-9 4

Apr-94

Jul-94 

O ct-94J an-9 5 Apr-95 Jul-95 O ct-95 Jan-9 6

Apr-96

Jul-96 

O ct-96 Jan-9 7 Apr-97 Jul-97 O ct-97Jan-9 8

Apr-98

Jul-98 

O ct-98 Jan-9 9 Apr-99 Jul-99 O ct-99Jan-0 0

Apr-00 Jul-00 

O ct-00 Jan-0 1

Apr-01

Jul-01 

O ct-01

J an-0 2 Apr-02 Jul-02 

O ct-02 Jan-0 3

Apr-03

Jul-03 

O ct-03 Jan-0 4 Apr-04 Jul-04 O ct-04 

time 

M o n t h l y E s t i m a t e s 

I NDI CA T O R* A nnua l LP I DP F D DA F D RA T I O TR E N D 

Figure 1-7b: Annual, Indicator and Interpolated Series 

Gasoline Consumption (E1GAS1 1993-2004) 

100000 

120000 

140000 

160000 

180000 

200000 

220000 

240000 

Jan-9 3 

Apr-93 Jul-93

O ct-93Jan-9 4 

Apr-94 Jul-94 Oct-94 Jan-9 5 Apr-95

Jul-95 

Oct-95 Jan-9 6 Apr-96 Jul-96 Oct-96Jan-9 7 

Apr-97 Jul-97

O ct-97 

Jan-9 8 

Apr-98 Jul-98 Oct-98

Jan-9 9 Apr-99

Jul-99 

Oct-99 Jan-0 0 

Apr-00 Jul-00

O ct-00Jan-0 1 

Apr-01 Jul-01

O ct-01

Jan-0 2 

Apr-02 Jul-02 Oct-02 Jan-0 3 Apr-03 Jul-03 

Oct-03 Jan-0 4 

Apr-04

Jul-04 

Oct-04 

time 

M o n t h ly E s t im a t e s 

I NDI CA T O R* A n n ual LP I DP F D RA T I O TR E N D 
56 

Figure 1-8a: Annual, Indicator and Interpolated Series 

Investment in Pre-Packaged Equipment & Software (SFTWRE 1997-2004) 

-1 000 

-500 

0 

500 

1000 

1500 

Jan-9 7 

Apr-97

Jul-97 

Oct-97 Jan-9 8 

Apr-98 

Jul-98 

Oct-98Jan-9 9 

Apr-99

Jul-99 

Oct-99 Jan-0 0 

Apr-00

Jul-00 

Oct-00 Jan-0 1 

Apr-01

Jul-01 

Oct-01

Jan-0 2 

Apr-02

Jul-02 

Oct-02 Jan-0 3 

Apr-03

Jul-03 

Oct-03 Jan-0 4

Apr-04

Jul-04 

Oct-04 

time 

Q u a r t e r ly E s t im a t e s 

I NDI CA T O R * A n nual LP I DP F D DA F D RA T I O TR E N D 

Figure 1-8b: Annual, Indicator and Interpolated Series 

Investment in Pre-Packaged Equipment & Software (SFTWRE 1997-2004) 

30 

35 

40 

45 

50 

55 

60 

Jan-9 7

Apr-97

Jul-97 

Oct-97

Jan-9 8

Apr-98

Jul-98

O ct-98 

Jan-9 9

Apr-99

Jul-99

Oct-99 

Jan-0 0 

Apr-00

Jul-00

Oct-00

Jan-0 1

Apr-01

Jul-01 

Oct-01

Jan-0 2

Apr-02

Jul-02

O ct-02 

Jan-0 3 

Apr-03

Jul-03

Oct-03

Jan-0 4 

Apr-04

Jul-04

Oct-04 

time 

Q u a rt e rly E s t im a t e s 

IN DIC A T O R* A n n ual LP I DP F D RA T I O TR E N D 
J ul-93

J ul-94J ul-95J ul-96J ul-97 J ul-98Jul-99Jul-00 

J ul-01

J ul-02J ul-03J ul-04

Jan-9 3 J an-9 4 J an-9 5 Jan-9 6 J an-9 7 J an-9 8 Jan-9 9 J an-0 0 Jan-0 1 J an-0 2 J an-0 3 Jan-0 4 O ct-93 O ct-94 O ct-95 O ct-96 O ct-97O ct-98O ct-99 O ct-00 O ct-01 O ct-02 O ct-03 O ct-04 

Apr-93Apr-94Apr-95Apr-96Apr-97Apr-98Apr-99Apr-00Apr-01

Apr-02

Apr-03Apr-04

Apr-93

Jul-93

J ul-94

J ul-95

Jul-96 

J ul-97

Jul-98

J ul-99J ul-00

Jul-01

Jul-02

Jul-03

J ul-04

J an-9 3J an-9 4Jan-9 5 Jan-9 6J an-9 7J an-9 8J an-9 9 Jan-0 0Jan-0 1J an-0 2J an-0 3J an-0 4 O ct-93O ct-94 O ct-95Oct-96O ct-97O ct-98O ct-99 O ct-00O ct-01

O ct-02

O ct-03O ct-04 

Apr-94

Apr-95

Apr-96

Apr-98

Apr-99

Apr-00

Apr-01

Apr-04

Apr-97

Apr-02

Apr-03

57

Figure 1-9: Annual, Indicator and Interpolated Series 

Gasoline Consumption (E1GAS1 1993-2004) 

300000 

250000 

M o n th ly Estim ates

200000 

150000 

100000 

50000 

0 

time 

 AR ( 1 ) M A X A n nual RA W K R A W K M MA X A R (1 ) M I N RA W K M M I N I N DI C A T O R* LP I

25000 

Figure 1-10: Annual, Indicator and Interpolated Series 

Nonresidential Telecommunications (CURTUM 1993-2004) 

20000 

M o n t h l y E s t i m a t e s 15000 

10000 

5000 

0 

time 

I NDI C A T O R* A nnual LP I A R (1 ) M A X RA W K R A W K M MA X A R (1 ) M I N R A W K M MI N
58 

Figure 2-1: Absolute difference in period-to-period growth rate 

Monetary Interest (MIRPERSON 1995-2004) 

0. 00 

0. 01 

0. 01 

0. 02 

0. 02 

0. 03 

0. 03 

0. 04 

0. 04 

0. 05 

0. 05 

Jan-9 5

Apr-95

Jul-95

O ct-95 Jan-9 6 Apr-96

Jul-96 Oct-96

Jan-9 7

Apr-97

Jul-97 

Oct-97 Jan-9 8 Apr-98 Jul-98O ct-98 Jan-9 9

Apr-99

J ul-99 

Oct-99 Jan-0 0

Apr-00

Jul-00

O ct-00 Jan-0 1 Apr-01

Jul-01 Oct-01 Jan-0 2

Apr-02

Jul-02 

Oct-02 Jan-0 3 Apr-03 Jul-03O ct-03 Jan-0 4

Apr-04

J ul-04 

Oct-04 

time 

P ro p o rt io n a l C o r re c t io n 

LP I DP F D RA T I O TR E N D 

Figure 2-2: Absolute difference in period-to-period growth rate 

Housing Owner-Occupied Closing Cost (XOTHOTHCCO 1995-2005) 

0.0 0 

0.0 5 

0.1 0 

0.1 5 

0.2 0 

0.2 5 

0.3 0 

Jan-9 5 Apr-95 Jul-95 Oct-95 

Jan-9 6 

Apr-96

Jul-96 

Oct-96 Jan-9 7 Apr-97

Jul-97 

Oct-97 Jan-98 

Apr-98

Jul-98 

Oct-98 Jan-9 9 Apr-99 Jul-99 

Oct-99 Jan-0 0 Apr-00 Jul-00 Oct-00 

Jan-0 1 Apr-01 Jul-01 

Oct-01 

Jan-0 2 

Apr-02

Jul-02 O ct-02 Jan-0 3 

Apr-03

Jul-03 O ct-03 Jan-0 4 

Apr-04

Jul-04 O ct-04 Jan-05 

Apr-05

Jul-05 

Oct-05 

time 

P ro p o rt io n a l C o rre c t io n 

DP F D RA T I O TR E N D 
59 

Figure 2-3: Absolute Difference in Period-to-Period Growth Rate 

Owner-Occupied Stationary Homes (EOSTP 1995-2004 revised data) 

0 

0. 000 5 

0. 00 1 

0. 001 5 

0. 00 2 

0. 002 5 

0. 00 3 

0. 003 5 

Jan-9 5 Apr-95

Jul-95

O ct-95 Jan-9 6 

Apr-96

Jul-96

O ct-96 Jan-9 7 

Apr-97

Jul-97 

O ct-97 Jan-9 8 

Apr-98 J ul-98 

O ct-98 Jan-9 9 Apr-99

Jul-99

O ct-99 Jan-0 0 

Apr-00

Jul-00 O ct-00 

Jan-0 1 

Apr-01 J ul-01 

O ct-01 Jan-0 2 Apr-02 J ul-02 O ct-02 Jan-0 3 Apr-03

Jul-03 O ct-03 Jan-0 4 

Apr-04 Jul-04 O ct-04 

time 

P r o p o r t i o n a l C o r r e c t i o n 

LP I DP F D DA F D RA T I O TR E N D 

Figure 2-4: Absolute Difference in Period-to-Period Growth Rate 

Tennant-Occupied Mobile Homes (ETMHP 1995-2004 revised data) 

0 

0.00 5 

0. 01 

0.01 5 

0. 02 

0.02 5 

Jan-9 5 Apr-95 Jul-95Oct-95 Jan-9 6 

Apr-96 Jul-96 Oct-96 Jan-9 7Apr-97

Jul-97

O ct-97 Jan-9 8 Apr-98 Jul-98 Oct-98 Jan-9 9 Apr-99 Jul-99Oct-99 Jan-0 0 Apr-00 Jul-00 Oct-00 Jan-0 1 Apr-01 Jul-01

Oct-01 Jan-0 2

Apr-02

Jul-02 Oct-02 Jan-0 3 Apr-03 Jul-03

Oct-03 Jan-0 4 Apr-04 Jul-04 Oct-04 

time 

P r o p o r t io n a l C o rr e c t io n 

LP I DP F D DA F D RA T I O TR E N D 
 

Jul-9

Jul-01

4

Jul-95 

Jul-96

Jul-97Jul-98

Jul-99

Jul-00

Jul-02

Jul-03

Jan-9

Oct-01 

4 

Jan-9 5 Jan-9 6 Jan-9 7 Jan-9 8 Jan-9 9 Jan-0 0 Jan-0 1 

Jan-0 2 

Jan-0 3 Oct-94 Oct-95 Oct-96 Oct-97 Oct-98 Oct-99 Oct-00 Oct-02 Oct-03 

Apr-94

Apr-95

Apr-96

Apr-97

Apr-98

Apr-99

Apr-00

Apr-01

Apr-02

Apr-03

60 

Figure 2-5: Absolute Difference in Period-to-Period Growth Rate 

Crop Output (GFOCXLOSS$ 1994-2004 revised data) 

0 

0. 005 

0. 01 

0. 015 

0. 02 

0. 025 

0. 03 

Jan-9 4

Apr-94

J ul-94 Oct-94 

Jan-9 5

Apr-95

Jul-95

O ct-95 Jan-9 6

Apr-96

J ul-96 Oct-96 

Jan-9 7

Apr-97

Jul-97 

O ct-97 Jan-9 8

Apr-98

J ul-98 Oct-98 

Jan-9 9

Apr-99

Jul-99

O ct-99 Jan-0 0

Apr-00

J ul-00 Oct-00 

Jan-0 1

Apr-01

Jul-01 

O ct-01 

Jan-0 2

Apr-02

J ul-02 Oct-02 

Jan-0 3

Apr-03

Jul-03 

O ct-03 Jan-0 4

Apr-04

J ul-04 Oct-04 

time 

P r o p o r t i o n a l C o r r e c t i o n 

LP I DP F D DA F D RA T I O TR E N D 

Figure 2-6: Absolute Difference in Period-to-Period Growth Rate 

Livestock Output (GFOL$ 1994-2004 revised data) 

0 . 000 

0 . 020 

0 . 040 

0 . 060 

0 . 080 

0 . 100 

0 . 120 

0 . 140 

0 . 160 

0 . 180 

P r o p o rt io n a l C o r re c t io n 

time 

LPI DPFD DAFD RATIO TREND 
61 

Figure 2-7: Absolute Difference in Period-to-Period Growth Rate 

Nonresidential Telecommunications (CURTUM 1993-2004) 

0 

0. 0 1 

0. 0 2 

0. 0 3 

0. 0 4 

0. 0 5 

0. 0 6 

0. 0 7 

0. 0 8 

0. 0 9 

Jan-9 3 

Apr-93

Jul-93

O ct-93 J an-9 4 

Apr-94

Jul-94

O ct-94 J an-9 5 Apr-95 Jul-95 O ct-95 Jan-9 6 

Apr-96

Jul-96

O ct-96 J an-9 7 

Apr-97

Jul-97

O ct-97 J an-9 8

Apr-98

Jul-98

O ct-98 J an-9 9 Apr-99 Jul-99 O ct-99 Jan-0 0 

Apr-00

Jul-00 

O ct-00 J an-0 1

Apr-01

Jul-01

O ct-01 J an-0 2 Apr-02 Jul-02 O ct-02 Jan-0 3 

Apr-03

Jul-03 

O ct-03 Jan-0 4 

Apr-04

Jul-04 

O ct-04 

tim e 

P r o p o r t i o n a l C o r r e c t i o n 

LP I DP F D DA F D RA T I O TR E N D 

Figure 2-8: Absolute Difference in Period-to-Period Growth Rate 

Nonresidential Manufacturing (CURMFG 1993-2004) 

0 

0. 01 

0. 02 

0. 03 

0. 04 

0. 05 

0. 06 

0. 07 

Jan-9 3 Apr-93 Jul-93 Oct-93 Jan-9 4 Apr-94 Jul-94 Oct-94 Jan-9 5 Apr-95 Jul-95 Oct-95 Jan-9 6 Apr-96 Jul-96 Oct-96 Jan-9 7 Apr-97 Jul-97 Oct-97 Jan-9 8 Apr-98 Jul-98 Oct-98 Jan-9 9 Apr-99 Jul-99 Oct-99 Jan-0 0 Apr-00 Jul-00 Oct-00 Jan-0 1 Apr-01 Jul-01

Oct-01 Jan-0 2 Apr-02 Jul-02 Oct-02 Jan-0 3 Apr-03 Jul-03 Oct-03 Jan-0 4 Apr-04 Jul-04 Oct-04 

time 

P ro p o r t io n a l C o r r e c t io n 

LP I DP F D DA F D RA T I O TR E N D 
62 

Figure 2-9a: Absolute Difference in Period-to-Period Growth Rate 

Gasoline Consumption (E1GAS1 1993-2004) 

0.0 0 

1.0 0 

2.0 0 

3.0 0 

4.0 0 

5.0 0 

6.0 0 

7.0 0 

8.0 0 

9.0 0 

10 . 0 0 

J an-9 3

Apr-93 Jul-93

O ct-93 Jan-9 4

Apr-94 Jul-94 O ct-94 

Jan-9 5

Apr-95 Jul-95

O ct-95 Jan-9 6 

Apr-96

Jul-96 

O ct-96 J an-9 7

Apr-97

Jul-97 

O ct-97 J an-9 8

Apr-98 Jul-98 O ct-98 

Jan-9 9 

Apr-99

Jul-99

O ct-99 Jan-0 0 

Apr-00

Jul-00

O ct-00 J an-0 1 

Apr-01

Jul-01

O ct-01 J an-0 2 Apr-02 Jul-02O ct-02 J an-0 3

Apr-03 Jul-03 O ct-03 

Jan-0 4 

Apr-04

Jul-04 O ct-04 

tim e 

P r o p o r t i o n a l C o r r e c t i o n 

LP I DP F D RA T I O TR E N D DA F D 

Figure 2-9b: Absolute Difference in Period-to-Period Growth Rate 

Gasoline Consumption (E1GAS1 1993-2004) 

0. 00 

0. 01 

0. 01 

0. 02 

0. 02 

0. 03 

0. 03 

0. 04 

0. 04 

0. 05 

0. 05 

Jan-9 3 

Apr-93

Jul-93 Oct-93

J an-9 4 

Apr-94 Jul-94 

Oct-94 Jan-9 5 Apr-95 Jul-95 Oct-95 Jan-9 6 

Apr-96 Jul-96 Oct-96

Jan-9 7 

Apr-97 J ul-97 Oct-97 Jan-9 8 Apr-98 Jul-98 Oct-98 

Jan-9 9 

Apr-99 Jul-99

Oct-99 Jan-0 0 

Apr-00

Jul-00 Oct-00 Jan-0 1 

Apr-01 Jul-01 

Oct-01 

Jan-0 2 

Apr-02

Jul-02 Oct-02 Jan-0 3 

Apr-03 Jul-03 Oct-03 Jan-0 4 Apr-04 Jul-04

O ct-04 

tim e 

P r o p o r t i o n a l C o r re c t io n 

LP I DP F D RA T I O TR E N D 
 

 

Jul-97 Jul-98 J ul-99 Jul-00 Jul-01 Jul-02 Jul-03 Jul-04 

J an-9 7 Jan-9 8 Jan-9 9 J an-0 0 Jan-0 1Jan-0 2 J an-0 3 Jan-0 4 O ct-97 O ct-98 O ct-99 O ct-00 O ct-01 O ct-02 O ct-03 O ct-04 

Apr-97 Apr-98 Apr-99 Apr-00 Apr-01 Apr-02 Apr-03 Apr-04 

J ul-97 J ul-98 J ul-99 J ul-00 

Apr-01

J ul-01 

J ul-02 J ul-03 J ul-04 

Jan-9 7 Jan-9 8 Jan-9 9Jan-0 0 Jan-0 1Jan-0 2 Jan-0 3Jan-0 4 O ct-97 O ct-98 O ct-99 O ct-00 O ct-01 O ct-02 O ct-03 O ct-04 

Apr-97 Apr-98 Apr-99 Apr-00 Apr-02 Apr-03 Apr-04 

63 

Figure 2-10a: Absolute Difference in Period-to-Period Growth Rate 

Investment in Pre-Packaged Equipment & Software (SFTWRE 1997-2004) 

0.00 

20.00 

40.00 

60.00 

80.00 

100.00 

120.00 

P r o p o r t i o n a l C o r r e c t i o n 

time 

LPI DPFD RATIO TREND DAFD 

Figure 2-10b: Absolute Difference in Period-to-Period Growth Rate 

Investment in Pre-Packaged Equipment & Software (SFTWRE 1997-2004) 

-0 . 0 1 

0. 00 

0. 01 

0. 01 

0. 02 

0. 02 

0. 03 

0. 03 

0. 04 

0. 04 

0. 05 

P r o p o r t i o n a l C o r r e c t i o n 

tim e 

LPI DPFD RATIO TREND 
 

 

Jan-0 1

Jul-01 

Jul-95

Jul-96

Jul-97Jul-98 Jul-99

Apr-00

Jul-00

Jul-02

Jul-03Jul-04

Jan-9 5 Oct-95 Jan-9 6 O ct-96

Jan-9 7 

Jan-9 8 

Jan-9 9 

Oct-99 Jan-0 0 

Jan-0 2 

Oct-02 Jan-0 3 

Jan-0 4 

O ct-97

Oct-98 

O ct-04 

O ct-00

Oct-01 

O ct-03

Apr-95

Apr-97

Apr-98

Apr-99

Apr-01

Apr-02

Apr-04

Apr-96

Apr-03

Jul-94

Jul-97 

Jan-9 9

Jul-99 

Jul-01

Jul-04 

Jan-9 4 

Jul-96

Jul-98

Jan-0 1 

Jul-03

Oct-94 

Jul-95 

Oct-96 

Oct-98 

Jul-00

Oct-01 

Jul-02 

Jan-95 

Jan-96 

Jan-97 

Jan-98 

Jan-00 

Jan-02 

Jan-03 

Oct-03 Jan-04 

Oct-00

Oct-95

Oct-97

Oct-99

Oct-02

Oct-04 

Apr-94

Apr-95

Apr-96

Apr-97

Apr-98

Apr-99

Apr-00

Apr-01

Apr-02

Apr-03

Apr-04

Proportional Correction Proportional Correction

64 

Figure 2-11: Absolute Difference in Period-to-Period Growth Rate 

Tennant-Occupied Mobile Homes (ETMHP 1995-2004 revised data) 

0.009 

0.008 

0.007 

0.006 

0.005 

0.004 

0.003 

0.002 

0.001 

0 

A R (1 ) M A X R A W K

time 

R A W K M MA X AR ( 1 ) M I N R A W K M M I N 

0. 0 3 

Figure 2-12: Absolute Difference in Period-to-Period Growth Rate 

Crop Output (GFOCXLOSS 1994-2004 revised data) 

0. 02 5 

0. 0 2 

0. 01 5 

0. 0 1 

0. 00 5 

0 

time 

AR(1) MAX RAWK RAWKM MAX AR(1) MIN RAWKM MIN LIP 
 

 

Jul-94

Jul-97

Jul-01

Jul-02

Jul-04

Jul-93

Jul-95Jul-96

Jul-98

Jul-99Jul-00

Jan-0 1 

Jul-03 

Jan-9 3 

Jan-9 4 

O c

Oct-00

t-94 

Jan-9 5 

O ct-95 Jan-9 6 

O ct-96 Jan-9 7 

O ct-97 Jan-9 8 Jan-9 9 Jan-0 0 

O ct-01 Jan-0 2 

Jan-0 3 

Jan-0 4 

O ct-93 

Oct-98 

Oct-99 

Oct-02

Oct-03

Oct-04 

Apr

Apr-95

Apr-96

Apr-97

Apr-01

Apr-02

-93

Apr-94

Apr-98

Apr-99

Apr-00

Apr-03

Apr-04

Jul-93 

Jul-94 

Jul-95 

Jan-9 3 

Jan-9 5 

Jan-9 6 

Jul-96

Jul-99

Jul-00

Jul-01 

Jul-02

Jul-03

Jan-0 4 

Jul-97 

Jul-98 

O ct-00 

Jul-04 

Jan-9 4 

Jan-9 7 

Jan-9 8 

Jan-99 

Jan-0 0 

Jan-0 1 

Jan-0 2 

Jan-0 3 

Oct-93

Oct-94

Oct-95

O ct-96

Oct-97

Oct-98

O ct-99

Oct-01

Oct-02

Oct-03

Oct-04 

Apr-93

Apr-94

Apr-95

Apr-96

Apr-97

Apr-98

Apr-99

Apr-00

Apr-01

Apr-02

Apr-03

Apr-04

Proportional Correction Proportional Correction

65 

Figure 2-13: Absolute Difference in Period-to-Period Growth Rate 

Gasoline Consumption (E1GAS1 1993-2004) 

0.007 

0.006 

0.005 

0.004 

0.003 

0.002 

0.001 

0 

time 

A R (1 ) M A X RA W K R A W K M MA X A R (1 ) M I N R A W K M MI N 

Figure 2-14: Absolute Difference in Period-to-Period Growth Rate 

Nonresidential Telecommunications (CURTUM 1993-2004) 

0. 0 1 4 

0. 0 1 2 

0. 0 1 

0. 0 0 8 

0. 0 0 6 

0. 0 0 4 

0. 0 0 2 

0 

time 

AR(1) MAX RAWK RAWKM MAX AR(1) MIN RAWKM MIN 
66 

Figure 3-1: Absolute Difference in Period-to-Period Growth Rate 

Taxes on Retail Trade (TPCRT 1997-2004) 

0. 000 

0. 002 

0. 004 

0. 006 

0. 008 

0. 010 

0. 012 

0. 014 

Jan-9 7 

Apr-97

Jul-97

Oct-97

Jan-9 8 

Apr-98

Jul-98

Oct-98 

Jan-9 9 

Apr-99

Jul-99 

Oct-99

Jan-0 0 

Apr-00

Jul-00

Oct-00 

Jan-0 1 

Apr-01

Jul-01 

Oct-01

Jan-0 2 

Apr-02

Jul-02

Oct-02 

Jan-0 3 

Apr-03

Jul-03

Oct-03 

Jan-0 4 

Apr-04

Jul-04

Oct-04 

time 

Pro p o rtio n al C o rrectio n 

LP I DP F D RA T I O TR E N D 

Figure 3-2: Absolute Difference in Period-to-Period Growth Rate 

Taxes on Wholesale Trade (TPCWT 1997-2004) 

0 . 000 

0 . 002 

0 . 004 

0 . 006 

0 . 008 

0 . 010 

0 . 012 

Jan-97 

Apr-97

Jul-97 

Oct-97 

Jan-98 

Apr-98

Jul-98 

Oct-98 

Jan-99 

Apr-99

Jul-99 

Oct-99 

Jan-00 

Apr-00 

Jul-00

Oct-00 

Jan-01 

Apr-01

Jul-01

Oct-01 

Jan-02 

Apr-02

Jul-02 

Oct-02 

Jan-03 

Apr-03 

Jul-03

Oct-03 

Jan-04 

Apr-04

Jul-04

Oct-04 

time 

Proportional Correction 

LP I DP F D RA T I O TR E N D 
 

67 

Figure 3-3: Absolute Difference in Period-to-Period Growth Rate 

Transfers from Mining (TRCMI 1997-2004) 

0 

0. 0 1 

0. 0 2 

0. 0 3 

0. 0 4 

0. 0 5 

0. 0 6 

Jan-97 

Apr-97

Jul-97 

Oct-97 

Jan-98 

Apr-98

Jul-98 

Oct-98 

Jan-99 

Apr-99

Jul-99 

Oct-99 

Jan-0 0

Apr-00

Jul-00

Oct-00

Jan-01 

Apr-01

Jul-01

Oct-01 

Jan-02 

Apr-02

Jul-02

Oct-02 

Jan-03 

Apr-03

Jul-03 

Oct-03 

Jan-04 

Apr-04

Jul-04 

Oct-04 

time 

Pro p o rtio n al C o rrectio n 

LP I DP F D RA T I O TR E N D 

Figure 3-4: Absolute Difference in Period-to-Period Growth Rate 

Transfers from Transportation and Warehousing (TRCTW 1997-2004) 

0. 00 

0. 01 

0. 02 

0. 03 

0. 04 

0. 05 

0. 06 

Jan-97 

Apr-97

Jul-97

Oct-97 

Jan-98 

Apr-98

Jul-98

Oct-98 

Jan-99 

Apr-99

Jul-99

Oct-99 

Jan-00 

Apr-00

Jul-00

Oct-00

Jan-01 

Apr-01

Jul-01 

Oct-01

Jan-02 

Apr-02

Jul-02

Oct-02 

Jan-03 

Apr-03

Jul-03

Oct-03 

Jan-04 

Apr-04

Jul-04

Oct-04 

time 

Pro p o rtio n al C o rrectio n 

LP I DP F D RA T I O TR E N D 
Jul-93

J ul-94J ul-95 Jul-96Jul-97Jul-98J ul-99Jul-00

Jul-01

Jul-02J ul-03Jul-04 

J an-9 3 J an-9 4 Jan-9 5 Jan-9 6J an-9 7 J an-9 8 J an-9 9 Jan-0 0Jan-0 1 J an-0 2 J an-0 3 Jan-0 4 O ct-93O ct-94 O ct-95O ct-96O ct-97O ct-98O ct-99 O ct-00O ct-01O ct-02O ct-03 O ct-04 

Apr-93Apr-94Apr-95Apr-96Apr-97Apr-98Apr-99Apr-00Apr-01

Apr-02

Apr-03Apr-04

68 

Figure 4-1: Final Sub-Annual Estimate 

TDI Benefits (CGNTRTD 1993-2004) 

0 

500 

1000 

1500 

2000 

2500 

3000 

3500 

4000 

4500 

J an-9 3 Apr-93 J ul-93 O ct-93 J an-9 4 Apr-94 J ul-94 O ct-94 J an-9 5 

Apr-95 J ul-95 O ct-95 J an-9 6 Apr-96 J ul-96 O ct-96 J an-9 7 Apr-97 J ul-97

O ct-97 J an-9 8 

Apr-98 Jul-98

O ct-98 J an-9 9 

Apr-99 Jul-99

O ct-99 J an-0 0 

Apr-00 Jul-00

O ct-00 Jan-0 1 

Apr-01 Jul-01

O ct-01Jan-0 2 

Apr-02 Jul-02

O ct-02J an-0 3 

Apr-03 J ul-03 

O ct-03 Jan-0 4 

Apr-04

Jul-04 

O ct-04 

time 

LP I DA F D BF L F D BF L S D RA T I O A nnual 

Figure 4-2: Final Sub-Annual Estimate 

Outer Continental Shelf Royalties (IBT811 1993-2004) 

0 

20 0 

40 0 

60 0 

80 0 

10 00 

12 00 

14 00 

16 00 

18 00 

20 00 

time 

LPI DAFD BFLFD BFLSD RATIO Annual 
0 

Jul-93 Jul-94 Jul-95 Jul-96Jul-97 Jul-98 Jul-99 Jul-00 

Jul-01 

J ul-02 J ul-03J ul-04

J an-9 3 J an-9 4 J an-9 5 J an-9 6 J an-9 7 J an-9 8 J an-9 9 Jan-0 0 Jan-0 1 Jan-0 2 Jan-0 3 Jan-0 4 O ct-93O ct-94O ct-95Oct-96O ct-97O ct-98O ct-99 O ct-00 O ct-01 O ct-02 Oct-03 Oct-04 

Apr-01

Apr-93

Apr-94Apr-95Apr-96Apr-97Apr-98Apr-99Apr-00Apr-02Apr-03Apr-04

J ul-93 

J ul-94 

J ul-95 

J ul-96 J ul-97 J ul-98J ul-99 J ul-00J ul-01J ul-02J ul-03J ul-04

Jan-9

Oct-01 

3 

Jan-9 4 Jan-9 5 Jan-9 6 Jan-9 7 Jan-9 8 Jan-9 9 Jan-0 0 Jan-0 1 

Jan-0 2 

Jan-0 3 Jan-0 4 O ct-93 O ct-94 O ct-95 O ct-96 O ct-97 Oct-98 O ct-99 Oct-00 Oct-02 Oct-03 Oct-04 

Apr-93

Apr-94

Apr-95

Apr-96

Apr-97

Apr-98

Apr-99

Apr-00

Apr-01

Apr-02

Apr-03

Apr-04

69 

Figure 4-3: Final Sub-Annual Estimate 

Nonresidential Other Power (CUROTHPOW 1993-2004) 

12000 

10000 

8000 

6000 

4000 

2000 

time 

LP I BF L F D BF L S D RA T I O DP F D A n nual 

Figure 4-4: Final Sub-Annual Estimate 

Maritime Operating (SUB420 1993-2004) 

35 0 

30 0 

25 0 

20 0 

15 0 

10 0 

50 

0 

time 

LPI DAFD BFLFD BFLSD RATIO Annual 
70 

Figure 4-5: Final Sub-Annual Estimate 

OPIC (ENT210 1993-2004) 

-1 5 

-1 0 

-5 

0 

5 

10 

15 

20 

Jan-9 3 Apr-93 Jul-93 O ct-93 Jan-9 4 Apr-94 J ul-94 O ct-94 Jan-9 5 Apr-95 J ul-95O ct-95 J an-9 6 Apr-96 J ul-96O ct-96 Jan-9 7 Apr-97 J ul-97O ct-97 J an-9 8 Apr-98 J ul-98O ct-98 J an-9 9

Apr-99 J ul-99

O ct-99 J an-0 0

Apr-00 J ul-00

O ct-00 J an-0 1 

Apr-01 J ul-01

O ct-01 J an-0 2 

Apr-02 J ul-02

O ct-02 J an-0 3

Apr-03 J ul-03

O ct-03 J an-0 4 

Apr-04

Jul-04 

O ct-04 

time 

LP I DA F D BF L F D BF L S D A nnu al 

Figure 4-6: Final Sub-Annual Estimate 

National Flood Insurance (ENT390 1993-2004) 

-2 0 0 

-1 5 0 

-1 0 0 

-5 0 

0 

50 

10 0 

15 0 

20 0 

Jan-9 3

Apr-93

Jul-93 Oct-93 Jan-9 4

Apr-94 Jul-94 Oct-94 Jan-9 5 Apr-95 Jul-95

Oct-95 Jan-9 6 

Apr-96 Jul-96

O ct-96 Jan-9 7

Apr-97

Jul-97 Oct-97 Jan-9 8

Apr-98 Jul-98 Oct-98 Jan-9 9 Apr-99 Jul-99

Oct-99 Jan-0 0 

Apr-00 Jul-00

O ct-00 Jan-0 1

Apr-01

Jul-01

Oct-01 Jan-0 2

Apr-02 Jul-02 Oct-02 Jan-0 3 Apr-03 Jul-03

Oct-03 Jan-0 4

Apr-04

Jul-04

O ct-04 

time 

LP I DA F D BF L F D BF L S D A nnual 
J ul-93

Apr-9J u4 

l-94 

Apr-9J u5 

l-95

J ul-96 

Apr-9 J u 7 

l-97

J ul-98

Apr-9J u9 

l-99 

AprJ u00 

l-00

Apr-0 J u 1 

l-01 

Apr-0J u2 

l-02

J ul-03 

Apr-0J u4 

l-04

J an-9 3J an-9 4 J an-9 5Jan-9 6 J an-9 7J an-9 8 J an-9 9J an-0 0Jan-0 1 J an-0 2Jan-0 3 J an-0 4O ct-93 O ct-94O ct-95O ct-96 O ct-97O ct-98 O ct-99O ct-00O ct-01O ct-02O ct-03 O ct-04 

Apr-93Apr-96Apr-98Apr-03

J ul-93 Jul-94J ul-95 Jul-96J ul-97 Jul-98J ul-99J ul-00

J ul-01

J ul-02J ul-03J ul-04

Jan-9 3J an-9 4 Jan-9 5 Jan-9 6 Jan-9 7 Jan-9 8 Jan-9 9 Jan-0 0 J an-0 1 J an-0 2 J an-0 3 J an-0 4O ct-93 O ct-94O ct-95 O ct-96O ct-97 O ct-98O ct-99 O ct-00 O ct-01O ct-02 O ct-03O ct-04 

Ap

Apr-01

r-93

Apr-94Apr-95Apr-96Apr-97Apr-98Apr-99Apr-00Apr-02Apr-03Apr-04

71 

Figure 4-7: Period-to-Period Growth Rate 

TDI Benefits (CGNTRD 1993-2004) 

-1 0 

-8 

-6 

-4 

-2 

0 

2 

4 

6 

8 

10 

g( t-1 ) 

time 

LPI DAFD BFLFD BFLSD RATIO 

Figure 4-8: Period-to-Period Growth Rate 

Outer Continental Shelf Royalties (IBT811 1993-2004) 

-2 0 

-1 0 

0 

10 

20 

30 

40 

g ( t - 1 ) 

time 

LPI DAFD BFLFD BFLSD RATIO 
Jul-93 

Jul-94Jul-95Jul-96Jul-97J ul-98 Jul-99Jul-00 

J ul-01

Jul-02 Jul-03Jul-04

J an-9 3J an-9 4 Jan-9 5J an-9 6Jan-9 7 Jan-9 8 J an-9 9 J an-0 0 J an-0 1 J an-0 2J an-0 3 Jan-0 4O ct-93O ct-94 O ct-95O ct-96 O ct-97 O ct-98O ct-99O ct-00O ct-01O ct-02O ct-03 O ct-04 

Ap

Apr-01

r-93

Apr-94Apr-95Apr-96Apr-97Apr-98Apr-99Apr-00Apr-02Apr-03Apr-04

Jul-93

Jul-95

Jul-96 

Jul-97

Jul-98

Jul-99

Jul-00 

Jul-01

Jul-02

Jan-95 

Jan-96 

Jan-99 

Jan-03 

Jul-03

Jan-04 

Jul-04 

Jan-93 

Oct-93 Jan-94 

Jul-94

Jan-97 

Oct-97 Jan-98 

Jan-00 

Jan-01 

Oct-01 J

Oct

an-02 

-94

Oct-95

Oct-98

Oct-96

Oct-99 

Oct-02

Oct-00

Oct-03

Oct-04 

Apr-93

Apr-94

Apr-95

Apr-96

Apr-97

Apr-98

Apr-99

Apr-00

Apr-01

Apr-02

Apr-03

Apr-04

72 

Figure 4-9: Period-to-Period Growth Rate 

Nonresidential Other Power (CUROTHPOW 1993-2004) 

50 

40 

30 

20 

10 

0 

-1 0 

-2 0 

-3 0 

-4 0 

time 

LP I BF L F D DF L S D RA T I O DP F D 

Figure 4-10: Period-to-Period Growth Rate 

Maritime Operating (SUB420 1993-2004) 

120 

100 

80 

60 

40 

20 

0 

-2 0 

-4 0 

-6 0 

-8 0 

time 

LPI DAFD BFLFD BFLSD RATIO 

g(t-1) g ( t - 1 ) 
Jul-93

Jul-94 

Jul-95

Jul-96 

Jul-97

Jul-98 

Jul-99

Jul-00

Jul-01 

JulJul-03 

Jul-04

02

Jan-9 3 

Oct-93 JanJan-9 5 

Oct-95 

Jan-9 7 

Oct-97 Jan-9 8 

Oct-99 Jan-0 0 

Oct-00 

Jan-0 2 

Oct-02 

Jan-0 4 

Oct-04 

9 4 

Jan-9 6 

Jan-9 9 

Jan-0 1 

Jan-0 3 

Oct-94 

Oct-96 

Oct-98 

Oct-01 

Oct-03 

Apr-93

Apr-94

Apr-95

Apr-96

Apr-97

Apr-98

Apr-99

Apr-00

Apr-01

Apr-02

Apr-03

Apr-04

73 

Figure 4-11: Absolute Period-to-Period Level Change 

OPIC (ENT210 1993-2004) 

0 

1 

2 

3 

4 

5 

6 

7 

8 

Jan-9 3 Apr-93

Jul-93

O ct-93 Jan-9 4

Apr-94 J ul-94

O ct-94 Jan-9 5 

Apr-95

Jul-95 

O ct-95 Jan-9 6 Apr-96

Jul-96 

O ct-96 Jan-9 7

Apr-97 Jul-97

O ct-97 J an-9 8

Apr-98

Jul-98 O ct-98 

Jan-9 9 Apr-99

Jul-99 

O ct-99 Jan-0 0 Apr-00

Jul-00

O ct-00 J an-0 1

Apr-01

Jul-01

O ct-01 Jan-0 2

Apr-02

Jul-02 O ct-02 

Jan-0 3 Apr-03

Jul-03

O ct-03 Jan-0 4

Apr-04

Jul-04

O ct-04 

time 

LP I DA F D BF L F D BF L S D 

Figure 4-12: Absolute Period-to-Period Level Change 

National Flood Insurance (ENT390 1993-2004) 

0 

10 

20 

30 

40 

50 

60 

70 

time 

LPI DAFD BFLFD BFLSD 
74 

Table 1: Correlation Coefficient ρ between Annual and Sub-annual Indicators 

NEA Series Taxes Transfer Payments 

Series ρ Series ρ Series ρ

MIRPERSON 0.8130 TPCAG 0.9516 TRCAG 0.9443 

XOTHOTHCOST 0.9236 TPCMI 0.8579 TRCMI 0.8968 

XOTHOTHCCO 0.9273 TPCUT 0.9844 TRCUT 0.9124 

GFOCXLOSS 0.9729 TPCAE 0.9918 TRCAE 0.9535 

EOMHP 0.8711 TPCCO 0.9970 TRCCO 0.9647 

EOSTP 0.9994 TPCDG 0.9971 TRCDG 0.9349 

ETMHP 0.9795 TPCED 0.9954 TRCED 0.9345 

ETSPP 0.9804 TPCFR 0.9941 TRCFR 0.8286 

EIGAS1 0.9988 TPCIN 0.9721 TRCIN 0.9698 

GFOLS 0.9927 TPCNG 0.9773 TRCNG 0.9503 

CURMFG 0.9320 TPCOS 0.9985 TRCOS 0.9536 

CURTUM 0.9996 TPCPF 0.9854 TRCPF 0.9757 

CURMFU 0.9999 TPCRT 0.9951 TRCRT 0.9473 

SFTWRE 0.9654 TPCTW 0.9550 TRCTW 0.9519 

TPCWT 0.9981 TRCWT 0.9460 

TPCGV -0.6598 

Table 2-1: Proportional Discrepancies of NEA Series from All Methods 

P Dz P Dx 

No. Series IND LPI DPFD DAFD RATIO TREND AR(1)MAX RAWK RAWKM MAX AR(1)MIN RAWKM MIN 

1 MIRPERSON 1.0502 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

2 XOTHOTHCOST 0.0399 1.0409 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

3 XOTHOTHCCO 0.0339 1.0332 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

4 GFOCXLOSS 1.0018 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

5 EOMHP 1.0240 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

6 EOSTP 1.0125 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

7 ETMHP 1.0050 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

8 ETSPP 0.9768 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

9 E1GAS1 0.1296 0.8997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

10 GFOL 1.0003 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

11 CURMFG 0.9098 0.9167 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

12 CURTUM 1.0008 0.9167 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

13 CURMFU 1.0002 0.8997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

14 SFTWRE 0.0021 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

MEAN 0.7276 0.9791 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 

Table 2-2: Proportional Discrepancies of Taxes and Transfers from All Methods 

P Dz 

P Dx P Dz P Dx 

No. Series IND LPI DPFD RATIO TREND No. Series IND LPI DPFD RATIO TREND 

1 TPCAG 0.0079 1.0000 1.0000 1.0000 1.0000 17 TRCAG 0.0051 1.0000 1.0000 1.0000 1.0000 

2 TPCMI 0.0181 1.0000 1.0000 1.0000 1.0000 18 TRCMI 0.0832 1.0000 1.0000 1.0000 1.0000 

3 TPCUT 0.0453 1.0000 1.0000 1.0000 1.0000 19 TRCUT 0.0528 1.0000 1.0000 1.0000 1.0000 

4 TPCAE 0.0565 1.0000 1.0000 1.0000 1.0000 20 TRCAE 0.0189 1.0000 1.0000 1.0000 1.0000 

5 TPCCO 0.0073 1.0000 1.0000 1.0000 1.0000 21 TRCCO 0.0363 1.0000 1.0000 1.0000 1.0000 

6 TPCDG 0.0193 1.0000 1.0000 1.0000 1.0000 22 TRCDG 0.0720 1.0000 1.0000 1.0000 1.0000 

7 TPCED 0.0122 1.0000 1.0000 1.0000 1.0000 23 TRCED 0.0718 1.0000 1.0000 1.0000 1.0000 

8 TPCFR 0.2877 1.0000 1.0000 1.0000 1.0000 24 TRCFR 0.1748 1.0000 1.0000 1.0000 1.0000 

9 TPCIN 0.0496 1.0000 1.0000 1.0000 1.0000 25 TRCIN 0.0429 1.0000 1.0000 1.0000 1.0000 

10 TPCNG 0.0381 1.0000 1.0000 1.0000 1.0000 26 TRCNG 0.2017 1.0000 1.0000 1.0000 1.0000 

11 TPCOS 0.0219 1.0000 1.0000 1.0000 1.0000 27 TRCOS 0.0156 1.0000 1.0000 1.0000 1.0000 

12 TPCPF 0.0299 1.0000 1.0000 1.0000 1.0000 28 TRCPF 0.0899 1.0000 1.0000 1.0000 1.0000 

13 TPCRT 0.1944 1.0000 1.0000 1.0000 1.0000 29 TRCRT 0.0598 1.0000 1.0000 1.0000 1.0000 

14 TPCTW 0.0230 1.0000 1.0000 1.0000 1.0000 30 TRCTW 0.0598 1.0000 1.0000 1.0000 1.0000 

15 TPCWT 0.1885 1.0000 1.0000 1.0000 1.0000 31 TRCWT 0.0444 1.0000 1.0000 1.0000 1.0000 

16 TPCGV 0.0002 1.0000 1.0000 1.0000 1.0000 

MEAN 0.0625 1.0000 1.0000 1.0000 1.0000 MEAN 0.0686 1.0000 1.0000 1.0000 1.0000 

P P D z x and D : proportional discrepancies with respect to the indicator and to the final estimates.
75 

Table 3-1: Average Absolute Change in Period-to-Period Growth Rates of the 14

NEA Series from All Methods Used in Estimation 

No. Series LPI DPFD DAFD RATIO TREND AR(1)MAX RAWK RAWKM MAX AR(1)MIN RAWKM MIN 

1 MIRPERSON 0.021167 0.019644 0.019731 0.019644 0.019583 0.021569 0.021677 0.021124 0.021570 0.020747 

2 XOTHOTHCOST 0.133133 0.073193 0.074254 0.073196 0.075050 0.091518 0.122100 0.134704 0.091679 0.134655 

3 XOTHOTHCCO 0.139077 0.072490 0.073897 0.072493 0.074296 0.090358 0.120557 0.133172 0.090591 0.133112 

4 GFOCXLOSS 0.005332 0.004029 0.004055 0.004030 0.004022 0.005208 0.004395 0.004614 0.004636 0.004339 

5 EOMHP 0.004309 0.004007 0.003991 0.004009 0.004002 0.004229 0.004312 0.004412 0.004235 0.004341 

6 EOSTP 0.000562 0.000453 0.000335 0.000456 0.000456 0.000517 0.000493 0.000523 0.000517 0.000519 

7 ETMHP 0.003792 0.002064 0.002042 0.002062 0.002068 0.001283 0.001202 0.001263 0.001284 0.001446 

8 ETSPP 0.002032 0.001540 0.001539 0.001540 0.001540 0.000992 0.000932 0.000979 0.001012 0.000987 

9 E1GAS1 0.007328 0.000842 0.338758 0.000840 0.000841 0.001474 0.001000 0.000919 0.001504 0.000962 

10 GFOL 0.036586 0.002612 0.002607 0.002612 0.002617 0.002124 0.002591 0.002714 0.002264 0.002715 

11 CURMFG 0.013898 0.002238 0.005462 0.002239 0.002241 0.005300 0.005270 0.005476 0.001294 0.001498 

12 CURTUM 0.010083 0.000770 0.000770 0.000772 0.000771 0.001103 0.001602 0.002184 0.000131 0.000192 

13 CURMFU 0.010561 0.000165 0.000174 0.000165 0.000165 0.000184 0.000279 0.000705 0.000185 0.000705 

14 SFTWRE 0.003651 0.004529 9.256027 0.008688 0.008688 0.004369 0.004883 0.004966 0.004397 0.004921 

MEAN 0.027965 0.013470 0.698832 0.013768 0.014024 0.017374 0.022031 0.024061 0.016992 0.023555 

STEDV 0.046776 0.025616 2.464566 0.025530 0.026180 0.033123 0.044423 0.049064 0.033385 0.049261 

Note: Average absolute change in period-to-period growth rate measures how strong the short-term

movement preservation is from final estimates generated from each method. Both mathematical

and regression methods are used in the temporal disaggregation of the 14 NEA series. 

Table 3-2: Average Absolute Change in Period-to-Period Growth Rates of taxes 

and Transfer Payment Series from All Methods Used in Estimation 

No. Series LPI DPFD RATIO TREND No. Series LPI DPFD RATIO TREND 

1 TPCAG 0.006431 0.011371 0.011368 0.011369 17 TRCAG 0.013351 0.014940 0.014993 0.015030 

2 TPCMI 0.028513 0.028179 0.028153 0.028008 18 TRCMI 0.019062 0.017891 0.017911 0.017824 

3 TPCUT 0.004098 0.004910 0.004919 0.004916 19 TRCUT 0.017733 0.017084 0.017081 0.017095 

4 TPCAE 0.003437 0.003621 0.003622 0.003619 20 TRCAE 0.011737 0.016984 0.016982 0.016821 

5 TPCCO 0.007254 0.008035 0.008061 0.008044 21 TRCCO 0.011490 0.016933 0.016879 0.016468 

6 TPCDG 0.001573 0.001938 0.001945 0.001945 22 TRCDG 0.014070 0.019178 0.019176 0.019165 

7 TPCED 0.004984 0.003499 0.003526 0.003526 23 TRCED 0.013455 0.018675 0.018663 0.018666 

8 TPCFR 0.004669 0.003052 0.003051 0.003050 24 TRCFR 0.029007 0.020065 0.020062 0.019831 

9 TPCIN 0.005971 0.004057 0.004063 0.004063 25 TRCIN 0.008618 0.013094 0.013120 0.012951 

10 TPCNG 0.007865 0.007526 0.007527 0.007531 26 TRCNG 0.029719 0.049168 0.049170 0.049002 

11 TPCOS 0.002999 0.001962 0.001965 0.001963 27 TRCOS 0.011520 0.016679 0.016801 0.016528 

12 TPCPF 0.004389 0.004072 0.004079 0.004078 28 TRCPF 0.008145 0.012897 0.012914 0.012550 

13 TPCRT 0.003302 0.002215 0.002315 0.002313 29 TRCRT 0.012417 0.017723 0.017709 0.017575 

14 TPCTW 0.013122 0.014782 0.014782 0.014796 30 TRCTW 0.011007 0.014489 0.014456 0.014437 

15 TPCWT 0.003304 0.002680 0.002680 0.002682 31 TRCWT 0.012330 0.017084 0.017073 0.016974 

16 TPCGV 0.032408 0.039481 0.041658 0.039787 

MEAN 0.008395 0.008836 0.008982 0.008856 MEAN 0.014911 0.018859 0.018866 0.018728 

STEDV 0.009045 0.010596 0.011012 0.010629 STEDV 0.006520 0.008635 0.008632 0.008631 

Note: Average absolute change in period-to-period growth rate measures how strong the short-term

movement preservation is from final estimates generated from each method. Only mathematical

methods are used in the temporal disaggregation of taxes and transfer payment series. LPI

estimates are from 2000 to 2004 and estimates from other methods are from 1997 to 2004. 
76 

Table 4-1a: Distortions at Breaks between Years (CB) of NEA Series

from All Methods Used in Estimation 

No. Series LPI DPFD DAFD RATIO TREND AR(1)MAX RAWK RAWKM MAX AR(1)MIN RAWKM MIN 

1 MIRPERSON NA 0.023008 0.023009 0.023008 0.023104 0.024147 0.024050 0.023277 0.024120 0.022932 

2 XOTHOTHCOST NA 0.087255 0.087255 0.087260 0.092153 0.103276 0.142011 0.158695 0.103421 0.158572 

3 XOTHOTHCCO NA 0.086506 0.086607 0.086512 0.091278 0.101939 0.139940 0.156539 0.102142 0.156395 

4 GFOCXLOSS 0.006340 0.004395 0.004387 0.004395 0.004417 0.005000 0.005246 0.005280 0.004983 0.005280 

5 EOMHP 0.004603 0.004479 0.004532 0.004481 0.004460 0.004839 0.004894 0.004567 0.004911 0.004450 

6 EOSTP 0.000740 0.000517 0.000357 0.000520 0.000520 0.000580 0.000543 0.000524 0.000581 0.000523 

7 ETMHP 0.002839 0.001467 0.001455 0.001485 0.001485 0.001858 0.001517 0.001333 0.001810 0.001591 

8 ETSPP 0.001236 0.000898 0.000910 0.000899 0.000899 0.001079 0.001007 0.000981 0.001104 0.000987 

9 E1GAS1 0.006609 0.000998 0.385888 0.000997 0.001008 0.001340 0.001015 0.001033 0.001367 0.001104 

10 GFOL 0.037537 0.003389 0.003403 0.003390 0.003391 0.003272 0.003095 0.003076 0.003216 0.003076 

11 CURMFG 0.016333 0.002544 0.006457 0.002546 0.002555 0.005676 0.005555 0.005475 0.005845 0.005495 

12 CURTUM 0.013909 0.001004 0.001004 0.001009 0.001015 0.001482 0.001914 0.002562 0.001482 0.002795 

13 CURMFU 0.009057 0.000210 0.000213 0.000208 0.000208 0.000247 0.000300 0.000647 0.000243 0.000647 

14 SFTWRE 0.003956 0.005271 4.795107 0.009219 0.009219 0.004939 0.006167 0.005808 0.004950 0.005362 

MEAN 0.009378 0.015853 0.473065 0.016138 0.016837 0.019595 0.025468 0.027999 0.019633 0.027988 

STEDV 0.010558 0.030632 1.438067 0.030546 0.032269 0.037360 0.051637 0.057835 0.037419 0.057769 

Note: Distortion at breaks between years is measured by the average absolute change in period-toPeriod growth rate between final estimates and indicator series of all sub-annual periods

at breaks (4th to 1st quarters for quarterly data, Nov. to Feb. for monthly data) of all

sample years. 

Table 4-1b: Average Absolute Change in Period-to-Period Growth Rates of All

Sub-annual Periods in the Middle of Each Sample Year (CM) from NEA Series 

No. Series LPI DPFD DAFD RATIO TREND AR(1)MAX RAWK RAWKM MAX AR(1)MIN RAWKM MIN 

1 MIRPERSON 0.015665 0.016433 0.016436 0.016433 0.016222 0.019109 0.019412 0.019068 0.019137 0.018660 

2 XOTHOTHCOST 0.089314 0.059769 0.059779 0.059771 0.058725 0.080294 0.103094 0.111804 0.080471 0.111825 

3 XOTHOTHCCO 0.098526 0.059110 0.059180 0.059111 0.058086 0.079303 0.102055 0.110867 0.079565 0.110887 

4 GFOCXLOSS 0.004127 0.003497 0.003554 0.003498 0.003463 0.003818 0.004010 0.004021 0.003725 0.004020 

5 EOMHP 0.004223 0.003833 0.003784 0.003836 0.003834 0.003939 0.004079 0.004390 0.003911 0.004343 

6 EOSTP 0.000485 0.000428 0.000329 0.000431 0.000431 0.000480 0.000472 0.000524 0.000480 0.000520 

7 ETMHP 0.001496 0.001029 0.001016 0.001047 0.001052 0.001024 0.001064 0.001245 0.001048 0.001389 

8 ETSPP 0.000911 0.000811 0.000811 0.000811 0.000811 0.000938 0.000894 0.000976 0.000953 0.000984 

9 E1GAS1 0.007737 0.000776 0.317487 0.000773 0.000769 0.001535 0.000992 0.000867 0.000272 0.000203 

10 GFOL 0.035682 0.001873 0.001851 0.001872 0.001881 0.001028 0.002110 0.002369 0.001356 0.002370 

11 CURMFG 0.012363 0.002114 0.005038 0.002115 0.002114 0.005089 0.005101 0.005431 0.005073 0.005470 

12 CURTUM 0.007751 0.000665 0.000665 0.000666 0.000660 0.000922 0.001464 0.002021 0.000959 0.002192 

13 CURMFU 0.011392 0.000145 0.000156 0.000146 0.000146 0.000152 0.000267 0.000722 0.000155 0.000722 

14 SFTWRE 0.003365 0.003833 13.438140 0.008191 0.008191 0.003835 0.003678 0.004177 0.003880 0.004508 

MEAN 0.020931 0.011023 0.993445 0.011336 0.011170 0.015202 0.018847 0.020331 0.015162 0.020276 

STEDV 0.032251 0.020917 3.582809 0.020834 0.020463 0.029097 0.037496 0.040682 0.029217 0.040706 

Note: Each entry contains the average absolute change in period-to-period growth rates of all

sub-annual periods in the middle of a year (2nd to 3rd quarter for quarterly data and Mar.

to Oct. for monthly data) of all sample years. 
77 

Table 4-2a: Distortion at Breaks between Years (CB) of Taxes and Transfer

Payments from All Methods Used in Estimation 

No. Series LPI DPFD RATIO TREND No. Series LPI DPFD RATIO TREND 

1 TPCAG 0.004006 0.014711 0.014748 0.014736 17 TRCAG 0.009569 0.017161 0.017387 0.017973 

2 TPCMI 0.017983 0.033139 0.033123 0.033240 18 TRCMI 0.014683 0.020406 0.020390 0.020362 

3 TPCUT 0.002694 0.005509 0.005517 0.005495 19 TRCUT 0.013601 0.019511 0.019466 0.019541 

4 TPCAE 0.001787 0.003891 0.003885 0.003876 20 TRCAE 0.006814 0.019793 0.019713 0.020096 

5 TPCCO 0.004287 0.008569 0.008542 0.008477 21 TRCCO 0.006817 0.020173 0.020119 0.020319 

6 TPCDG 0.000994 0.002303 0.002282 0.002293 22 TRCDG 0.008304 0.022325 0.022373 0.022963 

7 TPCED 0.002905 0.003724 0.003729 0.003705 23 TRCED 0.007538 0.021346 0.021356 0.022007 

8 TPCFR 0.002744 0.003491 0.003491 0.003474 24 TRCFR 0.020035 0.022859 0.022851 0.022560 

9 TPCIN 0.003072 0.004344 0.004351 0.004343 25 TRCIN 0.004809 0.015646 0.015757 0.015716 

10 TPCNG 0.005173 0.009191 0.009205 0.009199 26 TRCNG 0.017785 0.060204 0.060218 0.058322 

11 TPCOS 0.001776 0.002375 0.002377 0.002366 27 TRCOS 0.006660 0.019262 0.019646 0.019645 

12 TPCPF 0.001988 0.004352 0.004336 0.004336 28 TRCPF 0.005246 0.015206 0.015230 0.015187 

13 TPCRT 0.001707 0.002487 0.002486 0.002482 29 TRCRT 0.007027 0.020437 0.020451 0.020827 

14 TPCTW 0.010877 0.018076 0.018067 0.018105 30 TRCTW 0.006800 0.016670 0.016898 0.016860 

15 TPCWT 0.002663 0.003697 0.003697 0.003691 31 TRCWT 0.007491 0.020202 0.020205 0.020418 

16 TPCGV 0.018741 0.047274 0.047523 0.048617 

MEAN 0.005212 0.010446 0.010460 0.010527 MEAN 0.009545 0.022080 0.022137 0.022186 

STEDV 0.005618 0.012724 0.012772 0.013003 STEDV 0.004695 0.010788 0.010766 0.010258 

Note: Distortion at breaks between years is measured by the average absolute change in period-toPeriod growth rate between final estimates and indicator series of all sub-annual periods

at breaks (4th to 1st quarters for quarterly data, Nov. to Feb. for monthly data) of all

sample years. LPI estimates are from 2000 to 2004 and estimates from other methods are from

1997 to 2004. 

Table 4-2b: Average Absolute Change in Period-to-Period Growth Rates of All Subannual Periods in the Middle of Each Sample Year (CM) of Taxes and Transfers 

No. Series LPI DPFD RATIO TREND No. Series LPI DPFD RATIO TREND 

1 TPCAG 0.003881 0.008239 0.008199 0.008213 17 TRCAG 0.006884 0.012858 0.012749 0.012271 

2 TPCMI 0.017000 0.023228 0.023494 0.023093 18 TRCMI 0.008871 0.015533 0.015587 0.015445 

3 TPCUT 0.002341 0.004367 0.004359 0.004373 19 TRCUT 0.008307 0.014808 0.014844 0.014801 

4 TPCAE 0.002341 0.004367 0.004359 0.004373 20 TRCAE 0.007550 0.014350 0.014421 0.013751 

5 TPCCO 0.004595 0.007563 0.007610 0.007638 21 TRCCO 0.007254 0.013895 0.013842 0.012858 

6 TPCDG 0.000937 0.001596 0.001630 0.001620 22 TRCDG 0.008923 0.016227 0.016178 0.015604 

7 TPCED 0.003195 0.003288 0.003335 0.003358 23 TRCED 0.008912 0.016171 0.016138 0.015535 

8 TPCFR 0.002972 0.002641 0.002639 0.002652 24 TRCFR 0.015663 0.017446 0.017448 0.017274 

9 TPCIN 0.004211 0.003807 0.003793 0.003800 25 TRCIN 0.005726 0.010700 0.010648 0.010359 

10 TPCNG 0.004490 0.005965 0.005954 0.005968 26 TRCNG 0.018618 0.038822 0.038813 0.040264 

11 TPCOS 0.001897 0.001576 0.001578 0.001585 27 TRCOS 0.007436 0.014258 0.014133 0.013606 

12 TPCPF 0.003348 0.003830 0.003838 0.003836 28 TRCPF 0.004755 0.010733 0.010744 0.010077 

13 TPCRT 0.002320 0.002153 0.002154 0.002156 29 TRCRT 0.008158 0.015178 0.015138 0.014525 

14 TPCTW 0.005385 0.011694 0.011703 0.011694 30 TRCTW 0.006695 0.012445 0.012167 0.012166 

15 TPCWT 0.002335 0.003447 0.003446 0.003450 31 TRCWT 0.007619 0.014161 0.014137 0.013746 

16 TPCGV 0.020915 0.030505 0.036160 0.030524 

MEAN 0.005135 0.007392 0.007766 0.007396 MEAN 0.008758 0.015839 0.015799 0.015485 

STEDV 0.005563 0.008165 0.009306 0.008146 STEDV 0.003638 0.006638 0.006654 0.007131 

Note: Each entry contains the average absolute change in period-to-period growth rates of all

sub-annual periods in the middle of a year (2nd to 3rd quarter for quarterly data and March

to October for monthly data) of all sample years. 
 

78 

Table 5-1a: Absolute Change in Period-to-Period Growth Rate between Final and

Indicator Series in the Second Period (C2) of NEA Series from all Methods 

No. Series LPI DPFD DAFD RATIO TREND AR(1)MAX RAWK RAWKM MAX AR(1)MIN RAWKM MIN 

1 MIRPERSON 0.000443 0.001073 0.001073 0.001073 0.001039 0.017312 0.010937 0.006471 0.017072 0.004062 

2 XOTHOTHCOST 0.045531 0.009303 0.009303 0.009186 0.008289 0.116648 0.091610 0.087212 0.116653 0.087183 

3 XOTHOTHCCO 0.026249 0.009230 0.009230 0.009092 0.008624 0.116038 0.091041 0.086632 0.116044 0.086601 

4 GFOCXLOSS 0.007853 0.003692 0.004403 0.003701 0.003475 0.008857 0.009002 0.008804 0.008647 0.008878 

5 EOMHP 0.003782 0.001241 0.001470 0.001224 0.000747 0.004238 0.000760 0.000332 0.004480 0.000021 

6 EOSTP 0.003167 0.000832 0.000505 0.000828 0.000828 0.001041 0.000322 0.000354 0.001048 0.000319 

7 ETMHP 0.007607 0.002218 0.002198 0.002204 0.002204 0.000141 0.000231 0.000036 0.000169 0.000554 

8 ETSPP 0.003847 0.001287 0.001283 0.001283 0.001278 0.001998 0.001070 0.001209 0.002205 0.001280 

9 E1GAS1 0.001092 0.000078 0.212876 0.000011 0.000003 0.001804 0.001055 0.000699 0.001878 0.000524 

10 GFOL 0.033026 0.000581 0.000576 0.000587 0.000575 0.000922 0.003670 0.005313 0.001176 0.005315 

11 CURMFG 0.049258 0.000014 0.000403 0.000003 0.000003 0.008260 0.008403 0.009842 0.007807 0.010026 

12 CURTUM 0.057952 0.000106 0.000106 0.000029 0.000136 0.001060 0.000496 0.000473 0.000969 0.000511 

13 CURMFU 0.027046 0.000151 0.000221 0.000149 0.000149 0.000332 0.000492 0.001729 0.000335 0.001727 

14 SFTWRE 0.000002 0.003444 1.007543 0.015540 0.015540 0.001229 0.001381 0.000069 0.001554 0.000929 

MEAN 0.019061 0.002375 0.111962 0.003208 0.003064 0.021435 0.016853 0.016085 0.021422 0.015923 

STEDV 0.020461 0.003145 0.303743 0.004710 0.004594 0.042400 0.033263 0.031619 0.042395 0.031668 

Note: The statistics C2 measures distortion at the beginning of the final sub-annual series. 

Table 5-1b: Absolute Change in Period-to-Period Growth Rate between Final and

Indicator Values in the Last Period (CT) of NEA Series from All Methods 

No. Series LPI DPFD DAFD RATIO TREND AR(1)MAX RAWK RAWKM MAX AR(1)MIN RAWKM MIN 

1 MIRPERSON 0.043712 0.005955 0.005955 0.005956 0.005492 0.032176 0.044425 0.037230 0.032772 0.032371 

2 XOTHOTHCOST 0.012632 0.011093 0.011093 0.011327 0.012817 0.010289 0.052239 0.042231 0.010444 0.040693 

3 XOTHOTHCCO 0.012786 0.011717 0.011717 0.011993 0.013430 0.009981 0.050552 0.038861 0.010173 0.037164 

4 GFOCXLOSS 0.014604 0.000599 0.000894 0.000601 0.000631 0.004792 0.008068 0.008126 0.004230 0.008034 

5 EOMHP 0.007308 0.000617 0.000407 0.000613 0.000614 0.001133 0.004646 0.003704 0.001761 0.003943 

6 EOSTP 0.001078 0.000130 0.000115 0.000132 0.000132 0.000134 0.000359 0.001018 0.000140 0.001163 

7 ETMHP 0.000858 0.000061 0.000005 0.000152 0.000152 0.001022 0.000190 0.000815 0.000913 0.001466 

8 ETSPP 0.001090 0.000019 0.000192 0.000019 0.000019 0.000471 0.000399 0.000718 0.000693 0.000729 

9 E1GAS1 0.009177 0.000171 0.184967 0.000109 0.000126 0.000895 0.000002 0.001454 -0.000928 0.001946 

10 GFOL 0.054080 0.000869 0.000871 0.000869 0.000859 0.000307 0.000505 0.001064 0.000427 0.001081 

11 CURMFG 0.013799 0.000001 0.008517 0.000000 0.000000 0.007744 0.002638 0.002246 0.009797 0.002370 

12 CURTUM 0.006496 0.000102 0.000102 0.000152 0.000085 0.000750 0.000766 0.001297 0.000621 0.001356 

13 CURMFU 0.001781 0.000057 0.000053 0.000037 0.000086 0.000332 0.000492 0.001729 0.000335 0.001727 

14 SFTWRE 0.000905 0.002961 1.414260 0.007532 0.007532 0.004255 0.003198 0.003590 0.004628 0.002967 

MEAN 0.012879 0.002454 0.146398 0.002821 0.002998 0.005387 0.012714 0.010807 0.005491 0.010311 

STEDV 0.016236 0.004131 0.424106 0.004425 0.004862 0.008900 0.020920 0.016473 0.009195 0.015282 

Note: The statistics CT measures distortion at the end of the final sub-annual series. 
79 

Table 5-2a: Absolute Change in Period-to-Period Growth Rate between Final and 

Indicator Values in the Second Period (C2) of Taxes and Transfer Payments 

No. Series LPI DPFD RATIO TREND No. Series LPI DPFD RATIO TREND 

1 TPCAG 0.018493 0.009110 0.008296 0.008305 17 TRCAG 0.031642 0.007879 0.007591 0.007591 

2 TPCMI 0.072552 0.010632 0.010337 0.010510 18 TRCMI 0.029310 0.004831 0.004924 0.004927 

3 TPCUT 0.002654 0.002573 0.002400 0.002365 19 TRCUT 0.029292 0.005503 0.005590 0.005595 

4 TPCAE 0.012119 0.001877 0.002002 0.001971 20 TRCAE 0.022738 0.010853 0.011757 0.011757 

5 TPCCO 0.008943 0.005186 0.006252 0.005979 21 TRCCO 0.016262 0.009734 0.009696 0.009696 

6 TPCDG 0.008943 0.000145 0.000502 0.000502 22 TRCDG 0.025447 0.011808 0.011704 0.012568 

7 TPCED 0.011180 0.000902 0.001524 0.001524 23 TRCED 0.028500 0.011293 0.011294 0.011722 

8 TPCFR 0.007507 0.000319 0.000302 0.000296 24 TRCFR 0.023732 0.003871 0.003902 0.003703 

9 TPCIN 0.014039 0.000513 0.000380 0.000380 25 TRCIN 0.015417 0.007985 0.008165 0.007409 

10 TPCNG 0.022426 0.002826 0.002664 0.002582 26 TRCNG 0.121623 0.040431 0.060572 0.040703 

11 TPCOS 0.012504 0.000835 0.001127 0.001127 27 TRCOS 0.019182 0.010779 0.011391 0.011391 

12 TPCPF 0.019175 0.002064 0.002330 0.002211 28 TRCPF 0.019182 0.007345 0.007353 0.007348 

13 TPCRT 0.011990 0.000821 0.000855 0.000821 29 TRCRT 0.026207 0.011188 0.010822 0.011344 

14 TPCTW 0.017095 0.011550 0.011888 0.011775 30 TRCTW 0.028514 0.008688 0.008476 0.009471 

15 TPCWT 0.002261 0.000292 0.000262 0.000262 31 TRCWT 0.024032 0.010570 0.009975 0.010667 

16 TPCGV 0.129279 0.030823 0.030470 0.030647 

MEAN 0.023198 0.005029 0.005100 0.005079 MEAN 0.030739 0.010851 0.012214 0.011059 

STEDV 0.032526 0.007860 0.007727 0.007773 STEDV 0.025625 0.008557 0.013615 0.008649 

Note: The statistics C2 measures distortion at the beginning of the final sub-annual series. LPI

estimates are from 2000 to 2004 and estimates from other methods are from 1997 to 2004. 

Table 5-2b: Absolute Change in Period-to-Period Growth Rate between Final and

Indicator Values in the Last Period (CT) of Taxes and Transfer Payments 

No. Series LPI DPFD RATIO TREND No. Series LPI DPFD RATIO TREND 

1 TPCAG 0.010793 0.001024 0.001899 0.001899 17 TRCAG 0.009570 0.005301 0.001394 0.005049 

2 TPCMI 0.012411 0.001644 0.001232 0.001174 18 TRCMI 0.013677 0.004520 0.004478 0.003420 

3 TPCUT 0.000366 0.000668 0.000842 0.000791 19 TRCUT 0.012528 0.004429 0.004047 0.003193 

4 TPCAE 0.001566 0.000328 0.000230 0.000190 20 TRCAE 0.007930 0.003557 0.002308 0.003221 

5 TPCCO 0.003585 0.003445 0.002725 0.002579 21 TRCCO 0.006229 0.003172 0.003003 0.002934 

6 TPCDG 0.002355 0.000610 0.000205 0.000205 22 TRCDG 0.005519 0.003129 0.003178 0.002372 

7 TPCED 0.001004 0.000183 0.000371 0.000371 23 TRCED 0.004393 0.003063 0.002927 0.002345 

8 TPCFR 0.002443 0.000359 0.000347 0.000343 24 TRCFR 0.026826 0.022630 0.022577 0.013893 

9 TPCIN 0.004657 0.001246 0.001383 0.001359 25 TRCIN 0.001812 0.003684 0.004256 0.003320 

10 TPCNG 0.001705 0.001062 0.001272 0.001208 26 TRCNG 0.008472 0.004066 0.004085 0.003191 

11 TPCOS 0.001722 0.000700 0.000472 0.000366 27 TRCOS 0.006256 0.003355 0.002964 0.000610 

12 TPCPF 0.002068 0.000313 0.000055 0.000055 28 TRCPF 0.003099 0.003303 0.003419 0.002556 

13 TPCRT 0.002064 0.000202 0.000172 0.000172 29 TRCRT 0.005842 0.003085 0.002958 0.001965 

14 TPCTW 0.004204 0.002021 0.001716 0.001716 30 TRCTW 0.003715 0.003657 0.003694 0.002372 

15 TPCWT 0.004416 0.000441 0.000467 0.000467 31 TRCWT 0.006607 0.003163 0.002574 0.001663 

16 TPCGV 0.011953 0.035845 0.001698 0.001820 

MEAN 0.004207 0.003131 0.000943 0.000920 MEAN 0.008165 0.004941 0.004524 0.003474 

STEDV 0.003928 0.008765 0.000784 0.000775 STEDV 0.006106 0.004936 0.005060 0.003046 

Note: The statistics CT measures distortion at the end of the final sub-annual series. 
80 

Table 6-1: Absolute Change in Period-to-Period Growth Rate between the Final

and Indicator Values in the Linking Period (CL) Using Linking Alternative 1:

Setting Linking As An Initial Condition in the Optimization Model

 Alternative 1: Linking to Benchmarked Series Linking to Benchmarked Series

 Obtained Using the Same Method Obtained Using LPI 

Series DPFD** RATIO** TREND** DPFD* RATIO* TREND* 

MIRPERSON 0.019145 0.019975 0.019144 0.029351 0.028715 0.029350 

XOTHOTHCCO 0.101260 0.101252 0.108766 0.124225 0.124217 0.134156 

XOTHOTHCOST 0.103711 0.103699 0.111862 0.133326 0.133320 0.145946 

GFOCXLOSS 0.005340 0.005327 0.005389 0.003868 0.003868 0.003906 

GFOL 0.006112 0.006102 0.006065 0.023099 0.023099 0.023506 

EOMHP 0.011240 0.011241 0.011500 0.011225 0.011226 0.011225 

EOSTP 0.001217 0.001304 0.001304 0.001373 0.001380 0.001380 

ETMHP 0.007767 0.007776 0.006539 0.005021 0.005014 0.005014 

ETSPP 0.005756 0.005756 0.005756 0.005249 0.005249 0.005249 

MEAN 0.029061 0.029159 0.030703 0.037415 0.037343 0.039970 

Note: ** stands for linking to benchmarked sub-annual values estimated using the same method, and

* stands for linking to benchmarked sub-annual values estimated using the LPI procedure. 

Table 6-2: Absolute Change in Period-to-Period Growth Rate between the Final

and Indicator Values in the Linking Period (CL) Using Linking Alternative 2:

Replacing Previous Estimates with Revised Final Estimates in the Whole Sample 

Alternative 2: Linking to Benchmarked Series Linking to Benchmarked Series

 Obtained Using the Same Method Obtained Using LPI 

Series DPFD^ RATIO^ TREND^ DPFD RATIO TREND 

MIRPERSON 0.015850 0.015851 0.017117 0.033120 0.033121 0.033187 

XOTHOTHCCO 0.096004 0.095995 0.106978 0.138122 0.138113 0.143138 

XOTHOTHCOST 0.098423 0.098405 0.110324 0.152860 0.152852 0.158419 

GFOCXLOSS 0.006296 0.006271 0.006336 0.003744 0.003742 0.003807 

GFOL 0.006747 0.006731 0.006701 0.043753 0.043751 0.043780 

EOMHP 0.025510 0.025503 0.025254 0.025449 0.025439 0.024142 

EOSTP 0.003523 0.003836 0.003836 0.004163 0.004145 0.004145 

ETMHP 0.049781 0.049899 0.044998 0.038944 0.039000 0.039172 

ETSPP 0.017140 0.017142 0.017133 0.015085 0.015086 0.015078 

MEAN 0.035475 0.035515 0.037631 0.050582 0.050583 0.051652 

Note: ^ stands for using newly revised final estimates to replace previous estimates estimated

using the same method. 
 

 

81 

Table 7-1: Contemporaneous Discrepancy in Two-Stage Estimation for Taxes 

 Stage 1=LPI Stage 2=DPFD Stage 1=DPFD Stage 2=DPFD Stage 1=RATIO Stage 2=DPFD Stage 1=TREND Stage 2=DPFD 

Date Diff Diff % Diff Diff % Diff Diff % Diff Diff % Diff Diff % Diff Diff % Diff Diff % Diff Diff % 

1/1/1997 NA NA NA NA -66.66 0.0112 0.45 0.0000 -0.11 0.0000 0.02 0.0000 -47.22 0.0079 0.03 0.0000 

4/1/1997 NA NA NA NA -4.25 0.0007 4.02 0.0000 -1.44 0.0002 -0.01 0.0000 -28.41 0.0047 0.07 0.0000 

7/1/1997 NA NA NA NA 31.71 0.0051 0.91 0.0000 0.27 0.0000 -0.08 0.0000 15.20 0.0025 0.11 0.0000 

10/1/1997 NA NA NA NA 39.21 0.0063 -1.28 0.0000 1.28 0.0002 0.17 0.0000 60.43 0.0097 0.15 0.0000 

1/1/1998 NA NA NA NA 16.80 0.0027 -4.52 0.0000 -1.74 0.0003 0.44 0.0001 67.92 0.0108 0.19 0.0000 

4/1/1998 NA NA NA NA 1.04 0.0002 5.94 0.0000 0.39 0.0001 0.42 0.0001 16.60 0.0026 0.24 0.0000 

7/1/1998 NA NA NA NA -7.96 0.0012 10.68 0.0000 0.61 0.0001 0.38 0.0001 -36.21 0.0056 0.28 0.0000 

10/1/1998 NA NA NA NA -9.85 0.0015 -13.02 0.0000 0.74 0.0001 0.56 0.0001 -48.30 0.0074 0.32 0.0000 

1/1/1999 NA NA NA NA -4.29 0.0007 4.87 0.0000 -1.60 0.0002 0.37 0.0001 -4.55 0.0007 0.37 0.0001 

4/1/1999 NA NA NA NA -0.30 0.0000 -18.88 0.0000 -0.10 0.0000 0.03 0.0000 -11.27 0.0017 0.44 0.0001 

7/1/1999 NA NA NA NA 1.99 0.0003 4.28 0.0000 1.83 0.0003 -0.68 -0.0001 -6.91 0.0010 0.51 0.0001 

10/1/1999 NA NA NA NA 2.57 0.0004 3.23 0.0000 -0.13 0.0000 0.41 0.0001 22.73 0.0033 0.57 0.0001 

1/1/2000 7192.00 1.0310 1.96 0.0000 1.21 0.0002 3.43 0.0000 -0.28 0.0000 1.28 0.0002 55.85 0.0080 0.66 0.0001 

4/1/2000 343.00 0.0485 4.20 0.0000 0.18 0.0000 1.01 0.0000 -0.32 0.0000 0.12 0.0000 9.86 0.0014 0.78 0.0001 

7/1/2000 -3422.00 0.4805 6.58 0.0000 -0.50 0.0001 4.02 0.0000 -0.16 0.0000 0.14 0.0000 -29.75 0.0042 0.88 0.0001 

10/1/2000 -4113.00 0.5723 9.12 0.0000 -0.89 0.0001 -5.81 0.0000 0.75 0.0001 0.02 0.0000 -35.96 0.0050 1.00 0.0001 

1/1/2001 -1721.00 0.2373 11.37 0.0000 -0.88 0.0001 -10.20 0.0000 0.35 0.0000 0.11 0.0000 -4.08 0.0006 1.13 0.0002 

4/1/2001 -78.00 0.0107 13.21 0.0000 -0.52 0.0001 -25.36 0.0000 -0.22 0.0000 -0.53 -0.0001 -23.32 0.0032 1.25 0.0002 

7/1/2001 821.00 0.1132 14.75 0.0000 0.19 0.0000 14.36 0.0000 1.47 0.0002 1.31 0.0002 -13.91 0.0019 1.36 0.0002 

10/1/2001 978.00 0.1326 16.15 0.0000 1.25 0.0002 -6.84 0.0000 -1.61 0.0002 0.28 0.0000 41.32 0.0056 1.41 0.0002 

1/1/2002 324.00 0.0434 17.24 0.0000 2.76 0.0004 -11.96 0.0000 1.16 0.0002 0.12 0.0000 122.90 0.0165 1.47 0.0002 

4/1/2002 -65.00 0.0086 21.55 0.0000 2.27 0.0003 14.98 0.0000 0.93 0.0001 0.08 0.0000 28.07 0.0037 1.57 0.0002 

7/1/2002 -259.00 0.0336 23.83 0.0000 -0.19 0.0000 8.07 0.0000 -0.12 0.0000 0.87 0.0001 -69.49 0.0090 1.67 0.0002 

10/1/2002 0.00 0.0000 23.68 0.0000 -4.82 0.0006 -7.34 0.0000 -1.97 0.0003 -1.13 -0.0001 -81.48 0.0105 1.76 0.0002 

1/1/2003 120.00 0.0153 25.45 0.0000 -11.64 0.0015 -42.07 -0.0001 -1.37 0.0002 1.75 0.0002 3.43 0.0004 1.83 0.0002 

4/1/2003 68.00 0.0086 28.21 0.0000 -9.98 0.0013 -4.19 0.0000 1.11 0.0001 0.61 0.0001 17.11 0.0022 1.93 0.0002 

7/1/2003 -32.00 0.0040 32.96 0.0000 0.68 0.0001 -32.99 0.0000 -1.22 0.0002 -0.05 0.0000 -3.13 0.0004 2.01 0.0002 

10/1/2003 -156.00 0.0190 35.58 0.0000 20.96 0.0026 -17.05 0.0000 1.48 0.0002 -0.04 0.0000 -17.41 0.0021 2.09 0.0003 

1/1/2004 -845.00 0.1009 38.66 0.0000 51.65 0.0062 -10.29 0.0000 1.12 0.0001 0.78 0.0001 -4.80 0.0006 2.14 0.0003 

4/1/2004 -591.00 0.0697 40.09 0.0000 44.20 0.0052 -22.88 0.0000 -1.84 0.0002 -0.50 -0.0001 0.18 0.0000 2.14 0.0003 

7/1/2004 123.00 0.0144 40.74 0.0000 -3.22 0.0004 -11.31 0.0000 0.05 0.0000 -0.16 0.0000 1.01 0.0001 2.16 0.0003 

10/1/2004 1313.00 0.1508 41.78 0.0000 -92.62 0.0106 1.64 0.0000 0.67 0.0001 -0.54 -0.0001 3.62 0.0004 2.19 0.0003 

Mean 0.00 0.1547 22.35 0.0000 0.00 0.0019 -5.13 0.0000 0.00 0.0001 0.20 0.0000 0.00 0.0042 1.08 0.0001 

Std dev 2153.22 0.2589 12.83 0.0000 26.28 0.0030 13.20 0.0000 1.10 0.0001 0.59 0.0001 40.97 0.0040 0.76 0.0001 

Table 7-2: Contemporaneous Discrepancy in Two-Stage Estimation for Transfers 

 Stage 1=LPI Stage 2=DPFD Stage 1=DPFD Stage 2=DPFD Stage 1=RATIO Stage 2=DPFD Stage 1=TREND Stage 2=DPFD 

Date Diff Diff % Diff Diff % Diff Diff % Diff Diff % Diff Diff % Diff Diff % Diff Diff % Diff Diff % 

1/1/1997 NA NA NA NA -3.05 0.0109 0.00 0.0000 0.00 0.0000 0.00 0.0000 -149.00 0.5337 0.00 0.0000 

4/1/1997 NA NA NA NA -0.29 0.0010 0.00 0.0000 1.00 0.0035 0.00 0.0000 -88.00 0.3110 0.00 0.0000 

7/1/1997 NA NA NA NA 1.60 0.0049 0.00 0.0000 -2.00 0.0061 0.00 0.0000 0.00 0.0000 0.00 0.0000 

10/1/1997 NA NA NA NA 1.72 0.0058 0.00 0.0000 1.00 0.0034 0.00 0.0000 237.00 0.8050 0.00 0.0000 

1/1/1998 NA NA NA NA 0.86 0.0027 0.00 0.0000 0.00 0.0000 0.00 0.0000 317.00 1.0000 0.00 0.0000 

4/1/1998 NA NA NA NA 0.13 0.0004 0.00 0.0000 -1.00 0.0032 0.00 0.0000 130.00 0.4208 0.00 0.0000 

7/1/1998 NA NA NA NA -0.34 0.0010 0.00 0.0000 0.00 0.0000 0.00 0.0000 -127.00 0.3884 0.00 0.0000 

10/1/1998 NA NA NA NA -0.65 0.0015 0.00 0.0000 1.00 0.0022 0.00 0.0000 -320.00 0.7169 0.00 0.0000 

1/1/1999 NA NA NA NA -0.22 0.0006 0.00 0.0000 -1.00 0.0029 0.00 0.0000 -7.00 0.0200 0.00 0.0000 

4/1/1999 NA NA NA NA -0.02 0.0001 0.00 0.0000 0.00 0.0000 0.00 0.0000 -51.00 0.1447 0.00 0.0000 

7/1/1999 NA NA NA NA 0.09 0.0003 0.00 0.0000 -1.00 0.0028 0.00 0.0000 -28.00 0.0783 0.00 0.0000 

10/1/1999 NA NA NA NA 0.13 0.0003 0.00 0.0000 2.00 0.0053 0.00 0.0000 86.00 0.2281 0.00 0.0000 

1/1/2000 752.00 1.7735 0.00 0.0000 0.07 0.0002 0.00 0.0000 0.00 0.0000 0.00 0.0000 200.00 0.4717 0.00 0.0000 

4/1/2000 36.00 0.0834 0.00 0.0000 0.01 0.0000 0.00 0.0000 0.00 0.0000 0.00 0.0000 75.00 0.1738 0.00 0.0000 

7/1/2000 -357.00 0.8122 0.00 0.0000 -0.04 0.0001 0.00 0.0000 0.00 0.0000 0.00 0.0000 -114.00 0.2594 0.00 0.0000 

10/1/2000 -431.00 0.9527 0.00 0.0000 -0.06 0.0001 0.00 0.0000 0.00 0.0000 0.00 0.0000 -161.00 0.3559 0.00 0.0000 

1/1/2001 -179.00 0.3865 0.00 0.0000 -0.05 0.0001 0.00 0.0000 -1.00 0.0022 0.00 0.0000 26.00 0.0561 0.00 0.0000 

4/1/2001 -8.00 0.0170 0.00 0.0000 -0.02 0.0000 0.00 0.0000 2.00 0.0043 0.00 0.0000 23.00 0.0490 0.00 0.0000 

7/1/2001 86.00 0.1733 0.00 0.0000 0.02 0.0000 0.00 0.0000 1.00 0.0020 0.00 0.0000 -24.00 0.0484 0.00 0.0000 

10/1/2001 101.00 0.2136 0.00 0.0000 0.07 0.0001 0.00 0.0000 -2.00 0.0042 0.00 0.0000 -25.00 0.0529 0.00 0.0000 

1/1/2002 34.00 0.0727 0.00 0.0000 0.18 0.0004 0.00 0.0000 -2.00 0.0043 0.00 0.0000 26.00 0.0556 0.00 0.0000 

4/1/2002 -6.00 0.0129 0.00 0.0000 0.13 0.0003 0.00 0.0000 0.00 0.0000 0.00 0.0000 12.00 0.0258 0.00 0.0000 

7/1/2002 -29.00 0.0622 0.00 0.0000 -0.01 0.0000 0.00 0.0000 2.00 0.0043 0.00 0.0000 -15.00 0.0322 0.00 0.0000 

10/1/2002 1.00 0.0021 0.00 0.0000 -0.28 0.0006 0.00 0.0000 0.00 0.0000 0.00 0.0000 -23.00 0.0493 0.00 0.0000 

1/1/2003 30.00 0.0631 0.00 0.0000 -0.67 0.0014 0.00 0.0000 0.00 0.0000 0.00 0.0000 -19.00 0.0400 0.00 0.0000 

4/1/2003 16.00 0.0330 0.00 0.0000 -0.58 0.0012 0.00 0.0000 0.00 0.0000 0.00 0.0000 -16.00 0.0330 0.00 0.0000 

7/1/2003 -8.00 0.0162 0.00 0.0000 0.05 0.0001 0.00 0.0000 1.00 0.0020 0.00 0.0000 2.00 0.0040 0.00 0.0000 

10/1/2003 -38.00 0.0760 0.00 0.0000 1.19 0.0024 0.00 0.0000 -1.00 0.0020 0.00 0.0000 33.00 0.0660 0.00 0.0000 

1/1/2004 -191.00 0.3719 0.00 0.0000 2.98 0.0058 0.00 0.0000 0.00 0.0000 0.00 0.0000 59.00 0.1149 0.00 0.0000 

4/1/2004 -134.00 0.2571 0.00 0.0000 2.55 0.0049 0.00 0.0000 -1.00 0.0019 0.00 0.0000 24.00 0.0461 0.00 0.0000 

7/1/2004 23.00 0.0481 0.00 0.0000 -0.19 0.0004 0.00 0.0000 1.00 0.0021 0.00 0.0000 -26.00 0.0544 0.00 0.0000 

10/1/2004 302.00 0.5518 0.00 0.0000 -5.38 0.0098 0.00 0.0000 0.00 0.0000 0.00 0.0000 -57.00 0.1042 0.00 0.0000 

Mean 0.00 0.2990 0.00 0.0000 0.00 0.0018 0.00 0.0000 0.00 0.0018 0.00 0.0000 0.00 0.2106 0.00 0.0000 

Std dev 238.48 0.4406 0.00 0.0000 1.43 0.0028 0.00 0.0000 1.08 0.0019 0.00 0.0000 117.92 0.2560 0.00 0.0000 

Note: LPI estimates are from 2000 to 2004 and estimates from other methods are from 1997 to 2004. 
 

 

82 

Table 8-1a: Average Absolute Period-to-Period Growth Rates of 10 Nonnegative

Series with (with no Indicators) 

No Series LPI DAFD BFLFD BFLSD RATIO 

1 CGNTRMVGM 0.023405 0.021686 0.021686 0.022544 0.021698 

2 CGNTRD 0.022476 0.018115 0.018115 0.019001 0.018124 

3 CGNTRWC 0.012288 0.011966 0.011966 0.012215 0.011964 

4 CGNTRMVCHP 0.147960 0.152658 0.153884 0.136917 0.153983 

5 IBT811 0.070604 0.061159 0.061159 0.065522 0.061168 

6 SUB230 0.008720 0.007409 0.007407 0.007485 0.007364 

7 SUB270 0.021909 0.017439 0.017439 0.017856 0.017534 

8 SUB420 0.196998 0.170361 0.172380 0.178834 0.169732 

9 SUB370 0.084699 0.078003 0.078002 0.082908 0.078017 

10 CUROTHPOW 0.039285 0.019513 0.019513 0.021267 0.019513 

MEAN 0.062834 0.055831 0.056155 0.056455 0.055910 

STEDV 0.063841 0.060243 0.060889 0.059506 0.060345 

Table 8-1b: Average Absolute Period-to-Period Change in Level of 5 Series

with Negative Values (with no Indicators)

No Series LPI DAFD BFLFD BFLSD RATIO 

11 ENT210 1.276596 1.032340 1.032843 1.115596 NA 

12 ENT230 30.446809 31.502340 31.502088 33.923819 NA 

13 ENT260 1.000000 0.680000 0.684605 2.539333 NA 

14 ENT330 6.680851 6.362979 6.362781 6.788759 NA 

15 ENT390 84.454545 83.853636 83.852493 86.886250 NA 

MEAN 24.771760 24.686259 24.686962 26.250751 NA 

STEDV 35.49169395 35.42170396 35.42037586 36.42909631 NA 

Table 8-2a: Period-to-Period Growth Rate in the Second and Last Periods for 

10 Nonnegative Series (with no indicators)

Perod-to-period growth rate in second period Period-to-period growth rate in period T 

No Series LPI DAFD BFLFD BFLSD RATIO LPI DAFD BFLFD BFLSD RATIO 

1 CGNTRMVGM 0.032407 0.008242 0.008234 0.033392 0.008301 0.009831 0.001589 0.001594 0.001853 0.001581 

2 CGNTRD 0.076803 0.004719 0.004727 0.015832 0.004689 0.053591 0.010363 0.010368 0.039420 0.010215 

3 CGNTRWC 0.010700 0.002202 0.002208 0.005663 0.002197 0.006607 0.002997 0.003004 0.009008 0.002891 

4 CGNTRMVCHP 0.980769 0.855564 0.867715 0.705519 0.872928 0.031250 0.008663 0.008668 0.028051 0.008653 

5 IBT811 0.016692 0.006208 0.006200 0.051918 0.006024 0.044795 0.005834 0.005828 0.053421 0.005511 

6 SUB230 0.000000 0.002712 0.002688 0.006793 0.000000 0.000000 0.000258 0.000267 0.001478 0.000000 

7 SUB270 0.022989 0.003803 0.003750 0.005436 0.011236 0.015000 0.006155 0.006165 0.020230 0.005000 

8 SUB420 0.044118 0.013564 0.013675 0.061061 0.013699 0.130081 0.058041 0.058059 0.229874 0.061728 

9 SUB370 0.054217 0.012591 0.012593 0.064276 0.011457 0.026087 0.000289 0.000289 0.032888 0.000000 

10 CUROTHPOW 0.020180 0.002322 0.002331 0.025108 0.002331 0.000241 0.000269 0.000278 0.004725 0.000278 

MEAN 0.125887 0.091193 0.092412 0.097500 0.093286 0.031748 0.009446 0.009452 0.042095 0.009586 

STEDV 0.301219 0.268604 0.272445 0.214827 0.273975 0.039076 0.017457 0.017461 0.068245 0.018670 

Table 8-2b: Period-to-Period Change in Level in the Second and Last Period

for 5 Series with Negative Values (with no indicators)

 Perod-to-period level change in second period Period-to-period level change in period T 

No Series LPI DAFD BFLFD BFLSD RATIO LPI DAFD BFLFD BFLSD RATIO 

11 ENT210 7.000000 0.390000 0.392000 1.950306 NA 0.000000 0.380000 0.379384 1.381385 NA 

12 ENT230 21.000000 11.250000 11.242637 33.943410 NA 2.000000 1.750000 1.750832 1.334403 NA 

13 ENT260 1.000000 1.490000 1.489087 5.028955 NA 1.000000 0.680000 0.684605 2.539333 NA 

14 ENT330 10.000000 0.740000 0.744938 3.643498 NA 10.000000 5.090000 5.094136 18.364176 NA 

15 ENT390 3.000000 12.050000 12.051963 52.603882 NA 23.000000 8.850000 8.849186 34.293185 NA 

MEAN 8.400000 5.184000 5.184125 19.434010 NA 7.200000 3.350000 3.351629 11.582496 NA 

STEDV 7.861298 5.922743 5.920239 22.766691 NA 9.679876 3.598173 3.597543 14.599927 NA 
 

 

83 

Table 9-1: Comparative Evaluation of the Methods Used in the Estimation of

the 14 NEA Series According to the Test Criteria 

Mean of Statistics LPI DPFD DPAD RATIO TREND AR(1)MAX RAWK RAWKM MAX AR(1)MIN RAWKM MIN

p DX 0.979053 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

 Cp 0.027965 0.013470 0.874160 0.013768 0.014024 0.017374 0.022031 0.024061 0.016992 0.023555 

CB 0.009378 0.015853 0.473065 0.016138 0.016837 0.019595 0.025468 0.027999 0.019633 0.027988 

C2 0.019061 0.002375 0.111962 0.003208 0.003064 0.021435 0.016853 0.016085 0.021422 0.015923 

CT 0.012879 0.002454 0.146398 0.002821 0.002998 0.005387 0.012714 0.010807 0.005491 0.010311 

CL NA 0.029061 NA 0.029159 0.030703 NA NA NA NA NA 

Note: Each entry contains the mean of the test statistics of the 14 series from a particular

method used in estimation. 

Table 9-2: Comparative Evaluation of the Methods Used in Estimation of Series

on Taxes and Transfer Payments According to the Test Criteria 

16 series of taxes on production and imports 15 series of transfer payments 

Mean of Statistics LPI DPFD RATIO TREND LPI DPFD RATIO TREND

p DX 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

 Cp 0.008395 0.008836 0.008982 0.008856 0.014911 0.018859 0.018866 0.018728 

CB 0.005212 0.010446 0.010460 0.010527 0.009545 0.022080 0.022137 0.022186 

C2 0.023198 0.005029 0.005100 0.005079 0.030739 0.010851 0.012214 0.011059 

CT 0.004207 0.003131 0.000943 0.000920 0.008165 0.004941 0.004524 0.003474 

DX 

C

(%) 0.154714 0.024831 0.024834 0.024855 0.296904 0.001800 0.001835 0.000397 

Note: Each entry in the left panel contains the mean of the test statistics of the 16 series on

taxes from a particular method used in estimation, and each entry in the right panel

contains the corresponding mean statistics of the 15 series on transfer payments. Estimates

from LPI are from 2000 to 1997 and estimates from other methods are from 1997 to 2004. 

Table 9-3: Comparative Evaluation of the Methods Used in Estimation of Series

without Sub-annual Indicators According to the Test Criteria 

 Average statistics for 10 non-negative series Average statistics for 5 series with negative values 

Mean of Statistics LPI DAFD BFLFD BFLSD RATIO LPI DAFD BFLFD BFLSD RATIO 

gt 6.283447 5.583084 5.615520 5.645487 5.590992 24.771760 24.686259 24.686962 26.250751 NA 

g2 12.588737 9.119256 9.241199 9.749978 9.328617 8.400000 5.184000 5.184125 19.434010 NA 

gT 3.174829 0.944566 0.945199 4.209480 0.958577 7.200000 3.350000 3.351629 11.582496 NA 

Note: Each entry in the left panel contains the mean of the test statistics of the 10 nonnegative

series without indicators, and each entry in the right panel contains the corresponding

test statistics of the 5 series with negative values. 
 

84 

Appendix: Information on Series and Software Used in the Experiment 

Table A1: Series Used in Estimation Experiment 

Series Definition 

Series with Indicators 

1 MIRPERSON.a 

JTOTN.q 

Monetary Interest Paid by Persons 

2 XOTHOTHCOST.A 

MTGVALORIG.Q 

Total Closing Cost

Mortgage Originations, seasonally adjusted 

3 XOTHOTHCCO.A 

MTGVALORIG.Q 

Owner-Occupied Closing Cost

Mortgage Originations, seasonally adjusted 

4 GFOCXLOSS$.A 

JGFOCXLOSS$.Q 

Crop Output

Constant-Dollar Gross Crop Output Excluding Losses 

5 EOMHP 

EOMH 

Owner-Occupied Mobile Homes

Extrapolated using Mobile Home Housing Stock 

6 EOSTP 

EOST 

Owner-Occupied Stationary Homes

Extrapolated using Stationary Unit Housing Stock 

7 ETMHP 

ETMH 

Tennant-Occupied Mobile Homes

Extrapolated using Mobile Home Housing Stock 

8 ETSPP 

ETSP 

Tennant-Occupied Stationary Homes

Extrapolated using Stationary Unit Housing Stock 

9 E1GAS1.A 

E1GAS0A.M 

Consumption of Gasoline

Monthly Indicator of Gasoline Consumption 

10 GFOL$.A 

JGFOL$.Q 

Livestock output

Constant-Dollar Gross Livestock Output 

11 CURMFG.A 

VIPMFG.M 

Nonresidential Manufacturing

Monthly Value Estimates for Nonresidential Const. 

12 CURTUM.A 

VIPTCM.M 

Nonresidential Telecommunications 

Monthly Value Estimates for Telecommunications Const. 

13 CURMFU.A 

VIPMFU.M 

Residential Multi-Family

Monthly Value Estimates for Multi-Family Const. 

14 SFTWRE 

QIND_PRE_SA.Q 

Investment in Pre-Packaged Equipment & Software

Quarterly Indicator 

15 TPCAG 

TPCTL 

Taxes on Ag., Forestry, Fishing, Hunting

Total Taxes 

16 TPCMI 

TPCTL 

Taxes on Mining

Total Taxes 

17 TPCUT 

TPCTL 

Taxes on Utilities 

Total Taxes 

18 TPCAE 

TPCTL 

Taxes on Arts, Entertainment, etc.

Total Taxes 

19 TPCCO 

TPCTL 

Taxes on Construction 

Total Taxes 

20 TPCDG 

TPCTL 

Taxes on Durable Goods 

Total Taxes 

21 TPCED 

TPCTL 

Taxes on Education, Health, Social Assistance

Total Taxes 

22 TPCFR 

TPCTL 

Taxes on Fin., Ins., Real Est., Rental, Leasing

Total Taxes 

23 TPCIN 

TPCTL 

Taxes on Information 

Total Taxes 

24 TPCNG 

TPCTL 

Taxes on Nondurable Goods 

Total Taxes 

25 TPCOS 

TPCTL 

Taxes on Other Services, Except Government

Total Taxes 

26 TPCPF 

TPCTL 

Taxes on Professional & Business Services 

Total Taxes 

27 TPCRT 

TPCTL 

Taxes on Retail Trade 

Total Taxes 

28 TPCTW 

TPCTL 

Taxes on Transportation & Warehousing

Total Taxes 

29 TPCWT 

TPCTL 

Taxes on Wholesale Trade 

Total Taxes 

30 TPCGV 

TPCTL 

Taxes on Government Enterprises

Total Taxes 

31 TRCAG 

TRCTL 

Transfers from Ag., Forestry, Fishing, Hunting

Total Transfers 

32 TRCMI 

TRCTL 

Transfers from Mining

Total Transfers 
 

 

 

85 

Table A1: Series Used in Estimation Experiment (Cont.) 

Series Definition 

Series with Indicators 

33 TRCUT 

TRCTL 

Transfers from Utilities 

Total Transfers 

34 TRCAE 

TRCTL 

Transfers from Arts, Entertainment, etc.

Total Transfers 

35 TRCCO 

TRCTL 

Transfers from Construction 

Total Transfers 

36 TRCDG 

TRCTL 

Transfers from Durable Goods 

Total Transfers 

37 TRCED 

TRCTL 

Transfers from Education, Health & Social Assistance

Total Transfers 

38 TRCFR 

TRCTL 

Transfers from Fin., Ins., Real Est., Rental, Leasing

Total Transfers 

39 TRCIN 

TRCTL 

Transfers from Information 

Total Transfers 

40 TRCNG 

TRCTL 

Transfers from Nondurable Goods 

Total Transfers 

41 TRCOS 

TRCTL 

Transfers from Other Services, Except Government

Total Transfers 

42 TRCPF 

TRCTL 

Transfers from Professional & Business Services 

Total Transfers 

43 TRCRT 

TRCTL 

Transfers from Retail Trade 

Total Transfers 

44 TRCTW 

TRCTL 

Transfers from Transportation & Warehousing

Total Transfers 

45 TRCWT 

TRCTL 

Transfers from Wholesale Trade 

Total Transfers 

Series without Indicators 

46 CGNTRMVGM State & Local Transfers: General Medical Assistance 

47 CGNTRTD TDI Benefits 

48 CGNTRWC Workers Compensation Benefits 

49 CGNTRMVCHP State & Local Transfers: Child Health Program 

50 ENT210 OPIC 

51 ENT230 Bonneville Power 

52 ENT260 Colorado River Basins 

53 ENT370 TVA 

54 ENT390 National Flood Insurance 

55 IBT811 Outer Continental Shelf Royalties 

56 SUB230 Housing-235-236-Rent Supplement 

57 SUB270 Rental Assistance 

58 SUB310 Amtrak 

59 SUB420 Maritime Operating 

60 CUROTHPOW.A Nonresidential Other Power 

Note: The indictors and their definitions are in italic letters. 

TPC-series: Taxes on production & imports. TRC-series: Business transfers to government.

ENT-series: Current surplus of government enterprise. SUB-series: Subsidies. 

IBT-series: Indirect business Taxes.