Section B5: Benchmarking Methods for Time Series

B5.1. Overview of Temporal Disaggregation of Time Series

When we have a low frequency time series (e.g., annual), and we want a higher frequency times 

series (e.g., monthly or quarterly), we may apply temporal disaggregation methods to obtain an 

approximation of the desired series. Methods are available in statistical software packages. The 

basic steps in temporal disaggregation are as follows:

1) Determine a low-frequency series yy for which a higher-frequency series is desired.

2) Determine an appropriate preliminary high-frequency series pp, sometimes called a 

pattern series. The high-frequency movements of pp should be similar to the expected 

high-frequency movements of the target series.

3) Benchmark pp to force consistency between pp and yy at the lower-frequency level (e.g., the 

annual averages or sums).

Various methods of identifying or computing pattern series pp are discussed in the statistics 

literature. In the regression-based approaches discussed below, multiple high-frequency series of 

explanatory variables may be used. When we already have a high-frequency series that we wish 

to benchmark to a lower-frequency series, we skip step 1. Step 3 above proceeds as follows:

3a) Aggregate pp to the lower frequency (e.g., average or sum the quarterly values to obtain 

annual values). 

3b) Compute the differences between the low-frequency values of pp and the observed lowfrequency values of yy.

3c) Distribute the differences among the values of pp, to force consistency with yy.

B5.2. Types of Temporal Disaggregation Methods

Various methods involve different ways of distributing the differences (step 3c). Temporal 

disaggregation methods are categorized into (1) mathematical or operations research methods

and (2) regression-based or statistical methods.

Operations research methods use a single indicator or pattern series for the preliminary highfrequency series pp. To distribute the benchmarking adjustments, these methods optimize an 

objective function, subject to the benchmarking constraints. Operations research methods include 

the widely used Denton (1971) methods as well as the Causey-Trager method used at the U.S. 

Census Bureau.

Regression-based or statistical methods often use multiple indicator series to compute the 

preliminary series, and they use an estimated or assumed autocorrelation structure to distribute 

the benchmarking adjustments. Regression-based methods include the Chow-Lin (1971), 

Fernandez (1981), and Litterman (1983) methods.

Some benchmarking techniques are designed to work with stock variables, while others are 

designed to work with flow variables. Stock variables measure, at a specific point in time, a 

quantity that may have accumulated in the past, e.g., volume of natural gas in underground 
storage, total solar electricity generating capacity. Flow variables measure a quantity over an 

interval of time, e.g., natural gas consumption per month, solar electricity generation per year.

B5.3. Denton Benchmarking Methods

The Denton benchmarking methods are based on the concept of movement preservation, i.e., the 

benchmarked series θθ follows the same pattern of movement over time as the preliminary high 

frequency series pp. There are four basic variants of movement preservation:

1) Additive First Difference Method: Make the additive corrections (θθtt − pptt) as constant as 

possible over time so that the series θθ is roughly parallel to pp. In this approach, we 

minimize the sum of squared differences of the additive first differences:

min

θθ [(θθtt − pptt) − (θθtt−1 − pptt−1)] = minθθ [∆(θθtt − pptt)]. (B5.3.1)

The additive first difference method is Denton’s (1971) original method.

2) Additive Second Difference Method: Make the additive corrections as linear as possible 

by minimizing the sum of squared differences of the additive second differences:

min

θθ {[(θθtt − pptt) − (θθtt−1 − pptt−1)] − [(θθtt−1 − pptt−1) − (θθtt−2 − pptt−2)]}

= min

θθ [∆2(θθtt − pptt)]. (B5.3.2)

In this method, the additive correction (θθtt − pptt) is close to 2(θθtt−1 − pptt−1) −

(θθtt−2 − pptt−2), a linear function of the previous two corrections. This method is often 

used for stock variables.

3) Proportional First Difference Method: Make the multiplicative corrections θθtt

pptt

 as 

constant as possible by minimizing the sum of squared differences of the ratios θθtt

pptt

:

min

θθ �

θθtt

pptt

− θθtt−1

pptt−1

� = min

θθ �∆ �

θθtt

pptt

��. (B5.3.3)

In this method, θθtt

pptt

 is close to θθtt−1

pptt−1

. This method is the most widely used of the Denton 

methods, because it is robust to scale.

4) Proportional Second Difference Method: Make the multiplicative corrections θθtt

pptt

 as 

linear as possible by minimizing the sum of squared second differences of the ratios θθtt

pptt

:

min

θθ ��

θθtt

pptt

− θθtt−1

pptt−1

� − �

θθtt−1

pptt−1

− θθtt−2

pptt−2

�� = min

θθ �∆2 �

θθtt

pptt

��. (B5.3.4)
In this method, θθtt

pptt

 is close to 2θθtt−1

pptt−1

− θθtt−2

pptt−2

. This method is often used for stock variables.

The derivations of the Denton methods are similar, involving the following basic steps:

i. Write the objective function ff(θθ) and the benchmarking constraints in vector notation.

ii. Multiply the constraint term by a vector γγ of Lagrange multipliers, and add it to the 

objective function; the function becomes ff(θθ, γγ).

iii. Differentiate ff(θθ, γγ) with respect to θθ and γγ, and set the derivatives equal to 00. 

iv. Solve for θθ. Because ff(θθ, γγ) is quadratic, it can be shown that the solution θθ� is a global 

minimum.

The appendix provides the derivation of the additive first difference method. The original Denton 

methods implicitly incorporate an initial condition,

pp1 = θθ1. (B5.3.5)

This constraint has been found disadvantageous for many applications, because it can result in a 

spurious movement at the beginning of the benchmarked series. The Denton-Cholette (1984) 

methods remove this initial condition and are generally preferred over the original Denton 

methods.

B5.4. Regression-based Methods

The regression-based temporal disaggregation methods use the following basic steps:

1) Instead of a single indicator series pp, we assume we have a TT × kk matrix XX consisting of 

TT high frequency (HF) observations of kk ≥ 1 explanatory variables. We aggregate the 

HF series to the low-frequency (LF) level (e.g, compute annual averages from monthly 

data) to form the MM × kk matrix JJ , where JJ is an appropriate MM × TT aggregation matrix.

(See the appendix for a definition of JJ.)

2) We fit an ordinary (or generalized) least square regression model to the LF series yy, using 

JJ as the matrix of explanatory variables:

yy = JJ ββ + εε. (B5.4.1)

3) The main assumption behind the regression-based methods is that the linear relationship 

between the LF series yy and JJ also holds between the HF series θθ and XX. We form the 

preliminary series by setting

pp = XXββ̂. (B5.4.2)

4) To benchmark the preliminary series, the Chow-Lin method assumes that the true 

additive corrections follow an autoregressive process of order 1 (AR1) with a constant 

correlation coefficient ρρ, where |ρρ| < 1:
uutt = θθtt − pptt = ρρ tt−1 + εεtt, (B5.4.3)

where εεtt is a normally distributed random error term with variance σσεε

2.

5) Chow and Lin consider the variance-covariance matrix of the vector θθ� − EE(θθ), in terms 

of ρρ and σσεε

2. They use Lagrange multipliers to minimize the variance of θθ� − EE(θθ), 

subject to the benchmarking constraints. (The Chow-Lin estimator of θθ is a Best Linear 

Unbiased Estimator (BLUE)).

6) The solution takes the form 

θθ� = XXββ̂ + ΣΣJJ′(JJΣΣJJ′

)−11�JJ ββ̂ − yy�, (B5.4.4)

where ΣΣ = σσεε

2

1−ρρ2

⎣

⎢

⎢

⎡ 1 ρρ

ρρ 1

⋯ ρρTT−1

... ρρTT−2

⋮ ⋮

ρρTT−1 ρρTT−2

⋱ ⋮

... 1 ⎦

⎥

⎥

⎤

.

Calculating θθ� involves estimating the autoregressive parameter ρρ. Chow and Lin suggest an 

iterative approach. Other methods have also been presented in the literature, e.g., Dagum and 

Cholette (2006) developed a method that allows users to specify ρρ. The Fernandez and Litterman 

methods, useful for disaggregating stock variables, are similar to the Chow-Lin method, but they 

assume that the additive corrections follow a non-stationary autoregressive process:

uutt = θθtt − pptt = θθtt−1 − pptt−1 + vvtt, (B5.4.5)

where vvtt = ρρ tt−1 + εεtt. 

B5.5. Comparing Benchmarking Methods

The Denton proportional difference methods are generally robust to scale and provide good 

results when one good pattern series is available. They are, however, sensitive to outliers. The 

Chow-Lin method (and the other regression-based methods) assume that the strength of the 

correlation between the LF series holds for the HF series. If the LF series are weakly correlated, 

the regression-based methods don’t preserve movement. The main advantage of the regressionbased methods is that they can incorporate multiple pattern series. They are also fairly robust to 

outliers.

B5.6. Appendix: Deriving the Denton First Difference Method

Notation

Let MM represent the number of low-frequency (LF) time periods (e.g., years) in the data series.

Let LL represent the number of high-frequency (HF) periods in each LF period (e.g., for monthly 

data, LL = 12). Let mm = 1, ... , MM represent the LF time periods and denote the HF periods in each 

LF period mm by ttmm,1, ... ,ttmm,LL. Let TT = MM , the total number of HF time periods in the data 

series.
The summing (or averaging) weights jjmm,tt form an MMxTT matrix JJ:

JJ = �

jj11 jj12

jj21 jj22

... jj1TT

⋯ jj2TT

⋮ ⋮

jjMM1 jjMM2

⋮ ⋮

⋯ jjMM

�. (B5.6.1)

For example, when benchmarking two years of quarterly data to annual sums, we set

JJ = �

1 1

0 0

1 1

0 0

0 0

1 1

0 0

1 1

�. (B5.6.2)

For annual averages of quarterly benchmarked data,

JJθθ = �

1

4

1

4

0 0

1

4

1

4

0 0

0 0 1

4

1

4

0 0 1

4

1

4

�

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎡

θθ11

θθ12

θθ13

θθ14

θθ21

θθ22

θθ23

θθ24⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎤

=

⎣

⎢

⎢

⎢

⎢

⎡1

4�θθ1ll

4

ll=1

1

4�θθ2ll

4

ll=1 ⎦

⎥

⎥

⎥

⎥

⎤

. (B5.6.3)

The benchmarking constraints are therefore written as yy − JJ = 00.

All of the Denton methods use a single indicator series for the preliminary high-frequency series 

pp. The indicator series may be a HF series that we’d like to benchmark to a low-frequency series 

or, in the case of temporal disaggregation, a series whose short-term movements we believe to 

correlate with the target HF series.

For each HF time period tt, let pptt and θθtt represent the values of the preliminary and benchmarked 

series, respectively. Then

pptt = θθtt + eett, (B5.6.4)

where eett is an error term to be estimated. The benchmarking constraints may be expressed as 

yy − JJ = 00 or

yymm = � jjtt,mmθθtt

ttmm,LL

tt=ttmm,1

, (B5.6.5)

for mm = 1, ... , MM, where yymm is the LF series to which we wish to benchmark the pptt (e.g., yymm may 

represent annual averages), and the jjtt,mm are weights (e.g., 1

12 ,

1

4 , 1).
To optimize the objective function and solve for the vector θθ, we write the function as a 

quadratic form in matrix notation and add the linear benchmarking constraints with a vector γγ of 

Lagrange multipliers:

ff(θθ, γγ) = (θθ − pp)′DD′DD(θθ − pp) − 2γγ′(yy − JJ ), (B5.6.6)

where the matrix DD is the first-difference operator matrix of order TT:

DD = �

1

−1

0

⋮

0

1

−1

⋮

0

0

1

⋮

⋯

...

...

⋱

�. (B5.6.7)

The matrix DD′DD is symmetric, and the factor 2 in the second term of ff(θθ, γγ) is applied for 

computational convenience. We compute partial derivatives of ff(θθ, γγ) with respect to θθ and γγ

and set them equal to 0:

∂∂

∂∂θθ ff(θθ, γγ) = 2DD′

DD − 2DD′DD + 2JJ′γγ = 00

⇒ DD′DD + JJ′γγ = DD′DD .

(B5.6.8)

Similarly,

∂∂

∂∂γγ ff(θθ, γγ) = 22 − 2yy = 00

⇒ JJ = JJ + (yy − JJ ).

(B5.6.9)

The equations 

DD′DD + JJ′γγ = DD′DD and JJ = JJ + (yy − JJ ) (B5.6.10)

can be written in partitioned matrix form as

�

DD′DD JJ′

JJ 00��

θθ

γγ� = �

DD′DD 00

JJ II

�� pp

(yy − JJ )

�, (B5.6.11)

which gives the solution

�

θθ�

γγ�

� = �

DD′

DD JJ′

JJ 00�

−11

�

DD′DD 00

JJ II

�� pp

(yy − JJ )

�. (B5.6.12)

Statistical software packages perform the matrix inversion numerically to compute θθ�.
References

Cholette, P. A., 1984. “Adjusting Sub-annual Series to Yearly Benchmarks,” Survey 

Methodology, Vol. 10, pp. 35–49.

Chow, G. C. and Lin, A.L., 1971. Best linear unbiased interpolation, distribution, and 

extrapolation of time series by related series. The Review of Economics and Statistics, 

53(4):372–375, Nov. 1971. pp. 80-83. Available at http://www.oecd.org/std/21779751.pdf. 

Dagum, E. B. and Cholette, P. A. (2006). Benchmarking, Temporal Distribution, and 

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York.

Denton, F. T. (1971). Adjustment of monthly or quarterly series to annual totals: An approach 

based on quadratic minimization. Journal of the American Statistical Association, 66, pp. 99–

102, March 1971. 

Fernández, R. B. (1981). A methodological note on the estimation of time series. The Review of 

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