Smoothing Variance Estimates for

Price Indexes Over Time

Richard Valliant

U.S. Bureau of Labor Statistics

Room 2126, 441 G St. NW

Washington DC 20212

April 1992
1

ABSTRACT

This paper explores the possibility of developing generalized variances for price

indexes by applying nonparametric scatterplot smoothers to time series of point variance

estimates. The goal here is to formulate smoothed variances which are approximately

unbiased, which provide acceptable confidence interval coverage, and which are more

stable than the point variance estimates. Smoothing methods are applied to time series of

point variance estimates in a simulation study using data from the U.S. Consumer Price

Index program.

Key words: Generalized variance function, Laspeyres price index, linearization variance

estimator, loess, super smoother.

ACKNOWLEDGEMENT

Any opinions expressed are those of the author and do not constitute policy of the

Bureau of Labor Statistics. I thank the editors and referees for their useful comments.
2

1. INTRODUCTION

Price index series are some of the most important statistics published by national

governments. Having a measurement of inflation, for example, is a fundamental

requirement for tracking the health of an economy. When index series are estimated

from sample surveys, the indexes are subject to sampling error, and an important

statistical question is how to best estimate their variances. The importance of calculating

variances of indexes was illustrated by an experience in Sweden related by Andersson,

Forsman, and Wretman (1987). In January 1987 major Swedish labor agreements were

invalidated when the consumer price index increased 3.26% during 1986. If the increase

had been .02 percentage points less, the threshold specified in the labor agreements

would not have been exceeded and the agreements would have remained in effect.

Although the increase may have been within sampling error of the threshold, at the time,

procedures were not in place for variance estimation so that no testing of that possibility

could be done.

Index series are characterized by seasonal and irregular fluctuations in addition to

underlying trends. The literature is replete with methods for decomposing and

smoothing such time series. Point estimators of variance, obtained by linearization,

replication, or another method, may be subject to the same types of seasonal and irregular

variations as the index series themselves. The variable nature of point variance estimates

was illustrated by Leaver (1990) for indexes. This paper explores the possibility of

developing generalized variances for price indexes by applying nonparametric scatterplot

smoothers to series of point variance estimates. The goal here is to formulate smoothed

variances which are approximately unbiased, which provide acceptable confidence

interval coverage, and which, most importantly, are more stable than the point variance

estimates.

The approach taken here is somewhat different than that which is sometimes used

in household surveys for estimating generalized variance functions (GVF's). That
3

method is described in Wolter (1985) with some justifying theory given in Valliant

(1987). The general idea is to use models to approximate variances. Given a set of

survey variables whose variances all follow the same model, parameters of the model are

estimated by least squares. The parameter estimates are then provided to users rather

than individual variance estimates in order to condense survey publications. Ideally, the

models will also lead to more stable estimates of variance. Applications of GVF's in two

particular surveys can be found in Hanson (1978) and Johnson and King (1987). In the

case of price indexes, finding multiple indexes whose variances follow the same model

may be difficult. However, smoothing the variances of a particular index series over

time is a practical alternative. For a given index series this is a two-step process

consisting of estimating variances at a number of points in time and of smoothing the

series of point variance estimates. As will be illustrated, this approach can produce more

stable variance estimates that are approximately unbiased and that provide near nominal

confidence interval coverage.

Section 2 defines the population Laspeyres price index, a class of index

estimators, and a superpopulation model which is used to study the variance of the index

estimators. In section 3, an approximation to the variance of a long-term price change

estimator is discussed. The fourth section presents the methods that were tested for

estimating generalized variances. A simulation study, described in section 5, was

conducted using data from the U.S. Consumer Price Index to determine how well the

proposed variance estimators would work in practice. Finally, section 6 gives

conclusions.

2. INDEX ESTIMATORS AND A SUPERPOPULATION MODEL

The population is divided into H strata with stratum h containing Nh

establishments. Establishment (hi) contains Mhi items, and the total number of items in

all establishments in stratum h is Mh = Mhi i=1

Nh

Â . At time t the price of item j in
4

establishment (hi) is phij

t , and the price relative between time t and the base period time 0

is rhij

t,0 = phijt phij

0 . The quantity of item (hij) purchased in the base period is qhij0 . The

finite population value of the long-term fixed base Laspeyres price index for comparing

period t to period 0 is

I pq pq

W r

t

hij

t

hij

j

M

i

N

h

H

hij hij

j

M

i

N

h

H

hij hij

t

h i j

h hi hi h

,

, ,

0 0

1 1 1

0 0

1 1 1

0 0

=

=

= = = = = =

Â Â ÂÂ ÂÂ

Â Â Â (1)

where Whij

0 = phij0 qhij0 phij0 qhij0

Âh,i, j is the fraction of total base period cost or value

accounted for by item (hij). For later reference, it is also convenient to define the

stratum index Ih

t,0 = Whij

0 rhij

t,0

j=1

Mhi

i=1ÂNh

Â Wh

0 where Wh0 = Whij0

j=1

Mhi

i=1Â

Nh

Â . Using long-term

indexes, the population short-term index for comparing periods t2 and t1 (t

1 < t2 ) is

defined as I

t2 ,t1 = It2 ,0 It

1,0 . Monthly, quarterly, semiannual, and annual changes are

commonly published by index programs.

In order to analyze the properties of index estimators, we will consider the

superpopulation model defined below, which was also used in Valliant (1991).

rhij

t,0 = ath +wthi + e thij

e thij = rhe t-1,hij + x thij

(2)

where E wthi b g = E wt1hiwt2h¢i¢ d i = 0 for all t, h, i, and tb g 1hi π t2 b g h¢i¢ ; E wthi

2 c h = s wh2

;

E x thij c h = E x t

1hijx t2h¢i¢j¢ d i = 0 for all t, h, i, j and tb g 1hij π t2 b g h¢i¢j¢ ; E x thij

2 c h = sxh2 ; and

-1 < rh <1. By convention, define a0h ∫ 1 and e 0 0 hij ∫ . Considering times only back

to the base period and not beyond, (2) implies that e thij = rh

k

k=0

t-1

Â x t-k,hij . Using this

expression and the properties of x thij , the covariance structure implied by model (2) is

cov ,

, ,,

, ,,

, ,,

, , r r

t th hi i j j

t t h hi i j j

t th hi i j j

hij

t

hij

t

h h

t

h

h

t t

h

t

h

h

2 1

2

21 1

0 0

2 2 2

2 1

2 2

1 2

2

2 1

1

1

0

¢¢¢

-

=

+- = = ¢ = ¢ = ¢

- <= ¢ = ¢ = ¢

= = ¢ =¢ π ¢

R

S

|

|

T

|

|

|

d i

d i

d i

s r

r r

s

w

w

D

D

otherwise

(3)
5

where Dh

2

= sxh

2 1-rh c 2

h. Expression (3) implies that price relatives for a particular item

are correlated over time. At a given time period, items within a particular establishment

are also correlated, while other items are not.

The sample design addressed here is a rotating panel survey in which

establishments are sampled as the first-stage units. Establishments are retained in the

sample for a specified period of time and then rotated out and replaced by new units. At

each time t (t=1,...,T), we have a sample sth of nh establishments from the Nh

establishments in stratum h and a sample sthi of mhi items from the Mhi items in sample

establishment (thi). A two-stage sampling plan, often approximated in practice, is one in

which establishments are selected with probabilities proportional to Whi

0 = Whij0

j=1

Mhi

.

Items within establishments are then selected with probabilities proportional to Whij

0 .

Surrogate measures of size, felt to be closely related to Whi

0

 and Whij

0 , such as current sales

values or employment are often used in practice. At each time period, the total

establishment sample size is assumed to be constant at n = nh h with the total number

of sample items in stratum h being mh = mhi i ̨sth. At each time period a proportion dh

of the sample establishments is rotated out in stratum h and an equal number rotated in.

The size of the overlap, stuh = sth  ̇ suh, between samples from time t and u (t ‡ u) is

max , 0 1 n tu m h h - - a fd r.

The class of estimators considered here was introduced in Valliant and Miller

(1989) for one-stage sampling and generalized in Valliant (1991). For the long-term

index, define

$It,0 = zth

t

h

zuh

u

zu+1,h

u

L

N

M O

Q

P u=1

t-1 gh

tu

(4)

where zkh

u = lhirkhiu,0

i ̨skh

, rkhi

u,0 = rhiju,0

j ̨skhi

mhi for k=u or u+1 (u=1,...,t-1), and gh

tu is a

real number. The term lhi is a coefficient which does not depend on the model random

variables rhij

t,0 . For the two-stage probability-proportional-to-size design mentioned

above, for example, lhi = Wh

0 nh . We restrict consideration to cases where
6

lhi = Wh

0

i ̨suh

in which case E zkh

u -Wh0

Ih c u,0h= 0. When all within-stratum samples of establishments

are large, $It,0 is approximately model-unbiased under (2). Short-term estimators are

defined by taking ratios of long-term estimators. The price change from time t

1 to

t2 t

1 <t b 2g is estimated by $It2 ,t

1 = $It2 ,0 $It1,0 .

A number of estimators in class (4) are listed in Valliant (1991). Three are of

particular interest. If gh

tu ”1, then (4) is the product estimator, which can be written as

$I1

t,0 = zuh

u

zuh

u-1

L

N

M O

Q

P u=1

t

h

with z1h

0 ”1. If gh

tu ” 0, (4) reduces to the simple index estimator

$I2

t,0 = zth

t

h

.

A third choice of gh

tu is the one which minimizes the approximate variance of $It,0 under

model (2). The optimum is complicated in general, but in the special case of a constant

number of sample items per establishment, mhi = mh, and lhi a constant for all sample

establishments in stratum h, then the optimum reduces to

gh

tu

*

= 1

2

auh

ath

rh

t-u 1+mh

guh

1-guh

L

N

M O

Q

P

-1

for £ u £ t -1 and guh h h h

u = +- ss r w w h2 2 22 c h 1 D . The long-term estimator which uses

the optimal gh

tu

*

 will be denoted as $I3

t,0 .

3. APPROXIMATE VARIANCES UNDER THE MODEL

When the establishment sample size nh is large in each stratum, the long-term

index estimator can be approximated, as shown in Appendix A of Valliant (1991), by

$It,0 @ zth

t + ghtu ath

auh

zuh

u -zu+1,h c hu

u=1

R t-1

ST

UV

h W

. (5)

Using results in the appendix of that paper, we can write the approximate variance of the

long-term estimator as

var $Id it,0 @ atuh ih c ht,u 2+2 btuhih

t,u

u=1

t-1

+cth

u=1

Rt-1

ST

UV

h W (6)
7

where

ih

t,u = ath auh =E Ih ct,0hE Ih cu,0h,

atuh = gh c htu 2 lhi

2

mhi

vuhi

i ̨Cuh

+

lhi

2

mhi

vuhi

i ̨Duh

L

N

MM

O

Q

PP

,

btuh = gh

turht-u 1-rh c h2u Dh

2 lhi

2

mhi

- lhi

2

i ̨s mhi t,u+1,h i ̨stuh

L

N

MM

O

Q

PP

 ,

cth = lhi

2

mhi

vthi

i ̨sth

 ,

where vuhi = vuh 1+bmhi -1gguh , vuh = swh

2 + 1-rh c 2u

hDh

2 , Cuh = suh -su+1,h , i.e. the part of

suh that is not contained in su+1,h, and Duh = su+1,h -suh .

An expression similar to (6) can also be worked out for the approximate variance

of the short-term index estimator $It,u.

4. GENERALIZED VARIANCE FUNCTIONS FOR INDEXES

Judging from (6), the approximate variance is a second order polynomial in the

stratum superpopulation short-term indexes ih

t,u

. This is analogous to the relationship

between an estimator T$ of the population total T in two-stage sampling and its

approximate variance derived in Valliant (1987) for a particular class of models in which

the variance of a unit was a quadratic function of the unit's mean:

var $ejT aE T bE T @ + bg bg. 2(7)

The terms a and b are coefficients that depend on various quantities, such as intracluster

correlations, numbers of population and sample units within clusters, and coefficients in

the estimator T$. In fitting the GVF model defined by (7), the usual procedure is to select

a group of variables which all have the same a and b coefficients, calculate point

estimators of variance for each of the variables, and then to estimate a and b by some

form of least squares. Application of this course to (6) would be fraught with practical

difficulties. In (7) there are only two regression coefficients to be estimated −− a and b.

In (6) there are 2(t-1) + 1. As t increases so does the number of coefficients. The
8

components of the coefficients, atuh ,btuh ,and cth, are also complex, so that identifying

different indexes that all follow model (6) would be a problem.

An alternative approach is to work with a particular index series and attempt to

model the behavior of its variance over time. If ih

t,u

 is a smooth function of time, e.g. a

polynomial in t-u, then the variance (6) will also be a smooth function of time, say, f taf.

If an unbiased, or approximately unbiased, variance estimator is used for $It,0 , then its

expectation can also be described by f taf. As data are accumulated over time, a time

series of point variance estimates is developed and the function f taf can be fitted by a

scatterplot smoother without having to know the explicit form of the function. A number

of such smoothers are available, and we will consider two that have proved to be useful

in other situations.

The two smoothers used here are the super smoother (Friedman 1984) and loess

(Cleveland 1979, Cleveland, Cleveland, McRae, and Terpenning 1990). The two

algorithms are fairly complex to describe in detail, so that only rough sketches will be

given here. Both methods use local linear fits in neighborhoods around each point t. A

critical parameter in both algorithms is the span, the size of the neighborhood around t,

which is used to estimate f taf. In loess the span is fixed while for the super smoother,

spans can be variable. Of the two, loess explicitly incorporates features to reduce the

effects of outlying values and tends to produce a smoother looking curve of estimates.

The variable span used by super smoother allows it to adapt more readily to changing

curvature in f taf. Super smoother also has the advantage of being computationally

faster than loess.

5. AN EMPIRICAL STUDY

A simulation study, using a population derived from data collected for the U.S.

Consumer Price Index program by the BLS, was undertaken to test the usefulness of the

proposed method of calculating GVF's. The population was composed of establishments
9

and items and was described in detail in Valliant (1991). Its main features are briefly

recounted here. The population was divided into the five strata, which are listed in Table

1 along with various population and sample allocation numbers. The six hundred and

fifty-nine establishments contained an average of just under 10 items each. Each item

had prices for 42 consecutive months.

Two sets of 500 stratified two-stage samples were selected with the number of

sample establishments allocated to each stratum being roughly proportional to Wh

0

. The

total establishment sample sizes in the two sets of samples were n = 50 and 100.

Samples were selected in such a way that 20% of the sample establishments were rotated

in each 12-month period. This was done by first selecting a large systematic, randomstart sample of establishments in each stratum with probabilities proportional to Whi

0

. For

samples of size n = 50, the initial, large sample size was 84 and was 168 for the samples

of size n = 100. These initial samples were large enough to accommodate all 42 months

accounting for the amount of establishment rotation. The initial sample from each

stratum was then sorted in a random order. For a particular time period t, the stratum

establishment sample consisted of establishments +(t -1)dhnh,K ,nh +(t -1)dhnh,

where dh was the proportion of establishments rotated in a month. For both the cases of

n = 50 and n = 100, dh =1 60 which resulted in an annual turnover of 2bnh 60g= nh 5

establishments or 20%. From each sample establishment, mh = 2 sample items were

selected systematically with probability proportional to Whij

0 .

From each sample, the long-term estimators $Ij

t,0 (j=1,2,3; t=1,...,42), and the

short-term estimators of 1-month and 12-month change $Ij

t2 ,t

1

 (j=1,2,3; t1=t2 -1, t2 -12 for

t

1 ‡1) were computed. The special case of lhi = Wh

0 nh was used, which produces a

design-unbiased estimator under the simulation study sampling plan. The parameters

needed for the optimal estimator $I3

t,0 were approximated as described in Valliant (1991).

Empirical results for the simple estimator were similar to those for the optimal so that

only the latter is discussed subsequently.
10

Point variance estimates were obtained by the linearization method and were

described in detail in Valliant (1991). It should be emphasized that the results here do

not depend on the use of any particular method of point variance estimation. Estimates

obtained by balanced repeated replication, the jackknife, or another approach would

work just as well as long as consistent or approximately unbiased variance estimates were

used. For each sample the linearization variance estimate was computed for each of the

long-term and short-term index estimates and time periods named above. Two GVF's −−

super smoother and loess −− were then computed for each index series. For example, for

the product long-term index estimate, a series of 42 point variance estimates was

produced for each sample. The super smoother and loess estimates were calculated in

each sample by applying those methods to the series of 42 linearization estimates for

each index estimate. The simulation calculations were performed in double precision

using Borland's Turbo Pascal. GVF's were calculated with the software package S-PLUS

for DOS by Statistical Sciences Inc.

Summary statistics were then calculated across all 500 samples. The square roots

of the empirical mean squared errors were computed as $d i I - I

2 L 500 NM O

QP

1 2

 with the

summation being over the 500 samples, $I being one of the long- or short-term

estimators, and I being the population index defined in section 2. Square roots of the

average of the variance estimates were computed at each time period as v where

v = vs 500 s=1

500 and vs is one of the three types of variance estimate (linearization, super

smoother, or loess) at a particular time period from sample s. This was done separately

for the product and optimal estimates for long-term, 1-month, and 12-month price

change.

Summary results across all samples and time periods are listed in Tables 2 and 3.

The ratios (in percent) of the square root of the average variance estimate to the root of

the empirical mean squared error (RMSE) are generally somewhat less than 100 in all
11

cases, i.e. both the point variance estimate (v$) and the GVF's are underestimates, but the

problem is minor. The exception is the long-term optimal estimator for n=50, where the

variance estimates are slight overestimates. In all cases, for both the product and optimal

estimators, the GVF's are more stable than v$. For example, in Table 2, the standard

deviation of the super smoother GVF is 61% of that of v$ for 1-month change when

n=100. For the same case the loess GVF has a standard deviation which is 57% of that

of v$. The biggest gains in stability are for 1-month price change while the smallest gains

occur for long-term change. The loess estimates are generally more precise than the

super smoother estimates with the improvement compared to the linearization estimate

being somewhat less for the larger sample size. Tables 2 and 3 also list empirical

coverage of 95% confidence intervals across the 42 time periods. Normal approximation

confidence intervals were computed in the usual way as $I –1.96 v where $I is one of

the long- or short-term indexes and v is one of the variance estimates. Although all

variance estimates provide slightly less than the nominal 95% coverage, the smallest

percentage in Tables 2 and 3 is 92.0%, and the GVF's are quite competitive with v$.

Figures 1-4 are plots of summary statistics over the 500 samples by time period

for n=100. Figures are given only for the long-term and 1-month product and optimal

estimators. Plots for the 12-month price change estimators are qualitatively similar. The

upper left panel in each figure plots empirical RMSE's and the square root of the average

of each GVF versus time. The GVF's are much smoother than v$, as might be expected.

Although both smoothers are not inordinately influenced by outliers among the v$'s, the

super smoother does follow the fluctuations of the v$ curves more closely than does loess.

This leads to the super smoother's generally having a larger standard deviation than loess,

as shown in the upper right-hand panel of each figure. The lower left-hand panel shows

the ratio of the GVF standard deviation over the 500 samples to the standard deviation of

v$. This again simply illustrates that the two GVF's are more precise than the

linearization estimate with gains being especially large for 1-month change. The lower
12

right-hand panel of each figure charts the coverage of 95% confidence intervals over

time. The GVF's give reasonably good coverage which is almost equal to that of the

point variance estimates. The time periods where the GVF's provide noticeably poorer

coverage than v$ are ones where the smoothers do not closely follow upward fluctuations

in v$.

A further possibility, which we have not pursued, would be to calculate a

weighted average of a smoothed variance and the point variance estimate at each time

point. This could be advantageous when the point variance estimators are felt to be more

nearly unbiased than the smoothed estimates because of failure of the approximate

variance (6) to be a smooth function of time.

6. CONCLUSION

In continuing surveys which produce time series of estimates, the methods studied

here for smoothing variance estimates appear to be quite useful. For continuing surveys

in which the sample design and sample size are the same for long periods of time, users

expect variances to be smooth over time, a feature which point variance estimates

generally do not have. Such expectations by users may seem, at first, to be statistically

unreasonable since actual mean square errors may vary over time. However, for price

indexes we have shown, using large-sample theory and simulations, that smoothed,

approximately unbiased variance estimates can be obtained which are more stable than

point variance estimates for both long-term and short-term price change, and which also

provide near nominal confidence interval coverage. Thus, in the situation studied here,

smoothed variances have a statistical justification, in addition to having considerable

cosmetic appeal to data users.

The methods explored here also should apply to other types of panel surveys, like

labor force surveys, which publish time series of employment estimates. The linearized

form of the index estimator given by (5) in section 3 is similar to the composite
13

estimators described by Cantwell (1990) for household surveys. Since, in large samples,

the index estimator and the composite estimators have similar structure, the possibility of

smoothing variances of the latter over time appears to be worth pursuing.

The nonparametric approach may also have use in sample design problems for

continuing surveys. If components of variance are estimated at a number of points in

time, one alternative is to average the components over time. However, when

components of variance are subject to seasonal factors, averaging of variance

components across time periods may obscure the seasonality. Alternatively, the

smoothers can be used to obtain more stable estimates of components which could

subsequently be used for determining sample allocations. Smoothing can be done in

such a way that seasonal differences in components are preserved in order to study their

effects on computed allocations.

One useful feature of the GVF's, described in Wolter (1985), for household

surveys is the ability to publish a limited number of model parameters from which users

calculate their own standard errors. This feature is lost for the smoothed variances

because an explicit functional form is not estimated. The main use of the variance

smoothers, thus, may be to produce more stable estimates.

REFERENCES

Andersson, C., Forsman, G., and Wretman, J. (1987). On the Measurement of Errors in

the Swedish Consumer Price Index. Bulletin of the International Statistical Institute,

52, Book 3, 155-171.

Cantwell, P. (1990). Variance Formulae for Composite Estimators in Rotation Designs.

Survey Methodology, 16, 153-163.

Cleveland, W.S. (1979). Robust Locally Weighted Regression and Smoothing

Scatterplots. Journal of the American Statistical Association, 74, 829-836.
14

Cleveland, R.B., Cleveland, W.S., McRae, J.E., and Terpenning, I. (1990). STL: A

Seasonal Decomposition Procedure Based on Loess. Journal of Official Statistics, 6,

3-32.

Friedman, J.H. (1984). A Variable Span Smoother. Technical Report No. 5, Laboratory

for Computational Statistics, Stanford University.

Hanson, R.H. (1978). The Current Population Survey: Design and Methodology,

Technical Paper 40, Washington DC, U.S. Bureau of the Census.

Johnson, E.G. and King, B.F. (1987). Generalized Variance Functions for a Complex

Sample Survey. Journal of Official Statistics, 3, 235-250.

Leaver, S.G. (1990). Estimating Variances for the U.S. Consumer Price Index for 1978-

1986. Proceedings of the Section on Survey Research Methods, American Statistical

Association, 290-295.

Valliant, R. (1987). Generalized Variance Functions in Stratified Two-stage Sampling.

Journal of the American Statistical Association, 82, 499-508.

−−−−−−− (1991). Variance Estimation for Price Indexes from a Two-stage Sample with

Rotating Panels. Journal of Business and Economic Statistics, 9, 409-422.

Valliant, R. and Miller, S.M. (1989). A Class of Multiplicative Models for Laspeyres

Price Indexes. Journal of Business and Economic Statistics, 7, 387-394.

Wolter, K.M. (1985). Introduction to Variance Estimation, Springer-Verlag, New York.

FIGURE TITLES

Figure 1. Summary plots of simulation results for the long-term product estimator from

500 samples of size n=100 establishments. Legend for each of the panels is in the upper

left-hand panel. v denotes the linearization estimator; supsmu denotes the super

smoother.

Figure 2. Summary plots of simulation results for the 1-month product estimator from

500 samples of size n=100 establishments. Legend for each of the panels is in the upper
15

left-hand panel. v denotes the linearization estimator; supsmu denotes the super

smoother.

Figure 3. Summary plots of simulation results for the long-term optimal estimator from

500 samples of size n=100 establishments. Legend for each of the panels is in the upper

left-hand panel. v denotes the linearization estimator; supsmu denotes the super

smoother.

Figure 4. Summary plots of simulation results for the 1-month optimal estimator from

500 samples of size n=100 establishments. Legend for each of the panels is in the upper

left-hand panel. v denotes the linearization estimator; supsmu denotes the super

smoother.
16

Table 1. Universe and sample characteristics for the study population

fh

Stratum Wh

0 Nh Mh nh n n=50 n=100

1 Beef .32 154 1800 .32 .10 .20

2 Eggs .13 57 653 .12 .10 .21

3 Milk, other dairy .33 155 1800 .32 .10 .20

4 Fresh vegetables .10 193 1013 .12 .03 .06

5 Sugar .12 100 1175 .12 .06 .12

Total 1.00 659 6441 1.00

Table 2. Simulation results for the product estimator from 500 two-stage cluster samples

averaged over 42 time periods. All figures are in percent. v$ denotes the linearization

variance estimate.

v RMSE Std. dev. (GVF)

/Std. dev.(v$)

95% CI coverage

v$ Supsmu Loess Supsmu Loess v$ Supsmu Loess

n=50

LT 99.5 99.0 98.0 90.8 86.8 93.7 93.1 92.8

1-month 99.3 99.4 96.8 56.8 50.8 93.3 93.6 92.9

12-month 95.0 94.8 93.9 74.3 71.0 92.0 92.3 92.1

n=100

LT 99.8 99.2 98.5 96.1 92.1 94.2 93.4 93.3

1-month 99.3 99.8 98.1 61.0 57.0 93.8 93.6 93.3

12-month 96.1 95.8 96.0 79.1 79.6 93.2 93.4 93.3
17

Table 3. Simulation results for the optimal estimator from 500 two-stage cluster samples

averaged over 42 time periods. All figures are in percent. v$ denotes the linearization

variance estimate.

v RMSE Std. dev. (GVF)

/Std. dev.(v$)

95% CI coverage

v$ Supsmu Loess Supsmu Loess v$ Supsmu Loess

n=50

LT 104.2 103.7 102.5 86.6 82.6 94.4 93.9 93.7

1-month 98.8 99.1 96.1 50.4 42.9 93.1 93.6 92.9

12-month 95.3 94.9 94.3 71.8 69.4 92.1 92.9 92.6

n=100

LT 99.9 99.4 98.7 91.5 86.9 94.3 93.5 93.4

1-month 98.7 99.3 97.7 55.8 51.4 93.8 93.5 93.2

12-month 94.5 94.2 94.3 78.3 77.9 92.7 92.8 92.7