APPROXIMATION METHODS FOR COVARIANCE 

MATRIX ESTIMATORS USED IN ANALYSIS OF DIARY 

AND INTERVIEW DATA 

FROM THE U.S. CONSUMER EXPENDITURE SURVEY 

John L. Eltinge and Amang Sukasih 

U.S. Bureau of Labor Statistics and Texas A&M University 

Abstract: Multivariate analyses of complex survey data often make extensive use of estimators Vˆ of the 

variance-covariance matrix V of a random vector yˆ where the variances and covariances are evaluated 

with respect to the sample design. For example, Vˆ

 is often used in computation of quadratic-form test 

statistics, and also may be used in computation of generalized least squares point estimators. However, 

these analyses can be problematic when Vˆ is based on a small or moderate number of degrees of freedom. 

This paper considers methods for approximation of V, and for computation of associated modified 

estimators of V. Principal emphasis is placed on exploratory evaluation of modeling assumptions. The 

methods discussed in this paper are motivated and illustrated by analyses of interview and diary data from 

the U.S. Consumer Expenditure Survey. 

Keywords: Correlation matrix; Diagnostics; Eigenvalues and eigenvectors; Exploratory analysis; 

Generalized variance functions; Misspecification effect matrix. 

1. Introduction: Multivariate Analysis of Interview and Diary Data in the U.S. Consumer 

Expenditure Survey 

This paper describes some methods for approximation of covariance matrices used in the analysis of 

complex survey data. In the interest of space, we focus primarily on one specific application to data from 

the U.S. Consumer Expenditure Survey. A more general theoretical development will be considered 

elsewhere. 

The U.S. Consumer Expenditure Survey (U.S. Bureau of Labor Statistics, 1997) is a household 

survey based on a stratified multistage sample of consumer units, which are roughly equivalent to 

households. Within each selected primary sample unit, some consumer units are asked to provide data 

through a quarterly interview process, while other consumer units are asked to complete expenditure 

diaries. Interviewed consumer units are asked to report their expenditures separately for each of the 

previous three complete methods. For example, in an interview conducted on November 14, the 

interviewer would ask the consumer unit to report expenditures separately for October (the proximate 

month), September (the middle month) and August (the most distant month). In addition, consumer units 

assigned to the diary group are asked to complete a weekly expenditure diary in each of two consecutive 

weeks. For the discussion below, we will use the labels i = 1 through 5 for, respectively, data from the 

proximate interview month, the middle interview month, the distant interview month, the first diary week 

and the second diary week. 

Eltinge, Sukasih and Weber (2000) carried out a detailed analysis of the vectors 

yˆ 

p = ( yˆ p1

,...yˆ 

p5

)′ where each yˆ

pi is an estimator of mean annual expenditures based on source i , 

using data collected for each month or week, respectively, in a specific year p . These estimators use 

standard weighting adjustments to account for unequal selection probabilities, unit nonresponse and 

differences in the lengths of (monthly or weekly) reference periods. Thus, if there were no problems with 

nonsampling errors, one would anticipate that each estimator yˆ

pi would be approximately unbiased for the

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2. A Correlation-Based Approximation Method for Covariance Matrix Estimators 

For the years p = 1,..., P , let V equal the variance-covariance matrix of the approximate distribution eep

of the vector yˆ , evaluated with respect to the sample design. Consider the decomposition, p 

V = D R D (2.1) eep p p p 

1/ 2 1/ 2 1/ 2 where Dp = diag(Veep11,...,Veep55 ) , Veepii is the square root of the i -th diagonal element of Veep , and 

R is the 5 × 5 dimensional correlation matrix associated with V .

p eep

The decomposition (2.1) suggests the following two-step method for approximation of V , and eep

for construction of an alternative estimator of V that may be more stable than Vˆ . First, one may use eep eep

univariate generalized variance function methods to compute alternative estimators of Veepii based on the 

standard estimators Vˆ 

eep and available auxiliary information. (For some background on generalized 

variance function methods, see, e.g., Wolter (1985, Chapter 5), Johnson and King (1987), Valliant (1987) 

*

and references cited therein.) Let Veepii be the resulting modified univariate variance estimators and define 

* * 1/ 2 * 1/ 2 D = diag({V } ,...,{V } ) .

p eep11 eep55 

In addition, consider the assumption that the correlation matrices R are constant over all years p 

p , so that R = R , say. This assumption may be reasonable in cases like the Consumer Expenditure p 

Survey in which the basic sample design remains the same over multiple years. Under the assumption of a 

* −1

common correlation matrix R , a simple estimator of R is R = P ∑

P 

= 

Rˆ 

p , where Rˆp is the 

p 1 

*

correlation matrix computed directly from Vˆ . We then may use D and R

*

 to define the modified eep p 

* * * * *

estimator V = D R D . In general, the performance of V as an estimator of V will depend on the p p p p p 

true mean expenditure for year p . Conversely, differential patterns of underreporting or nonreporting of 

specific expenditures would cause one or more of the estimators yˆ

pi to have different expectations. 

Evaluation of these and related bias issues in Eltinge, Sukasih and Weber (2000) led to extensive 

use of a design-based covariance matrix estimator Vˆ

eep computed through standard balanced repeated

replication methods using 44 sets of replicate weights. For many national level estimands in this 

application, 40 ×Vˆ 

eep would be distributed approximately as a Wishart random matrix with 40 degrees of 

freedom. However, for estimands involving relatively rare types of expenditures or subpopulations 

concentrated in a moderate number of geographical areas, Vˆ

eep may be considerably less stable than would 

be suggested by the nominal 40 degrees of freedom term. Consequently, it is of interest to consider 

approximation methods that may lead to variance estimators that are more stable than Vˆ 

eep . Section 2

develops a class of estimators based on approximations for the correlation matrix and univariate variances 

associated with the matrix Vˆ 

eep . Section 3 applies the proposed methods to one specific example from the 

Consumer Expenditure Survey, with special emphasis on diagnostics to assess the adequacy of 

approximations used in Section 2. 

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relative magnitudes of the sampling variability of Vˆ 

eep , the approximation errors in the generalized

variance functions used to construct D *

p

, and the approximation errors Rp − R .

3. Application to Consumer Expenditure Survey Data 

We applied the general ideas of Section 2 to data from the Consumer Expenditure Survey for expenditures 

contained in one six-digit Universal Classification Code group, 360330 (men’s accessories, e.g., hats, ties 

and belts) for P = 11 years, 1987 through 1997. First, to develop appropriate estimators D *

p , we

explored the univariate variance-function patterns for each of the sources 1 through 5. Figure 1 displays a 

time plot of the sample coefficient of variation, se(yˆ

pi)/ yˆpi , with the plotting symbol set equal to the 

source label i . Due to differences in the numbers of respondents and the lengths of reference periods, the 

coefficients of variation for the diary sources ( i = 4 and 5) are larger than those for the interview sources 

(i = 1, 2 and 3). In addition, note that none of the five sources display any pronounced pattern of increase 

or decrease of the coefficients of variation over time. Figure 2 displays a plot of se( yˆ

pi) against yˆpi , 

with the plotting symbol again set equal to i . Taken across all five sources, this plot is roughly consistent 

with the simple linear regression model, 

se( yˆ

pi ) = γ 0 + γ1 y pi + error (3.1) 

where ypi is defined to equal the expectation of ypi and γ 0

 and γ1

ˆ are fixed coefficients. An ordinary 

least squares regression fit of the 55 points (5 sources across 11 years) displayed in Figure 2 resulted in a 

sample R

2

value equal to 0.83, suggesting a relatively good fit. Figure 2 also displays a fairly pronounced 

clustering of the yˆ

pi for the interview sources (1, 2 and 3) and diary sources (4 and 5), respectively. This 

is consistent with point estimation bias issues raised in Section 1 and discussed further in Eltinge, Sukasih 

and Weber (2000). Initial exploratory work with expansion of model (3.1) to include additional covariates 

did not lead to substantial improvements, and thus will not be considered further here. 

Second, recall that the multivariate decomposition (2.1) and the modified variance-covariance 

matrix estimator V

*

p

 were based on the assumption that the matrix R p is constant across p = 1,..., P . To

explore the consistency of this assumption with our data, Figures 3 and 4 present time plots of the sample 

correlation values Rˆ 

ijp , say, computed directly from the initial sample variance-covariance matrix Vˆ eep . 

Figure 3 displays results for R12 , the correlation between the interview sources 1 and 2. The symbol 

E represents the point estimate of this correlation. The symbols L and U represent lower and upper 95% 

pointwise confidence bounds for this correlation, based on a standard Fisher Z transformation (Snedecor 

and Cochran, 1967, p. 185) approach, under the assumption that 40 ×Vˆ 

eep is approximately distributed as

a Wishart random matrix with 40 degrees of freedom. Note that these confidence bounds are relatively 

wide, reflecting the relatively large amount of sampling variability encountered in the standard estimators 

Vˆ 

eep . Also, Figure 3 does not display any pronounced patterns of increase or decrease of R12 over time; 

and with the exception of 1988, 1993 and 1994, the confidence intervals for R12 contain the value zero. 

Figure 4 displays the corresponding results for R34 , the correlation between the interview source 3 and the 

diary source 4. Note especially that for all eleven years, the 95% confidence interval for R34 contains zero. 

Third, we considered the estimated misspecification effect matrix M = (V * ) −1/ 2 V ˆ

(V * )

−1/ 2

p p eep p 

3 
where (V

*

)

−1/ 2 equals the inverse of the symmetric square root of V*

p p . For some general background on

misspecification effects, see, e.g., Skinner (1989) and references cited therein. For the current discussion, it 

suffices to note that under correctly specified variance function models, the approximation Rp = R , and

additional regularity conditions, the matrix M p converges to the k × k dimensional identity matrix I

k ,

and each of its eigenvalues converge to one. Consequently, we can obtain a partial indication of the

adequacy of our estimator V

*

p through examination of the eigenvalues of M p . Figure 5 displays a time 

plot of the eigenvalues of M p for the years 1987 through 1997. The plotting symbols 1 through 5, 

respectively, correspond to the largest through the smallest eigenvalues for a given year p ; and the symbol 

m corresponds to the arithmetic average of the five eigenvalues. Figure 6 presents a similar plot for the 

case in which V

* * 

p is computed under the additional constraint R = I k ; cf. the relatively weak evidence 

of nonzero correlation displayed in Figures 3 and 4. Note especially that in Figure 6, the eigenvalues tend 

to be distributed more tightly around one, compared to the eigenvalues in Figure 5. To some degree, this 

may reflect the greater stability of V

* * 

p induced by the constraint R = I k . Finally, note that deviations of 

the observed eigenvalues from the value one reflect the combined effects of the sampling variability of 

Vˆ and V *

eep p , and the lack of fit in our approximations for Dp

 and R p . To identify deviations that are

not attributable to sampling variability alone (and thus indicate problems with lack of fit), it is useful to 

compare the observed eigenvalues with quantiles of the appropriate reference distributions. In general, 

these reference distributions may be obtained through simulation work, as outlined in Lee and Eltinge 

(2001). 

4. References 

Eltinge, J.L., Sukasih, A. and Weber, W. (2000). Feasibility of constructing combined estimators using 

consumer expenditure interview and diary data. Paper presented to the Bureau of Labor Statistics 

Conference on Issues in measuring Price Change and Consumption, June 8, 2000. 

Johnson, E.G. and King, B.F. (1987). Generalized variance functions for a complex sample survey. 

Journal of Official Statistics 3, 235-250. 

Lee, S.R. and Eltinge, J.L. (2001). Exploratory analysis of estimated design effect matrices computed from

complex survey data. Submitted for publication. 

Skinner, C.J. (1989). Introduction to Part A. In C.J. Skinner, D. Holt and T.M.F. Smith (eds.) Analysis of 

Complex Surveys. New York: Wiley. 

Snedecor, G.W. and Cochran, W.G. (1967). Statistical Methods, Sixth Edition. Ames, Iowa: Iowa State 

University Press. 

U.S. Bureau of Labor Statistics (1997). Chapter 16: Consumer expenditures and income. BLS Handbook 

of Methods. U.S. Department of Labor, Bureau of Labor Statistics Bulletin 2490, April, 1997. 

Washington, DC: U.S. Government Printing Office. 

Valliant, R. (1987). Generalized variance functions in stratified two-stage sampling. Journal of the 

American Statistical Association 82, 499-508. 

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

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