P. A. V. B. Swamy1

, Tamara S. Zimmerman2, Jatinder S. Mehta3 and Edwin L. Robison4 ∗

 

 

 

 

 

 

 

 

 

New Improved Small Area Models 

Abstract 

This paper provides the conditions under which the precisions of direct domain estimators produced within the standard 

design-based mode of inference used in survey analysis are improved by combining the direct estimators with matched 

statistical models. 

1. Introduction 

Estimates for small domains are produced in response to policy makers’ demands. Domains may be defined in terms of 

geographic areas, socio-demographic areas, industry groups, or other subpopulations. An estimator for a domain in a time 

period is referred to as “direct” if it is based on the domain- and time-specific sample data. A domain is considered as small if 

the domain sample size is not large enough to yield direct estimates of adequate precision. Reliable estimates for small 

domains can only be produced by moving away from the design-based estimation of conventional direct estimates to indirect 

model-dependent estimates. The essence of model-dependent approach is the use of auxiliary data available at the small area 

level. These data are used to construct predictor variables for use in a regression model that can be used to predict the 

variable of interest for all small areas. The problem here is that if this model is seriously misspecified, then inferences based 

on the model can be much worse than design-based inferences. The effectiveness of indirect model-dependent estimation 

depends on the availability of methods that can limit the effects of model misspecifications. Such methods are presented in 

this paper. 

We consider an area level model to supplement small domain direct estimates with area-specific auxiliary data. This 

model has two equations: the first says that for each domain in a time period, a direct survey estimator is the sum of the true 

value and a design-induced error. The second relates the true value to area-specific auxiliary variables. Following J.N.K. Rao 

(2003, p. 77), we call the first equation “the sampling model” and the second equation “the linking model.” To analyze the 

specification problems associated with the linking model, we begin with the conceptual idea that there is the “true” model 

linking the “true” variables involving the “true” coefficients. We then show that the linking model is an exact representation 

of the “true” model if its coefficients are correctly interpreted. In the absence of the conditions implied by the correct

interpretations, the linking model is misspecified. In this paper, we prove that not all linking models that yield designconsistent estimators limit the effects of model misspecifications in small samples. To do so, we present two linking models, 

one of which is misspecified and the other of which is correctly specified. Both these models yield design-consistent 

estimators, but the predictions from the misspecified model are affected by the misspecifications in small samples. 

In the next section, we exploit the connection between a linking model and the underlying “true” model. By so doing, we 

are able to derive explicit algebraic expressions for the effects of model misspecifications and to develop a method of 

analyzing such effects. The implications of our discussion are summarized in the final section. 

2. Basic Area Level Model 

1

Bureau of Labor Statistics, U.S. Department of Labor, 2 Massachusetts Avenue, NE, Washington, DC 20212/ paravastu_s@bls.gov 

2

Bureau of Labor Statistics, U.S. Department of Labor, 2 Massachusetts Avenue, NE, Washington, DC 20212/ 

Zimmerman_Tamara@bls.gov 

3

Temple University, Philadelphia, PA 19122/ mehta1007@comcast.net 

4

Bureau of Labor Statistics, U.S. Department of Labor, 2 Massachusetts Avenue, NE, Washington, DC 20212/ Robison_E@bls.gov 

∗

 Any opinions expressed in this paper are those of the authors and do not constitute policy of the Bureau of Labor Statistics 

or the Bureau of the Census. We thank Professor Roderick Little for giving us a chance to discuss with him some of the 

issues raised in this paper. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

We assume throughout that the available observations on variables are the sums of “true” values and sampling and/or 

nonsampling errors. In what follows, symbols with an asterisk denote “true” values and symbols with a hat or without either a 

hat or an asterisk denote observable variables measured with error. Let yˆ

it be a direct survey estimator of yit 

* , which, for 

domain i and time t, denotes the true value of a population characteristic. 

2.1 Sampling Model 

We assume that yˆ

it is design-unbiased (or p-unbiased) for yit 

* , in symbols, Ep 

( yˆ

it ) = yit 

* . The design variance of yˆ

it is 

denoted by 

p V y() ˆ . We assume further that it 

yˆ

it = yit 

*

 + eit , i = 1, 2, ..., n, t = 1, 2, ..., Ti (1) 

where the sampling errors eit are independent with 

E e(| y 

*

) = 0, V ( | *

) = ψ . (2) p it it p

eit yit it 

These assumptions may be quite restrictive in some applications. For example, some sample designs imply a nonzero 

autocorrelation structure for the sampling errors. Also, the absolute size of the sampling error changes over time because of 

redesigns, sample size changes, and variation in the levels of certain population characteristics. When modeling the sampling 

error, it is important to account for these types of sampling design features. We make use of the procedures suggested in the 

literature to estimate the variance ψ it and the parameters of the stochastic process eit may follow (see J.N.K. Rao (2003, p. 

166)). If the direct survey estimates given by yˆ

it contain nonsampling error, then we show below how we might quantify this 

error. 

2.2 Derivation and Properties of a Linking Model 

In superpopulation modeling, the population values, * 

y , are assumed to be a random sample from an infinite it 

“superpopulation” and are assigned a probability distribution. To derive this distribution, we adopt the following definition: 

Definition 1 (Swamy and Tavlas, 2001) Any variable or value that is not mismeasured is true and any economic relationship 

with the correct functional form, without any omitted explanatory variables and without mismeasured variables, is true. 

Accordingly, if the “true” relationship among * 

y and a set of observable area-specific auxiliary variables, denoted by it 

x1it ,..., xK −1, it , exists, then it is expressible in the form: 

K −1 mit 

* * * 

yit = α 0it + ∑α jit xjit + ∑α git xgit (3) 

j =1 g K=

where all the determinants (both observed and unobserved) of * 

y are included on the right-hand side, even though we may it 

know nothing about some of the unobserved determinants. In other words, there are no excluded explanatory variables in 

equation (3). To avoid the possibility of excluding from (3) any determinants of * 

y at any time for any domain, we assume it 

that the number of the determinants of * 

y may change across i at a point in time and through time. Hence is domain- it mit 

specific and time-dependent. 

Freedman and Navidi (1986, p. 7) criticized the modeling assumptions that had no foundation in theory or in fact. To 

make sure that (3) has such a foundation, we use only the determinants of yit 

* suggested by relevant economic theories. 

Equation (3) avoids all restrictions not implied by these theories. In this connection, it can be thought of as the relationship 

implied by economic theories, in which case, we can assume that a mechanism exists through which the right-hand side 

variables in (3) exactly determine yit 

* . If we exclude from (3) the variables, say xg

* 

it , g = K, ..., mit , then an explanation of 

* * * *

the relationship between yit and xjit , j = 1, ..., K-1, can be found in the dependence of both yit and xjit , j = 1, ..., K-1, on 

some of excluded variables, x

* 

git , g = K, ..., mit , a phenomenon known as spurious correlation (see Lehmann and Casella 

(1998, p. 107)). Thus, by virtue of it’s including all relevant explanatory variables in the right way, the formulation in (3) 

avoids all possible spurious correlations. 
 

 

 

 

 

 

 

 

 

 

 

 

 

Although the relationship in (3) corresponds to that suggested by economic theories, typically, its correct functional form 

is unknown. Consequently, any specific assumption concerning its functional form may be incorrect. 

Solution to the Unknown-Functional-Form Problem Equation (3) is linear if α jit , j = 1, ..., K-1, and α git , g = K, ..., mit , 

are constants; otherwise, the equation is nonlinear. Different time profiles of its coefficients assign different functional forms 

to (3). Restrictions on the pattern of variation in (e.g., the constancy of) its coefficients may force (3) to have an incorrect 

functional form. Any finite class of functional forms may not cover the “true” functional form of (3) as a special case. In our 

state of ignorance about the “true” functional form of (3), permitting all of its coefficients to differ among domains both at a 

point in time and through time gives an infinite class of functional forms that encompasses its “true” functional form. 

We make use of this solution by allowing all the coefficients of (3) to vary freely. The coefficients of (3) with the “true” 

time profiles are called ‘the “true” coefficients.’ They are denoted by α

* 

jit , j = 0, 1, ..., K-1, and α g

* 

it , g = K, ..., mit , the 

existence of which is assumed here. This assumption is equivalent to the assumption that the “true” functional form of (3) 

exists. 

Therefore, the “true” model that is a member of the class in (3) is

K −1 mit 

* * * * * * 

yit = α 0it + ∑α jit xjit + ∑α git xgit . (4) 

j =1 g K=

This equation satisfies Definition 1. Equation (4) differs from (3) in that the former equation includes the “true” values of 

both variables and coefficients, whereas the class of models represented by (3) includes only the “true” values of variables. 

* * *

The values of yit that are defined for the unrealized values of xjit , j = 1, ..., K-1, and xgit , g = K, ..., mit , will not be 

realized and Pratt and Schlaifer (1988, p. 28) call them potential values. The only values of yit 

* that are determined by the 

realized values of x

* 

jit , j = 1, ..., K-1, and xg

* 

it , g = K, ..., mit will be realized. It is the existence of these potential values that 

guarantees the existence of the “true” model. The “true” model cannot be a real-world relation unless it exists. Basmann 

(1988, p. 99) shows that causation is a real-world relation between events. Consequently, if the potential values of yit 

* do not 

exist, any version of (4) fitted to observations is a pure statistical artifact. 

Unfortunately, we cannot estimate the “true” model because it has certain unknown determinants. For example, data on 

some of the determinants of * are not available and even the data we have on the other determinants of * 

y contain errors. it yit 

Suppose that data on x

* 

jit , j = 1, ..., K-1, are available and data on xg

* 

it , g = K, ..., mit , are not available. The observed 

measurements, x jit = x

* 

jit + v jit , j = 1, ..., K-1, are the sums of “true” values, x

* 

jit , and measurement errors, v jit . We now 

show that excluding xg

* 

it , g = K, ..., mit , from the “true” model or mismeasuring x

* 

jit , j = 1, ..., K-1, introduces biases into 

α

* 

jit , j = 0, 1, ..., K-1. 

A model involving only the observable counterparts of the first K-1 explanatory variables of the “true” model can be 

written as 

K −1 

yit

*

 = γ 0it + ∑γ jit xjit (5) 

j =1 

We call this model “the time-varying coefficient (TVC) model.” Substituting the right-hand side of this equation for yit 

* in 

(1) gives a model with yˆ

it as its dependent variable and the design-induced errors, eit , as its errors. 

A Classification of the Explanatory Variables of the “True” Model The explanatory variables, xjit , j = 1, ..., K-1, are 

called the included explanatory variables because they are included in the TVC model. The variables, xg

* 

it , g = K, ..., mit , are 

called excluded variables because they are excluded from the TVC model. 

Connections between the Included and Excluded Explanatory Variables Pratt and Schlaifer (1988, p. 34) show that the 

condition that the included explanatory variables be (mean) independent of ‘the’ excluded variables themselves “is 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

meaningless unless the definite article is deleted and can then be satisfied only for certain ‘sufficient sets’ of excluded 

variables some if not all of which must be defined in a way that makes them unobservable as well as unobserved.” (p. 34) 

(see also Pratt and Schlaifer (1984, pp. 11-13)). From this result it follows that a meaningful assumption is 

K −1 

* * * * 

xgit = λ0git +∑λ jgit xjit , g = K, ..., mit (6) 

j =1 

where the coefficients with the correct time profiles are denoted by λ

* 

jgit , j = 0, 1, ..., K-1. 

Mapping between the Coefficients of the “True” Model and the Coefficients of the TVC Model Substituting xjit - v jit 

for x

* 

jit in the equation that results when the right-hand side of equation (6) is substituted for xg

* 

it in the “true” model, we 

obtain 

mit mit v

* * * * * * γ 0it = α 0it + ∑α git λ0 git and γ jit = ( α jit + ∑α git λ jgit )(1 - jit ) (j = 1, ..., K-1) (7) 

g K= gK x = jit

Correct Interpretations of the Coefficients of the TVC Model The first equation in (7) can be interpreted as implying that 

the intercept, γ , of the TVC model is the sum of (i) the intercept ( α

* ) of the “true” model, (ii) the joint effect 0it 0it 

mit * * *

( ∑gK 

α git λ0git ) on the dependent variable ( yit 

* ) of the portions of the “true” values, xgit , g = K, ..., mit , of excluded 

=

variables remaining after the effects of the “true” values, x

* 

jit , j = 1, ..., K-1, of the included explanatory variables have been 

removed. The last K-1 equations in (7) can be interpreted as implying that for j = 1, ..., K-1, γ jit is the sum of (i) the 

* * mit * * coefficient α jit on xjit of the “true” model, (ii) a term ( ∑ α git λ jgit ) capturing omitted-variables bias due to excluded 

g K=

* * * variables, and (iii) a measurement-error bias, -( α + 

mit α λ )( / ), due to mismeasuring the included jit ∑gK git jgit v jit x 

= jit

explanatory variable x jit (see Chang, Swamy, Hallahan and Tavlas (2000), Swamy and Tavlas (2001) and Swamy, Chang, 

Mehta and Tavlas (2003)). 

The omitted-variable biases are zero if the included explanatory variables are uncorrelated with every excluded variable 

and the measurement-error biases are zero if the included explanatory variables are measured without error. These conditions 

are rarely, if ever, satisfied. Thus, Freedman and Navidi’s (1986, pp. 6 and 7) point about omitted variables and measurement 

errors causing serious bias in the estimates of yit 

* is correct. The coefficients of the TVC model are called “the biased 

coefficients” because they contain omitted-variable and measurement-error biases. The coefficients of the “true” model are 

called “the bias-free coefficients” since they are not subject to any biases. The component α

* 

jit is called “the bias-free 

component” of γ jit . The relationship between yˆ

it and xjit is spurious if the bias-free component of γ jit is zero for all t. If an 

estimate of the coefficient on xjit in the regression of yˆ

it on x1it , ..., xK −1,it has a wrong sign, then omitted-variables and 

measurement-error bias contained in the coefficient and the incorrect functional-form of the regression must have caused this 

wrong sign. The significance (or insignificance) of an estimate of γ jit is not a good indicator of the significance (or 

insignificance) of the implied estimate of α

* 

jit . To avoid this difficulty, we suggest below a method of decomposing an 

efficient estimator of γ jit into the estimators of its components in (7). 

An Important Result Under the decomposition of its coefficients in (7), the TVC model is an exact representation of the 

“true” model because the right-hand sides of equations (4) and (5) are exactly equal to each other when the coefficients of (5) 

satisfy the equations in (7). 

One question that remains to be answered is that of parameterization: which features of the TVC model ought to be 

treated as constant parameters? To answer this question, we need the following implications: 

Implications of the Correct Interpretations of the Coefficients of the TVC Model (i) Variations in its components lead to 

variations in γ jit . It can be seen from (7) that the real-world sources of variations in its components are: (a) the nonlinearities 
 

 

 

 

 

 

 

 

 

 

of the “true” model resulting in variations in the α

* ’s, (b) the nonlinearities of the relationships among the “true” values of 

excluded and included explanatory variables resulting in variations in the λ

* ’s, (c) variation in the ratio of vjit to xjit , and 

(d) changes in mit . (ii) The measurement-error bias component of γ is a function of both and , implying that the jit x jit v jit 

included explanatory variable x jit is correlated with its own coefficient γ jit (i.e., in the TVC model, the included explanatory 

variables are correlated with their own coefficients). (iii) For j = 0, 1, ..., K-1, the γ jit ’s cannot be uncorrelated with each 

other because the coefficients, α g

* 

it , g = K, ..., mit , on excluded variables are their common determinants. 

In our work, these implications are the prime considerations guiding the selection of parameters. We decompose each 

coefficient of the TVC model into its components shown in (7) by assuming that γ jit is linearly related to some observable 

variables, denoted by zdit , d = 1, ..., p-1, plus a linear combination of stochastic errors. 

p−1 q−1 

γ = π + π z + μ + l ε (8) jit j0 ∑ jd dit ji ∑ jh hit 

d =1 h=0 

where none of the zdit is equal to 1 for all i and t, the l jh ’s are known positive constants, and the μ ji ’s and ε hit ’s are random 

variables. We give the z’s the new name, “the coefficient drivers.” The TVC model is called “the stochastic coefficients (SC) 

model” if its coefficients follow assumption (8). This assumption was made previously in Swamy and Mehta (1975), Swamy 

and Tinsley (1980), and Swamy, Chang, Mehta, and Tavlas (2003). We further assume that for j > 0, the sum of p1 (< p) 

*

specific terms in π j0

 + ∑

p−1

π jd zdit is equal to the bias-free component, α jit , of γ jit and the sum of the remaining p - p1 d =1 

q−1 

terms and μ + l ε is equal to the sum of omitted-variables and measurement-error bias components of γ (see ji ∑ jh hit jit 

h=0 

Swamy, Tavlas and Chang (2005)). From this assumption it follows that only those bias-free coefficients of the “true” model 

that are also the components of the coefficients of the TVC model are identifiable--subject to the restrictions implied by (8)--

on the basis of the available data, whereas the bias-free coefficients on excluded variables are not identifiable. The only way 

in which the bias-free coefficient on an excluded variable is identifiable is through converting the excluded variable into an 

included variable. Assumption (8) does not contradict the implications of the correct interpretations of the γ jit if (i) the 

p−1 function, π j0

 + ∑ π jd zdit , completely accounts for the correlation between xjit and γ jit so that the remainder, μ ji + d =1 

q−1

∑h=0 

l

jhε hit , obtained by subtracting the function from γ jit is independent of x jit and (ii) the right-hand side of equation (8) 

is expressible as the sum of two sums, one of which is equal to the bias-free component of γ jit and the other of which is 

equal to the sum of omitted-variables and measurement-error bias components of γ jit . The satisfaction of these conditions 

should underpin the selection of coefficient drivers. 

Linking Model Substituting the right-hand side of equation (8) for γ jit in the TVC model gives the reduced-form model: 

p−1 K −1 p−1 q−1 K −1 q−1 

* yit = π 00 +∑π 0d zdit +∑(π j0 +∑π jd zdit )xjit + μ0i +∑l

0h

ε hit +∑(μ ji +∑l

jhε hit )xjit (9) 

d =1 j=1 d =1 h=0 j =1 h=0 

This is our assumed linking model. It is a nonlinear regression model with heteroscedastic (and possibly serially correlated) 

error terms and coincides with the “true” model in (4) if the decompositions of the coefficients of the TVC model in (8) 

coincide with those in (7). Substituting the right-hand side of equation (9) for yit 

* in (1) gives an area level model, an analysis 

of which leads to a combination of design-based and model-based weighting. The virtue of this model-based weighting is that 

it has been derived from the “true” model using very weak parametric assumptions. Estimation of the area level model can be 

done using an Iteratively Re-Scaled Generalized Least Squares (IRSGLS) method of Swamy, Chang, Mehta and Tavlas 

(2003). 

Generalized Linear Mixed Model All the area level models that are covered in J.N.K. Rao’s (2003) survey are the special 

cases of the following general linear mixed (GLM) model: 
 

 

 

 

 

 

 

 

 

 

 

 

K −1 

yit

*

 = β0

 + ∑β j xjit + bi

vi + uit (10) 

j =1 

where the bi’s are known positive constants, the vi ’s are domain-specific random effects, and the uit ’s are area-by-time 

specific effects (see J.N.K. Rao (2003, p. 83)). The dependent variable of model (10) becomes observable if (10) is inserted 

into (1). Such a model involves both design-induced and model errors. Estimation of a GLM model with an observable 

dependent variable is discussed in Lehmann and Casella (1998, p. 518). 

2.3 Comparison of the “True”, TVC, SC, GLM, and Hierarchical Bayes Models 

* * *

(i) Rewriting (3) in terms of xjit , j = 1, ..., K-1, and a function of xjit , j = 1, ..., K-1, and xgit , g = K, ..., mit , leaves the 

coefficients of the TVC model invariant (see Swamy, Mehta and Singamsetti (1996)). The coefficients of (3), however, do 

not possess this invariance property. Consequently, in (3), excluded explanatory variables, x

* 

git , g = K, ..., mit , and the 

coefficients on the included explanatory variables are not unique, as shown originally by Pratt and Schlaifer (1984, p. 13). 

This non-uniqueness implies the non-uniqueness of the coefficients and the error terms of the GLM model. The non-unique 

coefficients of the GLM model do not measure the direct effects of its explanatory variables on its dependent variable (see 

Pratt and Schlaifer (1984, 1988)). They also cannot account for omitted-variable and measurement-error biases and hence the 

GLM model is misspecified. The linear functional form of the GLM model may also be incorrect. By contrast, the “true” 

model is unique if it is a real-world relation that remains invariant against changes in the language we use to describe it (see 

Basmann (1988, pp. 72-74)). For example, both adding and subtracting a term on the right-hand side of a representation of a 

real-world relation change only the representation but not the relation itself. The TVC model, but not the GLM model, shares

this invariance property with the real-world relations. 

(ii) The SC model coincides with the GLM model if for j = 1, ..., K-1, the distribution of γ jit is degenerate at β j and the 

distribution of γ 0it is the same as that of β0

 + bi vi + uit . Thus, the parametric assumptions underlying the GLM model are

much stronger than (8). 

(iii) The assumption that the x jit ’s in the GLM model are (mean) independent of vi and uit is, in the terminology of 

Pratt and Schlaifer, “meaningless” if vi and uit are used to denote ‘the’ excluded variables. Hence restricted maximum 

likelihood estimation discussed in Lehmann and Casella (1998, p. 518) may not lead to consistent estimators when it is 

applied to the GLM model. The area level model in (9) can be consistently estimated using an IRSGLS method if the 

coefficient drivers in (8) assign the correct functional form to the TVC model and if the included explanatory variables in the 

TVC model are conditionally independent of the error terms in (8) , given the coefficient drivers. 

(iv) A crucial distinction The SC model differs from hierarchical Bayes models analyzed in Lehmann and Casella 

(1998, pp. 253-262) in that the distribution of the coefficients of the former model is part of the likelihood function, whereas 

the distributions of the coefficients of the latter models are parts of the prior distributions. Two Bayesian statisticians, Pratt 

and Schlaifer (1988, p. 49), produce a very convincing argument to show that a Bayesian will do much better to search like a 

non-Bayesian for concomitants that absorb omitted-variable and measurement-error biases. Using (8), we do exactly what 

Pratt and Schlaifer suggest. The coefficient drivers in (8) are our concomitants. If some of them absorb the bias components 

of the coefficients of the TVC model, then they should appear as the explanatory variables of the coefficients, as in (8). The 

GLM model as well as hierarchical Bayes models completely ignores these biases. The prior distributions employed in their 

Bayesian analyses cannot distinguish between the bias-free and bias components of the coefficients of their TVC versions. 

Users of hierarchical Bayes models do not do what Pratt and Schlaifer suggest. Hierarchical Bayes models are not the devices 

capable of providing estimates (and, therefore, predictions) that would empirically be indistinguishably as good as those 

provided by good approximations to their (classical) TVC versions with unique coefficients. 

2.4 Effects of Omitted-Variable and Measurement-Error Biases and Incorrect Functional Forms on GLM Models 

A vector formulation of the GLM model without the uit ’s but with the sampling errors is 

yˆ = + b v + (11) it xit ′ β i i eit 

where xit = (1, x1it , ..., xK −1,it )′ , β = ( β0

, β1, ..., β K −1)′ . J.N.K. Rao (2003, p.116) assumes that (i) the vi ’s are 

2

identically and independently distributed with mean zero and constant variance, σ , (ii) the sampling errors, eit , are v 
 

 it ′ β

#

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

independently distributed with mean zero and known variance, ψ it , and (iii) vi is independent of eit for all i and t. Suppose 

that Ti = 1. Under these assumptions, the best linear unbiased predictor (BLUP) of the true value, yit 

* , from model (11) is 

y#

it 

*

 = ωit yˆ

it + (1 - ωit ) x (12) 

where 

ω = 

2 2 / (ψ + 2 2 σ b σ b ) (13) it v i it v i 

and 

−1 

β

# =

⎡

n 

22 ⎤ ⎡

n 

2 2 ⎤

∑ xit xit ′ /(ψ it +σ v 

bi ) ∑ xit yˆ

it /(ψ it +σ v 

bi ) . (14) ⎢ ⎥ ⎢ ⎥

⎣i=1 ⎦ ⎣i=1 ⎦

J.N.K. Rao’s (2003, p. 117) elegant method shows that the (average) mean square error (MSE) of (12) is 

−1

n 

2 ⎡ 2 2 ⎤

ω ψ + − (1 ω ) x′ x x′ /( ψ +σ b ) x (15) it it it it ⎢∑ it it it v i ⎥ it ⎣i=1 ⎦

where the second term arises as a direct consequence of using an estimator of β in (12). It can happen that predictor (12) is 

not more efficient than the direct estimator yˆ

it because (15) can be larger than ψ it in small samples. 

* To derive the BLUP of yit from the linking model in (9), we make the following assumptions: For i, i′ = 1, ..., n, and t, 

t′ = 1, ..., Ti , 

(A1) The K-vector μi = (μ0i , μ1i , ..., μ K −1,i )′ is distributed with Em

( μi | z1it , ..., z p−1, it ) = 0 and 

Δ if i i

Em

( μi μi ′

′

| z1it ,..., z p−1,it ) = { = ′, (16) 

0 if i i ≠ ′

where Em denotes the model expectation and Δ may not be diagonal. 

(A2) The q-vector ε it = (ε 0it , ε 1it , ..., ε q −1,it )′ follows the stochastic equation 

ε it = φii ε it −1

 + ait (17) 

where φii is a q× q diagonal matrix whose eigenvalues are less than 1 in absolute value, the q-vector ait = ( a0it , 

a1it , ..., aq−1,it )′ is distributed with Em( ait | z1it , ..., z p−1, it ) = 0 and 

σ

2Δ if i i = ′ and t=t ′

ii Em

( a ai t′ | z1it ,..., z p−1, ) = { i , (18) it ′ ′ it 0 if i ≠i′ and t ≠t′

where Δii may not be diagonal. 

(A3) Given zdit , d = 1, ..., p-1, the vectors, μi and ε it , are independent and each of them varies 

 independently across i. 

(A4) Given zdit , d = 1, ..., p-1, the vectors, μi and ε it , are independent of the xjit ’s. 

(A5 The eit ’s are independent of the μi ’s, ε it ’s, and x jit ’s. 

A vector formulation of the model obtained by substituting the linking model in (9) for yit 

* in the sampling model is 

yˆ

it = (zit ′ ⊗ xit ′ )π

Long + xit ′ (μi + Lε it ) + eit , (19) 

where xit is as defined in (11), μi is as defined in (16), ε it is as defined in (17), zit = (1, z1it ,..., z p−1, it )′ , Π = 

π

Long ⎡π ⎤ is a K × p matrix having π as its (j, d) element, is a Kp-vector denoting a column stack of Π , ⊗ ⎣ jd ⎦ jd

0≤≤ − j K 1,0 ≤ ≤ − d p 1 

denotes a Kronecker product, and L = ⎡ ⎤ l is a K × q matrix having as its (j, h) element. We call (19) “the jh h q 1 

l ⎣ ⎦ jh

0≤ ≤j K −1,0 ≤≤−

improved area level model.” When Ti = 1, the BLUP of the true value, yit 

* , from the improved area level model is 

Long * Long * * ) ˆ Long 

yˆ = (z′ ⊗ x′ πˆ + ω {yˆ − z′ ⊗ x′ )πˆ } = ω yˆ + −ω )( z′ ⊗ x′

it 

*

it it ) it it ( it it it it (1 it it it π (20) 

where 

* 2 ⎡ ′ ω

2 ⎤

 = x′{Δ + L(φ Γ φ′ +σ Δ )L′}x / x′{Δ + L(φ Γ φ +σ Δ )L′}x +ψ (21) it it ii ii ii i ii it it ii ii ii i ii it it ⎣ ⎦

−1 

πˆ Long ⎡

n 

2 ⎤

= (z ⊗ x )( z ′ ′ ′ ⊗ x ) /( x Δ + L( Γ φ′ ′ σ Δ )L }x

⎢∑ it it it it it { φii ii ii + i ii it +ψ it )

⎥

⎣i=1 ⎦
 

 

 

 

 

 

 

 

 

 

⎡

n 

2 ⎤

× ⎢∑(zit ⊗ xit ) yˆ

it /( xit ′{Δ + L(φii Γiiφii ′ +σ i Δii )L′}xit +ψ it )

⎥

(22)

⎣i=1 ⎦

with E ε ε ′ = Γ = φ Γ φ′ +σ

2 Δ .

it it ii ii ii ii i ii 

What are the properties of predictor (12) when in fact the improved area level model in (19) is appropriate? Predictor 

(12) can be still design-consistent. The direct survey estimator, yˆ

it , in (1) is design-consistent if its design bias goes to zero 

and if its design variance, ψ it , tends to zero as the sample size increases. Under these conditions, predictor (12) is designconsistent because ωit → 1 as ψ it → 0 . Of course, (12) no longer has minimum MSE and the formula in (15) for the MSE 

of (12) is no longer appropriate. Predictor (20) has minimum MSE and is also design-consistent because ωit

* →1 as ψ it → 0 . 

Of the two design-consistent predictors, (12) and (20), only (12) is affected by the misspecifications in the GLM model in 

(11). In small samples, i.e., when n is small, these effects can only be limited by replacing the GLM model by the improved 

area level model in (19), but not by using the design-consistent predictor (12). Thus, not all models that yield designconsistent predictors limit the effects of model misspecifications in small samples. 

The MSE of (20) about yit 

* is 

n ⎡ (z ⊗ x )( z′ ⊗ x′ ) ⎤

−1 

* * 2 it it it it 

it it (1 ωit ) ( zit ′ ⊗ xit ′ ) ⎢∑ 2

(zit ⊗ xit ω ψ + − ⎥) (23)

i=1 ( { x′ Δ + L(φ Γ φ′ +σ Δ )L′}x +ψ ) ⎣ it ii ii ii i ii it it ⎦

π

Long where the second term arises as a direct consequence of using an estimator of in predictor (20). 

This MSE is less than or equal to the MSE of (12) derived under the assumption that the improved area level model in 

(19) is appropriate. The inappropriate formula in (15) leads to the following conclusions: Under some regularity conditions 

stated in J.N.K. Rao (2003, p. 117), the second term in (15) goes to zero as n goes to ∞ . Therefore, this term can be ignored 

when n is large. Comparison of the first term, ωitψ it , in (15) with the design variance, ψ it , of the direct estimator, yˆ

it , shows 

that predictor (12) leads to large gains in efficiency when ωit is small, that is, when the variability of the GLM model’s error, 

bi vi , is small relative to the total variability of bi vi + eit . This result due to J.N.K. Rao (2003, p. 117)) arises as a direct

consequence of using the GLM model in (11) and ignoring the bias components contained in the coefficients of the TVC 

model. The MSE (15) of predictor (12) derived under the assumption that the GLM model in (11) holds without the u ’s it 

exaggerates the gains in efficiency resulting from the “strength” the GLM model “borrows” from the area-specific auxiliary 

variables, x , if (23) is larger than (15). jit 

We can conclude that in small samples, the linking model in (9) permits us to “borrow” strength from the area-specific 

auxiliary variables, x , if (23) is smaller than ψ . This condition is unlikely to hold in small samples. Therefore, the indirect it it 

model-dependent estimator (20) can have larger MSE than the direct estimator in (1) in small samples. Even the 

inappropriate MSE in (15) can be larger than ψ it . However, for large n, the second term in (23) can be close to zero and MSE 

(23) can be smaller than ψ , since ω

* < 1. Therefore, in large samples, the linking model in (9) permits us to “borrow” it it 

strength from the area-specific auxiliary variables. 

2.5 Empirical BLUP 

Predictor (20) depends on the distinct nonzero elements of Δ , φii , Δii , and σ i 

2 which are unknown in practical applications. 

Hence it is not operational. Let δ be a vector consisting of these elements. Swamy et al.’s (2003) IRSGLS method may be 

π

Long used to estimate simultaneously the parameter vectors and δ of the improved area level model in (19). The predictor 

of yit 

* obtained by using this estimate of δ in place of the “true” value of δ used in (20) is called “the empirical BLUP 

* ˆ * ˆ * * * ˆ

(EBLUP)” and is denoted by yˆ δ . The error in the EBLUP may be decomposed into δ - = ( - ) + ( δ - it ( ) yˆ

it ( ) yit yˆ

it 

* 

yit yˆ

it ( )

yˆ

it 

* ) where yˆ

it 

* is given in (20). Therefore, the MSE of EBLUP is 

* ˆ * ˆ* 2 *2 * ˆ* 2 MSE[ yˆ δ ] = E y[ () ˆ δ − y ] = E( - + E y[ () ˆ δ − yˆ ] (24) it ( ) it it yˆ

it 

* 

yit ) it it 

* * * ˆ * where use is made of the conditions under which Ey [ ˆ − y y ][ ˆ (δ ) − y ˆ ] = 0. These conditions are given in J.N.K. Rao it it it it 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(2003, p. 103). The first term on the right-hand side of the second equality sign in (24) is equal to (23). The last term in (24) 

accounts for the variability in the estimator of δ . Since this term is generally intractable except in special cases, J.N.K. Rao 

(2003, pp. 103-104) finds a second-order approximation to the term. The MSE of EBLUP is larger than (23). An estimator of 

* ˆ * ˆ

the MSE of yˆ δ as a measure of variability in δ is given in J.N.K. Rao (2003, pp. 104-105). it ( ) yˆ

it ( )

3. Conclusions 

In making estimates of population characteristics for small domains with adequate level of precision, it is often necessary to 

use models that relate the true values of population characteristics to auxiliary variables. The major weakness of this

approach is that if these models are seriously misspecified, they can yield inferences that are worse than design-based 

inferences. Misspecifications of models occur when relevant explanatory variables are omitted from the models, when 

included variables are measured with error, and/or when the unknown functional forms of the models are incorrectly 

specified. The biasing effects of measurement error, omitted variables, and misspecifications of functional forms are a 

pervasive problem in applied statistics. It has been shown that wrong inferences can be obtained if these biases are ignored. A 

method of accounting for these biases is proposed. The strength a statistical model “borrows” from auxiliary variables in 

making estimates for small areas may be largely offset by the effects of the model’s misspecifications. 

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