A Comparison of Alternative Estimators of Linearly Aggregated Macro Models

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 This paper deals with the linear aggregation problem. For the true underlying micro relations, which explain the micro behavior of the individuals, no restrictive rank conditions are assumed. Thus the analysis is presented in a framework utilizing generalized inverses of singular matrices. We investigate several estimators for certain linear transformations of the systematic part of the corresponding macro relations. Homogeneity of micro parameters is discussed. Best linear unbiased estimation for micro parameters is described.


Introduction
The linear aggregation problem considered consists of obtaining consistent and economically meaningful micro economic (from individuals) and macro economic (from groups of individuals) relationships.
The problem of aggregation over micro units has had a long tradition in the econometric literature, streching back to the pioneering work of (Theil 1954).(Kloek 1961) used matrix notation to explain aggregation over one set of individuals.(Misra 1969, Gupta 1970, Moriguchi 1970, Lutjohann 1972, Wu 1973, Akdeniz and Milliken 1975), Akdeniz and Werner (1999)also utilized matrix notation to study micro and macro economic relations.Sasaki (1978) has analyzed empirically the aggregation problems.Leeet al. (1990) considered the problem of 'aggregation bias' defined by the deviation of the macro parameters from the average of the corresponding micro parameters.In addition to an aggregation of a real units, one may also distinguish an aggregation of models or equations pertaining to a real units.In this regard, one may analyze the impact of a particular aggregation level of individual units upon the explanatory power of a model or relationship (see Blommestein and Nijkamp,1986).
In this paper, we considered a more general linear aggregation model by allowing for collinearity between the explanatory variables.Thus, the analysis is presented in a framework using generalized inverses of singular matrices.In this case, we obtained a consistent relationship between the micro and macro parameters.
We are concerned with a comparison of two estimators of the macro parameters (actually, due to the collinearity, of the structural portion Xγ).One obvious estimate is obtained directly from the application of ordinary least squares to the macro equation.It is well known (indeed, because of the specification error above) that the expectation of this estimate generally differs from the aggregate of the expectation of ordinary least squares estimates obtained from the micro equations.Therefore, comparing these two estimators will necessarily involve considerations of bias as well as efficiency.Nevertheless, this is the comparison which has traditionally been made.In the present study the direct ordinary least squares estimate (OLSE) of the macro equation is retained, but a rather different aggregate of the micro estimates is employed.Instead of simply aggregating the micro estimates, the projection into the space spanned by the macro explanatory variables is taken.This paper is organized as follows.The next section describes the general linear aggregation problem.A comparison of two estimators of the macro parameters is given in Section 3. The OLSE and weighted least squares estimator (WLSE) of the macro parameters is compared in Section 4. Homogeneous micro parameters and aggregation bias are discussed in Section 5. Section 6 describes the best linear unbiased estimation of micro parameters.

The general linear aggregation problem
Let the economic relationship for the i-th economic unit (general disaggregated model) be given by 2) where i y is a vector of observations on the micro dependent variable; i X is a T×K matrix of explanatory micro variables; i  is a vector of K micro parameters and i u is a disturbance vector, N is the number of economic units and 0 ) , cov(  j i y y for i≠j; i X and i V ≥ 0 are fixed and known matrices.
we can obviously combine our N individual micro relations in the single linear micro model The following assumptions are used throughout this paper.Assumption 1.The matrix i X has rank K r i  and consists of nonstochastic elements.Assumption 2. ' i u s have zero means, are independent of ) ,..., , ( where * V is block-diagonal matrix, consisting of matrices i V .Define the macro variables and (2.5) as unweighted aggregates of the corresponding micro variables over the N individuals.
For convenience, put , where T I is a T×T identity matrix N j is a N×1 column vector of unit elements and

Using
the aggregate observations the artificial macro model (aggregate model) can be written as The macro relations in (2.9) generally differ from the true macro relations, implying there is possible specification error in (2.9). can occur.We are trying essentially to describe the vector * *  X J T in the best possible way by  X .The best description is the projection of * *  X J T onto the vector space spanned by the columns of X.The projection of onto the vector space is where  X denotes the Moore-Penrose inverse of X (Styan, 1983)).Akdeniz and Milliken   (1975) has shown that the system of equations relating to  and *  , * *   X J XX X T   is consistent.We don't need the Moore-Penrose inverse of X to establish the relationship between the quantities in the micro and macro equations.We can use any generalized inverse, but we need the Moore-Penrose inverse in Section 3.
Thus the specification error u ~is in the vector space spanned by the columns of X, i.e., , ) ( The specification error is the vector between * *  X J T and its projection onto C(X).It is also called the "Aggregation bias" of u (Green, 1964).Where C(X) denotes the column space of X. Direct estimation of aggregate model parameters generates in general aggregation bias.
The macro disturbance vector will be equal to only if the "perfect aggregation" condition 0 : (Pesaranet al. 1989).Hence the macro disturbance vector u is expressed as follows: Where , : (Wu,1973;Terzi, 2009).S is a T×KN matrix of least squares residuals and P is a K×KN matrix of least squares "auxiliary regression" coefficients.The estimated auxiliary equation for the i-th micro unit is written in the form (Kloek,1961):


(2.17) The aggregation procedure complicates the structure of macro errors.In equation (2.15) there are two terms; the first term is the linear combination of all micro coefficients, weighted by components of the S matrix, and the second term is the sum of corresponding micro errors (Wu,1973;Maddala, 1977).When the macro relation is specified as the regression model (2.9), the macro parameter  is implicitly defined as a vector of constants independent of X (Lutjohann (1972)).For full rank model  is defined as a mathematical expectation of its least squares estimator (OLSE) (see Kloek, 1961).For less than full rank model X is defined as the expectation of the least squares estimates when y is regressed on where y X   ˆis the OLSE of  from (2.9).It can be shown that (2.11) and (2.18) are equivalent.

Comparison of two estimators of the macro parameters
By using the relationship between  and *  one can obtain two estimators of .
 X Therefore, we consider  X here rather than  , since we don't assume that X necessarily has full column rank.Many widespread uses of the linear model u X y    are such that  can not be estimated uniquely.But linear combinations of elements of  X can be estimated uniquely.Confining attention to estimating  X therefore avoid that lack of uniqueness.
The least squares estimator, to be denoted But another estimator exists by using the relationship .
be the vector of least squares estimate of the micro parameters obtained from (2.1).Then the other unbiased estimate of  Thus we have two unbiased estimators of  X , 1  and 2  .These two estimators will, in general, not identical.The problem of equality 1  = 2  is discussed in the following theorem.
, where denotes the column space of i X .

Proof of necessity
 for each i completing the proof on necessity.

Proof of sufficiency:
To that end, let (3.5) hold true for each The experimental situations where these two estimators are identical will be the exception rather than the rule, thus we wish to determine which of the two estimators is better when they are not equal.To do that we examine the covariance structure of each estimator to determine which estimator has the smaller variance.The following Theorem describes this relationship.
THEOREM 3.2.The variances of the components of 2  are smaller than or equal to the corresponding variances of the components of 1  , thus 2  is the better of the two estimators.
Proof.The proof consists of deriving the dispersion matrices of 1  and 2  and then examining the diagonal elements of the difference of the two dispersion matrices.The dispersion matrix of 1  is denoted by Σ 1 and is defined as

( ) HE uu H   
We use the expression for u given in (2.15) to evaluate where To compare the two estimators, we examine 1 2 where We assumed that each i V is positive definite, thus * V is also positive definite.To substantiate the theorem, it is sufficient to show that the matrix Q is nonnegative.As * V is positive definite there exists a non singular matrix  such that * Thus it is sufficient to show I N N   is nonnegative, where 1 .
has characteristic roots of +1 and 0, which implies I N N   is idempotent and hence a nonnegative matrix.This means that Q is a nonnegative matrix and thus .That completes the proof.

Comparison of OLSE and WLSE of the macro parameters
In model (2.9), X  is defined as a mathematical expectation of its weighted least squares estimator (WLSE) . The aggregation problems which arise from the use of macro equation (2.9) instead of the micro equations (2.1).The macro relations in (2.9) are generally different from the true macro relations in (2.7).The ''aggregate'' method presumably involves a specification error in (2.9).Essentially we are trying to describe the vector * *  X J T in the best possible way by  X weighted by the respective covariance matrices.This best description is the projection of * *  X J T onto the vector space spanned by the columns of X.The projection of * *  X J T onto the vector space weighted by V is , where is the orthogonal projector onto C(X), which is unique for any choice of the generalized inverse involved in (4.1) (Rao,1973)).It is shown that equations (4.1) and (4.2) are equivalent.
Thus the specification error 0 u is in the vector space orthogonal to the vector space spanned by the columns of a matrix X, i.e., Considering (2.7), (2.9) and (4.3) then the macro disturbance vector u can be written as By utilizing the relationship between  and *  two estimators of  X can be obtained.Proof.The proof consists of deriving the dispersion matrices of  X and * Xb and then examining the diagonal elements of the difference of the two dispersion matrices.
The dispersion matrix of  X is denoted by 1 W and is to be The equation (4.4) is then used to evaluate (4.9).The expectation of  X is where To compare the two estimators, we examine 1 2 Where M:=I-B(B′B) + B′ and .: It is well known that M is n.n.d., then M can be written as M :=CC′.Thus * * 1 2 ( )( ) is n.n.d.matrix.This completes the proof.Thus theorem implies that the better estimator of the macro relations is via the consistent equation expressing the relationship between  and *  .

Homogeneous micro parameters and aggregation bias
The problem of 'aggregation bias', as originally discussed by Theil (I954) is defined in terms of the deviations of macro parameters from the averages of the corresponding micro parameters.Theil (1971) discusses the nature of aggregation bias in estimating a linear model with aggregate data to make macro inferences when the micro models are linear.Lee, Pesaran and Pierse (1990) develop direct tests of aggregation bias in contrast to the indirect test proposed by Zellner (1962) which tests the hypothesis that all the disaggregated coefficients are equal.Offutt (1988) performed two tests to investigate the hypothesis of aggregation bias for the random coefficient model.Pieraccini (2005) shows that direct estimation of the aggregate model leads in general to aggregation bias and that even in presence of perfect aggregation the disaggregate model is to be preferred.Monteforte (2007) assess the existence and characteristics of the aggregation error in estimated macroeconometric models.It should be recalled that for any macro parameter the aggregation bias due to any micro parameter is absent when all micro parameters are equal among micro units.Suppose the null hypothesis that all micro parameters are equal among micro units is tested.
It should be noted that the homogeneity of micro regression coefficients is not a necessary condition but a sufficient condition for the absence of aggregation bias.Hence, the rejection the hypothesis of homogeneity does not necessarily mean the presence of aggregation bias (Sasaki,1978).Zellner (1962) developed the two alternative methods for test of homogeneity of micro regression coefficients.

Homogeneity of Micro Parameters
It is possible to establish test of a variety of hypothesis about coefficient vector in (2.2).Now, if we impose the restriction . .then the system in (2) becomes where In this case, the dispersion matrix of Finally, using the Theorm 3.2 one may shows that (5.7) is nonnegative definite.

Best linear unbiased estimation for micro parameters
Fundamental problem is to find the "best" estimate of the parameter vector *  in Eq.   .On the other hand, using the Eq.(2.6) we have the following  .Thus we have the following dispersion matrix of Since T′(V-AVA′)T = 0, T′(V-AVA′)U = T′VU, U′(V-AVA′)T = U′VT, and U′(V-AVA′)U=U′VU, (6.6) reduces to 0 0.
Evidently, the matrix (6.7) and so V-AVA′ is nonnegative definite if and only if 0 T VU   .The latter is equivalent to (6.5), and so the proof is complete.
for some K hold along with any one of the following two conditions: c 1 )

Conclusions
In this paper our primary concern has been with the problem of choice between macro and micro regression equations for the purpose of predicting macro variables.We are concerned with a comparison of two estimators of the macro parameters (actually, due to the collinearity, of the structural portion )  X .We consider  X rather than  , since we do not assume that X necessarily has full column rank.The analysis is presented in a framework utilizing generalized inverses.The OLSE and WLSE estimates of the macro parameters is compared.Homogeneity of micro parameters is discussed.Best linear unbiased estimation for micro parameters is described.
roots of N are either +1 or 0 as are the characteristic roots of .N N  The matrix N N  is nonnegative, thus there exists an orthogonal matrix  such that N N D       where D  has the characteristic roots of N N  on the diagonal, i being nonnegative implies that its diagonal elements are greater than or equal to zero, thus the variance of each component of1 ˆX  is greater than or equal to the variance of the corresponding component of 2 ˆXb 

. 8 ) 4 . 1 .
THEOREM The variances of the components of  X are greater than or equal to the corresponding variances of the components of * Xb , thus * Xb is the better of the two estimators.
., there is no aggregation bias.It is shown that

( 2 . 3 ) 1 .
based on the N y y y ,..., , 2 There are many equivalent forms of the best linear

THEOREM 6 . 1
4) is nonnegative definite matrix by Theorem 4.1.FollowingPuntanen et al. (2000) we will state the following Theorem 6.1.Let V be p×p a nonnegative definite and symmetric matrix.Moreover, let A be a p×p symmetric idempotent matrix of rank r ≤ p. Then Σ=V-AVA′ =V-AVA is nonnegative definite if and only if AV(I-A′)=0.(6.5)Proof.Let [T:U] be a block-partitioned orthogonal matrix such that the columns of T and U constitute a basis for the range of A and I-A, respectively.Then we have V-AVA′ ≥ 0 if and only if uniquely determined.But another estimator exists by utilizing the relationship  Using the dispersion matrices we have the following Lowner partial ordering: Using the Theorem 6.1 we get the following results for model Proof: .n.d. by Theorem 4.1, where is r is n