Double Autocorrelation in Two Way Error Component Models

In this paper, we extend the works by [1-5] accounting for autocorrelation both in the time specific effect as well as the remainder error term. Several transformations are proposed to circumvent the double autocorrelation problem in some specific cases. Estimation procedures are then derived.

1. Introduction

Following the works of [6], the regression model with error components or variance components has become a popular method for dealing with panel data. A summary of the main features of the model, together with a discussion of some applications, is available in [7-10] among others.

However, relatively little is known about the two way error component models in the presence of double autocorrelation, i.e, autocorrelation in the time specific effect and in the remainder error term as well.

This paper extends the works by [2-5] on the one-way random effect model in the presence of serial autocorrelation, and by [1] on the single autocorrelation two-way approach. It investigates some potential transformations to circumvent the double autocorrelation issue, along with some estimation procedures. In particular, we derive several transformations when the two disturbances follow various structures: from autoregressive and moving-average processes of order 1 to a general case of double serial correlation. We deduce several GLS estimators as well as their asymptotic properties and provide a FGLS version.

The remainder of this paper is organized as follows: Section 2 considers simple transformations on the presence of relatively manageable double autocorrelation structure. In Section 3, general transformations are considered when the double autocorrelation is more complex. GLS estimators are derived in Section 4. Asymptotic properties of the GLS estimators are considered in Section 5. Section 6 provides a FGLS counterpart approach. Finally, some concluding remarks appear in Section 7.

2. Simple Transformations

To circumvent the double autocorrelation issue, we first need to transform the model based on the variance-covariance matrix. The general regression model considered is,; where is the intercept and is a vector of slope coefficients, is a row vector of explanatory variables which are uncorrelated with the usual two-way error components disturbances (see [7]). In matrix form, we write.

2.1. When the Errors Follow AR(1) Structures

If the time specific term follows an AR(1) structure, , , with, and the remainder error term also follow an AR(1) structure, , with, we can define two transformation matrices of dimensions and respectively,

(1)

and since we have

and

(2)

the transformed errors and follow two different MA(1) processes, of parameters and respectively. Thus, by applying the appropriate transformation matrices, the autoregressive error structure can be changed into a moving-average one. The only cost is the loss of the initial and first pseudo-differences, which has no serious consequence for a long time dimension. As a result, we focus on the MA(1) error structure.

2.2. When the Errors Follow MA(1) Structures

Here, the time specific term follows an MA(1) structure, , with while the remainder error term, also follows an MA (1) structure, , with . For convergence purpose and assuming normality, the initial values are defined

and

The variance-covariance matrix of the three components error term is given by,

(3)

where and are positive definite matrices of order and where is defined by. The exact inverse of such matrices suggested by [11] and [1] does not involve the parameters and. Following [11], let be the Pesaran orthogonal matrix whose t-th row is given by,

where

, , , and with.

Pre-multiplying the model by yields the following variance-covariance matrix of,

(4)

where.

3. General Transformations

We are now in the context of a general case of double autocorrelation issue and lead to a suitable error covariance matrix similar to Equation (4) and its inverse.

3.1. First Transformation

Let denote the matrix such that. Such a matrix does exist for and is a positive-definite matrix. Transformation of the initial model by yields

(5)

and the variance-covariance of the transformed errors is

(6)

This transformation has removed the autocorrelation in the time-specific effect. Unfortunately, by doing so it has infected the and worsened the initial correlation in the remainder disturbances. An additional “treatment” is therefore needed.

3.2. Second Transformation

We now consider an orthogonal matrix and a diagonal matrix such that (diagonalization of). Thus, applying a second transformation yields,

(7)

The underlying variance-covariance matrix of the errors is,

(8a)

or,

(8b)

where, , , , , and, if.

Here, because of the choice of matrices and, we end up with since is an orthogonal matrix. Generally speaking, and just need to have zero off-diagonal elements, i.e., to be diagonal matrices. The double autocorrelation structure is thus absorbed, and one can easily accommodate with the non-spherical form of by means of an accurate inversion process.

3.3. Computing the Inverse

The inverse of is obtained using the procedure developed by [1]. After a bit of algebra, one gets

(9)

where

, , , , ,

and

with.

Proof: (see the Appendix)

4. GLS Estimation

We begin with the definition of the estimator followed by its interpretation and weighted average property.

4.1. The GLS Estimator

Proposition 1:

The GLS estimator is,

(10)

Proof: (Straightforward)

4.2. Interpretation

In classical two-way regression models, [12,13] provide an interpretation of the GLS estimator, which is appealing in view of the sources of variation in sample data. In the straight line of their work, the GLS estimator may be viewed as obtained by pooling three uncorrelated estimators: the covariance estimator (or within estimator), the between-individual estimator and the within-individual estimator. They are the same as those suggested by [1] except for the last one which was labeled between-time estimator. We have

1) The covariance estimator,

where;

2) The between-individual estimator,

where and

3) The within-individual estimator,

where.

It is important to note that these estimators are obtained from some transformations of the regression Equation (7), i.e.,

The covariance estimator, is obtained when Equation (7) is pre-multiplied by; the transformation annihilates the individualand time-effects as well as the column of ones in the matrix of explanatory variables. It is equivalent to the within estimator in the classical two-way error component model (see [1-7]).

The between-individual estimator comes from the transformation of Equation (7) by the matrix . This is equivalent to averaging individual equations for each time period.

The within-individual estimator is derived when Equation (7) is transformed by. The presence of the idempotent matrix indicates that this transformation wipes out the constant term as well as the time specific error term. However, the individual effect remains.

4.3. GLS as a Weighted Average Estimator

As in [14], the GLS estimator is a weighted average of the three estimators defined above.

Proposition 2:

(11)

with,

, and

Proof:

From Equation (10), it comes that

with

By definition, the estimators, and are respectively such that

and

Therefore,

Or,

Thus,

with, and defined according to Equation (11).

We should also note that the three estimators, and are uncorrelated. In fact,

and

because, while since . As a result,

(12)

Moreover, following [1], the fact that

(13)

gives evidence on the use of all available information from the sample. The estimators, and together use up the entire set of information to build the GLS estimator with no loss at all.

5. Asymptotic Properties

Under regular assumptions, the GLS and the three pseudo estimators of the coefficient vector, say, , and are all consistent and asymptotically equivalent. It is a result similar to the one obtained in the classical two-way error component model (see [15]).

5.1. Assumptions

We assume that the are weakly non-stochastic, i.e. do not repeat in repeated samples. We also state that the following matrices exist and are positive definite:

for the first transformation;

for the second transformation; and

for the third transformation. Furthermore, in the straight line of [1], we also assume that,

for the first transformation;

for the second transformation;

for the third transformation. In addition,

, so that the variance-components quantity denotes by remains infinite as. The limits and probabilities are taken as and. All along this section, following [1], we consider the “usual” assumptions regarding the error vector, as stated in [16] and [17], which ensures the asymptotic normality.

5.2. Asymptotic Property of the Covariance Estimator

Proposition 3:

The covariance estimator is consistent.

Proof:

Since,

Hence,

Making use of assumptions (a1) and (a2), we establish the consistency of the covariance estimator,.

Proposition 4:

The covariance estimator has an asymptotic normal distribution given by,

(14)

Proof:

Under the M₁-transformation, we have

Moreover its variance is given by and its inverse is equal to while assumption (a2) states the absence of correlation between regressors and disturbances under the M₁ transformation. We have

(15)

and,

from which we deduce that

Thus, the asymptotic normality of the covariance estimator immediately follows,

5.3. Asymptotic Property of the Between-Time Estimator

Proposition 5:

The between time estimator is consistent.

Proof:

Since,

Hence, according to assumptions (b1) and (b2),

Making use of assumptions (b1) and (b2), we establish the consistency of the between time estimator,

Proposition 6:

The between-time estimator has an asymptotic normal distribution given by,

(16)

Proof:

Under the M₂-transformation, we get

The variance of this error term is written as

Its inverse is. Again, assumption (b2) states the absence of correlation between regressors and disturbances under the M₂ transformation. We get

In addition, we have

from which we deduce that

Thus, the asymptotic normality of the between-time estimator immediately follows,

5.4. Asymptotic Property of the Within-Individual Estimator

Proposition 7:

The within individual estimator is a consistent estimator.

Proof:

Since,

Hence,

Making use of assumptions (c1) and (c2), we establish the consistency of the covariance estimator,

Proposition 8:

The within individual estimator has an asymptotic normal distribution given by,

(17)

Proof:

Under the M₃-transformation, we obtain

The variance of is obtained as

The inverse of this matrix is. Assumption (c2) states the absence of correlation between regressors and disturbances under the M₃ transformation. We have

and,

from which we deduce that

Thus, the asymptotic normality of the within individual estimator immediately follows,

5.5. Asymptotic Property of the GLS Estimator

Proposition 9:

The GLS estimator is asymptotically equivalent to the covariance estimator and therefore,

(18)

Proof:

From Equation (10), we get

On the one hand, we have

where, as. Therefore, from assumption (a1), we find that , when. Likewise, assumption (a2) leads us to, when. Hence,

On the other hand, we can write

Under the M₁ and M₂ transformations, we get

leading to

As a result,

i.e.,

Finally, has the same limiting distribution as. This shows the asymptotic equivalence of the two estimators and. We then deduce that,

Thus, the GLS estimator suggested under the double autocorrelation error structure has the desired asymptotic properties.

6. FGLS Estimation

In practice, the variance-covariance matrix is unknown, as well as all the parameters involved in its determination. Therefore, a FGLS approach is required. The method used consists in removing the time specific effect to obtain a one-way error component model where only carries the serial correlation (see [18] and [3]). This method has been directly applied to AR(1) and MA(1) processes in separate subsections.

6.1. Feasible Double AR(1) Model

We assume that, , , , ,. The within error term is,

(19)

The associated variance-covariance matrix is,

(20)

Since follows an AR(1) process of parameter, we define the matrix as the familiar [19] transformation matrix with parameter. This matrix is such that,

The resulting GLS estimator is given by

(21)

where and.

The covariance matrix of, using [20] trick, is

(22)

where

and

Following [21], another GLS estimator can be derived. We label this estimator the within-type estimator and is given by

(23)

with and. In order to get the estimates of numerous parameters involved in the model, we first need an estimate of the correlation coefficient. The autocorrelation function of the error term is given by

(24)

We deduce from it that. It then leads to a convergent estimator of (see [7]), i.e.,

where with defined as the OLS residuals of the within equation. Hence, we get

(25a)

and

(25b)

Furthermore, the BQU estimate of is also available as

(26)

being the OLS estimate of. As a consequence, we get

(27a)

and

(27b)

We now need to find and. The autocovariance function of the initial error term is given

for.

It comes that,

(28)

We immediately deduce a convergent estimator of the second correlation coefficient, i.e.,

(29)

where with denoting the OLS residuals of. The variances is estimated by,

(30)

In addition to the GLS estimators mentioned in Section 4, other GLS estimators such as the within estimator and the within-type estimator can all be performed as well. Actually, the knowledge of the AR(1) parameters and entitles us to build the matrices involved in the determination of, say matrices, , , , , , , , and.

6.2. Feasible Double MA(1) Model

We now state that, with and. Again, deviations from individual means lead to the model

with.

The variance-covariance matrix of is still given by Equation (20), with now

(31)

Here, we set, denoting the correlation correction matrix as defined by [8] in their orthogonalizing algorithm. We then transform the within model by. The new error term has the following covariance matrix,

(32)

Because of the moving average nature of the process, linear estimation of the correlation parameter is not easily obtainable. Instead, proves useful. The autocorrelation function of the within error term is given by,

(33)

with denoting the autocovariance function of. As a consequence, for some

and

(34)

where is the empirical autocovariance function and are the OLS residuals of the within equation. We also get, for some,

(35)

We then apply the [8] matrix to the data (for instance to the within transformed dependent vector). Moreover, will be applied to the vector of constants to get estimates of the. We have, in the straight line of [8], the following steps:

Step 1: Compute and

for

where for.

Step 2: Compute knowing that

. The estimates of the are obtained as and

for.

We then obtain the estimate of as. The autocovariance function of the initial composite error term and its empirical counterpart, (being the OLS residuals of the initial two-way model) permit the estimation of and,

(36a)

and

(36b)

The within estimator and the within-type one are now obtainable. However, the GLS estimator can be estimated, provided the MA(1) parameters and are known, especially under the conditions

and. In other words, the estimates and should both lie inside the open interval as a pre-requisite to a direct estimation of, , and.

7. Final Remarks

This paper has considered a complex but realistic correlation structure in the two-way error component model: the double autocorrelation case. It dealt with some parsimonious models, especially the AR(1) and MA(1) ones, as well as the general framework. Through a precise formula of the variance-covariance matrix of the errors, we derived the GLS estimator and related asymptotic properties. An investigation of the FGLS is also considered in the paper.

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Appendix: Computing the Inverse of

We established that

with and.

Setting, we can rewrite the variance covariance matrix as

where. By the means of an update formula, we deduce an expression of the inverse of,

We need to obtain and the inverse of the bracketed expression. On the one hand,

Let denote the matrix. At this step, the inverse of is required. Let be a orthogonal matrix. Then,

Therefore,

with

It is worth mentioning that for and are different columns of the same diagonal matrix. It is therefore obvious that has already been diagonalized. As a consequence, the inverse of is given by,