Pooled Estimation

846—Chapter 27. Pooled Time Series, Cross-Section Data

of the supported techniques is provided in “Estimation Background” beginning on page 859.

Estimating a Pool Equation

To estimate a pool equation specification, simply press the Estimate button on your pool object toolbar or select Proc/Estimate... from the pool menu, and the basic pool estimation dialog will open:

First, you should specify the estimation settings in the lower portion of the dialog. Using the Method combo box, you may choose between LS - Least Squares (and AR), ordinary least squares regression, TSLS - Two-Stage Least Squares (and AR), two-stage least squares (instrumental variable) regression. If you select the latter, the dialog will differ slightly from this example, with the provision of an additional tab (page) for you to specify your instruments (see “Instruments” on page 851).

You should also provide an estimation sample in the Sample edit box. By default, EViews will use the specified sample string to form use the largest sample possible in each crosssection. An observation will be excluded if any of the explanatory or dependent variables for that cross-section are unavailable in that period.

The checkbox for Balanced Sample instructs EViews to perform listwise exclusion over all cross-sections. EViews will eliminate an observation if data are unavailable for any crosssection in that period. This exclusion ensures that estimates for each cross-section will be based on a common set of dates.

848—Chapter 27. Pooled Time Series, Cross-Section Data

Note that EViews only allows specification by list for pool equations. If you wish to estimate a nonlinear specification, you must first create a system object, and then edit the system specification (see “Making a System” on page 843).

Fixed and Random Effects

You should account for individual and period effects using the Fixed and Random Effects combo boxes. By default, EViews assumes that there are no effects so that the combo boxes are both set to None. You may change

the default settings to allow for either Fixed or Random effects in either the cross-section or period dimension, or both.

There are some specifications that are not currently supported. You may not, for example, estimate random effects models with cross-section specific coefficients, AR terms, or weighting. Furthermore, while two-way random effects specifications are supported for balanced data, they may not be estimated in unbalanced designs.

Note that when you select a fixed or random effects specification, EViews will automatically add a constant to the common coefficients portion of the specification if necessary, to ensure that the effects sum to zero.

Weights

By default, all observations are given equal weight in estimation. You may instruct EViews to estimate your specification with estimated GLS weights using the combo box labeled

Weights.

If you select Cross section weights, EViews will estimate a feasible GLS specification assuming the presence of cross-section heteroskedasticity. If you select Cross-section SUR, EViews estimates a feasible GLS specification correcting for both cross-section heteroskedasticity and

contemporaneous correlation. Similarly, Period weights allows for period heteroskedasticity, while Period SUR corrects for both period heteroskedasticity and general correlation of observations within a given cross-section. Note that the SUR specifications are each examples of what is sometimes referred to as the Parks estimator.

850—Chapter 27. Pooled Time Series, Cross-Section Data

Coefficient Name

By default, EViews uses the default coefficient vector C to hold the estimates of the coefficients and effects. If you wish to change the default, simply enter a name in the edit field. If the specified coefficient object exists, it will be used, after resizing if necessary. If the object does not exist, it will be created with the appropriate size. If the object exists but is an incompatible type, EViews will generate an error.

Iteration Control

The familiar Max Iterations and Convergence criterion edit boxes that allow you to set the convergence test for the coefficients and GLS weights.

If your specification contains AR terms, the AR starting coefficient values combo box allows you to specify starting values as a fraction of the OLS (with no AR) coefficients, zero, or user-specified values.

If Display Settings is checked, EViews will display additional information about convergence settings and initial coefficient values (where relevant) at the top of the regression output.

The last set of radio buttons is used to determine the iteration settings for coefficients and GLS weighting matrices.

The first two settings, Simultaneous updating and Sequential updating should be employed when you want to ensure that both coefficients and weighting matrices are iterated to convergence. If you select the first option, EViews will, at every iteration, update both the coefficient vector and the GLS weights; with the second option, the coefficient vector will be iterated to convergence, then the weights will be updated, then the coefficient vector will be iterated, and so forth. Note that the two settings are identical for GLS models without AR terms.

If you select one of the remaining two cases, Update coefs to convergence and Update coefs once, the GLS weights will only be updated once. In both settings, the coefficients are first iterated to convergence, if necessary, in a model with no weights, and then the weights are computed using these first-stage coefficient estimates. If the first option is selected, EViews will then iterate the coefficients to convergence in a model that uses the first-stage weight estimates. If the second option is selected, the first-stage coefficients will only be iterated once. Note again that the two settings are identical for GLS models without AR terms.

By default, EViews will update GLS weights once, and then will update the coefficients to convergence.

852—Chapter 27. Pooled Time Series, Cross-Section Data

The pool identifiers for our data are “AR”, “CH”, “DM”, “GE”, “GM”, “GY”, “IB”, “UO”, “US”, “WH”.

We cannot possibly demonstrate all of the specifications that may be estimated using these data, but we provide a few illustrative examples.

First, we estimate a model regressing I? on the common regressors F? and K?, with a crosssection fixed effect. All regression coefficients are restricted to be the same across all crosssections, so this is equivalent to estimating a model on the stacked data, using the crosssectional identifiers only for the fixed effect.

The top portion of the output from this regression, which shows the dependent variable, method, estimation and sample information is given by:

854—Chapter 27. Pooled Time Series, Cross-Section Data

Effects Specification

Cross-section fixed (dummy variables)

A few of these summary statistics require discussion. First, the reported R-squared and F- statistics are based on the difference between the residuals sums of squares from the estimated model, and the sums of squares from a single constant-only specification, not from a fixed-effect-only specification. As a result, the interpretation of these statistics is that they describe the explanatory power of the entire specification, including the estimated fixed effects. Second, the reported information criteria use, as the number of parameters, the number of estimated coefficients, including fixed effects. Lastly, the reported Durbin-Wat- son stat is formed simply by computing the first-order residual correlation on the stacked set of residuals.

We may reestimate this specification using White cross-section standard errors to allow for general contemporaneous correlation between the firm residuals. The “cross-section” designation is used to indicate that covariances are allowed across cross-sections (contemporaneously). Simply click on the Options tab and select White cross-section as the coefficient covariance matrix, then reestimate the model. The relevant portion of the output is given by:

White cross-section standard errors & covariance (d.f. corrected)

The new output shows the method used for computing the standard errors, and the new standard error estimates, t-statistic values, and probabilities reflecting the robust calculation of the coefficient covariances.

Alternatively, we may adopt the Arellano (1987) approach of computing White coefficient covariance estimates that are robust to arbitrary within cross-section residual correlation. Select the Options page and choose White period as the coefficient covariance method.

856—Chapter 27. Pooled Time Series, Cross-Section Data

Alternatively, we may produce estimates for the two way random effects specification. First, in the Specification page, we set both the cross-section and period effects combo boxes to Random. Note that the dialog changes to show that weighted estimation is not available with random effects (nor is AR estimation).

Next, in the Options page we change the Random effects method to use the

Wansbeek-Kapteyn method of computing the estimates of the random component variances.

Lastly, we click on OK to estimate the model.

The top portion of the dialog displays basic information about the specification, including the method used to compute the component variances, as well as the coefficient estimates and associated statistics:

Dependent Variable: I?

Method: Pooled EGLS (Two-way random effects)

Date: 12/03/03 Time: 14:28

Sample: 1935 1954

Included observations: 20

Number of cross-sections used: 10

Total pool (balanced) observations: 200

Wansbeek and Kapteyn estimator of component variances

The middle portion of the output (not depicted) displays the best-linear unbiased predictor estimates of the random effects themselves.

The next portion of the output describes the estimates of the component variances:

858—Chapter 27. Pooled Time Series, Cross-Section Data

Note that EViews displays results for each of the cross-section specific K? series, labeled using the equation identifier followed by the series name. For example, the coefficient labeled “AR--KAR” is the coefficient of KAR in the cross-section equation for firm AR.

In our last example, we consider the use of the @INGRP pool function to estimate an specification containing group dummy variables (see “Pool Series” on page 830). Suppose we modify our pool definition so that we have define a group named “MYGROUP” containing the identifiers “GE”, “GM”, and “GY”. We may then estimate a pool specification using the common regressor list:

c f? k? @ingrp(mygrp)

where the latter pool series expression refers to a set of 10 implicit series containing dummy variables for group membership. The implicit series associated with the identifiers “GE”, “GM”, and “GY” will contain the value 1, and the remaining seven series will contain the value 0.

The results from this estimation are given by:

Dependent Variable: I?

Method: Pooled Least Squares

Date: 12/18/03 Time: 13:29

Sample: 1935 1954

Included observations: 20

Number of cross-sections used: 10

Total pool (balanced) observations: 200

We see that the mean value of I? for the three groups is substantially lower than for the remaining groups, and that the difference is statistically significant at conventional levels.

Pool Equation Views and Procedures

Once you have estimated your pool equation, you may examine your output in the usual ways:

Pooled Estimation—859

Representation

Select View/Representations to examine your specification. EViews estimates your pool as a system of equations, one for each cross-section unit.

Estimation Output

View/Estimation Output will change the display to show the results from the pooled estimation.

As with other estimation objects, you can examine the estimates of the coefficient covariance matrix by selecting View/Coef Covariance Matrix.

Testing

EViews allows you to perform coefficient tests on the estimated parameters of your pool equation. Select View/Wald Coefficient Tests… and enter the restriction to be tested. Additional tests are described in the panel discussion “Panel Equation Testing” on

page 922

Residuals

You can view your residuals in spreadsheet or graphical format by selecting View/Residuals/Table or View/Residuals/Graph. EViews will display the residuals for each cross-sec- tional equation. Each residual will be named using the base name RES, followed by the cross-section identifier.

If you wish to save the residuals in series for later use, select Proc/Make Resids. This procedure is particularly useful if you wish to form specification or hypothesis tests using the residuals.

Residual Covariance/Correlation

You can examine the estimated residual contemporaneous covariance and correlation matrices. Select View/Residual and then either Correlation Matrix or Covariance Matrix to examine the appropriate matrix.

Forecasting

To perform forecasts using a pool equation you will first make a model. Select Proc/Make Model to create an untitled model object that incorporates all of the estimated coefficients. If desired, this model can be edited. Solving the model will generate forecasts for the dependent variable for each of the cross-section units. For further details, see Chapter 26, “Models”, on page 777.

Estimation Background

The basic class of models that can be estimated using a pool object may be written as:

860—Chapter 27. Pooled Time Series, Cross-Section Data

t = 1, 2, …, T . The α parameter represents the overall constant in the model, while the δi and γt represent cross-section or period specific effects (random or fixed). Identification obviously requires that the β coefficients have restrictions placed upon them. They may be divided into sets of common (across cross-section and periods), cross-section specific, and period specific regressor parameters.

While most of our discussion will be in terms of a balanced sample, EViews does not require that your data be balanced; missing values may be used to represent observations that are not available for analysis in a given period. We will detail the unbalanced case only where deemed necessary.

We may view these data as a set of cross-section specific regressions so that we have M cross-sectional equations each with T observations stacked on top of one another:

for i = 1, …, M , where lT is a T -element unit vector, IT is the T -element identity matrix, and γ is a vector containing all of the period effects, γ′ = ( γ1, γ2, …, γT) .

Analogously, we may write the specification as a set of T period specific equations, each with M observations stacked on top of one another.

for t = 1, …, T , where lM is a M -element unit vector, IM is the M -element identity matrix, and δ is a vector containing all of the cross-section effects, δ′ = ( δ1, δ2, …, δM) .

For purposes of discussion we will employ the stacked representation of these equations. First, for the specification organized as a set of cross-section equations, we have:

where the matrices β and X are set up to impose any restrictions on the data and parameters between cross-sectional units and periods, and where the general form of the unconditional error covariance matrix is given by:

862—Chapter 27. Pooled Time Series, Cross-Section Data

EViews estimates these models by internally creating interaction variables, M for each regressor in the cross-section regressor list and T for each regressor in the period-specific list, and using them in the regression. Note that estimating models with cross-section or period specific coefficients may lead to the generation of a large number of implicit interaction variables, and may be computationally intensive, or lead to singularities in estimation.

AR Specifications

EViews provides convenient tools for estimating pool specifications that include AR terms. Consider a restricted version of Equation (27.2) on page 860 that does not admit period specific regressors or effects,

where the cross-section effect δi is either not present, or is specified as a fixed effect. We then allow the residuals to follow a general AR process:

further that there is no unit root. Note that we allow the autocorrelation coefficients ρ to be cross-section, but not period specific.

If, for example, we assume that it follows an AR(1) process with cross-section specific AR coefficients, EViews will estimate the transformed equation:

Yit = ρ1iYit − 1 + α( 1 − ρ1i) + ( Xit − ρ1iXit − 1) ′βi + δi( 1 − ρ1i) + ηit (27.15)

using iterative techniques to estimate ( α, βi, ρi ) for all i . See “Estimating AR Models” on page 497 for additional discussion.

We emphasize that EViews does place are restrictions on the specifications that admit AR errors. AR terms may not be estimated in specifications with period specific regressors or effects. Lastly, AR terms are not allowed in selected GLS specifications (random effects, period specific heteroskedasticity and period SUR). In those GLS specifications where AR terms are allowed, the error covariance assumption is for the innovations not the autoregressive error.

Fixed and Random Effects

The presence of cross-section and period specific effects terms δ and γ may be handled using fixed or random effects methods.

You may, with some restrictions, specify models containing effects in one or both dimension, for example, a fixed effect in the cross-section dimension, a random effect in the period dimension, or a fixed effect in the cross-section and a random effect in the period

864—Chapter 27. Pooled Time Series, Cross-Section Data

differ slightly from those obtained from other sources since EViews always uses the more complicated unbiased estimators involving traces of matrices that depend on the data (see Baltagi (2001) for discussion, especially “Note 2” on p. 27).

Once the component variances have been estimated, we form an estimator of the composite residual covariance, and then GLS transform the dependent and regressor data.

If instrumental variables estimation is specified with random effects, EViews will GLS transform both the data and the instruments prior to estimation. This approach to random effects estimation has been termed generalized two-stage least squares (G2SLS). See Baltagi (2001, pp. 111-115) and “Random Effects and GLS” on page 868 for additional discussion.

Generalized Least Squares

You may estimate GLS specifications that account for various patterns of correlation between the residuals. There are four basic variance structures that you may specify: crosssection specific heteroskedasticity, period specific heteroskedasticity, contemporaneous covariances, and between period covariances.

Note that all of the GLS specifications described below may be estimated in one-step form, where we estimate coefficients, compute a GLS weighting transformation, and then reestimate on the weighted data, or in iterative form, where to repeat this process until the coefficients and weights converge.

Cross-section Heteroskedasticity

Cross-section heteroskedasticity allows for a different residual variance for each cross section. Residuals between different cross-sections and different periods are assumed to be 0. Thus, we assume that:

E( it it Xi ) = σ2i

(27.16)

E( is jt Xi ) = 0

for all i , j , s and t with i ≠ j and s ≠ t , where Xi contains Xi and, if estimated by fixed effects, the relevant cross-section or period effects (δi, γ ).

Using the cross-section specific residual vectors, we may rewrite the main assumption as:

GLS for this specification is straightforward. First, we perform preliminary estimation to obtain cross-section specific residual vectors, then we use these residuals to form estimates of the cross-specific variances. The estimates of the variances are then used in a weighted least squares procedure to form the feasible GLS estimates.

Pooled Estimation—865

Period Heteroskedasticity

Exactly analogous to the cross-section case, period specific heteroskedasticity allows for a different residual variance for each period. Residuals between different cross-sections and different periods are still assumed to be 0 so that:

E( it jt Xt ) = σ2t

(27.18)

E( is jt Xt ) = 0

for all i , j , s and t with s ≠ t , where Xt contains Xt and, if estimated by fixed effects, the relevant cross-section or period effects (δ, γt ).

Using the period specific residual vectors, we may rewrite the first assumption as:

We perform preliminary estimation to obtain period specific residual vectors, then we use these residuals to form estimates of the period variances, reweight the data, and then form the feasible GLS estimates.

Contemporaneous Covariances (Cross-section SUR)

This class of covariance structures allows for conditional correlation between the contemporaneous residuals for cross-section i and j , but restricts residuals in different periods to be uncorrelated. More specifically, we assume that:

for all i , j , s and t with s ≠ t . Note that the contemporaneous covariances do not vary over t .

There does not appear to be a commonly accepted name for this variance structure, so we term it a Cross-section SUR specification since it involves covariances across cross-sections

866—Chapter 27. Pooled Time Series, Cross-Section Data

as in a seemingly unrelated regressions type framework (where each equation corresponds to a cross-section).

Cross-section SUR weighted least squares on this specification (sometimes referred to as the Parks estimator) is simply the feasible GLS estimator for systems where the residuals are both cross-sectionally heteroskedastic and contemporaneously correlated. We employ residuals from first stage estimates to form an estimate of ΩM . In the second stage, we perform feasible GLS.

Bear in mind that there are potential pitfalls associated with the SUR/Parks estimation (see Beck and Katz (1995)). For one, EViews may be unable to compute estimates for this model when you the dimension of the relevant covariance matrix is large and there are a small number of observations available from which to obtain covariance estimates. For example, if we have a cross-section SUR specification with large numbers of cross-sections and a small number of time periods, it is quite likely that the estimated residual correlation matrix will be nonsingular so that feasible GLS is not possible.

It is worth noting that an attractive alternative to the SUR methodology estimates the model without a GLS correction, then corrects the coefficient estimate covariances to account for the contemporaneous correlation. See “Robust Coefficient Covariances” on page 869.

Note also that if cross-section SUR is combined with instrumental variables estimation, EViews will employ a Generalized Instrumental Variables estimator in which both the data and the instruments are transformed using the estimated covariances. See Wooldridge (2002) for discussion and comparison with the three-stage least squares approach.

(Period Heteroskedasticity and Serial Correlation) Period SUR

This class of covariance structures allows for arbitrary period serial correlation and period heteroskedasticity between the residuals for a given cross-section, but restricts residuals in different cross-sections to be uncorrelated. Accordingly, we assume that:

E( is it Xi ) = σst

(27.23)

E( is jt Xi ) = 0

for all i , j , s and t with i ≠ j . Note that the heteroskedasticity and serial correlation does not vary across cross-sections i .

Using the cross-section specific residual vectors, we may rewrite this assumption as,

for all i , where,

We term this a Period SUR specification since it involves covariances across periods within a given cross-section, as in a seemingly unrelated regressions framework with period specific equations. In estimating a specification with Period SUR, we employ residuals obtained from first stage estimates to form an estimate of ΩT . In the second stage, we perform feasible GLS.

See “Contemporaneous Covariances (Cross-section SUR)” on page 865 for related discussion.

Instrumental Variables

All of the pool specifications may be estimated using instrumental variables techniques. In general, the computation of the instrumental variables estimator is a straightforward extension of the standard OLS estimator. For example, in the simplest model, the OLS estimator may be written as:

ii

while the corresponding IV estimator is given by:

There are, however, additional complexities introduced by instruments that require some discussion.

Cross-section and Period Specific Instruments

As with the regressors, we may divide the instruments into three groups (common instruments Z0it , cross-section specific instruments Z1it , and period specific instruments

Z2it ).

You should make certain that any exogenous variables in the regressor groups are included in the corresponding instrument groups, and be aware that each entry in the latter two groups generates multiple instruments.

868—Chapter 27. Pooled Time Series, Cross-Section Data

Fixed Effects

If instrumental variables estimation is specified with fixed effects, EViews will automatically add to the instrument list any constants implied by the fixed effects so that the orthogonal projection is also applied to the instrument list. Thus, if Q is the fixed effects transformation operator, we have:

Random Effects and GLS

Similarly, for random effects and other GLS estimators, EViews applies the weighting to the instruments as well as the dependent variable and regressors in the model. For example, with data estimated using cross-sectional GLS, we have:

In the context of random effects specifications, this approach to IV estimation is termed generalized two-stage least squares (G2SLS) method (see Baltagi (2001, pp. 111-115) for references and discussion). Note that in implementing the various random effects methods (Swamy-Arora, Wallace-Hussain, Wansbeek-Kapteyn), we have extended the existing results to derive the unbiased variance components estimators in the case of instrumental variables estimation.

More generally, the approach may simply be viewed as a special case of the Generalized Instrumental Variables (GIV) approach in which data and the instruments are both transformed using the estimated covariances. You should be aware that this has approach has the effect of altering the implied orthogonality conditions. See Wooldridge (2002) for discussion and comparison with a three-stage least squares approach in which the instruments are not transformed. See “GMM Details” on page 931 for an alternative approach.

AR Specifications

EViews estimates AR specifications by transforming the data to a nonlinear least squares specification, and jointly estimating the original and the AR coefficients.

870—Chapter 27. Pooled Time Series, Cross-Section Data

E( t t′ Xt ) can depend on Xt in arbitrary, unknown fashion. See Wooldridge (2002, pp. 148-153) and Arellano (1987).

Alternatively, the White period method is robust to arbitrary serial correlation and timevarying variances in the disturbances. The coefficient covariances are calculated using:

where, in contrast to Equation (27.32), the summations are taken over individuals and individual stacked data instead of periods.

The White period robust coefficient variance estimator is designed to accommodate arbitrary serial correlation and time-varying variances in the disturbances. The corresponding multivariate regression (with an equation for each period) allows the unconditional variance matrix E( i i′ ) = ΩT to be unrestricted, and the conditional variance matrix

E( i i′ Xi ) may depend on Xi in general fashion.

In contrast, the White (diagonal) method is robust to observation specific heteroskedasticity in the disturbances, but not to correlation between residuals for different observations. The coefficient asymptotic variance is estimated as:

fashion. Note that this method is both more general and more restrictive than the previous approaches. It is more general in that observations in the same cross-section or period may have different variances; it is more restrictive in that all off-diagonal variances are restricted to be zero.

EViews also allows you to compute non degree of freedom corrected versions of all of the robust coefficient covariance estimators. In these cases, the leading ratio term in the expressions above is dropped from the calculation. While this has no effect on the asymptotic validity of the estimates, it has the practical effect of lowering all of your standard error estimates.

PCSE Robust Covariances

The remaining methods are variants of the first two White statistics in which residuals are replaced by moment estimators for the unconditional variances. These methods, which are variants of the so-called Panel Corrected Standard Error (PCSE) methodology (Beck and Katz, 1995), are robust to unrestricted unconditional variances ΩM and ΩT , but place additional restrictions on the conditional variance matrices. A sufficient (though not necessary) condition is that the conditional and unconditional variances are the same.