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The Order of obs option allows you to order the data in Stacked form (stacking the data by series, which orders by crosssection), or in Interleaved format (stacked the data by interleaving series, which orders the data by period or date).

The default naming rule for series in the destination is to use the base name. For example, if you stack the pool series “SALES?” and the individual series GENDER, the corresponding stacked series will, by default, be named “SALES”, and “GENDER”. If use of the default naming convention will create problems in the destination workfile, you should use the

Name for stacked series field to specify

an alternative. If, for example, you enter “_NEW”, the target names will be formed by taking the base name, and appending the additional text, as in “SALES_NEW” and “GENDER_NEW”.

See “Stacking a Workfile” on page 257 of User’s Guide I for a more detailed discussion of the workfile stacking procedure.

Pooled Estimation

EViews pool objects allow you to estimate your model using least squares or instrumental variables (two-stage least squares), with correction for fixed or random effects in both the cross-section and period dimensions, AR errors, GLS weighting, and robust standard errors, all without rearranging or reordering your data.

We begin our discussion by walking you through the steps that you will take in estimating a pool equation. The wide range of models that EViews supports means that we cannot exhaustively describe all of the settings and specifications. A brief background discussion of the supported techniques is provided in “Estimation Background,” beginning on page 601.

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:

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Dependent Variable

List a pool variable, or an EViews expression containing ordinary and pool variables, in the

Dependent Variable edit box.

Regressors and AR terms

On the right-hand side of the dialog, you should list your regressors in the appropriate edit boxes:

•Common coefficients: — enter variables that have the same coefficient across all cross-section members of the pool. EViews will include a single coefficient for each variable, and will label the output using the original expression.

•Cross-section specific coefficients: — list variables with different coefficients for each member of the pool. EViews will include a different coefficient for each crosssectional unit, and will label the output using a combination of the cross-section identifier and the series name.

•Period specific coefficients: — list variables with different coefficients for each observed period. EViews will include a different coefficient for each period unit, and will label the output using a combination of the period identifier and the series name.

For example, if you include the ordinary variable TIME and POP? in the common coefficient list, the output will include estimates for TIME and POP?. If you include these variables in the cross-section specific list, the output will include coefficients labeled “_USA—TIME”, “_UK—TIME”, and “_USA—POP_USA”, “_UK—POP_UK”, etc.

Be aware that estimating your model with cross-section or period specific variables may generate large numbers of coefficients. If there are cross-section specific regressors, the number of these coefficients equals the product of the number of pool identifiers and the number of variables in the list; if there are period specific regressors, the number of corresponding coefficients is the number of periods times the number of variables in the list.

You may include AR terms in either the common or cross-section coefficients lists. If the terms are entered in the common coefficients list, EViews will estimate the model assuming a common AR error. If the AR terms are entered in the cross-section specific list, EViews will estimate separate AR terms for each pool member. See “Estimating AR Models” on page 89 for a description of AR specifications.

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 583).

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Options

Clicking on the Options tab in the dialog brings up a page displaying a variety of estimation options for pool estimation. Settings that are not currently applicable will be grayed out.

Coef Covariance Method

By default, EViews reports conventional estimates of coefficient standard errors and covariances.

You may use the combo box at the top of the page to select from the various

robust methods available for computing the coefficient standard errors. Each of the methods is described in greater detail in “Robust Coefficient Covariances” on page 611.

Note that the checkbox No d.f. correction permits to you compute robust covariances without the leading degree of freedom correction term. This option may make it easier to match EViews results to those from other sources.

Weighting Options

If you are estimating a specification that includes a random effects specification, EViews will provide you with a Random effects method combo box so that you may specify one of the methods for calculating estimates of the component variances. You may choose between the default Swamy-Arora, Wallace-Hussain, or Wansbeek-Kapteyn methods. See “Random Effects” on page 605 for discussion of the differences between the methods. Note that the default Swamy-Arora method should be the most familiar from textbook discussions.

Details on these methods are provided in Baltagi (2005), Baltagi and

Chang (1994), Wansbeek and Kapteyn (1989).

The checkbox labeled Keep GLS weights may be selected to require EViews to save all estimated GLS weights with the equation, regardless of their size. By default, EViews will not save estimated weights in system (SUR) settings, since the size of the required matrix may be quite large. If the weights are not saved with the equation, there may be some pool views and procedures that are not available.

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Instruments

To estimate a pool specification using instrumental variables techniques, you should select

TSLS - Two-Stage Least Squares (and AR) in the Method combo box at the bottom of the main (Specification) dialog page. EViews will respond by creating a three-tab dialog in which the middle tab (page) is used to specify your instruments.

As with the regression specification, the instrument list specification is divided into a set of Common, Cross-sec- tion specific, and Period specific instruments. The interpretation of these lists is the same as for the regressors; if there are cross-sec- tion specific instruments, the number of these instruments equals the product of the number of pool identifiers and the number of variables in the list; if there are period specific instruments, the number of corresponding

instruments is the number of periods times the number of variables in the list.

Note that you need not specify constant terms explicitly since EViews will internally add constants to the lists corresponding to the specification in the main page.

Lastly, there is a checkbox labeled Include lagged regressors for equations with AR terms that will be displayed if your specification includes AR terms. Recall that when estimating an AR specification, EViews performs nonlinear least squares on an AR differenced specification. By default, EViews will add lagged values of the dependent and independent regressors to the corresponding lists of instrumental variables to account for the modified differenced specification. If, however, you desire greater control over the set of instruments, you may uncheck this setting.

Pool Equation Examples

For illustrative purposes, we employ the balanced firm-level data from Grunfeld (1958) that have been used extensively as an example dataset (e.g., Baltagi, 2005). The workfile (“Grunfeld_Baltagi_pool.WF1”) contains annual observations on investement (I?), firm value (F?), and capital stock (K?) for 10 large U.S. manufacturing firms for the 20 years from 1935-54.

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EViews displays both the estimates of the coefficients and the fixed effects. Note that EViews automatically includes a constant term so that the fixed effects estimates sum to zero and should be interpreted as deviations from an overall mean.

Note also that the estimates of the fixed effects do not have reported standard errors since EViews treats them as nuisance parameters for the purposes of estimation. If you wish to compute standard errors for the cross-section effects, you may estimate a model without a constant and explicitly enter the C in the Cross-section specific coefficients edit field.

The bottom portion of the output displays the effects specification and summary statistics for the estimated model.

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(clustering by cross-section). Select the Options page and choose White period as the coefficient covariance method. The coefficient results are given by.

White period standard errors & covariance (d.f. corrected)

WARNING: estimated coefficient covariance matrix is of reduced rank

We caution that the White period results assume that the number of cross-sections is large, which is not the case in this example. In fact, the resulting coefficient covariance matrix is of reduced rank, a fact that EViews notes in the output.

AR Estimation

We may add an AR(1) term to the specification, and compute estimates using Cross-section SUR PCSE methods to compute standard errors that are robust to more contemporaneous correlation. EViews will estimate the transformed model using nonlinear least squares, will form an estimate of the residual covariance matrix, and will use the estimate in forming standard errors. The top portion of the results is given by:

Dependent Variable: I? Method: Pooled Least Squares Date: 08/17/09 Time: 14:45 Sample (adjusted): 1936 1954

Included observations: 19 after adjustments Cross-sections included: 10

Total pool (balanced) observations: 190

Cross-section SUR (PCSE) standard errors & covariance (d.f. corrected)

Convergence achieved after 14 iterations

Note in particular the description of the sample adjustment where we show that the estimation drops one observation for each cross-section when performing the AR differencing, as well as the description of the method used to compute coefficient covariances.

Pooled Estimation—597

Random Effects

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 estimate the coefficient covariance using the Ordinary

method and 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.

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

Here, we see that the estimated cross-section, period, and idiosyncratic error component standard deviations are 89.26, 15.78, and 51.72, respectively. As seen from the values of Rho, these components comprise 0.73, 0.02 and 0.25 of the total variance. Taking the crosssection component, for example, Rho is computed as:

In addition, EViews reports summary statistics for the random effects GLS weighted data used in estimation, and a subset of statistics computed for the unweighted data.

Cross-section Specific Regressors

Suppose instead that we elect to estimate a specification with I? as the dependent variable, C and F? as the common regressors, and K? as the cross-section specific regressor, using crosssection weighted least squares. The top portion of the output is given by:

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Dependent Variable: I?

Method: Pooled Least Squares

Date: 08/22/06 Time: 10:47

Sample: 1935 1954

Included observations: 20

Cross-sections included: 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:

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 668

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for i = 1, º, M , where lT is a T -element unit vector, IT is the T -element identity matrix, and g is a vector containing all of the period effects, g¢ = (g1, g2, º, gT).

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, º, T , where lM is a M -element unit vector, IM is the M -element identity matrix, and d is a vector containing all of the cross-section effects, d¢ = (d1, d2, º, dM).

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 b 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:

If instead we treat the specification as a set of period specific equations, the stacked (by period) representation is given by,

The remainder of this section describes briefly the various components that you may employ in an EViews pool specification.

Cross-section and Period Specific Regressors

The basic EViews pool specification in Equation (35.2) allows for b slope coefficients that are common to all individuals and periods, as well as coefficients that are either cross-sec-

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for all i , where the innovations hit are independent and identically distributed, assuming further that there is no unit root. Note that we allow the autocorrelation coefficients r to be cross-section, but not period specific.

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

using iterative techniques to estimate (a, bi, ri ) for all i . See “Estimating AR Models” on page 89 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 d and g 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 dimension. Note, in particular, however, that two-way random effects may only be estimated if the data are balanced so that every cross-section has the same set of observations.

Fixed Effects

The fixed effects portions of specifications are handled using orthogonal projections. In the simple one-way fixed effect specifications and the balanced two-way fixed specification, these projections involve the familiar approach of removing cross-section or period specific means from the dependent variable and exogenous regressors, and then performing the specified regression using the demeaned data (see, for example Baltagi, 2005). More generally, we apply the results from Davis (2002) for estimating multi-way error components models with unbalanced data.

Note that if instrumental variables estimation is specified with fixed effects, EViews will automatically add to the instrument list, the constants implied by the fixed effects so that the orthogonal projection is also applied to the instrument list.

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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(eiteit Xi ) = j2i

(35.16)

E(eisejt Xi ) = 0

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

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.

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(eitejt Xt ) = j2t

(35.18)

E(eisejt Xt ) = 0

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

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.

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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 611.

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.

Serial Correlation (Period SUR)

This class of covariance structures allows for arbitrary heteroskedasticity and serial correlation between the residuals for a given cross-section, but restricts residuals in different crosssections to be uncorrelated. This error structure is sometimes referred to as clustered by cross-section since observations in a given cross-section are correlated (form a cluster).

Accordingly, we assume that:

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

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 QT . In the second stage, we perform feasible GLS.

See “Contemporaneous Covariances (Cross-section SUR)” on page 607 for related discussion of errors clustered-by-period.

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In the context of random effects specifications, this approach to IV estimation is termed generalized two-stage least squares (G2SLS) method (see Baltagi (2005, p. 113-116) 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 677 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.

This transformation approach raises questions as to what instruments to use in estimation. By default, EViews adds instruments corresponding to the lagged endogenous and lagged exogenous variables introduced into the specification by the transformation.

For example, in an AR(1) specification, we have the original specification,

You may, however, instruct EViews not to add these additional instruments. Note, however, that the order condition for the transformed model is different than the order condition for the untransformed specification since we have introduced additional coefficients corresponding to the AR coefficients. If you elect not to add the additional instruments automati-

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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:

This method allows the unconditional variance matrix E(ee¢) = L to be an unrestricted diagonal matrix, with the conditional variances E(e2it Xit ) depending on Xit in general fashion.

EViews 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 variance matrices QM and QT , but place additional restrictions on the conditional variance matrices.

A sufficient (though not necessary) condition for use of PCSE is that the conditional and unconditional variances are the same. (Note also that as with the SUR estimators above, we require that QM and QT not vary with t and i , respectively.)

For example, the Cross-section SUR (PCSE) method handles cross-section correlation (period clustering) by replacing the outer product of the cross-section residuals in Equation (35.32) with an estimate of the (contemporaneous) cross-section residual covariance matrix QM :

Analogously, the Period SUR (PCSE) handles between period correlation (cross-section clustering) by replacing the outer product of the period residuals in Equation (35.33) with an estimate of the period covariance QT :

ance estimators are robust to heteroskedasticity across cross-sections or periods, respectively, but not to general correlation of residuals.