Chapter 23. Specification and Diagnostic Tests

140—Chapter 23. Specification and Diagnostic Tests

Variable Models,” on page 247), and the tests for cointegration (“Testing for Cointegration” on page 234).

Coefficient Diagnostics

These diagnostics provide information and evaluate restrictions on the estimated coefficients, including the special case of tests for omitted and redundant variables.

Scaled Coefficients

The Scaled Coefficients view displays the coefficient estimates, the standardized coefficient estimates and the elasticity at means. The standardized coefficients are the point estimates of the coefficients standardized by multiplying by the standard deviation of the dependent variable divided by the standard deviation of the regressor.

The elasticity at means are the point estimates of

the coefficients scaled by the mean of the dependent variable divided by the mean of the regressor.

Confidence Intervals and Confidence Ellipses

The Confidence Intervals view displays a table of confidence intervals for each of the coefficients in the equation.

The Confidence Intervals dialog allows you to enter the size of the confidence levels. These can be entered a space delimited list of decimals, or as the name of a scalar or vector in the workfile containing confidence levels. You can also choose how you would like to display the confidence intervals. By default they will be shown in pairs where the low and high values for each confidence level

are shown next to each other. By unchecking the Arrange in pairs checkbox you can choose to display the confidence intervals concentrically.

The Confidence Ellipse view plots the joint confidence region of any two functions of estimated parameters from an EViews estimation object. Along with the ellipses, you can choose to display the individual confidence intervals.

We motivate our discussion of this view by pointing out that the Wald test view (View/ Coefficient Diagnostics/Wald - Coefficient Restrictions...) allows you to test restrictions on the estimated coefficients from an estimation object. When you perform a Wald test, EViews provides a table of output showing the numeric values associated with the test.

142—Chapter 23. Specification and Diagnostic Tests

C(1)=0 (since it falls below the end

of the univariate confidence interval). If C(3)=.8, we cannot reject the joint null that C(1)=0, and C(3)=0 (since C(1)=-.65, C(3)=.8 falls within the confidence ellipse).

EViews allows you to display more than one size for your confidence ellipses. This feature allows you to draw confidence contours so that you may see how the rejection region changes at different probability values. To do so, simply enter a space delimited list of confidence levels. Note that while the coefficient restriction expressions must be separated by commas, the contour levels must be separated by spaces.

Here, the individual confidence intervals are depicted with shading. The individual intervals are based on the largest size confidence level (which has the widest interval), in this case, 0.9.

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The centered VIF is numerically identical to 1 § (1 – R2 ) where R2 is the R-squared from the regression of that regressor on all of the other regressors in the equation.

Note that since the VIFs are calculated from the coefficient variance-covariance matrix, any robust standard error options will be present in the VIFs.

Coefficient Variance Decomposition

The Coefficient Variance Decomposition view of an equation provides information on the eigenvector decomposition of the coefficient covariance matrix. This decomposition is a useful tool to help diagnose potential collinearity problems amongst the regressors. The decomposition calculations follow those given in Belsley, Kuh and Welsch (BKW) 2004 (Section 3.2). Note that although BKW use the singular-value decomposition as their method to decompose the variance-covariance matrix, since this matrix is a square positive semi-defi- nite matrix, using the eigenvalue decomposition will yield the same results.

In the case of a simple linear least squares regression, the coefficient variance-covariance matrix can be decomposed as follows:

where S is a diagonal matrix containing the eigenvalues of X¢X , and V is a matrix whose columns are equal to the corresponding eigenvectors.

The variance of an individual coefficient estimate is then:

146—Chapter 23. Specification and Diagnostic Tests

Coefficient Variance Decomposition

Date: 07/16/09 Time: 12:42

Sample: 1947 1962

Included observations: 16

The top line of the table shows the eigenvalues, sorted from largest to smallest, with the condition numbers below. Note that the final condition number is always equal to 1. Three of the four eigenvalues have condition numbers smaller than 0.001, with the smallest condition number being very small: 1.09E-11, which would indicate a large amount of collinearity.

The second section of the table displays the decomposition proportions. The proportions associated with the smallest condition number are located in the first column. Three of these values are larger than 0.5, indeed they are very close to 1. This indicates that there is a high level of collinearity between those three variables, YEAR, PRICE and GNP.

Wald Test (Coefficient Restrictions)

The Wald test computes a test statistic based on the unrestricted regression. The Wald statistic measures how close the unrestricted estimates come to satisfying the restrictions under the null hypothesis. If the restrictions are in fact true, then the unrestricted estimates should come close to satisfying the restrictions.

148—Chapter 23. Specification and Diagnostic Tests

c(2) + c(3) = 1

and click OK. EViews reports the following result of the Wald test:

Wald Test:

Equation: EQ1

Null Hypothesis: C(2) + C(3) = 1

Restrictions are linear in coefficients.

EViews reports an F-statistic and a Chi-square statistic with associated p-values. In cases with a single restriction, EViews reports the t-statistic equivalent of the F-statistic. See “Wald Test Details” on page 151 for a discussion of these statistics. In addition, EViews reports the value of the normalized (homogeneous) restriction and an associated standard error. In this example, we have a single linear restriction so the F-statistic and Chi-square statistic are identical, with the p-value indicating that we can decisively reject the null hypothesis of constant returns to scale.

To test more than one restriction, separate the restrictions by commas. For example, to test the hypothesis that the elasticity of output with respect to labor is 2/3 and the elasticity with respect to capital is 1/3, enter the restrictions as,

c(2)=2/3, c(3)=1/3

and EViews reports:

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Dependent Variable: LOG(Q)

Method: Least Squares

Date: 08/10/09 Time: 13:39

Sample: 1947 1971

Included observations: 25

Convergence achieved after 288 iterations

LOG(Q)=C(1)+C(2)*LOG( C(3)*K^C(4)+(1-C(3))*L^C(4) )

To test the nonlinear restriction b2 = 1 § b4 , choose View/Coefficient Diagnostics/WaldCoefficient Restrictions… from the equation toolbar and type the following restriction in the Wald Test dialog box:

c(2)=1/c(4)

The results are presented below:

Wald Test:

Equation: Untitled

Null Hypothesis: C(2) = 1/C(4)

Delta method computed using analytic derivatives.

We focus on the p-values for the statistics which show that we fail to reject the null hypothesis. Note that EViews reports that it used the delta method (with analytic derivatives) to compute the Wald restriction variance for the nonlinear restriction.

Coefficient Diagnostics—151

It is well-known that nonlinear Wald tests are not invariant to the way that you specify the

Delta method computed using analytic derivatives.

so that the test now decisively rejects the null hypothesis. We hasten to add that this type of inconsistency in results is not unique to EViews, but is a more general property of the Wald test. Unfortunately, there does not seem to be a general solution to this problem (see Davidson and MacKinnon, 1993, Chapter 13).

Wald Test Details

Consider a general nonlinear regression model:

where y and e are T -vectors and b is a k -vector of parameters to be estimated. Any restrictions on the parameters can be written as:

152—Chapter 23. Specification and Diagnostic Tests

where T is the number of observations and b is the vector of unrestricted parameter estimates, and where Vˆ is an estimate of the b covariance. In the standard regression case, Vˆ is given by:

where u is the vector of unrestricted residuals, and s2 is the usual estimator of the unrestricted residual variance, s2 = (u¢u) § (N – k), but the estimator of V may differ. For

ˆ

example, V may be a robust variance matrix estimator computing using White or NeweyWest techniques.

More formally, under the null hypothesis H0 , the Wald statistic has an asymptotic x2(q) distribution, where q is the number of restrictions under H0 .

where R is a known q ¥ k matrix, and r is a q -vector, respectively. The Wald statistic in Equation (23.12) reduces to:

where u˜ is the vector of residuals from the restricted regression. In this case, the F-statistic compares the residual sum of squares computed with and without the restrictions imposed.

We remind you that the expression for the finite sample F-statistic in (23.17) is for standard linear regression, and is not valid for more general cases (nonlinear models, ARMA specifications, or equations where the variances are estimated using other methods such as Newey-West or White). In non-standard settings, the reported F-statistic (which EViews always computes as W § q ), does not possess the desired finite-sample properties. In these cases, while asymptotically valid, F-statistic (and corresponding t-statistic) results should be viewed as illustrative and for comparison purposes only.

Coefficient Diagnostics—153

Omitted Variables

This test enables you to add a set of variables to an existing equation and to ask whether the set makes a significant contribution to explaining the variation in the dependent variable. The null hypothesis H0 is that the additional set of regressors are not jointly significant.

The output from the test is an F-statistic and a likelihood ratio (LR) statistic with associated p-values, together with the estimation results of the unrestricted model under the alternative. The F-statistic is based on the difference between the residual sums of squares of the restricted and unrestricted regressions and is only valid in linear regression based settings. The LR statistic is computed as:

where lr and lu are the maximized values of the (Gaussian) log likelihood function of the unrestricted and restricted regressions, respectively. Under H0 , the LR statistic has an asymptotic x2 distribution with degrees of freedom equal to the number of restrictions (the number of added variables).

Bear in mind that:

•The omitted variables test requires that the same number of observations exist in the original and test equations. If any of the series to be added contain missing observations over the sample of the original equation (which will often be the case when you add lagged variables), the test statistics cannot be constructed.

•The omitted variables test can be applied to equations estimated with linear LS, ARCH (mean equation only), binary, ordered, censored, truncated, and count models. The test is available only if you specify the equation by listing the regressors, not by a formula.

•Equations estimated by Two-Stage Least Squares and GMM offer a variant of this test based on the difference in J-statistics.

To perform an LR test in these settings, you can estimate a separate equation for the unrestricted and restricted models over a common sample, and evaluate the LR statistic and p- value using scalars and the @cchisq function, as described above.

How to Perform an Omitted Variables Test

To test for omitted variables, select View/Coefficient Diagnostics/Omitted Variables-Like- lihood Ratio… In the dialog that opens, list the names of the test variables, each separated by at least one space. Suppose, for example, that the initial regression specification is:

log(q) c log(l) log(k)

If you enter the list:

log(l)^2 log(k)^2

154—Chapter 23. Specification and Diagnostic Tests

in the dialog, then EViews reports the results of the unrestricted regression containing the two additional explanatory variables, and displays statistics testing the hypothesis that the coefficients on the new variables are jointly zero. The top part of the output depicts the test results (the bottom portion shows the estimated test equation):

Omitted Variables Test

Equation: EQ1

Specification: LOG(Q) C LOG(L) LOG(K)

Omitted Variables: LOG(L)^2 LOG(K)^2

The F-statistic has an exact finite sample F-distribution under H0 for linear models if the errors are independent and identically distributed normal random variables. The numerator degrees of freedom is the number of additional regressors and the denominator degrees of freedom is the number of observations less the total number of regressors. The log likelihood ratio statistic is the LR test statistic and is asymptotically distributed as a x2 with degrees of freedom equal to the number of added regressors.

In our example, neither test rejects the null hypothesis that the two series do not belong to the equation at a 5% significance level.

Redundant Variables

The redundant variables test allows you to test for the statistical significance of a subset of your included variables. More formally, the test is for whether a subset of variables in an equation all have zero coefficients and might thus be deleted from the equation. The redundant variables test can be applied to equations estimated by linear LS, TSLS, ARCH (mean equation only), binary, ordered, censored, truncated, and count methods. The test is available only if you specify the equation by listing the regressors, not by a formula.

How to Perform a Redundant Variables Test

To test for redundant variables, select View/Coefficient Diagnostics/Redundant Vari- ables-Likelihood Ratio… In the dialog that appears, list the names of each of the test vari-

156—Chapter 23. Specification and Diagnostic Tests

whether the demand function for energy differs between the different states of the USA. The test may be used with least squares and two-stage least squares regressions.

By default the Factor Breakpoint test tests whether there is a structural change in all of the equation parameters. However if the equation is linear EViews allows you to test whether there has been a structural change in a subset of the parameters.

To carry out the test, we partition the data by splitting the estimation sample into subsamples of each unique value of the classification variable. Each subsample must contain more observations than the number of coefficients in the equation so that the equation can be estimated. The Factor Breakpoint test compares the sum of squared residuals obtained by fitting a single equation to the entire sample with the sum of squared residuals obtained when separate equations are fit to each subsample of the data.

EViews reports three test statistics for the Factor Breakpoint test. The F-statistic is based on the comparison of the restricted and unrestricted sum of squared residuals and in the simplest case involving two subsamples, is computed as:

from subsample i , T is the total number of observations, and k is the number of parameters in the equation. This formula can be generalized naturally to more than two subsamples. The F-statistic has an exact finite sample F-distribution if the errors are independent and identically distributed normal random variables.

The log likelihood ratio statistic is based on the comparison of the restricted and unrestricted maximum of the (Gaussian) log likelihood function. The LR test statistic has an asymptotic x2 distribution with degrees of freedom equal to (m – 1)k under the null hypothesis of no structural change, where m is the number of subsamples.

The Wald statistic is computed from a standard Wald test of the restriction that the coefficients on the equation parameters are the same in all subsamples. As with the log likelihood ratio statistic, the Wald statistic has an asymptotic x2 distribution with (m – 1)k degrees of freedom, where m is the number of subsamples.

For example, suppose we have estimated an equation specification of lwage c grade age high

using data from the “Cps88.WF1” workfile.