12.1 Introduction

The early developments in econometrics were concerned with the problems of estimation of an econometric model once the model was specified. The major preoccupation of econometricians was with devising methods of estimation that produced consistent and efficient estimates of the parameters. These are the methods we discussed in the previous chapters. Sometimes more simplified methods like two-stage least squares were suggested because some other methods like limited-information maximum likelihood (LIML) were complicated to compute. With the recent developments in computer technology, such search for simplified methods of estimation is no longer worthwhile, at least in a majority of problems.

During recent years the attention of econometricians has been diverted to the problems of:

1. Checking the adequacy of the specification of the model. This is called "diagnostic checking" and "specification testing."

2. Choosing between alternative specifications of the model. This is called "model selection."

3. Devising methods of estimation based on weaker assumptions about the error distributions. This is called "semiparametric estimation."

The last area is well beyond the scope of this book and will not be discussed at all. The first two areas are also very vast in scope. Hence what we will do is to consider only a few of the major developments.

It is, of course, true that the problem of diagnostic checking has not been completely ignored. In fact, we discussed some tests for this purpose in the earlier chapters. For instance:

1. Tests for parameter stability in Chapter 4.

2. Tests for heteroscedasticity in Chapter 5.

3. Tests for autocorrelation in Chapter 6.

However, these tests are all based on least squares residuals, and during recent years some alternative residuals have been suggested. Also, tests for diagnostic checking have been more systematized (and incorporated in computer programs like the SAS regression program). The limitations of some standard tests have been noticed and further modifications have been suggested. For instance, in Chapter 6 we discussed the limitations of the DW test, which has been the most commonly used "diagnostic test" for many years.

We begin our discussion with diagnostic tests based on the least squares residuals. Here we summarize the tests discussed in the earlier chapters and discuss some other tests. We will then present some alternatives to least squares residuals and the diagnostic tests based on them. Next, we discuss some problems in model selection and specification testing. Many of the specification tests involve "expanded regressions," that is, the addition of residuals or some

Other constructed variables as regressors to the original model and testing for the significance of the coefficients of these added variables. Thus most of these tests can be easily implemented using the standard regression packages.

12.2 Diagnostic Tests Based on Least Squares Residuals

Diagnostic tests are tests that are meant to "diagnose" some problems with the models that we are estimating. The least squares residuals play an important role in many diagnostic tests. We have already discussed some of these tests, such as tests for parameter stability in Chapter 4, tests for heteroscedasticity in Chapter 5, and tests for autocorrelation in Chapter 6. Here we discuss two other tests using least squares residuals.

Tests for Omitted Variables

Consider the linear regression model

y, = fix, + u,

To test whether the model is misspecified by the omission of a variable z„ we have to estimate the model

y, = P-, + IZ, + u,

and test the hypothesis 7 = 0.

If data on z, are available, there is no problem. All we do is regress y, on x, and z, and test whether the coefficient of z, is zero. Suppose, on the other hand, that we use the following procedure:

1. Regress y, on x, and get the residual a,.

2. Regress a, on z,. Let the regression coefficient be -y. Test the hypothesis 7 = 0 using this regression equation.

What is wrong with this procedure?

The answer is that is an inconsistent estimator of 7 unless 7 = 0. Furthermore, the distribution of 7 is complex and the standard errors provided by the least squares estimation of step 2 will not be the correct ones. If the least squares residuals at step 1 are to be used, we should regress them on z, and x, and then test whether the coefficient of z, is zero.

Thus if we are to use the residuals , from a regression of y, on jc, for testing specification errors caused by omitted variables, it is advisable to regress , on z, and X, and not z, only, and test the hypothesis that the coefficient of z, is zero.

We are omitting the detailed proofs. They can be found in the paper by A. R. Pagan and A. D. Hall, "Diagnostic Tests as Residual Analysis" (with discussion). Econometric Reviews, Vol. 2, 1983, pp. 159-218.

-The only problem is that if z is uncorrelated with z, the test would have low power, but cases where z is uncorrelated with z are rare.

J. B. Ramsey, "Tests for Specification Errors in Classical Linear Least Squares Regression Analysis," Journal of the Royal Statistical Society, Series B. Vol. 31. 1969, pp. 350-371. "R. F. Engle, "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Vol. 50, 1982, pp. 987-1007. See also Section 6.11.

When observations on z are not available, we use a proxy i for z- The preceding results apply to this case as well. That is, an appropriate test for omitted variables is to estimate the model

y, = px, + yz, + V,

and test the hypothesis 7 = 0. Alternatively, if we are using the residuals , from a regression of y, on x„ we have to regress , on z, and x, and then test whether the coefficient of z, is zero. Of course, we need not go through this circuitous procedure, but the reason for discussing this is to make clear the distinction between tests for heteroskedasticity discussed in Chapter 5 and tests for omitted variables.

Ramsey suggests the use of y], y] and y\ as proxies for z„ where y, is the predicted value of y, from a regression of y, on x,.

The test procedure is as follows:

1. Regress y, on x, and get y,.

2. Regress y, on x„ y], y] and y and test the hypothesis that the coefficients of the powers of y, are zero.

Note that this test is slightly different from Ramseys test for heteroskedasticity discussed in Section 5.2. That test proceeds as follows:

1. Regress y, on x, and get the residual ,.

2. Regress u, on yj, yj, and y* and test that the coefficients of these variables are zero.

As explained earlier, if we want to use u, to test for omitted variables, we have to include x, as an extra explanatory variable. This is the difference between the two Ramsey tests.

Tests for ARCH Effects

In econometric models, the uncertainty in the economic relationship is captured by the variance cr of the error term u,. It has been found that it is important to model this error variance because it affects the behavior of economic agents. One such model is the ARCH model (Autoregressive Conditionally Heteroscedastic model) suggested by Engle."* In this model the unconditional variance E{uj) is constant but the conditional variance E(uj\x,) is not. Denoting this conditional variance by ct;, the model suggested by Engle is

aj = + 7M? , 7 > 0