EXERCISES 521

it amounts to is supplementing the 7-test with an F-test based on a comprehensive model.

Exercises

1. Explain the meaning of each of the following terms.

(a) Least squares residual.

(b) Predicted residual.

(c) Studentized residual.

(d) Recursive residual.

(e) BLUS residual.

(f) DFFITS.

(g) Bounded influence estimation.

(h) ARCH errors.

(i) Post-data model construction.

(j) Overparametrization with data-based simplification, (k) Posterior odds ratio. (1) Cross validation, (m) Selection of regressors. (n) Diagnostic checking, (o) Specification testing, (p) Hausmans test, (q) J-test.

(r) Encompassing test, (s) Differencing test.

2. Examine whether the following statements are true, false, or uncertain. Give a short explanation. If a statement is not true in general but is true under some conditions, state the conditions.

(a) Since the least squares residuals are heteroskedastic and autocorrelated, even when the true errors are independent and homoskedastic, tests for homoskedasticity and serial independence based on least squares residuals are useless.

(b) There is a better chance of detecting outliers with predicted residuals or studentized residuals than with least squares residuals.

(c) In tests of significance we should not use a constant level of significance. The significance level should be a decreasing function of the sample size.

(d) To test for omitted variables, all we have to do is get some proxies for them, regress the least squares residual il, on the proxies, and test whether the coefficients of these proxies are zero.

(e) The procedure of estimating the regression equation

where w, are the least squares residuals, does not necessarily test for ARCH effect.

(f) The R criterion of deciding whether to retain or delete certain regressors in an equation implies using a 5% significance level for testing that the coefficients of these regressors are zero,

(g) The criteria for selection of regressors proposed by Amemiya, Hocking, and Mallows imply using a higher significance level for testing the coefficients of regressors than the criterion.

(h) The R, Cp, and PC criteria for selection of regressors depend on the assumption that one of the models considered is a "true" model, whereas the criterion does not. Hence given the choice between these criteria, we should use only the Sp criterion.

(i) The maximum R criterion and the predictive criteria PC, Cp, and Sp for selection of regressors are not really comparable since they answer different questions.

0) For comparing different models we should always save part of the

sample for prediction purposes, (k) In the model

1 = 2 2 + + +

where x, is exogenous, to test whether and y, are endogenous, we estimate the equation

= 2 2 + + yi-l + «22 + «33 + w

{2 and v3 are the estimated residuals from the reduced form equations for yjandyj).

(1) In the model in part (k), if x, is specified as exogenous and as endogenous, to test whether is endogenous, we estimate the equation

= 2)2 + + + «33 +

and test a, = 0.

3. Comment on the following: In a study on the demand for money, an economist regressed real cash balances on permanent income. To test whether the interest rate has been omitted, she regressed the residual from this regression on the interest rate and found that the coefficient was not significantly different from zero. She therefore concluded that the interest rate does not belong in the demand for money function.

4. Explain how each of the following tests can be considered as a test for the coefficients of some extra variables added to the original equation.

(a) Test for serial correlation.

(b) Test for omitted variables.

(c) Test for errors in variables.

(d) Test for exogeneity.

(e) J-test.

(f) PSW differencing test.

Appendix to Chapter 12

(1) Least Squares Residuals

We have = X(XX)-Xy = . H = ih,) is known as the hat matrix. The least square residuals are u = (I - H)y = (I - H)u. This gives equation (12.1). Var(u) = (I - H)(j since I - H is idempotent. Hence var(w,) = (1 - h,)<j, which is equation (12.3) and cov(m„ u) = -h,fj. Thus the least squares residuals are heteroskedastic and correlated.

(2) The R Criterion

The R criterion or minimum " criterion for the choice of models is based on the following result: Suppose that

= xp -I- u is the true model. X is / x A.

= Z6 + v is the misspecified model. Z is / x r.

We estimate the misspecified model. Consider the estimate of the residual variance from this model. It is

a = -3-yNy where N = I - Z(ZZ)-Z

Since = xp + u and £(uu) = W, we have yNy = (Xp + u)N(Xp + u) = PXNXp -I- uNu -I- 2PXNu. Since £(u) = 0 the last term has expectation zero. Also, £(uNu) = {n - r)(j since N is idempotent of rank (n - r). Hence

we get E{a-) = a + -- PXNXp. Since the second term in the expression n - r

is > 0, we have

) >

Thus the estimate of the error variance from the misspecified equation is upward-biased. This is the basis of what is known as the "minimum s" or the "maximum R" rule. The rule says that if we are considering some alternative regression models, we should choose the one with the minimum estimated error variance. The idea behind it is that "on the average, the misspecified model has a larger estimated error variance than the "true" model. Of course, the suggested rule is based on the assumption that one of the models being considered is the "true" model. It should be noted, however, that £(&) = a even for a misspecified model if XN = 0. This will be the case if Z consists of the variables X and any number of irrelevant variables. This does not make these models any better than models with a few omitted variables for which £(ct) > ctI

In each case explain how you would interpret the results of the test and the actions you would take.