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173

-22 «-23

2

In our simple case we have

v2v3

C22 - - <23 - c32 - •

where A = vll vl - (Z <2<if-

3. Define

X2 = c22v2 + C2,i?3

x3 = c32v2 + c33v3

These are the missing variables we use.

4. Estimate the following equation by OLS.

1 = 2 2 + + aZ, + 72-2 + 73-3 + e

"This is pointed out in Alberto Holly, "A Simple Procedure for Testing Whether a Subset of Endogenous Variables Is Independent of the Disturbance Term in a Structural Equation," discussion paper. University of Lausanne, November 1982.

1 = + + iZ, + 722 + lA + e (12.24)

by OLS and test the hypothesis 72 = 7 = 0.

This procedure applies only if we are testing the exogeneity of all the variables under question and does not apply for testing the exogeneity of a subset of these variables. For instance, in equation (12.22) suppose we assume that is endogenous and Zi is exogenous. What we want to do is test the hypothesis that is exogenous. Now we cannot just estimate equations like (12.23) and (12.24) and just test 7, = 0. Nor does the equivalence between (12.23) and (12.24) apply anymore."

In this case we have to obtain the 2SLS estimates of ( 3 under two assumptions:

1. is exogenous. Call this estimate 3 with variance Vq.

2. is endogenous. Call this estimate p, with variance V,.

Then = - and var() = V, - V. We now apply the Hausman test as explained earlier. Note that both V, and V,, depend on = var(h,). However, when we get an estimate of var(q) we have to use the estimate of under , that is, from the 2SLS estimation treating as exogenous.

The omitted variable method in this case is somewhat complicated, but consists of the following steps:

1. First we get the reduced-form residuals for the endogenous variables and as before. Call these residuals and v3. However, we do not use these as the omitted variables. We construct linear combinations of these as explained in the next step.

2. Form the covariance matrix of these residuals and get its inverse. Denote this by



12.11 THE PLOSSER-SCHWERT-WHITE DIFFERENCING TEST 5 J 3

and test the hypothesis = 0. This is the required test for the hypothesis that is exogenous.

The test is actually not very complicated. In fact, step 1 has to be used in any 2SLS estimation. Only step 2 involves matrix inversion. Once this is done, step 3 is easy and step 4 is just OLS estimation.

12.11 The Plosser-Schwert-White Differencing Test

The Plosser-Schwert-White (PSW) differencing test" is, like the Hausman test, a general test for model misspecification, but is applicable for time-series data only. The test involves estimation of the regression models in levels and in first differences. If the model is correctly specified, the estimators from the differenced and undifferenced models have the same probability limits and hence the resufis should corroborate one another. On the other hand, if there are specification errors, the differenced regression should lead to different results. The PSW test, like the Hausman test, is based on a comparison from the differenced and undifferenced regressions.

Davidson, Godfrey, and MacKinnon" show that, like the Hausman test, the PSW test is equivalent to a much simpler omitted variables test, the omitted variables being the sum of the lagged and one-period forward values of the variables.

Thus if the regression equation we are considering is

y, = fitXt, + , + «/ the PSW test involves estimating the expanded regression equation

y, = pllr + , + JlZu + I2Z2, + u,

where

Z21 - 2./+1 + 2.(-l

and testing the hypothesis "Vi = 72 = 0 by the usual F-test.

If there are lagged dependent variables in the equation, the test needs a minor modification. Suppose that the model is

y, = ,-1 + tX, + u, Now the omitted variables would be defined as

Zl, = y, + y,-2

"C. L Plosser, G. W. Schwert, and H. White, "Differencing as a Test of Specification," International Economic Review, October 1982, pp. 535-552.

"R. Davidson, L. G. Godfrey, and J. G. MacKinnon, "A Simplified Version of the Differencing Test," International Economic Review, October 1985, pp. 639-647.



Zit ~ ,+ ] + x, x

There is no problem with Z2, but Zi, would be correlated with the error term u, because of the presence of y, in it. The solution would be simply to transfer it to the left-hand side and write the expanded regression equation as

(1 - yi)y, = 3iy,-i + ix, + 1 ,-2 + yiZi, + , This equation can be written as

y, = ,-! + + 1 ,-2 + 72Z2, + ,

where all the starred parameters are the corresponding unstarred ones divided by (1 - 7,).

The PSW test now tests the hypothesis 7I = 72 = 0. Thus, in the case where the model involves the lagged dependent variable y,-, as an explanatory variable, the only modification needed is that we should use y,2 the omitted variable, not (y, + y, 2). For the other explanatory variables, the corresponding omitted variables are defined as before. Note that it is only y,„, that creates a problem, not higher-order lags of y„ like y, 2. ,- . and so on.

12.12 Tests for Nonnested Hypotheses

Consider the problem of testing two hypotheses:

: = + Mo Mo ~ IN(0, crl) (12.25)

,: = + M, , ~ IN(0, a?) (12.26)

The hypotheses are said to be nonnested since the explanatory variables under one of the hypotheses are not a subset of the explanatory variables in the other.

It is a common occurrence in economics that there are many competing economic theories trying to explain the same variable (consumption, investment, etc.), and the explanatory variables in the different theories contain nonover-lapping variables. An extreme example is the paper by Friedman and Meiselman around which there was a considerable amount of controversy in the 1960s. The Keynesian and monetarist theories were formulated in their simpUfied form as

C, = tto + poA, + Mo, (Keynesian) C, = a, -I- PM, -f M„ (Monetarist)

where C, = consumption expenditure in constant dollars A, = autonomous expenditure in constant dollars M, = money supply

"M. Friedman and D. Meiselman, "The Relative Stability of Monetary Velocity and the Investment MultipUer in the U.S. 1897-1958." in Stabilization Policies (Commission on Money and Credit) (Englewood Cliffs, N.J.: Prentice Hall, 1963).



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