Ixl iYxzf 1 - r

where r is the squared correlation between x and the instrumental variable z.* The test statistic is

(1 - r)t>o which we use as with 1 d.f.

Some Illustrative Exarhples

As an illustration, let us consider the data on Australian wine industry in Chapter 9 (Table 9.2). Let us consider the equation (all variables in log form)

Q, = oi + + ,

We are interested in testing the hypothesis that is exogenous. The OLS estimator of (3 is

Using the index of storage costs s as an instrumental variable, we get the IV estimator as

= IS = = , - Po = 1-9777

The squared correlation coefficient between /?" and s, is r = 0.408. We also have

= 0.09217 and t>o = = 0.18164

The test statistic is

This is significant at the 1% level (from the x"-tables with 1 d.f.). Thus we cannot treat p" as exogenous and we have to consider a simultaneous equations model.

"Note that both V, and Vo depend on o--. But when we get an estimate of var(), the estimate of <r we use is the one obtained under „.

Let us see what happens if we use per capita disposable income y, as the instrumental variable. In this case we have

B, = = 4.8236

0.4592

= 0.624

q = 4.8236 - 3.5085 = 1.3151

The test statistic is now

(1.3151)0.624) = 0.376(0.18164) = -

Again, this is significant at the 1% level, indicating that we have to treat as endogenous.

As yet another example, consider the demand and supply model for pork considered in Chapter 9 (data in Table 9.1). Let us consider the hypothesis that Q is exogenous in the demand function (normalized with respect to P).

P = a+Q + SY+u

The OLS estimate of p is

P = -1.2518 with SE = 0.1032 The 2SLS estimate of p is

P = -1.2165

Thus

= P - P = 0.0353

Vo = (0.1032)2 = 0.01065

The instrumental variable that is implied by the use of 2SLS is Q, the predicted value of Q from the reduced form. The squared correlation between Q and Q is the from the reduced-form equation for Q, which is 0.898. Hence the test statistic is

(0.0353)2(0.898) (1 - 0.898)(0.01065)

which is not significant even at the 25% level (from the x-tables with 1 d.f.).

Thus we can treat Q as exogenous and estimate the demand equation by OLS. Note that this was our conclusion in Section 9.3.

An Omitted Variable Interpretation of the Hausman Test

There is an alternative way of implementing the Hausman test. Returning to equation (12,20), let x be the predicted value of x from a regression of x on the

I - r 1 - r where r is the squared correlation between x and z- Also,

Thus

var(y) (1 - r2)Vo

the statistic obtained earlier.

This omitted variable interpretation enables us to generalize the test to the case where we have more variables whose exogeneity we are interested in testing.

Suppose that one of the equations in a simultaneous equations model is

1 = 2 2 + + ctiZi + (12.22)

We want to test the hypothesis that and y, are exogenous (i.e., independent of M,), the alternative hypothesis being that they are not. The variable z, is considered exogenous.

To apply the test, we need to have at least two instrumental variables, say Zj and Z3. But there could be more in the system. Then what we do is regress y and on the other exogenous variables in the system and obtain the predicted values y, and y,. Now we estimate the expanded regression equation

y, = 2 2 + + aiZ, + 7252 + + e (12-23)

by ordinary least squares and test the hypothesis 72 = = 0 (by the methods described in Chapter 4). This is the Hausman test for exogeneity.

Let us define v2 = 2 ~ 2 and v3 = - - Then, instead of (12.23), we can also estimate

*For a proof, see Hausman, "Specification Tests," p. 1261.

instrumental variable z, and i? the estimated residual. That \s,\> = x-x. Then estimate by OLS the equation

= fix + yx + e

= fix + y\> +

and test the hypothesis 7 = 0. These equations are known as expanded regressions. This test is identically the same as the test based on the statistic m derived earlier.

We will not prove this result" but we will explicitly state what the OLS estimate will turn out to be. We have