Testing validity of instruments a statistic often

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TESTING VALIDITY OF INSTRUMENTS A statistic often used to test for validity of the chosen set of instruments is Sargan’s (1964) test. Call this statistic ST. The method of calculating this statistic is explained in the Microfit manual. However, it is outputted automatically when using IV or GIVE in Microfit (and several other packages as well). In effect, the ST statistic allows one to test the null hypothesis that the instruments are uncorrelated with the disturbance term in the regression model being estimated. If the null is true, ST has a chi-square distribution
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9 with p-k degrees of freedom, where p is the number of instruments selected and K is the number of regressors in the model being estimated. EXAMPLE OF INSTRUMENT CHOICE Suppose we wish to estimate the parameters of the model C Y Q Q Q u t t t t t t = + + + + + β β β β β 1 2 3 1 4 2 5 3 , , , where C and Y are consumers’ expenditure and disposable income respectively, and the model contains three quarterly dummy variables. Disposable income may be correlated with the disturbance term, and so is potentially “endogenous”. So in terms of our earlier notation, we have X = { X 1 ¦ X 2 } with X 1 = {intercept, Q 1,t , Q 2,t , Q 3,t } and X 2 = Y t In an IS/LM model, the money supply (M) and government expenditure (G) are often regarded as “exogenous” variables, and so are candidates as instruments for Y. Therefore, our tentative choice of instruments is W = { intercept, Q 1,t , Q 2,t , Q 3,t , M t , G t }. We use the term tentative here, as one should use Sargan’s 1964 test to test for the validity of our choice of instrument. Estimation with this set of instruments using GIVE should only proceed if we cannot reject the null of valid instruments using Sargan’s instrument validity test. 6 SOME MATTERS CONCERNING THE USE OF INSTRUMENTAL VARIABLES ESTIMATION IN PRACTICE 6.1 THE HAUSMAN TEST PRINCIPLE In practice, given that disturbances are unobservable and that we are working with finite-sized samples, we cannot know whether a variable is in fact correlated asymptotically with the disturbance term. This means that the choice between OLS and IV (or GIVE) will be uncertain. A common way of proceeding is to estimate our model by both OLS and GIVE and to test whether the differences in parameter estimates are statistically significant. This underlies the Hausman test procedure that we now explain. The arguments we have used so far suggest that if plim 1 T X u = 0 then both OLS and IV (or GIVE) will be consistent, but IV will be inefficient relative to OLS. We might therefore envisage the following hypotheses H 0 : plim 1 T X u = 0 H a : plim 1 T X u 0 to be a statement of the problem. Under the null hypothesis, one would not expect there to be a “large” difference between β GIVE and β OLS . If we do find a large difference, we may regard this as evidence that does not support the null hypothesis. The Hausman test operates by finding the distribution of the quantity (
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10 β GIVE - β OLS
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