A1 The Sargan Test For The Validity Of Instruments

Suppose we use an instrumental variable(s) to replace an explanatory variable^) that is correlated with the error term. How valid is the instrumental variable(s), that is, how do we know that the instruments chosen are independent of the error term? Sargan has developed a statistic, dubbed SARG, to test the validity of the instruments used in instrumental variable(s) (IV).t The steps involved in SARG are as follows*:

1. Divide the variables included in a regression equation into two groups, those that are independent of the error term (say, X1, X2, . . . , Xp) and those that are not independent of the error term (say, Z1, Z2, . . . , Zq).

2. Let W1, W2, . . . , Ws be the instruments chosen for the Z variables in 1, where s > q.

3. Estimate the original regression, replacing the Z's by the W's, that is, estimate the original regression by IV and obtain the residuals, say, U.

4. Regress U on a constant, all the X variables and all the W variables but exclude all the Z variables. Obtain R2 from this regression.

5. Now compute the SARG statistic, defined as:

where n = the number of observations and k = the number of coefficients in the original regression equation. Sargent has shown that (17A.1.1) follows the x2 distribution with r df, where r = s - q.

6. The null hypothesis is that all (w) instruments are valid. If the computed chi-square exceeds the critical chi-square value, we reject the null hypothesis, which means that at least one instrument is correlated with the error term and therefore the IV estimates based on the chosen instruments are not valid.

The choice between Granger and Sims causality tests is not clear. For further discussion of these tests, see G. Chamberlain, "The General Equivalence of Granger and Sims Causality," Econometrica, vol. 50, 1982, pp. 569-582.

^Sargan, J. D., "Wages and Prices in the United Kingdom: A Study in Econometric Methodology," in P. E. Hart, G. Mills, and J. K. Whitaker (eds.) Econometric Analysis for National Economic Planning, Butterworths, London, 1964.

*The following discussion leans on H. R. Seddighi, K. A. Lawler and A. V. Katos, Econometrics: A Practical Approach, Routledge, New York, 2000, pp. 155-156.

part FOUR

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