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Perform Pagan-Hall and related heteroskedasticity tests after IV and OLS estima> tion

ivhettest[varlist] [,ivlevivsqivcpfitlevfitsqphphnormnr2bpgall]

ivhettestis for use after ivreg, ivreg2 and regress.Weights are not compatible with the current implementation of

ivhettest.

Description

ivhettestperforms various flavors of Pagan and Hall's (1983) test of heteroskedasticity for instrumental variables (IV) estimation. It will also perform the related standard heteroskedasticity tests of Breusch-Pagan/Godfrey/Cook-Weisberg and White/Koenker after estimation by OLS or IV. These are all tests of whether there is heteroskedasticity in the estimated regression that is related to one or more indicator variables. The test flavors vary according to the choice of indicator variables and the choice of test statistics. Under the null of no heteroskedasticity, the test statistic is distributed as chi-sq with degrees of freedom = number of indicator variables.

Options

ivlevrequests that the set of indicator variables is the full set of instruments (exogenous variables) used in the original regression (excluding the constant). This is the default choice of indicator variables. In this and other options, if the original regression was OLS usingregress, the instruments are the regressors.

ivsqrequests that the set of indicator variables is the full set of instruments (excluding the constant) and their squares.

ivcprequests that the set of indicator variables is the full set of instruments (excluding the constant), their squares and cross-products.

fitlevrequests that the indicator variable is the "fitted value" of the dependent variable. In the case of IV regression, "fitted values" arenotcalculated directly from the regressors and the estimated coefficients. Rather, the endogenous regressors are first replaced with their "first stage" fitted values, i.e., with fitted values from regression of the endogenous regressors on the full set of instruments. The exogenous regressors, the fitted values of the endogenous regressors, and the estimated coefficients from the main regression are then used to calculate the "fitted values" of the dependent variable.

fitsqrequests that the indicator variables are the "fitted value" of the dependent variable and its square.A user-defined set of indicator variables can be specified in

varlist.If

varlist,ivlev,ivsq,ivcp,fitlevandfitsqare all omitted, the default set of indicator variables is the full set of instruments (excluding the constant) only.

phrequests Pagan and Hall's (1983) general test statistic for heteroskedasticity in an IV regression. This statistic is the default for regressions estimated by ivreg and ivreg2.

phnormrequests Pagan and Hall's (1983) test statistic for heteroskedasticity in an IV regression when the error term in the IV regression is in addition assumed to be normally distributed.

nr2requests the standard White/Koenker nR-sq test statistic for heteroskedasticity. For an IV regression, this test statistic will be distributed as chi-sq under the null only if the system covariance matrix is homoskedastic. This statistic is the default for regressions estimated by regress.

bpgrequests the standard Breusch-Pagan/Godfrey/Cook-Weisberg test statistic for heteroskedasticity in a linear regression when the error term is assumed to be normally distributed. For an IV regression, this test statistic will be distributed as chi-sq under the null only if the system covariance matrix is homoskedastic.

allrequests all applicable statistics be reported: White/Koenker, Breusch-Pagan/Godfrey/Cook-Weisberg, and, if the regression is estimated by IV, the two versions of the Pagan-Hall statistic.

RemarksThe Breusch-Pagan/Godfrey/Cook-Weisberg and White/Koenker statistics are standard tests of the presence of heteroskedasticity in an OLS regression. The principle is to test for a relationship between the residuals of the regression and indicator variables that are hypothesized to be related to the heteroskedasticity. Breusch & Pagan (1979), Godfrey (1978), and Cook & Weisberg (1983) separately derived the same test statistic. This statistic is distributed as chi-squared under the null of no heteroskedasticity, and under the maintained hypothesis that the error of the regression is normally distributed. Koenker (1981) showed that when the assumption of normality is removed, a version of the test is available that can be calculated as the sample size n times the centered R-sq from an artificial regression of the squared residuals from the original regression on the indicator variables. The degrees of freedom of all these chi-sq tests are equal to the number of indicator variables. When the indicator variables are the regressors, their squares and their cross-products, Koenker's test is identical to White's (1980) nR-sq general test for heteroskedasticity. These tests are available using

ivhettestas well as via hettest and whitetst.As Pagan and Hall (1983) point out, these tests will be valid tests for heteroskedasticity in an IV regression only if heteroskedasticity is present in that equation

and nowhere else in the system. They derive a test which relaxes this requirement. Under the null of homoskedasticity in the IV regression, the Pagan-Hall statistic is distributed as chi-sq with degrees of freedom equal to the number of indicator variables, irrespective of the presence of heteroskedasticity elsewhere in the system.The trade-off in the choice of indicator variables in all these tests is that a smaller set of indicator variables will conserve on degrees of freedom, at the cost of being unable to detect heteroskedasticity in certain directions. When the above tests are applied to an IV regression, the indicator variables must be functions of exogenous variables (instruments) only. If, e.g., the indicator variable is the fitted value of the dependent variable, this prediction is calculated as a linear combination of the instruments alone as described above.

A full discussion of these computations and related topics can be found in Baum, Schaffer, and Stillman (2003).

Saved results

ivhettestsaves the value of the test statistics, their p-values, and the degrees of freedom of the test. Seereturn list.

Examples(IV) . ivreg y x1 x2 x3 (y2 = z1 z2 z3) . ivhettest . ivhettest x1, ph nr2 . ivhettest, fitsq all

(OLS) . regress y x1 x2 x3 . ivhettest . ivhettest, ivcp bpg

ReferencesBaum, C. F., Schaffer, M. E., Stillman, S., 2003. Instrumental variables and GMM: Estimation and testing. The Stata Journal, Vol. 3, No. 1, pp. 1-31. Unpublished working paper version: Boston College Department of Economics Working Paper No. 545. http://fmwww.bc.edu/ec-p/WP545.pdf

Breusch, T.S., and Pagan, A.R. "A Simple Test for Heteroskedasticity and Random Coefficient Variation." Econometrica, Vol. 47, 1979, pp. 1287-1294.

Cook, R.D., and Weisberg, S. "Diagnostics for Heteroscedasticity in Regression." Biometrika, Vol. 70, 1983, pp. 1-10.

Godfrey, L.G. "Testing for Multiplicative Heteroskedasticity." Journal of Econometrics, Vol. 8, 1978, pp. 227-236.

Koenker, R. "A Note on Studentizing a Test for Heteroskedasticity." Journal of Econometrics, Vol. 17, 1981, pp. 107-112.

Pagan, A.R. and Hall, D. "Diagnostic Tests as Residual Analysis." Econometric Reviews, Vol. 2, No. 2, 1983, pp. 159-218.

White, H. "A Heteroskedasticity-consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity." Econometrica, Vol. 48, 1980, pp. 817-838.

AuthorMark E Schaffer, Heriot-Watt University, UK m.e.schaffer@hw.ac.uk

Also seeManual:

[R] ivreg,[R] regression diagnosticsOn-line: help for ivreg; ivreg2 (if installed)