4-16 Statistical Distributions for Diagnostic Tests Often, an F-and a -version of the test are available The F-test version (Wald test) involves estimating a restricted and an unrestricted version of a test regression and comparing the rss. The x-version is sometimes called an"LM test, and only has one degree of freedom parameter the number of restrictions being tested, m. Asymptotically, the 2 tests are equivalent since the z is a special case of the F-distribution: x n >F(m, T-k)as T-k For small samples, the F-version is preferable
4-16 Statistical Distributions for Diagnostic Tests • Often, an F- and a 2 - version of the test are available. • The F-test version(Wald test) involves estimating a restricted and an unrestricted version of a test regression and comparing the RSS. • The 2 - version is sometimes called an “LM” test, and only has one degree of freedom parameter: the number of restrictions being tested, m. • Asymptotically, the 2 tests are equivalent since the 2 is a special case of the F-distribution: • For small samples, the F-version is preferable. ( ) → F(m T − k) T − k → m m , as 2
4-17 5Assumption 1: E(u,=0 Assumption that the mean of the disturbances is zero For all diagnostic tests, we cannot observe the disturbances and so perform the tests of the residuals. The mean of the residuals will always be zero provided that there is a constant term in the regression 没有常数项时,可能导致斜率估计值出现偏差。 R2可能是负的
4-17 5 Assumption 1: E(ut ) = 0 • Assumption that the mean of the disturbances is zero. • For all diagnostic tests, we cannot observe the disturbances and so perform the tests of the residuals. • The mean of the residuals will always be zero provided that there is a constant term in the regression. • 没有常数项时，可能导致斜率估计值出现偏差。 • R2 可能是负的
4-18 6 Assumption 2: Var(u,=0<oo We have so far assumed that the variance of the errors is constant, o- this is known as homoscedasticity. If the errors do not have a constant variance, we say that they are heteroscedastic e.g. say we estimate a regression and l1+ calculate the residuals, i
4-18 6 Assumption 2: Var(ut ) = 2 < • We have so far assumed that the variance of the errors is constant, 2 - this is known as homoscedasticity. If the errors do not have a constant variance, we say that they are heteroscedastic e.g. say we estimate a regression and calculate the residuals, u $ .t t uˆ + - t x2
4-19 Detection of Heteroscedasticity Graphical methods Formal tests One of the best is white's general test for heteroscedasticity The test is carried out as follows: Assume that the regression we carried out is as follows r=B1+B2x2x+B33+ And we want to test Var(u=0. We estimate the model, obtaining the residuals, ut 2. Then run the auxiliary regression C1+2x2+ay3x3+a421+0x3+6x2x3t+v
4-19 Detection of Heteroscedasticity • Graphical methods • Formal tests: One of the best is White’s general test for heteroscedasticity. • The test is carried out as follows: 1. Assume that the regression we carried out is as follows yt = 1 + 2x2t + 3x3t + ut And we want to test Var(ut ) = 2 . We estimate the model, obtaining the residuals, 2. Then run the auxiliary regression u $ t t t t t t t t t u = + x + x + x + x + x x + v 6 2 3 2 5 3 2 1 2 2 3 3 4 2 2 ˆ
4-20 Performing White's Test for Heteroscedasticity 3. Obtain R from the auxiliary regression and multiply it by the number of observations .T. it can be shown that TR(m) where m is the number of regressors in the auxillary regression excluding the constant term (还可以对辅助回归方程做F检验) If the x test statistic from step 3 is greater than the corresponding value from the statistical table then reject the null hypothesis that the disturbances are homoscedastic. 例子:p150
4-20 Performing White’s Test for Heteroscedasticity 3. Obtain R2 from the auxiliary regression and multiply it by the number of observations, T. It can be shown that T R2 2 (m) where m is the number of regressors in the auxiliary regression excluding the constant term. (还可以对辅助回归方程做F 检验) 4. If the 2 test statistic from step 3 is greater than the corresponding value from the statistical table then reject the null hypothesis that the disturbances are homoscedastic. • 例子：p150