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Intercept-Only Model .

.13 Covariance Matrix Estimation Under Homoskedasticity For inference, we need an estimator of the covariance matrix V b of the least-squares estimator. In this section we consider the homoskedastic regression model (Assumption 4.3). Under homoskedasticity, the covariance matrix takes the relatively simple form V 0 b = X0X 1  2 which is known up to the unknown scale  2 . In Section 4.11 we discussed three estimators of  2 : The most commonly used choice is s 2 ; leading to the classic covariance matrix estimator Vb 0 b = X0X 1 s 2 : (4.30) Since s 2 is conditionally unbiased for  2 , it is simple to calculate that Vb 0 b is conditionally unbiased for V b under the assumption of homoskedasticity: E  Vb 0 b j X  = X0X 1 E s 2 j X  = X0X 1  2 = V b: This was the dominant covariance matrix estimator in applied econometrics for many years, and is still the default method in most regression packages. For example, Stata uses the covariance matrix estimator (4.30) by default in linear regression unless an alternative is speciÖed. If the estimator (4.30) is used, but the regression error is heteroskedastic, it is possible for Vb 0 b to be quite biased for the correct covariance matrix V b = (X0X) 1 (X0DX) (X0X) 1 : For example, suppose k = 1 and  2 i = x 2 i with E (xi) = 0: The ratio of the true variance of the least-squares estimator to the expectation of the variance estimator is V b E  Vb 0 b j X  = Pn i=1 x 4 i  2 Pn i=1 x 2 i ‘ E x 4 i  E x 2 i 2 def = : (Notice that we use the fact that  2 i = x 2 i implies  2 = E  2 i  = E x 2 i  🙂 The constant  is the standardized fourth moment (or kurtosis) of the regressor xi ; and can be any number greater than one. For example, if xi  N 0; 2  then  = 3; so the true variance V b is three times larger than the expected homoskedastic estimator Vb 0 b. But  can be much larger. CHAPTER 4. LEAST SQUARES REGRESSION 118 that xi   2 1 1: In this case  = 15; so that the true variance V b is Öfteen times larger than the expected homoskedastic estimator Vb 0 b. While this is an extreme and constructed example, the point is that the classic covariance matrix estimator (4.30) may be quite biased when the homoskedasticity assumption fail