Suppose we estimate the model y i u i , where u i N 0 , i 2 (a) Show that the OLS estimator of simplifies to y (b) Hence directly obtain the variance of y Show that this equals White 's heteroskedastic consistent estimate of the variance given in (4 21) VIBOLS (X'X) 'X'X(X'X) N N N (xx) ' (x)' i 1 (4 21)

Question

Suppose we estimate the model y i      u i , where u i  N   0 ,  i 2     (a) Show that the OLS estimator of  simplifies to      y    (b) Hence directly obtain the variance of y    Show that this equals White 's heteroskedastic consistent estimate of the variance given in (4 21)  VIBOLS    (X'X) 'X'X(X'X) N N N   (xx) '  (x)'  i 1 (4 21)

SolutionInn · Accepted Answer

a Show that the OLS estimator of simplifies to y In this modelwe have yi ui ui N0 i2 The OLS estimator for minimizes the sum of squared residuals min yi 2 Taking the derivative with respect to and setting it equal to zero 2 yi 0 Simplifying yi N Where N is the number of observationsThereforethe OLS estimator for is yi N y b Obtain the variance of y and show it equals Whites heteroskedastic consistent estimate The variance of y is given by Vary Varyi N 1N2 Varyi Since yi ui and ui N0i2we have Varyi Var ui Varui i2 Thereforethe variance of y becomes Vary 1N2 i2 Nowlets look at Whites heteroskedastic consistent estimate given in 421 VOLS XX1 ui2 Xi Xi XX1 In our simple modelX is a vector of ones since we only have an intercept termThereforeXX N and XX1 1N Substituting these values into the Whites estimate VOLS 1N ui2 1N 1N2 ui2 Since ui is the residual yi ywe have ui2 yi y2 i2 assuming the error terms are homoscedastic ThereforeWhites estimate simplifies to VOLS 1N2 i2 This is exactly the same as the variance of y we derived earlier Hence we have shown that the variance of y equals Whites heteroskedastic consistent estimate of the variance in this simple model Explanation of the Answer Part a OLS Estimator of Simplifies to y Model We start with a simple linear model yi ui where ui is the error term following a normal distribution with mean 0 and variance i2 OLS Estimator The Ordinary Least Squares OLS method finds the value of that minimizes the sum of squared residuals the differences between the observed yi and the predicted value Minimization To find this minimum we take the derivative of the sum of squared residuals with respect to and set ...

Question: Suppose we estimate the model y i = + u i , where u i N [ 0 , i 2 ]

Step by Step Solution

Students Have Also Explored These Related Microeconomics Questions!