Multiple regression analysis y-Bo+Bx+Bx2+... Bkk+u 23 Asymptotic Properties Economics 20- Prof anderson
Economics 20 - Prof. Anderson 1 Multiple Regression Analysis y = b0 + b1 x1 + b2 x2 + . . . bk xk + u 3. Asymptotic Properties
Consistency o Under the Gauss-Markov assumptionS OLS IS BLUE, but in other cases it wont al ways be possible to find unbiased estimators In those cases, we may settle for estimators that are consistent, meaning as n>o0, the distribution of the estimator collapses to the parameter value Economics 20- Prof anderson
Economics 20 - Prof. Anderson 2 Consistency Under the Gauss-Markov assumptions OLS is BLUE, but in other cases it won’t always be possible to find unbiased estimators In those cases, we may settle for estimators that are consistent, meaning as n → ∞, the distribution of the estimator collapses to the parameter value
Sampling Distributions as n t ni <n<n 2 Bi Economics 20- Prof anderson
Economics 20 - Prof. Anderson 3 Sampling Distributions as n b1 n1 n2 n3 n1 < n2 < n3
Consistency of ols o Under the Gauss-Markov assumptions, the OLS estimator is consistent(and unbiased) o Consistency can be proved for the simple regression case in a manner similar to the proof of unbiasedness o Will need to take probability limit(plim)to establish consistency Economics 20- Prof anderson 4
Economics 20 - Prof. Anderson 4 Consistency of OLS Under the Gauss-Markov assumptions, the OLS estimator is consistent (and unbiased) Consistency can be proved for the simple regression case in a manner similar to the proof of unbiasedness Will need to take probability limit (plim) to establish consistency
Proving Consistency B=(x1-元/∑(x1-元 =B+n∑(x1-元x((-) plim B,=B,+Cov(x, u / var(x=B because Cov(xi, u)=0 Economics 20- Prof anderson 5
Economics 20 - Prof. Anderson 5 Proving Consistency ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) ( ) ( ) because ( , ) 0 , ˆ plim ˆ 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 = = + = = + − − = − − − − Cov x u Cov x u Var x n x x u n x x x x y x x i i i i i i b b b b b
A Weaker Assumption For unbiasedness. we assumed a zero conditional mean -E(ux x, ., xk=0 e For consistency, we can have the weaker assumption of zero mean and zero correlation -(u=0 and Cov(x ) =0, for 2...k o Without this assumption, Ols will be biased and inconsistent Economics 20- Prof anderson 6
Economics 20 - Prof. Anderson 6 A Weaker Assumption For unbiasedness, we assumed a zero conditional mean – E(u|x1 , x2 ,…,xk ) = 0 For consistency, we can have the weaker assumption of zero mean and zero correlation – E(u) = 0 and Cov(xj ,u) = 0, for j = 1, 2, …, k Without this assumption, OLS will be biased and inconsistent!
Deriving the Inconsistency Just as we could derive the omitted variable bias earlier. now we want to think about the inconsistency, or asymptotic bias, in this case True model: y=o+Bx,+B,x,+v You think y=Bo+B,x+u, so that u=B,x,+v and, plim B,=B,+,8 where=Cov(xi,x,) var(x) Economics 20- Prof anderson 7
Economics 20 - Prof. Anderson 7 Deriving the Inconsistency Just as we could derive the omitted variable bias earlier, now we want to think about the inconsistency, or asymptotic bias, in this case ( ) ( ) 1 2 1 2 2 1 1 2 0 1 1 0 1 1 2 2 where , ~ and, plim You think : ,so that True model : Cov x x Var x u x v y x u y x x v = = + = + = + + = + + + b b b b b b b b b
Asymptotic Bias(cont) o So, thinking about the direction of the asymptotic bias is just like thinking about the direction of bias for an omitted variable Main difference is that asymptotic bias uses the population variance and covariance, while bias uses the sample counterparts o Remember, inconsistency is a large sample problem-it doesnt go away as add data Economics 20- Prof anderson 8
Economics 20 - Prof. Anderson 8 Asymptotic Bias (cont) So, thinking about the direction of the asymptotic bias is just like thinking about the direction of bias for an omitted variable Main difference is that asymptotic bias uses the population variance and covariance, while bias uses the sample counterparts Remember, inconsistency is a large sample problem – it doesn’t go away as add data
Large Sample Inference Recall that under the clm assumptions the sampling distributions are normal, so we could derive t and F distributions for testing o This exact normality was due to assuming the population error distribution was normal o This assumption of normal errors implied that the distribution of y, given the xs, was normal as well Economics 20- Prof anderson 9
Economics 20 - Prof. Anderson 9 Large Sample Inference Recall that under the CLM assumptions, the sampling distributions are normal, so we could derive t and F distributions for testing This exact normality was due to assuming the population error distribution was normal This assumption of normal errors implied that the distribution of y, given the x’s, was normal as well
Large Sample inference(cont) Easy to come up with examples for which this exact normality assumption will fail e Any clearly skewed variable, like wages arrests, savings, etc. cant be normal, since a normal distribution is symmetric o Normality assumption not needed to conclude ols iS BLue, only for inference Economics 20- Prof anderson 10
Economics 20 - Prof. Anderson 10 Large Sample Inference (cont) Easy to come up with examples for which this exact normality assumption will fail Any clearly skewed variable, like wages, arrests, savings, etc. can’t be normal, since a normal distribution is symmetric Normality assumption not needed to conclude OLS is BLUE, only for inference