Economics 326 Methods of Empirical Research in Economics. Lecture 18: The asymptotic variance of OLS and heteroskedasticity

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1 Ecoomics 326 Methods of Empirical Research i Ecoomics Lecture 8: The asymptotic variace of OLS ad heteroskedasticity Hiro Kasahara Uiversity of British Columbia December 24, 204

2 Asymptotic ormality I I the previous lecture, we showed that whe the data are iid ad the regressors are exogeous: Y i = β 0 + β X i + U i, EU i = E (X i U i ) = 0, the OLS estimator of β is asymptotically ormal: p ˆβ, β!d N (0, V ), V = E (X i EX i ) 2 Ui 2 (Var (X i )) 2. I For the purpose of hypothesis testig, we eed to obtai a cosistet estimator of the asymptotic variace V : ˆV! p V. /4

3 Homoskedastic errors I Let s assume that the errors are homoskedastic: E U 2 i jx i = σ 2 for all X i s. I I this case, the asymptotic variace ca be simpli ed usig the Law of Iterated Expectatio: i E (X i EX i ) 2 Ui 2 = EE h(x i EX i ) 2 Ui 2 jx i = E (X i EX i ) 2 E Ui 2 jx i = E (X i EX i ) 2 σ 2 = σ 2 E (X i EX i ) 2 = σ 2 Var (X i ). 2/4

4 Homoskedastic errors I Thus, whe the errors are homoskedastic with EUi 2 = σ 2, V = E (X i EX i ) 2 Ui 2 (Var (X i )) 2 = σ2 Var (X i ) (Var (X i )) 2 = σ 2 Var (X i ). I Let Û i = Y i ˆβ 0, ˆβ, X i, where ˆβ 0, ad ˆβ, are the OLS estimators of β 0 ad β. I A cosistet estimator for the asymptotic variace ca be costructed by usig the Method of Momets. ˆσ 2 = dvar (X i ) = i= Û 2 i, (X i X ) 2, ad i= ˆV = ˆσ 2 i= (X i X ) 2. 3/4

5 Homoskedastic errors ˆV = ˆσ 2 i= (X i X ) 2, ˆσ2 = Ûi 2, Û i = Y i ˆβ 0, ˆβ, X i. i= I Whe provig the cosistecy of OLS i Lecture 6, we showed that (X i X ) 2! p Var (X i ), i= ad to establish ˆV! p V, we eed to show that ˆσ 2! p σ 2. I Note that the LLN caot be applied directly to Ûi 2 i= because Û i s are ot iid: they are depedet through ˆβ 0, ad ˆβ,. 4/4

6 Proof of ˆσ 2! p σ 2 I First, write Û i = Y i ˆβ 0, ˆβ, X i = (β 0 + β X i + U i ) ˆβ 0, ˆβ, X i = U i ˆβ 0, β 0 ˆβ, β Xi. I Now, ˆσ 2 = i= Û 2 i = i= U i ˆβ 0, β 0 ˆβ, β Xi 2. 5/4

7 Proof of ˆσ 2! p σ 2 I We have ˆσ 2 = = i= i= U i ˆβ 0, β 0 ˆβ, β Xi 2 Ui 2 + ˆβ 0, β ˆβ, β 2 2 ˆβ 0, β 0 i= + 2 ˆβ 0, β 0 ˆβ, β U i 2 ˆβ, β i= X i. i= i= X 2 i U i X i I By the LLN, i= U 2 i! p EU 2 i = σ 2. I Because ˆβ 0, ad ˆβ, are cosistet, ˆβ 0, β 0! p 0 ad ˆβ, β! p 0. 6/4

8 Homoskedastic errors I Thus, whe the errors are homoskedastic, ˆV = ˆσ 2 i= (X i is a cosistet estimator of V = I Note that s 2 = 2 ad therefore ˆV = X ) 2, with ˆσ2 = σ2 Var (X i ). Ûi 2! p σ 2, i= s 2 i= (X i X ) 2 is also a cosistet estimator of V = σ2 Var (X i ). Ûi 2, i= I This versio has a advatage over the oe with ˆσ 2 : i additio to beig cosistet, s 2 is also a ubiased estimator of σ 2 if the regressors are strogly exogeous. 7/4

9 Homoskedastic errors: Asymptotic approximatio I Recall that p ˆβ, β!d N (0, V ) is used as the followig approximatio: a ˆβ, N β, V, where a deotes approximately i large samples. Thus, the variace of ˆβ, ca be take as approximately V /. I s Note that, with ˆV = 2 we have i=(x i X ) 2 ˆV = s 2 i= (X i X ) 2 = s 2 i= (X i X ) 2. 8/4

10 Homoskedastic errors: Asymptotic approximatio ˆV = s 2 i= (X i X ) 2 I Thus, i the case of homoskedastic errors we have the followig asymptotic approximatio:! s 2 a ˆβ, N β, i= (X i X ) 2 I I ite samples, we have the same result exactly, whe the regressors are strogly exogeous ad the errors are ormal.. 9/4

11 Asymptotic T-test I Cosider testig H 0 : β = β,0 vs H : β 6= β,0. I Cosider the behavior of T statistic uder H 0 : β = β,0. Sice p ˆβ, β!d N (0, V ) ad ˆV! p V, we have that T = ˆβ, β,0 p ˆV / = p ˆβ, β,0 p ˆV p H = 0 ˆβ, β p ˆV N (0, V )! d p = d N (0, ). V 0/4

12 Asymptotic T-test I We have that uder H 0 : β = β,0, T = ˆβ, β,0 p ˆV /! d N (0, ), provided that ˆV! p V (the asymptotic variace of ˆβ, ). I A asymptotic size α test rejects H 0 : β = β,0 agaist H : β 6= β,0 whe where z jt j > z α/2, α/2 is a stadard ormal critical value. I Asymptotically, the variace of the OLS estimator is kow - we behave as if the variace was kow. /4

13 Heteroskedastic errors I I geeral, the errors are heteroskedastic: E Ui 2 jx i is ot costat ad chages with X i. I s I this case, ˆV = 2 is ot a cosistet estimator i=(x i X ) 2 of the asymptotic variace V = E((X i EX i ) 2 Ui 2 ) : (Var (X i )) 2 s 2 i= (X i X ) 2! p EU 2 i Var (X i ) E (X i EX i ) 2 EU 2 i = (Var (X i )) 2 6= E (X i EX i ) 2 Ui 2 (Var (X i )) 2. 2/4

14 A heteroskedasticity cosistet (HC) estimator of the asymptotic variace of OLS I I the case of heteroskedastic errors, a cosistet estimator of V = E((X i EX i ) 2 Ui 2 ) ca be costructed as follows: (Var (X i )) 2 ˆV HC = i= (X i X ) 2 Ûi 2 i= (X i X ) 2 2. I Oe ca show that ˆV HC! p V whe the errors are heteroskedastic or homoskedastic. I We have the followig asymptotic approximatio: a N β,, ˆβ, ˆV HC ad the stadard errors ca be computed as q SE ˆβ, = ˆV HC /. 3/4

15 HC variace estimatio i Stata I I Stata, the HC estimator of stadard errors ca be obtaied by addig the optio robust to the regressio commad:. regress liver alcohol, robust Robust liver Coef. Std. Err. t P> t [95% Cof. Iterval] alcohol _cos I Compare with the o-hc stadard errors based o ˆV :. regress liver alcohol liver Coef. Std. Err. t P> t [95% Cof. Iterval] alcohol _cos /4