Econometrics-A Modern Introduction (17)

Review of Standard Errors (cont.)
• To derive the variance of a linear estimator, we need to add the two assumptions about the variance and covariances of the error term:
ei s n k 1
2 2
Review of Standard Errors (cont.)
• We can estimate the STANDARD ERROR of our OLS estimator (the cross-sample standard deviation).
ei (n k 1) ˆ e.s.e.( ) 2 ( X i X )
Gauss朚 arkov DGP Yi 0 1 X 1i i E ( i ) 0 Var ( i )
2
Cov( i , j ) 0, if i j X 's fixed across samples.
Review of Standard Errors (cont.)
• The model includes a stochastic component, the error term. • There are many possible realizations of the error term, but we get to observe only one. • The underlying process plus the particular realization of the error termour sample.
Gauss朚 arkov DGP Yi 0 1 X 1i i E ( i ) 0 X 's fixed across samples. These assumptions suffice to derive the expectation of a linear estimator: E (wiYi ) 0 wi 1wi X i
i 1 j i n
wi 2 Var (Yi ) wi 2 Var ( 0 1 X 1i i ) 2wi 2
Review of Standard Errors (cont.)
Variance of OLS with a Single Explanator Yi 0 1 X i i
OLS and Heteroskedasticity (Chapter 10.2)
• What are the implications of heteroskedasticity for OLS?
• Under the Gauss–Markov assumptions (including homoskedasticity), OLS was the Best Linear Unbiased Estimator. • Under heteroskedasticity, is OLS still Unbiased? • Is OLS still Best?
2
Review of Standard Errors (cont.)
• Estimated standard errors are used to calculate t-statistics, used for testing single hypotheses.
* ˆ ˆ t ˆ e.s.e.( )
Review of Standard Errors (cont.)
• Using our model and our sample, we can estimate NOT ONLY the underlying parameters BUT ALSO the crosssample distribution of our estimates.
2 2 ˆ Var ( 1 ) w ˆ
1i
(Xi X ) wˆ i 1 ( X j X ) 2
Review of Standard Errors (cont.)
Variance of OLS with a Single Explanator (Xi X ) 2 2 2 ˆ Var ( 1 ) wi ( X j X ) 2
Review of Standard Errors (cont.)
• We observe only one sample.
• Our estimator (OLS) provides a rule for going from one sample to one best guess of the underlying parameters. • If we could observe many samples, we could estimate many best guesses. • If we could observe many samples, we would have a distribution of estimates.
Heteroskedasticity (Chapter 10.1)
• HETEROSKEDASTICITY
– The variance of i is NOT a constant 2. – The variance of i is greater for some observations than for others.
• Wealthy consumers can choose to spend a lot of money on rent, or to spend less, depending on tastes. Var(i ) is high.
Figure 10.1 Rents and Incomes for a Sample of New Yorkers
Var ( i )
2
Cov( i , j ) 0, if i j
Review of Standard Errors (cont.)
Variance of a Linear Estimator ˆ ) Var (wY ) Var ( i i Var ( wY i i ) cov( wY i i , w jY j )
2
( X j X ) 2

2
xi2
Review of Standard Errors (cont.)
• Problem: we do not know 2
• Solution: estimate 2 • We do not observe the ACTUAL error terms, i • We DO observe the residual, ei
OLS and Heteroskedasticity (cont.)
• Is OLS still unbiased under this DGP? E ( wiYi ) E ( wiYi ) wi E (Yi )
wi ( 0 1 X1i .. k X ki ) E ( wiYi ) 1 if wi 0, wi X1i 1, wi X 2 i 0...wi X ki 0
OLS and Heteroskedasticity (cont.)
• A DGP with Heteroskedasticity
Yi 0 1 X 1i ... k X ki i E ( i ) 0 Var ( i )
2 i
Cov( i , j ) 0 for i j X 抯fixed across samples
• Estimating the cross-sample distribution lets us judge how likely our best guess is to be close to the real underlying parameters.
Review of Standard Errors (cont.)
C.I .
%
nk 1 垐 t (1 ) e.s.e.( ) 2
Review of Standard Errors (cont.)
• Our formula for Estimated Standard Errors relied on ALL the Gauss–Markov DGP assumptions. • For this lecture, we will focus on the assumption of homoskedasticity. • What happens if we relax the 2 assumption that Var(i ) ?
Lecture 13 Heteroskedasticity
(Chapter 10.1–10.4, 10.7)
Agenda for Today
• Review Standard Errors (Chapter 5.2)
• Heteroskedasticity (Chapter 10.1) • OLS and Heteroskedasticity (Chapter 10.2) • Tests for Heteroskedasticity (Chapter 10.3)
Var(i )
2 i
Heteroskedasticity (cont.)
• For example, consider a regression of housing expenditures on income. Renti 0 1 Incomei i
• Consumers with low values of income have little scope for varying their rent expenditures. Var(i ) is low.