Nonspherical Disturbances

1
Chapter 10

• Nonspherical Disturbances

2
Overview Part 2

• 10.3 Robust estimation of asymptotic covariance
  matrices
• Heteroscedastic
• Autocorrelated
• 10.5 Efficient estimation by generalized least
  squares estimate (GLS)
• GLS
• FGLS

3
Estimation with Nonspherical disturbances

• Discard OLS?
• 3 cases
• ? is completely unknown
• Use OLS or IV (lack of alternatives)
• Estimate the asymptotic covariance matrix of b
  (10.3)
• ? is known
• Simple and efficient estimate (10.5)
• ? is unknown but its structure is know
• we can estimate ? using sample info (also in
  10.5) then tread as if it was known
• Often favored over OLS

4
Robust est of asy cov matrices(II)

• Consider the first case ?2? is completely
  unknown
• If ?2? were known
• Est of asy cov matrix

5
Robust est of asy cov matrices(III)

• Nonlinear LSE
• IV Estimate

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Robust est of asy cov matrices(IV)

• Let
• Estimate by
• LS b is consistent for ?
• ei point wise consistent for ?i ? i
• ?Use X and e to create an estimator of Q

7
Robust est of asy cov matrices-Heteroscedasticity(
I)

• Heteroscedasticity Case
• Estimated by
• Under very general conditions
• Not an estimate but a function of sample data
  close to that of the actual outcomes for large
  n!!

8
Robust est of asy cov matrices-Heteroscedasticity(
II)

• By Thrms D2-4 (LLN), if Q has a plim
• consistency of b for ? justifies the replacement
  of ei with ?i in S0

9
Robust est of asy cov matrices-Heteroscedasticity(
III)

• Final result White heteroscedasticity consistent
  estimator
• Especially important if nature of
  heteroscedasticity is unknown (most of the time)
• Examples in Chapter 11!

10
Robust est of asy cov matrices- Autocorrelation(I)

• Autocorrelation case
• Natural counterpart to White
• Estimate
• With
• But there are a few problems!!

11
Robust est of asy cov matrices-
Autocorrelation(II)

• Problems
• is n-1 times the sum of n2 terms (different
  from heteroscedasticity)
• needs to be positive definite
• Newey and West came up with an estimator that
  solves this problem

12
Robust est of asy cov matrices-
Autocorrelation(III)

• Newely-West autocorrelation consistent covariance
  estimator
• But how large must L be?

13
Hypothesis Testing

• OLS and IV are asymptotically normal and
  asymptotic covariance matrix is estimated
• But distribution of the disturbances is not
  specified
• F-stat is approximate at best
• Likelihood ratio and Lagrange multiplier tests
  are unavailable
• Wald stat with asymptotic t ratios is main
  inference tool
• Applications in the next few chapter

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Efficient estimation by GLS(I)

• Ass ? is known, symmetric, positive definite
• b/c pd it can be factored into
• Let
• Then
• Premultiply the general linear regression model
  by P

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Efficient estimation by GLS(II)

• Transformed GLRM
• y and X are observed data, so CLRM applies to
  them!
• ?OLS is efficient

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Efficient estimation by GLS(III)

• generalized least squares(GLS) or Aitken
  estimator

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Efficient estimation by GLS(IV)

• Assume
• the disturbances and regressors are uncorrelated
• the transformed data X is well behaved
• By Aikens theorem GLS is
• Efficient
• Consistent
• Asymptotically normal
• MVLUE
• Gauss-Markov theorem is special case where ?I
• CLRM inference tools can be applied to the
  transformed model, but not R2

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Efficient estimation by GLS(V)

• F-test

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FGLS(I)

• ? is a n?n matrix containing unknown parameters
• cannot be estimated using the sample
• GLS is not feasable
• Typically there is a small set of parameters
• e.G. time series

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FGLS(II)

• Let
• Estimate by where
• is asymptotically equivalent to
• use this to obtain the feasible generalized least
  squares
• Same asymptotic properties as GLS
• But in most cases no finite sample properties!!

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The End

• Questions?