Econometrics

1
Econometrics

• Lecture Notes Hayashi, Chapter 2c
• Large Sample Theory

2
Assumption

• DGP (Data Generating Process) A stochastic
  process that generated the finite sample (y,X)
  must satisfies
• Assumption 1 Linearityyi xib ei
  (i1,2,,n)

3
Assumption

• Assumption 2 Ergodic Stationarityyi,xi is
  jointly stationary and ergodic
• yi,xi is asymptotic independent.
• ei yi-xib is ergodic stationary.
• E(xi xi) limn?? ?i xixi/n SxxE(xiyi)
  limn?? ?i xiyi/n Sxy
• The stationarity implies unconditional
  homoscedasticity, but allows for conditional
  heteroscedasticity.

4
Assumption

• Assumption 3 Predetermined RegressorsE(gi) 0,
  where gi xiei xi(yi-xib)or, E(xikei) 0
  for all i,k1,2,K.
• We do not assume E(eiX) 0 or E(eixi) 0,
  although E(eiX) 0 ? E(eixi) 0 ? E(xikei)
  0. Therefore, it is possible that E(xjkei) ? 0
  for some j,k and j?i.

5
Assumption

• Assumption 4 Rank ConditionSxx E(xixi) is
  nonsingular.
• Let Sxx ?i xixi/n XX/n
• limn?? Sxx S xx by Erogodic Theorem.
• Therefore Sxx is nonsingular (no multicolinearity
  in the limit).

6
Assumption

• Assumption 5 gi is a martingale difference
  sequence with finite second moments. That is,
  E(gi) 0 and S E(gigi) is nonsigular. CLT
  states that

7
Assumption

• Assumption 5 (Continued)
• By definition gi xiei, then S E(ei2xixi).
• If xi11 (constant), then gi1 ei. Because
  E(gigi-1,gi-2,,g1) 0 for i?2, we have
  E(eiei-1,ei-2,,e1) 0. Therefore, ei is an
  m.d.s and hence is serially uncorrelated.

8
Assumptions

• Assumption 6 Finite 4th Moments for Regressors.
  That is, E(xikxij)2 exists and is finite for
  all k,j (1,2,,K).

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Estimating S E(ei2xixi)

• Proposition 4 Suppose SE(ei2xixi) exists and
  is finite. Under A.1-2, and A.6,

10
Estimating S E(ei2xixi)

• Alternative Forms for the Estimator of S
• Degree of freedom correction
• Let pi xi(XX)-1xi xiSxx-1xi/n, then

11
Large Sample Theory

• Proposition 1 Asymptotic Distribution of the OLS
  Estimator
• Consistency Under A.1-4, plimn??b b.
• Asymptotic Normality Under A.1-5,where
  Avar(b) Sxx-1SSxx-1, Sxx E(xixi), S
  E(ei2xixi)

12
Large Sample Theory

• Proposition 1 (Continued)
• Consistent Estimate of Avar(b) Since b is
  consistent, the consistent estimator of S is
• Then , under A.2, Avar(b) is consistently
  estimated by where Sxx ?ixixi/n
• Denote Est(Avar(b))

13
Large Sample Theory

• Proposition 2 (Consistent Estimation of Error
  Variance) Let ei yi-xib be the OLS residual
  for observation i. Under A.1-4,provided
  E(ei2) exists and is finite.

14
Large Sample Theory

• Proposition 2 (Continued)
• Write
• We can prove ??iei2/n ?p E(ei2), therefores2 ?p
  E(ei2).
• If some consistent estimator, rather than the OLS
  estimator b, is used to form the residuals, the
  error variance estimator is still consistent for
  E(ei2).

15
Conditional Homoscedasticity

• Assumption 7 E(ei2xi) s2 gt 0
• S E(ei2xixi) EE(xixiei2xi)
  ExixiE(ei2xi) s2Exixi
• Avar(b) Exixi-1s2ExixiExixi-1
• ?ixixi/n (XX/n)-1 ?p Exixi
• s2?ixixi/n s2XX/n ?p s2ExixiS
• Est(Avar(b)) s2(XX/n)-1

16
Conditional Homoscedasticity

• Proposition 5 Suppose A.1-5 and 7 are satisfied.
  Then
• ?n(b-b) ?d N(0,Avar(b))
• Est(Avar(b)) s2(XX/n)-1 ?p Avar(b)