Econometrics

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Econometrics

• Lecture Notes Hayashi, Chapter 2d
• Hypothesis Testing

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Linear Hypothesis

• Statistical inference in large-sample theory is
  based on test statistics whose asymptotic
  distributions are known under the truth of null
  hypothesis.
• For OLS estimator b of b, a consistent estimator
  of S E(gigi) with gixiei, is

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Linear Hypothesis

• Probability of type I error is the probability of
  rejecting the null hypothesis when it is true.
• The finite-sample size of a test is the
  probability of type I error (actual or exact
  size), which equals the desired significance
  level a (nominal size).
• The exact size and nominal size of a test will
  equal only approximately for large samples. The
  difference is called size distortion.

4
Linear Hypothesis

• Probability of type II error is the probability
  of not rejecting the null hypothesis when it is
  false.
• The finite-sample power of a test is the
  probability of rejecting the null hypothesis when
  it is false given a finite sample. That is, the
  power is 1 minus the probability of type II
  error.
• Power depends on the DGPs considered as the
  alternatives as well as on the size (significance
  level) of the test.

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Linear Hypothesis

• A test is consistent against a set of
  alternatives (DGPs), none of which satisfies the
  null, if the power against any particular member
  of the set approaches unite as n??, for any
  assumed significance level.

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Linear Hypothesis (1)
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Linear Hypothesis (1)

• SE(bk) is called the heteroscedasticity-consisten
  t standard error, (heteroscedasticity) robust
  standard error, or Whites standard error.
• The statistic tk is called the robust t-ratio.

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Linear Hypothesis (2)

• Under the null hypothesis H0 Rb r, where R is
  JxK matrix of full row rank and J is the number
  of restrictions (the dimension of r),W
  n(Rb-r)REst(Avar(b))R-1(Rb-r) ?2(J)

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Linear Hypothesis (2)

• Under A.7, or conditional homoscedasticity, W
  n(Rb-r)Rns2(XX)-1R-1(Rb-r)
  (Rb-r)R(XX)-1R-1(Rb-r)/s2 JF (SSRr
  SSRur)/s2 ?2(J)

10
Nonlinear Hypothesis

• Under the null hypothesis with J restrictions H0
  a(b) 0 such that A(b) ?a(b)/?b, the JxK
  matrix of continuous first derivatives of a(b),
  is of full row rank, we have W n
  a(b)A(b)Est(Avar(b))A(b)-1a(b)?d ?2(J)

11
Testing for Conditional Homoscedasticity

• Robust Variance-Covariance Matrix?iei2xixi/n
  ?p E(ei2xixi)where ei yi xib
• OLS Variance-Covariance Matrixs2?ixixi/n, where
  s2 ee/(n-K)
• Under homoscedasticity, ?i(ei2-s2)xixi/n ?p 0

12
Testing for Conditional Homoscedasticity

• Proposition 6 In addition to A.1 (linearity),
  A.4 (rank condition), suppose that
• yi,xi is i.i.d. with finite E(ei2xixi)
  (stronger A.2 and A.5)
• ei is independent of xi (stronger A.3 and
  conditional homoscedasticity
• a certain condition holds on the moments of ei
  and xi
• Then nR2 ?d ?2(m) for the auxiliary
  regressionei2 on a constant and yi (xixi),
  and m is the dimension of yi .