Econometrics

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Econometrics

• Lecture Notes Hayashi, Chapter 6d
• Vector Processes

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Vector White Noise Process

• et, where et e1t,e2t,, eMt
• E(et) 0
• E(et et) ? (positive definite)
• E(et et-j) 0 for j ? 0

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Vector MA(?) Process

• yt m ?j0,, ? ?jet-j with ?0 I, where
  ?j are square matrices.
• ?j is absolutely summable ? ?j0,, ? ?klj lt
  ? for all (k,l)
• The vector MA(?) process yt defined above is
  covariance stationary.
• The mean of yt is m .

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Vector MA(?) Process

• The j-th order autocovariance matrix of yt is
  defined by ?j E(yt-m)(yt-m) ?k0,, ?
  ?jk??k(j0,1,2,)
• ?-j ?j
• ?j is absolutely summable.

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Multivariate Filter

• H(L) H0 H1L H2L2 where Hj is a
  sequence of (not necessarily square) matrices.
• Let xt be a vector covariance stationary
  process. Then the vector process yt defined by
  yt ?j0,, ? Hjxt-j is covariance stationary.

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Multivariate Filter

• Product of FilterA(L) B(L) D(L)mxrrxs
  mxs
• InversesB(L) A(L )-1 if A(L) B(L) I A(L)
  A(L)-1 A(L)-1 A(L)

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Lag Matrix Polynomial

• F(L) I - F1L - F2L2 -- FpLp where Fj is a
  sequence of rxr matrices with Fp? 0.
• Stability Condition All the roots of the
  determinantal equationI - F1z - F2z2 --
  Fpzpare greater than 1 in absolute value (i.e.,
  lie outside the unit circle)

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Lag Matrix Polynomial

• Let ?(L) F(L)-1.
• Following from the stability condition, each
  component of the coefficient matrix sequence ?j
  is bounded in absolute value by a geometrically
  declining sequence.

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Vector AR(p) Process

• yt c F1yt-1 Fpyt-p et,
• F(L)(yt-m) et where F(L) I - F1L - F2L2
  -- FpLp
• yt-m F(L)-1et Y(L)et

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Vector AR(p) Process

• Suppose the p-th degree lag matrix polynomial
  F(z) satisfies the stability condition. Then the
  unique covariance stationary solution to the
  vector AR(p) process has the vector MA(?)
  representationyt m F(L)-1et m Y(L)et
• The coefficient sequence Yj is bounded in
  absolute value by a geometrically declining
  sequence and hence is absolute summable.

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Vector AR(p) Process

• An M-variate vector AR(p) is a set of M equations
  with Mp1 common regressors.
• The mean of the vector AR(p) process is given by
  m c/F(1).
• The sequence Gj is bounded in absolute value by
  a sequence that declines geometrically with j.

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Vector ARMA(p,q) Process

• yt c ?j1,,p Fjyt-j ?j1,,q Qjet-j et
  
• F(L)yt c Q(L)et
• F(L)(ytm) Q(L)et, where m c/F(1)
• Q(L)1F(L)(ytm) Y(L)(ytm) et

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Vector ARMA(p,q) Process

• Suppose the p-th degree lag matrix polynomial
  F(z) satisfies the stability condition. Then the
  unique covariance stationary solution to the
  vector ARMA(p,q) process has the vector MA(?)
  representationQ(L)1F(L)(ytm) Y(L)(ytm)
  et
• The coefficient sequence Yj is bounded in
  absolute value by a geometrically declining
  sequence and hence is absolute summable.

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Vector ARMA(p,q) Process

• The mean of the vector ARMA(p,q) process is given
  by m c/F(1).
• The sequence Gj is bounded in absolute value by
  a sequence that declines geometrically with j.
  Hence the autocovariances are absolutely
  summable.

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Autocovariance-Generating Function

• For a vector covariance stationary process yt,
  the autocovariance-generating function is Gy(z)
  ?j -?,, ? Gjzj G0 ?j 1,, ? (Gjzj
  Gjz-j)
• The spectrum of yt is sy(w) Gy(e-iw)/(2p)

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Autocovariance-Generating Function

• Let xt be an s dimentional covariance
  stationary process with absolutely summable
  autocovariance.
• Gx(z) is the autocovariance-generating function
  of xt.
• H(L) is an rxs absolutely summable filter.
• Then the autocovaiance-generating function of
  ytH(L)xt is given by Gy(z) H(z)Gx(z)H(z-1)
  (rxr) (rxs) (sxs) (sxr)

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Autocovariance-Generating Function

• Vector white noise Gy(z) ?
• VMA(?) Gy(z) Y(z)?Y(z-1)
• VAR(p) Gy(z) F(z)-1?F(z-1)-1
• VARMA(p,q) Gy(z) F(z)-1Q(z)?Q(z-1)F(z-
  1)-1
• VARMA(0,1) yt et Q1et-1
• Gy(z) (IQ1z)?(IQ1 z -1)