Econometrics

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Econometrics

• Lecture Notes Hayashi, Chapter 2b

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Time Series Analysis Fundamental Concepts

• Time series is a stochastic process (a sequence
  of random variables) zi (i1,2,).
• Stationarity
• Ergodicity
• Martingales, Random Walks
• Martingale Differences

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Stationarity

• ziis (strictly) stationary if
• The distribution of zi does not depend on the
  absolute position i of zi .
• The joint distribution of zi, zi1, zi2,, zir
  depends only on i1-i, i2-i, , ir-i for any given
  i and for any set of subscripts i1, i2, , ir.
• Any transformation of a stationary process is
  itself stationary.

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Stationarity

• Examples
• A sequence of independent and identically
  distributed (i.i.d) random variables is
  stationary process that exhibits no serial
  correlation.
• The constant series is a stationary process with
  maximum serial dependence.

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Stationarity

• ziis weak (or covariance) stationary if
• E(zi) does not depend on i.
• gj Cov (zi,zi-j) exists, is finite, and
  depends only on j but not on i.
• Autocorrelation of the j-th order

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Stationarity

• A covariance stationary process zi is white
  noise if E(zi) 0 and Cov(zi,zi-j) 0 for j ?
  0.
• An i.i.d. process zi with E(zi) 0 Var(zi)
  s2 is a special case of white noise process,
  called independent white noise process.

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Ergodicity

• A stationary process zi is ergodic if, for any
  two bounded functions f Rk1?R and g Rl1?R,
  limn??Ef(zi,,zik)g(zin,,zinl)
  Ef(zi,,zik) Eg(zi,,zil)

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Ergodicity

• A stationary process is ergodic if it is
  asymptotically independent, that is, if any two
  random variables positioned far apart in the
  sequence are almost independently distributed.
• A stationary process that is ergodic will be
  called ergodic stationary.

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Ergodicity

• Ergodic Theorem (Generalization of Kolmogorovs
  LLN)
• Let zi be an ergodic stationary process with
  E(zi) m. Then

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Ergodicity

• f(zi) is ergodic stationary whenever zi is,
  for any function f(.).
• Therefore, any moment of an ergodic stationary
  process (e.g., E(zizi), if it exists and finite)
  is consistently estimated by the sample moment
  (e.g., ?izizi/n).

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Martingales

• Let xi be a scalar element of vector zi. The
  scalar process xi is called a martingale w.r.t.
  zi if E(xizi-1,zi-2,,z1) xi-1 for i?2.
• A vector process zi is called martingale if
  E(zizi-1,zi-2,,z1) zi-1 for i?2.

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Random Walks

• Let gi be a vector of independent white noise
  process. A random walk process zi is defined
  by
• z1 g1
• z2 g1 g2
• zi g1 gi

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Random Walks

• The first difference of random walk is
  independent white noise.
• A random walk is martingaleE(zizi-1,zi-2,,z1)
  E(g1gigi-1,,g1) E(gigi-1,,g1)(g1g
  i-1) 0(g1gi-1) zi-1

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Martingale Differences

• A vector process gi with E(gi) 0 is called a
  martingale difference sequence (m.d.s.) or
  martingale differences if E(gigi-1,gi-2,,g1)
  0 for i?2.
• The cumulative sum zi created from m.d.s. gi
  is a martingale conversly, if zi is
  martingale, the first differences are m.d.s.

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Martingale Differences

• A m.d.s. has no serial correlation. That
  is,Cov(gi,gi-j) E(gi,gi-j) 0 for all i and
  j ? 0.
• Example ARCH(1)
• gi (zagi-12)½ei, where ei i.i.d.(0,1).
• gi is an m.d.s. because E(gigi-1,gi-2,,g1)
  E((zagi-12)½ei gi-1,gi-2,,g1 (zagi-12)½
  E(ei gi-1,gi-2,,g1) 0

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Martingale Differences

• Example ARCH(1) continued
• E(gi2gi-1,gi-2,,g1) zagi-12 The conditional
  second moment depends on its own history of the
  process. The process exhibits own conditional
  heteroscedasticity.
• The process is strictly stationary and erogodic
  if alt1, provided that g1 is draw from an
  appropriate distribution.
• If gi is stationary, the unconditional second
  moment can be shown as E(gi2) z/(1-a).

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Martingale Differences

• Ergodic Stationary Martingale Differences CLT
  Let gi be a vector m.d.s. that is ergodic
  stationary with E(gi,gi) S. Then

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Discussions

• gi is independence white noise ? gi is
  stationary m.d.s. with finite variance? gi is
  white nose
• LLN and CLT are not just applicable to i.i.d.
  sequences.
• LLN for serially correlated processes, by Ergodic
  Theorem.
• CLT for ergodic stationary m.d.s. such as ARCH(1).