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ie_Slide11 - Introductory Econometrics ECON2206/ECON3209 S2...

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Introductory Econometrics ECON2206/ECON3209 S2, 2010 Slides11 Lecturer: Minxian Yang ie_Slides10 my, School of Economics, UNSW 1

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11. Further Issues with Time Series Data (Ch11) 11. Further Issues with Time Series Data – Stationary time series – Weakly dependent time series – Asymptotic properties of OLS – Dynamically completed models – Persistent time series E(u t |x t 1 ,..., x tk ) = 0, t = 1,2,. .., n . (z10) Var(u t |x t 1 ,..., x tk ) = σ 2 , t = 1,2,. .., n. (h10) E(u t u s | x t , x s ) = 0, for all t s . (u10) ie_Slides10 my, School of Economics, UNSW 2
11. Further Issues with Time Series Data (Ch11) • Stationary time series – Stochastic process (SP) • A time series is viewed as a SP, ie, a random variable that is a function of time index. • Once trends and seasonality are removed, a time series can often be described as a stationary SP. – Stationary SP • A SP is strictly stationary if its joint distribution at any set of points in time is invariant to any time-shift. • A SP with a finite variance is covariance stationary if its mean and autocovariances are independent of the time index. ie_Slides10 my, School of Economics, UNSW 3 γ( h ) = Cov( x t , x t-h ), h = 0, 1, 2, . ... Stationarity and WD are needed for applying LLN and CLT.

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11. Further Issues with Time Series Data (Ch11) • Stationary time series eg. Consider a SP: Y t – Strict stationary: the same (joint) distribution at t = 1, 2, . .. ; or t = (2,6), t = (4,8), t = (6,10),. .. ; or t = (1,3,5), t = (2,4,6), . .. – Covariance stationary: the mean, variance and autocovariance are the same at any t . ie_Slides10 my, School of Economics, UNSW 4
11. Further Issues with Time Series Data (Ch11) • Stationary time series – The relationship between the two “stationary” notions • Strict stationarity with finite variance implies covariance stationarity. • The reversal is generally not true. – Examples of stationary SP • An independent identically distributed (iid) random sequence { e t } is a strictly stationary SP. • For an iid SP { e t }, {f( e t )} is strictly stationary for any measurable function f(.). • For an iid SP { e t }, the finite “distributed lags” is a strictly stationary SP. ie_Slides10 my, School of Economics, UNSW 5 } { q j j t j e δ 0 { e t } = {. .., e -1 , e 0 , e 1 , e 2 , e 3 ,...}

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11. Further Issues with Time Series Data (Ch11) • Weakly dependent time series – Dependence in time series • Time series are generally dependent: the future is influenced by the present and the past. • When a time series is independent, it is equivalent to a random sample in the sense that the LLN and CLT apply to sample averages and OLS asymptotics hold. – Weak dependence
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ie_Slide11 - Introductory Econometrics ECON2206/ECON3209 S2...

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