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229-08-L12 - (229-2-1 Econ 229 Handout on time series...

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(229-2)-1 Econ 229 – Handout on time series econometrics • Objective: - Primer on time series econometrics relevant for empirical macroeconomics - Distinction between unit root and trend-stationary series; notion of cointegration. • Background reading (not required): Hamilton, Time Series Econometrics, ch.15-17. Stochastic Processes Example #1: Stationary AR(1) y t = " # y t $ 1 + % + & t with AR coefficient | " | < 1 and constant " . • Assume " t is white noise = mean zero, finite variance, uncorrelated over time. • Projection n steps ahead: y t + n = " # y t + n $ 1 + % + & t + n = " # [ " # y t + n $ 2 + % + & t + n $ 1 ] + % + & t + n ... => y t + n = " n # y t + " j j = 0 n $ 1 % ( & + t + n $ j ) = " n # y t + 1 $ " n 1 $ " & + " j j = 0 n $ 1 % t + n $ j • Forecast mean: E t y t + n = " n # y t + 1 $ " n 1 $ " % converges to the unconditional mean E [ y t + n ] = E [ y t ] = " 1 # $ • Forecast variance is Var t [ y t + n ] = E t [( y t + n " E t y t + n ) 2 ] = E t [( # j j = 0 n " 1 $ % t + n " j ) 2 ] = # 2 j j = 0 n " 1 $ & % 2 = 1 " # 2 n 1 " # 2 & % 2 converges to the unconditional variance Var [ y t ] = 1 1 " # 2 $ % 2 • Time-t disturbance has declining effect over time: E t y t + n " E t " 1 y t + n = # n $ t (“Impulse response” = graph against n) • Moving Average (MA) representation: y t = " n # y t $ n + 1 $ " n 1 $ " % + " j j = 0 n $ 1 & t $ j ( % 1 $ " + " j j = 0 ) & t $ j
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(229-2)-2 Example #2: Random walk y t = y t " 1 + # + $ t with drift coefficient " . - Interpret as limiting case of AR(1) with " # 1 . (Sums in the AR don’t converge for " = 1 .) - Projection n steps ahead: y t + n = y t + n " + # t + n $ j j = 0 n $ 1 % - Conditional expectation “drifts” with current value: E t y t + n = y t + n " # => No unconditional mean. - Time-t disturbance has a permanent effect: E t y t + n " E t " 1 y t + n = # t - Forecast error grows over time: Var t [ y t + n ] = E t [( " t + n # j j = 0 n # 1 $ ) 2 ] = n % " 2 - Taking differences: " y t
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