TimeSeriesNotes

TimeSeriesNotes - ECON3300/7360 TIME SERIES ECONOMETRICS...

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ECON3300/7360 TIME SERIES ECONOMETRICS Reference: Ch.21 of Gujarati, D.N., (1995) Basic Econometrics , McGraw Hill . Ln CPI t = α 0 + α 1 Ln (Money SS) t t = 1950 to 2000 R 2 = 0.83 DW = 0.2 - pure autocorrelation or misspecification dynamics? Let’s say DW is ok - But spurious regression if variables are not checked for stationarity . t May be unstationary Ln CPI 1
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Spurious Regression Due to Non-stationarity There is no constant mean as data has an explosive nature. constant mean Stationary process 2
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A process or random variable is stationary if E(X t ) = constant = μ Var(X t ) = constant = σ 2 cov(X t , X t+j ) = σ j If any one or more of the above is violated χ t is non-stationary (Problem) Fig. 5.1: Stochastic process with non-stationary mean Fig. 5.2: Stochastic process with stationary mean 3
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White Noise Error Term Y t = β 0 + β 1 X t + ε t where ε t ~ N (O, σ 2 ) cov( ε t , ε t+s ) = 0 (errors are not correlated) With some autocorrelation coeff of covariance i.e. ε t = ρε t-1 + u t - 1 < ρ < 1 c o v ( ε t , ε t+s ) = ρ s ε t , ε t-1 , ε t-2 , …. ρε t-1 ρ ρ ( ρε t-2 ) ρ 2 Since |ρ| < | , the further away the lag, the weaker the correlation between errors. ε t is covariance stationary i.e. time-invariant covariance. RECALL (a + b) 2 = a 2 + 2ab + b 2 Var( ε t ) = ( ) () [] 2 t 2 t E E ε ε not the same! 4
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Examples of Non-stationary Process 1) Random Walk without Drift Y t = Y t-1 + U t where U t ~N (O, σ 2 ) ( ) () ( ) ( ) [ ] 2 2 t 2 t 2 t t t 0 U E U E U E U Var 0 U E σ = = = = Cov (U t , U t+s ) = 0 i.e. E (U t U t+s ) = 0 t = 1 Y 1 = Y 0 + U 1 t = 2 Y 2 = Y 1 + U 2 = Y 0 + U 1 + U 2 = + 2 0 Y = 1 t t U Y t = Y 0 + = 1 t t U ± Find mean E(Y t ) Are these constants? ± Find variance Var(Y t ) () ( ) ( ) () () () [] ( ) . tan ... .. 2 2 0 2 2 2 0 2 1 0 2 1 0 0 0 first Y E Find Y Y E Y E Y E Y Var mean t cons Y U E U E U E Y U U U E Y U E Y E U Y E Y E t t t t t t t t t t = = = + + + + = + + + = + = + = 5
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( ) ( ) [ ] () [] () ()() ()() 2 2 0 2 2 2 2 1 2 0 3 1 2 1 2 2 2 2 1 2 0 3 1 2 1 2 2 2 2 1 2 0 2 2 1 2 0 2 0 2 0 2 0 2 0 2 0 2 .. ... 2 2 .. ... 2 2 .. .. 2 2 σ t Y U E U E U E Y U U E U U E U E U E U E Y U U U U U U U E Y U U U E Y U E U E Y Y E U U Y Y E U Y E Y E t t t t t t t t t t + = + + + + = + + + + + + + = + + + + + + = + + + = + + = + + = + = ∑∑ Var(Y t ) = 2 0 2 2 0 Y t Y σ + = t σ 2 (time-varying variance) 2) Random Walk with Drift Y t = δ + Y t-1 + U t drift parameter Show : E(Y t ) = Y 0 + t δ Var(Y t ) = t σ 2 That is, time-varying mean and variance non-stationary A situation of non-stationarity is called a unit root process Y t = ρ Y t-1 + U t -1 < ρ < 1 ρ = 1 : pure random walk (infinite memory) |ρ| < 1 : process becomes stationary as past effects fade off. 6
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Two Types of Stochastic Processes Trend Stationary Difference Stationary - need to subtract mean - need to take first o f Y t from Y t first differences i.e. Y t – E(Y t ) i . e . Y t – Y t-1 () [] ( ) [] () 1 t t t t 1 t t t t Y Y Var Y E Y Var Y Y E Y E Y E Two Types of Trends in Time Series Deterministic Trend Stochastic Trend - trend is completely - trend is not predictable predictable See Fig. 21.5 on page: 804 Eg. of Trend Stationary Process ~ N (0, σ 2 ) t t 2 1 t U where U t Y + β + β = 0 E(Y t ) = β 1 + β 2 t + E(U t ) non-constant mean Var(Y t ) = Var(U t ) = σ 2 constant variance constant 7
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To make Y t stationary (detrend) Y t – E(Y t ) = β 1 + β 2 t + U t – ( β 1 + β 2 t) = U t E[Y t – E(Y t )] = E(U t ) = 0
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This note was uploaded on 03/13/2010 for the course ECON econ1010 taught by Professor Margretfinch during the Three '08 term at Griffith.

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TimeSeriesNotes - ECON3300/7360 TIME SERIES ECONOMETRICS...

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