Time Series Econometrics 4 Vijayamohanan Pillai N

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1 Time Series Econometrics 4 Vijayamohanan Pillai N Vijayamohan: CDS MPhil: Time Series 5 1 Autoregressive Moving Average Process: ARMA(p, q) Vijayamohan: CDS MPhil: Time Series 5 2 1

2 Autoregressive Moving Average Process: ARMA(p, q) The general ARMA(p, q): AR of order pand MA of order q : Y t = Φ 1 Y t-1 + Φ 2 Y t Φ p Y t-p + ε t θ 1 ε t-1 θ 2 ε t-2. θ q ε t-q, where ε t is a white noise process. Vijayamohan: CDS MPhil: Time Series 5 3 Autoregressive Moving Average Process: ARMA(p, q) Y t = Φ 1 Y t-1 + Φ 2 Y t Φ p Y t-p + ε t θ 1 ε t-1 θ 2 ε t-2. θ q ε t-q, With the lag operator: Φ(L)Y t = θ(l)ε t, where Φ(L) = (1 Φ 1 L Φ 2 L 2. Φ p L p ); and θ(l) = (1 θ 1 L θ 2 L 2. θ q L q ). Properties of ARMA process: a mixture of those of AR and MA processes. Vijayamohan: CDS MPhil: Time Series 5 4 2

3 The simplest ARMA process: ARMA(1, 1): Y t = Φ 1 Y t-1 + ε t θ 1 ε t-1. (1)E(Y t ) = 0; With a constant, E(Y t ) = /(1 - Φ 1 ); Φ 1 < 1. Vijayamohan: CDS MPhil: Time Series 5 5 ARMA(1, 1) (2) Var(Y t ) = E(Φ 1 Y t-1 + ε t θ 1 ε t-1 ) 2 : = Φ 12 Var(Y t ) 2Φ 1 θ 1 E(Y t-1 ε t-1 ) + Var(ε t )+ θ 12 Var(ε t-1 ): Since Var(ε t )= Var(ε t-1 ) = σ 2 ; and E(Y t-1 ε t-1 ) = E(Y t ε t )= σ 2, Var(Y t ) = σ 2 (1 + θ 12 2Φ 1 θ 1 )/ (1 Φ 12 ), Φ 1 < 1. Vijayamohan: CDS MPhil: Time Series 5 6 3

4 ARMA(1, 1) (3) Cov(Y t ; Y t-k ) = γ k : The covariances are recursively estimated: with k= 1: γ 1 = E(Y t Y t-1 ) = E{(Φ 1 Y t-1 + ε t θ 1 ε t-1 ) Y t-1 } = Φ 1 Var(Y t ) θ 1 σ 2 = Φ 1 γ 0 θ 1 σ 2 = σ 2 (1 Φ 1 θ 1 )(Φ 1 θ 1 )/ (1 Φ 12 ), Vijayamohan: CDS MPhil: Time Series 5 7 Φ 1 < 1. ARMA(1, 1) (3) Cov(Y t ; Y t-k ) = γ k : with k= 2: γ 2 = E(Y t Y t-2 ) = E{(Φ 1 Y t-1 + ε t θ 1 ε t-1 ) Y t-2 } = Φ 1 E(Y t-1 Y t-2 ) = Φ 1 E(Y t Y t-1 ) = Φ 1 γ 1. Thus γ k = Φ 1 γ k-1, k 2. Vijayamohan: CDS MPhil: Time Series 5 8 4

5 (4) ACF: ARMA(1, 1) ρ k = γ k / γ 0 = (1 Φ 1 θ 1 )(Φ 1 θ 1 )/(1 + θ 12 2Φ 1 θ 1 ), k= 1; = Φ 1 ρ k-1, k 2. Thus ACF decays geometrically from the starting value, ρ 1, a function of both Φ 1 and θ 1. Vijayamohan: CDS MPhil: Time Series 5 9 (4) ACF: = ρ k = γ k / γ 0 ARMA(1, 1) = Φ 1 ρ k-1, k 2. Thus ACF decays geometrically from the starting value, ρ 1, a function of both Φ 1 and θ 1. Note: The decay is determined by Φ 1 only: the MA part has only one period memory. Note: When Φ 1 = θ 1, ρ k = 0, as Y t collapses to a white noise process: Y t = ε t +θ 1 (Y t-1 ε t-1 ). Vijayamohan: CDS MPhil: Time Series

6 ARMA(1, 1) 5. PACF: The first PAC coefficient = Φ 1 ; The second PACC = Φ 2 ly for other higher order PACF. PACF generally declines as the lag increases. Vijayamohan: CDS MPhil: Time Series 5 11 Y t = 0.4Y t-1 + ε t 0.2ε t-1. Vijayamohan: CDS MPhil: Time Series

7 Y t = 0.4Y t-1 + ε t 0.2ε t-1. Vijayamohan: CDS MPhil: Time Series 5 13 Y t = 0.4Y t-1 0.5Y t-2 + ε t 0.2ε t ε t-2. Vijayamohan: CDS MPhil: Time Series

8 Y t = 0.4Y t-1 0.5Y t-2 + ε t 0.2ε t ε t-2. Vijayamohan: CDS MPhil: Time Series 5 15 Theoretical ACF and PACF of ARMA(1, 1) process Φ 1 positive Φ 1 negative Vijayamohan: CDS MPhil: Time Series

9 Non-stationarity Non-stationarity due to (1)Integrated process: random walk: Y t = Y t-1 + ε t Difference stationary process Stochastic trend (2) Trend: Trend stationary process Both stochastic and deterministic trend Vijayamohan: CDS MPhil: Time Series 5 17 Consequences of Non-stationarity (1) With stationary series, possible to model the process via a fixed-coefficients equation estimated from past data. Not possible if the structural relationship changes over time(if non-stationary). (2) For non-stationary series, Varand Covsare functions of time: so the conventional asymptotic theory cannot be applied to these series. Vijayamohan: CDS MPhil: Time Series

10 Consequences of Non-stationarity (2) For non-stationary series, Varand Covsare functions of time: so the conventional asymptotic theory cannotbe applied to these series. For example: In a regression of Y t on X t : βˆ = Cov Y, X ) Var( X ) If X t is non-stationary, Var(X t ) infinitely, and dominates the Cov(Y t, X t ): Then the OLS estimatordoes not have an asymptotic distribution. Vijayamohan: CDS MPhil: Time Series 5 19 ( t t t Consequence of unit root (non-stationarity): Consider two unrelated RW processes: Y t = Y t 1 + U t ; U t IIN(0, σ U2 ) X t = X t 1 + V t ; V t IIN(0, σ V2 ); Cov(U t, V t ) = 0. Vijayamohan: CDS MPhil: Time Series

11 Consequence of unit root (non-stationarity): 20 y x Y t = Y t 1 + U t ; U t IIN(0, σ U2 ) X t = X t 1 + V t ; V t IIN(0, σ V2 ); Cov(U t, V t ) = 0. Vijayamohan: CDS MPhil: Time Series 5 21 Consequence of unit root (non-stationarity): 3 eps ep U t IIN(0, σ U2 ) V t IIN(0, σ V2 ) Vijayamohan: CDS MPhil: Time Series

12 Consequence of unit root (non-stationarity): Cross Correlation function CCF-eps x ep CCF-ep x eps Cov(U t, V t ) = 0. Vijayamohan: CDS MPhil: Time Series 5 23 Consequence of unit root (non-stationarity): Now consider the regression: Y t =β 0 + β 1 X t + ε t. We expect R 2 from this regression wouldtend to zero; BUT. Vijayamohan: CDS MPhil: Time Series

13 Consequence of unit root (non-stationarity): Vijayamohan: CDS MPhil: Time Series 5 25 Consequence of unit root (non-stationarity): Vijayamohan: CDS MPhil: Time Series

14 Consequence of unit root (non-stationarity): Granger and Newbold(1974): High R 2 and highly significant t, but a low DW statistic. When the regression was run in first differences? Vijayamohan: CDS MPhil: Time Series 5 27 Consequence of unit root (non-stationarity): Vijayamohan: CDS MPhil: Time Series

15 Consequence of unit root (non-stationarity): When the regression was run in first differences, R 2 close to zero; DW close to 2: No relationship between Y t and X t ; and the high R 2 obtained was spurious. R 2 > DW Spurious Regression. Vijayamohan: CDS MPhil: Time Series 5 29 Spurious or Nonsense Regression? Long-standing Puzzle over high correlations between what ought to be unrelated time series variables; e.g., High positive correlation between the murder rate and membership of the Church of England: Yule(1926). Yule(1926) classifies: (1): Nonsense regression; (2): Spurious regression. Vijayamohan: CDS MPhil: Time Series

16 Spurious or Nonsense Regression? (1): Nonsense regression: integrated, but mutually independent, time series; (high serial correlation in each series high correlation); (2): Spurious regression: Variables depending on common third factors (e.g., having a linear trend). Vijayamohan: CDS MPhil: Time Series 5 31 So, check for stationarity of time series: Classical: Autocorrelation function (ACF): Fast-decreasing ACF Stationarity Correlogram Modern: Unit root tests: (Augmented) Dickey-Fuller test; Phillips-Perron non-parametric test; etc. Vijayamohan: CDS MPhil: Time Series

17 If non-stationary series, run regression in (first) differences as in Classical (ARIMA) modelling or check for Cointegration among the series Vijayamohan: CDS MPhil: Time Series 5 33 Non-stationarity and differencing: Consider the random walk: Y t = Y t-1 + ε t where ε t is a white noise First difference: Y t = Y t Y t-1 = ε t : Stationary. If a non-stationary series is transformed into a stationary one by differencing once, the series in level = integrated of order one: I(1): one unit root; Vijayamohan: CDS MPhil: Time Series

18 Non-stationarity and differencing: Consider the random walk: Y t = Y t-1 + ε t First difference: where ε t is a white noise Y t = Y t Y t-1 = ε t : Stationary. The series in level = integrated of order one: I(1): one unit root; Vijayamohan: CDS MPhil: Time Series 5 35 Non-stationarity and differencing: Process of inverse of differencing = integration. With one unit root, first differencing, to make y t stationary. Then y t = integrated of order one: I(1): one unit root; The differenced, y t, stationary series = integrated of order zero: I(0); no unit root. Thus Y t I(1); and Y t I(0). Vijayamohan: CDS MPhil: Time Series

19 I(1) versus I(0) processes (1) I(1) series wanders widely; I(0) series = mean reverting: direct/oscillatory convergence to mean. (2) I(1) series has infinitely long memoryof its past behaviour: shocks permanently affectthe process: evident from slow decay of ACF. I(0) series has limited memory: shock effects are only transitory: Vijayamohan: CDS MPhil: Time Series 5 37 evident from fast decay of ACF. I(1): the wandererand I(0): the mean-reverter I(1): No tendency to return to zero I(0): Rarely drifts from zero time Vijayamohan: CDS MPhil: Time Series

20 Non-stationarity and differencing: If a non-stationary series, Y t into a stationary one by differencing d times, d, the series in level, Y t = integrated of order d: I(d): dunit roots. Then d Y t = stationary:i(0). Vijayamohan: CDS MPhil: Time Series 5 39 Non-stationarity and differencing: If d Y t = stationary process that can be represented by an ARMA(p, q) model, then Y t = an integrated autoregressive moving average processof order: d, p, and q: ARIMA(p, d, q), d= number of differences = order of integration. Vijayamohan: CDS MPhil: Time Series

21 Non-stationarity Non-stationarity due to (1)Integrated process: random walk: Y t = Y t-1 + ε t Difference stationary process Stochastic trend (2) Trend: Trend stationary process Both stochastic and deterministic trend Vijayamohan: CDS MPhil: Time Series 5 41 Non-stationarity: Trend stationary process: TSP: The determistic trend: Y t = a+bt+ u t, u t : white noise process. Stationary fluctuations around a linear trend. Non-stationary process? Vijayamohan: CDS MPhil: Time Series

22 Non-stationarity: Trend stationary process: TSP: Y t = a+bt+ u t, u t : white noise process. mean changes with time: E(Y t ) = a+bt. But constantvariance : Var(Y t ) = Var(u t ) = σ u2. Cov(Y t, Y t-k ) = Cov(u t, u t-k ) = 0. Vijayamohan: CDS MPhil: Time Series 5 43 To make a TSP stationary, detrend it: Y t E(Y t ) = = Y t (a+bt) Non-stationarity : = u t, stationary. A TSP: Y t = t+ u Detrended Series t Vijayamohan: CDS MPhil: Time Series

23 EQ( 1) Modelling y by OLS The estimation sample is: 1 to 100 Coefficient Std.Error t-value t-prob A TSP: Y t = t+ u t Constant Trend sigma RSS R F(1,98) = 2.083e+004 [0.000]** log-likelihood DW 1.71 no. of observations 100 no. of parameters 2 Fitted series: Y t est = t Detrended series: Y t Y t est Vijayamohan: CDS MPhil: Time Series 5 45 Differencing a TSP: Y t = a + bt+ε t ; then Y t-1 = a + b(t 1) +ε t-1 ; Y t = Y t Y t-1 = b+ ε t ε t-1. Y t has a MA unit root! Vijayamohan: CDS MPhil: Time Series

24 Differencing a TSP: Differencing detrends, but generates MA process that can show a cycle when there is none in the original series: spurious cycle : Slutzky effect(slutzky 1937). Differenced series: Y t = Y t Y t-1 = 5 + ε t ε t-1 Detrended series: Y t ( t) = ε t Vijayamohan: CDS MPhil: Time Series 4 47 Detrending: Time series regression on time: The classic result of RagnarFrisch and F. V. Waugh (1933): including a time trend in a regression = detrendingthe variablesby regressing them (1) Residuals interpretedas cyclical individually on time. components in business cycle theory; (2) Estimating trend growth rates (Crafts, et al., 1989) Vijayamohan: CDS MPhil: Time Series

25 Vijayamohan: CDS MPhil: Time Series