Stationary models MA, AR and ARMA Matthieu Stigler November 14, 2008 Version 1.1 ThisdocumentisreleasedundertheCreativeCommonsAttribution-Noncommercial2.5India license. MatthieuStigler () Stationarymodels November14,2008 1/65 Lectures list 1 Stationarity 2 ARMA models for stationary variables 3 Seasonality 4 Non-stationarity 5 Non-linearities 6 Multivariate models 7 Structural VAR models 8 Cointegration the Engle and Granger approach 9 Cointegration 2: The Johansen Methodology 10 Multivariate Nonlinearities in VAR models 11 Multivariate Nonlinearities in VECM models MatthieuStigler () Stationarymodels November14,2008 2/65 Outline 1 Last Lecture 2 AR(p) models Autocorrelation of AR(1) Stationarity Conditions Estimation 3 MA models ARMA(p,q) The Box-Jenkins approach 4 Forecasting MatthieuStigler () Stationarymodels November14,2008 3/65 Recall: auto-covariance Definition (autocovariance) Cov(X ,X ) ≡ γ (t) ≡ E[(X −µ)(X −µ)] t t−k k t t−k Definition (Autocorrelation) Corr(X ,X ) ≡ ρ (t) ≡ Cov(Xt,Xt−k) t t−k k Var(Xt) Proposition Corr(X ,X ) = Var(X ) t t−0 t Corr(X ,X ) = φj depend on the lage: plot its values at each lag. t t−j MatthieuStigler () Stationarymodels November14,2008 4/65 Recall: stationarity The stationarity is an essential property to define a time series process: Definition A process is said to be covariance-stationary, or weakly stationary, if its first and second moments are time invariant. E(Y ) = E[Y ] = µ ∀t t t−1 Var(Y ) = γ < ∞ ∀t t 0 Cov(Y ,Y ) = γ ∀t, ∀k t t−k k MatthieuStigler () Stationarymodels November14,2008 5/65 Recall: The AR(1) The AR(1): Y = c +ϕY +ε ε ∼ iid(0,σ2) t t−1 t t with |ϕ| < 1, it can be can be written as: t−1 c (cid:88) Y = + ϕiε t t−i 1−ϕ i=0 Its ’moments’ do not depend on the time: : E(X ) = c t 1−ϕ Var(X ) = σ2 t 1−ϕ2 Cov(X ,X ) = ϕj σ2 t t−j 1−ϕ2 Corr(X ,X ) = φj t t−j MatthieuStigler () Stationarymodels November14,2008 6/65 Outline 1 Last Lecture 2 AR(p) models Autocorrelation of AR(1) Stationarity Conditions Estimation 3 MA models ARMA(p,q) The Box-Jenkins approach 4 Forecasting MatthieuStigler () Stationarymodels November14,2008 7/65 Outline 1 Last Lecture 2 AR(p) models Autocorrelation of AR(1) Stationarity Conditions Estimation 3 MA models ARMA(p,q) The Box-Jenkins approach 4 Forecasting MatthieuStigler () Stationarymodels November14,2008 8/65 Autocorrelation function A usefull plot to understand the dynamic of a process is the autocorrelation function: Plot the autocorrelation value for different lags. MatthieuStigler () Stationarymodels November14,2008 9/65 ff ==0 ff ==0.5 8 8 0. 0. k k rr 0.4 rr 0.4 0 0 0. 0. 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Lag k Lag k ff ==0.9 ff ==1 8 8 0. 0. k k rr 0.4 rr 0.4 0 0 0. 0. 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Lag k Lag k MatthieuStigler () Stationarymodels November14,2008 10/65