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Cointegration, Stationarity and Error Correction Models. From the notes you need to know: The definition of weak or covariance stationarity on page 2 of the notes plus an intuitive understanding of the term. Testing for unit roots, the Dickey Fuller test and the Augmented Dickey Fuller test The concept of cointegration (pages 2 & 3). Testing for cointegration (Page 5-6) The Engle Granger representation theorem Error Correction Models – including superconsistency of the long run equilibrium parameters (page 4-5) The Engle Granger two step method The Role of cointegration in economics To supplement these we have: Trend Stationarity The following is an AR(1) (autoregressive with lag depth or order 1) with a deterministic linear trend term Yt = θYt-1 + δ + γt + εt (1) Where |θ| < 1 The moving average (MA) representation of this on past error terms is Yt = θtY0 + μ0 + μ1t + εt + θεt-1 + θ2εt-2 + θ3εt-3 + ….. (2) E(Yt) = θtY0 + μ0 + μ1t → μ0 + μ1t as t → ∞ (3) This has a finite, unchanging variance, but no constant mean (because of μ1t). Thus the process is not stationary. However, the deviation from the mean Yt = Yt – E[Yt] = Yt - μ0 - μ1t (4) is stationary and hence Yt is called a trend stationary process. Thus the shocks to the process are transitory and the process is ‘mean reverting’, with the mean μ0 + μ1t being the ‘attractor’. UNIT ROOT PROCESSES The following is an AR(1) model with a unit root θ=1 Yt = θYt-1 + δ + εt = Yt-1 + δ + εt (5) It is clearly non-staionary with no specified mean. But the process ∆Yt is stationary and we term Yt a difference stationary process (in this case I(1)) Yt = Yt-1 + δ + εt (6) Yt = Yt-2 + δ + εt + δ + εt-1 (substituting for Yt-1) (7) Yt = Yt-3 + δ + εt + δ + εt-1 + δ + ε-2 (substituting for Yt-2) Yt-3 + δ + δ + δ + εt + εt-1 + ε-2 (7) And so on until: Yt = Y0 + δt + Σεi (8) Hence E[Yt] = Y0 + δt (9) The effect of the initial value Y0 stays in the process. We can also see from (8) that shocks have permanent effects and the figure below shows the impact of a large shock. The process has no attractor. DETERMINISTIC TRENDS AND TREND STATIONARITY A time series that is stationary around a deterministic trend is called a trend stationary process. Examples of Different Processes To test for trend stationarity we include a trend term and typically a constant term in the Dickey Fuller/ADF regressions. However this changes the asymptotic distribution relating to the test shifting the distribution to the left, the inclusion of a time trend shifts it still further to the left which is why we need different critical values for the DF and ADF tests depending on whether a constant and/or a time trend is included in the specification. But apart from this the test is the same and relates to the t statistic on Yt-1. We are also interested in whether the coefficient on the constant term is significant.