Behavior of constant terms and general ARIMA modelsUnit-root nonstationaryPowerPoint PresentationSlide 4Random Walk with DriftSlide 6Slide 7Unit Root TestsUnit root tests continuedUnit root testsStationary TestsK. Ensor, STAT 4211Spring 2005Behavior of constant terms and general ARIMA models•MA(q) – the constant is the mean•AR(p) – the mean is the constant divided by the coefficients of the characteristic polynomial•Random walk with drift – constant is the slope over time of the drift•As we have seen – differencing can be used to derive a stationary process•ARIMA models – r(t) is an ARIMA model if the first difference of r(t) is an ARMA model.K. Ensor, STAT 4212Spring 2005Unit-root nonstationary•Random walk p(t)=p(t-1)+a(t) p(0)=initial value a(t)~WN(0,2)•Often used as model for stock movement (logged stock prices).•Nonstationary•The impact of past shocks never diminishes – “shocks are said to have a permanent effect on the series”.•Prediction?–Not mean reverting–Variance of forecast error goes to infinity as the prediction horizon goes to infinityK. Ensor, STAT 4213Spring 2005Time0 50 100 150 200 2500 5 10 15Simulated Random Walk0 5 10 15 200 10 20 30 40 50 60HistogramSimulated Random Walk LagACF0 5 10 15 20-1.0 -0.5 0.0 0.5 1.0ACFLagACF0 5 10 15 20-1.0 -0.5 0.0 0.5 1.0PACFK. Ensor, STAT 4214Spring 20050 50 100 150 200 2500 10 20 30 4050 Simulated Random Walk Paths with Starting Unit of 20K. Ensor, STAT 4215Spring 2005Random Walk with Drift•Include a constant mean in the random walk model.–Time-trend of the log price p(t) and is referred to as the drift of the model.–The drift is multiplicative over time p(t)=t + p(0) + a(t) + … + a(1)–What happens to the variance?K. Ensor, STAT 4216Spring 2005Time0 50 100 150 200 25020 60 100 160Simulated Random Walk with Drift0 50 100 1500 10 20 30 40HistogramSimulated Random Walk with Drift LagACF0 5 10 15 20-1.0 -0.5 0.0 0.5 1.0ACFLagACF0 5 10 15 20-1.0 -0.5 0.0 0.5 1.0PACFDrift parameter= 0.5Standard Deviation of shocks=2.0K. Ensor, STAT 4217Spring 20050 50 100 150 200 2500 50 100 150 20050 Simulated Random Walk Paths with DriftDrift parameter= 0.5Standard Deviation of shocks=2.0K. Ensor, STAT 4218Spring 2005Unit Root Tests•The classic test was derived by Dickey and Fuller in 1979. The objective is to test the presence of a unit root vs. the alternative of a stationary model. •The behavior of the test statistics differs if the null is a random walk with drift or if it is a random walk without drift (see text for details).K. Ensor, STAT 4219Spring 2005Unit root tests continued•There are many forms. The easiest to conceptualize is the following version of the Augmented Dickey Fuller test (ADF):•The test for unit roots then is simply a test of the following hypothesis: against•Use the usual t-statistic for testing the null hypothesis. Distribution properties are different.tpjjtjtttarrXr 110: oH 0: aH1K. Ensor, STAT 42110Spring 2005Unit root tests•In finmetrics use the following command•Without finmetrics you will need to simulate the distribution under the null hypothesis – see the Zivot manual for the algorithm.unitroot(rseries,trend="c",statistic="t",method="adf",lags=6)K. Ensor, STAT 42111Spring 2005Stationary Tests•Null hypothesis is that of stationarity. •Alternative is a non-stationary process.•Null hypothesis is that the variance of ε is 0.•In finmetrics use command tttttttrXy 1stationaryTest(x, trend="c", bandwidth=NULL,