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Behavior 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. Spring 2005 K. Ensor, STAT 421 1 Unit-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 infinity Spring 2005 K. Ensor, STAT 421 2 0 5 10 15 Simulated Random Walk 0 50 100 150 200 250 Time Histogram PACF 0.0 ACF 0.0 0 5 10 15 Simulated Random Walk Spring 2005 20 -1.0 0 -1.0 10 -0.5 -0.5 20 ACF 30 40 0.5 0.5 50 1.0 1.0 60 ACF 0 5 10 15 20 Lag K. Ensor, STAT 421 0 5 10 15 20 Lag 3 0 10 20 30 40 50 Simulated Random Walk Paths with Starting Unit of 20 0 Spring 2005 50 100 150 K. Ensor, STAT 421 200 250 4 Random 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? Spring 2005 K. Ensor, STAT 421 5 20 60 100 160 Simulated Random Walk with Drift 0 50 100 150 200 250 Time PACF 0 50 100 150 Simulated Random Walk with Drift Spring 2005 0.5 0.0 ACF 0.0 -0.5 -1.0 0 -1.0 10 -0.5 20 ACF 30 0.5 40 1.0 ACF 1.0 Histogram 0 5 10 15 20 Lag 0 5 10 15 20 Lag Drift parameter= 0.5 Standard Deviation of shocks=2.0 K. Ensor, STAT 421 6 0 50 100 150 200 50 Simulated Random Walk Paths with Drift 0 50 100 150 200 250 Drift parameter= 0.5 Standard Deviation of shocks=2.0 Spring 2005 K. Ensor, STAT 421 7 Unit 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). Spring 2005 K. Ensor, STAT 421 8 Unit root tests continued 1 • There are many forms. The easiest to conceptualize is the following version of the Augmented Dickey Fuller p test (ADF): rt X t rt 1 j rt j at j 1 • The test for unit roots then is simply a test of the following hypothesis: Ho : 0 against Ha : 0 • Use the usual t-statistic for testing the null hypothesis. Distribution properties are different. Spring 2005 K. Ensor, STAT 421 9 Unit root tests • In finmetrics use the following command unitroot(rseries,trend="c",statistic="t", method="adf",lags=6) • Without finmetrics you will need to simulate the distribution under the null hypothesis – see the Zivot manual for the algorithm. Spring 2005 K. Ensor, STAT 421 10 Stationary Tests • Null hypothesis is that of stationarity. • Alternative is a non-stationary process. yt X t t rt t t 1 t • Null hypothesis is that the variance of ε is 0. • In finmetrics use command stationaryTest(x, trend="c", bandwidth=NULL, na.rm=F) Spring 2005 K. Ensor, STAT 421 11