1. K. Ensor, STAT 421 1 Spring 2005 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.

2. K. Ensor, STAT 421 2 Spring 2005 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

3. K. Ensor, STAT 421 3 Spring 2005

4. K. Ensor, STAT 421 4 Spring 2005

5. K. Ensor, STAT 421 5 Spring 2005 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?

6. K. Ensor, STAT 421 6 Spring 2005 Drift parameter= 0.5 Standard Deviation of shocks=2.0

7. K. Ensor, STAT 421 7 Spring 2005 Drift parameter= 0.5 Standard Deviation of shocks=2.0

8. K. Ensor, STAT 421 8 Spring 2005 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).

9. K. Ensor, STAT 421 9 Spring 2005 Unit 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.

10. K. Ensor, STAT 421 10 Spring 2005 Unit 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)

11. K. Ensor, STAT 421 11 Spring 2005 Stationary 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 stationaryTest(x, trend="c", bandwidth=NULL, na.rm=F)