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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.

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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

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K. Ensor, STAT 421 3 Spring 2005

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K. Ensor, STAT 421 4 Spring 2005

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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?

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

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

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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).

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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.

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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)

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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)

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