K. Ensor, STAT 4211

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.

K. Ensor, STAT 4212

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

K. Ensor, STAT 4213

Spring 2005

Time

0 50 100 150 200 250

05

1015

Simulated Random Walk

0 5 10 15 20

010

2030

4050

60

Histogram

Simulated Random Walk Lag

AC

F

0 5 10 15 20

-1.0

-0.5

0.0

0.5

1.0

ACF

LagA

CF

0 5 10 15 20

-1.0

-0.5

0.0

0.5

1.0

PACF

K. Ensor, STAT 4214

Spring 2005

0 50 100 150 200 250

010

2030

40

50 Simulated Random Walk Paths with Starting Unit of 20

K. Ensor, STAT 4215

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?

K. Ensor, STAT 4216

Spring 2005

Time

0 50 100 150 200 250

2060

100

160

Simulated Random Walk with Drift

0 50 100 150

010

2030

40

Histogram

Simulated Random Walk with Drift Lag

AC

F

0 5 10 15 20

-1.0

-0.5

0.0

0.5

1.0

ACF

LagA

CF

0 5 10 15 20

-1.0

-0.5

0.0

0.5

1.0

PACF

Drift parameter= 0.5Standard Deviation of shocks=2.0

K. Ensor, STAT 4217

Spring 2005

0 50 100 150 200 250

050

100

150

200

50 Simulated Random Walk Paths with Drift

Drift parameter= 0.5Standard Deviation of shocks=2.0

K. Ensor, STAT 4218

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).

K. Ensor, STAT 4219

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.

t

p

jjtjttt arrXr

11

0: oH 0: aH

1

K. Ensor, STAT 42110

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)

K. Ensor, STAT 42111

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

ttt

tttt rXy

1

stationaryTest(x, trend="c", bandwidth=NULL, na.rm=F)