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

10

15

Simulated Random Walk

0 5 10 15 20

01

02

03

04

05

06

0

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

01

02

03

04

0

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

20

60

10

01

60

Simulated Random Walk with Drift

0 50 100 150

01

02

03

04

0

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

05

01

00

15

02

00

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)