#### Transcript Document

```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
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",
• 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
```