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.
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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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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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Standard Deviation of shocks=2.0
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Standard Deviation of shocks=2.0
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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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Unit root tests continued
  1
• There are many forms. The easiest to
conceptualize is the following version of the
Augmented Dickey Fuller
p test (ADF):
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.
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Unit root tests
• In finmetrics use the following
command
unitroot(rseries,trend="c",statistic="t",
method="adf",lags=6)
• Without finmetrics you will need to
simulate the distribution under the null
hypothesis – see the Zivot manual for
the algorithm.
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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)
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