Introductory Econometrics for Finance Chris Brooks ch5.ppt

‘Introductory Econometrics for Finance’© Chris Brooks 2002
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Chapter 5
Univariate time series modelling and forecasting
‘Introductory Econometrics for Finance’© Chris Brooks 2002
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Where we attempt to predict returns using only information contained in their past values.
Some Notation and Concepts
A Strictly Stationary Process
A strictly stationary process is one where

. the probability measure for the sequence {yt} is the same as that for {yt+m}  m.
A Weakly Stationary Process
If a series satisfies the next three equations, it is said to be weakly or covariance
stationary
1. E(yt) = , t = 1,2,...,
2.
3.  t1 , t2
Univariate Time Series Models
‘Introductory Econometrics for Finance’© Chris Brooks 2002
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So if the process is covariance stationary, all the variances are the same and all the covariances depend on the difference between t1 and t2. The moments
, s = 0,1,2, ...
are known as the covariance function.
The covariances, s, are known as autocovariances.

However, the value of the autocovariances depend on the units of measurement of yt.
It is thus more convenient to use the autocorrelations which are the autocovariances normalised by dividing by the variance:
, s = 0,1,2, ...
If we plot s against s=0,1,2,... then we obtain the autocorrelation function or
correlogram.
Univariate Time Series Models (cont’d)
‘Introductory Econometrics for Finance’© Chris Brooks 2002
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A white noise process is one with (virtually) no discernible structure. A definition of a white noise process is

Thus the autocorrelation function will be zero apart from a single peak of 1 at s = 0. s  approximately N(0,1/T) where T = sample size

We can use this to do significance tests for the autocorrelation coefficients by constructing a confidence interval.

For example, a 95% confidence interval would be given by . If the sample autocorrelation coefficient, , falls outside this region for any value of s, then we reject the null hypothesis that the true va