The following exercise assumes knowledge of Chapter 9 of the book.
Exercise. We construct an AR(p) model from the time series and use, in order to construct
the predictors, the first 500 values of the time series for the construction and the last 500
to test its quality. The time series is given by the data file exercise 9.m which requires
that you first enter in MATLAB a vector called ‘personalnumbers’ that consists of two
natural numbers.
Questions
1. Use the sample covariances of the time series for t = 1, . . . , 500 and the Yule-Walker
equations (9.11), with for instance p = 12, to construct a reasonable predictor of p0
for which all coefficients ak in the AR(p) representation could be zero for k > p0 .
2. Now determine from the Yule-Walker equations with p = p0 the AR(p) model for the
time series of outputs. So
y(t) =
p
X
ak y(t − k) + ²(t).
k=1
3. We will investigate how well the AR(p)Pmodel predicts the series of outputs for
t = 501, . . . 1000. Compute σ 2 = R0 − pj=1 aj Rj , where the numbers Rj are the
sample covariances for the given time series. Then σ 2 is the theoretical value of the
variance of the white noise in the AR(p) model.
4. P
Determine the predicted values for the data. For a time series y this is ŷ(t) =
p
j=1 aj y(t − j). Compute the difference between the given time series and the predicted time series. This difference should be the white noise. Make an estimation of
the variance of this difference-series. How well does this correspond to the theoretical
value for the variance found in the previous part?
(Warning: make sure that the predicted value at time t − 1 for time t is compared
with the observed value at time t).
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