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Chapter5 UNDERSTANDING LINEAR DEPENDENCE: A LINK TO ECONOMIC MODELS_solutions University of Alabama EC 410
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CHAPTER 5. UNDERSTANDING LINEAR DEPENDENCE: A LINK TO ECONOMIC MODELS SOLUTIONS by Wei Lin and Yingying Sun (University of California, Riverside) Exercise 1 We simulate 1000 observations of the process pt = 6.43 + 0.55pt−1 + εt and plot 100 observations. Compare Figures 1, 2, and 3 with...

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  • February 4, 2023
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  • chapter5solutions university of alabama ec 410
  • chapter 5 understanding linear dependence a link to economic models solutions by wei lin and yingying sun university of california
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Gloria González-Rivera Forecasting For Economics and Business 2013


CHAPTER 5.

UNDERSTANDING LINEAR DEPENDENCE:
A LINK TO ECONOMIC MODELS

SOLUTIONS
by
Wei Lin and Yingying Sun
(University of California, Riverside)



Exercise 1
We simulate 1000 observations of the process pt = 6.43 + 0.55pt−1 + εt and plot 100 observations.
Compare Figures 1, 2, and 3 with Figures 5.2 and 5.3 in the textbook. The price oscillates around
an average price of $14.3 in all these graphs. The time series of the simulated prices in Figures 1, 2
and 3 exhibit smooth dynamics similar to those of the time series in Figure 5.3, in contrast to the
zig-zag behavior of the simulated price in Figure 5.2. This is due to the sign of the autoregressive
parameter, which is positive, i.e. φ = 0.55. When the variance of the error term εt increases,
the time series become noisier and more volatile, so that it tends to ‘hide’ the time dependence.
However, the autocorrelation functions in the three Figures 4, 5 and 6 deliver the same message.
The profile of the three ACF and PACFs is the same: a smooth decay of the autocorrelations
towards zero in the ACFs, and only a significant spike, partial autocorrelation of order one, in the
PACFs. Different variances in the error term do not affect the autocorrelation functions because
the effect of the error variance in the numerator and denominator of the autocorrelation coefficients
cancel each other out. Observe that these autocorrelation functions are similar to that in Figure
5.3 of the textbook. The main difference with the time series and the ACF and PCF in Figure
5.2 of the textbook is the sign of the autoregressive parameter. In Figure 5.2, the sign is negative
(φ = −0.6), which produces the ziz-zag behavior of the time series and the alternating signs of the
autocorrelation coefficients.




1

, Gloria González-Rivera Forecasting For Economics and Business 2013



16.0
Simulated price with variance = 0.25
15.5

15.0

14.5

14.0

13.5

13.0

12.5

12.0
200 225 250 275 300


Figure 1: Simulated price pt = 6.43 + 0.55pt−1 + εt with σε2 = 0.25


17
Simulated price with variance = 1
16


15


14

13

12

11
200 225 250 275 300


Figure 2: Simulated price pt = 6.43 + 0.55pt−1 + εt with σε2 = 1


18
Simulated price with variance = 2
17

16

15

14

13

12

11

10
200 225 250 275 300


Figure 3: Simulated price pt = 6.43 + 0.55pt−1 + εt with σε2 = 2


2

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