STA457H1/STA2202HF: Time Series Analysis Assignment 2 solution

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Q1 (1 points): For the Box and Cox power transformations, use calculus to show that, for any
fixed x > 0, as λ → 0, (x
λ − 1)/λ → log(x).
1

Q2 (1 points): Use two different smoothing techniques (e.g., Kernel, Lowess, spline, etc.) to
estimate the trend in the global temperature series globtemp. Comment!

Q3 (8 points for undergraduate students and 6 points for graduate students): Consider
the two weekly time series oil and gas. The oil series is in dollars per barrel, while the gas series is
in cents per gallon.
1. Plot the data on the same graph. Do you believe the series are stationary (explain your answer)?
2. In economics, it is often the percentage change in price (termed growth rate or return), rather
than the absolute price change, that is important. Argue that a transformation of the form
yt = ∇ log xt might be applied to the data, where xt
is the oil or gas price series.
3. Transform the data as described in part (2), plot the data on the same graph, look at the
sample ACFs and CCF of the transformed data, and comment.
4. Exhibit scatterplots of the oil and gas growth rate series for up to three weeks of lead time of
oil prices; include a nonparametric smoother in each plot and comment on the results (e.g.,
Are there outliers? Are the relationships linear?).
5. There have been a number of studies questioning whether gasoline prices respond more quickly
when oil prices are rising than when oil prices are falling (“asymmetry”). We will attempt to
explore this question here with simple lagged regression; we will ignore some obvious problems
such as outliers and autocorrelated errors, so this will not be a definitive analysis. Let Gt and
Ot denote the gas and oil growth rates.
(i) Fit the regression (and comment on the results)
Gt = β0 + β1It + β2Ot + β3Ot−1 + wt
where wt ∼ wn(0, σ2
w) and It = 1 if Ot ≥ 0 (i.e., It
is the indicator of increase in the oil
price) and 0 otherwise (no positive growth in the oil price).
(ii) What is the fitted model when there is negative growth in oil price at time t? What is
the fitted model when there is no or positive growth in oil price? Do these results support
the asymmetry hypothesis?
(iii) Fit the regression
Gt = β0 + β1Ot + wt
and decide (using AIC/BIC) whether it is better than the previous one or not?
(iv) Test the full model as specified in part (i) against the reduced model as as specified in
part (iii).
(v) Analyze the residuals from the best fit and comment.

Q4 (2 points): Quarterly earnings per share for the Johnson & Johnson Company are given in
the data file named jj. The earnings per share data, say yt
, covers the years from 1960 through
1980. In this problem, you are going to fit a special type of structural model, xt = log(yt), where
xt = Tt + St + Et
, where Tt
is a trend component, St
is a seasonal component, and Et
is error (noise)
term. Note that the time t is in quarters (1960.00, 1960.25, · · ·) so one unit of time is a year.
1. Fit the regression model
xt = βt
|{z}
trend
+ α1Q1(t) + α2Q2(t) + α3Q3(t) + α4Q4(t)
| {z }
seasonal
+ wt
|{z}
noise
where Qi(t) = 1 if time t corresponds to quarter i = 1, 2, 3, 4, and zero otherwise. The Qi(t)’s
are called indicator variables. We will assume for now that wt
is a Gaussian white noise
sequence.
2. If the model is correct, what is the estimated average annual increase in the logged earnings
per share?
3. If the model is correct, does the average logged earnings rate increase or decrease from the
third quarter to the fourth quarter? And, by what percentage does it increase or decrease?
4. What happens if you include an intercept term in the model in (a)? Explain why there was a
problem.
5. Graph the data, xt
, and superimpose the fitted values, say ˆxt
, on the graph. Examine the
residuals, xt − xˆt
, and state your conclusions. Does it appear that the model fits the data well
(do the residuals look white)?