GR5221/GU4221
Summer 2021
Homework 3
Due date : Friday May 28 to Canvas
The submission link will be closed at 11:59 pm (US Eastern Time) on the due day.
1. An AR(2) process has ρ(1) = 0.75, ρ(2) = 0.475 and ρ(3) = 0.2775. Calculate its
partial autocorrelation function (pacf). (A number must be given for the pacf at any
lag.)
2. Consider the following AR(2) model:
Xt −Xt−1 + 1
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Xt−2 = Zt, Zt ∼WN(0, 1).
(a) Show that Xt is causal.
(b) Find the first four coefficients (ψ0, . . . , ψ3) of the MA(∞) representation of Xt.
(c) Find the pacf at lag 3, φ33, of the AR(2) model.
3. Use simulated data sets to compare the performance of AIC, BIC, and AICC.
(a) Generate 1,000 data sets from MA(2) with θ1 = 0.8 and θ2 = −0.1. For each data
set, calculate the AIC, BIC, and AICC selection of q (set the maximum possible
value of q to 4). Present the estimated probability of q = 0, 1, 2, 3, 4 in a table
and summarize your finding.
(b) Generate 1,000 data sets from AR(3) with φ1 = 0.8, φ2 = −0.2, and φ3 = 0.1. For
each data set, calculate the AIC, BIC, and AICC selection of p (set the maximum
possible value of p to 5). Present the estimated probability of in a table and
summarize your finding.
4. The daily closing IBM stock price data is recorded in ibmclose.csv.
(a) Plot the series, ACFs and PACFs. Do you think the data is stationary?
(b) Transform and then difference the data. Choose the best transformation among
logXt,
√
Xt and X
2
t . Keeping the mean included, fit an ARMA(p,q) model and
select (p, q) by AIC, BIC and AICC. Report the MLE of model parameters for
each criterion.
(c) For the (p, q) selected by AIC in (c), fit the transformed data with or without the
mean. Do you think the mean term should be included in the estimation? Hint:
you may think about the confidence interval of the mean or AIC (or BIC, AICC).
(d) Conduct all necessary model goodness of fit diagnostics.
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5. The file unemployment.csv contains the U.S. Civilian Unemployment Rate, monthly,
from December, 1948 to the Present.
(a) For the log of the unemployment series, use time series plots, ACFs, and PACFs
to choose d for an ARIMA model. Explain your reasoning.
(b) With your choice of d from part (a), consider all 9 possible models with p ≤ 2
and q ≤ 2 with a nonzero mean. Use the AIC to identify p, q.
(c) Estimate the parameters under the model identified in (a). Construct asymptotic
confidence intervals for the AR and/or MA operator coefficients under confidence
level 95%. Note, the asymptotic variances are returned if you use the arima
function in R.
(d) Plot the residuals from the fitted model, as well as the ACF and PACF of the
residuals. Do these plots indicate any inadequacies in the model?
(e) Examine the Ljung-Box statistics for lack of fit for lags equal to 12, 24, 36, and 48.
According to these statistics, does the model seem to be adequate? The model is
declared to be inadequate if the p-value is less than 0.05 (in this case, we reject
the null hypothesis that the model is adequate).
(f) Obtain the one-step-ahead prediction.
6. Consider the Motor Vehicle Retail Sales data in motor.csv retrieved from Federal
Reserve Bank of St. Louis. Take logarithm of the data and conduct the following
analysis.
(a) Obtain the time plot, the ACF and PACF plots. Do you think the process is
stationary?
(b) Differnce the data once. Repeat the data visualization on the differenced data.
(c) Does the differenced data have seasonal behavior? If so, what is the seasonal
period? Do you think a seasonal difference should be conducted? If so, take a
seaonsal difference of your choice and visualize the seasonally differenced data.
(d) Based on (a)-(c), if we would like to fit an ARIMA(p, d, q)× (P,D,Q)s model to
the log-transformed data, what is the appropriate selection of (d,D, s)?
(e) With your selection of (d,D, s), fit ARIMA(p, d, q)×(P,D,Q)s to the log-transformed
data and select p, q, P,Q by AIC. You may set the maximum possible values of
p, q, P,Q to 3 and exclude the mean term.
(f) Conduct all necessary model goodness of fit diagnostics.
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