Stat 551, Fall 2009
Homework 2
Due date: September 24, 2009
Problem 1: Let {Xt } be a stationary process with mean 0 and autocovariance function γX . Let a and b
be constants, and let B denote the backshift operator.
(a) If Yt = a + b t + Xt , show that Yt is not stationary, but that ∇Yt = (1 − B)Yt is stationary.
(b) In general, show that if Yt = µt +Xt where µt is a polynomial in t of degree d, then ∇m Yt = (1−B)m Yt
is stationary for m ≥ d and nonstationary for 0 ≤ m < d.
(c) If Yt = a + b t + st + Xt , where st is a seasonal component with period 12, show that ∇∇12 Yt =
(1 − B)(1 − B 12 )Yt is stationary and express its autocovariance function in terms of γX .
(d) If Yt = (a + b t)st + Xt , where st is a seasonal component with period 12, show that ∇212 Yt =
(1 − B 12 )2 Yt is stationary and express its autocovariance function in terms of γX .
Problem 2: Let Zt , t = 0, ±1, ±2, . . . be independent normal random variables each with mean 0 and
finite variance σz2 , and c a finite constant. Which of the two processes below is stationary? For each
stationary process specify the mean and autocovariance function.
(a) Xt = Z1 cos(c t) + Z2 sin(c t)
(b) Xt = Zt cos(c t) + Zt−1 sin(c t)
Problem 3: Let {Xt } be a series in which we are interested. However, because the measurement process
itself is not perfect, we actually observe Yt = Xt + Wt . We assume that {Xt } and {Wt } are independent
processes, that {Wt } is a mean 0 white noise process, and that {Xt } is stationary. We call {Xt } the signal
2 /σ 2 the signal-to-noise ratio, or
and {Wt } the measurement noise or error process. We call σX
W
SNR. Show that the larger the SNR, the closer the autocorrelation function of the observed process {Yt }
is to the autocorrelation function of the desired signal {Xt }.
(Hint: express the autocorrelation function of the process {Yt } (denoted as ρY (k)) as a function of ρX (k),
2 and σ 2 ).
σX
W
Problem 4: Estimation of constant mean. Consider the simple situation where a constant mean
function is assumed for a time series model: Yt = µ + Xt , where E(Xt ) = 0 for all t. The most common
estimate of µ is the sample mean defined as
Ȳ =
n
1X
Yt .
n t=1
(a) Show that Ȳ is an unbiased estimate of µ.
(b) Suppose that {Xt } is a stationary time series with autocorrelation function ρX . Express V ar(Ȳ ) as
2 and ρ .
a function of σX
X
(c) Argue that for a moving average process of the form Yt = Wt − 12 Wt−1 , the negative correlation at
lag 1 improves the estimation of the mean compared with the estimation obtained by ignoring the
dependence in the data (i.e. assuming Yt is i.i.d.) for large sample sizes. Back your argument with
the theoretical derivations necessary. (Hint: use the results in part (b)).
Problem 5: Austrian Industrial Production Index. The data file austrian.dat gives the quarterly
Industrial Production index starting with the first quarter in 1980.
(a) Display and identify the main features in the time series plot of the data. Which assumptions of
stationarity seem to be violated by this series?
(b) Display and comment on the aspect of the differenced series. Which assumptions of stationarity seem
to be violated by this series?
(c) Suggest and implement a transformation of the original series which alleviates the heteroskedasticity
of the data. Plot the time series plot of the transformed data. Which assumptions of stationarity
seem to be violated by the transformed series?
For the following analyses, set aside the last four values of the series. We will use the first part of the data
as the training data set and the last four as validation points.
(d) Use least-squares to fit a seasonal-means trend to this (transformed) time series. Plot the fitted
values (using a different symbol) along with the original time series plot of the transformed data.
Comment on the model fit.
(e) Use least-squares to fit a polynomial trend to the transformed data. Plot the fitted values (using a
different symbol) along with the original time series plot of the transformed data. Comment on the
model fit.
(f) Construct a model for the trend which combines the two models in (d) and (e). Plot the fitted values
(using a different symbol) along with the original time series plot of the transformed data. Comment
on the model fit.
(g) Construct a cosine wave +polynomial model for the trend. Plot the fitted values (using a different
symbol) along with the original time series plot of the transformed data. Comment on the model fit.
(h) Compare the two models fit in (f) and (g). Report any visual differences, as well as MAPE (mean
absolute prediction error). Which model seems to fit the data better?
Validation
(i) Using the four values set aside as validation points, asses which of the two models provide a better
forecast for the data. For this, you will need to use the above regression models to predict at the
last four time points, do a simple visual comparison and then use a measure such as MAPE as your
criterion for comparison.
Residual analysis
(j) Using the residuals from the models fit in (f) and (g), construct and plot the residual time series and
ACF plots. Comment on the appropriateness of the stationarity assumption of the residuals.