Statistics 451
Spring 2015
Final Examination
Name
You must show all of your work
1. Let Zt denote the amount of sales in month t, measured in millions of dollars. Consider the
seasonal time series model for sales defined by
Wt = θ0 + (1 − Θ1 B12 )at ,
at ∼ nid(0, σa2 )
with the differencing scheme
Wt = (1 − B12 )Zt .
(a) Write down the unscrambled equation for Wt (note, that Z should not appear in this
unscrambled equation).
(b) Derive Var(Wt ).
(c) Derive ρ12 and ρ13 for Wt .
(d) Write down the unscrambled equation for Zt (note that W should not appear in this
unscrambled equation).
(e) Is Zt stationary or not? Explain briefly.
(f) What is the interpretation of θ0 in this model? Be precise by including the units of θ0 in
your answer.
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2. Consider the SARIMA(1, 1, 0)(1, 1, 1)12 model.
(a) Write down the unscrambled form of the model.
(b) Under what conditions will the model be invertible?
3. List two of the advantages or disadvantages of using VAR versus VARIMA to model multivariate time series.
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4. We saw that JMP has some useful features and capabilities that are not available in RTSERIES. List two of these.
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5. Derive Cov(Zt , at−2 ) for an ARMA(1,1) model.
6. One common application of Box-Jenkins time series forecasting is to generate forecasts for
exogenous X variables in a regression model so that forecasts for the regression response
variables can be generated. Briefly explain why this can lead to wrong answers when the
regression program is used to construct prediction intervals.
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7. A company manufactures a product, but does not use television advertising. A marketing
analyst argues that sales of their product (yt is total sales in week t) depends on past television
advertising levels for a competing product (xt is the competitor’s expenditures for advertising
in week t). The company’s sales have been stationary (no important seasonality or trend).
Weekly time series data are available for both x and y for the last 52 weeks. The marketing
managers are interested in developing a time series model to provide short term forecasts
for future values of y so that steps can be taken to avoid problems of stock-out (not having
product available when requested by a customer) and carrying excessive inventory. The model
for x has been identified as an AR(3).
(a) List the steps that you would use to identify the impulse-response function relating x
and y.
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The marketing analyst has identified the transfer function model as
yt = ν(B)xt + (1 − θ1 B)at =
(ω1 B − ω2 B2 )
xt + (1 − θ1 B)at
(1 − δ1 B)
(1)
(b) Based on (1), derive an expression for the first four terms of the impulse-response function relating x and y.
(c) Based on (1), write down the unscrambled model for y (only a finite number of lags lags
allowed on the right-hand side of the equation).
(d) Model (1) implies no feedback. Is it possible that there is a feedback relationship between
x and y in the actual problem? Explain why or why not and explain how the feedback
could be detected if it exists.
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8. A time series Zt can be described by a random walk Zt = Zt−1 + at . Assume that a realization
Z1 , Z2 , . . . Z100 is available to compute forecasts and that a forecast is needed for Z103 , using
Z100 as the forecast origin.
(a) Give an expression for Z103 , based on this random walk model.
(b) Give an expression that can be used to compute Zb100 (3), the forecast for Z103 .
(c) Derive e100 (3), the forecast error for Zb100 (3).
(d) Give an expression for the variance of e100 (3).
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