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628 Chapter 14 Introduction to Time Series Regression and Forecasting

c i. Run an OLS regression of ∆Inflt on ∆Inflt-1. Does knowing the
change in inflation this quarter help predict the change in inflation
next quarter? Explain.

ii. Estimate an AR(2) model for ∆Infl. Is the AR(2) model better
than an AR(1) model? Explain.

iii. Estimate an AR(p) model for p = 0, c, 8. What lag length is
chosen by BIC? What lag length is chosen by AIC?

iv. Use the AR(2) model to predict the change in inflation from
2012:Q4 to 2013:Q1—that is, predict the value of ∆Infl2013:Q1.

v. Use the AR(2) model to predict the level of the inflation rate in
2013:Q1—that is, Infl2013:Q1.

d. i. Use the ADF test for the regression in Equation (14.31) with two
lags of ΔInfl to test for a stochastic trend in Infl.

ii. Is the ADF test based on Equation (14.31) preferred to the test
based on Equation (14.32) for testing for stochastic trend in Infl?
Explain.

iii. In (i) you used two lags of ΔInfl. Should you use more lags? Fewer
lags? Explain.

iv. Based on the test you carried out in (i), does the AR model for
Infl contain a unit root? Explain carefully. (Hint: Does the failure
to reject a null hypothesis mean that the null hypothesis is true?)

e. Use the QLR test with 15% trimming to test the stability of the coeffi-
cients in the AR(2) model for ΔInfl. Is the AR(2) model stable? Explain.

f. i. Using the AR(2) model for ΔInfl with a sample period that
begins in 1963:Q1, compute pseudo out-of-sample fore-
casts for the change in inflation beginning in 2003:Q1 and
going through 2012:Q4. (That is, compute ∆Infl2003:Q1͉2002:Q4,
∆Infl2003:Q2͉2003:Q1, c, ∆Infl2012:Q4͉2012:Q3.)

ii. Are the pseudo out-of-sample forecasts biased? That is, do the
forecast errors have a nonzero mean?

iii. How large is the RMSFE of the pseudo out-of-sample forecasts?
Is this consistent with the AR(2) model for ΔInfl estimated over
the 1963:Q1–2002:Q4 sample period?

iv. There is a large outlier in 2008:Q4. Why did inflation fall so much
in 2008:Q4? (Hint: Collect some data on oil prices. What happened
to oil prices during 2008?)