Page 625 -

624 Chapter 14 Introduction to Time Series Regression and Forecasting

d. Worried about a potential break, she computes a QLR test (with 15%
trimming) on the constant and AR coefficients in the AR(4) model.
The resulting QLR statistic was 3.94. Is there evidence of a break?
Explain.

e. Worried that she might have included too few or too many lags in the
model, the forecaster estimates AR(p) models for p = 0, 1, c, 6
over the same sample period. The sum of squared residuals from each
of these estimated models is shown in the table. Use the BIC to estimate
the number of lags that should be included in the autoregression. Do
the results differ if you use the AIC?

AR Order 0 1 2 3 4 56
SSR 19,533 18,643 17,377 16,285 15,842 15,824 15,824

14.3. Using the same data as in Exercise 14.2, a researcher tests for a stochastic
trend in ln(IPt), using the following regression:

∆ln(IPt) = 0 .030 + 0 .000014t - 0 .0085 ln(IPt - 1) + 0 .050∆ln(IPt - 1)
(0.015) (0.000009) (0.0044) (0 .054)

+ 0 .186∆ln(IPt - 2) + 0 .240∆ln(IPt - 3) + 0 .173∆ln(IPt - 4),
1 0 .0532 1 0 .0532 1 0 .0542

where the standard errors shown in parentheses are computed using the
homoskedasticity-only formula and the regressor t is a linear time trend.

a. Use the ADF statistic to test for a stochastic trend (unit root) in ln(IP).

b. Do these results support the specification used in Exercise 14.2?
Explain.

14.4 The forecaster in Exercise 14.2 augments her AR(4) model for IP growth to
include four lagged values of ∆Rt, where Rt is the interest rate on 3-month
U.S. Treasury bills (measured in percentage points at an annual rate).

a. The Granger-causality F-statistic on the four lags of ∆Rt is 4.16. Do
interest rates help predict IP growth? Explain.

b. The researcher also regresses ∆Rt on a constant, four lags of ∆Rt and
four lags of IP growth. The resulting Granger-causality F-statistic
on the four lags of IP growth is 1.52. Does IP growth help to predict
interest rates? Explain.