## Concluding Examples

We conclude this chapter by considering a few additional examples that illustrate some of the points we have made in this chapter.

THE RELATIONSHIP BETWEEN THE HELP-WANTED INDEX (HWI) AND THE UNEMPLOYMENT RATE (UN) FROM JANUARY 1969 TO JANUARY 2000

To study causality between HWI and UN, two indicators of labor market conditions in the United States, Marc A. Giammatteo considered the following regression model28:

25 25

25 25

To save space we will not present the actual regression results, but the main conclusion that emerges from this study is that there is bilateral causality between the two labor market indicators and this conclusion did not change when the lag length was varied. The data on HWI and UN are given in the data disk.

26T. Bollerslev, "Generalized Autoregressive Conditional Heteroscedasticity," Journal of Econometrics, vol. 31, 1986, pp. 307-326.

27For details, see Davidson and MacKinnon, op. cit., pp. 558-560.

28Marc A. Giammatteo (West Point, Class of 2000), "The Relationship between the Help Wanted Index and the Unemployment Rate," unpublished term paper. (Notations altered to conform to our notation.)

CHAPTER TWENTY-TWO: TIME SERIES ECONOMETRICS: FORECASTING 863

ARIMA MODELING OF THE YEN/DOLLAR EXCHANGE RATE: JANUARY 1971 TO DECEMBER 199829

The yen/dollar exchange rate (¥/\$) is a key exchange rate. From the logarithms of the monthly ¥/\$, it was found that in the level form this exchange rate showed the typical pattern of a nonstationary time series. But examining the first differences, it was found that they were stationary; the graph here pretty much resembles Figure 22.8.

Unit root analysis confirmed that the first differences of the logs of ¥/\$ were stationary. After examining the correlogram of the log first differences, we estimated the following ARIMA(1, 0, 2) model:

Yt =-0.0034 + 0.9678 Yt-1 + -0.5866ut-1 - 0.4057ut-2

t = (-4.3638) (67.5439) (-11.4361) (-7.9532) (22.11.3)

where Yt = first differences of the logs of ¥/\$ and u is a white noise error term.

To save space, we have provided the data underlying the preceding analysis in the data disk. Using these data, the reader is urged to try other models and compare their forecasting performances.

ARCH MODEL OF THE U.S. INFLATION RATE: JANUARY 1947 TO JANUARY 2001

To see if the ARCH effect is present in the U.S. inflation rate as measured by the CPI, we obtained CPI data from January 1947 to January 2001. The plot of the logarithms of the CPI showed that the time series was nonstationary. But the plot of the first differences of the logs of the CPI, as shown in Figure 22.9, show considerable volatility even though the first differences are stationary.