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204

(b) Since the data are quarterly, regress the data on seasonal dummies, and compute the residuals. Repeat the analysis with these residuals (assuming that they are the observations).

5. Consider the VAR models with one and two lags in Exercise 4.

(a) Estimate the characteristic roots and vectors for the relevant matrices discussed in Section 14.8. Apply tests for unit roots and tests for cointegration.

(b) If there is a cointegrating regression, estimate it from the characteristic vectors and also from the static regression as suggested by Granger and Engle.

(c) Are the results different for the VAR models with one and two lags? Are they different from those from the static regressions? What do you conclude from these results?

(d) Repeat parts (a) to (c) with the seasonally adjusted data (residuals from the regression on seasonal dummies).

6. Repeat Exercises 4 and 5 using the data in Table 13.5.

7. Repeat Exercises 4 and 5 using the data in Table 13.7. This time, when adjusting for seasonality, you have to regress the original series on a constant and 11 monthly dummies.

8. For the data in Table 4.10:

(a) Estimate a regression of HS on and RR.

(b) Estimate a VAR model with one lag. Compute the characteristic roots. Test for cointegration and estimate the cointegrating vectors, if any.

(c) What sense can you make of the multiple regression estimated in part (a)?

(d) Repeat the analysis with residuals from a regression of the raw data on quarterly dummies.

9. In the data set in Table 13.8, examine which of the time series are trend stationary and which are difference stationary. Also investigate whether there are any cointegrating relationships among the series.

10. Consider the data in Table 13.7. Take yearly data for each month (e.g.. Mar 52, Mar 53, Mar 54, ... , Mar 82; similarly, Nov 52, Nov 53, ... , Nov 82). We thus have 12 annual time series.

(a) For each of these series, apply unit root tests.

(b) For each of these series, compute a VAR model with one lag. Apply unit root tests and check for any cointegrating regressions.

(c) Compare the results of this analysis with those in the Exercise 7.



Sample Size

K-Test

t-Test

F-Test"

AR(1)

-11.9

-1.

-2.66

-1.95

-12.9

-1.1

-2.62

-1.95

-13.3

-7.9

-2.60

-1.95

-13.6

-8.0

-2.58

-1.95

-13.7

-8.0

-2.58

-1.95

-13.8

-8.1

-2.58

-1.95

AR(1)

with constant

-17.2

-12.5

-3.75

-3.00

-18.9

-13.3

-3.58

-2.93

-19.8

-13.7

-3.51

-2.89

-20.3

-14.0

-3.46

-2.88

-20.5

-14.0

-3.44

-2.87

-20.7

-14.1

-3.43

-2.86

AR(1)

with constant and trend

-22.5

-17.9

-4.38

-3.60

7.24

10.61

-25.7

-19.8

-4.15

-3.50

6.73

9.31

-27.4

-20.7

-4.04

-3.45

6.49

8.73

-28.4

-21.3

-3.99

-3.43

6.34

8.43

-28.9

-21.5

-3.98

-3.42

6.30

8.34

-29.5

-21.8

-3.96

-3.41

6.25

8.27

"A:=7"(p-1),/ = (p-l)/SE(p) and F-test is for-y = Oand p=l iny, = a + - / + , , + ,. Source: W. A. Fuller, Introduction to Statistical Time Series (New York: Wiley, 1976), p. 371 for the *:-test and p. 373 for the r-test; D. A. Dickey and W. A. Fuller, "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Vol. 49, No. 4, 1981, p. 1063 for the F-test.

Table 14.1 Critical Values for Unit Root Tests



CRDW

ADF"

0.78

-3.67

-3.29

0.39

-3.37

-3.17

0.20

-3.37

-3.25

0.99

-4.11

-3.75

0.55

-3.93

-3.62

0.39

-3.78

-3.78

1.10

-4.35

-3.98

0.65

-4.22

-4.02

0.48

-4.18

-4.13

1.28

-4.76

-4.15

0.76

-4.58

-4.36

0.57

-4.48

-4.43

"CRDW = X - «, CRDW means

"cointegrating regression Durbin-Watson" statistic; DF = f-test for a = 0 in «, = a«, + i),; ADF = Mest for

a = 0 , = a«, i + 2j . , + 1),. In all these tests I

, is the residual from the cointegrating regression. Source - R. F. Engle and S. Yoo, "Forecasting and Testing in Cointegrated Systems," Journal of Econometrics, Vol. 35, 1987.

Table 14.2 Critical Values (5%) for the Cointegration Tests



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