June 8, 2005 FINAL Econ 240C 1 Mr. Phillips

Answer all questions. They are equally weighted.

1.  (50 points ) Monthly retail and food services sales, not seasonally adjusted, in millions of dollars, is available at FRED from January 1992 through April 2005 and is illustrated in the following figure., Figure 1-1.

a.  Is the series evolutionary or stationary? Why? Evolutionary, i.e. time dependent because of trend in mean and seasonality

A trend model was estimated using seasonal dummies as shown in table 1-1.

Table 1-1

Dependent Variable: RETAILNFOOD
Method: Least Squares
Sample: 1992:01 2005:03
Included observations: 159
Variable / Coefficient / Std. Error / t-Statistic / Prob.
TREND / 1071.044 / 10.62762 / 100.7793 / 0.0000
JAN / 133492.2 / 1839.284 / 72.57836 / 0.0000
FEB / 132605.6 / 1844.098 / 71.90810 / 0.0000
MAR / 160588.7 / 1848.961 / 86.85348 / 0.0000
APR / 156291.4 / 1881.088 / 83.08563 / 0.0000
MAY / 168399.3 / 1885.616 / 89.30730 / 0.0000
JUNE / 163033.1 / 1890.193 / 86.25209 / 0.0000
JULY / 161853.5 / 1894.818 / 85.41902 / 0.0000
AUG / 166950.1 / 1899.491 / 87.89202 / 0.0000
SEPT / 150534.3 / 1904.213 / 79.05332 / 0.0000
OCT / 158947.1 / 1908.981 / 83.26279 / 0.0000
NOV / 159524.2 / 1913.797 / 83.35478 / 0.0000
DEC / 202759.7 / 1918.660 / 105.6778 / 0.0000
R-squared / 0.987521 / Mean dependent var / 243866.7
Adjusted R-squared / 0.986495 / S.D. dependent var / 52864.68
S.E. of regression / 6143.387 / Akaike info criterion / 20.36236
Sum squared resid / 5.51E+09 / Schwarz criterion / 20.61328
Log likelihood / -1605.808 / F-statistic / 962.8034
Durbin-Watson stat / 1.332533 / Prob(F-statistic) / 0.000000

The representation of the estimated equation is:

Estimation Equation:

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RETAILNFOOD = C(1)*TREND + C(2)*JAN + C(3)*FEB + C(4)*MAR + C(5)*APR + C(6)*MAY + C(7)*JUNE + C(8)*JULY + C(9)*AUG + C(10)*SEPT + C(11)*OCT + C(12)*NOV + C(13)*DEC

A Wald test was used to test whether the October and November dummies were equal., and is reported in Table 1-2.

Table 1-2

Wald Test:
Equation: Untitled
Null Hypothesis: / C(11) =C(12)
F-statistic / 0.057344 / Probability / 0.811080
Chi-square / 0.057344 / Probability / 0.810744

b.  Are the October and November dummies significantly different? No, F-statistic on the restriction is not significant at 5% level.

The trend equation was re-estimated dropping the October dummy and adding a constant. The results are reported in Table 1-3.

Table1-3

Dependent Variable: RETAILNFOOD
Method: Least Squares
Sample: 1992:01 2005:03
Included observations: 159
Variable / Coefficient / Std. Error / t-Statistic / Prob.
TREND / 1071.044 / 10.62762 / 100.7793 / 0.0000
JAN / -25454.92 / 2366.429 / -10.75668 / 0.0000
FEB / -26341.53 / 2366.310 / -11.13190 / 0.0000
MAR / 1641.566 / 2366.238 / 0.693745 / 0.4889
APR / -2655.736 / 2410.478 / -1.101746 / 0.2724
MAY / 9452.143 / 2410.221 / 3.921692 / 0.0001
JUNE / 4085.946 / 2410.010 / 1.695406 / 0.0921
JULY / 2906.363 / 2409.846 / 1.206037 / 0.2298
AUG / 8003.011 / 2409.728 / 3.321126 / 0.0011
SEPT / -8412.802 / 2409.658 / -3.491284 / 0.0006
NOV / 577.0328 / 2409.658 / 0.239467 / 0.8111
DEC / 43812.60 / 2409.728 / 18.18155 / 0.0000
C / 158947.1 / 1908.981 / 83.26279 / 0.0000
R-squared / 0.987521 / Mean dependent var / 243866.7
Adjusted R-squared / 0.986495 / S.D. dependent var / 52864.68
S.E. of regression / 6143.387 / Akaike info criterion / 20.36236
Sum squared resid / 5.51E+09 / Schwarz criterion / 20.61328
Log likelihood / -1605.808 / F-statistic / 962.8034
Durbin-Watson stat / 1.332533 / Prob(F-statistic) / 0.000000

c.  Is there any substantial difference in theses two regressions? Does R2 differ? Does the standard error of the regression differ? No. They are two different specifications of the same regression, with identical R2 and identical ser’s, with the constant dropped in the first, but including all 12 monthly dummies.

d.  Relate the value of the constant = 158,947.1 to which of the estimated dummy variables in the first equation, Tale 1-1. October, this dummy was dropped and the added constant picks it up. With the constant included, each monthly dummy is now the value of that monthly dummy in Table 1-1 minus October.

The dummies for March and November were dropped from the estimated equation and the results are reported in Table 1-4.

Table 1-4

Dependent Variable: RETAILNFOOD
Method: Least Squares
Sample: 1992:01 2005:03
Included observations: 159
Variable / Coefficient / Std. Error / t-Statistic / Prob.
TREND / 1071.000 / 10.57315 / 101.2943 / 0.0000
JAN / -26217.13 / 1898.263 / -13.81111 / 0.0000
FEB / -27103.70 / 1898.118 / -14.27925 / 0.0000
APR / -3418.082 / 1952.350 / -1.750753 / 0.0821
MAY / 8689.841 / 1952.037 / 4.451679 / 0.0000
JUNE / 3323.687 / 1951.780 / 1.702900 / 0.0907
JULY / 2144.148 / 1951.581 / 1.098672 / 0.2737
AUG / 7240.840 / 1951.440 / 3.710512 / 0.0003
SEPT / -9174.929 / 1951.355 / -4.701824 / 0.0000
DEC / 43050.61 / 1951.445 / 22.06088 / 0.0000
C / 159712.8 / 1291.112 / 123.7017 / 0.0000
R-squared / 0.987478 / Mean dependent var / 243866.7
Adjusted R-squared / 0.986632 / S.D. dependent var / 52864.68
S.E. of regression / 6112.150 / Akaike info criterion / 20.34062
Sum squared resid / 5.53E+09 / Schwarz criterion / 20.55293
Log likelihood / -1606.079 / F-statistic / 1167.153
Durbin-Watson stat / 1.328835 / Prob(F-statistic) / 0.000000

The graph of the actual, fitted, and residual is shown in Figure 1-2. The graph of the correlogram of the residuals is shown in Figure 1-3.

e. Suggest how this trend regression might be improved The trace of the residuals in Fig.1-2 and the correlogram, Fig. 1-3, show there is still structure that needs to be fitted by adding an ARMA model to trend model plus seasonal dummies.

2.  (50 points) Exports of services for travel, in billions of nominal dollars, is available quarterly at FRED from the first quarter of 1960 through the fourth quarter of 2004. Its trace is plotted as Figure 2-1.This trace suggests that 9/11 may have had a once and for all impact.

Figure 2-1

To control for inflation and provide context, exports of travel services was measured as a percent of exports of services, resulting in exprtrvpct, whose trace follows as Figure 2-2.

This series was prewhitened by first differencing, resulting in a stationary but kurtotic series, dexprtrvpct, whose correlogram follows as Figure 2-3

a.  What is the approximate standard deviation for these autocorrelations and partial autocorrelations? ~ 1/1691/2 = 0.075

b.  Which partial autocorrelations, i.e. at which lags, are significant? Pacf’s at lags 1 and 4.

An autoregressive model was estimated. The results are reported in Table 2-1, and the correlogram of the residuals is displayed in Figure 2-4. These residuals are kurtotic.

Table 2-1
Dependent Variable: D(EXPRTRVPCT)
Method: Least Squares
Sample(adjusted): 1961:2 2001:2
Included observations: 161 after adjusting endpoints
Convergence achieved after 3 iterations
Variable / Coefficient / Std. Error / t-Statistic / Prob.
C / 0.068821 / 0.089065 / 0.772707 / 0.4409
AR(1) / -0.306182 / 0.072883 / -4.201005 / 0.0000
AR(4) / 0.249443 / 0.071177 / 3.504520 / 0.0006
R-squared / 0.174559 / Mean dependent var / 0.068463
Adjusted R-squared / 0.164110 / S.D. dependent var / 1.306059
S.E. of regression / 1.194090 / Akaike info criterion / 3.211104
Sum squared resid / 225.2846 / Schwarz criterion / 3.268522
Log likelihood / -255.4939 / F-statistic / 16.70640
Durbin-Watson stat / 1.967596 / Prob(F-statistic) / 0.000000
Inverted AR Roots / .64 / -.07+.69i / -.07 -.69i / -.80

c.  Explain why an ar(1) ar(4) model may have been chosen. Based on the significant pacf’s at lags 1 and 4 in Fig. 2-3

d.  Why was the model estimated only through 2001.2? Precaution, so the estimated model would not be affected by the event 9/11 in the third quarter of 2001.

e.  Based on the evidence presented is this a satisfactory model? Yes. Based on the Fig. 2-4 there is no significant structure in the residuals. No info is provided about conditional heteroskedasticity other than the residuals are kurtotic, so one should check for it.

3.  (50 points) To test for the effect of a once and for all effect from 9/11, a step function was postulated in levels, equal to zero though 2001.2 and one thereafter. The model from problem 2, above, was estimated in differences, with the intervention of 9/11modeled as well. The results are displayed as Table 3-1. Exports of travel services as a percent of exports of services was 26.4 % in the second quarter of 2001 and has averaged 21.8% since the fourth quarter of 2001.

a. Why is the intervention effect captured or modeled using dstep in Table 3-1? Although the event is modeled with a step function in levels, the equation is estimated in first differences so dstep is include in table 3-1.

b. By how much does exports of travel services as a percent of exports of services, decline according to this intervention model? By 3.27 percentage points.

c. Explain whether this estimate seem reasonable, compared to the data numbers provided in the paragraph above question a? Yes, the decline from the second quarter of 2001 to the period beginning with the fourth quarter of 2001 was 4.6 percentage points.

d. does the decline caused by 9/11 exceed two times the standard error of the regression? Yes, 3.27 compared to ser = 1.23

Table 3-1
Dependent Variable: D(EXPRTRVPCT)
Method: Least Squares
Sample(adjusted): 1961:2 2004:4
Included observations: 175 after adjusting endpoints
Convergence achieved after 6 iterations
Variable / Coefficient / Std. Error / t-Statistic / Prob.
C / 0.057836 / 0.083267 / 0.694585 / 0.4883
DSTEP / -3.270758 / 1.153243 / -2.836140 / 0.0051
AR(1) / -0.322245 / 0.070438 / -4.574871 / 0.0000
AR(4) / 0.204567 / 0.068936 / 2.967468 / 0.0034
R-squared / 0.167897 / Mean dependent var / 0.039497
Adjusted R-squared / 0.153299 / S.D. dependent var / 1.333701
S.E. of regression / 1.227223 / Akaike info criterion / 3.269977
Sum squared resid / 257.5390 / Schwarz criterion / 3.342314
Log likelihood / -282.1229 / F-statistic / 11.50114
Durbin-Watson stat / 2.011656 / Prob(F-statistic) / 0.000001
Inverted AR Roots / .60 / -.08+.66i / -.08 -.66i / -.77

The model reported in Table 3-1 was used to forecast the eight quarters of 2005 and 2006, as displayed in Figure 3-1, along with approximate 95% confidence intervals.

e. What would have to happen to exports of services, travel as a percent of exports of services in the next eight months to cause you to think maybe the effect of 9/11 was beginning to fade? If exports of travel services as a percent of exports of services was to rise up to or beyond the upper bound then one would begin to question the once and for all effect since the forecast and bounds are based on an estimated model including the step function as a model of the event.

4.  (50 points) About 85% of natural gas consumption in the United Sates is from domestic production, and about 95% of the imports are from Canada. In contrast, for crude oil, only about 39% of domestic consumption is met by domestic production.

The question is whether the price of one source of energy affects the price of the other. The monthly price of natural gas, natural(t), in dollars per million BTU, is available since November 1993. The price of West Texas Intermediate, westex(t), in dollars per barrel, is compared in Figure 4-1.

To place the two prices on a comparable footing, and to pre-whiten, they were both converted to fractional changes, dlnnat and dlnoil, and their traces follow in Figure 4-2.

The correlogram of dlnoil is shown in Figure 4-3. Its histogram shows it is normally distributed.

The correlogram of dlnnat, Figure 4-4, shows structure and its histogram show it is kurtotic.

The cross-correlation between dlnnat and dlnoil follows as Figure 4-5 and a Granger test is reproduced in Table 4-1.

Table 4-1
Pairwise Granger Causality Tests
Sample: 1993:11 2005:05
Lags: 12
Null Hypothesis: / Obs / F-Statistic / Probability
DLNOIL does not Granger Cause DLNNAT / 126 / 2.19417 / 0.01726
DLNNAT does not Granger Cause DLNOIL / 0.98476 / 0.46863

a.  Does causality appear to be two-way, one-way or no-way? One-way from dlnoil to dlnnat.

b.  What kind of model seems most appropriate, distributed lag or VAR? Distributed lag since causality is one-way.

c.  Does further work need to be done or does the cross-correlation function in Fig. 4-5 meet the Box-Jenkins requirement that the lag structure be revealed by using an orthogonal explanatory variable? From Fig.4-3, the correlogram of dlnoil shows it is already orthogonal so it does not have to be modeled further but can be cross-correlated with dlnnat., as shown in Fig. 4-5.

d.  Based on Figure 4-5 what lags would you include in your model as a starting point? Would you expect their coefficients to be positive or negative? Lags nine and ten look promising, cross-correlations of 0.1826 and 0.1429, which are both positive. Furthermore, this makes economic sense since natural gas and oil are substitutes for producing energy so the elasticity between the two prices should be positive.

The estimation results are reported in Table 4-2., and the correlogram of the residuals follows in Figure 4-6 with the correlogram of the residuals squared in Figure 4-8. These residuals are normally distributed, not kurtotic.

e. Is this a satisfactory model? What else might you do? It is satisfactory in the sense that the residuals are orthogonal and normal. Nonetheless, Fig. 4-7 shows evidence of conditional heteroskedasticity so estimation of an ARCH-Garch model should be pursued.