June 12, 2002FINAL Econ 240C 1Mr. Phillips

  1. (40 points) The monthly sales of light weight vehicles: autos and light trucks, in millions of units at an annual rate, and seasonally adjusted, is available at FRED from January 1976 through April 2002. The trace, histogram, correlogram, and Dickey-Fuller test are shown, respectively, as Figures 1.1-1.3 and Table 1.1.
  1. In the previous two recessions, with troughs in March of 1991 and November of 1982, what happened to the sales of light vehicles?
  2. Any sign of a similar phenomenon in this recession? (the most recent boom peaked in March 2001)
  3. Is this time series stationary? Discuss the evidence.
  4. Is this series normally distributed? Discuss the evidence.
  5. What do we mean by stationary? Would a trended time series be stationary? Would a seasonal time series be stationary?


Table 3.1 Dickey-Fuller Unit Root Test of Light Vehicle Sales

ADF Test Statistic / -2.423867 / 1% Critical Value* / -3.4531
5% Critical Value / -2.8709
10% Critical Value / -2.5717
*MacKinnon critical values for rejection of hypothesis of a unit root.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(VEHICLES)
Method: Least Squares
Sample(adjusted): 1976:04 2002:04
Included observations: 313 after adjusting endpoints
Variable / Coefficient / Std. Error / t-Statistic / Prob.
VEHICLES(-1) / -0.068606 / 0.028304 / -2.423867 / 0.0159
D(VEHICLES(-1)) / -0.362993 / 0.056225 / -6.456075 / 0.0000
D(VEHICLES(-2)) / -0.296130 / 0.054464 / -5.437164 / 0.0000
C / 0.999948 / 0.406949 / 2.457181 / 0.0146
R-squared / 0.203563 / Mean dependent var / 0.013163
Adjusted R-squared / 0.195831 / S.D. dependent var / 1.049949
S.E. of regression / 0.941546 / Akaike info criterion / 2.730111
Sum squared resid / 273.9315 / Schwarz criterion / 2.777986
Log likelihood / -423.2624 / F-statistic / 26.32606
Durbin-Watson stat / 2.107880 / Prob(F-statistic) / 0.000000
  1. (40 points) Instead of looking at the sales of light vehicles, we could look at fractional changes from month to month, i.e. the first difference of the natural logarithm of light vehicle sales, denoted dlnveh(t). The trace, histogram, correlogram, and Dickey-Fuller test are shown, respectively, as Figures 2.1-2.3 and Table 2.1.
  1. Is the fractional change in vehicle sales, dlnveh(t), stationary?
  2. Is the fractional change in vehicle sales, dlnveh(t), normally distributed?
  3. What would be your first guess for specifying a pure moving average model?
  4. What would be your first guess for specifying a two-parameter ARMA model?


Table 2.1 Dickey-Fuller Unit Root Test of Fractional Changes in Light Vehicle Sales

ADF Test Statistic / -18.88973 / 1% Critical Value* / -3.4531
5% Critical Value / -2.8709
10% Critical Value / -2.5717
*MacKinnon critical values for rejection of hypothesis of a unit root.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(DLNVEH)
Method: Least Squares
ADF Test Statistic
Sample(adjusted): 1976:04 2002:04
Included observations: 313 after adjusting endpoints
Variable / Coefficient / Std. Error / t-Statistic / Prob.
DLNVEH(-1) / -1.644999 / 0.087084 / -18.88973 / 0.0000
D(DLNVEH(-1)) / 0.285479 / 0.054422 / 5.245682 / 0.0000
C / 0.001411 / 0.003678 / 0.383739 / 0.7014
R-squared / 0.669073 / Mean dependent var / 7.81E-05
Adjusted R-squared / 0.666938 / S.D. dependent var / 0.112722
S.E. of regression / 0.065054 / Akaike info criterion / -2.617672
Sum squared resid / 1.311911 / Schwarz criterion / -2.581766
Log likelihood / 412.6656 / F-statistic / 313.3815
Durbin-Watson stat / 2.112347 / Prob(F-statistic) / 0.000000
  1. (40 points) A pure autoregressive model was estimated for the fractional changes in light vehicle sales, dlnveh(t). The estimation results are reported in Table 3.1. A plot of the actual and fitted series, and the residual, is shown in Figure 3.1. The correlogram of the residuals is shown in Figure 3.2
  1. Is this a satisfactory model? Discuss the evidence.
  2. The values for dlnveh(t) for the first four months of 2002 are:

2002:01 -0.04136

2002:02 0.05235

2002:03 0.01566

2002:04 0.02829

Ignoring the constant, which is insignificantly different from zero, what is your forecast for the fractional change in light vehicle sales for May of this year?

  1. What is the standard error of your forecast?
  2. Lastly, the trace of the square of the residual from this autoregressive model is shown in Figure 3.3.What does the evidence in this figure imply about the characteristics of fractional changes in light vehicle sales and about modeling dlnveh(t).
  1. Might the evidence in Figure 3.3 affect your answer to part c.

Table 3.1 Autoregressive Model of Fractional Changes in the Sales of Light Vehicles

Dependent Variable: DLNVEH
Method: Least Squares
Sample(adjusted): 1976:06 2002:04
Included observations: 311 after adjusting endpoints
Convergence achieved after 3 iterations
Variable / Coefficient / Std. Error / t-Statistic / Prob.
C / 0.000857 / 0.001577 / 0.543241 / 0.5874
AR(1) / -0.447769 / 0.056522 / -7.922078 / 0.0000
AR(2) / -0.412746 / 0.060216 / -6.854450 / 0.0000
AR(3) / -0.264149 / 0.060259 / -4.383583 / 0.0000
AR(4) / -0.153989 / 0.056561 / -2.722532 / 0.0068
R-squared / 0.206317 / Mean dependent var / 0.000990
Adjusted R-squared / 0.195942 / S.D. dependent var / 0.070683
S.E. of regression / 0.063381 / Akaike info criterion / -2.663353
Sum squared resid / 1.229255 / Schwarz criterion / -2.603228
Log likelihood / 419.1514 / F-statistic / 19.88605
Durbin-Watson stat / 2.036670 / Prob(F-statistic) / 0.000000
Inverted AR Roots / .22 -.65i / .22+.65i / -.44+.37i / -.44 -.37i


  1. (40 points) Autos and light trucks are a consumer durable and are likely to be affected by the interest rate if consumers take out an auto loan. As an indicator of the cost of credit, the prime loan rate from banks is available in a monthly series from FRED beginning in January 1949. This interest rate data from January 1976 is combined with the data for light vehicle sales. The change in the prime rate, dprime, is related to the fractional change in light vehicle sales. The trace, histogram and correlogram of dprime are shown in Figures 4.1-4.3, respectively. This dprime series is stationary. The cross-correlation between dlnveh and dprime is shown in Figure 4.4. The Granger Causality Test, using 12 lags, is displayed in Table 4.1.
  1. Is there any evidence of either variable causing the other? Discuss.
  2. What kind of bivariate relationship do you think should be investigated? Why?

i No model

ii A distributed lag model, with which variable, dlnveh or dprime, as the dependent?

iii A VAR model

A distributed lag model was estimated and the results are reported in Table 4.2, with the residuals from this model shown in Figure 4.5, and their histogram in Figure 4.6.

  1. Is this a satisfactory model? Discuss.
  2. The prime rate is currently 4.75 %. If the prime rate went up to 5.75% in May, what would be the resulting fractional increase in light vehicle sales in June?
  3. The prime rate has been at 4.75% for the first four months of 2002. If it stays at this level for the rest of this year what would you expect to happen to fractional changes in light vehicle sales over the next 18 months?
  4. Can you think of any way to improve on the model reported in Table 4.2?


Table 4.1 Granger Causality Test Between dlnveh(t) and dprime(t)

Pairwise Granger Causality Tests
Sample: 1976:01 2002:04
Lags: 12
Null Hypothesis: / Obs / F-Statistic / Probability
DPRIME does not Granger Cause DLNVEH / 303 / 2.94865 / 0.00070
DLNVEH does not Granger Cause DPRIME / 1.28634 / 0.22592

Table 4.2 Distributed Lag Model of Fractional Changes in Light Vehicle Sales on the Change in the Prime Rate

Dependent Variable: DLNVEH
Method: Least Squares
Sample(adjusted): 1977:12 2002:04
Included observations: 293 after adjusting endpoints
Convergence achieved after 7 iterations
Variable / Coefficient / Std. Error / t-Statistic / Prob.
C / 0.000571 / 0.001454 / 0.392464 / 0.6950
DPRIME(-1) / -0.008167 / 0.003891 / -2.099322 / 0.0367
DPRIME(-5) / -0.013596 / 0.003882 / -3.502310 / 0.0005
DPRIME(-12) / -0.009564 / 0.003902 / -2.451058 / 0.0148
DPRIME(-18) / -0.012526 / 0.003976 / -3.150578 / 0.0018
AR(1) / -0.526696 / 0.058626 / -8.983979 / 0.0000
AR(2) / -0.479627 / 0.064322 / -7.456622 / 0.0000
AR(3) / -0.312396 / 0.064378 / -4.852499 / 0.0000
AR(4) / -0.171491 / 0.059058 / -2.903778 / 0.0040
R-squared / 0.288297 / Mean dependent var / 0.000608
Adjusted R-squared / 0.268249 / S.D. dependent var / 0.072374
S.E. of regression / 0.061910 / Akaike info criterion / -2.696030
Sum squared resid / 1.088534 / Schwarz criterion / -2.582987
Log likelihood / 403.9684 / F-statistic / 14.38039
Durbin-Watson stat / 2.051818 / Prob(F-statistic) / 0.000000
Inverted AR Roots / .20 -.67i / .20+.67i / -.46+.37i / -.46 -.37i