June 14, 2006 FINAL Econ 240C 1 Mr. Phillips

Answer all Four questions.

1.  (40) A longer data set for the import and export price and freight rate exist for #2 Dark Northern Spring 14%, than we examined in class. It is available from September 1971 though December 1990 for the difference, DIFF(t), in $/metric ton between the Japanese import price minus the sum of the Gulf export price (lagged 1) plus the Gulf freight rate (lagged 2).

Diff(t) = Import Price – [Export Price(t-1) + Freight Rate(t-2)]

The time series, DIFF(t), is plotted against time in Figure 1-1.

Notice there may be a trend in the price differential and also a seasonal component. The latter shows a little bit in the correlogram of Diff(t) shown in Figure 1-2, in the cycle peaking at lag 12.

To investigate these possibilities, a trend model was estimated for Diff(t) and twelve seasonal dummies were added for each month. The results are displayed in Figure 1-3. The correlogram of the residuals from this regression are shown in Figure 1-4

a.  Is there a significant trend? Explain. Yes coefficient on the time variable is significantly negative.

b.  Are there significant seasonal effects? Explain. Yes five of the monthly dummies are quite significant.

c.  Is the regression as a whole significant? Yes, the F-statistic is significant.

d.  Are the necessary assumptions for least squares met? No. The regression residuals are not independent of one another as indicated by the correlogram of the residuals.

e.  What might you suggest as a means to improve this regression and deal with any problems that you have detected, or are you done? You could model the residual by adding an AR(1) term to the regression.

2.  (40) The residual from the trend regression with seasonal dummies in question 1 above was saved as RESDIFF(t). RESDIFF(t) is the difference between the observations of the price differential, DIFF(t), and the fitted trend, adjusting for the seasonal pattern in the monthly data. A first order autoregressive process was estimated for RESDIFF(t) and the results are displayed in Figure 2-1.

Diagnostics were conducted for this ARONE model including a plot of the actual, fitted and residuals, Figure 2-2, the correlogram of the residuals, Figure 2-3, the correlogram of the residuals squared, Figure 2-4, and the histogram of the residuals, Figure 2-5.

  1. Express the estimated ARONE model for RESDIFF(t). RESDIFF(t) = 0.40*RESDIFF(t-1) + N(t), where N(t) is orthogonal
  1. Suppose there is a random shock to RESDIFF(t) at time t, say of magnitude one, and RESDIFF equals one at time t, and there are no more shocks, what value will RESDIFF equal in time t+1? In time t+2? Advance RESDIFF(t) by one: RESDIFF(t+1) = 0.4*RESDIFF(t) + N(t+1) At time t, given RESDIFF(t) = 1, and N(t+1) = 0, since no more shocks aftertime t. So RESDIFF(t+1) = 0.4, and the following period RESDIFF(t+2) = 0.16
  1. Is RESDIFF(t) stationary? Yes. You can fit a stable ARONE model to it.
  1. Is this a satisfactory model? Explain? It is except for the conditional heteroskedasticity.
  1. Is the residual from the ARONE model for RESDIFF(t) significantly skewed? Is it significantly kurtotic? It is not skewed but it is kurtotic.

3.  The residual from the ARONE Model for RESDIFF was squared and is plotted in Figure 3-1. An ARONE, ARCH-GARCH model was estimated through December 1989. The estimation results are shown in Figure 3-2. The correlogram for the standardized residuals is shown in Figure 3-3. The Arch-LM test for the standardized residuals is shown in Figure 3-4, and the histogram of the standardized residuals is shown in Figure 3-5. The within sample forecast for the twelve months of 1990 is shown in Figure 3-6.

  1. What appears to be the noisiest period for RESDIFF(t) ? 1973 and 1974
  1. What is the formula for the conditional variance, h(t), in this case? h(t) = 8.896 + 0.127*[e(t-1)]2 + 0.757*h(t-1)
  1. What is meant by the standardized residual? How is it calculated? The residual is partitioned into two parts; e(t) = WN(t)*h(t)1/2 . WN(t) is referred to as the standardized residual because it has variance one, should be orthogonal, homoskedastic, and hopefully normal. It is calculated by dividing the estimated residual from the model, e(t), by the square root of the estimated conditional variance h(t), as indicated in the formula above.
  1. Is this a satisfactory model? Explain. Yes. The standardized residuals are orthogonal and homoskedastic but are not normal, still kurtotic.
  1. What is the point of doing a forecast of the twelve months of 1990, since we already know what happened from the observations? To see how well the model fits the data.

Figure 3-4: ARCH-LM test for the Standardized Residuals

4.  (40) The import price was first differenced and called DDNSM. The export price (lagged one) plus freight rate (lagged two) was also first differenced and called DDNSX. A plot of these two time series are shown in Figure 4-1. Their cross-correlation function was estimated and shown in Figure 4-2. The Granger causality test is shown in Figure 4-3. Similar results were obtained using six lags and twelve lags. Lastly a distributed lag model was estimated and the results are shown in Figure 4-4. The correlogram for the residuals from this model are shown in Figure 4-5.

  1. Do these series look like prewhitened by eye? Yes.
  1. Is the causality one-way, two-way or no-way? One-way
  1. From the cross-correlation function, which way does the causality appear to run? Is this confirmed by the Granger test? From DDNSM to DDNSX. Yes
  1. Does this appear to be a satisfactory model? Yes the residuals fro the distributed lag model appear to be orthogonal.
  1. What other evidence might you want? Is there any conditional heteroskedasticity. Are the residuals kurtotic?
Figure 4-3
Pairwise Granger Causality Tests
Sample: 1971:09 1990:12
Lags: 3
Null Hypothesis: / Obs / F-Statistic / Probability
DDNSX does not Granger Cause DDNSM / 228 / 1.29032 / 0.27857
DDNSM does not Granger Cause DDNSX / 1791.07 / 0.00000