This question from Diebold book (1997) "Elements of Forecasting":

You are working for a London-based hedge fund, Thompson Energy Investors, and your

boss has assigned you to assess a model used to forecast U.S. crude oil imports. On

the last day of each quarter, the model is used to forecast oil imports at horizons

of 1-quarter-ahead through 4-quarter-ahead. Thompson has done this for each of the

past n = 80 quarters and has kept the corresponding four forecast error series, denote

EHAT1Q,...,EHAT4Q.

1. Based on a correlogram analysis in Fig 1 assess whether the 1-quarter-ahead forecast

errors are white noise. Be sure to discuss all parts of the correlogram: sample

autocorrelations, sample partial autocorrelations, standard errors and Ljung-Box

statistics. Why care? Note that in Fig 1 Q-Stat is the Ljung-Box statistic and

Prob is p-value of the statistic.

2. Table 1 report the result for the Sign test and the Wilcoxon signed rank for the 1-

step-ahead forecasts. State clearly which are the null and the alternative hypothesis

in both tests. Is the null rejected? Why are these test useful to assess the optimality

of the forecasts?

3. Tables 2 to 5 below report the results for the regression of each of the four forecast

error series on constants, in each case allowing for a MA(5) disturbances. Com-

ment on the signicance of the MA coecients in each of the four cases and use

the results to assess the optimality of the forecasts at each of the four horizons.

Does your 1-step-ahead MA(5)-based assessment match the correlogram-based as-

sessment obtained in part 1? Do the multi-step forecasts appear optimal? At the

bottom-left of each table you nd the F-statistic, that is the statistic for the null

hypothesis that all the slopes in the regression model are jointly zero. The column

labelled by Prob report the p-values for the t?statistics.

\fHypothesis Testing for EHAT1Q Included observations: 80 Test of Hypothesis: Median = 0.000000 Sample Median = -0.066231 Method Value Probability Sign (exact binomial) 41 0.9111 Sign (normal approximation) 0.111803 0.9110 Wilcoxon signed rank (normal approximation) 0.007194 0.9943 Table 1: Sign and Wilcoxon signed rank tests.Dependent Variable: EHAT1Q Included observations: 80 Variable Coefficient Std. Error t-Statistic Prob C 0.019423 0.122240 0.158892 0.8742 MA(1) 0.188542 D. 130726 1.442264 0.1535 MA(2) -0.118638 0. 121159 -0.979196 0.3307 MA(3) -0.02684G D.123065 -0.218141 0.8279 MA(4) 0.219670 0.103253 2.127499 0.0368 MA(5) -0.196911 D.118965 -1.655194 0.1022 R-squared 0.088414 Mean dependent var 0.022187 Adjusted R-squared 0.013489 S.D. dependent var 1.043690 S.E. of regression 1.036627 Akaike info criterion 3.001206 Sum squared resid 78.44552 Schwarz criterion 3.209633 Log likelihood -113.0482 Hannan-Quinn criter. 3.084771 F-statistic 1.180029 Durbin-Watson stat 2.006758 Prob(F-statistic) 0.326470 Table 2: 1-step-ahead Dependent Variable: EHAT2Q Included observations: 80 Variable Coefficient Std. Error t-Statistic Prob. C -0.021213 D.295211 -0.07185G 0.9429 MA(1) 1.103135 D.116691 9.453482 MA(2) 0.071319 D.169953 0.419641 0.6760 MA(3) -0.078919 0.175519 -0.449630 0.6543 MA(4) 0.239480 0.165937 1.446213 0.1524 MA(5) 0.185745 0.112600 1.649598 0.1033 R-squared 0.531 065 Mean dependent var D.ODG833 Adjusted R-squared 0.493172 S.D. dependent var 1.481816 S.E. of regression 1.054933 Akaike info criterion 3.055342 Sum squared resid 81.24044 Schwarz criterion 3.263770 Log likelihood -115.2137 Hannan-Quinn criter. 3.138907 F-statistic 13.81 189 Durbin-Watson stat 2.016917 Prob(F-statistic) 0.000000 Table 3: 2-step-aheadDependent Variable: EHAT3Q Included observations: 80 Variable Coefficient Std. Error t-Statistic Prob. 0.009GGG 0.346437 0.027901 0.9778 MA(1) 1.051045 0.119473 8.797311 0.DDOO MA(2) 0.798556 0.176488 4.524714 0.ODDO MA(3) 0.013386 0.203454 0.065794 0.9477 MA(4) 0.000175 D.170305 -0.001028 0.9992 MA(5) -0.048586 0.121368 -0.40 0320 0.6901 R-squared 0.639927 Mean dependent var -0.004287 Adjusted R-squared 0.610332 S.D. dependent var 1.698889 S.E. of regression 1.060504 Akaike info criterion 3.066492 Sum squared resid 82.10079 Schwarz criterion 3.274919 Log likelihood -115.6597 Hannan-Quinn criter. 3.150056 F-statistic 21.62279 Durbin-Watson stat 2.010739 Prob(F-statistic) 0.000000 Table 4: 3-step-ahead Dependent Variable: EHAT4Q Included observations: 80 Variable Coefficient Std. Error t-Statistic Prob. C -0.048GOD 0.447724 -0.108550 0.9139 MA(1) 1.025734 0.118676 8.643171 D.DDDO MA(2) 0.845217 0.169532 4.985580 0.00DO MA(3) 0.819713 0.185803 4.411737 0.0000 MA(4) 0.163041 0.180216 0.904697 D.3686 MA(5) -0.058953 0.120113 -0.490809 D.6250 R-squared 0.717039 Mean dependent var -0.012624 Adjusted R-squared 0.693782 S.D. dependent var 1.881788 S.E. of regression 1.041324 Akaike info criterion 3.034432 Sum squared resid 79.15801 Schwarz criterion 3.242859 Log likelihood -114.3773 Hannan-Quinn criter. 3.117996 F-statistic 30.83102 Durbin-Watson stat 1.965328 Prob(F-statistic) 0.000100 Table 5: 4-step-ahead