157hwk2keytopost_f11

157hwk2keytopost_f11 - Question 2(18 pts Time series...

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Question 2 (18 pts) Time series decomposition a. (12) Monthly data on US furniture and home furnishings sales are shown below, for the years 2008 and 2009. source: http://www.census.gov/retail/ Monthly Retail and Food Service sales, item category 442 not seasonally adjusted Use time series decomposition to forecast monthly sales for 2010, using both the multiplicative and additive models. Follow the steps and formating of the forecasting handout, page 1, to make your results easier for the readers to grade. Be sure to show your trend and seasonal components for full credit. 2008 2009 mult. seasonal additive seasonal mult forecast additive forecast Jan 8,205 6,902 0.971 -220 6,079 5,897 Feb 8,157 6,698 0.954 -346 5,973 5,771 Mar 8,451 7,025 0.994 -36 6,225 6,081 Apr 8,113 6,727 0.953 -354 5,969 5,763 May 8,704 7,103 1.014 130 6,354 6,247 Jun 8,225 7,082 0.984 -120 6,165 5,997 Jul 8,610 7,367 1.027 215 6,433 6,332 Aug 8,583 7,339 1.024 187 6,411 6,304 Sep 7,795 7,146 0.963 -303 6,031 5,814 Oct 7,831 7,036 0.957 -340 5,997 5,777 Nov 8,238 7,577 1.019 134 6,384 6,251 Dec 9,001 8,655 1.140 1,054 7,139 7,171 mean 8,326 7,221 multiplicative trend 0.8673 decline of over 13% additive trend -1,105 decline of $1105 b. (6) Now, suppose that later you found out what actual 2010 furniture sales values were (as shown below) Evaluate your forecasts as shown on the last step of the example in the handout. Which model (multiplicative or additive) had a better fit, based on MSE? Which model (multiplicative or additive) had a better fit, based on MAD? Did either model show evidence of bias, and if so, was the bias positive (upward), or negative (downward?) Multiplicative model Additive model 2010 error error^2 |error| error error^2 |error| Jan 6,525 446 198693 446 629 395012 629 Feb 6,684 711 505906 711 914 834482 914 Mar 7,531 1,306 1705328 1306 1450 2102500 1450 Apr 6,980 1,011 1022614 1011 1217 1481089 1217 May 7,194 840 705426 840 948 897756 948 Jun 7,168 1,003 1006345 1003 1172 1372412 1172 Jul 7,519 1,086 1178887 1086 1188 1410156 1188 Aug 7,611 1,200 1440152 1200 1307 1708249 1307 Sep 7,357 1,326 1758667 1326 1544 2382392 1544 Oct 7,026 1,029 1059479 1029 1250 1561250 1250 Nov 7,872 1,488 2212999 1488 1622 2629262 1622 Dec 8,762 1,623 2634594 1623 1591 2531281 1591 Means 1089 1285757 1089 1236 1608820 1236 MFE MSE MAD MFE MSE MAD The multiplicative model had a better goodness of fit, based on both MSE and MFE It is unusual to see that the MFE and MAD actually are the same number, because every error was positive for both models (thus they matched their absolute values) The reason for this is that we only had one observation to base our trend component on, which was the change in mean from 2008 to 2009. That change was very large and negative, thus both our models predicted sales would decline sharply again in 2010, when in fact, sales recovered somewhat. (Notice we would have gotten a better fit if we had simply averaged 2008 and 2009 sales by month to predict 2010, since 2010 sales look to be in between their high 2008 levels, and severely depressed 2009 levels. But, of course we wouldn't have known that ahead of time!!)
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157hwk2keytopost_f11 - Question 2(18 pts Time series...

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