* SEASONAL ARIMA MODELS * * EXAMPLE 1 * * Reference: G.E.P. Box and G.M. Jenkins, * TIME SERIES ANALYSIS FORECASTING AND CONTROL, Revised Edition, * Prentice Hall, 1976, Chapter 9 (pp. 300-333) * * Data set: Box and Jenkins, Series G, page 531 * International airline passengers, monthly totals * (thousands of passengers) January 1949 to December 1960 SAMPLE 1 144 TIME 1949 12 DATE READ Z / BYVAR 112 118 132 129 121 135 148 148 136 119 104 118 115 126 141 135 125 149 170 170 158 133 114 140 145 150 178 163 172 178 199 199 184 162 146 166 171 180 193 181 183 218 230 242 209 191 172 194 196 196 236 235 229 243 264 272 237 211 180 201 204 188 235 227 234 264 302 293 259 229 203 229 242 233 267 269 270 315 364 347 312 274 237 278 284 277 317 313 318 374 413 405 355 306 271 306 315 301 356 348 355 422 465 467 404 347 305 336 340 318 362 348 363 435 491 505 404 359 310 337 360 342 406 396 420 472 548 559 463 407 362 405 417 391 419 461 472 535 622 606 508 461 390 432 * NOTE: Box and Jenkins suggest that the log transformation is appropriate * for analyzing this data set. (See Section 9.3.5, p. 328) * The log transformation is requested by using the LOG option on the * ARIMA command. * ===> Identification * Reference: Box and Jenkins, Section 9.2.3, p. 313 * Table 9.3 p. 314 gives the estimated autocorrelations for various * differences of the logged airline data. These are computed in the * identification phase of the ARIMA command. ARIMA Z / LOG PLOTAC NLAG=48 PLOTPAC NLAGP=48 ARIMA Z / LOG PLOTAC NLAG=48 PLOTPAC NLAGP=48 NDIFF=1 ARIMA Z / LOG PLOTAC NLAG=48 PLOTPAC NLAGP=48 NSDIFF=1 NSPAN=12 ARIMA Z / LOG PLOTAC NLAG=48 PLOTPAC NLAGP=48 NDIFF=1 NSDIFF=1 NSPAN=12 * ===> Estimation and Diagnostic Checking * Reference: Box and Jenkins, Sections 9.2.4 & 9.2.5 * Estimation of an ARIMA(p,d,q)(P,D,Q)_s process * where p = AR order * d = number of nonseasonal differences * q = MA order * P = seasonal AR order * D = number of seasonal differences * Q = seasonal MA order * s = seasonal period * * The model specification considered is an ARIMA(0,1,1)(0,1,1)_12 process. * The estimates and standard errors are reported at the top of p. 319 of * Box and Jenkins. ARIMA Z / LOG NDIFF=1 NSDIFF=1 NSPAN=12 NOCONSTANT NMA=1 NSMA=1 COEF=BETA GEN1 S=SQRT($SIG2) * ===> Forecasting * Reference: Box and Jenkins * Origin date is July 1957 * The standard errors of the log forecasts can be compared with * the values reported in Table 9.2, p. 311 ARIMA Z / LOG NDIFF=1 NSDIFF=1 NSPAN=12 NOCONSTANT NMA=1 NSMA=1 COEF=BETA & SIGMA=S FBEG=1957.7 FEND=1960.12 GRAPHFORC DELETE / ALL COMPRESS * EXAMPLE 2 * * Reference: Walter Enders, APPLIED ECONOMETRIC TIME SERIES, Wiley, * 1995, Chapter 2 (pp. 111 - 118). * Data set: Monthly tourists visiting Spain, January 1970 to March 1989. * * NOTE: * The SHAZAM results have differences from Enders due to differences * in the estimation algorithm - also note that Enders uses a * different calculation method for the AIC and SC statistics. * SAMPLE 1 231 TIME 1970 12 DATE READ SPAIN / BYVAR 820671 796113 1297542 1188844 1671536 2302318 4204587 5270484 2496770 1538107 1101932 1163845 983136 980217 1236050 1898694 1937907 2586122 4689122 5223043 2853964 1753837 1254621 1360902 1113857 1117236 1689481 1925956 2163178 2943452 6237912 6774008 3367080 1902552 1414334 1797537 1202174 1126462 1477043 2388930 2126606 3148788 6637510 7236634 3655213 2177020 1538605 1753958 1241184 1144593 1524258 2225725 1981724 2814555 5521607 6464274 3205520 1589380 1180996 1451959 1204519 1013728 1547030 1493496 2118609 2939469 5950696 6697118 3265465 1396028 1043701 1452619 1201802 1054301 1334284 1969020 2002472 2775668 5561911 6159885 3060935 1959262 1338885 1595662 1329367 1239682 1548567 2157524 2234009 2995528 6423660 7037003 3421103 2266724 2134599 2002474 1606495 1482481 2223206 2124732 2789463 3363263 7017592 7968573 4242651 2747310 2050642 2354083 2041841 1708735 2137271 2868814 2897433 3867278 5878686 6746902 4029229 2514236 1837401 2374483 1926865 1601051 1995573 2469649 2589818 3138548 6365971 7918659 3930044 2249793 1671046 2132434 1808070 1463234 1748698 2571631 2719622 3343539 7100990 7877330 4562962 2770367 1870463 2323017 1956456 1656160 2000723 2769373 2997200 3791947 6953878 8029299 4665631 2875148 1910816 2357866 1861374 1660730 2199483 2671888 2943001 3797188 6645627 7674030 4590932 2972699 1939699 2306683 1893032 1727653 2057087 3124812 3197066 3966746 7365809 8277600 4765686 3129638 2028559 2413525 1809238 3489028 2149235 2919590 3079927 4019152 6837385 8002015 4696554 3385040 2073529 2875588 1984935 1760252 2626673 2645591 3501433 4248450 7403956 9102854 5253947 3279344 2398096 3183039 2178426 2047278 2372760 3337538 3864519 4672446 7911943 9376459 5368966 3602793 2411988 3394269 2412604 2362577 2920339 3505098 4000187 4515161 8735355 9684267 5818146 4243678 2624818 3356166 2597707 2335168 3194855 * ===> Identification ARIMA SPAIN / PLOTAC NLAG=48 PLOTPAC NLAGP=48 * Inspect the autocorrelation function of the logarithmic 12th differences ARIMA SPAIN / LOG PLOTAC NLAG=48 PLOTPAC NLAGP=48 NSDIFF=1 NSPAN=12 * Now consider 1st differences of the seasonally differenced data. ARIMA SPAIN / LOG PLOTAC NLAG=48 PLOTPAC NLAGP=48 NDIFF=1 NSDIFF=1 NSPAN=12 * ===> Estimation and Diagnostic Checking * Model 1 ARIMA SPAIN / LOG NDIFF=1 NSDIFF=1 NSPAN=12 NSAR=1 NMA=1 NOCONSTANT * Model 2 ARIMA SPAIN / LOG NDIFF=1 NSDIFF=1 NSPAN=12 NMA=1 NSMA=1 NOCONSTANT * Model 3 ARIMA SPAIN / LOG NDIFF=1 NSDIFF=1 NSPAN=12 NMA=12 NOCONSTANT START RESTRICT .1 0 0 0 0 0 0 0 0 0 0 .1 STOP