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Unobserved Components Model with Seasonality

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Accepted Solution

Unobserved Components Model with Seasonality

Hi

I have monthly frequency data from January 1960 to August 2016 (680 data points). I have to apply an Unobserved Components Model to decompose the data into separte components and then forecast.

As my data is monthly I attempt to include a trend, seasonal, cycle term and error term.

However I am not sure whether the trigonometric or the dummy seasonal term is suitable for the data.

I am attaching the data.

 

data WORK.AONE;
  infile datalines dsd truncover;
  input date:$8. VAR1:32. VAR2:32.;
datalines4;
1960M01,2815.009988,12000.36325
1960M02,2823.514977,12036.61994
1960M03,2798.154332,11892.40053
1960M04,2798.154332,11916.472
1960M05,2806.557623,11867.20712
1960M06,2806.557623,11988.47465
1960M07,2789.801214,12265.43789
1960M08,2798.154332,12060.90082
1960M09,2798.154332,12181.80525
1960M10,2773.245208,12073.36302
1960M11,2773.245208,11953.53488
1960M12,2781.497819,11905.35752
1961M01,2773.245208,11726.3494
1961M02,2773.245208,11858.10613
1961M03,2781.497819,12228.93178
1961M04,2806.557623,12665.59739
1961M05,2815.009988,13080.1978
1961M06,2815.009988,13577.73626
1961M07,2806.557623,13827.24151
1961M08,2798.154332,14256.32833
1961M09,2806.557623,14431.93298
1961M10,2806.557623,14299.14229
1961M11,2798.154332,14593.876
1961M12,2798.154332,14328.54275
1962M01,2781.497819,14195.39317
1962M02,2745.530168,14267.17771
1962M03,2705.626162,14466.31136
1962M04,2713.727247,14304.47128
1962M05,2721.876989,13899.6728
1962M06,2721.876989,13246.69381
1962M07,2713.727247,13001.87648
1962M08,2705.626162,12818.51985
1962M09,2681.611984,12704.74721
1962M10,2697.5733,12828.22394
1962M11,2697.5733,13091.97775
1962M12,2705.626162,12903.06393
1963M01,2705.626162,12818.51985
1963M02,2713.727247,12856.90058
1963M03,2721.876989,12943.79936
1963M04,2730.074327,13359.23762
1963M05,2721.876989,13693.90211
1963M06,2713.727247,13676.97181
1963M07,2705.626162,13540.14453
1963M08,2713.727247,13652.90034
1963M09,2713.727247,14075.24523
1963M10,2765.751199,14129.22591
1963M11,2765.751199,14658.30826
1963M12,2774.032308,15220.82839
1964M01,2757.519385,15621.83903
1964M02,2765.751199,16690.63901
1964M03,2870.519905,16185.87552
1964M04,2894.642931,15751.49489
1964M05,2894.642931,15920.54226
1964M06,2886.001772,17808.80273
1964M07,2877.412052,18774.92066
1964M08,2877.412052,19099.03874
1964M09,2877.412052,21388.53057
1964M10,2877.412052,23774.82277
1964M11,2877.412052,22323.90491
1964M12,2937.357306,19758.15135
1965M01,2928.642265,18743.07328
1965M02,2928.642265,18384.67328
1965M03,2928.642265,19436.55832
1965M04,2911.363274,21259.08366
1965M05,2902.800013,22653.51124
1965M06,2885.825272,22073.84627
1965M07,2885.825272,21190.14443
1965M08,2885.825272,21861.73646
1965M09,2877.411366,22420.21794
1965M10,2869.047905,21301.19934
1965M11,2860.731405,20235.26421
1965M12,2844.243713,20374.03207
1966M01,2836.070187,20616.34816
1966M02,2819.863259,20245.33087
1966M03,2819.863259,19692.9814
1966M04,2819.863259,19647.03929
1966M05,2819.863259,19255.48721
1966M06,2819.863259,18358.04984
1966M07,2803.841966,18242.32675
1966M08,2788.000258,17695.35111
1966M09,2772.337978,17335.71491
1966M10,2780.147775,17304.76931
1966M11,2780.147775,17089.61781
1966M12,2788.000258,17217.37739
1967M01,2780.147775,16998.51298
1967M02,2788.000258,17080.59225
1967M03,2795.898672,17175.05094
1967M04,2803.841966,17395.15023
1967M05,2788.000258,17330.93486
1967M06,2764.571936,17230.85144
1967M07,2756.849282,17159.75003
1967M08,2764.571936,16846.97683
1967M09,2756.849282,16675.88795
1967M10,2756.849282,16743.77155
1967M11,2756.849282,16833.60249
1967M12,2741.534082,16203.07982
1968M01,2733.939424,15823.62094
1968M02,2718.877015,15658.41924
1968M03,2718.877015,15669.49344
1968M04,2718.877015,15636.27083
1968M05,2703.979667,15440.46089
1968M06,2799.008103,15356.31245
1968M07,2791.401721,15270.38977
1968M08,2799.008103,15246.27954
1968M09,2783.836569,15207.21627
1968M10,2783.836569,15403.80582
1968M11,2768.829981,16352.59589
1968M12,2768.829981,16048.21952
1969M01,2807.984476,15811.29129
1969M02,2861.984638,15897.50976
1969M03,2846.719953,15801.99896
1969M04,2846.719953,16102.64137
1969M05,2824.127178,16219.92662
1969M06,2809.263959,16251.4095
1969M07,2794.555032,16450.92919
1969M08,2801.890191,16642.30342
1969M09,2794.555032,16598.73499
1969M10,2863.925205,16816.35155
1969M11,2862.541321,17195.91396
1969M12,2847.82449,17936.10755
1970M01,2833.258208,17957.92666
1970M02,2833.258208,17607.43235
1970M03,2826.030145,17696.29349
1970M04,2871.509098,18034.8891
1970M05,2884.77243,17463.02869
1970M06,2877.412946,16514.17885
1970M07,2862.806057,16225.82497
1970M08,2877.412946,16874.03394
1970M09,2862.806057,16910.80921
1970M10,2855.559406,16970.23582
1970M11,2841.172994,16641.26016
1970M12,2841.172994,16093.08722
1971M01,2819.864424,15831.69262
1971M02,2805.835822,15742.93136
1971M03,2798.873099,15993.59422
1971M04,2798.873099,16153.64805
1971M05,2785.050812,15884.83226
1971M06,2778.191988,15518.06856
1971M07,2778.191988,15557.67384
1971M08,2820.853607,15618.66554
1971M09,2877.412856,15865.51487
1971M10,2877.412856,15706.6437
1971M11,2880.240287,15866.03927
1971M12,2967.243612,16359.17724
1972M01,2972.394836,16084.17381
1972M02,2958.03603,16025.89566
1972M03,2958.03603,16726.33896
1972M04,2965.198707,17011.06301
1972M05,2640.285993,16579.48002
1972M06,2482.717441,16393.48572
1972M07,2340.294604,16238.09845
1972M08,2349.885845,16334.24623
1972M09,2334.736445,16505.9546
1972M10,2345.880313,16392.4203
1972M11,2338.749403,16027.78103
1972M12,2316.789216,15999.95882
1973M01,2299.259376,16066.60155
1973M02,2332.217888,16841.57886
1973M03,2349.886546,17646.88937
1973M04,2334.115663,17456.10119
1973M05,2363.315186,17653.20218
1973M06,2422.35052,18234.19867
1973M07,2549.830183,19929.40014
1973M08,2408.45528,19456.25133
1973M09,2358.494703,19565.17071
1973M10,2427.355869,20925.06154
1973M11,2528.448787,21667.60426
1973M12,2428.874092,24804.82853
1974M01,2252.60305,24252.36842
1974M02,2268.484966,27165.13664
1974M03,2503.692898,29915.83033
1974M04,2714.775611,33246.07585
1974M05,2730.169944,33210.50594
1974M06,2693.436921,32203.29203
1974M07,2632.178355,29579.53681
1974M08,2888.129514,29901.46156
1974M09,2824.323424,30904.01006
1974M10,2789.996834,24483.40728
1974M11,2738.149805,24069.86963
1974M12,2790.527667,23376.05772
1975M01,2879.43825,24695.12705
1975M02,2921.99173,24311.98504
1975M03,2960.786019,23835.84251
1975M04,2876.402605,22855.60962
1975M05,2877.412621,21959.70897
1975M06,2852.564159,21860.32919
1975M07,2694.750098,21234.99813
1975M08,2607.80851,21088.94381
1975M09,2551.759052,20243.8197
1975M10,2488.31349,19361.76431
1975M11,2481.588411,19160.12692
1975M12,2551.305536,18805.68334
1976M01,2593.979852,18975.66149
1976M02,2595.916243,19852.58813
1976M03,2488.31349,21056.61648
1976M04,2485.168124,21584.80485
1976M05,2494.392215,22536.2015
1976M06,2426.332928,23419.17056
1976M07,2716.504385,25535.12229
1976M08,2827.164494,24336.05142
1976M09,2989.862495,23885.87682
1976M10,2812.016058,23647.07631
1976M11,2802.916056,24013.76168
1976M12,2858.014777,24635.81978
1977M01,3088.8511,27690.32373
1977M02,3116.892929,29568.96138
1977M03,3109.95812,29801.01722
1977M04,3085.699145,27356.48088
1977M05,3061.816568,27843.11729
1977M06,3113.120083,27190.05844
1977M07,3290.874719,29283.12181
1977M08,3328.059232,31793.19203
1977M09,3312.769688,31221.51928
1977M10,3332.566663,33648.41287
1977M11,3413.775504,34989.77627
1977M12,3465.010418,35128.68464
1978M01,3584.115571,33007.47944
1978M02,3569.523589,32840.91468
1978M03,3499.578064,30783.11649
1978M04,3348.185696,29271.79965
1978M05,3261.067342,30860.90326
1978M06,3266.563755,32348.22755
1978M07,3347.444213,32221.25842
1978M08,3437.318874,33823.52214
1978M09,3572.548738,35896.41005
1978M10,3635.66679,38933.55342
1978M11,3543.739402,38210.73479
1978M12,3540.760714,35573.93914
1979M01,3135.709265,34347.71376
1979M02,3531.796987,35580.29546
1979M03,3751.05819,36550.46598
1979M04,3777.714466,36441.10763
1979M05,3769.483879,36740.50419
1979M06,3803.523864,38440.24414
1979M07,3617.866189,37206.68031
1979M08,3658.92459,34654.37572
1979M09,3727.412404,35528.11717
1979M10,4054.482749,36345.56822
1979M11,4123.780339,36816.02429
1979M12,4263.927015,37852.01734
1980M01,4496.682202,37229.81142
1980M02,4597.25043,36890.89912
1980M03,4228.046167,37336.24267
1980M04,4091.618725,36030.78662
1980M05,3742.590308,35916.18875
1980M06,3493.237942,36050.69313
1980M07,3617.601931,35167.8471
1980M08,3637.412071,34706.20637
1980M09,3375.23215,35297.40937
1980M10,3265.247641,33517.06163
1980M11,3009.868227,31252.26404
1980M12,2850.771892,29305.63363
1981M01,2811.262244,28160.57107
1981M02,2829.268276,26558.14374
1981M03,2781.249292,26277.22482
1981M04,2614.341356,25518.39673
1981M05,2466.105392,24078.35433
1981M06,2336.003001,23889.6891
1981M07,2216.284679,25189.18287
1981M08,2325.774349,27370.85764
1981M09,2208.774609,28221.36665
1981M10,2130.254872,28235.57986
1981M11,2019.199745,29557.29201
1981M12,2104.277679,29681.53867
1982M01,2054.818034,29776.86592
1982M02,2007.568109,29517.54504
1982M03,1903.440174,23610.46278
1982M04,1842.408765,23336.37627
1982M05,1796.420006,23506.6532
1982M06,1683.923152,20319.60817
1982M07,1743.160942,20595.79778
1982M08,1741.693535,22151.60272
1982M09,1749.740593,23115.35887
1982M10,1718.00582,22414.06269
1982M11,1737.591308,21570.29144
1982M12,1774.061085,21547.63742
1983M01,1943.702854,21816.05583
1983M02,2221.581042,23534.24221
1983M03,2355.153876,24289.67271
1983M04,2469.404091,24977.99702
1983M05,2621.634346,24392.52724
1983M06,2635.834217,24016.73726
1983M07,2728.659396,23680.57059
1983M08,2872.713326,22976.69064
1983M09,2902.187278,22939.99467
1983M10,2792.206067,22730.55042
1983M11,2711.900771,22836.07107
1983M12,2767.30045,22051.06029
1984M01,2742.458549,21455.59055
1984M02,2630.101576,21690.04838
1984M03,2562.717755,21839.08006
1984M04,2412.207817,21907.55768
1984M05,2271.455395,22171.49092
1984M06,2246.610525,22311.39749
1984M07,2039.077407,21812.41154
1984M08,1999.405706,21656.17967
1984M09,1792.008714,21408.31594
1984M10,1807.722293,20655.23646
1984M11,2019.787889,21027.67888
1984M12,1924.227735,20595.49156
1985M01,1887.624719,19436.88261
1985M02,1925.795329,19194.82396
1985M03,1922.554728,19677.7066
1985M04,1936.282768,20663.95089
1985M05,1926.974434,20765.36384
1985M06,1799.695088,21595.29492
1985M07,1758.743385,22176.92652
1985M08,1780.601523,22093.22435
1985M09,1742.514361,21801.68308
1985M10,1688.894937,21563.38594
1985M11,1644.519126,17135.94466
1985M12,1793.079459,15725.67955
1986M01,1938.505904,13699.99437
1986M02,1957.962251,13914.64244
1986M03,2084.713611,13053.62237
1986M04,2086.833597,10190.25842
1986M05,2080.208167,9780.53384
1986M06,2113.690175,9609.193152
1986M07,2020.931994,9693.117268
1986M08,2034.121243,9734.953406
1986M09,2167.634578,9702.241913
1986M10,2075.99165,9903.098712
1986M11,2020.102321,10873.79452
1986M12,2023.083008,11473.14455
1987M01,2073.886224,11884.39138
1987M02,2263.110786,11793.76534
1987M03,2410.911966,11693.43263
1987M04,2453.09558,11731.33598
1987M05,2461.881044,11767.70768
1987M06,2560.161616,11494.49671
1987M07,2863.125625,11093.18508
1987M08,3126.273567,11321.53213
1987M09,3024.529225,11095.60985
1987M10,3375.967684,11634.39935
1987M11,2885.826506,11892.31764
1987M12,3134.265469,11735.73141
1988M01,3430.972509,11664.8542
1988M02,3656.097486,11440.33161
1988M03,4311.854532,11559.39042
1988M04,4256.189576,11457.86901
1988M05,5045.93531,11444.94644
1988M06,6018.55699,11835.38855
1988M07,4318.595811,11956.69245
1988M08,4509.181577,12224.24011
1988M09,3984.968393,12367.47141
1988M10,3837.121264,12050.05841
1988M11,3938.54653,12091.99401
1988M12,4126.785047,12099.12397
1989M01,3913.024461,12093.15595
1989M02,3549.732516,12802.97241
1989M03,3353.294207,14166.17266
1989M04,3409.16654,16351.65054
1989M05,3600.326487,16164.50677
1989M06,3050.31224,15832.72938
1989M07,2803.256697,15334.23384
1989M08,2878.788703,13952.11688
1989M09,2742.591548,13202.17315
1989M10,2881.972337,12544.99272
1989M11,2749.070535,10826.75435
1989M12,2581.580691,10659.93065
1990M01,2379.160963,10262.34521
1990M02,2274.094822,9626.450981
1990M03,2448.785161,9796.609237
1990M04,2377.066972,9958.111703
1990M05,2368.6991,9811.238449
1990M06,2428.98192,9456.615372
1990M07,2434.84908,9182.296104
1990M08,2728.231954,9040.726443
1990M09,3120.246657,8616.979995
1990M10,2890.542665,9004.849314
1990M11,2400.933833,8878.334181
1990M12,2279.113397,8406.030809
1991M01,2274.066104,8439.610156
1991M02,2271.790451,8430.809742
1991M03,2276.437066,8395.731194
1991M04,2116.133599,8453.22289
1991M05,1967.058505,8653.661107
1991M06,1935.337875,8667.168935
1991M07,1971.577406,8619.854766
1991M08,1905.307961,8550.798305
1991M09,1837.614899,8446.923423
1991M10,1733.442924,8364.925135
1991M11,1709.018643,8291.330751
1991M12,1660.044437,8334.026563
1992M01,1781.854353,8290.958149
1992M02,1911.137817,8471.836503
1992M03,1930.055443,8489.126848
1992M04,1976.701621,8763.344668
1992M05,1949.628541,9132.202775
1992M06,1898.178579,9849.605565
1992M07,1955.642125,10409.93494
1992M08,1947.032597,10115.82684
1992M09,1894.158887,9911.135168
1992M10,1740.825804,8923.330281
1992M11,1720.434225,8494.926942
1992M12,1796.316383,8565.948443
1993M01,1788.226801,8744.197295
1993M02,1773.461717,8545.109172
1993M03,1693.217512,8323.06566
1993M04,1624.825429,8194.849479
1993M05,1646.066669,8060.009174
1993M06,1710.894506,7506.470728
1993M07,1767.927152,7313.01486
1993M08,1733.996473,7114.601702
1993M09,1652.816143,6658.802041
1993M10,1597.418034,6878.551706
1993M11,1529.209263,6825.196208
1993M12,1616.118952,7056.250332
1994M01,1727.425115,7269.189127
1994M02,1859.843272,7971.991267
1994M03,1886.241367,7909.737622
1994M04,1868.038517,7867.785787
1994M05,1927.312406,8021.154257
1994M06,2040.961454,8028.148941
1994M07,2165.798946,7707.610031
1994M08,2101.587714,7453.375788
1994M09,2273.490199,7709.224136
1994M10,2452.040057,7904.058454
1994M11,2723.994146,8867.636588
1994M12,2707.884106,8573.262775
1995M01,2948.804951,8871.673355
1995M02,2731.019617,7808.02378
1995M03,2566.44671,7865.960753
1995M04,2614.077848,8301.853182
1995M05,2484.047791,8351.842486
1995M06,2510.528763,9408.153827
1995M07,2623.428866,9405.615162
1995M08,2663.229501,9861.728755
1995M09,2485.41916,8932.140518
1995M10,2348.196841,8724.394435
1995M11,2319.796663,8957.486266
1995M12,2319.832599,8806.805611
1996M01,2216.611293,8746.837925
1996M02,2221.459504,8647.577743
1996M03,2245.314475,8628.241352
1996M04,2196.210667,8967.460112
1996M05,2188.66301,8830.891539
1996M06,2046.428957,8547.355419
1996M07,2013.602828,8627.833273
1996M08,2015.278748,8414.714223
1996M09,1936.665434,8396.693724
1996M10,1828.867821,8131.874899
1996M11,1986.861537,8212.841679
1996M12,2056.452132,7999.834419
1997M01,2158.007732,8050.29293
1997M02,2167.417988,8070.425824
1997M03,2239.898102,8112.435178
1997M04,2145.298703,7849.635642
1997M05,2234.718834,7852.900294
1997M06,2162.159105,7677.454496
1997M07,2201.060995,7524.993675
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1998M02,2039.115749,7294.184007
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1998M05,1894.304704,8154.399325
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1998M08,1825.376281,7925.188118
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2016M07,1561.840864,17090.77957
2016M08,1579.705151,17713.77865
;;;;

Thank you


Accepted Solutions
Solution
‎10-11-2016 11:43 PM
SAS Employee
Posts: 57

Re: Unobserved Components Model with Seasonality

I am glad you were able to run the programs.  Yes, in this case, dropping the seasonal component is reasonable.  By the way, it might be good to have some good book handy for such modeling questions.  There is a new book in the market, Pelagatti, M. M. (2015). Time Series Modelling with Unobserved Components. Boca Raton, FL: CRC Press, which might be worth a look.  The book's web site is: http://www.ucmbook.info/ 

View solution in original post


All Replies
SAS Super FREQ
Posts: 93

Re: Unobserved Components Model with Seasonality

I think the best way to determine which method is more suitable to model the seasonality for this particular data is to try both and validate the forecast against the holdout periods. In general the trig model is much smoother than the dummy model. Furthermore, I did a simple test of seasonality on the data you provided with diffing of 0, 1, and 2 and adding ar 0 and 1, neither var1 nor var2 are seasonal (with sig level 0.05). You will need some other variables to explain the variation in the data in addition to the seasonal component.

thanks

Alex

SAS Employee
Posts: 57

Re: Unobserved Components Model with Seasonality

Posted in reply to alexchien

As you requested, I am going to show you how to specify a y = trend + season + irregular model for your data using PROC UCM.  You can add a cycle component also by simply adding a cycle statement.  You can choose a trigonometric season or dummy season.  Usually the fits are similar.  I will also show you how to withhold data for model checking using the back= option in the ESTIMATE and FORECAST statements.  I ran this code for var1.  As noted by Alex in the earlier post, the seasonal component in the model is not very significant (at least for var1).  For var1, I have used simple random walk trend because the plot of the data does not seem to indicate a local linear model (you can get local linear trend by adding a slope statement).

 

*add a numeric date variable;

data AONE;

set AONE;

date1 = intnx('month', '01jan1960'd, _n_-1);

format date1 date.;

run;

 

*y = rw trend + season + irregular; 

proc ucm data=aone;

id date1 interval=month;

model var1;

irregular;

level;

season length=12 type=trig plot=smooth;

estimate back=24 plot=panel;

forecast back=24 lead=24 plot=decomp;

run;

 

/*

Dropping the season component seems reasonable.

Two alternate models are shown (just by trial and error).

First one:

y = rw trend + AR(1) irregular; 

Second one:

y = rw trend + second-order cycle + irregular;

(the higher order cycle (order=2) is an undocumented feature, that produces smoother cycles).

For your data it produces a cyle of about 7 years.  See if it makes business/domain sense.

*/

proc ucm data=aone;

id date1 interval=month;

model var1;

irregular p=1;

level;

estimate back=24 plot=panel;

forecast back=24 lead=24 plot=decomp;

run;

 

proc ucm data=aone;

id date1 interval=month;

model var1;

irregular;

level;

cycle order=2 plot=smooth;

estimate back=24 plot=panel;

forecast back=24 lead=24 plot=decomp;

run;

Occasional Contributor
Posts: 5

Re: Unobserved Components Model with Seasonality

Thank you very much for the suggestions.

I ran the models mentioned in the solutions above.

As the seasonal component is not significnat is it better to just drop it?

Solution
‎10-11-2016 11:43 PM
SAS Employee
Posts: 57

Re: Unobserved Components Model with Seasonality

I am glad you were able to run the programs.  Yes, in this case, dropping the seasonal component is reasonable.  By the way, it might be good to have some good book handy for such modeling questions.  There is a new book in the market, Pelagatti, M. M. (2015). Time Series Modelling with Unobserved Components. Boca Raton, FL: CRC Press, which might be worth a look.  The book's web site is: http://www.ucmbook.info/ 

Occasional Contributor
Posts: 5

Re: Unobserved Components Model with Seasonality

Thank you for the above suggestion. I will go ahead with the discussed procedure.

I was able to acquire the book by Matteo M. Pelagatti. its a very helpful reference thankx.

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