Results: souvenir_forecasting.sas

The Arima Procedure

Identification 1

Examining ARIMA(2,0,0)(0,1,1)_12 on Y=log(X)

The ARIMA Procedure

Descriptive Statistics

Name of Variable = y
Period(s) of Differencing 12
Mean of Working Series 0.288627
Standard Deviation 0.221759
Number of Observations 72
Observation(s) eliminated by differencing 12

Autocorrelation Check for White Noise

Autocorrelation Check for White Noise
To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 44.53 6 <.0001 0.513 0.461 0.214 0.152 0.148 0.137
12 55.12 12 <.0001 0.128 0.194 0.198 0.117 -0.010 -0.134
18 56.54 18 <.0001 -0.073 -0.069 -0.022 -0.031 -0.015 -0.058
24 69.09 24 <.0001 -0.077 -0.065 -0.195 -0.093 -0.150 -0.194

Series Correlation Panel

Trend and Correlation Analysis for y(12)

Estimation 1

Maximum Likelihood Estimation

Maximum Likelihood Estimation
Parameter Estimate Standard
Error		 t Value Approx
Pr > |t|		 Lag
MU 0.28562 0.03657 7.81 <.0001 0
MA1,1 0.52048 0.13629 3.82 0.0001 12
AR1,1 0.34695 0.11029 3.15 0.0017 1
AR1,2 0.35167 0.11058 3.18 0.0015 2

Fit Statistics

Constant Estimate 0.086079
Variance Estimate 0.029522
Std Error Estimate 0.17182
AIC -41.0894
SBC -31.9828
Number of Residuals 72

Correlations of Parameter Estimates

Correlations of Parameter Estimates
Parameter MU MA1,1 AR1,1 AR1,2
MU 1.000 -0.063 0.043 0.046
MA1,1 -0.063 1.000 -0.062 0.167
AR1,1 0.043 -0.062 1.000 -0.524
AR1,2 0.046 0.167 -0.524 1.000

Autocorrelation Check of Residuals

Autocorrelation Check of Residuals
To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 2.74 3 0.4341 0.038 0.049 -0.084 -0.134 0.002 -0.078
12 9.08 9 0.4300 -0.054 0.092 0.105 0.166 -0.076 0.133
18 11.10 15 0.7454 -0.020 -0.049 0.002 -0.066 0.091 -0.074
24 15.48 21 0.7983 -0.060 0.069 -0.163 0.070 -0.031 -0.031

Residual Correlation Panel

Residual Correlation Diagnostics for y(12)

Residual Normality Panel

Residual Normality Diagnostics for y(12)

Residual Plot

Residuals for y(12)

Model Filters

Model for variable y

Model for variable y
Estimated Mean 0.285617
Period(s) of Differencing 12

Autoregressive Factors

Autoregressive Factors
Factor 1: 1 - 0.34695 B**(1) - 0.35167 B**(2)

Moving Average Factors

Moving Average Factors
Factor 1: 1 - 0.52048 B**(12)

Forecast 1

Forecasts for y

Forecasts for variable y
Obs Forecast Std Error 95% Confidence Limits
85 9.5783 0.1718 9.2415 9.9150
86 9.7548 0.1819 9.3984 10.1113
87 10.2862 0.1991 9.8959 10.6765
88 10.0286 0.2051 9.6267 10.4306
89 9.9509 0.2101 9.5391 10.3627
90 10.1169 0.2127 9.7001 10.5338
91 10.3692 0.2145 9.9488 10.7895
92 10.4601 0.2155 10.0377 10.8825
93 10.5356 0.2162 10.1119 10.9593
94 10.5860 0.2166 10.1615 11.0105
95 11.0178 0.2168 10.5928 11.4428
96 11.7960 0.2170 11.3707 12.2213
97 9.8409 0.2346 9.3812 10.3006
98 10.0156 0.2370 9.5511 10.4800
99 10.5551 0.2409 10.0830 11.0272
100 10.2997 0.2424 9.8247 10.7747
101 10.2256 0.2436 9.7481 10.7030
102 10.3937 0.2442 9.9149 10.8724
103 10.6478 0.2447 10.1683 11.1274
104 10.7402 0.2450 10.2601 11.2203
105 10.8169 0.2451 10.3364 11.2973
106 10.8682 0.2452 10.3875 11.3488
107 11.3006 0.2453 10.8199 11.7814
108 12.0794 0.2453 11.5986 12.5603
109 10.1248 0.2600 9.6153 10.6343
110 10.2998 0.2619 9.7866 10.8131
111 10.8396 0.2651 10.3201 11.3592
112 10.5845 0.2663 10.0626 11.1064
113 10.5105 0.2673 9.9866 11.0344
114 10.6787 0.2678 10.1538 11.2037
115 10.9330 0.2682 10.4074 11.4587
116 11.0254 0.2684 10.4993 11.5515
117 11.1022 0.2686 10.5759 11.6286
118 11.1536 0.2686 10.6270 11.6801
119 11.5861 0.2687 11.0594 12.1127
120 12.3649 0.2687 11.8382 12.8916
121 10.4103 0.2821 9.8574 10.9632
122 10.5853 0.2838 10.0291 11.1416
123 11.1252 0.2868 10.5631 11.6873
124 10.8700 0.2879 10.3058 11.4343
125 10.7961 0.2888 10.2300 11.3621
126 10.9643 0.2893 10.3973 11.5314
127 11.2186 0.2897 10.6509 11.7863
128 11.3110 0.2899 10.7429 11.8791
129 11.3878 0.2900 10.8195 11.9562
130 11.4392 0.2901 10.8707 12.0077
131 11.8717 0.2901 11.3031 12.4403
132 12.6505 0.2901 12.0819 13.2192
133 10.6959 0.3026 10.1029 11.2889
134 10.8710 0.3042 10.2748 11.4671
135 11.4108 0.3069 10.8092 12.0124
136 11.1557 0.3080 10.5521 11.7592
137 11.0817 0.3088 10.4764 11.6870
138 11.2499 0.3093 10.6437 11.8561
139 11.5043 0.3096 10.8974 12.1111
140 11.5967 0.3098 10.9895 12.2039
141 11.6735 0.3099 11.0660 12.2809
142 11.7248 0.3100 11.1172 12.3324
143 12.1573 0.3100 11.5496 12.7650
144 12.9362 0.3101 12.3284 13.5439
145 10.9815 0.3217 10.3510 11.6121
146 11.1566 0.3232 10.5231 11.7901
147 11.6964 0.3258 11.0578 12.3350
148 11.4413 0.3268 10.8008 12.0818
149 11.3673 0.3276 10.7252 12.0094
150 11.5356 0.3281 10.8926 12.1785
151 11.7899 0.3284 11.1463 12.4334
152 11.8823 0.3285 11.2384 12.5262
153 11.9591 0.3286 11.3149 12.6032
154 12.0104 0.3287 11.3661 12.6547
155 12.4429 0.3288 11.7986 13.0873
156 13.2218 0.3288 12.5774 13.8662
157 11.2672 0.3398 10.6012 11.9332
158 11.4422 0.3412 10.7734 12.1110
159 11.9820 0.3437 11.3084 12.6557
160 11.7269 0.3446 11.0515 12.4023
161 11.6529 0.3454 10.9760 12.3299
162 11.8212 0.3458 11.1434 12.4989
163 12.0755 0.3461 11.3972 12.7538
164 12.1679 0.3463 11.4892 12.8465
165 12.2447 0.3464 11.5658 12.9236
166 12.2960 0.3464 11.6170 12.9750
167 12.7285 0.3465 12.0495 13.4076
168 13.5074 0.3465 12.8283 14.1865
169 11.5528 0.3570 10.8531 12.2524
170 11.7278 0.3583 11.0255 12.4301
171 12.2677 0.3607 11.5608 12.9745
172 12.0125 0.3615 11.3039 12.7211
173 11.9385 0.3623 11.2285 12.6486
174 12.1068 0.3627 11.3959 12.8176
175 12.3611 0.3630 11.6497 13.0725
176 12.4535 0.3631 11.7418 13.1652
177 12.5303 0.3632 11.8184 13.2422
178 12.5817 0.3633 11.8696 13.2937
179 13.0142 0.3633 12.3021 13.7263
180 13.7930 0.3634 13.0809 14.5052
181 11.8384 0.3733 11.1067 12.5701
182 12.0134 0.3746 11.2791 12.7477
183 12.5533 0.3769 11.8146 13.2919
184 12.2981 0.3777 11.5578 13.0385

Forecasts Only Plot

Forecasts Only Plot

Forecast Plot

Forecast Plot

Contents of the file RESIDS

The Print Procedure

Data Set WORK.RESIDS

Obs y FORECAST STD L95 U95 RESIDUAL
80 10.2607 10.1536 0.17184 9.8168 10.4904 0.10708
81 10.3257 10.2974 0.17184 9.9606 10.6342 0.02824
82 10.3360 10.3763 0.17184 10.0395 10.7132 -0.04038
83 10.7501 10.7904 0.17184 10.4536 11.1272 -0.04033
84 11.5585 11.5020 0.17184 11.1652 11.8388 0.05645
85 . 9.5783 0.17182 9.2415 9.9150 .
86 . 9.7548 0.18187 9.3984 10.1113 .
87 . 10.2862 0.19913 9.8959 10.6765 .
88 . 10.0286 0.20510 9.6267 10.4306 .
89 . 9.9509 0.21010 9.5391 10.3627 .
90 . 10.1169 0.21269 9.7001 10.5338 .

Back transform forecasts to the original scale

The Print Procedure

Data Set WORK.OSCALE

Obs osMEDIAN osMEAN osL95 osU95
85 14447.66 14662.50 10316.79 20232.55
86 17237.35 17524.78 12068.82 24619.32
87 29325.19 29912.43 19848.99 43325.47
88 22666.62 23148.41 15163.76 33881.82
89 20970.46 21438.43 13892.25 31655.06
90 24759.10 25325.47 16319.13 37564.07
91 31861.99 32603.16 20928.03 48508.46
92 34894.57 35714.28 22873.14 53234.09
93 37632.81 38522.43 24635.63 57487.00
94 39577.87 40517.08 25888.02 60507.04
95 60946.84 62396.67 39845.23 93223.63
96 132719.24 135881.19 86740.39 203070.29
97 18786.64 19310.60 11863.01 29751.12
98 22371.71 23008.83 14059.78 35597.55
99 38372.31 39501.68 23932.93 61523.34
100 29723.89 30609.74 18484.81 47796.50
101 27599.50 28430.54 17122.36 44487.58
102 32651.68 33640.24 20230.55 52699.13
103 42101.91 43381.35 26062.84 68011.42
104 46173.52 47579.80 28568.36 74627.80
105 49855.60 51376.20 30836.14 80606.12
106 52479.61 54081.64 32452.37 84866.22
107 80873.30 83343.46 50003.91 130799.58
108 176211.75 181595.75 108942.71 285017.51
109 24954.38 25812.06 14991.97 41536.97
110 29727.07 30763.97 17793.19 49665.00
111 51002.77 52826.46 30336.25 85748.32
112 39516.52 40942.62 23448.75 66594.39
113 36698.75 38033.44 21733.50 61968.75
114 43422.67 45008.38 25688.38 73400.02
115 55996.51 58047.14 33102.98 94722.86
116 61417.24 63670.13 36291.90 103937.15
117 66319.54 68754.86 39177.86 112264.46
118 69813.92 72379.25 41235.11 118199.84
119 107590.88 111545.98 63540.96 182178.50
120 234433.74 243053.90 138442.45 396982.11
121 33200.44 34548.14 19099.40 57712.27
122 39551.10 41176.71 22675.83 68984.88
123 67859.03 70707.60 38681.39 119045.55
124 52577.31 54801.84 29905.42 92437.21
125 48828.75 50908.35 27722.61 86003.69
126 57775.61 60244.87 32770.41 101860.85
127 74506.11 77698.05 42231.86 131444.87
128 81719.09 85225.02 46302.03 144227.15
129 88242.25 92031.51 49985.32 155779.64
130 92892.05 96883.23 52610.97 164013.96
131 143157.17 149310.22 81071.42 252789.16
132 311931.10 325341.21 176638.97 550846.81
133 44175.67 46244.64 24414.27 79932.37
134 52625.76 55117.42 28992.62 95523.29
135 90291.72 94646.31 49475.93 164779.01
136 69958.27 73355.55 38256.41 127930.42
137 64970.55 68143.92 35468.31 119012.50
138 76875.10 80641.45 41929.13 140946.91
139 99136.42 104003.64 54037.24 181875.14
140 108733.91 114079.00 59246.72 199556.42
141 117413.53 123189.93 63960.81 215537.25
142 123600.49 129684.28 67321.26 226928.03
143 190482.39 199861.13 103740.09 349754.30
144 415050.05 435489.75 226030.56 762138.29
145 58779.38 61901.37 31288.03 110426.11
146 70022.93 73778.15 37162.37 131940.19
147 120140.60 126690.08 63437.64 227526.84
148 93085.27 98191.05 49057.82 176625.60
149 86448.70 91214.96 45486.99 164297.05
150 102288.70 107943.70 53775.58 194567.46
151 131909.23 139215.48 69307.12 251057.12
152 144679.49 152701.99 75990.30 275458.23
153 156228.45 164897.55 82037.75 297513.37
154 164460.72 173590.65 86348.68 313233.83
155 253452.64 267526.83 133061.48 482771.12
156 552258.55 582930.73 289917.42 1051987.50
157 78210.85 82858.92 40181.96 152230.94
158 93171.32 98756.75 47733.64 181861.14
159 159857.05 169582.72 81504.89 313530.60
160 123857.68 131434.96 63035.90 243364.90
161 115027.18 122097.02 58452.48 226359.13
162 136103.61 144489.52 69106.66 268052.20
163 175516.19 186348.78 89068.82 345866.62
164 192508.08 204401.32 97659.34 379475.82
165 207874.92 220725.86 105432.46 409854.64
166 218828.64 232362.13 110973.52 431508.10
167 337239.79 358101.69 171008.61 665058.17
168 734825.87 780289.87 372598.55 1449197.97
169 104066.03 110911.94 51696.64 209486.31
170 123972.18 132192.19 61420.64 250226.97
171 212703.08 226997.27 104899.28 431295.63
172 164802.93 175934.06 81136.03 334746.55
173 153053.22 163434.63 75241.89 311333.05
174 181097.15 193408.41 88959.65 368663.52
175 233538.87 249439.70 114659.53 475672.68
176 256147.99 273604.18 125720.16 521887.61
177 276594.85 295455.60 135728.10 563661.58
178 291169.69 311031.50 142862.23 593437.36
179 448725.55 479341.89 220149.50 914626.74
180 977746.86 1044467.63 479669.30 1993016.69
181 138468.50 148462.69 66613.87 287830.83
182 164955.28 176947.66 79153.02 343767.60
183 283019.12 303850.28 135210.95 592406.32
184 219283.99 235498.94 104588.83 459757.21

The SGPlot Procedure

The SGPlot Procedure

The SGPlot Procedure