Why does auto.arima drop my seasonality component when stepwise=FALSE and approximation=FALSE?

Question

I started with this.

> auto.arima(mntm)
Series: mntm 
ARIMA(2,0,0)(2,0,0)[12] with non-zero mean 

Coefficients:
         ar1     ar2    sar1    sar2  intercept
      0.0966  0.0883  0.5115  0.4622   139.5995
s.e.  0.0365  0.0358  0.0316  0.0319    19.8640

sigma^2 estimated as 380.2:  log likelihood=-3440.66
AIC=6893.32   AICc=6893.42   BIC=6921.27

And next

> auto.arima(mntm, stepwise=FALSE, approximation=FALSE)
Series: mntm 
ARIMA(4,0,1) with non-zero mean 

Coefficients:
         ar1      ar2      ar3      ar4      ma1  intercept
      1.0353  -0.0871  -0.1914  -0.2790  -0.5642   133.7108
s.e.  0.0417   0.0541   0.0513   0.0394   0.0290     0.5935

sigma^2 estimated as 391.5:  log likelihood=-3438.04
AIC=6890.07   AICc=6890.22   BIC=6922.69

Doesn't stepwise=FALSE, approximation=FALSE sacrifice time for a more accurate model?

mntm is clearly seasonal.

> mntm
   Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
1   64  63  77 118 174 229 262 242 185 165  82  51
2   89  38  51 103 164 217 239 227 188 156  83  19
3   42  39  66 117 166 219 249 233 199 154  68  49
4   45  41  64 130 165 233 258 236 197 119  84  39
5   55  50  77 120 196 222 250 236 196 149  84  52
6   21  58  64 139 162 221 245 227 211 159  75  29
7    8  30  79 135 178 201 265 252 200 146  73   3
8    9  50  55 107 158 222 242 236 192 152  89  80
9    0  48  66 146 178 239 242 225 212 122  91  55
10   2  -2  46 126 170 204 258 235 195 142  99 -14
11  15  36  69 133 192 232 248 254 212 158  82  54
12  33  38  11 152 167 221 234 249 203 142  95   3
13  -6  47  84 106 159 217 255 240 230 144  96  29
14  20  23  58 125 185 219 227 233 185 142  70   9
15   4  -3  92 125 164 219 241 227 179 147  96   0
16  38  22  76 111 181 220 245 224 198 121  98  56
17   8  30  47 101 186 201 235 235 211 130  87  45
18   2  21  81 103 162 211 247 246 198 133  98  37
19  53  15  59 121 141 216 247 240 180 129  55  40
20  -1  -2  88 125 176 238 259 250 191 147  96  22
21   6  13  41 128 171 233 248 237 199 134  70  27
22 -19  20  46 117 180 219 242 238 216 157  93  30
23  -5  35  56 106 161 229 243 235 218 183  90  78
24  42  27  68 115 174 207 249 235 210 127  89  80
25  31  28 106 133 160 231 238 242 210 144  88  48
26  52  18  77 131 164 202 240 237 194 122  84  48
27  41  43  62  94 184 224 241 249 201 160 116  46
28  10  78  96 137 166 235 247 237 196 121  51  15
29 -45  19  93 134 180 216 264 263 229 140 115  42
30  11 -26  60 127 177 235 249 268 201 131  98  42
31  16 -31  83 118 182 202 238 240 209 134 112  58
32  27   4  61 137 187 214 258 256 221 134  74  26
33 -19  44  53 138 164 234 243 219 197 129  88  32
34 -12  33  70 110 193 217 253 229 201 137 102  69
35  26  30  84 114 164 214 252 247 210 161 110  45
36  13  77  58 120 172 234 243 246 190 177  79  79
37 -15  29  86 147 186 211 249 238 206 161 133  24
38  12  24  80 121 186 226 264 228 203 153  90  45
39  10  10  71 111 181 232 260 242 213 114  99  51
40  -4  32  75 114 174 223 259 256 192 113  97  31
41  45  30  77 117 170 242 244 239 212 154  83 -24
42  63  68  90 124 166 227 257 240 190 161  99  68
43  34  49  85 135 202 225 254 246 197 143  91  52
44  30  41  62 119 154 204 249 225 207 123  95  46
45  42   7  54 119 180 225 269 247 208 132  90  23
46  -4  25  77 153 156 243 270 229 197 130 111  66
47  46  23  88 131 180 230 270 254 211 155  62  11
48  14  24  46 122 164 227 238 230 204 142  56  57
49  22  59  80 110 157 210 252 233 205 147  90  48
50  63  63  84 121 168 216 247 246 226 147  87  57
51  49  45  63 124 177 219 268 246 209 136 110  54
52  16  49  98 121 186 232 230 235 197 146  71   9
53  26  46  58 126 167 222 216 239 177 126  96  59
54  38  40  78 134 161 217 244 244 204 143  75  24
55 -16   8  76 110 144 209 241 241 205 124 104  31
56 -14  18  74 122 204 208 241 227 200 128  84  35
57  17  26  41 114 135 215 249 244 206 144  93  17
58  57  22  61 122 159 211 249 239 182 128 102  57
59  43 -11  70 106 162 212 238 239 196 173  70  40
60  18  41  78 127 155 231 242 217 203 123  71  57
61  -5  33  61 125 178 217 237 252 195 146 109  36
62   8  -1  89 142 190 252 266 250 216 149  88   0
63  -2  47  71 151 196 244 275 249 225 149 116  75
64  53  59 122 135 206 232 282 260 212 163  80  83
65  45  40  57 140 188 244 272 241 208 169  88  63

Sorry. I tried insert the plot to show the seasonality, but I need 10 more points.

Is it because the AIC and AICc is smaller for ARIMA(4,0,1)? And, how are they smaller? Any help will be appreciated.

Also, input how I could have made this a higher quality question, will be greatly appreciated.

Answer 1

auto.arima will return the best model (according to AICc) that it can find given the constraints provided. When stepwise=FALSE, it looks through more models than using the default stepwise procedure. However, the number of models are restricted depending on the number of parameters allowed. In the default settings, no more than five total parameters are allowed (to save time looking at too many models). In your case, it found a non-seasonal model -- the ARIMA(4,0,1) -- that has the minimum AICc amongst all models (seasonal or otherwise) with 5 or fewer parameters. This model mimics the seasonality quite well using cyclic behaviour. Have a look at the forecasts and you will see that the results look seasonal even if they are not.

If you want to find a better truly seasonal model, just increase the number of parameters allowed by setting max.order=10 for example. There is not too much gained using approximation=FALSE. What that does is force it to evaluate the likelihood more accurately for each model, but the approximation is quite good and much faster, so is usually acceptable.