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Two examples
Canadian Lynx data1821-1934
Annual trappings of Canadian Lynx
The Data (an example of a prey-predator relationship). Note sharp peaks and wide minima
Try Log Scale
Its ACF. MA models not likely. Looks like AR model with complex roots
PACF is confusing though. Should we try AR(2) or AR(4) or AR(7) or even AR(11)?
AR(2)?
• m = 6.686 p-value 0.0000• a1 = 1.39 p-value 0.0000
• a2 = -0.7528 p-value 0.0000• Portmanteau test: p-value 0.0999
ACF for residuals in AR(2) model. Looks good? But-
AR(4)?
• m = 6.684 p-value 0.0000• a1 = 1.272 p-value 0.0000
• a2 = -0.7005 p-value 0.0000
• a3 = 0.1413 p-value 0.3604
• a4 = -0.2061 p-value 0.0318
• Portmanteau test: p-value 0.0350
ACF for residuals in AR(4) model, Portmanteau test is no good any longer.
AR(7)?
• m = 6.699 p-value 0.0000• a1 = 1.269 p-value 0.0000
• a2 = -0.6901 p-value 0.0000
• a3 = 0.2776 p-value 0.1018
• a4 = -0.3588 p-value 0.0335
• a5 = 0.1836 p-value 0.2778
• a6 = -0.213 p-value 0.1728
• a7 = 0.2316 p-value 0.0177• Portmanteau test: p-value 0.0184
For AR(7) model, Portmanteau test is even worse: p-value 0.018
AR(10)-first model with good Portmanteau test
• m = 6.697 p-value 0.0000• a1 = 1.243 p-value 0.0000
• a2 = -0.6605 p-value 0.0000
• a3 = 0.2863 p-value 0.0881
• a4 = -0.3504 p-value 0.0398
• a5 = 0.2112 p-value 0.2192
• a6 = -0.2084 p-value 0.2269
• a7 = 0.174 p-value 0.3072
• a8 = -0.1253 p-value 0.4589
• a9 = 0.3683 p-value 0.0194
• a10 = -0.2184 p-value 0.0335• Portmanteau test: p-value 0.2812
ACF for residuals in AR(10) model
Since φ(11) looked significant, we try AR(11) as well
• m = 6.697 p-value 0.0000• a1 = 1.168 p-value 0.0000
• a2 = -0.5346 p-value 0.0005
• a3 = 0.2515 p-value 0.1121
• a4 = -0.2963 p-value 0.0661
• a5 = 0.1409 p-value 0.3881
• a6 = -0.1397 p-value 0.3938
• a7 = 0.05 p-value 0.7618
• a8 = -0.0288 p-value 0.8595
• a9 = 0.1458 p-value 0.3633
• a10 = 0.2216 p-value 0.1503
• a11 = -0.3758 p-value 0.0003• Portmanteau test: p-value 0.8189
ACF for residuals in AR(11) model
AR(11) has the best AIC value, followed by AR(12)-AR(19) (??)Remember, the data set is 114
points long.
AIC values
Sunspots data
As of Yesterday …
We have seen this series before.We clearly see 11 years cycle. The data is too short to
see longer (100+ years) cycles
Here is another view:
and another …
Yet another view (reversed in time)
Let’s consider this, once again
ACF. Typical behavior for AR(2) model with complex roots.
Its PACF. AR(2)?
Now, let us look at monthly data,778 points, since December 1944. Note steep accents and not as
steep drops. This is a clear evidence of non-linearity in the model.
Its ACF (200 points, about 19 years)
Its PACF. First model that has reasonable Portmanteau test, is AR(13)
Since the data is only 60+ years long, we can see only the 11 years cycle here though longer
cycles definitely exist. The model below predicts next grand minimum around 2050
Have a Nice Break!