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!