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## CORE OF WHAT IS REALLY PREDICTABLE ABOUT EQUITY MARKETS

Rather than getting too caught up in the complicated abstract issues of long memory models, we can ask the larger question of what we mean when we say that we don’t think returns are predictable but that there is something predictable about volatility?  One can construct various complicated schemes to exploit ‘long memory’ in financial markets but it’s good to have a concrete model that is simple and shows us concretely that in fact some sort of long memory does describe volatility in equity markets.  Here is a simple approach that fits very well the volatility series if one considers the standard errors of the coefficients of the model: aggregate the volatility by averaging of $v(t) = \log(r_t^2)$ and fit say an AR(25) model and check whether the coefficient standard deviations are small compared to the coefficients.  I did this with 1899 stocks only averaging over stocks with some return information per day.  One can consider this series a ‘baseline volatility of the market’ which can be used with a similar sort of idea as Sharpe etc. CAPM model and consider this the ‘volatility of the market’.  This is not an optimal model yet because the Box-Ljung test produces residual autocorrelations:

```Call:
arima(x = daggvol, order = c(25, 1, 0))

Coefficients:
ar1      ar2      ar3      ar4      ar5      ar6      ar7      ar8
-1.6928  -2.2377  -2.6436  -2.9080  -3.0832  -3.1564  -3.1561  -3.1289
s.e.   0.0086   0.0170   0.0257   0.0342   0.0422   0.0495   0.0560   0.0615
ar9     ar10     ar11     ar12     ar13     ar14     ar15     ar16
-3.0279  -2.8894  -2.7299  -2.5196  -2.3180  -2.0912  -1.8674  -1.6586
s.e.   0.0660   0.0696   0.0721   0.0737   0.0742   0.0737   0.0722   0.0696
ar17     ar18     ar19     ar20     ar21     ar22     ar23     ar24
-1.4225  -1.1577  -0.9010  -0.6922  -0.5205  -0.3642  -0.1918  -0.0912
s.e.   0.0661   0.0615   0.0561   0.0496   0.0422   0.0342   0.0257   0.0170
ar25
-0.0211
s.e.   0.0086

sigma^2 estimated as 0.1727:  log likelihood = -7268.06,  aic = 14588.13

Box-Ljung test

data:  fit\$residuals
X-squared = 444.3672, df = 50, p-value < 2.2e-16
```

Increasing the autoregressive order to AR(35) reduces the Box-Ljung statistic a bit but the residuals are still quite far from white noise.

```arima(x = daggvol, order = c(35, 1, 0))

Coefficients:
ar1      ar2      ar3      ar4      ar5      ar6      ar7      ar8
-1.7125  -2.2928  -2.7461  -3.0709  -3.3137  -3.4615  -3.5410  -3.5965
s.e.   0.0086   0.0171   0.0262   0.0353   0.0440   0.0523   0.0601   0.0670
ar9     ar10     ar11     ar12     ar13     ar14     ar15     ar16
-3.5845  -3.5381  -3.4719  -3.3528  -3.2406  -3.1020  -2.9666  -2.8394
s.e.   0.0734   0.0790   0.0840   0.0883   0.0918   0.0947   0.0970   0.0986
ar17     ar18     ar19     ar20     ar21     ar22     ar23     ar24
-2.6787  -2.4801  -2.2766  -2.1090  -1.9616  -1.8100  -1.6188  -1.4672
s.e.   0.0997   0.1001   0.0997   0.0986   0.0969   0.0947   0.0918   0.0883
ar25     ar26     ar27     ar28     ar29     ar30     ar31     ar32
-1.3101  -1.1603  -0.9750  -0.8086  -0.6233  -0.4724  -0.3545  -0.2350
s.e.   0.0840   0.0791   0.0735   0.0671   0.0601   0.0524   0.0441   0.0353
ar33     ar34     ar35
-0.1286  -0.0504  -0.0193
s.e.   0.0262   0.0171   0.0086

sigma^2 estimated as 0.1687:  log likelihood = -7108.36,  aic = 14288.71

Box-Ljung test

data:  fit\$residuals
X-squared = 336.358, df = 50, p-value < 2.2e-16
```

```arima(x = daggvol, order = c(40, 1, 0))

Coefficients:
ar1      ar2      ar3      ar4      ar5      ar6      ar7      ar8
-1.7252  -2.3245  -2.7997  -3.1489  -3.4188  -3.5969  -3.7111  -3.8040
s.e.   0.0086   0.0171   0.0262   0.0354   0.0443   0.0528   0.0607   0.0681
ar9     ar10     ar11     ar12     ar13     ar14     ar15     ar16
-3.8317  -3.8277  -3.8059  -3.7325  -3.6623  -3.5660  -3.4700  -3.3834
s.e.   0.0749   0.0811   0.0867   0.0917   0.0960   0.0997   0.1028   0.1054
ar17     ar18     ar19     ar20     ar21    ar22     ar23     ar24
-3.2630  -3.1062  -2.9424  -2.8161  -2.7114  -2.602  -2.4486  -2.3292
s.e.   0.1075   0.1091   0.1101   0.1105   0.1105   0.110   0.1091   0.1074
ar25     ar26     ar27     ar28     ar29     ar30     ar31     ar32
-2.2019  -2.0803  -1.9204  -1.7726  -1.5992  -1.4493  -1.3219  -1.1801
s.e.   0.1053   0.1028   0.0997   0.0960   0.0918   0.0868   0.0812   0.0750
ar33     ar34     ar35     ar36     ar37     ar38     ar39     ar40
-1.0348  -0.8995  -0.7872  -0.6585  -0.5149  -0.3617  -0.2026  -0.0819
s.e.   0.0682   0.0609   0.0529   0.0443   0.0354   0.0262   0.0171   0.0086

sigma^2 estimated as 0.1659:  log likelihood = -6997.09,  aic = 14076.18

Box-Ljung test

data:  fit\$residuals
X-squared = 518.3599, df = 120, p-value < 2.2e-16
```

In this case it seems that differencing volatility is not necessary as Dickey-Fuller tests show that the undifferenced series is already stationary and in fact the Ljung-Box statistic is smaller for the AR(40) model without differencing.

```Call:
arima(x = aggvol, order = c(30, 0, 0))

Coefficients:
ar1     ar2     ar3     ar4     ar5     ar6     ar7     ar8     ar9
0.2266  0.0857  0.0877  0.0955  0.0563  0.0719  0.0508  0.0072  0.0553
s.e.  0.0086  0.0088  0.0089  0.0089  0.0089  0.0090  0.0090  0.0090  0.0090
ar10    ar11    ar12     ar13    ar14     ar15     ar16    ar17    ar18
0.0253  0.0127  0.0441  -0.0104  0.0237  -0.0038  -0.0112  0.0293  0.0371
s.e.  0.0090  0.0090  0.0090   0.0090  0.0090   0.0090   0.0090  0.0090  0.0090
ar19    ar20     ar21    ar22    ar23     ar24    ar25    ar26    ar27
0.0069  -0.032  -0.0178  0.0063  0.0427  -0.0328  0.0112  0.0020  0.0475
s.e.  0.0090   0.009   0.0090  0.0090  0.0090   0.0090  0.0090  0.0089  0.0089
ar28    ar29     ar30  intercept
-0.0022  0.0388  -0.0095    -9.3699
s.e.   0.0089  0.0089   0.0086     0.0619

sigma^2 estimated as 0.1589:  log likelihood = -6707.81,  aic = 13479.62

Box-Ljung test

data:  fit\$residuals
X-squared = 422.1478, df = 120, p-value < 2.2e-16

```