Autocorrelation and ARCH/GARCH

[Mathias Saeten]

Hi,

I have times series of a stock return, and I want to detect if there is any effect on the month January on this stock return (January effect). I have created dummies for the days in January.
If I want to perform regular OLS, I have to detect if there are any heteroskedasticity and autocorrelation:

First I fin out if there is any heteroskedasticity:

reg RetOSEBX JanOSEBX
estat hettest 

Breusch-Pagan / Cook-Weisberg test for heteroskedasticity 
         Ho: Constant variance
         Variables: fitted values of RetOSEBX

         chi2(1)      =     0.21
         Prob > chi2  =   0.6463

Okey, fine, no heteroskedasticity. Then I test for autocorrelation:

estat bgodfrey

Breusch-Godfrey LM test for autocorrelation
---------------------------------------------------------------------------
    lags(p)  |          chi2               df                 Prob > chi2
-------------+-------------------------------------------------------------
       1     |          0.651               1                   0.4198
---------------------------------------------------------------------------
                        H0: no serial correlation

As we observe, no autocorrelation either. But when I do the test for ARCH effects I get the following:

estat archlm
LM test for autoregressive conditional heteroskedasticity (ARCH)
---------------------------------------------------------------------------
    lags(p)  |          chi2               df                 Prob > chi2
-------------+-------------------------------------------------------------
       1     |        462.568               1                   0.0000
---------------------------------------------------------------------------
         H0: no ARCH effects      vs.  H1: ARCH(p) disturbance

Here we get that there is ARCH effect. How can this be the case when there is no autocorrelation? Doesnt ARCH implicit tell that there is also autocorrelation in the model?

[Jeff Wooldridge]

Mathias: You need to set aside Stata for awhile and do some reading. The difference between serial correlation and ARCH is fundamental in the asset pricing literature. In fact, it was the basic observation that while asset returns cannot be predicted the squared returns can be.

Serial correlation is fundamentally about the mean, and ARCH is about the variance. To say that E[y(t)|y(t-1),y(t-2),...] does not depend on the lags means that y(t) is not predictable. That means there will be no serial correlation. That is very different from whether Var[y(t)}y(t-1),y(t-2),...] depends on the past. That's what ARCH is about.

Efficient markets basically tell us there will be no serial correlation in asset returns. But the conditional variance can depend on past returns.

JW