# Are some stock prices not ARIMA(0,1,0) processes?

I am studying stock prices. Let Pt be price of stock at time t. While Pt is non stationary, the return, rt=log(Pt/Pt-1) is stationary.

However, when I study on rt, I decide on an ARMA(0,1) without intercept process rather than ARMA(0,0).

I am confused with this result. I was expecting rt to be an ARMA(0,0) process which supports Pt to be a white noise.

However, I obtained ARMA(0,1). Does not this result contadict with effciciency market hypothesis?

I use BIC criterion to select the model. The BIC criterions are as below:

MODEL       BIC-model contains intercept    BIC-model does not contain intercept
ARMA(0,0)   -11,936.6                       -11,941.2
ARMA(1,0)   -11,936.9                       -11,941.9
ARMA(0,1)   -11,937.1                       -11,942.0
ARMA(1,1)   -11,929.3                       -11,934.2
ARMA(2,0)   -11,929.5                       -11,934.4
ARMA(0,2)   -11,929.3                       -11,934.2
ARMA(2,1)   -11,921.9                       -11,926.5
ARMA(1,2)   -11,921.5                       -11,926.4
ARMA(2,2)   -11,922.7                       -11,927.5


According to BIC, I decide MA(1) process. The statistics of MA(1) process are as below:

Coefficient     0.058409111
Standard Error  0.019695617
P-value         0.003021036


I can't explain the obtained result.

I am also confused with the constant term. Even though BIC does not select a model with constant, I suspect the model include a constant term since in long term the price will go up becouse of the inflation rate.

I haven't studied in finance so much. I will be very glad for an explanation. Thanks a lot.

• I'm voting to close this question as off-topic because it is more about finance than statistics and thus better fits at Quantitative Finance Stack Exchange. – Richard Hardy May 1 '17 at 5:27
• @oercim this question is less likely to be closed if you tell us how you selected this model, and if you are asking for help with interpreting the statistics behind the thing you used. – Taylor May 1 '17 at 5:31

## 1 Answer

It is clear that in reality stock prices are a quite complex process that cannot be completely described by anything as simple as an ARMA(0, 0) or ARMA(0, 1) process. So, whatever model fitting algorithm you used had to decide between two (very) imperfect data generating models for the data and did so according to its criteria. E.g., if you used maximum likelihood, then the you may simply get ARMA(0, 1) rather than ARMA(0, 0) according to e.g. a likelihood ratio test or AIC or BIC (or whatever criteria you used), simply because it is the more complex model and presumably could approximate your data substantially better.

Whether one almost certainly wrong model fits the data substantially better than another almost certainly wrong model, does not necessarily tell you too much about the true underlying data generating model (e.g. whether the efficient market hypothesis is true or not and all information was "priced in"), because the extra moving average component may simply provide a better approximation that compensates for other failures of the model (e.g. error term distribution etc.). And let's be honest the efficient market hypothesis is almost certainly not exactly true and at best it might be a good approximation.

• To add to a sensible answer, one must remember that all models are wrong, but some are useful. Thus a model being wrong need not be a big problem, though it might. – Richard Hardy May 1 '17 at 9:03