Why an ARMA(1) Fit Does Not Refute Market Efficiency
Summary
The document considers why a stock’s log returns might select an MA(1) model rather than white noise under the Bayesian information criterion. The response stresses that this comparison is between simplified candidate models, neither of which is likely to capture every feature of real stock returns. A slightly better fit for the more complex model can reflect its ability to approximate omitted structure or compensate for misspecification, including an unsuitable error distribution.
The reported BIC values favor the MA(1) specification among the models considered, and its moving-average coefficient has a low reported p-value. Those findings describe the sample and model-selection exercise; they do not identify the true return-generating process. In particular, the result alone cannot determine whether the efficient market hypothesis holds, nor does it show that prices are white noise. The question of including a constant also cannot be settled by assuming long-run inflation implies a drift in the particular return series. The response offers a caution about interpretation rather than a full diagnostic procedure or empirical test of market efficiency.
Key ideas
- BIC selects among the candidate models supplied, not among every possible return process.
- A more complex model may fit better because it approximates omitted structure or other model errors.
- An MA(1) selection is not by itself evidence against market efficiency.
- A statistically significant coefficient does not establish that the model is a true data-generating process.
- The response does not provide a separate test for whether returns should include a constant.
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Full text
# Are some stock prices not ARIMA(0,1,0) processes? # 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. ## Answer by Björn (score 1) https://quant.stackexchange.com/a/33990 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.
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