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