# Why do I have a statistically significant slope regressing R(t) on R(t-1)

I am reading Cochrane's lecture note here

He mentioned that when you regress annual return on time t on that of time t-1, you will have neither statistically significant nor economically significant slope.

I performed a quick test with python as follows:

import statsmodels.formula.api as smf
import pandas as pd
import pandas.io.data as web
import datetime as dt
ts_spy = web.get_data_yahoo("^GSPC", start="1/1/1929")
ts_ret = ts_spy.Close.pct_change()

df_reg = pd.concat([ts_ret.shift(1), ts_ret], axis=1)

df_reg.columns =["prev", "cur"]
results = smf.ols("cur ~ prev", data=df_reg).fit()
print results.summary()


The result I got was not as claimed in the note.

     OLS Regression Results
==============================================================================
Dep. Variable:                    cur   R-squared:                       0.001
Method:                 Least Squares   F-statistic:                     12.60
Date:                Mon, 28 Apr 2014   Prob (F-statistic):           0.000387
Time:                        08:50:08   Log-Likelihood:                 52035.
No. Observations:               16180   AIC:                        -1.041e+05
Df Residuals:                   16178   BIC:                        -1.041e+05
Df Model:                           1
==============================================================================
coef    std err          t      P>|t|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
Intercept      0.0003   7.64e-05      4.306      0.000         0.000     0.000
prev           0.0279      0.008      3.549      0.000         0.012     0.043
==============================================================================
Omnibus:                     4891.255   Durbin-Watson:                   1.998
Prob(Omnibus):                  0.000   Jarque-Bera (JB):           298584.422
Skew:                          -0.614   Prob(JB):                         0.00
Kurtosis:                      24.009   Cond. No.                         103.
==============================================================================

• Btw, upvoted for useful reference and nice code Apr 28, 2014 at 23:44

• @zsljulius Google statistical power for that answer. Just because there's statistical significance doesn't mean you can make money. There are transaction costs, overfitting and other issues you need to be careful of. In addition, with sufficient power, a truly tiny in sample relationship can be significant, but even if you had known this ahead of time (!) you may be looking at an extremely tiny revenue from trading (because of the high power). Apr 29, 2014 at 13:54