# Odd shift of statistical significance when some variables are dropped from model

I am testing the statistical significance of some explanatory variables in a factor model, using the Fama-MacBeth two-pass regression (also known as the Cross-Sectional Approach).

Assuming my dependent variables are stock returns, and that my explanatory variables are macro factors, such as Inflation, Oil prices, Industrial Production:

• I first run a rolling-time-series regression (60 months window) of each stock with respect to all my explanatory variables. This will result in a time-series of sensitivities of each stock to the macro variables.

• Then, I run a cross-sectional regression for each month of my sample period, using stock returns as dependent variables, and the estimated sensitivities as independent variable. When the cross-sections are ran, we obtain the time-series factor values. For each factor, I test whether significantly different from zero.

Consider the following cross-sectional regression:

$R_i,t = \alpha_i,t + \beta_{i,1}(t)F_1,t + \beta_{i,2}(t)F_{2,t} + ... + \beta_{i,n}(t)F_{n,t} + \epsilon_i,t$

where the betas are estimated in previous periods, using time-series regression. The purpose of the cross-sectional regressions is to get time-series values of $F_1$, ..., $F_n$.

After I test for significance, I only collect the k significant variables, and include them in a regression. I redo the same test as above (time-series and cross-sectional regressions), only using my selected k variables.

BUT NOW, none of my k variables are significant!!

I am not sure how to construct my final model based on these results. Any help would be appreciated! Thank you!

• Can you perhaps describe your design a bit more? What do you mean by a factor model? – Behacad Sep 5 '13 at 16:33
• What does "t" represent? Should it be a subscript? – Alecos Papadopoulos Sep 5 '13 at 18:29
• Yes, it is a subscript. The syntax was too convoluted, that's why I put it in parenthesis. t shows that those betas vary through time. – Mayou Sep 5 '13 at 18:31
• This is sometimes a sign of severe multicollinearity among the $\beta$'s that are used as regressors in the cross-sectinal regressions. Maybe you should run some relevant tests. – Alecos Papadopoulos Sep 5 '13 at 18:32
• I made sure to only include variables with correlation < 0.6 with any other variable. DO you think that threshold is too high? Also, sometimes, even if two variables are highly correlated, they do not convey the same information, and should both be included.. – Mayou Sep 5 '13 at 18:34

The problem is coming from the fact that Fama-MacBeth standard errors do not include corrections regarding the fact that $\beta$'s are estimated (from the first stage TS regression). At least what you can do is compute the Shanken correction factors and check that the corrections in this case are indeed large. And that's why you are having the problem.