Is it permissible to aggregate results from multiple cross sectional regressions?

Question

Lets consider the following approach:

(1) We estimate a simple linear regression model y = b0 + b1*x + error. The input data in our model estimation shall be: y = 1 month stock returns (cross sectional) for month n x = market cap for each stock (cross sectional) for month n

(2) We test if our coefficient b1 is statistically significant. And record the result for month n. We also record the R squared.

(3) We do step (1) & (2) for a total of N month.

(4) Finally we have recorded the results of N significance tests for b1, and calculate the percentage of times our coefficient was significant. For R squared we calculate the average of all N R squared.

Are the resulting 'synthetic' measures of significance & average R squared sound from a statistical perspective? Is this approach of aggregating over multiple cross sectional regressions violating any best practices in statistical research?

Answer 1

I quite do not see the aim of this procedure. If it is to test that the stock return is proportional to the market cap, why don't you make only one regression and test the significance of $b_0$ ?

If you are looking for some seasonality in the $b-0$, then you should test if your b0 is a constant (i.e. var=0). The percentage of time it is not significant will give you a number but you have no other number to compare with. So unless on rare occasion (it is 100%, or there are not significant during winter for example), you will be able to deduce something.

The $R^2$ is the cosine of an angle, so I do not see what the average of cosine can be!

Answer 2

So you want to test if $y \approx a x + b$ with a $a$ that may depend on the month.

Define 12 variables:

• $x_1=x$ for January and $x_1=0$ for the rest of the year,
• $x_2=x$ for February and $x_2=0$ for the rest of the year,
• ...
• $x_{12}=1$ for December and $x_{12}=0$ for the rest of the year.

Technically, it is no more than adding a dozen of column full of 0 or copy of $x$ value.

You can now consider the multiple regression $z = a_1 x_1 + a_2 x_2 +...+ a_{12} x_{12} + b$.

Before to run on your software, let's make a few math, as multiplying by 0 is particularity easy:

• In January $z=a_1 x_1 + b=a_1 x + b$,
• in February $z=a_2 x_2 + b = a_2 x + b$,
• ..,
• in December $z = a_{12} x_{12} + b = a_{12} x + b$.

So the regression $y \approx z = a_1 x_1 + a_2 x_2 +...+ a_{12} x_{12} + b$, is in facts a regression $y \approx a_n x + b$ with $a_n$, varying over the month.

Your software will give you a global $R^2$ with the usual interpretation.

It will give you a significance level for $b$, telling you if the stock return is proportional to the market cap, or if there is an extra factor in it.

The significance of the $a_n$ (as a global factor) can be read on the Fischer $F$ near by the $SST = SSE+SSR$ summary.

You can plot the $a_n$ to see if it varies from month to month, and you can test this hypothesis by plotting their confidence intervals.

This model is a straightforward adaptation of the so called dummy variable technique, which does the same this time with a varying intercept $b$. There is a simplification though as the correlation between the variables $x_1$, $x_2$,..., $x_{12}$ will probably not be sufficient to create the multicolinearity pitfall.