Degrees of freedom for an F-test using the Fama-MacBeth procedure

mvikenes

New Member
I am currently writing my master thesis and in the process use the Fama-MacBeth (1973) procedure to obtain risk premia for my factors of interest.

Data structure: I have panel data with excess returns for 50 portfolios (my cross-section) for each point in time (time-series). In the same panel I have the factors for the Fama-French 3-factor model, i.e market excess return, SMB-returns and HML-returns. In addition, I have manually calculated rolling 5-year comoments of order 3 to 10 for portfolio return and market return (so the covariance, coskewness, cokurtosis and so on, between these two). Similarly, I have the corresponding standard univariate moments in the same structure.

Hypothesis: The Fama-French factors, SMB and HML, proxy for higher-order comoments. This is based on the rationale that there is no reason that only covariance (comoment of 2nd order), represented by the market beta, should be the only comoment of interest for the rational investor. Thus, other factors that research find significant, may just proxy for these higher-order comoments.

Methodology: I follow Chung et al. (2006) suit, and first obtain factor loadings (regression coefficients) for the three factors. This is done through rolling regressions, each time using 5 years of previous return data to estimate the coefficients. This is done for each portfolio. These coefficients are merged with the initial panel data so they can be used as variables for the next step. In the second step I run a cross-sectional (across the N = 50 portfolios) regression for each point in time. To illustrate, if I had 100 years of annual return data I would run T = 100 – 5 = 95 cross-sectional regressions. The last step of the Fama-MacBeth procedure is to average the 95 hypothetical coefficient estimates, resulting in the risk premia for each of the three factors. Fama-MacBeth (1973) outline that the correct standard errors of the coefficients are given by the standard deviation of the coefficient (i.e the standard deviation in the T = 95 sample), divided by the square root of T, where T obviously is 95. To further investigate the hypothesis I stepwise include comoments, starting by adding comoment of order 3, to the second step of the procedure, ending with the inclusion of all ten comoments in addition to the 3 factors. To test whether the joint significance of the SMB and HML risk premia are reduce by adding comoments, I also perform the abovementioned procedure from step 2 with a “restricted” model (i.e coefficients for SMB and HML are not included as explanatory variables). Thus, I can perform an F-test using the R-squared from both the “restricted” and “unrestricted” model. By also testing with standard univariate moments I aim to see whether it is the general inclusion of variables, or the systematic part of the moments, that actually reduce the joint significance.

Question: After completing the procedure by calculating the mean coefficients and the corresponding standard errors I would like to obtain the correct t-statistic. I assume the formula for the t-statistic, where the (mean) coefficient (minus zero) is divided by the standard error, is correct here. When calculating the p-value of this t-statistic, I similarly use T-1 as the degrees of freedom. The question arises for the F-test. I am confident as per the q and k parameters, but the N in “N-k-1” raise uncertainty as to which “N” I should use, N=50 or T=95. Or maybe neither is correct?

Does anyone have any good answers to this? It is very possible that this is completely clear for others more intellectually gifted than me. Personally, I immediately think that T = 95 should be used, as it follows analogously from the calculation of the p-value of t-statistic (as I have confirmed to be correct). At the same time, it is intuitive that F-statistic uses the number of observations in a regression, which frankly uses N = 50 observations (at each point in time the regression is run), although the coefficients are subsequently an average of 95 observations.