Example: Fama

The asymptotic variance requires computing the covariance of the demeaned returns and the weighted pricing errors. The problem is formulated using 2-step GMM where the moment conditions are \begin{equation} g_{t}\left(\theta\right)=\left[\begin{array}{c} \epsilon_{1t}\\ \epsilon_{1t}f_{t}\\ \epsilon_{2t}\\ \epsilon_{2t}f_{t}\\ \vdots\\ \epsilon_{Nt}\\ \epsilon_{Nt}f_{t}\\ \beta u_{t} \end{array}\right] \end{equation}

where $\epsilon_{it}=r_{it}^{e}-\alpha_{i}-\beta_{i}^{\prime}f_{t}$, $\beta_{i}$ is a $K$ by 1 vector of factor loadings, $f_{t}$ is a $K$ by 1 set of factors, $\beta=\left[\beta_{1}\,\beta_{2}\ldots\beta_{N}\right]$ is a $K$ by $N$ matrix of all factor loadings, $u_{t}=r_{t}^{e}-\beta'\lambda$ are the $N$ by 1 vector of pricing errors and $\lambda$ is a $K$ by 1 vector of risk premia. The vector of parameters is then $\theta= \left[\alpha_{1}\:\beta_{1}^{\prime}\:\alpha_{2}\:\beta_{2}^{\prime}\:\ldots\:\alpha_{N}\,\beta_{N}^{\prime}\:\lambda'\right]'$ To make inference on this problem, the derivative of the moments with respect to the parameters, $\partial g_{t}\left(\theta\right)/\partial\theta^{\prime}$ is needed. With some work, the estimator of this matrix can be seen to be

where $X_{t}=\left[1\: f_{t}^{\prime}\right]'$ and $\Sigma_{X}=E\left[X_{t}X_{t}^{\prime}\right]$. $G_{21}$ is a matrix with the structure

where

and where $E\left[u_{i}\right]$ is the expected pricing error. In estimation, all expectations are replaced with their sample analogues.