Incorporating idiosyncratic risk as a pricing factor with GMM

Question

Originally I posted this on Quantitative Finance SE here but got no response. Months later, I am posting it here hoping for better luck.

Suppose we are given a dataset with $T$ time periods and $N$ assets or portfolios. We are interested in estimating and testing an augmented CAPM or a multifactor model with an additional factor: the asset's idiosyncratic volatility. In the case of the augmented Capital Asset Pricing Model (CAPM), the model is $$ \mu^*_{i}=\beta_i \mu^*_{m}+\gamma\sigma_i^2 \tag{4} $$ where $\mu^*_i:=(\mu_{i,t}-r_{f,t})$ is firm's $i$ expected excess return, $\mu^*_{m}:=(\mu_{m,t}-r_{f,t})$ is the market's expected excess return and $\sigma_i^2$ is the idiosyncratic risk of asset $i$ w.r.t. to the CAPM.$\color{red}{^*}$ That is, $\sigma_i^2$ is the error variance of a time series regression $$ r^*_{i,t}=\alpha_i+\beta_i r^*_{m,t}+\varepsilon_{i,t}. \tag{2} $$ We could estimate the model Fama-MacBeth style as discussed in this thread. However, I want to do that using the GMM. How would I set that up?

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$\color{red}{^*}$For simplicity, $\mu^*_i$, $\mu^*_{m}$ and $\sigma_i^2$ are assumed to be constant over time.

Answer 1

Here is my guess:

If we were to estimate a simple CAPM by GMM, we could use the equation $(12.23)$ from Cochrane "Asset Pricing" (2005) section 12.2 (p. 241). In his notation, the moments are $$ g_T(b) = \begin{bmatrix} E(R^e_t-a-\beta f_t) \\ E[(R^e_t-a-\beta f_t)f_t] \\ E(R^e-\beta \lambda) \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \tag{12.23} $$ where $R^e_t=(R^e_{1,t},\dots,R^e_{N,t})'$ is a vector of individual assets' excess returns, $\beta=(\beta_1,\dots,\beta_N)'$ is a vector of betas, $f_t$ is factor's excess return and $\lambda=E(f)$ is the expected value of the factor's excess return. The first two rows correspond to time series regressions for $N$ assets (one regression per asset) that facilitate estimation of the $\beta$ vector, so there are actually $2N$ conditions. If I understand correctly, the third row corresponds to another $N$ conditions (one per asset) of time-averaged returns that are used for testing the model: $$ E_T(R^{ei})=\beta_i' \lambda, \quad i=1,2,\dots,N. \tag{12.10'} $$ ($(12.10)$ is specified for potentially many factors, but $(12.23)$ considers the simple case of a single factor, so vector $\beta'$ turns into scalar $\beta$, and the same holds for $\lambda$.)

If we incorporate idiosyncratic risk as another factor, we obtain the following: $$ \tilde g_T(b) = \begin{bmatrix} E(R^e_t-a-\beta f_t) \\ E[(R^e_t-a-\beta f_t)f_t] \\ E[(R^e_t-a-\beta f_t)^2-\sigma^2] \\ E(R^e-\beta\lambda-\gamma\sigma^2) \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \tag{*} $$ where $\sigma^2=(\sigma^2_1,\dots,\sigma^2_N)'$ is a vector of idiosyncratic variances. The third row facilitates estimation of the $\sigma^2$ vector and the fourth row again is a set of $N$ conditions for testing the model. Does that make sense?