Incorporating idiosyncratic risk as a pricing factor Fama-MacBeth style

Suppose we are given a dataset with $$T$$ time periods and $$N$$ assets or portfolios. We are interested in estimating and testing the CAPM or a multifactor model. Take the CAPM: $$r^*_{i,t}=\alpha_i+\beta_i \mu^*_{m}+\varepsilon_{i,t} \tag{1}$$ where $$r^*_i:=(r_{i,t}-r_{f,t})$$ is firm's $$i$$ excess return and $$\mu^*_{m}:=(\mu_{m,t}-r_{f,t})$$ is the market's expected excess return (assumed to be constant over time for simplicity). According to the CAPM, $$\alpha_i=0$$ for each $$i$$.

We could estimate the model Fama-MacBeth style. That is, we would first obtain estimates of the betas from time series regressions $$r^*_{i,t}=\alpha_i+\beta_i r^*_{m,t}+\varepsilon_{i,t} \tag{2}$$ for each asset (with $$r^*_{m,t}:=(r_{m,t}-r_{f,t})$$ where $$r_{m,t}$$ is the market return) and then estimate the alphas from cross sectional regressions $$r^*_{i,t}=\alpha_i+\lambda\hat\beta_i+\varepsilon_{i,t} \tag{3}$$ for each time period. Or we could do it using GMM – see my GMM question. (I suppose there are other alternatives, too.)

Now I would like to add idiosyncratic risk as a candidate pricing factor: $$r^*_{i,t}=\alpha_i+\beta_i \mu^*_{m}+\gamma\sigma_i^2+\varepsilon_{i,t} \tag{4}$$ where $$\sigma_i^2$$ is the idiosyncratic risk of asset $$i$$. (This is just an example. I am not saying I think the idiosyncratic risk is priced in reality.)

I think I have an idea about how we could incorporate it Fama-MacBeth style. $$\sigma_i^2$$ would be estimated alongside $$\beta_i$$ in the time series regressions $$(2)$$ (the first step) and then appended to the cross-sectional regressions $$(3)$$ (the second step) to yield $$r^*_{i,t}=\alpha_i+\lambda\hat\beta_i+\gamma\hat\sigma_i^2+\varepsilon_{i,t} \tag{3'}.$$ Does that look alright?

Update: The question about GMM estimation has been moved to a separate thread. This is because this thread seems to have gotten off track due to a highly upvoted answer to a slightly different question. While the answer is great and I appreciate it, I am still interested in GMM estimation and testing of the model, hence the split into two separate threads.

• The original post had some fairly important details messed up, hence the comprehensive edit. Commented Feb 13, 2023 at 15:03
• It is still messed up, I think, as $\alpha_i$ and $\lambda$ in $(3)$ are not individually identified from a single regression equation. I need to check how the second stage of Fama-MacBeth actually works... Commented Feb 15, 2023 at 17:32

• Thank you, that makes sense. I suppose the suggested approach suffers from the usual problem of Fama-MacBeth style analysis; factor-specific betas are estimated with error in the first step. Therefore, I am interested in a GMM solution; it seems cleaner. Also, your first sentence makes a lot of sense; accordingly, I have now elaborated on my description of how I would do my thing Fama-MacBeth style. According to this, it seems they used $\sigma_i$ while I was going to use $\sigma_i^2$. Commented Jan 31, 2023 at 9:33