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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.

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  • $\begingroup$ The original post had some fairly important details messed up, hence the comprehensive edit. $\endgroup$ Feb 13, 2023 at 15:03
  • $\begingroup$ 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... $\endgroup$ Feb 15, 2023 at 17:32

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By definition idiosyncratic volatility needs to be computed against a candidate asset pricing model. See for example this paper.

So my suggestion is:

  1. Run your favorite asset pricing model (e.g. the Cahart 4-factor model)
  2. Estimate idiosyncratic volatility from that model
  3. Create a long-short portfolio (based on idiosyncratic volatility)
  4. Use the return of that portfolio as a factor, just as you use the market.
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  • $\begingroup$ 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 $\sigma_i^2$. $\endgroup$ Jan 31, 2023 at 9:33

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