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I noticed that during the recent decade most of papers, which use Fama-MacBeth regressions compute Newey-West standard errors. I tried to find detailed description of this procedure in the books on empirical asset pricing (Campbell, Lo and MacKinlay; Cochrane; Bali, Engle, Murray), but none of them clearly decribes how to compute Newey-West standard errors in Fama-MacBeth (FM) regression.

As far as I understand, in the first step of FM procedure we run cross-sectional regression of returns on characteristics for each month: $R_i = \alpha + a_1\beta_i + a_2Ch_{1i} + a_3Ch_{2i} + ... + \epsilon_i$.

In the second step for each characteristic we find sample mean of the time series of its coefficients from step 1 and find the standard error, assuming that the coefficient estimates are normal iid. That is in time series $\{\hat{a}_{11}, \hat{a}_{12} ... \hat{a}_{1t}\}$ we find its mean $\bar{a_1}$ and its standard error as a square root of $\frac{1}{t}\sigma^2(\{\hat{a}_{11}, \hat{a}_{12} ... \hat{a}_{1t}\})$, where by $\sigma^2(X)$ I mean a variance of X.

Now we want to find Newey-West standard errors. Does it mean that now standard error (suppose 3 lags) is a square root of $\frac{1}{t}(\sigma^2(\{\hat{a}_{11}, \hat{a}_{12} ... \hat{a}_{1t}\})+\frac{4}{3}\gamma_1(\{\hat{a}_{11}, \hat{a}_{12} ... \hat{a}_{1t}\})+\frac{2}{3}\gamma_2(\{\hat{a}_{11}, \hat{a}_{12} ... \hat{a}_{1t}\}))$,

where $\gamma_1(X)$ means autocovariance of X at lag 1? I used Newey-West equation from Econometrics by Hayashi, page 409.

Is it correct?

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