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I am trying to wrap my head around GMM estimation of a single factor model such as the CAPM. I started by asking How come the cross-sectional CAPM equation produces $N$ moment conditions (not $1$)? and am now following it up. Equation $(12.23)$ in Cochrane "Asset Pricing" (2005) section 12.2 (p. 241) says 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), so there are actually $2N$ conditions. If I understand correctly, the third row corresponds to a cross-sectional regression of time-averaged returns: $$ E_T(R^{ei})=\beta_i' \lambda+\alpha_i, \quad i=1,2,\dots,N. \tag{12.10} $$ ($(12.10)$ 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 $\lambda$.)

Since we are trying to estimate (among other things) the expected market excess return $\lambda$, an obvious approach would be to use the sample average of the market excess returns over the $T$ time periods. In a GMM, the corresponding moment condition would be $\frac{1}{T}\sum_{t=1}^T (f_t-\lambda)=0$. In the notation of Cochrane, it would be $\color{blue}{E_T(f_t-\lambda)=0}$. Yet this is not what we see in the GMM estimator for the single-factor model in equation $(12.23)$.

What I would find intuitive is to have $$ \tilde g_T(b) = \begin{bmatrix} E(R^e_t-a-\beta f_t) \\ E[(R^e_t-a-\beta f_t)f_t] \\ \color{blue}{E(f_t-\lambda)} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ \color{blue}{0} \end{bmatrix} \tag{12.23'} $$ or perhaps $$ \tilde{\tilde g}_T(b) = \begin{bmatrix} E(R^e_t-a-\beta f_t) \\ [E(R^e_t-a-\beta f_t)f_t] \\ \color{red}{E(R^e-\beta \lambda)} \\ \color{blue}{E(f_t-\lambda)} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ \color{red}{0} \\ \color{blue}{0} \end{bmatrix}. \tag{12.23''} $$ Again, since we are interested in estimating $\lambda$ (among other things), is there a reason for excluding the obvious moment condition in blue from $(12.23)$? I do not have well developed intuition around GMM, but I think the blue condition is pretty informative about $\lambda$, probably more so than the red one.

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Notation note: I like using bold letters for vectors.

The vector equation: $$ \operatorname{E}[\mathbf{R}^e - \boldsymbol{\beta} \lambda] = \mathbf{0}$$

basically states that for all assets (or test assets), the expected return $\operatorname{E}[R^e_i ]$ of an asset is equal to its regression coefficient $\beta_i$ on the factor (i.e. quantity of risk) times the price of risk $\lambda$.

In the general case, the factor $f$ need not be a return (eg. $f$ is aggregate consumption from GDP numbers).

Special case: $f$ itself is a return

Then your blue line isn't excluded! It's in the 3rd line of (12.23).

If $f$ is a return, then substituting into $\operatorname{E}[R_i^e - \beta_i \lambda] = 0$ gives your blue equation (because the beta of $f$ on itself is 1):

$$ \operatorname{E}[f - \lambda ] = 0$$

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  • $\begingroup$ That is helpful! Does that mean if $f$ itself is a return, then the vector equation $ \operatorname{E}[\mathbf{R}^e - \boldsymbol{\beta} \lambda] = \mathbf{0}$ includes the scalar equation $\operatorname{E}[f - \lambda ] = 0$ as one of its components? $\endgroup$ Feb 15, 2023 at 16:10
  • $\begingroup$ @RichardHardy Yeah, the $E[R^e−\beta \lambda]=0$ eqn. applies to returns so if $f$ is a return, it applies to it too! If all the factors in a factor model are returns, then you can: (1) substitute expected return of a factor $\operatorname{E}[f]$ for the price of risk $\lambda$ associated with the factor (2) algebra works out such that a risk-based asset pricing model with factors that are returns will have alpha as the pricing error: hence 0 alpha is the same as the model explaining expected returns (if factor is something NOT a return like consumption, then alpha need not be zero). $\endgroup$ Feb 15, 2023 at 16:34
  • $\begingroup$ Unfortunately, algebra does not work out for me! I am quite confused by treating alpha as pricing errors in $R^{ei}_t=\alpha_i+\beta_i f_t+\varepsilon^i_t$ and using that as a basis for the GRS test, hence my question. $\endgroup$ Feb 15, 2023 at 17:26
  • $\begingroup$ I suppose it makes a difference whether I include $f_t$ among the test assets or not. (By default, I was not thinking of including it. I am interested in pricing assets, not their sum, and I can always obtain the sum from the elements.) If I do, then indeed the blue equation is part of the third row of $(12.23)$. However, if I do not, then it is not. If that is so, excluding $f_t$ from test assets would allow us to estimate the price of risk without pushing it closer to the market's realized mean excess return. On the other hand, including $f_t$ would push it that way. Does that make sense? $\endgroup$ Jun 2, 2023 at 10:25
  • $\begingroup$ However, I am not entirely convinced by my own argument. Including $f_t$ seems like adding a collinear data point, as the sum (the market's return) can be obtained from the individual elements (the assets' returns). So perhaps the estimate of $\lambda$ is pushed towards the market's realized mean excess return $E_T[f_t]=\bar f$ regardless. $\endgroup$ Jun 2, 2023 at 10:30
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A guess: In the context of the book, we are probably not trying to estimate the market's excess return per se. When the theoretical quantities are replaced by their empirical counterparts in the GMM estimator specified in $(12.23)$, $\lambda$ probably gets replaced by the empirical mean of $f$. Thus, we do not need the additional row in blue; it would be superfluous.

Then we can use the first two rows ($2N$ conditions) for estimating $\beta$. The third row (another $N$ conditions) is used to test the CAPM; if the CAPM holds, the empirical deviations from equality in the third row should be small. If they are large, the CAPM is unlikely to hold.

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  • $\begingroup$ Yeah, he's got a far more general framework in mind where there's some stochastic discount factor that explains asset prices and covariance with that stochastic discount factor gives a payoff $\lambda$. The CAPM is a specific theory of that form that makes the remarkably bold (and it turns out empirically false) assertion that the stochastic discount factor is just the market return. $\endgroup$ Feb 15, 2023 at 14:53

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