Why the CAPM GMM Moments May Already Identify the Market Premium

# GMM estimation of the CAPM: why not include sample mean of the market excess return as a moment?


# GMM estimation of the CAPM: why not include sample mean of the market excess return as a moment?












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.

## Answer by Matthew Gunn (score 2, accepted)

https://quant.stackexchange.com/a/74650

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):

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

## Answer by Richard Hardy (score 1)

https://quant.stackexchange.com/a/74539

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.