# Is CAPM a cross sectional or time series model?

Given that CAPM is an equilibrium model, it prices the assets in absolute terms. Asset pricing studies use CAPM/ICAPM/CCAPM in a cross-sectional framework i.e. stocks with higher betas will have higher returns in a cross section (or relative to other stocks with lower betas). My question is that given CAPM is equilibrium model, can it be used as an absolute pricing tool in a time series i.e. to predict tomorrows return for instance of Apple? Also please compare the FF 3 factor model in the same light !

To answer your question directly: CAPM is a cross-sectional model, and is NOT a time series model.

CAPM aims at explaining variance of single asset's return by overall market return of the same period. This makes it impossible to predict return because once you have observed the market return, you will also observe the asset's return

On the other hand, a (predictive) time series model involves predicting future values at any point in time based on information up to that time.

FF model is similar. It is also cross-sectional but NOT time series model

• Exactly! The CAPM is a model for expected returns and not for returns! You could run a time-series regression of asset $i$ excess returns on market excess returns, obtain an $R^2$ of zero and still don't reject the CAPM if the intercept is not statistically significant! – fni May 21 '18 at 21:46
• No. If $R^2$ is zero you must reject the CAPM. No only intercept is relevant but also the slope coefficient ($\beta$). See Fama and MacBeth procedure here it.wikipedia.org/wiki/Regressione_Fama-MacBeth Its true that CAPM is "static model" no time series one, however time series concepts are involved in the "CAPM framework". The two dimension can be related. In fact the Fama MacBeth procedure sound like panel. – markowitz Feb 19 '19 at 16:13

CAPM is neither a cross sectional model, nor a time series model!

The classic CAPM formula $$$$\operatorname{E}(R_{i})-R_{0}=\beta_{i}(\operatorname{E}(R_{M})-R_{0})\qquad i=1,2,\cdots,N \label{E:CAPM}%$$$$ is a relation on expected return, not on return (random variable) itself.

In Econometrics, a cross sectional or a time series model is talking about conditional expectation function, say $$\operatorname{E}\left( \left. Y\,\right\vert X\right) =\alpha+\beta X$$ or equivalently $$Y=\alpha+\beta X+\epsilon$$ with $$\operatorname{E}\left( \left. \epsilon\,\right\vert X\right) =0$$ (for consistent estimators, the mean independence is relaxed to orthogonality $$\operatorname{E}(\epsilon X) =0$$)

$$\operatorname{E}\left( Y\right)$$ is a number, but $$\operatorname{E}\left( \left. Y\,\right\vert X\right)$$ is a random variable: Let $$Y=a+bX+\epsilon$$ and (joint normal distribution) $$\begin{bmatrix} X\\ \epsilon \end{bmatrix} \sim\mathrm{N}\left( \begin{bmatrix} \mu_{X}\\ 0 \end{bmatrix} , \begin{bmatrix} \sigma_{X}^{2} & \rho\sigma_{\epsilon}\sigma_{X}\\ \rho\sigma_{\epsilon}\sigma_{X} & \sigma_{\epsilon}^{2}% \end{bmatrix} \right)$$ with $$\rho>0$$. Then $$\operatorname{E}\left( Y\right) =\mu_{Y}=a+b\mu _{X}=a+b\operatorname{E}\left( X\right)$$ However \begin{align*} \operatorname{E}\left( \left. Y\,\right\vert X\right) & =a+bX+\operatorname{E}\left( \left. \epsilon\,\right\vert X\right) \\ & =a+bX+\left( X-\operatorname{E}\left( X\right) \right) \rho \sigma_{\epsilon}^{\,}/\sigma_{X}^{\,}% \end{align*} Note that OLS estimator is NOT consistent because of endogeneity. Say $$\mathrm{cov}\left( \epsilon,X\right) =\rho\sigma_{X}\sigma_{\epsilon}>0$$.

• A warm welcome to Quantitative Finance SE... I appreciate your detailed answer (+1); but the CAPM actually describes a cross-sectional relationship of expected returns which contradicts your initial sentence. Actually it is an economic theory; but the empirical test setting requires a cross-sectional analysis. – skoestlmeier Apr 29 at 12:00
• TO: @skoestlmeier "the CAPM actually describes a cross-sectional relationship of expected returns". Speaking of economic theory, the returns are endogenous in CAPM. The beta value is calculated from the equilibrium return, using beta value to explain the expected return is a circular argument. For more on CAPM, see Arbitrage Opportunity, Impossible Frontier, and Logical Circularity in CAPM Equilibrium – Chen Deng-Ta Apr 29 at 23:26
• I appreciate your valuable comment; thanks for the linked paper. I would like to note that you have to distinguish returns and expected returns. While the former are used to calculate beta, the latter are estimated within the CAPM framework. I do not see a circular argument here, as long as actual returns (exogenous) and expected returns (endogenous) do not have to match (e.g. in the current recession, actual returns are highly negative; so stocks are more risky and investors demand higher (positive) expected returns) and their values can be far away from each other. – skoestlmeier Apr 30 at 7:13

The standard CAPM is in the same time period.

If you are trying to predict a share price for example, you will have to use lags of variables.

RM(t) = c + DY(t-1) + e

for example.

I think a really good way to learn things like this is to practice yourself using econometrics software. Eviews is great for beginners, you can download data from Kenneth Frenchs data library and practice.

In reviews, if you're truing to make a prediction about the market index using the dividend yield, we can simple write,

RM c DY(-1)

Now you can see that there is a possible lead lag relationship if DY is significant.

I think the previous answers are not correct.

CAPM can be either a time-series or cross-sectional model, depending on your specification.

For example, the Fama-Macbeth procedure estimates the beta using a time-series data, and then estimate the equity risk premium with a cross-section regression and the betas obtained in the first step.

The real question is: should CAPM hold in cross-sectional or time-series data?

Research has shown that neither holds. The closest that it could get is conditional CAPM - which assumes beta changes over time.