# Calculating the pricing error in Fama-Macbeth Regression for Fama/French 5 Factor model

I'm very much new to this area and I need to know on how to calculate the pricing error in Fama/French 5-Factor model. The evaluation was done using the Fama-Macbeth approach.

I did everything as shown in this answer. Fama-Macbeth second step confusion. The calculations were done in excel.

Now I'm having this with me,

That is the averaged, lambda values of Mkt-RF, SMB , HML , RMW , CMA. What is the pricing error in this case? and how to calculate it?

And how to estimate the SML line?

As I understood, is this the pricing error? But in Fama French 5 factor approch how can I calculate this?

Because there are 5 slopes I can be calculated since there are 4 beta values

John Cochrane (in Asset Pricing) p. 244:

Sampling error is, after all, about how a statistic would vary from one sample to the next if we repeated the observations.

## Clarification on linear regression

Any linear regression $$y = X \beta + \epsilon$$ involves the following parameters and variables:

• The unknown parameters, denoted as $$\beta$$ , a $$(\mathrm{p} \times 1)$$ vector
• The dependent variables $$Y$$, a $$(\mathrm{n} \times 1)$$ vector
• The independent variables $$X$$, a $$(\mathrm{n} \times \mathrm{p})$$ matrix
• The residuals $$\epsilon$$, a $$(\mathrm{n} \times 1)$$ vector

What you are trying to get, are point estimates for your regression-coefficients $$\beta$$. However, this estimates are tied to the sample you are analyzing. If you calculate $$\beta$$ for another sample, you will get different coefficients (see the cite above). So in fact, you obtain expected values $$X$$, i.e. $$\operatorname{E}(X)$$, and as a measure of uncertainty of this estimate, you use $$\sigma_{X}$$.

## Fama-MacBeth Regression

The Fama-MacBeth approach is a cross-sectional regression at each period of time: $$R_{t}^{ei}= \beta_{i}^{'}\lambda_t+a_{it}$$

where $$R_{t}^{ei}$$ is the excess-return of asset $$i$$ at time $$t$$ and $$\beta_{i}^{'}$$ denotes the estimated beta-factor of the stock.

What is the pricing error?

The pricing error is the part of the return $$R_{t}^{ei}$$, unexplained by your factors $$\beta$$, i.e. the pricing error is $$a_{it}$$.

You get a pricing error $$\hat{a}_{it}$$ for each cross-sectional regression, i.e. if you have e.g. a time-series of 120 month, you obtain 120 values for $$\hat{a}_{it}$$. After that, you just calculate the time-series average of these cross-sectional estimates:

$$\hat{a}_i = \frac{1}{T} \sum_{t=1}^{T}{\hat{a}}_{it}$$

How significant is this value $$\hat{a}_i$$?

You notice the hat on $$\hat{a}_i$$? That is because your estimate for $$a_i$$ is tied to the specific sample you are analyzing. How much would your $$a_i$$ differ, if you would e.g. have used other 120 month for your analysis?

We are used to deducing the sampling variance of the sample mean of a series $$x_t$$ by looking at the variation of $$x_t$$ through time in the sample. The estimate for the (squared) sampling error of $$\hat{a}_i$$ under the Fama-MacBeth assumptions is:

$$\sigma^2(\hat{a}_i) = \frac{1}{T^2} \sum_{t=1}^{T}{\left( \hat{a}_{it} - \hat{a}_i \right)^2}$$

, i.e. you divide the variance of $$\hat{a}_{it}$$ by $$T$$ (see here). The standard error $$SE$$ is then:

$$SE(\hat{a}_i) = \sqrt{\sigma^2(\hat{a}_i)}$$

Why do you need the standard error?

To test the statistical significance of you estimated pricing error $$\hat{a}_i$$. Under the null-hypothesis $$a_i = 0$$, your test-statistic is:

$$t_{score} = \frac{\hat{a}_i}{SE(\hat{a}_i)} \sim\mathcal{T}_{k}$$

$$t_{score}$$ has a t-distribution with $$k = T-p$$ (i.e. the number of observations $$T$$ minus the amount $$p$$ of estimated parameters $$\beta_i$$ in your regression) degrees of freedom if the null hypothesis is true.

• I have one question left though. Is the alpha the residual of the Fama Mac Beth Regression or an intercept I estimate? And also, is this average alpha calculated for every asset i= 1,...,n? If I had 20 stocks I regress e.g. do I have to calculate an alpha (and test statistic) for every stock? – J.Pop Feb 13 '19 at 13:00
• Alpha $a_t$ denotes the average error of a cross-sectional regression estimate you run at each point of time $t$. If you have e.g. 120 monthly returns of e.g. 100 stocks, you obtain an $a_t$ for each of the 120 cross-sectional regressions. The test statistic is derived from this time series of 120 $a_i$'s. – skoestlmeier Feb 13 '19 at 17:15
• @skoestlmeier So $\hat{a}_{it}$ is the average pricing errors across assets at each time t? And how is the test of $t_{score} = \frac{\hat{a}_i}{SE(\hat{a}_i)} \sim\mathcal{T}_{k}$ different from the $\hat{\alpha}^{'}cov(\hat{\alpha})^{-1}\hat{\alpha}\sim\chi_{N-1}^2$ as in Cochrane (2011)? – Janys Apr 25 '19 at 12:33
• The main difference is that a simple t-test as a univariat method just holds for the one specific time-series for $\hat{\alpha}_i$ you are looking for. What you describe is the GRS-test where i recommend you to read this wonderful answer. In fact, it is a jointly test for the significance of the estimated values across all $\hat{\alpha}_i$ simultaneously. – skoestlmeier Apr 26 '19 at 11:14
• @skoestlmeier Many thanks for your reply and the link! – Janys Apr 30 '19 at 20:14