Example: Fama-MacBeth regression

Estimating the Risk Premia using Fama-MacBeth Regressions

This example highlights how to implement a Fama-MacBeth 2-stage regression to estimate factor risk premia, make inference on the risk premia, and test whether a linear factor model can explain a cross-section of portfolio returns. This example closely follows [Cochrane::2001] (See also [JagannathanSkoulakisWang::2010]). As in the previous example, the first segment contains the imports.

In [1]:
from numpy import mat, cov, mean, hstack, multiply,sqrt,diag, \
    squeeze, ones, array, vstack, kron, zeros, eye, savez_compressed
from numpy.linalg import inv
from scipy.stats import chi2
from pandas import read_csv
import statsmodels.api as sm

Next, the data are imported. I formatted the data downloaded from Ken French's website into an easy-to-import CSV which can be read by pandas.read_csv. The data is split using named columns for the small sets of variables and ix for the portfolios. The code uses pure NumPy arrays, and so values is used to retrieve the array from the DataFrame. The dimensions are determined using shape. Finally the risk free rate is forced to have 2 dimensions so that it will be broadcastable with the portfolio returns in the construction of the excess returns to the Size and Value-weighted portfolios. asmatrix is used to return matrix views of all of the arrays. This code is linear algebra-heavy and so matrices are easier to use than arrays.

In [2]:
data = read_csv('FamaFrench.csv')

# Split using both named colums and ix for larger blocks
dates = data['date'].values
factors = data[['VWMe', 'SMB', 'HML']].values
riskfree = data['RF'].values
portfolios = data.iloc[:, 5:].values

# Use mat for easier linear algebra
factors = mat(factors)
riskfree = mat(riskfree)
portfolios = mat(portfolios)

# Shape information
T,K = factors.shape
T,N = portfolios.shape
# Reshape rf and compute excess returns
riskfree.shape = T,1
excessReturns = portfolios - riskfree

The next block does 2 things:

  1. Compute the time-series $\beta$s. This is done be regressing the full array of excess returns on the factors (augmented with a constant) using lstsq.
  2. Compute the risk premia using a cross-sectional regression of average excess returns on the estimates $\beta$s. This is a standard regression where the step 1 $\beta$ estimates are used as regressors, and the dependent variable is the average excess return.
In [3]:
# Time series regressions
X = sm.add_constant(factors)
ts_res = sm.OLS(excessReturns, X).fit()
alpha = ts_res.params[0]
beta = ts_res.params[1:]
avgExcessReturns = mean(excessReturns, 0)
# Cross-section regression
cs_res = sm.OLS(avgExcessReturns.T, beta.T).fit()
riskPremia = cs_res.params

The asymptotic variance requires computing the covariance of the demeaned returns and the weighted pricing errors. The problem is formulated using 2-step GMM where the moment conditions are \begin{equation} g_{t}\left(\theta\right)=\left[\begin{array}{c} \epsilon_{1t}\\ \epsilon_{1t}f_{t}\\ \epsilon_{2t}\\ \epsilon_{2t}f_{t}\\ \vdots\\ \epsilon_{Nt}\\ \epsilon_{Nt}f_{t}\\ \beta u_{t} \end{array}\right] \end{equation}

where $\epsilon_{it}=r_{it}^{e}-\alpha_{i}-\beta_{i}^{\prime}f_{t}$, $\beta_{i}$ is a $K$ by 1 vector of factor loadings, $f_{t}$ is a $K$ by 1 set of factors, $\beta=\left[\beta_{1}\,\beta_{2}\ldots\beta_{N}\right]$ is a $K$ by $N$ matrix of all factor loadings, $u_{t}=r_{t}^{e}-\beta'\lambda$ are the $N$ by 1 vector of pricing errors and $\lambda$ is a $K$ by 1 vector of risk premia. The vector of parameters is then $\theta= \left[\alpha_{1}\:\beta_{1}^{\prime}\:\alpha_{2}\:\beta_{2}^{\prime}\:\ldots\:\alpha_{N}\,\beta_{N}^{\prime}\:\lambda'\right]'$ To make inference on this problem, the derivative of the moments with respect to the parameters, $\partial g_{t}\left(\theta\right)/\partial\theta^{\prime}$ is needed. With some work, the estimator of this matrix can be seen to be

\begin{equation} G=E\left[\frac{\partial g_{t}\left(\theta\right)}{\partial\theta^{\prime}}\right]=\left[\begin{array}{cc} -I_{n}\otimes\Sigma_{X} & 0\\ G_{21} & -\beta\beta^{\prime} \end{array}\right]. \end{equation}

where $X_{t}=\left[1\: f_{t}^{\prime}\right]'$ and $\Sigma_{X}=E\left[X_{t}X_{t}^{\prime}\right]$. $G_{21}$ is a matrix with the structure

\begin{equation} G_{21}=\left[G_{21,1}\, G_{21,2}\,\ldots G_{21,N}\right] \end{equation}

where

\begin{equation} G_{21,i}=\left[\begin{array}{cc} 0_{K,1} & \textrm{diag}\left(E\left[u_{i}\right]-\beta_{i}\odot\lambda\right)\end{array}\right]\end{equation}

and where $E\left[u_{i}\right]$ is the expected pricing error. In estimation, all expectations are replaced with their sample analogues.

In [4]:
# Moment conditions
X = sm.add_constant(factors)
p = vstack((alpha, beta))
epsilon = excessReturns - X @ p
moments1 = kron(epsilon, ones((1, K + 1)))
moments1 = multiply(moments1, kron(ones((1, N)), X))
u = excessReturns - riskPremia[None,:] @ beta
moments2 = u * beta.T
# Score covariance
S = mat(cov(hstack((moments1, moments2)).T))
# Jacobian
G = mat(zeros((N * K + N + K, N * K + N + K)))
SigmaX = (X.T @ X) / T
G[:N * K + N, :N * K + N] = kron(eye(N), SigmaX)
G[N * K + N:, N * K + N:] = -beta @ beta.T
for i in range(N):
    temp = zeros((K, K + 1))
    values = mean(u[:, i]) - multiply(beta[:, i], riskPremia)
    temp[:, 1:] = diag(values)
    G[N * K + N:, i * (K + 1):(i + 1) * (K + 1)] = temp

vcv = inv(G.T) * S * inv(G) / T

The $J$-test examines whether the average pricing errors, $\hat{\alpha}$, are zero. The $J$ statistic has an asymptotic $\chi_{N}^{2}$ distribution, and the model is badly rejected.

In [5]:
vcvAlpha = vcv[0:N * K + N:4, 0:N * K + N:4]
J = alpha @ inv(vcvAlpha) @ alpha.T
J = J[0, 0]
Jpval = 1 - chi2(25).cdf(J)

The final block using formatted output to present all of the results in a readable manner.

In [6]:
vcvRiskPremia = vcv[N * K + N:, N * K + N:]
annualizedRP = 12 * riskPremia
arp = list(squeeze(annualizedRP))
arpSE = list(sqrt(12 * diag(vcvRiskPremia)))
print('        Annualized Risk Premia')
print('           Market       SMB        HML')
print('--------------------------------------')
print('Premia     {0:0.4f}    {1:0.4f}     {2:0.4f}'.format(arp[0], arp[1], arp[2]))
print('Std. Err.  {0:0.4f}    {1:0.4f}     {2:0.4f}'.format(arpSE[0], arpSE[1], arpSE[2]))
print('\n\n')

print('J-test:   {:0.4f}'.format(J))
print('P-value:   {:0.4f}'.format(Jpval))

i = 0
betaSE = []
for j in range(5):
    for k in range(5):
        a = alpha[i]
        b = beta[:, i]
        variances = diag(vcv[(K + 1) * i:(K + 1) * (i + 1), (K + 1) * i:(K + 1) * (i + 1)])
        betaSE.append(sqrt(variances))
        s = sqrt(variances)
        c = hstack((a, b))
        t = c / s
        print('Size: {:}, Value:{:}   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)'.format(j + 1, k + 1))
        print('Coefficients: {:>10,.4f}  {:>10,.4f}  {:>10,.4f}  {:>10,.4f}'.format(a, b[0], b[1], b[2]))
        print('Std Err.      {:>10,.4f}  {:>10,.4f}  {:>10,.4f}  {:>10,.4f}'.format(s[0], s[1], s[2], s[3]))
        print('T-stat        {:>10,.4f}  {:>10,.4f}  {:>10,.4f}  {:>10,.4f}'.format(t[0], t[1], t[2], t[3]))
        print('')
        i += 1
        Annualized Risk Premia
           Market       SMB        HML
--------------------------------------
Premia     6.6642    2.8731     2.8080
Std. Err.  0.5994    0.4010     0.4296



J-test:   95.2879
P-value:   0.0000
Size: 1, Value:1   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:    -0.8354      1.3099      1.2892      0.3943
Std Err.          0.1820      0.1269      0.1671      0.2748
T-stat           -4.5904     10.3196      7.7127      1.4348

Size: 1, Value:2   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:    -0.3911      1.0853      1.6100      0.3317
Std Err.          0.1237      0.0637      0.1893      0.1444
T-stat           -3.1616     17.0351      8.5061      2.2971

Size: 1, Value:3   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:    -0.1219      1.0747      1.1812      0.4648
Std Err.          0.0997      0.0419      0.0938      0.0723
T-stat           -1.2225     25.6206     12.5952      6.4310

Size: 1, Value:4   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:     0.0388      0.9630      1.2249      0.5854
Std Err.          0.0692      0.0232      0.1003      0.0353
T-stat            0.5614     41.5592     12.2108     16.5705

Size: 1, Value:5   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:     0.0918      0.9850      1.3453      0.9052
Std Err.          0.0676      0.0255      0.0818      0.0610
T-stat            1.3580     38.5669     16.4489     14.8404

Size: 2, Value:1   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:    -0.2397      1.0691      1.0520     -0.2647
Std Err.          0.0725      0.0318      0.0609      0.0591
T-stat           -3.3052     33.6540     17.2706     -4.4768

Size: 2, Value:2   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:    -0.0194      1.0416      0.9880      0.1877
Std Err.          0.0615      0.0170      0.0776      0.0350
T-stat           -0.3162     61.1252     12.7393      5.3646

Size: 2, Value:3   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:     0.0898      0.9590      0.8619      0.3553
Std Err.          0.0517      0.0170      0.0733      0.0320
T-stat            1.7359     56.4856     11.7528     11.0968

Size: 2, Value:4   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:     0.0482      0.9788      0.8178      0.5562
Std Err.          0.0495      0.0138      0.0454      0.0281
T-stat            0.9733     70.7006     18.0210     19.8055

Size: 2, Value:5   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:    -0.0109      1.0502      0.9373      0.8493
Std Err.          0.0596      0.0182      0.0281      0.0263
T-stat           -0.1830     57.7092     33.3971     32.2980

Size: 3, Value:1   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:    -0.1556      1.1416      0.7883     -0.1980
Std Err.          0.0591      0.0190      0.0445      0.0411
T-stat           -2.6320     60.1173     17.6973     -4.8171

Size: 3, Value:2   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:     0.0889      1.0133      0.5151      0.0720
Std Err.          0.0553      0.0179      0.0340      0.0334
T-stat            1.6068     56.6380     15.1651      2.1546

Size: 3, Value:3   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:     0.1118      1.0129      0.4130      0.3379
Std Err.          0.0578      0.0267      0.0324      0.0321
T-stat            1.9344     37.9790     12.7488     10.5399

Size: 3, Value:4   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:     0.0818      0.9615      0.4646      0.5068
Std Err.          0.0568      0.0141      0.0475      0.0301
T-stat            1.4399     68.3360      9.7754     16.8580

Size: 3, Value:5   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:    -0.0526      1.1447      0.4970      0.9143
Std Err.          0.0687      0.0197      0.0509      0.0390
T-stat           -0.7655     58.0319      9.7690     23.4302

Size: 4, Value:1   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:     0.0902      1.0661      0.2857     -0.3692
Std Err.          0.0498      0.0151      0.0444      0.0323
T-stat            1.8127     70.4710      6.4268    -11.4334

Size: 4, Value:2   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:    -0.0104      1.0308      0.2430      0.1328
Std Err.          0.0534      0.0217      0.0300      0.0294
T-stat           -0.1952     47.5567      8.0926      4.5183

Size: 4, Value:3   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:     0.0392      1.0096      0.2214      0.2980
Std Err.          0.0572      0.0209      0.0436      0.0486
T-stat            0.6862     48.3271      5.0836      6.1333

Size: 4, Value:4   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:     0.0148      1.0437      0.2016      0.5857
Std Err.          0.0593      0.0224      0.0343      0.0484
T-stat            0.2497     46.5053      5.8694     12.0922

Size: 4, Value:5   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:    -0.1762      1.2284      0.2974      0.9834
Std Err.          0.0803      0.0224      0.0490      0.0378
T-stat           -2.1927     54.8427      6.0726     26.0265

Size: 5, Value:1   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:     0.0794      1.0310     -0.1507     -0.2508
Std Err.          0.0372      0.0095      0.0247      0.0168
T-stat            2.1369    108.0844     -6.1067    -14.9673

Size: 5, Value:2   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:     0.0535      0.9576     -0.1893     -0.0107
Std Err.          0.0457      0.0170      0.0243      0.0239
T-stat            1.1690     56.3228     -7.7765     -0.4458

Size: 5, Value:3   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:    -0.0236      0.9753     -0.2173      0.3127
Std Err.          0.0559      0.0178      0.0309      0.0256
T-stat           -0.4225     54.6936     -7.0217     12.2061

Size: 5, Value:4   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:    -0.1978      1.0546     -0.1732      0.7115
Std Err.          0.0587      0.0230      0.0300      0.0316
T-stat           -3.3679     45.7933     -5.7749     22.5339

Size: 5, Value:5   Alpha   Beta(VWM)   Beta(SMB)   Beta(HML)
Coefficients:    -1.2737      1.1045      0.0076      0.8527
Std Err.          0.3557      0.1143      0.1594      0.1490
T-stat           -3.5805      9.6657      0.0477      5.7232

The final block converts the standard errors of $\beta$ to be an array and saves the results.

In [7]:
betaSE = array(betaSE)
savez_compressed('fama-macbeth-results', alpha=alpha, beta=beta,
                 betaSE=betaSE, arpSE=arpSE, arp=arp, J=J, Jpval=Jpval)

Save Results

Save the estimated values for use in the $\LaTeX$ notebook.

In [8]:
from numpy import savez
savez('fama-macBeth-results.npz', arp=arp, beta=beta, arpSE=arpSE,
      betaSE=betaSE, J=J, Jpval=Jpval)