Fama Macbeth Regression in Python (Pandas or Statsmodels)
An update to reflect the library situation for Fama-MacBeth as of Fall 2018. The fama_macbeth function has been removed from pandas for a while now. So what are your options?
If you're using python 3, then you can use the Fama-MacBeth method in LinearModels: https://github.com/bashtage/linearmodels/blob/master/linearmodels/panel/model.py
If you're using python 2 or just don't want to use LinearModels, then probably your best option is to roll you own.
For example, suppose you have the Fama-French industry portfolios in a panel like the following (you've also computed some variables like past beta or past returns to use as your x-variables):
In [1]: import pandas as pd
import numpy as np
import statsmodels.formula.api as smf
In [4]: df = pd.read_csv('industry.csv',parse_dates=['caldt'])
df.query("caldt == '1995-07-01'")
In [5]: Out[5]:
industry caldt ret beta r12to2 r36to13
18432 Aero 1995-07-01 6.26 0.9696 0.2755 0.3466
18433 Agric 1995-07-01 3.37 1.0412 0.1260 0.0581
18434 Autos 1995-07-01 2.42 1.0274 0.0293 0.2902
18435 Banks 1995-07-01 4.82 1.4985 0.1659 0.2951
Fama-MacBeth primarily involves computing the same cross-sectional regression model month by month, so you can implement it using a groupby. You can create a function that takes a dataframe (it will come from the groupby) and a patsy formula; it then fits the model and returns the parameter estimates. Here is a barebones version of how you could implement it (note this is what the original questioner tried to do a few years ago ... not sure why it didn't work although it's possible back then statsmodels result object method params wasn't returning a pandas Series so the return needed to be converted to a Series explicitly ... it does work fine in the current version of pandas, 0.23.4):
def ols_coef(x,formula):
return smf.ols(formula,data=x).fit().params
In [9]: gamma = (df.groupby('caldt')
.apply(ols_coef,'ret ~ 1 + beta + r12to2 + r36to13'))
gamma.head()
In [10]: Out[10]:
Intercept beta r12to2 r36to13
caldt
1963-07-01 -1.497012 -0.765721 4.379128 -1.918083
1963-08-01 11.144169 -6.506291 5.961584 -2.598048
1963-09-01 -2.330966 -0.741550 10.508617 -4.377293
1963-10-01 0.441941 1.127567 5.478114 -2.057173
1963-11-01 3.380485 -4.792643 3.660940 -1.210426
Then just compute the mean, standard error on the mean, and a t-test (or whatever statistics you want). Something like the following:
def fm_summary(p):
s = p.describe().T
s['std_error'] = s['std']/np.sqrt(s['count'])
s['tstat'] = s['mean']/s['std_error']
return s[['mean','std_error','tstat']]
In [12]: fm_summary(gamma)
Out[12]:
mean std_error tstat
Intercept 0.754904 0.177291 4.258000
beta -0.012176 0.202629 -0.060092
r12to2 1.794548 0.356069 5.039896
r36to13 0.237873 0.186680 1.274230
Improving Speed
Using statsmodels for the regressions has significant overhead (particularly given you only need the estimated coefficients). If you want better efficiency, then you could switch from statsmodels to numpy.linalg.lstsq. Write a new function that does the ols estimation ... something like the following (notice I'm not doing anything like checking the rank of these matrices ...):
def ols_np(data,yvar,xvar):
gamma,_,_,_ = np.linalg.lstsq(data[xvar],data[yvar],rcond=None)
return pd.Series(gamma)
And if you're still using an older version of pandas, the following will work:
Here is an example of using the fama_macbeth function in pandas:
>>> df
y x
date id
2012-01-01 1 0.1 0.4
2 0.3 0.6
3 0.4 0.2
4 0.0 1.2
2012-02-01 1 0.2 0.7
2 0.4 0.5
3 0.2 0.1
4 0.1 0.0
2012-03-01 1 0.4 0.8
2 0.6 0.1
3 0.7 0.6
4 0.4 -0.1
Notice, the structure. The fama_macbeth function expects the y-var and x-vars to have a multi-index with date as the first variable and the stock/firm/entity id as the second variable in the index:
>>> fm = pd.fama_macbeth(y=df['y'],x=df[['x']])
>>> fm
----------------------Summary of Fama-MacBeth Analysis-------------------------
Formula: Y ~ x + intercept
# betas : 3
----------------------Summary of Estimated Coefficients------------------------
Variable Beta Std Err t-stat CI 2.5% CI 97.5%
(x) -0.0227 0.1276 -0.18 -0.2728 0.2273
(intercept) 0.3531 0.0842 4.19 0.1881 0.5181
--------------------------------End of Summary---------------------------------
Note that just printing fm calls fm.summary
>>> fm.summary
----------------------Summary of Fama-MacBeth Analysis-------------------------
Formula: Y ~ x + intercept
# betas : 3
----------------------Summary of Estimated Coefficients------------------------
Variable Beta Std Err t-stat CI 2.5% CI 97.5%
(x) -0.0227 0.1276 -0.18 -0.2728 0.2273
(intercept) 0.3531 0.0842 4.19 0.1881 0.5181
--------------------------------End of Summary---------------------------------
Also, note the fama_macbeth function automatically adds an intercept (as opposed to statsmodels routines). Also the x-var has to be a dataframe so if you pass just one column you need to pass it as df[['x']].
If you don't want an intercept you have to do:
>>> fm = pd.fama_macbeth(y=df['y'],x=df[['x']],intercept=False)
EDIT: New Library
An updated library exists which can be installed via the following command:
pip install finance-byu
Documentation here: https://fin-library.readthedocs.io/en/latest/
The new library includes Fama Macbeth regression implementations and a Regtable class that can be helpful for reporting results.
This page in the documentation outlines the Fama Macbeth functions: https://fin-library.readthedocs.io/en/latest/fama_macbeth.html
There is an implementation which is very similar to Karl D.'s implementation above with numpy's linear algebra functions, an implementation that utilizes joblib for parallelization to increase performance when a large number of time periods in the data, and an implementation using numba for optimization that shaves off an order of magnitude on small data sets.
Here is an example with a small simulated data set as in the documentation:
>>> from finance_byu.fama_macbeth import fama_macbeth, fama_macbeth_parallel, fm_summary, fama_macbeth_numba
>>> import pandas as pd
>>> import time
>>> import numpy as np
>>>
>>> n_jobs = 5
>>> n_firms = 1.0e2
>>> n_periods = 1.0e2
>>>
>>> def firm(fid):
>>> f = np.random.random((int(n_periods),4))
>>> f = pd.DataFrame(f)
>>> f['period'] = f.index
>>> f['firmid'] = fid
>>> return f
>>> df = [firm(i) for i in range(int(n_firms))]
>>> df = pd.concat(df).rename(columns={0:'ret',1:'exmkt',2:'smb',3:'hml'})
>>> df.head()
ret exmkt smb hml period firmid
0 0.766593 0.002390 0.496230 0.992345 0 0
1 0.346250 0.509880 0.083644 0.732374 1 0
2 0.787731 0.204211 0.705075 0.313182 2 0
3 0.904969 0.338722 0.437298 0.669285 3 0
4 0.121908 0.827623 0.319610 0.455530 4 0
>>> result = fama_macbeth(df,'period','ret',['exmkt','smb','hml'],intercept=True)
>>> result.head()
intercept exmkt smb hml
period
0 0.655784 -0.160938 -0.109336 0.028015
1 0.455177 0.033941 0.085344 0.013814
2 0.410705 -0.084130 0.218568 0.016897
3 0.410537 0.010719 0.208912 0.001029
4 0.439061 0.046104 -0.084381 0.199775
>>> fm_summary(result)
mean std_error tstat
intercept 0.506834 0.008793 57.643021
exmkt 0.004750 0.009828 0.483269
smb -0.012702 0.010842 -1.171530
hml 0.004276 0.010530 0.406119
>>> %timeit fama_macbeth(df,'period','ret',['exmkt','smb','hml'],intercept=True)
123 ms ± 117 µs per loop (mean ± std. dev. of 7 runs, 10 loops each
>>> %timeit fama_macbeth_parallel(df,'period','ret',['exmkt','smb','hml'],intercept=True,n_jobs=n_jobs,memmap=False)
146 ms ± 16.9 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
>>> %timeit fama_macbeth_numba(df,'period','ret',['exmkt','smb','hml'],intercept=True)
5.04 ms ± 5.2 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
Note: Turning off the memmap makes for fair comparison without generating new data at each run. With the memmap, the parallel implementation would simply pull cached results.
Here are a couple simple implementations of the table class also using simulated data:
>>> from finance_byu.regtables import Regtable
>>> import pandas as pd
>>> import statsmodels.formula.api as smf
>>> import numpy as np
>>>
>>>
>>> nobs = 1000
>>> df = pd.DataFrame(np.random.random((nobs,3))).rename(columns={0:'age',1:'bmi',2:'hincome'})
>>> df['age'] = df['age']*100
>>> df['bmi'] = df['bmi']*30
>>> df['hincome'] = df['hincome']*100000
>>> df['hincome'] = pd.qcut(df['hincome'],16,labels=False)
>>> df['rich'] = df['hincome'] > 13
>>> df['gender'] = np.random.choice(['M','F'],nobs)
>>> df['race'] = np.random.choice(['W','B','H','O'],nobs)
>>>
>>> regformulas = ['bmi ~ age',
>>> 'bmi ~ np.log(age)',
>>> 'bmi ~ C(gender) + np.log(age)',
>>> 'bmi ~ C(gender) + C(race) + np.log(age)',
>>> 'bmi ~ C(gender) + rich + C(gender)*rich + C(race) + np.log(age)',
>>> 'bmi ~ -1 + np.log(age)',
>>> 'bmi ~ -1 + C(race) + np.log(age)']
>>> reg = [smf.ols(f,df).fit() for f in regformulas]
>>> tbl = Regtable(reg)
>>> tbl.render()
Produces the following:
>>> df2 = pd.DataFrame(np.random.random((nobs,10)))
>>> df2.columns = ['t0_vw','t4_vw','et_vw','t0_ew','t4_ew','et_ew','mktrf','smb','hml','umd']
>>> regformulas2 = ['t0_vw ~ mktrf',
>>> 't0_vw ~ mktrf + smb + hml',
>>> 't0_vw ~ mktrf + smb + hml + umd',
>>> 't4_vw ~ mktrf',
>>> 't4_vw ~ mktrf + smb + hml',
>>> 't4_vw ~ mktrf + smb + hml + umd',
>>> 'et_vw ~ mktrf',
>>> 'et_vw ~ mktrf + smb + hml',
>>> 'et_vw ~ mktrf + smb + hml + umd',
>>> 't0_ew ~ mktrf',
>>> 't0_ew ~ mktrf + smb + hml',
>>> 't0_ew ~ mktrf + smb + hml + umd',
>>> 't4_ew ~ mktrf',
>>> 't4_ew ~ mktrf + smb + hml',
>>> 't4_ew ~ mktrf + smb + hml + umd',
>>> 'et_ew ~ mktrf',
>>> 'et_ew ~ mktrf + smb + hml',
>>> 'et_ew ~ mktrf + smb + hml + umd'
>>> ]
>>> regnames = ['Small VW','','',
>>> 'Large VW','','',
>>> 'Spread VW','','',
>>> 'Small EW','','',
>>> 'Large EW','','',
>>> 'Spread EW','',''
>>> ]
>>> reg2 = [smf.ols(f,df2).fit() for f in regformulas2]
>>>
>>> tbl2 = Regtable(reg2,orientation='horizontal',regnames=regnames,sig='coeff',intercept_name='alpha',nobs=False,rsq=False,stat='se')
>>> tbl2.render()
Produces the following:
The documentation for the Regtable class is here: https://byu-finance-library-finance-byu.readthedocs.io/en/latest/regtables.html
These tables can be exported to LaTeX for easy incorporation into writing:
tbl.to_latex()