Rolling Regression¶
Rolling OLS applies OLS across a fixed windows of observations and then rolls (moves or slides) the window across the data set. They key parameter is window
which determines the number of observations used in each OLS regression. By default, RollingOLS
drops missing values in the window and so will estimate the model using the available data points.
Estimated values are aligned so that models estimated using data points \(i, i+1, ... i+window\) are stored in location \(i+window\).
Start by importing the modules that are used in this notebook.
[1]:
import pandas_datareader as pdr
import pandas as pd
import statsmodels.api as sm
from statsmodels.regression.rolling import RollingOLS
import matplotlib.pyplot as plt
import seaborn
seaborn.set_style('darkgrid')
pd.plotting.register_matplotlib_converters()
%matplotlib inline
pandas-datareader
is used to download data from Ken French’s website. The two data sets downloaded are the 3 Fama-French factors and the 10 industry portfolios. Data is available from 1926.
The data are monthly returns for the factors or industry portfolios.
[2]:
factors = pdr.get_data_famafrench('F-F_Research_Data_Factors', start='1-1-1926')[0]
print(factors.head())
industries = pdr.get_data_famafrench('10_Industry_Portfolios', start='1-1-1926')[0]
print(industries.head())
Mkt-RF SMB HML RF
Date
1926-07 2.96 -2.30 -2.87 0.22
1926-08 2.64 -1.40 4.19 0.25
1926-09 0.36 -1.32 0.01 0.23
1926-10 -3.24 0.04 0.51 0.32
1926-11 2.53 -0.20 -0.35 0.31
NoDur Durbl Manuf Enrgy HiTec Telcm Shops Hlth Utils Other
Date
1926-07 1.45 15.55 4.69 -1.18 2.90 0.83 0.11 1.77 7.04 2.16
1926-08 3.97 3.68 2.81 3.47 2.66 2.17 -0.71 4.25 -1.69 4.38
1926-09 1.14 4.80 1.15 -3.39 -0.38 2.41 0.21 0.69 2.04 0.29
1926-10 -1.24 -8.23 -3.63 -0.78 -4.58 -0.11 -2.29 -0.57 -2.63 -2.85
1926-11 5.21 -0.19 4.10 0.01 4.71 1.63 6.43 5.42 3.71 2.11
The first model estimated is a rolling version of the CAP-M that regresses the excess return on Technology sector firms on the excess return on the market.
The window is 60 months, and so results are available after the first 60 (window
) months. The first 59 (window - 1
) estimates are all nan
filled.
[3]:
endog = industries.HiTec - factors.RF.values
exog = sm.add_constant(factors['Mkt-RF'])
rols = RollingOLS(endog, exog, window=60)
rres = rols.fit()
params = rres.params
print(params.head())
print(params.tail())
/home/travis/miniconda/envs/statsmodels-test/lib/python3.7/site-packages/numpy/core/fromnumeric.py:2495: FutureWarning: Method .ptp is deprecated and will be removed in a future version. Use numpy.ptp instead.
return ptp(axis=axis, out=out, **kwargs)
const Mkt-RF
Date
1926-07 NaN NaN
1926-08 NaN NaN
1926-09 NaN NaN
1926-10 NaN NaN
1926-11 NaN NaN
const Mkt-RF
Date
2019-06 0.422862 1.101891
2019-07 0.407324 1.110808
2019-08 0.408477 1.116786
2019-09 0.381874 1.117752
2019-10 0.417572 1.122855
We next plot the market loading along with a 95% point-wise confidence interval. The alpha=False
omits the constant column, if present.
[4]:
fig = rres.plot_recursive_coefficient(variables=['Mkt-RF'], figsize=(14,6))
Next, the model is expanded to include all three factors, the excess market, the size factor and the value factor.
[5]:
exog_vars = ['Mkt-RF', 'SMB', 'HML']
exog = sm.add_constant(factors[exog_vars])
rols = RollingOLS(endog, exog, window=60)
rres = rols.fit()
fig = rres.plot_recursive_coefficient(variables=exog_vars, figsize=(14,18))
Formulas¶
RollingOLS
and RollingWLS
both support model specification using the formula interface. The example below is equivalent to the 3-factor model estimated previously. Note that one variable is renamed to have a valid Python variable name.
[6]:
joined = pd.concat([factors, industries], axis=1)
joined['Mkt_RF'] = joined['Mkt-RF']
mod = RollingOLS.from_formula('HiTec ~ Mkt_RF + SMB + HML', data=joined, window=60)
rres = mod.fit()
print(rres.params.tail())
Intercept Mkt_RF SMB HML
Date
2019-06 0.266000 1.125214 -0.185912 -0.412686
2019-07 0.263789 1.130699 -0.179602 -0.412318
2019-08 0.241410 1.145853 -0.173493 -0.380134
2019-09 0.283362 1.148361 -0.186190 -0.350970
2019-10 0.312366 1.148031 -0.163546 -0.359405
RollingWLS
: Rolling Weighted Least Squares¶
The rolling
module also provides RollingWLS
which takes an optional weights
input to perform rolling weighted least squares. It produces results that match WLS
when applied to rolling windows of data.
Fit Options¶
Fit accepts other optional keywords to set the covariance estimator. Only two estimators are supported, 'nonrobust'
(the classic OLS estimator) and 'HC0'
which is White’s heteroskedasticity robust estimator.
You can set params_only=True
to only estimate the model parameters. This is substantially faster than computing the full set of values required to perform inference.
Finally, the parameter reset
can be set to a positive integer to control estimation error in very long samples. RollingOLS
avoids the full matrix product when rolling by only adding the most recent observation and removing the dropped observation as it rolls through the sample. Setting reset
uses the full inner product every reset
periods. In most applications this parameter can be omitted.
[7]:
%timeit rols.fit()
%timeit rols.fit(params_only=True)
311 ms ± 4.37 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
63.5 ms ± 660 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)