Using Formulas¶
The asset pricing model estimators all all formulas. Since the models have multiple dependent variables (test portfolios) as well as multiple independent variables (factors), the standard formulaic syntax needs to be modified. There are two methods to use formulas. The first specified both the test portfolio and the factors. The second specifies only the factors and the test portfolios are passed using an optional keyword argument. The second syntax exists since in many models the number of test portfolios might be large and interest is usually in modifying the factors.
Available Syntax¶
Test Portfolios and Factors¶
The first syntax can be expressed as
"port1 + port2 + port3 + port4 + ... + portN ~ factor1 + ... + factorK"
so that both the test portfolios and the factors are separated using +. The two sets are separated using the usual separator between left-hand side and right-hand side variables, ~.
Factors Only¶
The second syntax specifies only factors and uses the keyword argument portfolios to pass the matrix of portfolio returns.
formula = "factor1 + ... + factorK"
LinearFactorModel.from_formula(formula, portfolios=portfolios)
Import data and transform to be excess returns¶
The data used comes from Ken French”s website and includes 4 factor returns, the excess market, the size factor, the value factor and the momentum factor. The available test portfolios include the 12 industry portfolios, a subset of the size-value two way sort, and a subset of the size-momentum two way sort.
[1]:
from linearmodels.datasets import french
data = french.load()
print(french.DESCR)
data.iloc[:, 6:] = data.iloc[:, 6:].values - data[["RF"]].values
Data from Ken French's data library
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
dates Year and Month of Return
MktRF Market Factor
SMB Size Factor
HML Value Factor
Mom Momentum Factor
RF Risk-free rate
NoDur Industry: Non-durables
Durbl Industry: Durables
Manuf Industry: Manufacturing
Enrgy Industry: Energy
Chems Industry: Chemicals
BusEq Industry: Business Equipment
Telcm Industry: Telecoms
Utils Industry: Utilities
Shops Industry: Retail
Hlth Industry: Health care
Money Industry: Finance
Other Industry: Other
S1V1 Small firms, low value
S1V3 Small firms, medium value
S1V5 Small firms, high value
S3V1 Size 3, value 1
S3V3 Size 3, value 3
S3V5 Size 3, value 5
S5V1 Large firms, Low value
S5V3 Large firms, medium value
S5V5 Large Firms, High value
S1M1 Small firms, losers
S1M3 Small firms, neutral
S1M5 Small firms, winners
S3M1 Size 3, momentum 1
S3M3 Size 3, momentum 3
S3M5 Size 3, momentum 5
S5M1 Large firms, losers
S5M3 Large firms, neutral
S5M5 Large firms, winners
First Syntax¶
This example shows the first syntax. The test portfolios are a combination of the industry, size-value, and size-momentum sorted portfolios. The factors are the market, value and momentum factors. This model is not adequate to price the assets.
[2]:
from linearmodels.asset_pricing import LinearFactorModel, TradedFactorModel
formula = "NoDur + Chems + S1V1 + S5V5 + S1M1 + S5M5 ~ MktRF + HML + Mom"
mod = LinearFactorModel.from_formula(formula, data)
res = mod.fit(cov_type="kernel", kernel="parzen", bandwidth=20)
print(res)
LinearFactorModel Estimation Summary
================================================================================
No. Test Portfolios: 6 R-squared: 0.7229
No. Factors: 3 J-statistic: 9.9450
No. Observations: 819 P-value 0.0190
Date: Tue, Apr 16 2024 Distribution: chi2(3)
Time: 17:36:16
Cov. Estimator: kernel
Risk Premia Estimates
==============================================================================
Parameter Std. Err. T-stat P-value Lower CI Upper CI
------------------------------------------------------------------------------
MktRF 0.0056 0.0016 3.4051 0.0007 0.0024 0.0088
HML 0.0044 0.0016 2.6929 0.0071 0.0012 0.0075
Mom 0.0081 0.0021 3.7457 0.0002 0.0038 0.0123
==============================================================================
Covariance estimator:
KernelCovariance, Kernel: parzen, Bandwidth: 20.0
See full_summary for complete results
Second Syntax¶
The second syntax contains only the factors and omits the test portfolios. The test portfolios are passed as an array or DataFrame using a keyword input. This syntax simplifies experimenting with alternative factors when there are many test portfolios. This model also appears to be inadequate, even allowing the risk-free rate to be a free parameter.
[3]:
ports = ["S{0}V{1}".format(i, j) for i in (1, 3, 5) for j in (1, 3, 5)]
ports += ["S{0}M{1}".format(i, j) for i in (1, 3, 5) for j in (1, 3, 5)]
portfolios = data[ports]
formula = "MktRF + HML + Mom"
mod = LinearFactorModel.from_formula(
formula, data, portfolios=portfolios, risk_free=True
)
res = mod.fit()
print(res)
LinearFactorModel Estimation Summary
================================================================================
No. Test Portfolios: 18 R-squared: 0.7723
No. Factors: 3 J-statistic: 86.435
No. Observations: 819 P-value 0.0000
Date: Tue, Apr 16 2024 Distribution: chi2(14)
Time: 17:36:16
Cov. Estimator: robust
Risk Premia Estimates
==============================================================================
Parameter Std. Err. T-stat P-value Lower CI Upper CI
------------------------------------------------------------------------------
risk_free 0.0024 0.0040 0.5940 0.5525 -0.0055 0.0103
MktRF 0.0050 0.0044 1.1337 0.2569 -0.0036 0.0136
HML 0.0042 0.0010 4.0918 0.0000 0.0022 0.0063
Mom 0.0081 0.0014 5.7058 0.0000 0.0053 0.0109
==============================================================================
Covariance estimator:
HeteroskedasticCovariance
See full_summary for complete results
Comparing results¶
To verify the results, the model is estimated using the standard interface. The J-statistic and \(R^2\) are identical.
[4]:
portfolios = data[ports]
factors = data[["MktRF", "HML", "Mom"]]
mod = LinearFactorModel(portfolios, factors, risk_free=True)
print(mod.fit())
LinearFactorModel Estimation Summary
================================================================================
No. Test Portfolios: 18 R-squared: 0.7723
No. Factors: 3 J-statistic: 86.435
No. Observations: 819 P-value 0.0000
Date: Tue, Apr 16 2024 Distribution: chi2(14)
Time: 17:36:16
Cov. Estimator: robust
Risk Premia Estimates
==============================================================================
Parameter Std. Err. T-stat P-value Lower CI Upper CI
------------------------------------------------------------------------------
risk_free 0.0024 0.0040 0.5940 0.5525 -0.0055 0.0103
MktRF 0.0050 0.0044 1.1337 0.2569 -0.0036 0.0136
HML 0.0042 0.0010 4.0918 0.0000 0.0022 0.0063
Mom 0.0081 0.0014 5.7058 0.0000 0.0053 0.0109
==============================================================================
Covariance estimator:
HeteroskedasticCovariance
See full_summary for complete results