Examples

General Setup

Models include factors and returns. Factors can be traded or non-traded. Common examples of traded assets include the excess return on the market and the size and value factors. Examples of non-traded assets include macroeconomic shocks and measures of uncertainty.

Models are tested using a set of test portfolios. Test portfolios are often excess returns although they do not need to be.

Import data

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 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

Transform the portfolios to be excesses

Subtract the risk-free rate from the test portfolios since these are not zero-investment.

[2]:
data.iloc[:, 6:] = data.iloc[:, 6:].values - data[["RF"]].values

1-Step Estimation using Seemingly Unrelated Regression (SUR)

When the factors are traded assets, they must price themselves, and so the observed factor returns can be used to consistently estimate the expected factor returns. This also allows a set of seemingly unrelated regressions where each test portfolio is regressed on the factors and a constant to estimate factor loadings and \(\alpha\)s.

Note that when using this type of model, it is essential that the test portfolios are excess returns. This is not a requirement of the other factor model estimators.

This specification is a CAP-M since only the market is included. The J-statistic tests whether all model \(\alpha\)s are 0. The CAP-M is unsurprisingly unable to price the test portfolios.

[3]:
from linearmodels.asset_pricing import TradedFactorModel

portfolios = data[
    ["S1V1", "S1V3", "S1V5", "S3V1", "S3V3", "S3V5", "S5V1", "S5V3", "S5V5"]
]
factors = data[["MktRF"]]
mod = TradedFactorModel(portfolios, factors)
res = mod.fit()
print(res)
                      TradedFactorModel Estimation Summary
================================================================================
No. Test Portfolios:                  9   R-squared:                      0.6910
No. Factors:                          1   J-statistic:                    70.034
No. Observations:                   819   P-value                         0.0000
Date:                  Thu, Apr 08 2021   Distribution:                  chi2(9)
Time:                          09:30:13
Cov. Estimator:                  robust

                            Risk Premia Estimates
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
MktRF          0.0065     0.0015     4.3553     0.0000      0.0035      0.0094
==============================================================================

Covariance estimator:
HeteroskedasticCovariance
See full_summary for complete results

The factor set is expanded to include both the size and the value factors.

While the extra factors lower the J statistic and increases the \(R^2\), the model is still easily rejected.

[4]:
factors = data[["MktRF", "SMB", "HML"]]
mod = TradedFactorModel(portfolios, factors)
res = mod.fit()
print(res)
                      TradedFactorModel Estimation Summary
================================================================================
No. Test Portfolios:                  9   R-squared:                      0.8971
No. Factors:                          3   J-statistic:                    53.271
No. Observations:                   819   P-value                         0.0000
Date:                  Thu, Apr 08 2021   Distribution:                  chi2(9)
Time:                          09:30:13
Cov. Estimator:                  robust

                            Risk Premia Estimates
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
MktRF          0.0065     0.0015     4.3553     0.0000      0.0035      0.0094
SMB            0.0016     0.0010     1.6021     0.1091     -0.0004      0.0035
HML            0.0035     0.0009     3.6993     0.0002      0.0016      0.0053
==============================================================================

Covariance estimator:
HeteroskedasticCovariance
See full_summary for complete results

Changing the test portfolios to include only the industry portfolios does not allow factors to price the assets.

[5]:
indu = [
    "NoDur",
    "Durbl",
    "Manuf",
    "Enrgy",
    "Chems",
    "BusEq",
    "Telcm",
    "Utils",
    "Shops",
    "Hlth",
    "Money",
    "Other",
]
portfolios = data[indu]
mod = TradedFactorModel(portfolios, factors)
res = mod.fit()
print(res)
                      TradedFactorModel Estimation Summary
================================================================================
No. Test Portfolios:                 12   R-squared:                      0.7118
No. Factors:                          3   J-statistic:                    61.617
No. Observations:                   819   P-value                         0.0000
Date:                  Thu, Apr 08 2021   Distribution:                 chi2(12)
Time:                          09:30:13
Cov. Estimator:                  robust

                            Risk Premia Estimates
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
MktRF          0.0065     0.0015     4.3553     0.0000      0.0035      0.0094
SMB            0.0016     0.0010     1.6021     0.1091     -0.0004      0.0035
HML            0.0035     0.0009     3.6993     0.0002      0.0016      0.0053
==============================================================================

Covariance estimator:
HeteroskedasticCovariance
See full_summary for complete results

The estimated factor loadings (\(\beta\)s) can be displayed using the betas property. There is reasonable dispersion in the factor loadings for all of the factors, except possibly the market which are all close to unity.

[6]:
print(res.betas)
          MktRF       SMB       HML
NoDur  0.803334 -0.029383  0.080556
Durbl  1.176659  0.104124  0.466510
Manuf  1.129601  0.094600  0.197135
Enrgy  0.913425 -0.234012  0.264609
Chems  0.970844 -0.179466  0.092021
BusEq  1.152119  0.182299 -0.543462
Telcm  0.782811 -0.158395  0.044042
Utils  0.605203 -0.175549  0.260051
Shops  0.942997  0.135802 -0.010064
Hlth   0.864135 -0.213336 -0.315180
Money  1.112368 -0.053364  0.378365
Other  1.109851  0.304442  0.237830
[7]:
import seaborn as sns

%matplotlib inline
sns.heatmap(res.betas)
[7]:
<AxesSubplot:>
../../_images/asset-pricing_examples_examples_14_1.png

Similarly the \(\alpha\)s can be displayed. These are monthly returns, and so scaling by 12 shows annualized returns. Healthcare has the largest pricing error.

[8]:
12 * res.alphas
[8]:
NoDur    0.023360
Durbl   -0.030919
Manuf   -0.010643
Enrgy    0.012009
Chems    0.002783
BusEq    0.024215
Telcm    0.009726
Utils    0.017078
Shops    0.009952
Hlth     0.050760
Money   -0.015197
Other   -0.033345
Name: alpha, dtype: float64

2-Step Estimation

When one of more factors are not returns on traded assets, it is necessary to use a two-step procedure (or GMM). In the 2-step estimator, the first estimates factor loadings and the second uses the factor loadings to estimate the risk premia.

Here all four factors are used to attempt to price the size-momentum portfolios.

[9]:
from linearmodels.asset_pricing import LinearFactorModel

factors = data[["MktRF", "SMB", "HML", "Mom"]]
portfolios = data[
    ["S1M1", "S1M3", "S1M5", "S3M1", "S3M3", "S3M5", "S5M1", "S5M3", "S5M5"]
]
mod = LinearFactorModel(portfolios, factors)
res = mod.fit()
print(res)
                      LinearFactorModel Estimation Summary
================================================================================
No. Test Portfolios:                  9   R-squared:                      0.9051
No. Factors:                          4   J-statistic:                    36.929
No. Observations:                   819   P-value                         0.0000
Date:                  Thu, Apr 08 2021   Distribution:                  chi2(5)
Time:                          09:30:15
Cov. Estimator:                  robust

                            Risk Premia Estimates
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
MktRF          0.0070     0.0015     4.5385     0.0000      0.0040      0.0100
SMB            0.0005     0.0014     0.3381     0.7353     -0.0023      0.0033
HML            0.0084     0.0025     3.3085     0.0009      0.0034      0.0133
Mom            0.0084     0.0014     5.8072     0.0000      0.0056      0.0112
==============================================================================

Covariance estimator:
HeteroskedasticCovariance
See full_summary for complete results
[10]:
print(res.betas)
         MktRF       SMB       HML       Mom
S1M1  1.092658  1.224223  0.244844 -0.691191
S1M3  0.874285  0.881880  0.459326 -0.082546
S1M5  1.047256  1.147949  0.239957  0.297941
S3M1  1.156849  0.623684  0.059730 -0.760059
S3M3  0.948556  0.467783  0.333619 -0.135465
S3M5  1.128861  0.713403  0.051098  0.413705
S5M1  1.138597 -0.112936 -0.062397 -0.755032
S5M3  0.946214 -0.200052  0.095270 -0.100067
S5M5  1.078098 -0.046531 -0.069780  0.467172

The identification requirements for this model are the the \(\beta\)s have unique variation – this requires some cross-sectional differences in exposures and that the loadings are not excessively correlated. Since these test portfolios do not attempt to sort on value, it is relatively non-dispersed and also correlated with both the market and size. This might make the inference unreliable.

[11]:
print(res.betas.corr())
          MktRF       SMB       HML       Mom
MktRF  1.000000 -0.001914 -0.733916 -0.253675
SMB   -0.001914  1.000000  0.638056 -0.015628
HML   -0.733916  0.638056  1.000000  0.002744
Mom   -0.253675 -0.015628  0.002744  1.000000

The size factor was insignificant and so is dropped. This does not have much of an effect on the estimates.

[12]:
from linearmodels.asset_pricing import LinearFactorModel

factors = data[["MktRF", "HML", "Mom"]]
portfolios = data[
    ["S1M1", "S1M3", "S1M5", "S3M1", "S3M3", "S3M5", "S5M1", "S5M3", "S5M5"]
]
mod = LinearFactorModel(portfolios, factors)
print(mod.fit())
                      LinearFactorModel Estimation Summary
================================================================================
No. Test Portfolios:                  9   R-squared:                      0.7915
No. Factors:                          3   J-statistic:                    36.406
No. Observations:                   819   P-value                         0.0000
Date:                  Thu, Apr 08 2021   Distribution:                  chi2(6)
Time:                          09:30:15
Cov. Estimator:                  robust

                            Risk Premia Estimates
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
MktRF          0.0073     0.0017     4.3856     0.0000      0.0040      0.0105
HML            0.0091     0.0027     3.4361     0.0006      0.0039      0.0143
Mom            0.0084     0.0015     5.7507     0.0000      0.0055      0.0113
==============================================================================

Covariance estimator:
HeteroskedasticCovariance
See full_summary for complete results

The risk-free rate can be optionally estimated. This is useful in case excess returns are not available of if the risk-free rate used to construct the excess returns might be misspecified.

Here the estimated risk-free rate is small and insignificant and has little impact on the model J-statistic.

[13]:
from linearmodels.asset_pricing import LinearFactorModel

factors = data[["MktRF", "HML", "Mom"]]
portfolios = data[
    ["S1M1", "S1M3", "S1M5", "S3M1", "S3M3", "S3M5", "S5M1", "S5M3", "S5M5"]
]
mod = LinearFactorModel(portfolios, factors, risk_free=True)
print(mod.fit())
                      LinearFactorModel Estimation Summary
================================================================================
No. Test Portfolios:                  9   R-squared:                      0.7915
No. Factors:                          3   J-statistic:                    36.006
No. Observations:                   819   P-value                         0.0000
Date:                  Thu, Apr 08 2021   Distribution:                  chi2(5)
Time:                          09:30:15
Cov. Estimator:                  robust

                            Risk Premia Estimates
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
risk_free     -0.0045     0.0079    -0.5607     0.5750     -0.0200      0.0111
MktRF          0.0111     0.0076     1.4636     0.1433     -0.0038      0.0259
HML            0.0110     0.0053     2.0946     0.0362      0.0007      0.0214
Mom            0.0086     0.0014     5.9315     0.0000      0.0058      0.0114
==============================================================================

Covariance estimator:
HeteroskedasticCovariance
See full_summary for complete results

The default covariance estimator allows for heteroskedasticity but assumes there is no autocorrelation in the model errors or factor returns. Kernel-based HAC estimators (e.g. Newey-West) can be used by setting cov_type="kernel".

This reduces the J-statistic indicating there might be some serial correlation although the model is still firmly rejected.

[14]:
mod = LinearFactorModel(portfolios, factors)
print(mod.fit(cov_type="kernel"))
                      LinearFactorModel Estimation Summary
================================================================================
No. Test Portfolios:                  9   R-squared:                      0.7915
No. Factors:                          3   J-statistic:                    25.841
No. Observations:                   819   P-value                         0.0002
Date:                  Thu, Apr 08 2021   Distribution:                  chi2(6)
Time:                          09:30:15
Cov. Estimator:                  kernel

                            Risk Premia Estimates
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
MktRF          0.0073     0.0017     4.3094     0.0000      0.0040      0.0106
HML            0.0091     0.0031     2.9499     0.0032      0.0031      0.0152
Mom            0.0084     0.0014     5.8371     0.0000      0.0056      0.0112
==============================================================================

Covariance estimator:
KernelCovariance, Kernel: bartlett, Bandwidth: 12
See full_summary for complete results

GMM Estimation

The final estimator is the GMM estimator which is similar to estimating the 2-step model in a single step. In practice the GMM estimator is estimated at least twice, once to get an consistent estimate of the covariance of the moment conditions and the second time to efficiently estimate parameters.

The GMM estimator does not have a closed form and so a non-linear optimizer is required. The default output prints the progress every 10 iterations. Here the model is fit twice, which is the standard method to implement efficient GMM.

[15]:
from linearmodels.asset_pricing import LinearFactorModelGMM

mod = LinearFactorModelGMM(portfolios, factors)
res = mod.fit()
print(res)
Iteration: 0, Objective: 47.75791582821261
Iteration: 10, Objective: 28.102587276803362
Iteration: 20, Objective: 26.36138467237513
Iteration: 30, Objective: 26.019666406527822
Iteration: 40, Objective: 22.325343252981494
Warning: Desired error not necessarily achieved due to precision loss.
         Current function value: 22.325343
         Iterations: 41
         Function evaluations: 2493
         Gradient evaluations: 73
Iteration: 0, Objective: 22.519350308870816
Iteration: 10, Objective: 22.303730007936863
Iteration: 20, Objective: 22.226426483844115
Iteration: 30, Objective: 22.193086067171414
Iteration: 40, Objective: 22.066871359322747
Warning: Desired error not necessarily achieved due to precision loss.
         Current function value: 22.066871
         Iterations: 41
         Function evaluations: 2244
         Gradient evaluations: 66
                    LinearFactorModelGMM Estimation Summary
================================================================================
No. Test Portfolios:                  9   R-squared:                      0.7904
No. Factors:                          3   J-statistic:                    22.067
No. Observations:                   819   P-value                         0.0012
Date:                  Thu, Apr 08 2021   Distribution:                  chi2(6)
Time:                          09:30:18
Cov. Estimator:                  robust

                            Risk Premia Estimates
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
MktRF          0.0067     0.0015     4.4335     0.0000      0.0037      0.0097
HML            0.0135     0.0023     5.8706     0.0000      0.0090      0.0180
Mom            0.0094     0.0014     6.5139     0.0000      0.0066      0.0123
==============================================================================

Covariance estimator:
HeteroskedasticCovariance
See full_summary for complete results

Kernel HAC estimators can be used with this estimator as well. Using a kernel HAC covariance also implies a Kernel HAC weighting matrix estimator.

Here the GMM estimator along with the HAC estimator indicates that these factors might be able to price this set of 9 test portfolios. disp=0 is used to suppress iterative output.

[16]:
res = mod.fit(cov_type="kernel", kernel="bartlett", disp=0)
print(res)
                    LinearFactorModelGMM Estimation Summary
================================================================================
No. Test Portfolios:                  9   R-squared:                      0.7901
No. Factors:                          3   J-statistic:                    13.833
No. Observations:                   819   P-value                         0.0316
Date:                  Thu, Apr 08 2021   Distribution:                  chi2(6)
Time:                          09:30:20
Cov. Estimator:                  kernel

                            Risk Premia Estimates
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
MktRF          0.0074     0.0014     5.1450     0.0000      0.0046      0.0102
HML            0.0137     0.0030     4.5918     0.0000      0.0078      0.0195
Mom            0.0078     0.0013     6.0849     0.0000      0.0053      0.0103
==============================================================================

Covariance estimator:
KernelCovariance, Kernel: bartlett, Bandwidth: 20
See full_summary for complete results

Iterating until convergence

The standard approach is efficient and uses 2-steps. The first consistently estimates parameters using a sub-optimal weighting matrix, and the second uses the optimal weighting matrix conditional using the first stage estimates.

This method can be repeated until convergence, or for a fixed number of steps using the steps keyword argument.

[17]:
res = mod.fit(steps=10, disp=25)
print(res)
Iteration: 0, Objective: 47.75791582821261
Iteration: 25, Objective: 26.28031986241902
Warning: Desired error not necessarily achieved due to precision loss.
         Current function value: 22.325343
         Iterations: 41
         Function evaluations: 2493
         Gradient evaluations: 73
Iteration: 0, Objective: 22.519350308870816
Iteration: 25, Objective: 22.222217187824842
Warning: Desired error not necessarily achieved due to precision loss.
         Current function value: 22.066871
         Iterations: 41
         Function evaluations: 2244
         Gradient evaluations: 66
Iteration: 0, Objective: 22.094157061998075
Iteration: 25, Objective: 22.091266591316497
Warning: Desired error not necessarily achieved due to precision loss.
         Current function value: 22.089908
         Iterations: 37
         Function evaluations: 2868
         Gradient evaluations: 84
Iteration: 0, Objective: 22.088798287143284
Iteration: 25, Objective: 22.088771428335885
Warning: Desired error not necessarily achieved due to precision loss.
         Current function value: 22.088753
         Iterations: 38
         Function evaluations: 2800
         Gradient evaluations: 82
Iteration: 0, Objective: 22.088622395402776
Warning: Desired error not necessarily achieved due to precision loss.
         Current function value: 22.088622
         Iterations: 4
         Function evaluations: 1473
         Gradient evaluations: 43
Iteration: 0, Objective: 22.088622318666037
Warning: Desired error not necessarily achieved due to precision loss.
         Current function value: 22.088622
         Iterations: 3
         Function evaluations: 1100
         Gradient evaluations: 32
                    LinearFactorModelGMM Estimation Summary
================================================================================
No. Test Portfolios:                  9   R-squared:                      0.7904
No. Factors:                          3   J-statistic:                    22.089
No. Observations:                   819   P-value                         0.0012
Date:                  Thu, Apr 08 2021   Distribution:                  chi2(6)
Time:                          09:30:26
Cov. Estimator:                  robust

                            Risk Premia Estimates
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
MktRF          0.0067     0.0015     4.4231     0.0000      0.0037      0.0096
HML            0.0135     0.0023     5.8726     0.0000      0.0090      0.0180
Mom            0.0094     0.0014     6.5017     0.0000      0.0066      0.0123
==============================================================================

Covariance estimator:
HeteroskedasticCovariance
See full_summary for complete results

Continuously Updating Estimator

The Continuously Updating Estimator (CUE) is optionally available using the flag use_cue. CUE jointly minimizes the J-statistic as a function of the moment conditions and the weighting matrix, rather than iterating between minimizing the J-statistic for a fixed weighting matrix and updating the weighting matrix.

Here the results are essentially the same as in the iterative approach.

[18]:
res = mod.fit(use_cue=True)
print(res)
Iteration: 0, Objective: 47.75791582821261
Iteration: 10, Objective: 28.102587276803362
Iteration: 20, Objective: 26.36138467237513
Iteration: 30, Objective: 26.019666406527822
Iteration: 40, Objective: 22.325343252981494
Warning: Desired error not necessarily achieved due to precision loss.
         Current function value: 22.325343
         Iterations: 41
         Function evaluations: 2493
         Gradient evaluations: 73
Iteration: 0, Objective: 22.524479922365927
Iteration: 10, Objective: 22.386060509865903
Iteration: 20, Objective: 22.339812399705494
Iteration: 30, Objective: 22.308318957561163
Warning: Desired error not necessarily achieved due to precision loss.
         Current function value: 22.077574
         Iterations: 38
         Function evaluations: 2732
         Gradient evaluations: 80
                    LinearFactorModelGMM Estimation Summary
================================================================================
No. Test Portfolios:                  9   R-squared:                      0.7903
No. Factors:                          3   J-statistic:                    22.719
No. Observations:                   819   P-value                         0.0009
Date:                  Thu, Apr 08 2021   Distribution:                  chi2(6)
Time:                          09:30:35
Cov. Estimator:                  robust

                            Risk Premia Estimates
==============================================================================
            Parameter  Std. Err.     T-stat    P-value    Lower CI    Upper CI
------------------------------------------------------------------------------
MktRF          0.0067     0.0015     4.4157     0.0000      0.0037      0.0096
HML            0.0136     0.0023     5.8881     0.0000      0.0090      0.0181
Mom            0.0094     0.0014     6.4984     0.0000      0.0066      0.0123
==============================================================================

Covariance estimator:
HeteroskedasticCovariance
See full_summary for complete results