BLP-Python

Introduction

BLP-Python provides a Python implementation of random coefficient logit model of Berry, Levinsohn and Pakes (1995). The specific implementation follows the model described in Nevo (2000b).

This code uses tight tolerances for the contraction mapping (Dube et al. 2012). With BFGS method, it quickly converges to the optimum (See Nevo (2000b) Example below).

I would like to thank Prof. Nevo and others for making their MATLAB available, which this package is originally based on. Also, I would like to thank Wingware, who generously provided a free license of WingIDE for this non-commercial open source project.

Notes on the code

• Use global states only for read-only variables
• Avoid inverting matrices whenever possible for numerical stability
• Use a tight tolerance for the contraction mapping
• Use greek unicode symbols whenever possible for readability
• μ and individual choice probability calculations are implemented in Cython, and it is parallelized across the simulation draws via openMP
• Use n-dimensional arrays (via xarray) to represent data more naturally

Installation

Dependencies

• Python 3.5 (for @ operator and unicode variable names). I recommend Anaconda Python Distribution, which comes with many of the scientific libraries, as well as conda, a convenient script to install many packages.
• numpy and scipy for array operations and linear algebra
• cython for parallelized market share integration
• xarray for multidimensional labeled arrays (does not come with Anaconda, install with conda install xarray)
• pandas for result printing

Download

• With git:

  git clone https://github.com/joonro/BLP-Python.git
      

• Or you can download the master branch as a zip archive

Compiling the Cython Module

• I include the compiled Cython module (_BLP.cp3X-win_amd64.pyd) for Python 3.5 and 3.6 64bit, so you should be able to run the code without compiling the module in Windows. You have to compile it if you want to change the Cython module (_BLP.pyx) or if you are on GNU/Linux or Mac OS. GNU/Linux distributions come with gcc so it should be straightforward to compile the module.
• cd into the BLP-Python directory, and compile the cython module with the following command:

  python setup.py build_ext --inplace
      

Windows

• For Windows users, to compile the cython module with the openMP (parallelization) support with 64-bit Python, you have to install Microsoft Visual C++ compiler following instructions at https://wiki.python.org/moin/WindowsCompilers. For Python 3.5 and 3.6, you either install Microsoft Visual C++ 14.0 standalone, or you can install Visual Studio 2015 which contains Visual C++ 14.0 compiler.

Nevo (2000b) Example

examples/Nevo_2000b.py replicates the results from Nevo (2000b). In the examples folder, you can run the script as:

python ./Nevo_2000b.py

It evaluates the objective function at the starting values and creates the following results table:

               Mean        SD      Income  Income^2       Age     Child
Constant  -1.833294  0.377200    3.088800  0.000000  1.185900   0.00000
           0.257829  0.129433    1.212647  0.000000  1.012354   0.00000
Price    -32.446922  1.848000   16.598000 -0.659000  0.000000  11.62450
           7.751913  1.078371  172.776110  8.979257  0.000000   5.20593
Sugar      0.142915 -0.003500   -0.192500  0.000000  0.029600   0.00000
           0.012877  0.012297    0.045528  0.000000  0.036563   0.00000
Mushy      0.801608  0.081000    1.468400  0.000000 -1.514300   0.00000
           0.203454  0.206025    0.697863  0.000000  1.098321   0.00000
GMM objective: 14.900789417017275
Min-Dist R-squared: 0.2718388379589566
Min-Dist weighted R-squared: 0.0946528053333926

This code uses a tight tolerance for the contraction mapping, and it minimizes the GMM objective function to the correct minimum of 4.56. (With BFGS, it only needs 45 iterations).

After running the code, you can try the full estimation with:

BLP.estimate(θ20=θ20)

For example, in an IPython console:

%run Nevo_2000b.py
BLP.estimate(θ20=θ20)

You should get the following results:

Optimization terminated successfully.
         Current function value: 4.561515
         Iterations: 45
         Function evaluations: 50
         Gradient evaluations: 50

               Mean        SD      Income   Income^2       Age      Child
Constant  -2.009919  0.558094    2.291972   0.000000  1.284432   0.000000
           0.326997  0.162533    1.208569   0.000000  0.631215   0.000000
Price    -62.729902  3.312489  588.325237 -30.192021  0.000000  11.054627
          14.803215  1.340183  270.441021  14.101230  0.000000   4.122563
Sugar      0.116257 -0.005784   -0.384954   0.000000  0.052234   0.000000
           0.016036  0.013505    0.121458   0.000000  0.025985   0.000000
Mushy      0.499373  0.093414    0.748372   0.000000 -1.353393   0.000000
           0.198582  0.185433    0.802108   0.000000  0.667108   0.000000
GMM objective: 4.5615146550344186
Min-Dist R-squared: 0.4591043336106454
Min-Dist weighted R-squared: 0.10116438381046189

You can check the gradient at the optimum:

>>> BLP._gradient_GMM(BLP.results['θ2']['x'])
contraction mapping finished in 0 iterations

array([  1.23888940e-07,   1.15056001e-08,   1.58824491e-08,
        -4.45649242e-08,  -9.61452074e-08,  -1.75233503e-08,
        -9.94539619e-07,   9.60900497e-08,  -3.30553299e-07,
         1.24174991e-07,   4.17569410e-07,   1.33642515e-07,
         1.94273594e-09])

I verified that the optimum is achieved with Nelder-Mead (simplex), BFGS, TNC, and SLSQP =scipy.optimize= methods. BFGS and SLSQP were the fastest, and BFGS is the default.

Unit Testing

I use pytest for unit testing. You can run them with:

python -m pytest

References

Berry, S., Levinsohn, J., & Pakes, A. (1995). Automobile Prices In Market Equilibrium. Econometrica, 63(4), 841.

Dubé, J., Fox, J. T., & Su, C. (2012). Improving the Numerical Performance of BLP Static and Dynamic Discrete Choice Random Coefficients Demand Estimation. Econometrica, 1–34.

Nevo, A. (2000). A Practitioner’s Guide to Estimation of Random-Coefficients Logit Models of Demand. Journal of Economics & Management Strategy, 9(4), 513–548.

License

BLP-Python is released under the GPLv3.

Changelog

0.5.0 ([2017-09-23 Sat])

• Change data structure to xarray.
• Major improvements on various aspects of the code.

0.4.2 ([2017-06-30 Fri])

• Fix setup.py for the Cython module for non-windows operating systems (thanks to Cheng Nie)

0.4.0 ([2016-12-18 Sun])

• Use global state only for read-only variables; now gradient-based optimization (such as BFGS) works and it converges quickly
• Use pandas.DataFrame to show results cleanly
• Implement estimation of parameter means
• Implement standard error calculation
• Use greek letters whenever possible
• Add Nevo (2000b) example
• Add a unit test
• Improve README

0.3.0 ([2014-11-28 Fri])

• Implement GMM objective function and estimation of \( θ₂ \)

0.1.0 ([2013-03-28 Thu])

• Initial release