Dearing and Blevins (2025) Replication Files

Dearing and Blevins (2025). Efficient and Convergent Sequential Pseudo-Likelihood Estimation of Dynamic Discrete Games. Review of Economic Studies 92, 981–1021.

https://doi.org/10.1093/restud/rdae050

Overview

This repository contains the replication code for all Monte Carlo experiments and the Application of Dearing and Blevins (2025). The replication files for each section are contained in separate subdirectories. Here we give a brief overview of the contents of the package.

• The mc-am2007 subdirectory contains the replication code for the Monte Carlo experiments in Section 4 of the paper. The simulations are based on the dynamic game of entry and exit in Example 1 of the paper, parameterized to match a model with five heterogeneous firms from Aguirregabiria and Mira (2007).

• The application subdirectory contains the replication code for the application to U.S. wholesale club store competition in Section 5 of the paper. The simulations are based on the dynamic game of entry and exit in Example 1 of the paper.

• The mc-psd2008 contains the replication code for the Monte Carlo experiments in Appendix B.1 of the paper. The simulations are based on the model of Pesendorfer and Schmidt-Dengler (2008), a dynamic game with multiple equilibria.

• The mc-psd2010 subdirectory contains the replication code for the Monte Carlo experiments in Appendix B.2 of the paper. The simulations are based on the model of Pesendorfer and Schmidt-Dengler (2010), a static game with an unstable equilibrium leading to non-convergence of NPL.

Data Availability

The Monte Carlo experiments in this paper do not involve analysis of external data (i.e., the only data used are generated via simulation in the code).

The application in the paper uses data from several sources:

• IPUMS NHGIS Table CL8: Total Population (1990, 2000, 2010, 2020)
• U.S. Department of Housing and Urban Development Zip Code to County Crosswalk Files
• Data Axle Historical Business Database

We have included in this replication package the NHGIS population data extract from Table CL8: Total Population (application/nhgis0012_ts_geog2010_zcta.csv) as well as the HUD crosswalk file (application/ZIP_COUNTY_122021.xlsx).

The source files from the Data Axle Historical Business Database cannot be made publicly available due to the terms of use. While we cannot redistribute the original dataset, we have provided our anonymized analysis sample which users can use to replicate the tables in the paper.

The full raw dataset can be obtained freely through certain university libraries or directly from Data Axle directly for a fee. Researchers interested in access to the data may contact Data Axle at https://www.data-axle.com/contact-us/.

The data preparation programs are also provided and can be used to generate the analysis sample once certain data files have been obtained from Data Axle. These files are named following the pattern BUSINESS_HISTORICAL_YYYY.zip for the years 1997 through 2020 and YYYY_Business_Academic_QCQ.txt.gz for the year 2021.

Details on each Data Source

Data Name	Data Files	Location	Provided	Citation
Zip Code Population	nhgis0012_ts_geog2010_zcta.csv	application/nhgis0012_csv/	True	Manson, et al. (2022)
Zip Code to County Crosswalk	ZIP_COUNTY_122021.xlsx	application/	True	HUD (2021)
Historical Business Database	Not available	application/	False	Data Axle (2022)

Software Requirements

Matlab is required to replicate all Monte Carlo experiments and the application. All code has been verified to be compatible with Matlab 2022b.

Python and Stata were used to prepare the analysis sample for the application.

The Python script application/extract.py imports the following Python modules: zipfile, gzip, and pandas. Although zipfile and gzip are part of the standard library, Pandas is an external package. Depending on your Python installation, it can likely be installed using the following command:

pip install pandas

The Stata scripts used for the application require two additional packages to be installed: xttrans2 and mat2txt. These can be installed using the following commands:

net install xttrans2
ssc install mat2txt

The analysis sample used for the application in the paper was produced using Python 3.11.6, Pandas 2.0.3, Stata MP 16.1, xttrans 1.2.0, and mat2txt 1.1.2.

Hardware Requirements

All of the programs in this replication package can be completed on a modern laptop (e.g., a circa 2020 MacBook Pro) without issue.

• The complete series of Monte Carlo experiments reported in Section 4 (mc-am2007 subdirectory) can be completed in approximately three to five hours.

• The estimation and counterfactual simulations for the application (application subdirectory), including 250 bootstrap replications can be completed in approximately 30 minutes to one hour.

• The complete series of Monte Carlo experiments reported in Appendix B.1 (mc-psd2008 subdirectory) can be completed in approximately one to two hours.

• The complete series of Monte Carlo experiments reported in Appendix B.2 (mc-psd2010 subdirectory) can be completed on a modern laptop in approximately five minutes.

Description of Code

Monte Carlo Experiments (Section 4)

Below is a complete list of program and function files used in the Monte Carlo simulations reported in Section 4. These files are contained in the mc-am2007 subdirectory.

Monte Carlo experiments:

• mc_main.m: Control program that runs all Monte Carlo simulations (across all experiments and sample sizes) and saves the results.
• mc_epl.m: The primary function which carries out each NPL and EPL Monte Carlo experiment for a fixed sample size and experiment number.
• Gfunc.m: Evaluates the equilibrium conditions of the dynamic entry/exit oligopoly model.
• epldygam.m: Estimates the structural parameters using EPL.
• npldygam.m: Estimates the structural parameters using NPL.
• freqprob.m: Frequency probability estimation.
• milogit.m: Maximum Likelihood Estimation of Logit Model.
• clogit.m: Maximum Likelihood estimation of McFadden's Conditional Logit model.
• loglogit.m: Log likelihood and gradient functions for multinomial logit.
• simdygam.m: Simulates data from the dynamic game.
• mpestat.m: Sample summary statistics.

Post-estimation programs to generate figures:

• iter_histograms.m: Produces histograms of the number of iterations taken by the EPL and NPL estimators across Monte Carlo replications.
• parameter_histograms.m: Produces histograms of the parameter estimates across Monte Carlo replications.
• time_histograms.m: Produces histograms of the runtimes of the EPL and NPL estimators across Monte Carlo replications.

Application (Section 5)

Below is a complete list of data, program, and function files used for the application described in Section 5 of the paper.

Raw Data

• ZIP_COUNTY_122021.xlsx - U.S. Department of Housing and Urban Development Zip Code to County Crosswalk File.

• nhgis0012_csv/ - This subdirectory contains the Zip code population data and codebook from IPUMS NHGIS.

• Data from the Data Axle Historical Business Database (BUSINESS_HISTORICAL_YYYY.zip and YYYY_Business_Academic_QCQ.txt.gz) could not be included in the replication package due to restrictions on redistribution.

Data Processing

The following Python and Stata scripts were used to process the raw data and produce the included analysis sample clubstore_county.csv:

1. extract.py - Extracts records related to wholesale club stores from the Data Axle Historical Business Database files. Produces extract.dta.

2. population_zip.do - Generate zip-code population data in Stata. Produces population_zip.dta.

3. population_county.do - Generate county population data in Stata. Produces zip_county_crosswalk.dta and population_county.dta.

4. clubstore_county.do - Generate final analysis dataset using Stata. Produces the analysis sample clubstore_county.csv and ptrans.txt (both included).

Analysis Sample

• clubstore_county.csv - This file contains the county-level analysis sample used in the application. This file was produced following the steps in the previous Data Processing section.

• ptrans.txt - This file contains the state-to-state transition matrix for market size used in the application.

Estimation

• main.m - This is the main Matlab control file for the application. This file produces Tables 7-11 in the paper.
• Gfunc.m: Evaluates the equilibrium conditions of the dynamic entry/exit oligopoly model.
• epldygam.m: Estimates the structural parameters using EPL.
• npldygam.m: Estimates the structural parameters using NPL.
• freqprob.m: Frequency probability estimation.
• clogit.m: Maximum Likelihood estimation of McFadden's Conditional Logit model.
• milogit.m: Maximum Likelihood Estimation of multinomial logit model.
• loglogit.m: Log likelihood and gradient functions for multinomial logit model.
• mpestat.m: Sample summary statistics.
• forwardsim.m: Simulates data on state transition and decisions, used for counterfactual simulations.

Monte Carlo Experiments (Appendix B.1)

Below is a complete list of program and function files used in the Monte Carlo simulations reported in Appendix B.1 of the paper. These files are contained in the mc-psd2008 subdirectory.

Main program:

• main.m - Main control program that executes all Monte Carlo experiments for multiple sample sizes (nobs), multiple equilibria (eqm_dgp), and multiple levels of noise in the estimates (c). Produces log files named according to the pattern psd2008_estimate_eqmN.log where N is the equilibrium index (eqm_dgp which can be 1, 2, or 3). Results are also stored in .mat files for further processing if desired.

Main support functions:

• psd2008_estimate.m - Estimates the model using the generated data and stores results. Requires setting nsims (= 1000) before running. Produces

• generate_data.m - Solves for equilibria and generates data for the Pesendorfer and Schmidt-Dengler (2008) model. Requires setting nsims (number of simulations; 1000 in the paper) nobs (number of observations; 250 and 1000 in the paper), and eqm_dgp (equilibrium number; 1, 2, or 3) in the Matlab session before running directly. Produces psd2008_data.mat.

Other support functions:

• EqmDiff.m - Equilibrium conditions for the model.
• Gderiv.m - Derivative of G function with respect to v.
• LL_EPL.m - Log likelihood for EPL estimation.
• LL_NPL.m - Log likelihood for NPL estimation.
• normsurplus.m - Social surplus function under the normal distribution.

Monte Carlo Experiments (Appendix B.2)

Below is a complete list of program and function files used in the Monte Carlo simulations reported in Appendix B.2 of the paper. These files are contained in the mc-psd2010 subdirectory.

• main.m - Main Monte Carlo simulation and estimation program.
• LL_MLE.m - Log likelihood function for MLE.
• LL_EPL.m - Log likelihood function for EPL.
• LL_NPL.m - Log likelihood function for NPL.
• myCDF.m - Error CDF specified in Pesendorfer and Schmidt-Dengler (2010).

To replicate the Monte Carlo experiments, run main in Matlab.

List of Tables and Figures

• Tables 1-6 were produced by mc-am2007/mc_main.m.

• Tables 7-11 in the paper were produced by application/main.m.

• Tables 12-16 were produced by mc-psd2008/psd2008_estimate_all.m. This script loops over all equilibria and sample sizes and estimates the model under several scenarios:

  • Table 12 corresponds to the case where eqm_dgp = 1, c = 0.0, nstart = 1, and nobs = [ 250, 1000 ].

  • Table 13 corresponds to the case where eqm_dgp = 2, c = 0.0, nstart = 1, and nobs = [ 250, 1000 ].

  • Table 14 corresponds to the case where eqm_dgp = 3, c = 0.0, nstart = 1, and nobs = 1000.

  • Table 15 corresponds to the case where eqm_dgp = [ 1, 2 ], c = 0.5, nstart = 1, and nobs = 250.

  • Table 16 corresponds to the case where eqm_dgp = [ 1, 2 ], c = 1.0, nstart = 5, and nobs = [ 250, 1000 ].

• Table 17 was produced by mc-psd2010/main.m.

• Figure 1 was produced by mc-am2007/parameter_histograms.m. The subfigures shown correspond to the following PDF files:

  • mc_epl_exper_1_6400_obs_param_7_hist.pdf
  • mc_epl_exper_2_6400_obs_param_7_hist.pdf
  • mc_epl_exper_3_6400_obs_param_7_hist.pdf
  • mc_epl_exper_1_6400_obs_param_8_hist.pdf
  • mc_epl_exper_2_6400_obs_param_8_hist.pdf
  • mc_epl_exper_3_6400_obs_param_8_hist.pdf

• Figure 2 was produced by mc-am2007/iter_histograms.m and mc-am2007/time_histograms.m. The subfigures shown correspond to the following PDF files:

  • mc_epl_exper_1_6400_obs_iter_hist.pdf
  • mc_epl_exper_2_6400_obs_iter_hist.pdf
  • mc_epl_exper_3_6400_obs_iter_hist.pdf
  • mc_epl_exper_1_6400_obs_time_hist.pdf
  • mc_epl_exper_2_6400_obs_time_hist.pdf
  • mc_epl_exper_3_6400_obs_time_hist.pdf

Results

Monte Carlo Experiments (Section 4)

The mc-am2007/results subdirectory contains the output of all Monte Carlo experiments reported in Section 4 of the paper. Equilibria for each experiment 1-3 are computed by the mc_epl.m program and stored in the files mc_epl_eq1.mat, mc_epl_eq2.mat, and mc_epl_eq3.mat respectively, if they do not already exist. If they exist, the equilibrium choice probabilities are loaded reused for computational efficiency.

Each of the log files named mc_epl_exper_J_N_obs.log, contain the results of 1000 replications single Monte Carlo replication for experiment J (1, 2, or 3) and sample size N (1600 or 6400). These log files are large, so they have been gzip compressed to save space. The raw Matlab data files containing the parameter estimates, number of iterations, and runtimes from each replication are saved in .mat files named mc_epl_exper_J_N_obs.mat. The parameter, time, and iteration histograms are saved in PDF files with similar names.

Application (Section 5)

The results in Section 5 of the paper were produced by running application/main.m described above using Matlab R2022b. The log file and results are included in the replication package:

• application/clubstore.log - Contains the estimation log file for the application, including estimation with the observed sample (replication 1) as well as 250 bootstrap replications (replications 2-251).

• application/clubstore.mat - The raw Matlab data file for the application, including estimation with the observed sample and all bootstrap replications

Monte Carlo Experiments (Appendix B.1)

The results in this section were produced by running mc-psd2008/main.m described above using Matlab 2018a. The log files are included in the replication package in the mc-psd2008/results subdirectory:

• psd2008_estimate_eqm1.log
• psd2008_estimate_eqm2.log
• psd2008_estimate_eqm3.log

Monte Carlo Experiments (Appendix B.2)

The results in the paper were produced by running mc-psd2010/main.m described above. A log file produced by Matlab 2018b is included in the replication package: mc-psd2010/mc-psd2010.log.

References

• Aguirregabiria, V., and P. Mira (2007). Sequential Estimation of Dynamic Discrete Games. Econometrica 75, 1-53.

• Data Axle. U.S. Historical Businesses, 1997-2021 [Data set]. Retrieved November 15, 2022 from The Ohio State University Libraries Research Commons. https://library.ohio-state.edu/record=e1002559~S7

• Manson, S., J. Schroeder, D. Van Riper, T. Kugler, and S. Ruggles. IPUMS National Historical Geographic Information System: Version 17.0 [Data set]. Minneapolis, MN: IPUMS. 2022. Retrieved March 9, 2023 from IPUMS NHGIS. http://doi.org/10.18128/D050.V17.0

• Pesendorfer, M. and P. Schmidt-Dengler (2008). Asymptotic least squares estimators for dynamic games. Review of Economic Studies 75, 901-928.

• Pesendorfer, M. and P. Schmidt-Dengler (2010). Sequential estimation of dynamic discrete games: A comment. Econometrica 78, 833-842.

• U.S. Department of Housing and Urban Development (HUD). HUD-USPS ZIP Code Crosswalk data 2021-Q4 [Data set]. Retrieved May 16, 2023 from the HUD Office of Policy Development and Research (PD&R). https://www.huduser.gov/portal/datasets/usps_crosswalk.html