Package actyr. August 5, 2015

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1 Type Package Package actyr August 5, 2015 Title Replication Package for Abbring, Campbell, Tilly, Yang (2014): Very Simple Markov Perfect Industry Dynamics Version 1.0 Date Author Jan Tilly URL Maintainer Jan Tilly This is the replication package for Abbring, Campbell, Tilly, Yang (2014): Very Simple Markov Perfect Industry Dynamics. It contains an example program, the Monte Carlo simulation, and the empirical illustration. The computationally intensive functions are all implemented in C++. -handling, model specification, and optimization is done in R. In addition to the nested fixed-point estimation (NFXP) algorithm that is presented in the paper, this package also includes an implementation of an MPEC (Mathematical Programming with Equilibrium Constraints) estimation algorithm. For this part only, a Knitro license is required. Knitro can then be used from R via the package knitror. License GPL-2 Imports Rcpp LinkingTo Rcpp, RcppArmadillo Depends gaussquad, tikzdevice, foreach, tictoc Suggests testthat R topics documented: actyr-package computemixingdensity computemodelmeans consolidatephis cov

2 2 actyr-package dpidmu dpidsigma ergodic firms getextractc getextractn getmodeltransitionmatrix likelihood likelihoodgradient mixingdensitygradient mixingprobabilities mpecconstraint mpecconstraintgradient mpeclikelihood mpeclikelihoodgradient mpecmixingdensitygradient N nfxp.covariance.step nfxp.gradient.step nfxp.gradient.step nfxp.gradient.step nfxp.likelihood.step nfxp.likelihood.step nfxp.likelihood.step onestepaheadforecast onestepaheadforecastoneovern population simulate tauchen valuefunction Index 23 actyr-package R / C++ programs for Abbring, Campbell, Tilly Yang (2014): Very Simple Markov Perfect Industry Dynamics This is the replication package for Abbring, Campbell, Tilly, Yang (2014): Very Simple Markov Perfect Industry Dynamics. It contains an example program, the Monte Carlo simulation, and the empirical illustration. The computationally intensive functions are all implemented in C++. -handling, model specification, and optimization is done in R. In addition to the nested fixedpoint estimation (NFXP) algorithm that is presented in the paper, this package also includes an implementation of an MPEC (Mathematical Programming with Equilibrium Constraints) estimation algorithm. For this part only, a Knitro license is required. Knitro can then be used from R via the package knitror.

3 computemixingdensity 3 Introduction This package contains an Rcpp implementation for the paper Very Simple Industry Dynamics. The C++ code is kept as generic as possible. The user only gets to choose the parameter ncheck. For a given ncheck, the C++ code assumes that there is a vector of surplus parameters k of length ncheck, a vector of entry cost parameters phi of length ncheck, and a scalar valued parameters omega. Any model restrictions are processed in R. E.g. in all of our specifications used in the paper, we require that phi(1)=...=phi(5). Similarly, a specification with covariates is simply changing the surplus parameters k. Thus, using the code and running different specifications will only involve changing the R code, not the C++ code. Example demo(example_nfxp) runs a simple example that generates a dataset and then estimates the model parameters using the Nested Fixed Point Algorithm. This code is stored in demo/example_nfxp.r. Replication The Monte Carlo results from the paper can be replicated by running demo(montecarlo). This code is stored in demo/montecarlo.r. The empirical illustration from the paper can be replicated by running demo(empirical). This code is stored in demo/empirical.r. Author(s) Jan Tilly See Also The paper is available at A Matlab sandbox is available at An extensive documentation of the Matlab sandbox is available at computemixingdensity Compute the mixing density This function computes the mixing density and returns it as a cube. computemixingdensity(,, vs) vs is an R list of settings is an R list of of parameters is a matrix with the value functions

4 4 consolidatephis A cube with the mixing density. computemodelmeans Compute the model means for Tforward periods Compute the model means for Tforward periods computemodelmeans(prime,, Tforward, initialc) Prime Tforward initialc structure structure integer vector with the initial distribution for the demand state matrix with means consolidatephis Consolidates the phis Consolidates the phis consolidatephis(, constraintgradientin) is an R list with constraintgradientin is a matrix with the constraint gradient constraint gradient in which the phis are consolidated into one

5 cov 5 cov Covariates for musas in the year 2000 This data set contains covariates for the musas in the year 2000 Format A data frame with 573 rows and 12 variables id CBSA id name CBSA name inc median income in 2000 dummy1 Middle Atlantic Census Region Dummy dummy2 East North Central Census Region Dummy dummy3 West North Central Census Region Dummy dummy4 South Atlantic Census Region Dummy dummy5 East South Central Census Region Dummy dummy6 West South Central Census Region Dummy dummy7 Mountain Census Region Dummy dummy8 Pacific Census Region Dummy dummy9 Low Diversity Dummy Source dpidmu Compute the gradient of the transition probability matrix with respect to mu Compute the gradient of the transition probability matrix with respect to mu dpidmu(pi, loggrid, mu, sigma)

6 6 dpidsigma Pi loggrid is the transition probability matrix is the demand grid mu is the mean of the innovations times 100 sigma is the standard deviation of the innovations times 100 A matrix with the derivative of the transition probability matrix with respect to mu. dpidsigma Compute the gradient of the transition probability matrix with respect to sigma Compute the gradient of the transition probability matrix with respect to sigma dpidsigma(pi, loggrid, mu, sigma) Pi loggrid mu is the transition probability matrix is the demand grid is the mean of the innovations sigma is the standard deviation of the innovations times 100 A matrix with the derivative of the transition probability matrix with respect to sigma.

7 ergodic 7 ergodic Compute the ergodic distribution of a Markov Chain Computes the ergodic distribution of a discrete Markov chain ergodic(transmat) transmat the transition probability matrix of the Markov chain a vector with the ergodic distribution firms Firm count data for musas from 2000 to 2009 Format This data set contains firm counts for Movie theaters for musas from 2000 to 2009 in panel data format A data frame with 573 rows and 12 variables id CBSA id name CBSA name firms2000 firms in 2000 firms2001 firms in 2001 firms2002 firms in 2002 firms2003 firms in 2003 firms2004 firms in 2004 firms2005 firms in 2005 firms2006 firms in 2006 firms2007 firms in 2007 firms2008 firms in 2008 firms2009 firms in 2009

8 8 getextractn Source getextractc Create extractc matrix Create extractc matrix getextractc(, ) structure structure extractc matrix getextractn Create extractn matrix Create extractn matrix getextractn() structure extractn matrix

9 getmodeltransitionmatrix 9 getmodeltransitionmatrix Computes the model s transition matrix Computes the model s implied transition matrix getmodeltransitionmatrix(, ) is an R list with settings is an R list with parameters square matrix with ccheck * (ncheck+1) rows and columns likelihood Compute likelihood contributions This function computes the vector of likelihood contributions for the data set in. The likelihood contributions are computed as follows: Use the function valuefunction to compute the equilibrium value functions and the entry and sure survival probabilities Compute the mixing density Assemble the likelihood contributions for each observations likelihood(,, ) is an R list of settings is an R list of parameters is an R list of the data and includes the matrices N and C A column vector of likelihood contributions for each observation

10 10 mixingdensitygradient likelihoodgradient Compute the gradient of the likelihood function This function computes the gradient of the likelihood function. It also computes the equilibrium and the likelihood function and then returns a list with the likelihood and gradient contributions. likelihoodgradient(,, ) is an R list of settings is an R list of parameters an R list of the data with matrices C and N An R list with a vector of likelihood contributions and a matrix of gradient contributions. mixingdensitygradient Compute the gradient of the mixing density for a variable of interest This function computes the gradient of the mixing density. Note that all parameters (except for omega) enter the mixing density only through the value functions. The derivative of the value function is given as argument to the function. mixingdensitygradient(,, Eq, D_vS, OMEGA_FLAG = FALSE) Eq D_vS OMEGA_FLAG is an R list of settings is an R list of parameters is an R list of equilibrium objects is a matrix of the gradient of the value function with respect to the variable of interest is a flag whether the variable in question is omega or not A cube of the mixing density p^{-1}(a_s, c, n)

11 mixingprobabilities 11 mixingprobabilities Compute mixing probabilities The function computes the mixing probability for a given state (N,C,W). mixingprobabilities(n, C, W, vs) N C W vs is an integer with the number of firms post-entry is an integer with the index of the demand grid is the a scalar with the current realization of the cost shock is the post-survival value function a scalar with the mixing probability mpecconstraint Compute MPEC constraint This function computes the equilibrium constraint for a given value function guess vs. mpecconstraint(,, vs) vs is an R list of settings is an R list of parameters is a matrix with the current guess of the value function A column vector of length equal to length(vs) with the distance between vs and the result from evaluating the Bellman operator at vs

12 12 mpeclikelihood mpecconstraintgradient Compute the gradient of the equilibrium constraint (MPEC) This function computes the gradient of the equilibrium constraint with respect to all structural parameters and the value functions. It returns the Jacobian. mpecconstraintgradient(,,, vs, CONSOLIDATE_PHIS) is an R list of settings is an R list of parameters an R list of the data with matrices C and N vs is a matrix with the current guess of the value function CONSOLIDATE_PHIS is an indicator whether or not the phis need to be consolidated A matrix with the constraint Jacobian. mpeclikelihood Compute MPEC likelihood contributions This function computes the vector of likelihood contributions for the data set in. The likelihood contributions are computed as follows: The function uses the value function guess vs to construct entry and sure survival probabilities Compute the mixing density Assemble the likelihood contributions for each observations mpeclikelihood(,,, vs)

13 mpeclikelihoodgradient 13 vs is an R list of settings is an R list of parameters is an R list of the data and includes the matrices N and C is a matrix with the current guess of the value function A column vector of likelihood contributions for each observation mpeclikelihoodgradient Compute the gradient of the likelihood function (MPEC) This function computes the gradient of the likelihood function. with respect to all the structural parameters and the value functions. mpeclikelihoodgradient(,,, vs) vs is an R list of settings is an R list of parameters an R list of the data with matrices C and N is a matrix with the current guess of the value function A vector with the gradient of the likelihood function

14 14 mpecmixingdensitygradient mpecmixingdensitygradient Compute the gradient of the mixing density (MPEC) This function computes the gradient of the mixing density with respect to all model parameters and with respect to the value functions. Note that omega is the only structural parameter that shows up in the mixing density directly. mpecmixingdensitygradient(,, Eq, D_vS, OMEGA_FLAG = FALSE) Eq D_vS OMEGA_FLAG is an R list of settings is an R list of parameters is an R list of equilibrium objects is a matrix of the gradient of the value function with respect to the variable of interest is a flag whether the variable in question is omega or not Details Note that with MPEC, we only need to compute the gradient of the mixing density for two scenarios (all other gradients are zero): We need the derivative of the mixing density with respect to omega (in which case D_vS is zero) We need the derivative of the mixing density with respect to vs(n,c), in which case D_vS(n,c) is equal to 1 for this particular (n,c) and zero elsewhere. A cube of the mixing density p^{-1}(a_s, c, n)

15 N30 15 N30 Compute the model means at time 30 as a function of demand at time zero Compute the model means at time 30 as a function of demand at time zero N30(,.factual, extractn).factual extractn structure structure matrix vector with means nfxp.covariance.step3 Third Step Variance-Covariance Matrix This function computes the variance-covariance matrix that corresponds to the third-step likelihood. The function uses the outer-product-of-the-gradient estimator of the hessian. nfxp.covariance.step3(x,,, ) x is a vector of parameters is an R list that contains a matrix C with indices of the demand state and a matrix N with number of active firms is an R list with settings is an R list with parameters Returns a vector with likelihood contributions

16 16 nfxp.gradient.step2 nfxp.gradient.step1 First Step Gradient Function This function computes the gradient used in the first step of the estimation procedure. nfxp.gradient.step1(mu_sigma,,, Pi = NULL) mu_sigma Pi is a vector whose elements are mu and sigma is an R list that contains a matrix C with indices of the demand state is an R list with program, most notably ccheck is the transition probabillity matrix; if Pi is not provided the transition probability matrix is computed based on mu_sigma times length(mygrid). Returns the vector valued gradient of the negative log-likelihood nfxp.gradient.step2 Second Step Gradient Function This function computes the gradient of the second step likelihood function nfxp.gradient.step2(x,,, ) x is a vector of parameters is an R list that contains a matrix C with indices of the demand state and a matrix N with number of active firms is an R list with settings is an R list with parameters Returns a vector of gradients

17 nfxp.gradient.step3 17 nfxp.gradient.step3 Second Step Gradient Function This function computes the gradient of the second step likelihood function nfxp.gradient.step3(x,,,, Pi = NULL) x Pi is a vector of parameters is an R list that contains a matrix C with indices of the demand state and a matrix N with number of active firms is an R list with settings is an R list with parameters is the transition probabillity matrix; if Pi is not provided the transition probability matrix is computed based on x times length(mygrid). Returns a vector of gradients nfxp.likelihood.step1 First Step Likelihood Function This function computes the likelihood used in the first step of the estimation procedure. nfxp.likelihood.step1(mu_sigma,,, Pi = NULL) mu_sigma Pi is a vector whose elements are mu and sigma is an R list that contains a matrix C with indices of the demand state is an R list with program, most notably ccheck is the transition probabillity matrix; if Pi is not provided the transition probability matrix is computed based on x times length(mygrid).

18 18 nfxp.likelihood.step3 Returns the scalar valued negative log-likelihood nfxp.likelihood.step2 Second Step Likelihood Function This function computes the second step likelihood function nfxp.likelihood.step2(x,,, ) x is a vector of parameters is an R list that contains a matrix C with indices of the demand state and a matrix N with number of active firms is an R list with settings is an R list with parameters Returns a vector with likelihood contributions nfxp.likelihood.step3 Third Step Likelihood Function This function computes the third step likelihood function nfxp.likelihood.step3(x,,,, Pi = NULL) x Pi is a vector of parameters is an R list that contains a matrix C with indices of the demand state and a matrix N with number of active firms is an R list with settings is an R list with parameters is the transition probabillity matrix; if Pi is not provided the transition probability matrix is computed based on x times length(mygrid).

19 onestepaheadforecast 19 Returns a vector with likelihood contributions onestepaheadforecast One-step-ahead forecast This function computes the one-step-ahead forecasts for the general model onestepaheadforecast(,, Row) Row is an R list with settings is an R list with parameters is a list containt the row-vectors C and N with data on one particular market Returns a vector with one-step-ahead forecasts onestepaheadforecastoneovern One-step-ahead forecast of 1/n This function computes the one-step-ahead forecasts for the general model for 1/n onestepaheadforecastoneovern(,, Row) Row is an R list with settings is an R list with parameters is a list containt the row-vectors C and N with data on one particular market Returns a vector with one-step-ahead forecasts

20 20 simulate population Population data for musas from 2000 to 2009 This data set contains population data for musas from 2000 to 2009 in panel data format Format A data frame with 573 rows and 12 variables id CBSA id name CBSA name pop2000 population in 2000 pop2001 population in 2001 pop2002 population in 2002 pop2003 population in 2003 pop2004 population in 2004 pop2005 population in 2005 pop2006 population in 2006 pop2007 population in 2007 pop2008 population in 2008 pop2009 population in 2009 Source simulate Generation This function generates a data set and returns a list with matrices N and C simulate(,, rcheck, tcheck, tburn = 25, initc = NULL, initn = NULL)

21 tauchen 21 rcheck tcheck tburn initc initn is an R list of settings is an R list of of params is an integer with the number of markets for which to generate data is an integer with the number of time periods for which to generate data burn in period initial C initial N an R list with the matrices N and C tauchen Tauchen Discretization of an AR(1) Tauchen computes the transition probability matrix for a discretized AR(1) with normally distributed innovations tauchen(mygrid, mu, sigma) mygrid mu sigma A vector with the equidistant support of the stochastic process The mean of the underlying normal distribution The standard deviation of the underlying normal distribution a square transition probability matrix with dimension length(mygrid) times length(mygrid). Examples tauchen( c(1,2,3), 0, 100)

22 22 valuefunction valuefunction Compute equilibrium value functions This function computes the equilibrium value functions and the probabilities of entry and certain survival. functions are computed via backward recursion. For this function to work, the following must have been set: ncheck The maximum number of firms with non-zero flow profits in some state of the world. loggrid The log of the grid tolinner The convergence tolerance for the inner loop maxiter The maximum number of iterations for the inner loop The following elements in must have been set k A vector of length ncheck that determines the flow surplus phi A vector of length ncheck that governs the entry costs omega A scalar that governs the variance of the cost shocks rho The discount factor Pi A transition probability matrix for the demand grid, which is computed via tauchen valuefunction(, ) is an R list of settings is an R list of of parameters List with value functions, entry probabilities and certain survival probabilities.

23 Index actyr-package, 2 computemixingdensity, 3 computemodelmeans, 4 consolidatephis, 4 cov, 5 simulate, 20 tauchen, 21 valuefunction, 22 dpidmu, 5 dpidsigma, 6 ergodic, 7 firms, 7 getextractc, 8 getextractn, 8 getmodeltransitionmatrix, 9 likelihood, 9 likelihoodgradient, 10 mixingdensitygradient, 10 mixingprobabilities, 11 mpecconstraint, 11 mpecconstraintgradient, 12 mpeclikelihood, 12 mpeclikelihoodgradient, 13 mpecmixingdensitygradient, 14 N30, 15 nfxp.covariance.step3, 15 nfxp.gradient.step1, 16 nfxp.gradient.step2, 16 nfxp.gradient.step3, 17 nfxp.likelihood.step1, 17 nfxp.likelihood.step2, 18 nfxp.likelihood.step3, 18 onestepaheadforecast, 19 onestepaheadforecastoneovern, 19 population, 20 23