Package 'prodest'

Title: Production Function Estimation
Description: TFP estimation with the control function approach.
Authors: Gabriele Rovigatti [aut,cre]
Maintainer: Gabriele Rovigatti <[email protected]>
License: GPL-3
Version: 0.1.1
Built: 2025-02-24 03:16:45 UTC
Source: https://github.com/gabrielerovigatti/prodest

Help Index

Data: Chilean firm-level production data 1986-1996

Description

Sectoral subsample of Chilean firm-level production data 1986-1996.

Usage

data("chilean")

Format

A data.frame object containing 9 variables with production-related data.

Value

Y

vector of log(outcome) - Value added.
sX

vector of log(capital).
fX

matrix of log(skilled labor) and log(unskilled labor).
cX

vector of log(water).
pX

vector of log(electricity).
inv

vector of log(investment).
idvar

vector of panel identifier.
timevar

vector of time.

References

http://www.ine.cl/canales/chile_estadistico/estadisticas_economicas/industria/series_estadisticas/series_estadisticas_enia.php

Simulate Panel dataset

Description

panelSim() produces a N*T balanced panel dataset of firms' production. In particular, it returns a data.frame with free, state and proxy variables aimed at performing Monte Carlo simulations on productivity-related models.

Usage

panelSim(N = 1000, T = 100, alphaL = .6, alphaK = .4, DGP = 1,
           rho = .7, sigeps = .1, sigomg = .3, rholnw = .3, seed = 123456)

Arguments

N

the number of firms. By default N=1000
T

the total time span to be simulated. Only a fraction (the last 10% of observations) will be returned. By default T=100
alphaL

the parameter of the free variable. By default alphaL=.6
alphaK

the parameter of the state variable. By default alphaK=.4
DGP

Type of DGP; accepts 1, 2 or 3. They differ in terms of shock to wages (0 or 0.1), Δ\Delta (0 or 0.5) and shock to labor (0 or 0.37). See detailsdetails. By default DGP=1.
rho

the AR(1) coefficient for omega. By default rho=0.7
sigeps

the standard deviation of epsilon. See detailsdetails. By default sigeps = .1.
sigomg

the standard deviation of the innovation to productivity ω\omega. By default sigomg = .3.
rholnw

AR(1) coefficient for log(wage). By default rholnw=.3.
seed

seed set when the routine starts. By default seed = 123456.

Details

panelSim() is the R implementation of the DGP written by Ackerberg, Caves and Frazer (2015).

Value

panelSim()panelSim() returns a data.frame with 7 variables:

  • idvaridvar ID codes from 1 to N (by default N=1000N = 1000).

  • timevartimevar time variable ranging 1 to round(T∗0.1)round(T*0.1) (by default T=100T = 100 and max(timevar)=10max(timevar) = 10).

  • YY log output value added variable

  • sXsX log state variable

  • fXfX log free variable

  • pX1pX1 log proxy variable - no measurement error

  • pX2pX2 log proxy variable - σmeasurementerror=.1\sigma_{measurementerror}= .1

  • pX3pX3 log proxy variable - σmeasurementerror=.2\sigma_{measurementerror}= .2

  • pX4pX4 log proxy variable - σmeasurementerror=.5\sigma_{measurementerror}= .5

Author(s)

Gabriele Rovigatti

References

Ackerberg, D., Caves, K. and Frazer, G. (2015). "Identication properties of recent production function estimators." Econometrica, 83(6), 2411-2451.

Examples

require(prodest)

  ## Simulate a dataset with 1000 firms (T = 100). \code{Panelsim()} delivers the last 10% of usable time per panel.

  panel.data <- panelSim()
  attach(panel.data)

  ## Estimate various models
  ACF.fit <- prodestACF(Y, fX, sX, pX2, idvar, timevar, theta0 = c(.5,.5))
  LP.fit <- prodestLP(Y, fX, sX, pX2, idvar, timevar)
  WRDG.fit <- prodestWRDG(Y, fX, sX, pX3, idvar, timevar, R = 5)
  
  ## print results in lateX tabular format 
  printProd(list(LP.fit, ACF.fit, WRDG.fit))

Print to lateX - prod objects

Description

The printProd() function accepts a list of prod class objects and returns a screen printed tabular in lateX format of the results.

Usage

printProd(mods, modnames = NULL, parnames = NULL, outfile = NULL, ptime = FALSE, nboot = FALSE)

Arguments

mods

a list of prod objects.
modnames

an optional vector of model names. By default, model names are the @ModelMethod values in prod objects.
parnames

an optional vector of parameter names. By default, parameter names are the names() vector of @Estimatespars in prod objects.
outfile

optional string with the path and directory to store a text file (.txt, .tex, etc. depending on the specified extension) with the tabular. By default outfile = NULL.
ptime

add a row showing the computational time. By default ptime = FALSE.
nboot

add a row showing the number of bootstrap repetitions. By default nboot = FALSE.

Value

The output of the function printProd is either a screen printed tabular in lateX format of prod object results or a text file tabular in lateX format of prod object results.

Author(s)

Gabriele Rovigatti

Examples

data("chilean")
    WRDGfit <- prodestWRDG_GMM(d$Y, fX = cbind(d$fX1, d$fX2), d$sX, d$pX, d$idvar, d$timevar)
    OPfit <- prodestOP(d$Y, fX = cbind(d$fX1, d$fX2), d$sX, d$pX, d$idvar, d$timevar)
    printProd(list(OPfit, WRDGfit), modnames = c('Olley-Pakes', 'Wooldridge'), parnames = c('bunsk', 'bsk', 'bk'))

Class for Prodest Fitted object

Description

Class for prodest fitted objects.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

Model:

Object of class list. Contains information about the model and the optimization procedue:

  • method: string The method used in estimation.

  • FSbetas: numeric First-stage estimated parameters.

  • boot.repetitions: numeric Number of bootstrap repetitions.

  • elapsed.time: numeric Time - in seconds - required for estimation.

  • theta0: numeric Vector of Second-stage optimization starting points.

  • opt: string Optimizer used for the Second-stage.

  • seed: numeric seed set.

  • opt.outcome: list Optimization outcome (depends on optimizer choice).

Data:

Object of class list. Contains:

  • Y: numeric Dependent variable - Value added.

  • free: matrix Free variable(s).

  • state: matrix State variable(s).

  • proxy: matrix Proxy variable(s).

  • control: matrix Control variable(s).

  • idvar: numeric Panel identifiers.

  • timevar: numeric Time identifiers.

  • FSresiduals: numeric First-Stage residuals.

Estimates:

Object of class list. Contains:

  • pars: numeric Estimated parameters for the variables of interest.

  • std.errors: numeric Estimated standard errors for the variables of interest.

Methods

  • show signature(object = 'prod'): Show table with the method, the estimated parameters and their standard errors.

  • summary signature(object = 'prod'): Show table with method, parameters, std.errors and auxiliary information on model and optimization.

  • FSres signature(object = 'prod'): Extract First-Stage residual vector.

  • omega signature(object = 'prod'): Extract estimated productivity vector.

  • coef signature(object = 'prod'): Extract estimated coefficients.

Author(s)

Gabriele Rovigatti

Estimate productivity - Ackerberg-Caves-Frazer correction

Description

The prodestACF() function accepts at least 6 objects (id, time, output, free, state and proxy variables), and returns a prod object of class S3 with three elements: (i) a list of model-related objects, (ii) a list with the data used in the estimation and estimated vectors of first-stage residuals, and (iii) a list with the estimated parameters and their bootstrapped standard errors .

Usage

prodestACF(Y, fX, sX, pX, idvar, timevar, zX = NULL, control = c('none','fs','2s'), dum = F, G = 3, A = 3, R = 20, orth = F, opt = 'optim',
                       theta0 = NULL, seed = 123456, cluster = NULL)

Arguments

Y

the vector of value added log output.
fX

the vector/matrix/dataframe of log free variables.
sX

the vector/matrix/dataframe of log state variables.
pX

the vector/matrix/dataframe of log proxy variables.
idvar

the vector/matrix/dataframe identifying individual panels.
timevar

the vector/matrix/dataframe identifying time.
zX

the vector/matrix/dataframe of (input price) control variables.
control

the way in which the control variables should be included. By default control = 'none'. Specifying control = 'fs' only includes controls in the first-stage polynomial, as in De Loecker and Warzynski (2012). Specifying control = 'fs' includes controls in both first and second stage, as in De Loecker, Goldberg, Khandelwal and Pavcnik (2016). Note that this is not desirable when estimating a revenue gross ouput production function, as in that case not controlling for input prices has the coincidental benefit that the input price bias partially cancels out the output price bias.
dum

whether time dummies should be included in the first stage. By default dum = F.
G

the degree of the first-stage polynomial in fX, sX and pX. By default G = 3.
A

the degree of the polynomial for the Markov productivity process. By default A = 3.
R

the number of block bootstrap repetitions to be performed in the standard error estimation. By default R = 20.
orth

a Boolean that determines whether first-stage polynomial should be orthogonal or raw. By default, orth = F. It is recommended to set orth to T if degree of polynomial is high.
opt

a string with the optimization algorithm to be used during the estimation. By default opt = 'optim'.
theta0

a vector with the second stage optimization starting points. By default theta0 = NULL and the optimization is run starting from the first stage estimated parameters + N(0,0.01)N(0,0.01) noise.
cluster

an object of class "SOCKcluster" or "cluster". By default cluster = NULL.
seed

seed set when the routine starts. By default seed = 123456.

Details

Consider a Cobb-Douglas production technology for firm ii at time tt

  • yit=α+witβ+kitγ+ωit+ϵity_{it} = \alpha + w_{it}\beta + k_{it}\gamma + \omega_{it} + \epsilon_{it}

where yity_{it} is the (log) output, w_it a 1xJ vector of (log) free variables, k_it is a 1xK vector of state variables and ϵit\epsilon_{it} is a normally distributed idiosyncratic error term. The unobserved technical efficiency parameter ωit\omega_{it} evolves according to a first-order Markov process:

  • ωit=E(ωit∣ωit−1)+uit=g(ωit−1)+uit\omega_{it} = E(\omega_{it} | \omega_{it-1}) + u_{it} = g(\omega_{it-1}) + u_{it}

and uitu_{it} is a random shock component assumed to be uncorrelated with the technicalefficiency, the state variables in kitk_{it} and the lagged free variables wit−1w_{it-1}. ACF propose an estimation algorithm alternative to OP and LP procedures claiming that the labour demand and the control function are partially collinear. It is based on the following set of assumptions:

  • a) pit=p(kit,lit,ωit)p_{it} = p(k_{it} , l_{it} , \omega_{it}) is the proxy variable policy function;

  • b) pitp_{it} is strictly monotone in ωit\omega_{it};

  • c) ωit\omega_{it} is scalar unobservable in pit=m(.)p_{it} = m(.) ;

  • d) The state variable are decided at time t-1. The less variable labor input, litl_{it}, is chosen at t-b, where 0<b<10 < b < 1. The free variables, witw_{it}, are chosen in t when the firm productivity shock is realized.

Under this set of assumptions, the first stage is meant to remove the shock ϵit\epsilon_{it} from the the output, yity_{it}. As in the OP/LP case, the inverted policy function replaces the productivity term ωit\omega_{it} in the production function:

  • yit=kitγ+witβ+litμ+h(pit,kit,wit,lit)+ϵity_{it} = k_{it}\gamma + w_{it}\beta + l_{it}\mu + h(p_{it} , k_{it} ,w_{it} , l_{it}) + \epsilon_{it}

which is estimated by a non-parametric approach - First Stage. Exploiting the Markovian nature of the productivity process one can use assumption d) in order to set up the relevant moment conditions and estimate the production function parameters - Second stage.

Value

The output of the function prodestACF is a member of the S3 class prod. More precisely, is a list (of length 3) containing the following elements:

Model, a list with elements:

  • method: a string describing the method ('ACF').

  • boot.repetitions: the number of bootstrap repetitions used for standard errors' computation.

  • elapsed.time: time elapsed during the estimation.

  • theta0: numeric object with the optimization starting points - second stage.

  • opt: string with the optimization routine used - 'optim', 'solnp' or 'DEoptim'.

  • seed: the seed set at the beginning of the estimation.

  • opt.outcome: optimization outcome.

  • FSbetas: first stage estimated parameters.

Data, a list with elements:

  • Y: the vector of value added log output.

  • free: the vector/matrix/dataframe of log free variables.

  • state: the vector/matrix/dataframe of log state variables.

  • proxy: the vector/matrix/dataframe of log proxy variables.

  • control: the vector/matrix/dataframe of log control variables.

  • idvar: the vector/matrix/dataframe identifying individual panels.

  • timevar: the vector/matrix/dataframe identifying time.

  • FSresiduals: numeric object with the residuals of the first stage.

Estimates, a list with elements:

  • pars: the vector of estimated coefficients.

  • std.errors: the vector of bootstrapped standard errors.

Members of class prod have an omega method returning a numeric object with the estimated productivity - that is: ωit=yit−(α+witβ+kitγ)\omega_{it} = y_{it} - (\alpha + w_{it}\beta + k_{it}\gamma). FSres method returns a numeric object with the residuals of the first stage regression, while summary, show and coef methods are implemented and work as usual.

Author(s)

Gabriele Rovigatti

References

Ackerberg, D., Caves, K. and Frazer, G. (2015). "Identification properties of recent production function estimators." Econometrica, 83(6), 2411-2451. De Loecker, J., Goldberg, P. K., Khandelwal, A. K., & Pavcnik, N. (2016). "Prices, markups, and trade reform." Econometrica, 84(2), 445-510. De Loecker, J., & Warzynski, F. (2012). "Markups and firm-level export status." American Economic Review, 102(6), 2437-71.

Examples

require(prodest)

    ## Chilean data on production.The full version is Publicly available at
    ## http://www.ine.cl/canales/chile_estadistico/estadisticas_economicas/industria/series_estadisticas/series_estadisticas_enia.php

    data(chilean)

    # we fit a model with two free (skilled and unskilled), one state (capital) and one proxy variable (electricity)

    ACF.fit <- prodestACF(d$Y, fX = cbind(d$fX1, d$fX2), d$sX, d$pX, d$idvar, d$timevar, theta0 = c(.5,.5,.5), seed = 154673)
    ACF.fit.solnp <- prodestACF(d$Y, fX = cbind(d$fX1, d$fX2), d$sX, d$pX, d$idvar, d$timevar, theta0 = c(.5,.5,.5), opt = 'solnp', seed = 154673)

    # run the same regression in parallel
    nCores <- as.numeric(Sys.getenv("NUMBER_OF_PROCESSORS"))
    cl <- makeCluster(getOption("cl.cores", nCores - 1))
    ACF.fit.par <- prodestACF(d$Y, fX = cbind(d$fX1, d$fX2), d$sX, d$pX, d$idvar, d$timevar, theta0 = c(.5,.5,.5), cluster = cl, seed = 154673)
    stopCluster(cl)

    # show results
    coef(ACF.fit)
    coef(ACF.fit.solnp)
    coef(ACF.fit.par)

     # show results in .tex tabular format
     printProd(list(ACF.fit, ACF.fit.solnp, ACF.fit.par))

Estimate productivity - Levinsohn-Petrin method

Description

The prodestLP() The prodestWRDG() function accepts at least 6 objects (id, time, output, free, state and proxy variables), and returns a prod object of class S3 with three elements: (i) a list of model-related objects, (ii) a list with the data used in the estimation and estimated vectors of first-stage residuals, and (iii) a list with the estimated parameters and their bootstrapped standard errors.

Usage

prodestLP(Y, fX, sX, pX, idvar, timevar, R = 20, G = 3, orth = F, cX = NULL,
            opt = 'optim', theta0 = NULL, seed = 123456, cluster = NULL, tol = 1e-100)

Arguments

Y

the vector of value added log output.
fX

the vector/matrix/dataframe of log free variables.
sX

the vector/matrix/dataframe of log state variables.
pX

the vector/matrix/dataframe of log proxy variables.
cX

the vector/matrix/dataframe of control variables. By default cX= NULL.
idvar

the vector/matrix/dataframe identifying individual panels.
timevar

the vector/matrix/dataframe identifying time.
R

the number of block bootstrap repetitions to be performed in the standard error estimation. By default R = 20.
G

the degree of the first-stage polynomial in fX, sX and pX. By default G = 3.
orth

a Boolean that determines whether first-stage polynomial should be orthogonal or raw. By default, orth = F. It is recommended to set orth to T if degree of polynomial is high.
opt

a string with the optimization algorithm to be used during the estimation. By default opt = 'optim'.
theta0

a vector with the second stage optimization starting points. By default theta0 = NULL and the optimization is run starting from the first stage estimated parameters + N(μ=0,σ=0.01)N(\mu=0,\sigma=0.01) noise.
cluster

an object of class "SOCKcluster" or "cluster". By default cluster = NULL.
seed

seed set when the routine starts. By default seed = 123456.
tol

optimizer tolerance. By default tol = 1e-100.

Details

Consider a Cobb-Douglas production technology for firm ii at time tt

  • yit=α+witβ+kitγ+ωit+ϵity_{it} = \alpha + w_{it}\beta + k_{it}\gamma + \omega_{it} + \epsilon_{it}

where yity_{it} is the (log) output, w_it a 1xJ vector of (log) free variables, k_it is a 1xK vector of state variables and ϵit\epsilon_{it} is a normally distributed idiosyncratic error term. The unobserved technical efficiency parameter ωit\omega_{it} evolves according to a first-order Markov process:

  • ωit=E(ωit∣ωit−1)+uit=g(ωit−1)+uit\omega_{it} = E(\omega_{it} | \omega_{it-1}) + u_{it} = g(\omega_{it-1}) + u_{it}

and uitu_{it} is a random shock component assumed to be uncorrelated with the technicalefficiency, the state variables in kitk_{it} and the lagged free variables wit−1w_{it-1}. The LP method relies on the following set of assumptions:

  • a) firms immediately adjust the level of inputs according to demand function m(ωit,kit)m(\omega_{it}, k_{it}) after the technical efficiency shock realizes;

  • b) mitm_{it} is strictly monotone in ωit\omega_{it};

  • c) ωit\omega_{it} is scalar unobservable in mit=m(.)m_{it} = m(.) ;

  • d) the levels of kitk_{it} are decided at time t−1t-1; the level of the free variable, witw_{it}, is decided after the shock uitu_{it} realizes.

Assumptions a)-d) ensure the invertibility of mitm_{it} in ωit\omega_{it} and lead to the partially identified model:

  • yit=α+witβ+kitγ+h(mit,kit)+ϵit=α+witβ+ϕ(mit,kit)+ϵity_{it} = \alpha + w_{it}\beta + k_{it}\gamma + h(m_{it}, k_{it}) + \epsilon_{it} = \alpha + w_{it}\beta + \phi(m_{it}, k_{it}) + \epsilon_{it}

which is estimated by a non-parametric approach - First Stage. Exploiting the Markovian nature of the productivity process one can use assumption d) in order to set up the relevant moment conditions and estimate the production function parameters - Second stage. Exploting the resisual νit\nu_{it} of:

  • yit−witβ^=α+kitγ+g(ωit−1,χit)+νity_{it} - w_{it}\hat{\beta} = \alpha + k_{it}\gamma + g(\omega_{it-1}, \chi_{it}) + \nu_{it}

and g(.)g(.) is typically left unspecified and approximated by a nthn^{th} order polynomial and χit\chi_{it} is an inidicator function for the attrition in the market.

Value

The output of the function prodestLP is a member of the S3 class prod. More precisely, is a list (of length 3) containing the following elements:

Model, a list containing:

  • method: a string describing the method ('LP').

  • boot.repetitions: the number of bootstrap repetitions used for standard errors' computation.

  • elapsed.time: time elapsed during the estimation.

  • theta0: numeric object with the optimization starting points - second stage.

  • opt: string with the optimization routine used - 'optim', 'solnp' or 'DEoptim'.

  • seed: the seed set at the beginning of the estimation.

  • opt.outcome: optimization outcome.

  • FSbetas: first stage estimated parameters.

Data, a list containing:

  • Y: the vector of value added log output.

  • free: the vector/matrix/dataframe of log free variables.

  • state: the vector/matrix/dataframe of log state variables.

  • proxy: the vector/matrix/dataframe of log proxy variables.

  • control: the vector/matrix/dataframe of log control variables.

  • idvar: the vector/matrix/dataframe identifying individual panels.

  • timevar: the vector/matrix/dataframe identifying time.

  • FSresiduals: numeric object with the residuals of the first stage.

Estimates, a list containing:

  • pars: the vector of estimated coefficients.

  • std.errors: the vector of bootstrapped standard errors.

Members of class prod have an omega method returning a numeric object with the estimated productivity - that is: ωit=yit−(α+witβ+kitγ)\omega_{it} = y_{it} - (\alpha + w_{it}\beta + k_{it}\gamma). FSres method returns a numeric object with the residuals of the first stage regression, while summary, show and coef methods are implemented and work as usual.

Author(s)

Gabriele Rovigatti

References

Levinsohn, J. and Petrin, A. (2003). "Estimating production functions using inputs to control for unobservables." The Review of Economic Studies, 70(2), 317-341.

Examples

require(prodest)

    ## Chilean data on production.
    ## Publicly available at http://www.ine.cl/canales/chile_estadistico/estadisticas_economicas/industria/series_estadisticas/series_estadisticas_enia.php

    data(chilean)

    ## we fit a model with two free (skilled and unskilled), one state (capital) and one proxy variable (electricity)

    LP.fit <- prodestLP(d$Y, fX = cbind(d$fX1, d$fX2), d$sX, d$pX, d$idvar, d$timevar, seed = 154673)
    LP.fit.solnp <- prodestLP(d$Y, fX = cbind(d$fX1, d$fX2), d$sX, d$pX, d$idvar, d$timevar, opt = 'solnp')

    # run the same model in parallel
    require(parallel)
    nCores <- as.numeric(Sys.getenv("NUMBER_OF_PROCESSORS"))
    cl <- makeCluster(getOption("cl.cores", nCores - 1))
    LP.fit.par <- prodestLP(d$Y, fX = cbind(d$fX1, d$fX2), d$sX, d$pX, d$idvar, d$timevar, cluster = cl, seed = 154673)
    stopCluster(cl)

    # show results
    summary(LP.fit)
    summary(LP.fit.solnp)
    summary(LP.fit.par)

    # show results in .tex tabular format
     printProd(list(LP.fit, LP.fit.solnp, LP.fit.par))

Estimate productivity - Olley-Pakes method

Description

The prodestOP() function accepts at least 6 objects (id, time, output, free, state and proxy variables), and returns a prod object of class S4 with three elements: (i) a list of model-related objects, (ii) a list with the data used in the estimation and estimated vectors of first-stage residuals, and (iii) a list with the estimated parameters and their bootstrapped standard errors .

Usage

prodestOP(Y, fX, sX, pX, idvar, timevar, R = 20, G = 3, orth = F, cX = NULL,
            opt = 'optim', theta0 = NULL, seed = 123456, cluster = NULL, tol = 1e-100)

Arguments

Y

the vector of value added log output.
fX

the vector/matrix/dataframe of log free variables.
sX

the vector/matrix/dataframe of log state variables.
pX

the vector/matrix/dataframe of log proxy variables.
cX

the vector/matrix/dataframe of control variables. By default cX= NULL.
idvar

the vector/matrix/dataframe identifying individual panels.
timevar

the vector/matrix/dataframe identifying time.
R

the number of block bootstrap repetitions to be performed in the standard error estimation. By default R = 20.
G

the degree of the first-stage polynomial in fX, sX and pX. By default, G = 3.
orth

a Boolean that determines whether first-stage polynomial should be orthogonal or raw. By default, orth = F. It is recommended to set orth to T if degree of polynomial is high.
opt

a string with the optimization algorithm to be used during the estimation. By default opt = 'optim'.
theta0

a vector with the second stage optimization starting points. By default theta0 = NULL and the optimization is run starting from the first stage estimated parameters + N(μ=0,σ=0.01)N(\mu=0,\sigma=0.01) noise.
cluster

an object of class "SOCKcluster" or "cluster". By default cluster = NULL.
seed

seed set when the routine starts. By default seed = 123456.
tol

optimizer tolerance. By default tol = 1e-100.

Details

Consider a Cobb-Douglas production technology for firm ii at time tt

  • yit=α+witβ+kitγ+ωit+ϵity_{it} = \alpha + w_{it}\beta + k_{it}\gamma + \omega_{it} + \epsilon_{it}

where yity_{it} is the (log) output, w_it a 1xJ vector of (log) free variables, k_it is a 1xK vector of state variables and ϵit\epsilon_{it} is a normally distributed idiosyncratic error term. The unobserved technical efficiency parameter ωit\omega_{it} evolves according to a first-order Markov process:

  • ωit=E(ωit∣ωit−1)+uit=g(ωit−1)+uit\omega_{it} = E(\omega_{it} | \omega_{it-1}) + u_{it} = g(\omega_{it-1}) + u_{it}

and uitu_{it} is a random shock component assumed to be uncorrelated with the technicalefficiency, the state variables in kitk_{it} and the lagged free variables wit−1w_{it-1}. The OP method relies on the following set of assumptions:

  • a) iit=i(kit,ωit)i_{it} = i(k_{it},\omega_{it}) - investments are a function of both the state variable and the technical efficiency parameter;

  • b) iiti_{it} is strictly monotone in ωit\omega_{it};

  • c) ωit\omega_{it} is scalar unobservable in iit=i(.)i_{it} = i(.) ;

  • d) the levels of iiti_{it} and kitk_{it} are decided at time t−1t-1; the level of the free variable, witw_{it}, is decided after the shock uitu_{it} realizes.

Assumptions a)-d) ensure the invertibility of iiti_{it} in ωit\omega_{it} and lead to the partially identified model:

  • yit=α+witβ+kitγ+h(iit,kit)+ϵit=α+witβ+ϕ(iit,kit)+ϵity_{it} = \alpha + w_{it}\beta + k_{it}\gamma + h(i_{it}, k_{it}) + \epsilon_{it} = \alpha + w_{it}\beta + \phi(i_{it}, k_{it}) + \epsilon_{it}

which is estimated by a non-parametric approach - First Stage. Exploiting the Markovian nature of the productivity process one can use assumption d) in order to set up the relevant moment conditions and estimate the production function parameters - Second stage. Exploting the resisual eite_{it} of:

  • yit−witβ^=α+kitγ+g(ωit−1,χit)+ϵity_{it} - w_{it}\hat{\beta} = \alpha + k_{it}\gamma + g(\omega_{it-1}, \chi_{it}) + \epsilon_{it}

and g(.)g(.) is typically left unspecified and approximated by a nthn^{th} order polynomial and χit\chi_{it} is an inidicator function for the attrition in the market.

Value

The output of the function prodestOP is a member of the S3 class prod. More precisely, is a list (of length 3) containing the following elements:

Model, a list containing:

  • method: a string describing the method ('OP').

  • boot.repetitions: the number of bootstrap repetitions used for standard errors' computation.

  • elapsed.time: time elapsed during the estimation.

  • theta0: numeric object with the optimization starting points - second stage.

  • opt: string with the optimization routine used - 'optim', 'solnp' or 'DEoptim'.

  • seed: the seed set at the beginning of the estimation.

  • opt.outcome: optimization outcome.

  • FSbetas: first stage estimated parameters.

Data, a list containing:

  • Y: the vector of value added log output.

  • free: the vector/matrix/dataframe of log free variables.

  • state: the vector/matrix/dataframe of log state variables.

  • proxy: the vector/matrix/dataframe of log proxy variables.

  • control: the vector/matrix/dataframe of log control variables.

  • idvar: the vector/matrix/dataframe identifying individual panels.

  • timevar: the vector/matrix/dataframe identifying time.

  • FSresiduals: numeric object with the residuals of the first stage.

Estimates, a list containing:

  • pars: the vector of estimated coefficients.

  • std.errors: the vector of bootstrapped standard errors.

Members of class prod have an omega method returning a numeric object with the estimated productivity - that is: ωit=yit−(α+witβ+kitγ)\omega_{it} = y_{it} - (\alpha + w_{it}\beta + k_{it}\gamma). FSres method returns a numeric object with the residuals of the first stage regression, while summary, show and coef methods are implemented and work as usual.

Author(s)

Gabriele Rovigatti

References

Olley, S G and Pakes, A (1996). "The dynamics of productivity in the telecommunications equipment industry." Econometrica, 64(6), 1263-1297.

Examples

require(prodest)

    ## Chilean data on production.The full version is Publicly available at
    ## http://www.ine.cl/canales/chile_estadistico/estadisticas_economicas/industria/series_estadisticas/series_estadisticas_enia.php

    data(chilean)

    # we fit a model with two free (skilled and unskilled), one state (capital) and one proxy variable (electricity)

    OP.fit <- prodestOP(d$Y, fX = cbind(d$fX1, d$fX2), d$sX, d$inv, d$idvar, d$timevar)
    OP.fit.solnp <- prodestOP(d$Y, fX = cbind(d$fX1, d$fX2), d$sX, d$inv, d$idvar, d$timevar, opt='solnp')
    OP.fit.control <- prodestOP(d$Y, fX = cbind(d$fX1, d$fX2), d$sX, d$inv, d$idvar, d$timevar, cX = d$cX)

    # show results
    summary(OP.fit)
    summary(OP.fit.solnp)
    summary(OP.fit.control)

    # show results in .tex tabular format
     printProd(list(OP.fit, OP.fit.solnp, OP.fit.control))

Estimate productivity - Wooldridge method

Description

The prodestWRDG() function accepts at least 6 objects (id, time, output, free, state and proxy variables), and returns a prod object of class S4 with three elements: (i) a list of model-related objects, (ii) a list with the data used in the estimation and estimated vectors of first-stage residuals, and (iii) a list with the estimated parameters and their bootstrapped standard errors.

Usage

prodestWRDG(Y, fX, sX, pX, idvar, timevar, R = 20, G = 3, orth = F, cX = NULL, seed = 123456,
                tol = 1e-100, theta0 = NULL, cluster = NULL)

Arguments

Y

the vector of value added log output.
fX

the vector/matrix/dataframe of log free variables.
sX

the vector/matrix/dataframe of log state variables.
pX

the vector/matrix/dataframe of log proxy variables.
cX

the vector/matrix/dataframe of control variables. By default cX= NULL.
idvar

the vector/matrix/dataframe identifying individual panels.
timevar

the vector/matrix/dataframe identifying time.
R

the number of block bootstrap repetitions to be performed in the standard error estimation. By default R = 20.
G

the degree of the polynomial for productivity in sX and pX. By default, G = 3
orth

a Boolean that determines whether first-stage polynomial should be orthogonal or raw. By default, orth = F. It is recommended to set orth to T if degree of polynomial is high.
theta0

a vector with the second stage optimization starting points. By default theta0 = NULL and the optimization is run starting from the first stage estimated parameters + N(μ=0,σ=0.01)N(\mu=0,\sigma=0.01) noise.
cluster

an object of class "SOCKcluster" or "cluster". By default cluster = NULL.
seed

seed set when the routine starts. By default seed = 123456.
tol

optimizer tolerance. By default tol = 1e-100.

Details

Consider a Cobb-Douglas production technology for firm ii at time tt

  • yit=α+witβ+kitγ+ωit+ϵity_{it} = \alpha + w_{it}\beta + k_{it}\gamma + \omega_{it} + \epsilon_{it}

where yity_{it} is the (log) output, w_it a 1xJ vector of (log) free variables, k_it is a 1xK vector of state variables and ϵit\epsilon_{it} is a normally distributed idiosyncratic error term. The unobserved technical efficiency parameter ωit\omega_{it} evolves according to a first-order Markov process:

  • ωit=E(ωit∣ωit−1)+uit=g(ωit−1)+uit\omega_{it} = E(\omega_{it} | \omega_{it-1}) + u_{it} = g(\omega_{it-1}) + u_{it}

and uitu_{it} is a random shock component assumed to be uncorrelated with the technical efficiency, the state variables in kitk_{it} and the lagged free variables wit−1w_{it-1}. Wooldridge method allows to jointly estimate OP/LP two stages jointly in a system of two equations. It relies on the following set of assumptions:

  • a) ωit=g(xit,pit)\omega_{it} = g(x_{it} , p_{it}): productivity is an unknown function g(.)g(.) of state and a proxy variables;

  • b) E(ωit∣ωit−1)=f[ωit−1]E(\omega_{it} | \omega_{it-1)}=f[\omega_{it-1}], productivity is an unknown function f[.]f[.] of lagged productivity, ωit−1\omega_{it-1}.

Under the above set of assumptions, It is possible to construct a system gmm using the vector of residuals from

  • r1it=yit−alpha−witβ−xitγ−g(xit,pit)r_{1it} = y_{it} - alpha - w_{it}\beta - x_{it}\gamma - g(x_{it} , p_{it})

  • r2it=yit−alpha−witβ−xitγ−f[g(xit−1,pit−1)]r_{2it} = y_{it} - alpha - w_{it}\beta - x_{it}\gamma - f[g(x_{it-1} , p_{it-1})]

where the unknown function f(.)f(.) is approximeted by a n-th order polynomial and g(xit,mit)=λ0+c(xit,mit)λg(x_{it} , m_{it}) = \lambda_0 + c(x_{it} , m_{it})\lambda. In particular, g(xit,mit)g(x_{it} , m_{it}) is a linear combination of functions in (xit,mit)(x_{it} , m_{it}) and citc_{it} are the addends of this linear combination. The residuals eqnr_it are used to set the moment conditions

  • E(Zit∗rit)=0E(Z_{it}*r_{it}) =0

with the following set of instruments:

  • Z1it=(1,wit,xit,cit)Z1_{it} = (1, w_{it}, x_{it}, c_{it})

  • Z2it=(wit−1,cit,cit)Z2_{it} = (w_{it-1}, c_{it}, c_{it})

Value

The output of the function prodestWRDG is a member of the S3 class prod. More precisely, is a list (of length 3) containing the following elements:

Model, a list containing:

  • method: a string describing the method ('WRDG').

  • elapsed.time: time elapsed during the estimation.

  • seed: the seed set at the beginning of the estimation.

  • opt.outcome: optimization outcome.

Data, a list containing:

  • Y: the vector of value added log output.

  • free: the vector/matrix/dataframe of log free variables.

  • state: the vector/matrix/dataframe of log state variables.

  • proxy: the vector/matrix/dataframe of log proxy variables.

  • control: the vector/matrix/dataframe of log control variables.

  • idvar: the vector/matrix/dataframe identifying individual panels.

  • timevar: the vector/matrix/dataframe identifying time.

Estimates, a list containing:

  • pars: the vector of estimated coefficients.

  • std.errors: the vector of bootstrapped standard errors.

Members of class prod have an omega method returning a numeric object with the estimated productivity - that is: ωit=yit−(α+witβ+kitγ)\omega_{it} = y_{it} - (\alpha + w_{it}\beta + k_{it}\gamma). FSres method returns a numeric object with the residuals of the first stage regression, while summary, show and coef methods are implemented and work as usual.

Author(s)

Gabriele Rovigatti

References

Wooldridge, J M (2009). "On estimating firm-level production functions using proxy variables to control for unobservables." Economics Letters, 104, 112-114.

Examples

data("chilean")

    # we fit a model with two free (skilled and unskilled), one state (capital) and one proxy variable (electricity)

    WRDG.fit <- prodestWRDG(d$Y, fX = cbind(d$fX1, d$fX2), d$sX, d$pX, d$idvar, d$timevar)

    # show results
    WRDG.fit

    ## Not run: 
      # estimate a panel dataset - DGP1, various measurement errors - and run the estimation
      sim <- panelSim()

      WRDG.sim1 <- prodestWRDG(sim$Y, sim$fX, sim$sX, sim$pX1, sim$idvar, sim$timevar)
      WRDG.sim2 <- prodestWRDG(sim$Y, sim$fX, sim$sX, sim$pX2, sim$idvar, sim$timevar)
      WRDG.sim3 <- prodestWRDG(sim$Y, sim$fX, sim$sX, sim$pX3, sim$idvar, sim$timevar)
      WRDG.sim4 <- prodestWRDG(sim$Y, sim$fX, sim$sX, sim$pX4, sim$idvar, sim$timevar)

      # show results in .tex tabular format
      printProd(list(WRDG.sim1, WRDG.sim2, WRDG.sim3, WRDG.sim4), parnames = c('Free','State'))
    
## End(Not run)

Estimate productivity - Wooldridge method

Description

The prodestWRDG_GMM() function accepts at least 6 objects (id, time, output, free, state and proxy variables), and returns a prod object of class S3 with three elements: (i) a list of model-related objects, (ii) a list with the data used in the estimation and estimated vectors of first-stage residuals, and (iii) a list with the estimated parameters and their bootstrapped standard errors.

Usage

prodestWRDG_GMM(Y, fX, sX, pX, idvar, timevar, G = 3, orth = F, cX = NULL, seed = 123456, tol = 1e-100)

Arguments

Y

the vector of value added log output.
fX

the vector/matrix/dataframe of log free variables.
sX

the vector/matrix/dataframe of log state variables.
pX

the vector/matrix/dataframe of log proxy variables.
cX

the vector/matrix/dataframe of control variables. By default cX= NULL.
G

the degree of the polynomial for productivity in sX and pX. By default, G = 3.
orth

a Boolean that determines whether first-stage polynomial should be orthogonal or raw. By default, orth = F. It is recommended to set orth to T if degree of polynomial is high.
idvar

the vector/matrix/dataframe identifying individual panels.
timevar

the vector/matrix/dataframe identifying time.
seed

seed set when the routine starts. By default seed = 123456.
tol

optimizer tolerance. By default tol = 1e-100.

Details

Consider a Cobb-Douglas production technology for firm ii at time tt

  • yit=α+witβ+kitγ+ωit+ϵity_{it} = \alpha + w_{it}\beta + k_{it}\gamma + \omega_{it} + \epsilon_{it}

where yity_{it} is the (log) output, w_it a 1xJ vector of (log) free variables, k_it is a 1xK vector of state variables and ϵit\epsilon_{it} is a normally distributed idiosyncratic error term. The unobserved technical efficiency parameter ωit\omega_{it} evolves according to a first-order Markov process:

  • ωit=E(ωit∣ωit−1)+uit=g(ωit−1)+uit\omega_{it} = E(\omega_{it} | \omega_{it-1}) + u_{it} = g(\omega_{it-1}) + u_{it}

and uitu_{it} is a random shock component assumed to be uncorrelated with the technical efficiency, the state variables in kitk_{it} and the lagged free variables wit−1w_{it-1}. Wooldridge method allows to jointly estimate OP/LP two stages jointly in a system of two equations. It relies on the following set of assumptions:

  • a) ωit=g(xit,pit)\omega_{it} = g(x_{it} , p_{it}): productivity is an unknown function g(.)g(.) of state and a proxy variables;

  • b) E(ωit∣ωit−1)=f[ωit−1]E(\omega_{it} | \omega_{it-1)}=f[\omega_{it-1}], productivity is an unknown function f[.]f[.] of lagged productivity, ωit−1\omega_{it-1}.

Under the above set of assumptions, It is possible to construct a system gmm using the vector of residuals from

  • r1it=yit−alpha−witβ−xitγ−g(xit,pit)r_{1it} = y_{it} - alpha - w_{it}\beta - x_{it}\gamma - g(x_{it} , p_{it})

  • r2it=yit−alpha−witβ−xitγ−f[g(xit−1,pit−1)]r_{2it} = y_{it} - alpha - w_{it}\beta - x_{it}\gamma - f[g(x_{it-1} , p_{it-1})]

where the unknown function f(.)f(.) is approximated by a n-th order polynomial and g(xit,pit)=λ0+c(xit,pit)λg(x_{it} , p_{it}) = \lambda_0 + c(x_{it} , p_{it})\lambda. In particular, g(xit,pit)g(x_{it} , p_{it}) is a linear combination of functions in (xit,pit)(x_{it} , p_{it}) and citc_{it} are the addends of this linear combination. The residuals eqnr_it are used to set the moment conditions

  • E(Zit∗rit)=0E(Z_{it}*r_{it}) =0

with the following set of instruments:

  • Z1it=(1,wit,xit,cit)Z1_{it} = (1, w_{it}, x_{it}, c_{it})

  • Z2it=(1,wit−1,xit,cit−1,qit)Z2_{it} = (1, w_{it-1}, x_{it}, c_{it-1}, q_{it})

where qit−1q_{it-1} is a set of non-linear functions of c_it-1.

Value

The output of the function prodestWRDG is a member of the S3 class prod. More precisely, is a list (of length 3) containing the following elements:

Model, a list containing:

  • method: a string describing the method ('WRDG').

  • elapsed.time: time elapsed during the estimation.

  • seed: the seed set at the beginning of the estimation.

  • opt.outcome: optimization outcome.

Data, a list containing:

  • Y: the vector of value added log output.

  • free: the vector/matrix/dataframe of log free variables.

  • state: the vector/matrix/dataframe of log state variables.

  • proxy: the vector/matrix/dataframe of log proxy variables.

  • control: the vector/matrix/dataframe of log control variables.

  • idvar: the vector/matrix/dataframe identifying individual panels.

  • timevar: the vector/matrix/dataframe identifying time.

Estimates, a list containing:

  • pars: the vector of estimated coefficients.

  • std.errors: the vector of bootstrapped standard errors.

Members of class prod have an omega method returning a numeric object with the estimated productivity - that is: ωit=yit−(α+witβ+kitγ)\omega_{it} = y_{it} - (\alpha + w_{it}\beta + k_{it}\gamma). FSres method returns a numeric object with the residuals of the first stage regression, while summary, show and coef methods are implemented and work as usual.

Author(s)

Gabriele Rovigatti

References

Wooldridge, J M (2009). "On estimating firm-level production functions using proxy variables to control for unobservables." Economics Letters, 104, 112-114.

Examples

data("chilean")

    # we fit a model with two free (skilled and unskilled), one state (capital) and one proxy variable (electricity)

    WRDG.GMM.fit <- prodestWRDG_GMM(d$Y, fX = cbind(d$fX1, d$fX2), d$sX, d$pX, d$idvar, d$timevar)

    # show results
    WRDG.GMM.fit

    # estimate a panel dataset - DGP1, various measurement errors - and run the estimation
    sim <- panelSim()

    WRDG.GMM.sim1 <- prodestWRDG_GMM(sim$Y, sim$fX, sim$sX, sim$pX1, sim$idvar, sim$timevar)
    WRDG.GMM.sim2 <- prodestWRDG_GMM(sim$Y, sim$fX, sim$sX, sim$pX2, sim$idvar, sim$timevar)
    WRDG.GMM.sim3 <- prodestWRDG_GMM(sim$Y, sim$fX, sim$sX, sim$pX3, sim$idvar, sim$timevar)
    WRDG.GMM.sim4 <- prodestWRDG_GMM(sim$Y, sim$fX, sim$sX, sim$pX4, sim$idvar, sim$timevar)

    # show results in .tex tabular format
    printProd(list(WRDG.GMM.sim1, WRDG.GMM.sim2, WRDG.GMM.sim3, WRDG.GMM.sim4), parnames = c('Free','State'))