prodestOP: Estimate productivity - Olley-Pakes method
In prodest: Production Function Estimation
Description
Usage
Arguments
Details
Value
Author(s)
References
Examples
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
1
2
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.
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) noise.
cluster
an object of class "SOCKcluster" or "cluster". By default cluster = NULL.
tol
optimizer tolerance. By default tol = 1e-100.
exit
Indicator for attrition in the data - i.e., if firms exit the market. By default exit = FALSE; if exit = TRUE, an indicator function for firms whose last appearance is before the last observation's date is generated and used in the second stage. The user can even specify an indicator variable/matrix/dataframe with the exit years.
Details
Consider a Cobb-Douglas production technology for firm i at time t
-
y_{it} = α + w_{it}β + k_{it}γ + ω_{it} + ε_{it}
where y_{it} is the (log) output, w_it a 1xJ vector of (log) free variables, k_it is a 1xK vector of state variables and ε_{it} is a normally distributed idiosyncratic error term.
The unobserved technical efficiency parameter ω_{it} evolves according to a first-order Markov process:
-
ω_{it} = E(ω_{it} | ω_{it-1}) + u_{it} = g(ω_{it-1}) + u_{it}
and u_{it} is a random shock component assumed to be uncorrelated with the technical efficiency, the state variables in k_{it} and the lagged free variables w_{it-1}.
The OP method relies on the following set of assumptions:
a) i_{it} = i(k_{it},ω_{it}) - investments are a function of both the state variable and the technical efficiency parameter;
b) i_{it} is strictly monotone in ω_{it};
c) ω_{it} is scalar unobservable in i_{it} = i(.) ;
d) the levels of i_{it} and k_{it} are decided at time t-1; the level of the free variable, w_{it}, is decided after the shock u_{it} realizes.
Assumptions a)-d) ensure the invertibility of i_{it} in ω_{it} and lead to the partially identified model:
-
y_{it} = α + w_{it}β + k_{it}γ + h(i_{it}, k_{it}) + ε_{it} = α + w_{it}β + φ(i_{it}, k_{it}) + ε_{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.
Exploiting the residual e_{it} of:
-
y_{it} - w_{it}\hat{β} = α + k_{it}γ + g(ω_{it-1}, χ_{it}) + ε_{it}
and g(.) is typically left unspecified and approximated by a n^{th} order polynomial and χ_{it} is an indicator 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'.
-
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: ω_{it} = y_{it} - (α + w_{it}β + k_{it}γ).
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
1
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14
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17
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30
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(chilean$Y, fX = cbind(chilean$fX1, chilean$fX2), chilean$sX,
chilean$inv, chilean$idvar, chilean$timevar)
OP.fit.solnp <- prodestOP(chilean$Y, fX = cbind(chilean$fX1, chilean$fX2),
chilean$sX, chilean$inv, chilean$idvar,
chilean$timevar, opt='solnp')
OP.fit.control <- prodestOP(chilean$Y, fX = cbind(chilean$fX1, chilean$fX2),
chilean$sX, chilean$inv, chilean$idvar,
chilean$timevar, cX = chilean$cX)
OP.fit.attrition <- prodestOP(chilean$Y, fX = cbind(chilean$fX1, chilean$fX2),
chilean$sX, chilean$inv, chilean$idvar,
chilean$timevar, exit = TRUE)
# 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, OP.fit.attrition))
Example output
Loading required package: dplyr
Attaching package: 'dplyr'
The following objects are masked from 'package:stats':
filter, lag
The following objects are masked from 'package:base':
intersect, setdiff, setequal, union
Loading required package: parallel
Loading required package: Matrix
-------------------------------------------------------------
- Production Function Estimation -
-------------------------------------------------------------
Method : OP
-------------------------------------------------------------
fX1 fX2 sX1
Estimated Parameters : 0.314 0.256 0.168
(0.025) (0.017) (0.039)
-------------------------------------------------------------
N : 2544
-------------------------------------------------------------
Bootstrap repetitions : 20
1st Stage Parameters : 0.314 0.256 -0.95
Optimizer : optim
-------------------------------------------------------------
Elapsed Time : 0.02 mins
-------------------------------------------------------------
-------------------------------------------------------------
- Production Function Estimation -
-------------------------------------------------------------
Method : OP
-------------------------------------------------------------
fX1 fX2 sX1
Estimated Parameters : 0.314 0.256 0.168
(0.04) (0.032) (0.032)
-------------------------------------------------------------
N : 2544
-------------------------------------------------------------
Bootstrap repetitions : 20
1st Stage Parameters : 0.314 0.256 -0.95
Optimizer : solnp
-------------------------------------------------------------
Elapsed Time : 0.03 mins
-------------------------------------------------------------
-------------------------------------------------------------
- Production Function Estimation -
-------------------------------------------------------------
Method : OP
-------------------------------------------------------------
fX1 fX2 sX1 cX1
Estimated Parameters : 0.314 0.256 0.168 0.311
(0.04) (0.032) (0.028) (0.284)
-------------------------------------------------------------
N : 2544
-------------------------------------------------------------
Bootstrap repetitions : 20
1st Stage Parameters : 0.314 0.256 0.311 -0.95
Optimizer : optim
-------------------------------------------------------------
Elapsed Time : 0.01 mins
-------------------------------------------------------------\begin{tabular}{ccccccccc}\hline\hline
& & OP & & OP & & OP & & OP \\\hline
fX1 & & 0.314 & & 0.314 & & 0.314 & & 0.314 \\
& & (0.025) & & (0.04) & & (0.04) & & (0.036) \\
& & & & \\
fX2 & & 0.256 & & 0.256 & & 0.256 & & 0.256 \\
& & (0.017) & & (0.032) & & (0.032) & & (0.026) \\
& & & & \\
sX1 & & 0.168 & & 0.168 & & 0.168 & & 0.202 \\
& & (0.039) & & (0.032) & & (0.028) & & (0.038) \\
& & & & \\
& & & & \\
N & & 2544 & & 2544 & & 2544 & & 2544 \\\hline\hline
\end{tabular}
prodest documentation built on May 2, 2019, 8:34 a.m.
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Description Usage Arguments Details Value Author(s) References Examples
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
1 2 |
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 |
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 |
opt |
a string with the optimization algorithm to be used during the estimation. By default |
theta0 |
a vector with the second stage optimization starting points. By default |
cluster |
an object of class |
tol |
optimizer tolerance. By default |
exit |
Indicator for attrition in the data - i.e., if firms exit the market. By default |
Details
Consider a Cobb-Douglas production technology for firm i at time t
-
y_{it} = α + w_{it}β + k_{it}γ + ω_{it} + ε_{it}
where y_{it} is the (log) output, w_it a 1xJ vector of (log) free variables, k_it is a 1xK vector of state variables and ε_{it} is a normally distributed idiosyncratic error term. The unobserved technical efficiency parameter ω_{it} evolves according to a first-order Markov process:
-
ω_{it} = E(ω_{it} | ω_{it-1}) + u_{it} = g(ω_{it-1}) + u_{it}
and u_{it} is a random shock component assumed to be uncorrelated with the technical efficiency, the state variables in k_{it} and the lagged free variables w_{it-1}. The OP method relies on the following set of assumptions:
a) i_{it} = i(k_{it},ω_{it}) - investments are a function of both the state variable and the technical efficiency parameter;
b) i_{it} is strictly monotone in ω_{it};
c) ω_{it} is scalar unobservable in i_{it} = i(.) ;
d) the levels of i_{it} and k_{it} are decided at time t-1; the level of the free variable, w_{it}, is decided after the shock u_{it} realizes.
Assumptions a)-d) ensure the invertibility of i_{it} in ω_{it} and lead to the partially identified model:
-
y_{it} = α + w_{it}β + k_{it}γ + h(i_{it}, k_{it}) + ε_{it} = α + w_{it}β + φ(i_{it}, k_{it}) + ε_{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. Exploiting the residual e_{it} of:
-
y_{it} - w_{it}\hat{β} = α + k_{it}γ + g(ω_{it-1}, χ_{it}) + ε_{it}
and g(.) is typically left unspecified and approximated by a n^{th} order polynomial and χ_{it} is an indicator 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'. -
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: ω_{it} = y_{it} - (α + w_{it}β + k_{it}γ).
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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | 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(chilean$Y, fX = cbind(chilean$fX1, chilean$fX2), chilean$sX,
chilean$inv, chilean$idvar, chilean$timevar)
OP.fit.solnp <- prodestOP(chilean$Y, fX = cbind(chilean$fX1, chilean$fX2),
chilean$sX, chilean$inv, chilean$idvar,
chilean$timevar, opt='solnp')
OP.fit.control <- prodestOP(chilean$Y, fX = cbind(chilean$fX1, chilean$fX2),
chilean$sX, chilean$inv, chilean$idvar,
chilean$timevar, cX = chilean$cX)
OP.fit.attrition <- prodestOP(chilean$Y, fX = cbind(chilean$fX1, chilean$fX2),
chilean$sX, chilean$inv, chilean$idvar,
chilean$timevar, exit = TRUE)
# 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, OP.fit.attrition))
|
Example output
Loading required package: dplyr
Attaching package: 'dplyr'
The following objects are masked from 'package:stats':
filter, lag
The following objects are masked from 'package:base':
intersect, setdiff, setequal, union
Loading required package: parallel
Loading required package: Matrix
-------------------------------------------------------------
- Production Function Estimation -
-------------------------------------------------------------
Method : OP
-------------------------------------------------------------
fX1 fX2 sX1
Estimated Parameters : 0.314 0.256 0.168
(0.025) (0.017) (0.039)
-------------------------------------------------------------
N : 2544
-------------------------------------------------------------
Bootstrap repetitions : 20
1st Stage Parameters : 0.314 0.256 -0.95
Optimizer : optim
-------------------------------------------------------------
Elapsed Time : 0.02 mins
-------------------------------------------------------------
-------------------------------------------------------------
- Production Function Estimation -
-------------------------------------------------------------
Method : OP
-------------------------------------------------------------
fX1 fX2 sX1
Estimated Parameters : 0.314 0.256 0.168
(0.04) (0.032) (0.032)
-------------------------------------------------------------
N : 2544
-------------------------------------------------------------
Bootstrap repetitions : 20
1st Stage Parameters : 0.314 0.256 -0.95
Optimizer : solnp
-------------------------------------------------------------
Elapsed Time : 0.03 mins
-------------------------------------------------------------
-------------------------------------------------------------
- Production Function Estimation -
-------------------------------------------------------------
Method : OP
-------------------------------------------------------------
fX1 fX2 sX1 cX1
Estimated Parameters : 0.314 0.256 0.168 0.311
(0.04) (0.032) (0.028) (0.284)
-------------------------------------------------------------
N : 2544
-------------------------------------------------------------
Bootstrap repetitions : 20
1st Stage Parameters : 0.314 0.256 0.311 -0.95
Optimizer : optim
-------------------------------------------------------------
Elapsed Time : 0.01 mins
-------------------------------------------------------------\begin{tabular}{ccccccccc}\hline\hline
& & OP & & OP & & OP & & OP \\\hline
fX1 & & 0.314 & & 0.314 & & 0.314 & & 0.314 \\
& & (0.025) & & (0.04) & & (0.04) & & (0.036) \\
& & & & \\
fX2 & & 0.256 & & 0.256 & & 0.256 & & 0.256 \\
& & (0.017) & & (0.032) & & (0.032) & & (0.026) \\
& & & & \\
sX1 & & 0.168 & & 0.168 & & 0.168 & & 0.202 \\
& & (0.039) & & (0.032) & & (0.028) & & (0.038) \\
& & & & \\
& & & & \\
N & & 2544 & & 2544 & & 2544 & & 2544 \\\hline\hline
\end{tabular}
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