prodestWRDG_GMM: Estimate productivity - Wooldridge method
In prodest: Production Function Estimation

Description Usage Arguments Details Value Author(s) References Examples

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.

1	prodestWRDG_GMM(Y, fX, sX, pX, idvar, timevar, cX = NULL, tol = 1e-100)

`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.

tol

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

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}. 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) ω_{it} = g(x_{it} , p_{it}): productivity is an unknown function g(.) of state and a proxy variables;
b) E(ω_{it} | ω_{it-1)}=f[ω_{it-1}], productivity is an unknown function f[.] of lagged productivity, ω_{it-1}.

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

r_{1it} = y_{it} - α - w_{it}β - x_{it}γ - g(x_{it} , p_{it})
r_{2it} = y_{it} - α - w_{it}β - x_{it}γ - f[g(x_{it-1} , p_{it-1})]

where the unknown function f(.) is approximated by a n-th order polynomial and g(x_{it} , m_{it}) = λ_0 + c(x_{it} , m_{it})λ. In particular, g(x_{it} , m_{it}) is a linear combination of functions in (x_{it} , m_{it}) and c_{it} are the addends of this linear combination. The residuals r_{it} are used to set the moment conditions

E(Z_{it}*r_{it}) =0

with the following set of instruments:

Z1_{it} = (1, w_{it}, x_{it}, c_{it})
Z2_{it} = (w_{it-1}, c_{it}, c_{it})

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.
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: 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.

Gabriele Rovigatti

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

    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(chilean$Y, fX = cbind(chilean$fX1, chilean$fX2),
                              chilean$sX, chilean$pX, chilean$idvar, chilean$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'))