prodest (version 1.0.1)

prodestWRDG: Estimate productivity - IV 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 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(Y, fX, sX, pX, idvar, timevar, cX = 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.

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.

• 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: ${\omega }_{it}=y_{it}-(\alpha +w_{it}\beta +k_{it}\gamma )$. 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.

Details

Consider a Cobb-Douglas production technology for firm $i$ at time $t$

• $y_{it}=\alpha +w_{it}\beta +k_{it}\gamma +{\omega }_{it}+{ϵ}_{it}$

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 ${\omega }_{it}$ evolves according to a first-order Markov process:

• ${\omega }_{it}=E({\omega }_{it}|{\omega }_{it-1})+u_{it}=g({\omega }_{it-1})+u_{it}$

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) ${\omega }_{it}=g(x_{it},p_{it})$: productivity is an unknown function $g(.)$ of state and a proxy variables;

• b) $E({\omega }_{it}|{\omega }_{it-1)}=f[{\omega }_{it-1}]$, productivity is an unknown function $f[.]$ of lagged productivity, ${\omega }_{it-1}$.

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

• $r_{1it}=y_{it}-\alpha -w_{it}\beta -x_{it}\gamma -g(x_{it},p_{it})$

• $r_{2it}=y_{it}-\alpha -w_{it}\beta -x_{it}\gamma -f[g(x_{it-1},p_{it-1})]$

where the unknown function $f(.)$ is approximated by a n-th order polynomial and $g(x_{it},m_{it})={\lambda }_0+c(x_{it},m_{it})\lambda$. 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

• $E(Z_{it}\ast r_{it})=0$

with the following set of instruments:

• $Z{1}_{it}=(1,w_{it},x_{it},c_{it})$

• $Z{2}_{it}=(w_{it-1},c_{it},c_{it})$

Following previous assumptions, being $f(\omega )={\delta }_0+{\delta }_1(c_{it}\lambda )+{\delta }_2(c_{it}\lambda {)}^2+...+{\delta }_G(c_{it}\lambda {)}^G$, one can set ${\delta }_1=G=1$ and estimate the model in a linear fashion - i.e., using a linear 2SLS model.

References

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

Examples

# NOT RUN {
    data("chilean")

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

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

    # show results
    WRDG.IV.fit

    
# }
# NOT RUN {
      # estimate a panel dataset - DGP1, various measurement errors - and run the estimation
      sim <- panelSim()

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

      # show results in .tex tabular format
      printProd(list(WRDG.IV.sim1, WRDG.IV.sim2, WRDG.IV.sim3, WRDG.IV.sim4),
                parnames = c('Free','State'))
    
# }
# NOT RUN {
  
# }