prodest (version 1.0.2)

prodestOP: 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, cX = NULL,
            opt = 'optim', theta0 = NULL, cluster = NULL, tol = 1e-100, exit = FALSE)

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: \(\omega_{it} = y_{it} - (\alpha + w_{it}\beta + k_{it}\gamma)\).

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.

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

a one-column vector/matrix/dataframe identifying individual panels; numeric, character and factor labels are accepted.

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 + \(N(\mu=0,\sigma=0.01)\) 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.

Author

Gabriele Rovigatti

Details

Consider a Cobb-Douglas production technology for firm \(i\) at time \(t\)

• \(y_{it} = \alpha + w_{it}\beta + k_{it}\gamma + \omega_{it} + \epsilon_{it}\)

where \(y_{it}\) is the (log) output, \(w_{it}\) is a \(1 \times J\) vector of (log) free variables, \(k_{it}\) is a \(1 \times K\) vector of state variables, and \(\epsilon_{it}\) 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 \(w_{it-1}\). The OP method relies on the following set of assumptions:

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

• b) \(i_{it}\) is strictly monotone in \(\omega_{it}\);

• c) \(\omega_{it}\) 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 \(\omega_{it}\) and lead to the partially identified model:

• \(y_{it} = \alpha + w_{it}\beta + k_{it}\gamma + h(i_{it}, k_{it}) + \epsilon_{it} = \alpha + w_{it}\beta + \phi(i_{it}, k_{it}) + \epsilon_{it} \)

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. Exploiting the residual \(e_{it}\) of:

• \(y_{it} - w_{it}\hat{\beta} = \alpha + k_{it}\gamma + g(\omega_{it-1}, \chi_{it}) + \epsilon_{it} \)

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

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)

    ## Source: Instituto Nacional de Estadisticas de Chile (ENIA).

    data(chilean)

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

    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.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)

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