Transportation Equipment Manufacturing Data — Equipment • AER

Statewide data on transportation equipment manufacturing for 25 US states.

Usage

data("Equipment")

Format

A data frame containing 25 observations on 4 variables.

valueadded: Aggregate output, in millions of 1957 dollars.
capital: Capital input, in millions of 1957 dollars.
labor: Aggregate labor input, in millions of man hours.
firms: Number of firms.

Source

Journal of Applied Econometrics Data Archive.

http://qed.econ.queensu.ca/jae/1998-v13.2/zellner-ryu/

Online complements to Greene (2003), Table F9.2.

https://pages.stern.nyu.edu/~wgreene/Text/tables/tablelist5.htm

References

Greene, W.H. (2003). Econometric Analysis, 5th edition. Upper Saddle River, NJ: Prentice Hall.

Zellner, A. and Revankar, N. (1969). Generalized Production Functions. Review of Economic Studies, 36, 241–250.

Zellner, A. and Ryu, H. (1998). Alternative Functional Forms for Production, Cost and Returns to Scale Functions. Journal of Applied Econometrics, 13, 101–127.

See also

Examples

## Greene (2003), Example 17.5
data("Equipment")

## Cobb-Douglas
fm_cd <- lm(log(valueadded/firms) ~ log(capital/firms) + log(labor/firms), data = Equipment)

## generalized Cobb-Douglas with Zellner-Revankar trafo
GCobbDouglas <- function(theta)
 lm(I(log(valueadded/firms) + theta * valueadded/firms) ~ log(capital/firms) + log(labor/firms), 
     data = Equipment)

## yields classical Cobb-Douglas for theta = 0
fm_cd0 <- GCobbDouglas(0)

## ML estimation of generalized model
## choose starting values from classical model
par0 <- as.vector(c(coef(fm_cd0), 0, mean(residuals(fm_cd0)^2)))

## set up likelihood function
nlogL <- function(par) {
  beta <- par[1:3]
  theta <- par[4]
  sigma2 <- par[5]

  Y <- with(Equipment, valueadded/firms)
  K <- with(Equipment, capital/firms)
  L <- with(Equipment, labor/firms)

  rhs <- beta[1] + beta[2] * log(K) + beta[3] * log(L)
  lhs <- log(Y) + theta * Y

  rval <- sum(log(1 + theta * Y) - log(Y) +
    dnorm(lhs, mean = rhs, sd = sqrt(sigma2), log = TRUE))
  return(-rval)
}

## optimization
opt <- optim(par0, nlogL, hessian = TRUE)
#> Warning: NaNs produced
#> Warning: NaNs produced

## Table 17.2
opt$par
#> [1] 2.91468783 0.34997794 1.09232479 0.10666219 0.04274876
sqrt(diag(solve(opt$hessian)))[1:4]
#> [1] 0.36055500 0.09671393 0.14079052 0.05849648
-opt$value
#> [1] -8.939045

## re-fit ML model
fm_ml <- GCobbDouglas(opt$par[4])
deviance(fm_ml)
#> [1] 1.068552
sqrt(diag(vcov(fm_ml)))
#>        (Intercept) log(capital/firms)   log(labor/firms) 
#>         0.12533876         0.09435337         0.11497723 

## fit NLS model
rss <- function(theta) deviance(GCobbDouglas(theta))
optim(0, rss)
#> Warning: one-dimensional optimization by Nelder-Mead is unreliable:
#> use "Brent" or optimize() directly
#> $par
#> [1] -0.03164062
#> 
#> $value
#> [1] 0.765549
#> 
#> $counts
#> function gradient 
#>       26       NA 
#> 
#> $convergence
#> [1] 0
#> 
#> $message
#> NULL
#> 
opt2 <- optimize(rss, c(-1, 1))
fm_nls <- GCobbDouglas(opt2$minimum)
-nlogL(c(coef(fm_nls), opt2$minimum, mean(residuals(fm_nls)^2)))
#> [1] -13.62126