In lachlandeer/hayashir: R Companion to "Econometrics" by Fumio Hayashi

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

Application: Returns to Scale in Electricity Supply

Load the hayashir package

library(hayashir)

And other libraries we need for this chapter

library(skimr) # summary stats
library(dplyr) # data manipulation
library(systemfit) # for fitting system of regressions

The Data

Let's get a quick look at our data by looking at the first 10 rows:

head(greene, 10)

Table 4.3: Summary Statistics

# assign the greene data to the object 'df'
df <- greene

df %>%
    mutate(output = output / 1e3) %>%
    select(output, labor_share, capital_share, fuel_share) %>%
    skim()

Table 4.4: Regression parameters from 2 regression set up

Following the text, the two equations we want to estimate are:

First, specify the equations we want to estimate:

eq1 <- labor_share ~ log(price_labor / price_fuel) + log(price_capital / price_fuel) + log(output) 
eq2 <- capital_share ~ log(price_labor / price_fuel) + log(price_capital / price_fuel) + log(output)

As Hayashi tells us, the unique symmetry restriction is then

restrict <- c(
    "labor_log(price_capital/price_fuel) = capital_log(price_labor/price_fuel)"
)

We estimate with the systemfit function from the package with the same name:

sur_results <- systemfit(list(labor = eq1, capital = eq2), 
                         method = "SUR", 
                         restrict.matrix = restrict,
                         data= df)
summary(sur_results)

Which yields approximately the same coefficients as in the text.

Then we can calculate estimates of the remaining coefficients using adding up restrictions, homogeneity and symmetry. For example: $$ \hat{\gamma}{33} = \hat{\gamma}{11} + 2 \hat{\gamma}{12} + \hat{\gamma}{22} $$

So that

gamma_11 <- sur_results$coefficients["labor_log(price_labor/price_fuel)"]
gamma_12 <- sur_results$coefficients["labor_log(price_capital/price_fuel)"]
gamma_22 <- sur_results$coefficients["capital_log(price_capital/price_fuel)"]

gamma_33 <- gamma_11 + 2 * gamma_12 + gamma_22

print(paste("gamma_33 is: ", round(gamma_33,3)))

Getting the Covariance Matrix, $\Sigma$ py pooled OLS

Run the three regressions via pooled OLS. The equations are:

eq1a <- labor_share ~ log(price_labor) + log(price_capital) + log(price_fuel) + log(output) 
eq2a <- capital_share ~ log(price_labor) + log(price_capital) + log(price_fuel) + log(output)
eq3a <- fuel_share ~ log(price_labor) + log(price_capital) + log(price_fuel) + log(output)

Then estimate:

pooled_ols <- systemfit(list(labor = eq1a, capital = eq2a, fuel = eq3a), 
                         method = "OLS", 
                         data= df)
summary(pooled_ols)

And extract the residual covariance matrix:

pooled_ols$residCov

Table 4.5: Substitution Elasticities

The cross elasticity (from eq 4.7.9) is:

$$ \eta_{jk} = \frac{\gamma_{jk} + s_j s_k}{s_j s_k} $$

So we calculate these for each data point, and then find the mean in our data. Let's do this for the labor-capital elasticity, $\eta_{12}$:

# the coefficient we are after
gamma_12 <- sur_results$coefficients["labor_log(price_capital/price_fuel)"]

# the fitted values
fitted_vals <- fitted( sur_results)

elasticity <- fitted_vals %>%
    mutate(eta_12 = (gamma_12 + (labor * capital)) / (labor * capital) ) %>%
    summarise(eta_12 = mean(eta_12)) 

print(paste0("Capital-Labor elasticity is: ", round(elasticity$eta_12, 2)))