Microeconomic Theory and Linear Regression (Part 2)

Introduction

In the first part of this article I explained and compared four different functional forms to model apples' production.

This is what I need to retake the previously explained examples:

# Libraries

#install.packages(c("micEcon","lmtest","bbmle","miscTools"))
library(micEcon)
library(lmtest)
library(stats4) #this is a base package so I don't install this
library(bbmle)
library(miscTools)

# Load data

data("appleProdFr86", package = "micEcon")
data = appleProdFr86
rm(appleProdFr86)

data$qCap = data$vCap/data$pCap
data$qLab = data$vLab/data$pLab
data$qMat = data$vMat/data$pMat

# Fit models

## Linear specification
prodLin = lm(qOut ~ qCap + qLab + qMat, data = data)
data$qOutLin = fitted(prodLin)

## Quadratic specification
prodQuad = lm(qOut ~ qCap + qLab + qMat + I(0.5*qCap^2) + I(0.5*qLab^2) + I(0.5*qMat^2)
               + I(qCap*qLab) + I(qCap*qMat) + I(qLab*qMat), data = data)
data$qOutQuad = fitted(prodQuad)

## Cobb-Douglas specification
prodCD = lm(log(qOut) ~ log(qCap) + log(qLab) + log(qMat), data = data)
data$qOutCD = fitted(prodCD)

## Translog specification
prodTL = lm(log(qOut) ~ log(qCap) + log(qLab) + log(qMat) + I(0.5*log(qCap)^2) 
             + I(0.5*log(qLab)^2) + I(0.5*log(qMat)^2) + I(log(qCap)*log(qLab))
             + I(log(qCap)*log(qMat)) + I(log(qLab)*log(qMat)), data = data)
data$qOutTL = fitted(prodTL)

These four specifications presented different problems that suggested model misspecification. Ideally, I should find a model with positive predicted output and positive factor elasticity to make sense of the context of production I'm working with.

Model misspectification

RESET test (the name stands for Regression Equation Specification Error Test) is a test for functional form misspecification.

As a general test, RESET helps detecting ommited variables. Package lmtest provides resettest, a function that provides one of the many versions of the test, and it fits this model by default:

$$y_i = \beta_0 + \beta_1 x_1 + \ldots + \beta_p x_p + \gamma_1 \hat{y}_i^2 + \gamma_2 \hat{y}_i^3 + e_i$$

The key idea behind fitting another model that includes the output is to detect if the model ignores an important nonlinearity.

The hypothesis testing is:

$$H_0:\: \gamma_i = 0 \text{ for any } i \\ H_1:\: \gamma_i \neq 0 \text{ for some } i$$

If there is evidence to reject the null hypthosis, the test does not provide an alternative model and it just states that the model is misspecified.

The statistic will be a t statistic if we just include \(\gamma_1 \hat{y}_i^2\), and it will change to a F statistic if we include \(\gamma_2 \hat{y}_i^3\) or a higher power.

In the context of the worked specifications:

resettest(prodLin)
    RESET test

data:  prodLin
RESET = 17.639, df1 = 2, df2 = 134, p-value = 1.584e-07
resettest(prodQuad)
    RESET test

data:  prodQuad
RESET = 7.3663, df1 = 2, df2 = 128, p-value = 0.0009374
resettest(prodCD)
    RESET test

data:  prodCD
RESET = 2.9224, df1 = 2, df2 = 134, p-value = 0.05724
resettest(prodTL)
    RESET test

data:  prodTL
RESET = 1.2811, df1 = 2, df2 = 128, p-value = 0.2813

\(\Longrightarrow\) The null hypothesis is rejected both for Linear and Quadratic specifications while Cobb-Douglas specification is rejected at 10% of significance. The null hypothesis can't be rejected for the Translog Specification.

I can also modify the powers in the model:

resettest(prodTL, power = 2)
    RESET test

data:  prodTL
RESET = 2.358, df1 = 1, df2 = 129, p-value = 0.1271
resettest(prodTL, power = 2:4)
    RESET test

data:  prodTL
RESET = 1.3475, df1 = 3, df2 = 127, p-value = 0.262

\(\Longrightarrow\) The null hypothesis can't be rejected for Translog Specification, not even by adding a higher power.

Quasiconcavity

A function

$$ \begin{align*} f: \mathbb{R}^n &\to \mathbb{R} \cr x &\mapsto f(x) \end{align*} $$

is quasiconcave if for \(0 \leq t \leq 1\)

$$f(tx_1 + (1-t)x_2) \geq \min\{f(x_1),f(x_2)\}$$

In particular, any concave and twice continuously differentiable function is quasiconcave.

In really raw terms, quasiconcavity guarantees there is a production plan where the company is minimizing cost. Adding differentiability, you can find an optimal production plan by using derivatives.

For the details please check Advanced Microeconomic Theory by Geoffrey A. Jehle.

I'll check the quasiconcavity in an indirect way, by checking the concavity at each observation using the hessian method:

# Quadratic Specification

## Regression coefficients
b1  = coef(prodQuad)["qCap"]
b2  = coef(prodQuad)["qLab"]
b3  = coef(prodQuad)["qMat"]
b11 = coef(prodQuad)["I(0.5 * qCap^2)"]
b22 = coef(prodQuad)["I(0.5 * qLab^2)"]
b33 = coef(prodQuad)["I(0.5 * qMat^2)"]
b12 = b21 = coef(prodQuad)["I(qCap * qLab)"]
b13 = b31 = coef(prodQuad)["I(qCap * qMat)"]
b23 = b32 = coef(prodQuad)["I(qLab * qMat)"]

## Marginal productivity
data$mpCapQuad = with(data, b1 + b11*qCap + b12*qLab + b13*qMat)
data$mpLabQuad = with(data, b2 + b21*qCap + b22*qLab + b23*qMat)
data$mpMatQuad = with(data, b3 + b31*qCap + b32*qLab + b33*qMat)

## Quasiconcavity
data$quasiConcQuad = NA

for(obs in 1:nrow(data)) {
  hmLoop = matrix(0, nrow = 3, ncol = 3)
  hmLoop[1,1] = b11
  hmLoop[1,2] = hmLoop[2,1] = b12
  hmLoop[1,3] = hmLoop[3,1] = b13
  hmLoop[2,2] = b22
  hmLoop[2,3] = hmLoop[3,2] = b23
  hmLoop[3,3] = b33

  data$quasiConcQuad[obs] = hmLoop[1,1] < 0 & det(hmLoop[1:2,1:2] > 0) & det(hmLoop < 0)
}

sum(!data$quasiConcQuad)
[1] 140
# Translog Specification

## Regression coefficients
b1  = coef(prodTL)["log(qCap)"]
b2  = coef(prodTL)["log(qLab)"]
b3  = coef(prodTL)["log(qMat)"]
b11 = coef(prodTL)["I(0.5 * log(qCap)^2)"]
b22 = coef(prodTL)["I(0.5 * log(qLab)^2)"]
b33 = coef(prodTL)["I(0.5 * log(qMat)^2)"]
b12 = b21 = coef(prodTL)["I(log(qCap) * log(qLab))"]
b13 = b31 = coef(prodTL)["I(log(qCap) * log(qMat))"]
b23 = b32 = coef(prodTL)["I(log(qLab) * log(qMat))"] 

## Scale elasticity
data$eCapTL = with(data, b1 + b11*log(qCap) + b12*log(qLab) + b13*log(qMat))
data$eLabTL = with(data, b2 + b21*log(qCap) + b22*log(qLab) + b23*log(qMat))
data$eMatTL = with(data, b3 + b31*log(qCap) + b32*log(qLab) + b33*log(qMat))

## Marginal productivity
data$mpCapTL = with(data, eCapTL * qOutTL / qCap)
data$mpLabTL = with(data, eLabTL * qOutTL / qLab)
data$mpMatTL = with(data, eMatTL * qOutTL / qMat)

## Derivatives of marginal productivity
data$dmpCapCapTL = with(data, (qOutTL / qCap^2) * (b11 + eCapTL^2 - eCapTL))
data$dmpLabLabTL = with(data, (qOutTL / qLab^2) * (b22 + eLabTL^2 - eLabTL))
data$dmpMatMatTL = with(data, (qOutTL / qLab^2) * (b33 + eMatTL^2 - eMatTL))

data$dmpCapLabTL = with(data, (qOutTL / (qCap * qLab)) * (b12 + eCapTL * eLabTL))
data$dmpCapMatTL = with(data, (qOutTL / (qCap * qMat)) * (b13 + eCapTL * eMatTL))

data$dmpLabMatTL = with(data, (qOutTL / qLab * qMat) * (b23 + eLabTL * qMat))

## Quasiconcavity
data$quasiConcTL = NA

for(obs in 1:nrow(data)) {
  hmLoop = matrix(0, nrow = 3, ncol = 3)
  hmLoop[1,1] = data$dmpCapCapTL[obs]
  hmLoop[1,2] = hmLoop[2,1] = data$dmpCapLabTL[obs]
  hmLoop[1,3] = hmLoop[3,1] = data$dmpCapMatTL[obs]
  hmLoop[2,2] = data$dmpLabLabTL[obs]
  hmLoop[2,3] = hmLoop[3,2] = data$dmpLabMatTL[obs]
  hmLoop[3,3] = data$dmpMatMatTL[obs]

  data$quasiConcTL[obs] = hmLoop[1,1] < 0 & det(hmLoop[1:2,1:2] > 0) & det(hmLoop < 0)
}

sum(!data$quasiConcTL)
[1] 126

With this information I can extend the information from the last table in the first part of the post:

Quadratic Translog
\(R^2\) on \(y\) 0.85 0.77
\(R^2\) on \(ln(y)\) 0.55 0.63
Obs. with negative predicted output 0 0
Obs. that violate the monotonicity condition 39 48
Obs. with negative scale elasticity 22 0
Obs. that violate the quasiconcavity condition 140 126

\(\Longrightarrow\) Now I have more evidence in favour of a different approach as it was mentioned at the end of the first part of the post.

Non-parametric regression

A non-parametric model is of the form:

$$Y_i = m(X_i) + \varepsilon_i$$

The advantages are: The functional form is not defined beforehand Obtains a best fit function \(m\) from the data * Obtains marginal effects

The disadvantages are: Higher computational cost There are no regression coefficients (the focus is the regression function) * Hard to interpret

Single variable non-parametric regression

I can predict the output by using capital as the only input:

library(KernSmooth)

qCap = as.vector(data$qCap)
qOut = as.vector(data$qOut)

bw = dpill(qCap,qOut)
fit1 = locpoly(qCap, qOut, bandwidth = bw, degree = 1) # 1st degree polynomial
plot(fit1$x, fit1$y, type = "l")

plot of chunk microeconomics_6

Here are different specifications:

# 2nd degree polynomial
fit2.1 <- locpoly(qCap, qOut, bandwidth = bw, degree = 2)
fit2.2 <- locpoly(qCap, qOut, bandwidth = bw*2, degree = 2) # less variance / more bias
fit2.3 <- locpoly(qCap, qOut, bandwidth = bw/2, degree = 2) # more variance / less bias

# linear regression
fit.lm <- lm(qOut ~ qCap)

# plots
plot(fit1$x, fit1$y, type = "l");par(mfrow=c(2,2))
plot(fit1$x, fit1$y, type = "l");lines(fit2.1, col = "blue")
plot(fit1$x, fit1$y, type = "l");lines(fit2.2, col = "red")
plot(fit1$x, fit1$y, type = "l");lines(fit2.3, col = "green")
plot(fit1$x, fit1$y, type = "l");abline(fit.lm, col = "purple")

plot of chunk microeconomics_7

Checking the models:

# model 1
sum(is.na(fit1$y))
[1] 50
# models 2, 3 y 4
sum(is.na(fit2.1$y))
[1] 50
sum(is.na(fit2.2$y))
[1] 9
sum(is.na(fit2.3$y))
[1] 84

\(\Longrightarrow\) I should use model 2.2 and adjust it.

I'll try with a neoclassical function with diminishing marginal returns:

# se ajusta la banda y el grado
fit.sqrt = locpoly(qCap, qOut, bandwidth = bw*3, degree = 0.5)
sum(is.na(fit.sqrt$y)) 
[1] 0
sum(fit.sqrt$y < 0)
[1] 0

\(\Longrightarrow\) In this model there is no indetermined or negative predicted output.

plot(fit.sqrt$x, fit.sqrt$y, type = "l")

plot of chunk microeconomics_10

I can obtain the median productivity:

median(fit.sqrt$y)/median(fit.sqrt$x)
[1] 18.05637

\(\Longrightarrow\) In median term each capital unit increases the output in 18 units.

If I adjust the bandwidth the function will look more and more like a neoclassical function from textbooks, with an initial part of increasing marginal returns and from the point where marginal and median productivities are equal (the same point where marginal productivity is at maximum) the return becomes increasing at decreasing scale.

This is an example of the last paragraph:

fit.sqrt.2 = locpoly(qCap, qOut, bandwidth = bw*5, degree = 0.5)
plot(fit.sqrt.2$x, fit.sqrt.2$y, type = "l")

plot of chunk microeconomics_12

Multivariable non-parametric regression

Among many alternatives, Epanechnikov's method is a non-parametric method that weights the contribution of each observation and iterating obtains the best fit function.

I'll fit a regression applying logs to the variables:

library(np)
options(np.messages=FALSE)

prodNP = npreg(log(qOut) ~ log(qCap) + log(qLab) + log(qMat), regtype = "ll", 
                bwmethod = "cv.aic", ckertype = "epanechnikov",  data = data, 
                gradients = TRUE)
summary(prodNP)
Regression Data: 140 training points, in 3 variable(s)
              log(qCap) log(qLab) log(qMat)
Bandwidth(s):  1.039647    332644 0.8418465

Kernel Regression Estimator: Local-Linear
Bandwidth Type: Fixed
Residual standard error: 0.6227669
R-squared: 0.6237078

Continuous Kernel Type: Second-Order Epanechnikov
No. Continuous Explanatory Vars.: 3

These are the regression plots: plot of chunk microeconomics_14 the variables that do not appear in a plot are considered to be fixed to a value equal to their median.

I can test the significance for this model:

options(np.messages=FALSE)
npsigtest(prodNP)
Kernel Regression Significance Test
Type I Test with IID Bootstrap (399 replications, Pivot = TRUE, joint = FALSE)
Explanatory variables tested for significance:
log(qCap) (1), log(qLab) (2), log(qMat) (3)

              log(qCap) log(qLab) log(qMat)
Bandwidth(s):  1.039647    332644 0.8418465

Individual Significance Tests
P Value: 
log(qCap) 0.11779  
log(qLab) < 2e-16 *** 
log(qMat) < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

\(\Longrightarrow\) at 10% of significance the only variable with non-statistical significant effects is capital.

Finally I can obtain information about Product-Factor Elasticity and Scale Elasticity:

summary(gradients(prodNP)[,1]) # capital elasticity
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
-0.10300  0.08598  0.14360  0.14500  0.18810  0.30060 
summary(gradients(prodNP)[,2]) # labour elasticity
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
0.02208 0.63050 0.69880 0.70970 0.80670 0.97890 
summary(gradients(prodNP)[,3]) # materials elasticity
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.3873  0.4546  0.5669  0.5850  0.6502  1.3480 
summary(rowSums(gradients(prodNP))) # scale elasticity
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  1.267   1.363   1.416   1.440   1.479   1.880 
par(mfrow=c(2,2))
hist(gradients(prodNP)[,1], main = "Capital")
hist(gradients(prodNP)[,2], main = "Labour")
hist(gradients(prodNP)[,3], main = "Materials")
hist(rowSums(gradients(prodNP)), main = "Scale")

plot of chunk microeconomics_16