setwd("C:/Users/StatLab1PC5/Documents/Jasmine")
library(tidyverse)
library(ggplot2)cbd<- read.csv("Cobb_Douglas.csv")Is the Cobb-Douglas production function (1) linear in the parameters?
Explain your answer by writing the appropriate “linearized” version of
the model.
Yes, the production function is linear in parameter.
\[Y_i=\beta_0X_{1i}^{\beta_{1}}X_{2i}^{\beta_{2}}e^{u_i}\] \[log(Y_i)=\beta_0+log(X_{1i}^{\beta_{1}})+log(X_{2i}^{\beta_{2}})+u_i\] \[log(Y_i)=\beta_0+\beta_{1}log(X_{1i})+\beta_{2}log(X_{2i})+u_i\]
Fit the “linearized” model you have identified in No. 1 to the Cobb-Douglas data. Indicate the summary of the regression results.
mod<- lm(log(Output) ~ log(lab)+log(cap), data = cbd)
summary(mod)##
## Call:
## lm(formula = log(Output) ~ log(lab) + log(cap), data = cbd)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.45645 -0.12112 -0.05319 0.04518 1.21579
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.88760 0.39623 9.812 4.70e-13 ***
## log(lab) 0.46833 0.09893 4.734 1.98e-05 ***
## log(cap) 0.52128 0.09689 5.380 2.18e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2668 on 48 degrees of freedom
## Multiple R-squared: 0.9642, Adjusted R-squared: 0.9627
## F-statistic: 645.9 on 2 and 48 DF, p-value: < 2.2e-16Write the sample regression equation.
\[\hat{log(output)}=3.888+0.468(log(labor))+0.521(log(capital))\]
Is it logical to interpret the y-intercept estimate? Why and why not?
It is not logical to interpret the y-intercept as there is no production output when there is no labor and capital at all.
Interpret the estimates of the regression slope parameters.
\(\beta_1:\)
For an additional percent of labor input, then it is predicted to increase in percentage on production output by 0.468%, holding the effect of capital input constant.
\(\beta_2:\)
For a percent increase in capital input, then it is predicted to increase in percentage on production output by 0.521%, holding the effect of labor input constant.
Does the model provide a good fit to the data? Justify your answer.
Since the RMSE is close to zero with 0.267, this means that the model
does fit well to the data. For the coefficient of determination of
0.964, this means that 96.4% of the sample variation in log (output) is
explained by log(labor) and log(capital).
With RMSE is close to zero and high value of \(R^2\) close to 100%, it implies that the
model does provide a good fit to the data.
What are the estimated variances of the estimates of the regression slope parameters?
\[se(\hat{\beta_1)}= 0.09893 \] \[\hat{\sigma}^2_1=0.0098\] \[se(\hat{\beta_2)}= 0.09689 \] \[\hat{\sigma}^2_2=0.0094\]