Introduction of Econometrics

Empirical Project: Semester One 2022

Due date: Tuesday, 14 June 2022, 11:55 pm.

Submission: PDF Electronic submission via Canvas.

Marks: Total of 40 marks contributing 40% towards the final grade.

Notes:

i. You must paste regression estimation outputs from R-studio under each question and R-code if explicitly asked.

ii. A brief review of production functions and an explanation of the variables used in Task 2 are provided in the Appendix. This is not an exhaustive review, so if you have questions related to that topic, please consult any textbook on Microeconomics.

Task:1 (18 marks)

This task tests your understanding of the data generating process in the context of regression models.

1. Consider a Cobb-Douglas production function with three logged inputs ??1, ??2 and ??3 that can be represented as:

?? = ??0 + ??1??1 + ??2??2 + ??3??3 + ??

To answer the questions that follow, you need to use the observations on three inputs in the file ????????????????????_????????. ???????? and create the synthetic dependent variable ??. You will need to select the values of ??1, ??2 and ??3 such that your production function follows either constant, increasing or decreasing returns to scale (refer to Appendix A1 for an explanation of returns to scale). You also need to simulate residual term ??.

a. Select the values of ??0, ??1, ??2, ??3 and residual term ?? (column name this term ??????1)from a relatively tight normal distribution, say ??(0,0.07), to create the dependent variable ?? (column name this term ????????????1). Paste the corresponding R-code. (2 marks)

b. Now using ?? as a dependent variable and ??1, ??2 and ??3 as independent variables, run the regression. Paste the regression output and R-code below. (2 marks)

c. Has your regression model from (b.) successfully recovered the returns to scale coefficients you initially chose? Comment on the statistical significance of each input variable. (2 marks)

d. Use the same values of ??0, ??1, ??2 and ??3 but this time, draw residual term ?? (column name this term ??????2) from a relatively wide normal distribution, say ??(0,0.25), to create the dependent variable ?? (column name this term ????????????2). Run the regression and paste the output and R-code below. (3 marks)

e. Do you notice any difference in the values of coefficients and their statistical significance between the regression output in (b.) and (d.)? Explain in a sentence or two the reasoning behind such difference. (3 marks)

f. Now use the same values of ??0, ??1, ??2 and ??3 but make the residual3 term heteroscedastic and create the dependent variable ?? (column name this term ????????????3). Run the regression and paste the regression output and R-code below. (4 marks)

g. Do a Breusch Pagan Test of heteroskedasticity for the regression model estimated in (f.). Paste the test result. (1 mark)

h. Plot the residual against the fitted value of the regression from (f.) to test for heteroskedasticity. Make sure you paste R-code for the plot. (1 mark)

Task:2 (22 marks)

This task tests your ability to utilize real-world data to run and interpret regression models.

a. FARMS.xlsx contains data on the logged input and outputs of 558 Chinese Farms (refer to Appendix A2 for an explanation of inputs and output).

Use this data, and estimate a Cobb-Douglas production function of the following form:

???????????????????? = ??0 + ??1???????????? + ??1??????h?????????? + ??2???????????????????? + ??3 ??????????????????????

Paste the R code and regression output—comment on whether the production function follows an increasing, decreasing or constant return to scale. (4 marks)

b. Plot the residual against the fitted value of the regression model estimated in (a.)— comment on whether residuals are homoscedastic or not. Make sure you paste Rcode for the plot. (4 marks)

c. Now conduct the Breusch Pagan Test of heteroskedasticity. Make sure you start by explicitly specifying the null and alternative hypotheses for the test. Paste the result and comment on the result (use the chi-square value test at a 5% significance level, not the p-value test). State clearly why you reject or fail to reject the null hypothesis. Make sure you paste R-code and output for the Breusch Pagan Test. (7 marks)

d. Extract the residuals of the regression model estimated in (a.) and name it ??????????????. Run the following regression

???????????????????? = ??0 + ??1??????????????

What does the coefficient of ?????????????? and its statistical significance/insignificance implies? Paste the R-code and regression output below. (7 marks)

Appendix

Appendix A1: Production Function Overview

A production function represents the technological relationship between the quantities of inputs (e.g.capital, labour, and materials) and the quantity of output that a firm can produce. One traditional formulation is the so-called Cobb-Douglas production function:

?? = ?? ????1????2 ????3,

where ?? represents a firm's production output, ?? the firm's capital, ?? the firm's labour, and ?? the amount of materials input used in the production by the firm. The parameter ??1, ??2 and ??3 measures the output elasticities of capital, labour, and materials, respectively.

For econometric estimation, the Cobb-Douglas production function can be estimated using OLS by log-linearizing the model and adding an error term:

????(??) = ????(??) + ??1????(??) + ??2????(??) + ??3????(??) + ??

When ??1 +??2 +??3 1, the production function displays decreasing returns to scale, and when ??1 +??2 +??3 1, increasing returns to scale. Also, notice that when ??1 +??2 +??3 =1, the Cobb-Douglas production function implies constant returns to scale, i.e., when all inputs increase by, say, a multiplying factor c, the production also increases by c

Appendix A2: Explanation of inputs and output for data in FARMS.xlsx

• Farm Output (????????????????????) is measured as the gross output value of agriculture (100 million yuan in constant 2010 prices).

Inputs

• Labour is measured by the total number of employed persons in the primary sector (in 10,000?s of people).

• Machinery is measured by the sum of all agricultural machinery power (in 10,000?s of kW) and measures the mechanical power available in agriculture.

• Pesticides are measured by the annual use of agricultural pesticides (in 10,000?s of tons).

• Fertilizers is measured as the applied quantity of effective fertilizer components (in 10,000?s of tons)—i.e., gross weight is converted to the weight of fertilizer's 100% effective components (e.g., the 100% potassium oxide content in potash fertilizer). ***********************************THE END ***********************************

Empirical Project: Semester One 2022

Due date: Tuesday, 14 June 2022, 11:55 pm.

Submission: PDF Electronic submission via Canvas.

Marks: Total of 40 marks contributing 40% towards the final grade.

Notes:

i. You must paste regression estimation outputs from R-studio under each question and R-code if explicitly asked.

ii. A brief review of production functions and an explanation of the variables used in Task 2 are provided in the Appendix. This is not an exhaustive review, so if you have questions related to that topic, please consult any textbook on Microeconomics.

Task:1 (18 marks)

This task tests your understanding of the data generating process in the context of regression models.

1. Consider a Cobb-Douglas production function with three logged inputs ??1, ??2 and ??3 that can be represented as:

?? = ??0 + ??1??1 + ??2??2 + ??3??3 + ??

To answer the questions that follow, you need to use the observations on three inputs in the file ????????????????????_????????. ???????? and create the synthetic dependent variable ??. You will need to select the values of ??1, ??2 and ??3 such that your production function follows either constant, increasing or decreasing returns to scale (refer to Appendix A1 for an explanation of returns to scale). You also need to simulate residual term ??.

a. Select the values of ??0, ??1, ??2, ??3 and residual term ?? (column name this term ??????1)from a relatively tight normal distribution, say ??(0,0.07), to create the dependent variable ?? (column name this term ????????????1). Paste the corresponding R-code. (2 marks)

b. Now using ?? as a dependent variable and ??1, ??2 and ??3 as independent variables, run the regression. Paste the regression output and R-code below. (2 marks)

c. Has your regression model from (b.) successfully recovered the returns to scale coefficients you initially chose? Comment on the statistical significance of each input variable. (2 marks)

d. Use the same values of ??0, ??1, ??2 and ??3 but this time, draw residual term ?? (column name this term ??????2) from a relatively wide normal distribution, say ??(0,0.25), to create the dependent variable ?? (column name this term ????????????2). Run the regression and paste the output and R-code below. (3 marks)

e. Do you notice any difference in the values of coefficients and their statistical significance between the regression output in (b.) and (d.)? Explain in a sentence or two the reasoning behind such difference. (3 marks)

f. Now use the same values of ??0, ??1, ??2 and ??3 but make the residual3 term heteroscedastic and create the dependent variable ?? (column name this term ????????????3). Run the regression and paste the regression output and R-code below. (4 marks)

g. Do a Breusch Pagan Test of heteroskedasticity for the regression model estimated in (f.). Paste the test result. (1 mark)

h. Plot the residual against the fitted value of the regression from (f.) to test for heteroskedasticity. Make sure you paste R-code for the plot. (1 mark)

Task:2 (22 marks)

This task tests your ability to utilize real-world data to run and interpret regression models.

a. FARMS.xlsx contains data on the logged input and outputs of 558 Chinese Farms (refer to Appendix A2 for an explanation of inputs and output).

Use this data, and estimate a Cobb-Douglas production function of the following form:

???????????????????? = ??0 + ??1???????????? + ??1??????h?????????? + ??2???????????????????? + ??3 ??????????????????????

Paste the R code and regression output—comment on whether the production function follows an increasing, decreasing or constant return to scale. (4 marks)

b. Plot the residual against the fitted value of the regression model estimated in (a.)— comment on whether residuals are homoscedastic or not. Make sure you paste Rcode for the plot. (4 marks)

c. Now conduct the Breusch Pagan Test of heteroskedasticity. Make sure you start by explicitly specifying the null and alternative hypotheses for the test. Paste the result and comment on the result (use the chi-square value test at a 5% significance level, not the p-value test). State clearly why you reject or fail to reject the null hypothesis. Make sure you paste R-code and output for the Breusch Pagan Test. (7 marks)

d. Extract the residuals of the regression model estimated in (a.) and name it ??????????????. Run the following regression

???????????????????? = ??0 + ??1??????????????

What does the coefficient of ?????????????? and its statistical significance/insignificance implies? Paste the R-code and regression output below. (7 marks)

Appendix

Appendix A1: Production Function Overview

A production function represents the technological relationship between the quantities of inputs (e.g.capital, labour, and materials) and the quantity of output that a firm can produce. One traditional formulation is the so-called Cobb-Douglas production function:

?? = ?? ????1????2 ????3,

where ?? represents a firm's production output, ?? the firm's capital, ?? the firm's labour, and ?? the amount of materials input used in the production by the firm. The parameter ??1, ??2 and ??3 measures the output elasticities of capital, labour, and materials, respectively.

For econometric estimation, the Cobb-Douglas production function can be estimated using OLS by log-linearizing the model and adding an error term:

????(??) = ????(??) + ??1????(??) + ??2????(??) + ??3????(??) + ??

When ??1 +??2 +??3 1, the production function displays decreasing returns to scale, and when ??1 +??2 +??3 1, increasing returns to scale. Also, notice that when ??1 +??2 +??3 =1, the Cobb-Douglas production function implies constant returns to scale, i.e., when all inputs increase by, say, a multiplying factor c, the production also increases by c

Appendix A2: Explanation of inputs and output for data in FARMS.xlsx

• Farm Output (????????????????????) is measured as the gross output value of agriculture (100 million yuan in constant 2010 prices).

Inputs

• Labour is measured by the total number of employed persons in the primary sector (in 10,000?s of people).

• Machinery is measured by the sum of all agricultural machinery power (in 10,000?s of kW) and measures the mechanical power available in agriculture.

• Pesticides are measured by the annual use of agricultural pesticides (in 10,000?s of tons).

• Fertilizers is measured as the applied quantity of effective fertilizer components (in 10,000?s of tons)—i.e., gross weight is converted to the weight of fertilizer's 100% effective components (e.g., the 100% potassium oxide content in potash fertilizer). ***********************************THE END ***********************************

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