1. Consider a Cobb-Douglas production function with three logged1 inputs 1, 2 and 3 that can be represented as: = 0 + 11 + 22 + 33 + 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 values2 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.

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.

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.

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.

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.

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.

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

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.

What the Data looks like (only took a bit of the data as it is over 500 columns long):

obs	x1	x2	x3
1	0.724888	1.528823	-0.25618
2	0.706865	1.538306	-0.28172
3	0.709251	1.432015	-0.30767
4	0.76593	1.529903	-0.34675
5	0.925107	1.571177	-0.35578
6	0.899868	1.521407	-0.40766
7	0.880109	1.486143	-1.07582
8	0.893287	1.558533	-1.19158
9	0.897461	1.595225	-0.42353

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: = 12 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