ARE 212 - Problem Set 2

Posted 2021-02-12 newpythont

ARE 212 - Problem Set 2
Due March 10th
Part I: Theory (For your practice only. Not required)
1. Derive the sampling error of the GLS estimator by showing that bGLS ?
2. I used concentrated likelihood to derive the ML estimate for the CLRM. We could also set the likelihood
equations (first derivatives of the log likelihood) equal to zero. Confirm that the ML estimator proposed in
class satisfies these first order conditions.
3. Show that V (s2|X) = 2σ4
n?k and indicate which of our assumptions you need to get this result. [Hint: Look up
what the mean and variance of a random variable ～ χ2
m are.]
Part II: Applied - Returns to Scale in Electricity Supply
Read Nerlove’s (1955) paper ”Returns to Scale in Electricity Supply”. The goal of this problem set is to replicate
some of the results from this classic paper. The paper and data are available on the class website.
1. Read the data into R. Print out the data. Read it. Plot the series and make sure your data are read in correctly.
Make sure your data are sorted by size (kwh). [Hint: Check for obvious typos in the data and if you find any
fix them!]
2. Replicate regression I (page 176) in the paper. (You won’t be able to exactly, since the original data contained
an error that has been fixed). Your estimate for the price differences will differ slightly, but the R2 will be the
same.
3. Conduct the hypothesis test using constant returns to scale (βy = 1) as your null hypothesis. What is the pvalue
associated with you test statistic? What is your point estimate of returns to scale? Constant? Increasing?
Decreasing?
4. Plot the residuals against output. What do you notice? What does this potentially tell you from an economic
perspective? Compute the correlation coefficient of the residuals with output for the entire sample? What does
this tell you?
5. Nerlove tried to remedy his ”residual problem” by running separate models for different sized industries. Divide
your sample into 5 subgroups of 29 firms each according to the level of output. Estimate the regression model
again for each group separately. Can you replicate Equations IIA - IIIE? Calculate the point estimates for
returns to scale for each sample. Is there a pattern relating to size of output?
6. Create ”dummy variables” for each industry [which you may have done in the previous part]. Interact them with
the output variable to create five ”slope coefficients”. Run a model, letting the intercept and slope coefficient
on output differ across plants, but let the remainder of the coefficients be pooled across plants. Are there any
noticeable changes in returns to scale from the previous part?
7. Conduct a statistical test comparing the first model you estimate to the last model you estimated. (Hint: Is
one model a restricted version of the other?). Would separate t-test have given you the same results?
8. To see whether returns to scale declined with output, Nerlove tested a nonlinear specification by including
(ln(y))2 as a regressor. Conduct a statistical test you feel is appropriate to test this hypothesis.

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