计量经济学代写｜ECON 4261 – Homework 2

Question 1

Using data from 1988 for houses sold in Andover, Massachusetts, from Kiel and McClain (1995), the following equation relates housing price (price

) to the distance from a recently built garbage incinerator (dist

):

Q1-1 Interpret the coefficient on log(dist

). Is the sign of this estimate what you expect it to be?

Q1-2 Do you think simple linear regression provides an unbiased estimator of the ceteris paribus elasticity of price

 with respect to dist

? (Think about the city’s decision on where to put the incinerator)

Q1-3 What other factors about a house might affect its price? Might these be correlated with distance from the incinerator?

Question 2

Consider the savings function

where ε

 is a random variable with E[ε]=0

 and V[ε]=σ2

. Assume that ε

 is independent of inc

.

Q2-1 Show that E[ε|inc]=0

, so that the key zero conditional mean assumption (A3) is satisfied.

Q2-2 Show that V[u|inc]=σ2inc

, so that the homoscedasticity assumption (A4) is violated. In particular, the variance of sav

 increases with inc

.

Q2-3 Provide a discussion that supports the assumption that the variance of savings increases with family income.

Question 3

We are interested in the birth weight (bwght

) of infants and the number of cigarettes the mother smoked per day during pregnancy (cigs

). The following simple regression was estimated using data on n=1388

 births

Q3-1 What is the predicted birth weight when cigs

 = 0? What about when cigs=20

 (one pack a day)? Comment on the difference

Q3-2 Does this simple regression necessarily capture a causal relationship between the child’s birth weight and the mother’s smoking habits? Explain.

Q3-3 To predict a birth weight of 125 ounces, what would cigs

 have to be? Comment.

Q3-4 The proportion of women in the sample who did not smoke while pregnant is about 0.85. Does this help reconcile your finding from part (3)?

Question 4

In this question we will compare the difference between the finite sample properties and large sample properties of OLS. Let’s say the population regression is

where

• β0
  ${\beta }_0$

   = 3

• β1
  ${\beta }_1$

   = 5

• Xi∼N(2,1)
  $X_i\sim N(2,1)$

• εi∼N(0,1)
  ${\epsilon }_i\sim N(0,1)$

Q4-1 Simulate {(Yi,Xi)}5000i=1

 (i.e., 5000 data points), save it as a data frame, and plot the histogram of Yi

 and Xi

. Properly label your graphs (you will lose points if you don’t – you can add + xlab("appropriate label for X") + ylab("appropriate label for Y") to your line of code)

Q4-2 Now let’s show the unbiasedness of β̂ 

.

Do the following steps in R.

1. Create a function that will calculate β̂ 0
   ${\hat{\beta }}_0$

    and β̂ 1

   ${\hat{\beta }}_1$

    from a sample of size N

• Initiate your function using

 regOLS <- function(N){
   
   
 }

• Now inside your function, use samp <- df[sample(nrow(df), N), ] to draw a sample of size N from your data frame and save it as samp
• Calculate the OLS β̂ 1
  ${\hat{\beta }}_1$

   and β̂ 0

  ${\hat{\beta }}_0$

   based your samp data

• Have your function return data.frame(b0 = __, b1 = __)

2. Create 4 empty data frames to store your values of β̂ 1
   ${\hat{\beta }}_1$

    and β̂ 0

   ${\hat{\beta }}_0$

val1 <- data.frame(b0 = double(), b1 = double())
val2 <- data.frame(b0 = double(), b1 = double())
val3 <- data.frame(b0 = double(), b1 = double())
val4 <- data.frame(b0 = double(), b1 = double())

3. Using a for loop, run your regOLS function for 100, 500, 1000, 5000 times, saving β̂ 1
   ${\hat{\beta }}_1$

    and β̂ 0

   ${\hat{\beta }}_0$

    each time into your val1,val2,val3,val4 dataframes, respectively. Use N=5

   $N=5$

    for your sample size, so that you are running the regression on a sample of size 5 each time. val1 should have size 100, val2 should have size 500, and so on.

4. Report the average of β̂ 1
   ${\hat{\beta }}_1$

    and β̂ 0

   ${\hat{\beta }}_0$

    for each of your val data frames by running the following code as is:

results = data.frame(n= double(), beta0_avg = double(), beta1_avg = double())
results[1:4,'N'] =c(100,500,1000,5000)
results[1,2:3] = colMeans(val1)
results[2,2:3] = colMeans(val2)
results[3,2:3] = colMeans(val3)
results[4,2:3] = colMeans(val4)

print(results)

Show the output of print(results) for credit.

Q4-3 Interpret your results. Does having a small sample size of 5 matter in terms of expected values?

Q4-4 Since we simulated the unbiasedness of the OLS estimator, now let’s simulate the consistency of it. Using the same regOLS function from before, run the function four times with N=10,50,500,5000

 each. You can just run the code below.

results = data.frame(n= double(), beta0_avg = double(), beta1_avg = double())
results[1:4,'n'] =c(10,50,500,5000)
results[1,2:3] = colMeans(regOLS(10))
results[2,2:3] = colMeans(regOLS(50))
results[3,2:3] = colMeans(regOLS(500))
results[4,2:3] = colMeans(regOLS(5000))

print(results)

Show the output of print(results) for credit.

Q4-5 Interpret your results. What happens to your estimators as N

 increases?

Q4-6 Explain the difference between unbiasedness (finite sample property) and consistency (large sample / asymptotic property).