Analytical Problems (100 Points)

Problem Set 3
ECONOMETRICS / Prof. Oscar Jorda
Due Date: Thursday, February 14

Instructions: The goal of the problem set is to understand what you are doing rather than just getting the correct result. Please show your work clearly and neatly. No credit will be given to late homework, regardless of the excuse. Please write your answers in the space provided.

Analytical Problems (100 points)

1. [20 points] Suppose you have a sample of T observations on the variable Yt and that you want to calculate the sample mean. Unfortunately, a memory lapse has caused you to forget the formula for the sample mean. Being an astute and resourceful econometrician, you realize that this calculation can be interpreted as estimating the following regression

(a) Set up the least squares estimation procedure for this problem, which will consist in minimizing the sum of squared residuals

and compute the first order condition. Express the formula for the estimator.

(b) Alternatively, calculate by the method of moments, the estimator for m by realizing that you only need one moment condition, namely E(et) = 0.

2. [30 points] An economic model relating food expenditures (Y) and income level (X) of households postulates that this relationship can be modeled as,

In order to calculate this empirical relationship for the U.S. economy, you collect a sample of T individual’s food expenditures and income levels.

(a) Based on the above theoretical model, set up the corresponding statistical model that would allow you to compute b from the sample of data you collected. Hint: take into consideration that the relationship will not hold exactly for the data due to sampling errors, approximations errors, or other sources.

(b) Using the Least Squares principle, derive the estimator of b from your statistical model. Hint: Depending on the way you set up the statistical model, you may have a nonlinear regression problem. Worry not because this does not make the computations any harder.

(d) Being a clever econometrician, you suddenly realize that this economic relationship could be modified to look like a linear regression model. Write down the linear regression that would allow you to compute an estimate of b with the conventional, least squares estimator.

(e) Comment on the economic appropriateness of the economic model postulated above by discussing the properties of the elasticity of food expenditures to income. Hint:The elasticity in question is defined as:

Hints: For any W, and Z, log(WZ) = log(W) + log(Z), and

3. [25 points] Suppose a sample of data of size T was generated by the model: Yt = a + bXt + et with the usual assumptions. Now, suppose that a researcher executes the following transformations of the data:

where denote the sample means of Y and X respectively. With these variables, the researcher estimates the following regression

(a) Use the normal equations to show that

(b) Suppose Xt = {1,2,3,4,5} and Yt = { 5,5,9,9,12}. Use two graphs to represent the regression lines corresponding to the model that relates Y and X, and the model that relates Z and W. Observing what the two graphs look like and the results in (a), explain in a few words what is going on and whether it makes a difference or not to estimate the model with the constructed variables Z and W.

(c) Do you expect that the fit achieved by the regression of Z on W will be better/the same/worse than the fit achieved in the regression of Y on X? Explain why.

4. [25 points] The following questions are based on the following output from EViews

Regression 1

Dependent Variable: CRIME_RATE
Method: Least Squares
Date: 02/01/01 Time: 13:59
Sample(adjusted): 1 90
Included observations: 87
Excluded observations: 3 after adjusting endpoints
Variable / Coefficient / Std. Error / t-Statistic / Prob.
C / 4.820928 / 3.747441 / 0.0003
UR / 0.374783 / 0.192015 / 0.0542
R-squared / 0.042897 / Mean dependent var / 7.222108
Adjusted R-squared / 0.031637 / S.D. dependent var / 3.566183
S.E. of regression / 3.509318 / F-statistic / 3.809688
Sum squared resid / 1046.801 / Prob(F-statistic) / 0.054250

Regression 2

Dependent Variable: CRIME_RATE
Method: Least Squares
Date: 02/01/01 Time: 14:01
Sample(adjusted): 1 90
Included observations: 87
Excluded observations: 3 after adjusting endpoints
Variable / Coefficient / Std. Error / t-Statistic / Prob.
C / 27.63285 / 6.594709
HSPERCENT / -0.260769 / 0.053357 / -4.887215
R-squared / 0.219359 / Mean dependent var / 7.222108
Adjusted R-squared / 0.210175 / S.D. dependent var / 3.566183
S.E. of regression / 3.169342 / F-statistic / 23.88487
Sum squared resid / 853.8018 / Prob(F-statistic) / 0.000005

The data used in these regressions consists on county-level data for 87 counties sampled randomly across all states in 1991. The variable CRIME_RATE is the total number of crimes divided by the total population in that county and converted to a percentage; the variable UR is the county’s unemployment rate in percent, the variable HSPERCENT is the percentage of the population that finished high-school.

(a) Fill the missing cells and compute the mean of the independent variables.

(b) Using regression 1, interpret the effects that an increase of 1% in the unemployment rate has on the crime rate. Similarly, using regression 2, explain what would happen to the crime rate if the percent of people that finished high school went up 5 percentage points.

(d) In one paragraph, prepare an executive summary with your recommendations to fight crime based on these regressions. Adjusts your comments to the evidence contained in the regression output only.

EViews Exercise (50 points)

Use EViews to answer the following question. It is typically argued that high levels of unemployment benefits help explain why unemployment rates in Europe are higher than in the U.S. Below are the average unemployment rates of OECD countries for the period 1983-1996; the benefit replacement rates (i.e., percentage of your salary paid by unemployment benefits); and the benefit duration (number of years of paid benefits).

Country / Unemployment Rate / Benefits (%) / Duration
Austria / 3.8 / 50 / 2
Belgium / 9.7 / 60 / 4
Denmark / 9.9 / 90 / 2.5
Finland / 9.1 / 63 / 2
France / 10.4 / 57 / 3
Germany / 6.2 / 63 / 4
Ireland / 15.1 / 37 / 4
Italy / 7.6 / 20 / 0.5
Netherlands / 8.4 / 70 / 2
Norway / 4.2 / 65 / 1.5
Portugal / 6.4 / 65 / 0.8
Spain / 19.7 / 70 / 3.5
Sweden / 4.3 / 80 / 1.2
Switzerland / 1.8 / 70 / 1
U.K. / 9.7 / 38 / 4
Canada / 9.8 / 59 / 1
U.S. / 6.5 / 50 / 0.5
Japan / 2.6 / 60 / 0.5
Australia / 8.7 / 36 / 4
New Zealand / 6.8 / 30 / 4

These data are contained in the file ps3.xls, which is in EXCEL format. Answer the following questions:

(a) Test the assertion that the U.S. unemployment rate is lower than in Europe. Find out the average Unemployment rate in Europe (excluding Canada, U.S., Japan, Australia and New Zealand) and test H0: Average unemployment rate in Europe = Unemployment rate in the U.S. at a 90\% confidence level.

(b) What are the basic statistics and correlations between UR and the Benefit Level BL? UR and Benefit Duration BR? BL and BD? What do you conclude from this exploratory analysis?

(d) Regress UR on BD. What can you say about the relationship between them?

(e) Regress UR on BL. How do this results compare to (d)?

(f) Regress BD on BL. What is the relationship?

(g) Some practitioners prefer to use the logarithm of UR in their regressions. Do your results for (d) and (e) change when you transform UR to log(UR)?

(h) How do your answers change for countries with UR<8%?

(i) Prepare an executive summary based on the previous empirical analysis (1/2 page maximum) with your recommendations to the President of a country with 15% Unemployment, 90% of Benefit Levels that can last up to 5 years. The goal is to reach a 10% Unemployment rate.