Ph.D. Econometrics I Heinz School, Carnegie Mellon University 90-906, Spring 2006

Final

Instructions You may use any books, notes, calculators, and other aids you like. You may not converse, nor may you cooperate. Please complete all questions. Please show all relevant work. Please interpret your results in plain English. You have until 8:20 to complete the test.

1. Consider the hospital cost dataset we used in homework 4. Remember that this is a dataset of California hospitals with data measured quarterly from 1991 to 1995. The data elements are:

C  is hospital costs for the quarter
D  is number of inpatient days for the quarter
V  is number of outpatient visits for the quarter
time is number of quarters since Q1, 1991

Let’s think about the two regressions:

ln C_{it} = β_{1} + β_{2} ln D_{it} + β_{3} ln V_{it} + β_{4}t + ϵ_{it}  (1)
ln C_{it} = β_{1} + β_{2} ln D_{it} + β_{3} ln V_{it} + β_{4}t + β_{5}(ln D_{it})^{2} + ϵ_{it} (2)

(3)

(a) Considering equation 1 and assuming that the assumptions of the CLRM hold, what is your best estimate and 95% confidence interval for the effect of a 1% increase in inpatient days on hospital costs?

(b) How does your answer change if you suspect heteroskedasticity?

(c) Still on equation 1 and worrying about heteroskedasticity, what is the annual growth rate of hospital costs in CA (holding constant days and visits)? Estimate and 95% confidence interval.

(d) Still considering equation 1 and worrying about heteroskedasticity, we are interested in assessing whether hospitals are a constant returns to scale business. A business is constant returns to scale if a simultaneous 1% increase in all outputs leads to exactly a 1% increase in costs. Test the theory that California hospitals are a constant returns business.

(e) Now, considering equation 2 and continuing to worry about heteroskedasticity, at what level of days (not log days) are California hospitals exactly constant returns to scale (that is \frac{∂ ln C}{∂ ln D} + \frac{∂ ln C}{∂ ln V} = 1)? Please set up completely but do not calculate the standard error of your estimate (You don’t have to multiply, invert, etc matrixes, just set up the relevant equation(s)).

2. When people talk about the repeated measures problem, they often talk in terms of how many observations you “really” have in your data, as opposed to how many observations you seem to have. The intuition is that if you have,say, two obs on each person, but those obs are highly correlated with each other, that there is not really 2 obs worth of information in them.

Consider a panel dataset with 2 observations for each person. Suppose further that each person’s X variables are the same for each of their 2 observations, so that we can write:

Y_{it} = X^{i}β + ϵ_{it}  (4)

Suppose further that each individual’s two observations ϵ_{i1} and ϵ_{i2} have correlation equal to 1 > ρ > 0. Otherwise, suppose the usual CLRM assumptions hold.

Now, X is a 2N × K matrix. However, since each individual is in the matrix twice with the exact same values, we can write the X matrix as:

X = [ X̃ ; X̃ ]  (5)

Here the data are sorted by time and then individual, so that X̃ is an N × K matrix with one obs for each individual.

(a) Prove that the usual (and correct) OLS variance for a regression using only one observation per individual is: σ̂^{2}_{OLS}(X̃′X̃)^{−1}

(b) Prove that the usual OLS variance matrix for the regression using two observations per individual is: \frac{σ̂^{2}_{OLS}}{2} (X̃′X̃)^{−1}

(c) Calculate the correct variance matrix for this regression and relate it to the discussion of the “effective” number of observations above. How does the number of effective observations change with ρ?

3. Suppose that you are interested in the effect of a person turning 65 on income. Suppose further that, the true model is:

I_{it} = β_{1} + β_{2}age_{it} + β_{3}(age_{it})^{2} + β_{4}senior_{it} + ϵ_{it}  (6)

but that you estimate:

I_{it} = β_{1} + β_{2}age_{it} + β_{4}senior_{it} + ϵ_{it}  (7)

Here, I am assuming you have a panel of people followed over time and that senior is a dummy variable =1 for age ≥ 65.

To make things easy, assume that, in truth, β_{4} = 0, β_{2} > 0, and β_{3} < 0.

Please discuss how β_{4} is likely to be biased. First, draw a picture and discuss in words how you think the bias will work. Then, do the necessary algebra.

If it helps you to make some assumptions about the dataset, that’s OK. For example, you might want to consider only a single person in the panel or you might want to make particular assumptions about which ages for the person are in the data (say they are observed from age 30-70 or something).

Ultimately, however, a correct answer is one that proves your claims with algebra. Just pictures and words will not get you full credit.