Economics 405 - Oakland University



Economics 405 Problem Set #1 Solutions

1. Wooldridge states “After data on the relevant variables have been collected, econometric methods are used to estimate the parameters in the econometric model…” What does he mean by parameters in this context?

Given the population model [pic], then the parameters we want to estimate are the population slope ([pic]) and population intercept ([pic]).

2.

a. Determine the first derivative of the function [pic].

Recognize that [pic], then apply the power function rule and the addition rule to get [pic].

b. Determine the partial derivative of y with respect to z of the function [pic].

Use the chain rule. [pic].

3. Suppose that the random variables X and Y have the marginal and joint probability density functions given in the tables below:

|X |fx(X) |

|2 |.5 |

|4 |.3 |

|6 |.2 |

|Y |fy(Y) |

|5 |.2 |

|10 |.5 |

|15 |.3 |

| |[pic] |

| |Y |

|X |5 |10 |15 |

|2 |.15 |.2 |.15 |

|4 |.03 |.2 |.07 |

|6 |.02 |.1 |.08 |

a. Are X and Y independent? Why or why not?

No, since [pic]in the joint pdf table.

b. Calculate the unconditional expected value of X.

[pic]

c. Calculate the relevant conditional expected values for X (i.e., [pic]. Compare and comment on the conditional expected values obtained here versus the unconditional expected value obtained in b.

There’s a typo in the first line of this question (credit to Naved for spotting it!). Last part of the first sentence should read: “…(i.e., [pic].”

Accordingly, the correct answer is obtained by taking:

[pic].

Therefore, [pic]

[pic]

[pic]

Note the moral of the story here. Unconditional expected value of X is 3.4. Nevertheless, the conditional expected values of X all depend on the value of Y. This is a consequence of the dependence between X and Y noted in part a.

d. Does [pic]? (Hint: Determine E(XY) using the joint pdf and determine E(X) and E(Y) using the marginal pdfs.)

We know E(X) = 3.4.

Furthermore, [pic]

Therefore: [pic]

To calculate E(XY), take:

[pic]

Hence [pic]. This comes as no surprise since X and Y are not independent. I claimed in class that they would be equal if X and Y are independent.

4. Suppose that the random variable X and the random variable Y have the following probability density functions:

|X |fx(X) |

|-5 |.4 |

|5 |.3 |

|10 |.3 |

|Y |fy(Y) |

|10 |.2 |

|15 |.5 |

|20 |.3 |

a. Assume that X and Y are independent. Construct the joint probability density function table for X and Y. Briefly explain what you’ve done here.

If X and Y are independent then the joint probability density function equals the product of the unconditional pdfs, i.e., [pic]. Accordingly,

| |[pic] |

| |Y |

|X |10 |15 |20 |

|-5 |.08 |.2 |.12 |

|5 |.06 |.15 |.09 |

|10 |.06 |.15 |.09 |

b. Determine the unconditional expected value of Y.

E(Y) = 15.5.

c. Determine the expected value of Y conditional on X = 10. Compare your answer in c with that in b and comment.

[pic]

Therefore: [pic]

Note that [pic], which follows from the fact that X and Y are independent. As a result, the particular level of X has no influence on the expected value of Y.

d. Show mathematically whether [pic].

Since X and Y are independent, it should be true that [pic]. We know that E(Y)=15.5. Using the table of marginal densities for X above, I calculate E(X) = 2.5. Therefore: E(X)E(Y) = 38.75. To calculate E(XY), take

[pic]Hence E(XY) = E(X)E(Y). We should have expected this since X and Y are independent.

5. A professor decides to run an experiment to measure the effect of time pressure on final exam scores. He gives each of the 400 students in his course the same final exam, but some students get 90 minutes to complete the exam while others have 120 minutes. Each student is randomly assigned one of the examination times based on the flip of a coin. Let Yi denote the number of points scored on the final exam and let Xi denote the amount of time that the student has to complete the exam. Consider the regression model: [pic].

a. Explain what the term ui represents. Why will different students have different values of ui?

ui is the error term. It will contain other unobserved factors that also affect performance on the exam such as hours studied, quality of study, student ability, and so forth.

b. Will the zero conditional mean assumption, [pic], be valid for this model? Why or why not?

The professor has set up what is known as a controlled experiment. As a consequence, the zero conditional mean assumption should be valid. In other words, given the random nature by which students are assigned test taking time (X), there is no way that the factors mentioned in part a could be correlated with amount of time the student has to take the test. With this said, then, it should seem reasonable that [pic], i.e., more time improves test score.

Usually in econometric problems we do not have the luxury of using such nice experimental data, however. Rather, we work with observational data. With observational data, the researcher/econometrician has no control over X, unlike the professor in the hypothetical. This leads to a useful thought experiment. Suppose we now assume that the professor merely records the time it takes each student to take the test (X) and then, once the exam is graded, records the student test scores. In this scenario, the professor has observed, rather than controlled via random assignment, the value of X. Now one could well argue that ZCMA is violated. For example, it might be the case that students who have studied harder and are better prepared actually take less time to finish the test! If this is the case, when the regression is run the estimated effect of X could well be negative or at least could be much closer to 0 (though positive) because test taking time and preparation are negatively correlated. The problem here is that the effect of being well prepared will be confounded with the effect of additional time and, as a result, it will be impossible to determine the pure effect of additional time on test score since other things (like study time) are not being held equal.

c. The estimated regression is [pic]. Compute the regression’s prediction for the average score of students given 90 minutes to complete the exam. Compute the regression’s prediction for the average score of students given 120 minutes to complete the exam. What is the estimated gain in score for a student who is given an additional 10 minutes on the exam?

[pic]

[pic]

[pic]. Hence an extra 10 minutes of time is predicted to improve exam score by 3.3 points.

6. a. Show algebraically that [pic].

(1) [pic]

(2) [pic]

Hence both sides of the statement solve to [pic]

b. Convince me that the result is true for the ACT scores given in problem 2.3 on p. 61 of the text.

| |ACT = X | | | |

| |1 |2 |3 |4 |

| |X |(Xi - Xbar) |X*(Xi-Xbar) |(Xi-Xbar)^2 |

| |21 |-4.875 |-102.375 |23.765625 |

| |24 |-1.875 |-45 |3.515625 |

| |26 |0.125 |3.25 |0.015625 |

| |27 |1.125 |30.375 |1.265625 |

| |29 |3.125 |90.625 |9.765625 |

| |25 |-0.875 |-21.875 |0.765625 |

| |25 |-0.875 |-21.875 |0.765625 |

| |30 |4.125 |123.75 |17.015625 |

|Total |207 | |56.875 |56.875 |

|Xbar |25.875 | | | |

Notice that while the individual entries in columns (3) and (4) are very different, they both add up to 56.875!

7. A researcher wished to investigate the effect of the federally funded school lunch program on 10th grade student test scores. The researcher hypothesized that students participating in the lunch program would have better scores than those not participating. (That is, full stomachs make for better performance and empty stomachs lead to worse performance.) Seems like common sense but when the researcher ran the regression for a randomly selected sample of 408 10th grade classes from the State of Michigan, the computer spit out the following result:

[pic]=32.14 – 0.319lnchprg n = 408,

where math10 is the school’s average score on a standardized test; and

lnchprg is the percentage of students at the school who are eligible for the free lunch program (eligibility is determined on the basis of family income).

a. What does the fitted model predict will be the effect on test scores of a 10 percentage point increase in eligibility for the free lunch program?

The model predicts that a 10 percentage point increase in free lunch program eligibility lowers test scores by 3.19 points (= -.319 x 10).

b. What is the predicted average test score at a school for which 50% of the students are eligible for the free lunch program? What is the predicted average test score at a school for which 10% of the students are eligible for the free lunch program?

32.14 - .319(50) = 16.19 in the first case and 32.14 - .319(10) = 28.95 in the second instance.

c. Contrary to common sense, the negative sign on the slope coefficient in the fitted model suggests that better nutrition has an adverse effect on exam performance! Discuss why the estimated coefficient is negative. [Hint: Base your discussion on the relation [pic], which we derived from the population regression model.]

What factors are in the error term? Family income, poverty level, quality of instruction, and so forth. As free lunch eligibility (x) increases, I would expect u to decrease, i.e., free lunch eligibility will be associated with higher levels of poverty, lower levels of income, perhaps poorer quality of instruction, all of which lead to poorer performance on the exam. The fitted model suggests that these adverse correlations between x and u overwhelm whatever positive causal effect ([pic]) the free lunch program has on test scores. In other words, [pic]and larger in absolute value than [pic], which presumably is positive.

8. The formula for the least squares slope coefficient is [pic]. Show step-by-step how this formula is obtained using the method of moments approach.

If the zero conditional mean assumption is valid, then [pic]. This then implies two restrictions on the moments of the u distribution:

1) [pic]

2) [pic] (follows from the independence of x and

u implied by zcma).

To derive the method of moments coefficient estimators, we impose conditions (1) and (2) on the sample data and solve for the sample values of the coefficients, which will be our estimators of the population coefficients. In other words, if the zero conditional mean assumption is valid or reasonable, then the sample data should be subject to the logical implications of conditions (1) and (2). Now note that condition (1) says that the average difference between y and the systematic part of the population regression function is 0. Imposing this requirement on our randomly chosen sample of n, implies:

3) [pic]

Note that (3) is the average difference between observed y and predicted y. Furthermore note that I’ve put ^s over the betas indicating that they will be estimators of the parameters.

Now, since we’re not asked to prove the following, I’ll simply note that (3) solves out to [pic]. We’ll use this result in what follows.

To get the slope estimator, impose condition (2) on the sample. This yields:

4) [pic]. Substituting for[pic] in (4) gives us:

5) [pic]. Re-arranging terms inside the rounded brackets results in:

6) [pic]. Doing the multiplication implied inside the square brackets gives:

7) [pic]. Now apply Property Sum.3 (p. 696) to the sum to get:

8) [pic]. Multiply both sides by n then move the second sum to the other side of the equation to get:

9) [pic]. Using Property Sum.2 (p. 695), results in:

10) [pic]. Solving for [pic]yields:

11) [pic]. We know from class discussion and from problem 6 above that [pic] and[pic]

respectively. Substituting into (11) gives:

12) [pic].

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