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Below in Table 1, the joint distribution of Nobel Prize winners in Economics and Non- Economics Disciplines (X), and citizenship (Y) for the period 1969-2000 is Treat the table as the distribution of a population.
Table 1: Distribution of Nobel prize winners.
So for example, is the probability that a Nobel prize winner is a US citizen and won the prize in Economics.
Columbia University Econometrics Tutors Lecture 1
Columbia University Econometrics Tutors Lecture 2
New York Econometrics Tutors online Lecture 3
PhD Tutor Econometrics Lecture 4
- Compute E[Y] and interpret the resulting number.
- Calculate and interpret
- A randomly selected Nobel Prize winner reports that he is a non-U.S. citizen. What is the probability that this genius has won the Economics Nobel Prize? A Nobel Prize in the other five disciplines?
- Show what the joint distribution would look like if the two categories (X and Y) were independent
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You are hired by the governor to study whether a tax on liquor has decreased average liquor consumption in New York. You obtain from a random sample of n individuals in New York, each person’s liquor consumption both for the year before and for the year after the introduction of the tax. From this data, you compute Yi ="change in liquor consumption" for individual i = 1,…. n. Yi is measured in ounces so if, for example, Yi = 10, then individual i increased his liquor consumption by 10 ounces. Let the parameters μy and
You are interested in testing the hypothesis H0 that there was no change in liquor consumption due to the tax. State this formally in terms of the population parameters.
The alternative, H1, is that there was a decline in liquor consumption; state the alternative in terms of the population parameters.
Suppose that your sample size is n = 900 and you obtain estimates μy = -32.8 and σy= 466.4. Report the t-statistic for testing H0 against H1. Obtain the p-value for the test [use Table 1 in Stock and Watson, p. 749-750]. Do you reject at a 5% level? At 1% level?
Would you say that the estimated fall in consumption is large in magnitude? Comment on the practical versus statistical significance of this estimate.
In your analysis, what has been implicitly assumed about other determinants of liquor consumption over the two-year period in order to infer causality from the tax change to liquor consumption?
Let Y be a Bernoulli random variable with success probability Pr(Y=1) = p, and let Y1,...,Yn
be i.i.d. draws from this distribution. Let pˆ be the fraction of successes (1s) in this sample.
Show that pˆ = Y
Show that pˆ is an unbiased estimator of p.
Show that var( pˆ ) = p(1-p)/n
Let Y1, Y2, Y3, Y4, be independently, identically distributed random variables from a population with mean and variance . Let Y = (1/4) (Y1+Y2+Y3+Y4) denote the average of these four random variables.
What are the expected value and variance of
Now, consider a different estimator of : Ỹ =(1/8)Y1+(1/8)Y2,+(1/4)Y3+(1/2)Y4. This is an example of a weighted average of the Yi.’s. Show that Ỹ is also an unbiased estimator of
. Find the variance of Ỹ.
Based on your answer to parts (a) and (b), which estimator of do you prefer, Y or Ỹ?
Suppose Y1, Y2, Y3, Y4 follow a Normal distribution with mean 5 and variance 2=3. What is the distribution of Y and Ỹ?
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Suppose at Columbia University, grade point average (GPA) and SAT scores are related by the conditional expectation E(GPA|SAT) = .90 + .001 SAT.
Find the expected GPA when SAT = 1600.
Find E(GPA|SAT=2200)
If the average SAT in the university is 2000, what is the average GPA?
Let u and X be two random variables that satisfy E [u|X] = 0 and E[u2|X]=2.
Find the unconditional mean and variance of u.
What is the covariance between u and X?
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Following questions will not be graded, they are for you to practice and will be discussed at the recitation:
[Practice question, not graded] SW 2.3
| Rain (X=0) | No Rain (X=1) | Total |
Long Commute (Y=0) | 0.15 | 0.07 | 0.22 |
Short Commute (Y=1) | 0.15 | 0.63 | 0.78 |
Total | 0.30 | .70 | 1.00 |
Using the random variables X and Y from Table 2.2 (given above), consider two new random variables W = 3 + 6X and V = 20 – 7Y. Compute:
E(W) and E(V).
σ²W and σ²V.
σW,V and Corr(W,V).
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[Practice question, not graded] SW 2.6
The following table gives the joint probability distribution between employment status and college graduation among those either employed or looking for work (unemployed) in the working age US population, based on the 1990 US Census.
| Unemployed (Y=0) | Employed (Y=1) | Total |
Non-college grads (X=0) | 0.045 | 0.709 | 0.754 |
College grads (X=1) | 0.005 | 0.241 | 0.246 |
Total | 0.050 | 0.950 | 1.000 |
Compute E(Y).
The unemployment rate is the fraction of the labor force that is unemployed. Show that the unemployment rate is given by 1-E(Y).
Calculate the E(Y|X=1) and E(Y|X=0).
Calculate the unemployment rate for (i) college graduates and (ii) non-college graduates.
A randomly selected member of this population reports being unemployed. What is the probability that this worker is a college graduate? A non-college graduate?
Are educational achievement and employment status independent? Explain.
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[Practice question, not graded] SW 2.14 [Hint: Use SW Appendix Table 1.]
In a population E[Y] = 100 and Var(Y) = 43. Use the central limit theorem to answer the following questions:
In a random sample of size n = 100, find Pr(
In a random sample of size n = 165, find Pr(
In a random sample of size n = 64, find Pr(101 ≤
[Practice question, not graded] SW 3.12
To investigate possible gender discrimination in a firm, a sample of 100 men and 64 women with similar job descriptions are selected at random. A summary of the resulting monthly salaries are:
| Avg. Salary ( | Stand Dev (of Y) | n |
Men | $3100 | $200 | 100 |
Women | $2900 | $320 | 64 |
What do these data suggest about wage differences in the firm? Do they represent statistically significant evidence that wages of men and women are different? (To answer this question, first state the null and alternative hypothesis; second, compute the relevant t-statistic; and finally, use the p-value to answer the equation.)
Do these data suggest that the firm is guilty of gender discrimination in its compensation politics? Explain.
[Practice question, not graded] SW 3.2
Let Y be a Bernoulli random variable with success probability Pr(Y=1) = p, and let Y1,...,Yn
be i.i.d. draws from this distribution. Let pˆ be the fraction of successes (1s) in this sample.
Show that pˆ =
Show that pˆ is an unbiased estimator of p.
Show that var( pˆ ) = p(1-p)/n
[Practice question, not graded] SW 2.10 [Hint: Use SW Appendix Table 1.]
Compute the following probabilities:
If Y is distributed N(1,4), find Pr(Y≤3).
If Y is distributed N(3,9), find Pr(Y>0).
If Y is distributed N(50,25), find Pr(40≤Y≤52).
If Y is distributed N(5,2), find Pr(6≤Y≤8)
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[Practice question, not graded] SW 3.3
In a survey of 400 likely voters, 215 responded that they would vote for the incumbent and 185 responded that they would vote for the challenger. Let p denote the fraction of all likely voters that preferred the incumbent at the time of the survey, and let pˆ be the fraction of
survey respondents that preferred the incumbent.
Use the survey results to estimate p.
Use the estimator of the variance of pˆ , pˆ (1 - pˆ )/n to calculate the standard error of your estimator.
What is the p-value for the test H0: p=0.5 vs. H1:p≠0.5?
What is the p-value for the test H0: p=0.5 vs. H1:p>0.5?
Why do the results from (c) and (d) differ?
Did the survey contain statistically significant evidence that the incumbent was ahead of the challenger at the time of the survey? Explain.
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