计量经济学 作业1

Econometrics
ASSIGNMENT 1
100 POINTS TOTAL DUE: Thursday, January 26, 3:20 p.m.
**IMPORTANT REMINDER: LATE ASSIGNMENTS CANNOT BE ACCEPTED – NO EXCEPTIONS**
Note : Questions 1~3 are simple and straightforward. Questions 4 & 5 are also simple, albeit long. Questions 7 & 8 are on hypothesis testing.
1. [5] Let X be the random variable distributed as Normal (5,4). Find the probabilities of the following events: (i) P(X ≤ 6)
(ii) P(X > 4)
(iii) P(|X – 5| > 1)
2. [5] Let X denote the prison sentence, in years, for people convicted of auto theft in California. Suppose that the pdf of X is given by
219()(,0 3.=<<f x x x
Use integration to find the expected prison sentence. [hint : requires calculus.]
3.[5] Let X denote the annual salary on university professors in the United States, measured in thousands of dollars . Suppose that the average salary is 52.3, with a standard deviation of 1
4.6. Find the mean and standard deviation when salary is measured in dollars .
4. [20] Suppose a fair coin is tossed 3 times and let a random variable X denote the number of heads. Define Y, a Bernoulli random variable, as Y = 1 if the first tossed coin turns up head, and Y=0 otherwise.
(i) Determine and sketch the pdf of X and cdf of X .
(ii) Determine the joint probability density function of (X, Y )
(iii) Are random variables X and Y independent? Explain.
(iv) Compute P(X = 2|Y = 1), Cov (X,Y ) and Corr (X, Y )
5. [10] Question 2.3. pp52. [answers are provided for (a),(c) and (e). Credits are awarded for (b),(d) and (f)]
6. [15] Let Y denote the sample average from a random sample with mean µ and variance σ2. Consider two alternative estimators of µ: W 1 = [(n – 1)/n ]Y and W 2 = Y /2.
(i) Show that W 1 and W 2 are both biased estimators of µ and find the biases. What happens to the biases as n → ∞? Comment on any important difference in bias for the two estimators as the sample size gets large.
(ii) Find the probability limits of W 1 and W 2. Which estimator is consistent?
[hint : If plim (T n ) = a and plim (U n ) = b , then plim (T n + U n ) = a + b and plim (T n U n ) = ab ; also plim [(n – 1)/n ] = 1 ] (iii) Find Var(W 1) and Var(W 2).
7. [20] You are hired by the governor to study whether a tax on liquor has decreased average liquor consumption in your state. You are able to obtain, for a sample of individuals selected at random, the difference in liquor consumption (in ounces) for the years before and after the tax. For person i who is sampled randomly from the population, Y i denotes the change in liquor consumption. Treat these as a random sample from a Normal (µ, σ2) distribution.
(i) The null hypothesis is that there was no change in average liquor consumption. State this formally in terms of µ.
(ii) The alternative is that there was a decline in liquor consumption; state the alternatives in terms of µ.
(iii) Now, suppose your sample size is n = 900 and you obtain the estimates y= -32.8 and s = 466.4. Calculate the test statistic for testing H0against H1. Do you reject H0at the 5% level? At the 1% level?
(iv) Would you say that the estimated fall in consumption is large in magnitude? Comment on the practical versus statistical significance of this estimate.
8. [20] A random sample of 104 executives was asked about their views on the economy. 40 percent said it would improve (the rest saying it would not improve). Test the hypothesis that the true fraction of all executives who believe the economy will improve is 50 percent. Do you reject the null hypothesis - why or why not? Use both a one-side and a two-sided test, and state which makes more sense.。