PS 1 Notes 1. Please submit your problem set zip files which contains all related material into CANVAS by the deadline. Late submissions will not be accepted. 2. Hand in your problem set together with the i) R codes that you used to generate the results, ii) the associated R log file, and iii) your written solution. 4. Each student needs to write his/her own solutions, even though discussions of the assignments between students are encouraged. 5. If not specifically specified, use 5% significance level (the associated critical value is 1.96 for standard normal distribution) to draw conclusions in this problem set. 6. fBasics, quantmod packages of R are helpful in doing problems 6. 1. Let Y1, · · · , Yn be n i.i.d. random samples with common mean µ and common variance σ2. Let Y¯ denote the sample average. (a) Define the class of linear estimators of µ by Wa = a1Y1 + a2Y2 + · · ·+ anYn, where ai are constants. What restrictions on the ai needed for Wa to be an unbiased estimator of µ? (b) Explain what does var(Wa) tell you. (c) Compute var(Wa) and var(Y¯ ). (d) Based on part (b), which estimator of µ, Wa or Y¯ is more efficient, in a sense that has smaller variance? Page 1 PS 1 2. An investment firm offers its customers municipal bonds that mature after varying number of years. Given that F (t), cumulative distribution of T , the number of years to maturity for a randomly selected bond is F (t) =  0 t < 1, 1 5 , 1 ≤ t < 3, 1 2 , 3 ≤ t < 5, 3 5 , 5 ≤ t < 7, 1, t ≥ 7. (a) Find P (T = 5) and P (T = 1.5), that the probability that the bond has 5 and one and half year to maturity. (b) Find P (T > 6), that the probability that the bond has at least 6 years to maturity. (c) Construct the probability (Mass) density function for T . 3. You are hired by the governor to study whether a tax on liquor has decreased average liquor consumption in your region. You are able to obtain, for a sample of individuals selected at random, the difference in liquor consumption for the years before and after the tax. For person i, who is sampled randomly from the population, Yi, denotes the change in liquor consumption. Treat these as a random sample from a Normal N(µ, σ2) distribution. (a) The null hypothesis is that there was no change in average liquor consumption. State this formally in terms of µ (b) The alternative is that there was a change in liquor consumption. State the alternative in terms of µ. (c) Now, suppose your sample size is n = 300 and you obtain the estimates y¯ = −22.8 and s = 455.4. Calculate the t statistic for testing Ho against H1. At 5% confidence level, do you think the taxation policy affects liquor consumption? (d) Now, suppose your sample size is n = 9 and you obtain the estimates y¯ = −22.8 and s = 455.4. Calculate the t statistic for testing Ho against the alternative that liquor consumption is higher after the tax. At 5% confidence level, do you think the taxation policy affects liquor consumption? 4. Suppose we have Y1, ..., Yn taken from a normally-distributed population. If Y¯ = 4, test whether µ = 3.5 against a two-sided alternative when n = 100 and s2 = 1, at a 5% significance level. Page 2 PS 1 5. In a recent poll of 725 registered voters, 47% indicated that they approve of the job Donald Trump is doing as president. Public Policy Polling, January 30-31, 2017 could be accessed at http://www.publicpolicypolling.com/pdf/2017/PPP_Release_National_ 2217.pdf. Test whether the approval in the population is 50%, at the 5% level and against a two-sided alternative. 6. Consider the Pfzier stock, Starbucks stock and 3M stock from 2009-01-02 to 2021-09-30. (a) Use quanmod package of R to download the data. (b) Use each stock’s adjusted price to compute their associated daily simple returns and log returns. (c) Compute the sample mean, sample variance, skewness and excess kurtosis (compare with 3), minimum and maximum of each stock’s simple returns. (d) Compute the sample mean, sample variance, skewness and excess kurtosis (compare with 3), minimum and maximum of each stock’s log returns. (e) Compute pairwise correlations of these three stocks. Which two stocks are more likely to move in the same way (the so-called co-movement)? (f) Which stock is more likely to encounter big losses? (g) Which stock is more likely to generate returns higher than its mean level? (h) Plot densities of each stock using histogram and density curves. (i) Create a time series of monthly log returns for all three stocks during the same sampling period. (j) The null hypothesis is that the monthly simple return of Pfzier stock is zero and the alternative is that the monthly simple return of Amazon stock is not zero. State these hypothesis in terms of µ. (k) Based on part (g), perform your test and draw your conclusion (with the help of R, you could compute sample moments to construct your test statistic). Page 3