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d. LY, for some integer k> 0 5. The following table contains quarterly nominal GDP in U.S. (billions of dol- lars). Let Y, denote the GDP at time t and let y, = ln (Y). (Show your calculations in a spreadsheet, e.g., in Microsoft Excel.) a. Plot the time series (Y). Can the underlying stochastic process be weakly b. Calculate the growth rate of nominal GDP by computing the percentage c. Plot the natural logarithm of the series (y,) and compare with part i., com- d. Repeat part i. by taking the first log-differences (in percentage), that is, e. Do you observe any significant differences between gır and gar computed in stationary of any order? Explain why or why not. changes of the series, that is, g,-100 × (Yt-Y,-)/ menting on stationarity and smoothness g21 100 X (V1--1). ii. and iv, respectively? Date GDP 2001-01-01 10021.5 10128.9 10135.1 10226.3 10338.2 10445.7 10546.5 10617.5 10744.6 10884.0 11116.7 11270.9 2004-01-0111472.6 11657.5 2001-04-01 2001-07-01 2001-10-01 2002-01-01 2002-04-01 2002-07-01 2002-10-01 2003-01-01 2003-04-01 2003-07-01 2003-10-01 2004-04-01 2004-07-01 11814.9 2004-10-0111994.8

CHAPTER 3 Statistics and Time Series 6. Following with the same data as in Exercise 5, a. Compute the sample moments A (mean) and Yo (variance) of g2r b. Compute the autocorrelation function of g2t, that is, pk for k = 1, 2, 3, 4 Interpret the autocorrelations by plotting 82i against the lagged values of Give an economic interpretation. 7. Download the daily S&P500 Index from January 2, 2006, and continuing to the most recent date. Let P denote the SP500 time series and p,In (P). a. Compute the daily return (i.e., R, p,-P1). b. Compute the sample moments of returns: mean, variance, skewness, and kurtosis. Plot the histogram. c. Plot R, against R,-1, R,-2, Re-3, and R,-4. Can you discern any pattern in any of the four graphs? 8. Following with the same data as in Exercise 7, and using the EViews software, a. Calculate the sample autocorrelation functions, ACF and PACF of R, for k- b. Compute the following conditional means (assume linearity): E(RIR, E (R R,-1, R-2, and E(RI R-1, R-2, R4). Do you think that it will be possible to predict future returns based on linear combinations of past returns? Why or why not? 9. Analyze the ACF and PACF that you calculated in Exercise 8b. Are the autocor- relation and partial autocorrelation coefficients statistically different from zero? State a single hypothesis and a joint hypothesis, and implement t-ratios and Q-statistics. Interpret your results 110. For the four series that you downloaded in Exercise 3, calculate the ACF and PACF using EViews. Comment on the shapes of these functions. Are the auto correlation and partial autocorrelation coefficients statistically different from zero? State a single hypothesis and a joint hypothesis, and implement t-ratio and Q-statistics. Interpret your results.

Exercise 6,7,8, and 9

use data from exercise 5 for exercise 6.

use data from the graph for 7,8,9

math   Statistics-And-Probability   
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