Financial Econometrics 1st Edition Peijie Wang - Solutions

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2.13 X is a Bernoulli random variable with Pr(X = 1) = 0.90, Y is distributed N(0, 4), W is distributed N(0, 16), and X, Y, and W are independent. Let S = XY + (1 - X)W. (That is, S = Y when X = 1, and S = W when X = 0.)a. Show that E(Y 2) = 4 and E(W 2) = 16.b. Show that E(Y3) = 0 and E(W3) = 0.
2.12 Compute the following probabilities:a. If Y is distributed t12, find Pr(Y … 1.36).b. If Y is distributed t30, find Pr(-1.70 … Y … 1.70).c. If Y is distributed N(0, 1), find Pr(-1.70 … Y … 1.70).d. When do the critical values of the normal and the t distribution coincide?e. If Y is
2.11 Compute the following probabilities:a. If Y is distributed x23, find Pr(Y … 6.25).b. If Y is distributed x28, find Pr(Y … 15.51).c. If Y is distributed F8,, find Pr(Y … 1.94).d. Why are the answers to (b) and (c) the same?e. If Y is distributed x21, find Pr(Y … 0.5). (Hint: Use the
2.10 Compute the following probabilities:a. If Y is distributed N(4, 9), find Pr(Y … 5).b. If Y is distributed N(5, 16), find Pr(Y 7 2).c. If Y is distributed N(1, 4), find Pr(2 … Y … 5).d. If Y is distributed N(2, 1), find Pr(1 … Y … 4).
2.9 X and Y are discrete random variables with the following joint distribution:That is, Pr(X = 3, Y = 2) = 0.04, and so forth.a. Calculate the probability distribution, mean, and variance of Y.b. Calculate the probability distribution, mean, and variance of Y given X = 6.c. Calculate the
2.8 The random variable Y has a mean of 4 and a variance of 1/9. Let Z =3(Y - 4). Find the mean and the variance of Z.
2.7 In a given population of two-earning male-female couples, male earnings have a mean of $50,000 per year and a standard deviation of $15,000.Female earnings have a mean of $48,000 per year and a standard deviation of $13,000. The correlation between male and female earnings for a couple is 0.90.
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 population of country A.a. Compute E(Y).b. The unemployment rate is a fraction of the labor force that is
2.5 In July, Fairtown’s daily maximum temperature has a mean of 65°F and a standard deviation of 5°F. What are the mean, standard deviation, and variance in °C?
2.4 Suppose X is a Bernoulli random variable with P(X = 1) = p.a. Show E(X3) = p.b. Show E(Xk) = p for k 7 0.c. Suppose that p = 0.3. Compute the mean, variance, skewness, and kurtosis of X. (Hint: You might find it helpful to use the formulas given in Exercise 2.21.)
2.3 Using the random variables X and Y from Table 2.2, consider two new random variables W = 4 + 8X and V = 11 - 2Y. Compute (a) E(W) and E(V); (b) J2 W and J2 V; and (c) JWV and corr(W, V).
2.2 Use the probability distribution given in Table 2.2 to compute (a) E(Y) and E(X); (b) s2 X and s2 Y; and (c) sXY and corr(X, Y).
2.1 Let Y denote the number of “heads” that occur when two loaded coins are tossed. Assume the probability of getting “heads” is 0.4 on either coin.a. Derive the probability distribution of Y.b. Derive the mean and variance of Y.
2.7. Y is a random variable with mY = 0, sY = 1, skewness = 0, and kurtosis = 100. Sketch a hypothetical probability distribution of Y.Explain why n random variables drawn from this distribution might have some large outliers.
2.6. Suppose that Y1,c, Yn are i.i.d. random variables with the probability distribution given in Figure 2.10a. You want to calculate Pr( Y … 0.1).Would it be reasonable to use the normal approximation if n = 5? What about n = 25 or n = 100? Explain.
2.5. Suppose that Y1,c, Yn are i.i.d. random variables with a N(1, 4) distribution.Sketch the probability density of Y when n = 2. Repeat this for n = 10 and n = 100. In words, describe how the densities differ. What is the relationship between your answer and the law of large numbers?
2.4. An econometrics class has 80 students, and the mean student weight is 145 lb. A random sample of 4 students is selected from the class, and their average weight is calculated. Will the average weight of the students in the sample equal 145 lb? Why or why not? Use this example to explain why
2.3. Suppose that X denotes the amount of rainfall in your hometown during a randomly selected month and Y denotes the number of children born in Los Angeles during the same month. Are X and Y independent?Explain.
2.2. Suppose that the random variables X and Y are independent and you know their distributions. Explain why knowing the value of X tells you nothing about the value of Y.
2.1. Examples of random variables used in this chapter included (a) the gender of the next person you meet, (b) the number of times a computer crashes,(c) the time it takes to commute to school, (d) whether the computer you are assigned in the library is new or old, and (e) whether it is raining or
1.3 You are asked to study the causal effect of hours spent in remedial classes at schools by students who are struggling in mathematics on their final test scores and performance in the subject. Describe:a. an ideal randomized controlled experiment to measure this causal effect;b. an observational
1.2 Design a hypothetical ideal randomized controlled experiment to study the effect of the consumption of alcohol on long-term memory loss. Suggest some impediments to implementing this experiment in practice.
1.1 Design a hypothetical ideal randomized controlled experiment to study the effect of reading on the improvement of a person’s vocabulary. Suggest some impediments to implementing this experiment in practice.
18.7 In Exercise 17.5 we asked you to estimate alternative ARIMA models for the 3 month Treasury bill rate. Now use your models to generate forecasts comparable to those in Example 18.1. Have you been able to construct a model whose forecasting perfor mance is better?
18.6 Suppose that a particular nonstationary time series y, can be modeled as a stochastic process that is ARIMA(1, 1, 1). (4) After you have estimated the model's parameters, how would you forecast y, one period ahead? Express this one-period forecast, y,(1), as a function of observable data. In
18.5 Repeat Exercise 18.4 for the ARMA(2, 1) process.
18.4 Derive expressions for the one-, two-, and three-period forecasts for the second- order autoregressive process AR(2). What are the error variances of these forecasts?
18.3 Derive expressions for the one-, two-, and three-period forecasts, (1), (2), and (3), for the second-order moving average process MA(2). What are the variances of the errors for these forecasts? What is the variance of the error for the I-period forecast, with 1>3?
18.2 Does it seem reasonable that for any ARIMA specification the forecast error vari ance one period ahead is always the variance of the error term? Offer an intuitive explanation for why Eq. (18.22) must always hold.
18.1 Write the equation that determines the forecast () in terms of wr(1), wy(2)... for a third-order homogeneous nonstationary process; Le., derive the equivalent of Eq (18.18) for an ARIMA model with d = 3.
17.5 Using data for the 3-month Treasury bill rate (Table 17.1 on pages 514-515) (or some other short-term interest rate), specify and estimate alternative models to those in Example 17.1. Experiment with higher-order ARIMA (p. 1, q) models, and also with ARIMA (p. 2. q) models. How sensitive are
17.4 Suppose that a particular homogeneous nonstationary time series y, can be modeled as a stochastic process that is ARIMA(1, 1, 1). (a) How would you calculate the sample autocorrelation functions for y, and its differences and use them to verify that ARIMA(1, 1, 1) is indeed a proper
17.3 Repeat Exercise 17.2 for an ARMA(0, 2) model estimated for a time series that has been generated by an ARMA(2, 3) process.
17.2 Suppose that an ARMA(0, 2) model has been estimated for a time series that has been generated by an ARMA(1, 2) process. (a) How would the diagnostic test indicate that the model has been misspecified? (b) What will the residual autocorrelationsf, look like? What characteristics of these
17.1 Following the example in Appendix 17.1, show how the estimation of o. 2, and 8, for an ARMA(2, 1) model would be carried out. Go through the steps of the Taylor series expansion, show how the data series are generated, and indicate how the lineari regressions are performed.
16.8 Refer to the time series for nonfarm inventory investment in Fig. 15.3, its sample autocorrelation function in Fig. 15.4, and its partial autocorrelation function in Fig. 16.8. Can you suggest one or more ARMA(p. 4) processes that might have generated that time series?
16.7 Relate the summation operator to the backward shift operator by showing that
16.6 Suppose that y, is first-order homogeneous nonstationary, and that w, Ay, can be represented by the ARMA(1, 1) model w9w-1.6-1+1 If y, = 0 for = 0, what is E(y,) as a function of time?
16.5 Show that the autocorrelation function for the general ARMA(p. 4) process is given by = + +...+ - as in Eq. (16.63).
16.4 Derive the autocorrelation function for the ARMA(2, 1) process y=-1+ 12+880-1 = that is, determine P. Pz, etc., in terms ofd. 2, and 0. Draw this autocorrelation function for p, .6, 2.3, and 6, 9. Repeat for d = 6, 6, 3, and 6,--.9. Repeat for =.6, 2.3, and = -.9.
16.3 Show that the covariances y, of MA(q), the moving average process of ordera, are given by + Yx- +84-28)0 k=1.9 k> q and that the autocorrelation function for MA(4) is given by P = 1+7++++++ 0 as in Eq. (16.17). k-1.9 k>q
16.2 What are the characteristics that one would expect of a realization of the following MA(1) process? How would these characteristics differ from those of a realization of the following MA(1) process? y 18-1
16.1 Calculate the covariances y for MA(3), the moving average process of order 3. Determine the autocorrelation function for this process. Plot the autocorrelation function. for the MA(3) process y=1+8+88-1-581-2 +381-3
15.5 Calculate the sample autocorrelation function for retail auto sales. (Use the data in Table 14.2 at the end of Chapter 14.) Does the sample autocorrelation function indicate seasonality?
15.4. Specifi- cally, do the sample autocorrelation functions for crude oil and copper prices exhibit stationarity? Does the sample autocorrelation function for the price of lumber indicate that the series is nonstationary? (b) How robust are the Dickey-Fuller test results to the sample size?
15.3 The data series for the prices of crude oil, copper, and lumber are printed in Table 15.4.TABLE 15.4 PRICES OF CRUDE OIL, COPPER, AND LUMBER (in 1967 constant dollars) Obs. Oil Copper Lumber Obs. Oil Copper Lumber 1870 8.64 41.61 9.13 1929 3.63 36.86 32.65 1871 10.16 47.54 9.70 1930 3.64 29.21
15.2 Consider the time series 1, 2, 3, 4, 5, 6, the sample autocorrelation function of this function? 20. Is this series stationary? Calculate for k = 1, 2,....5. Can you explain the shape
15.1 Show that the random walk process with drift is first-order homogeneous nonsta- tionary.
14.6 Monthly data for retail auto sales are shown in Table 14.2 on page 438. The data are also plotted in Fig. 14.13. (a) Use a 6-month centered moving average to smooth the data. Is a seasonal pattern evident? Would you expect auto sales to exhibit seasonal regularities? (b) Using the original
14.5 Monthly data for the Standard & Poor 500 Common Stock Price Index are shown in Table 14.1. The data are also plotted in Fig. 14.12. (a) Using all but the last three data points (ie., April, May, and June of 1988). exponentially smooth the data using a value of .9 for the smoothing parametera.
14.4 Show that the EWMA forecast / periods ahead is the same as the forecast one period ahead, i.e.. Iria (1 - a)'y-
14.3 Show that the exponentially weighted moving average (EWMA) model will gener ate forecasts that are adaptive in nature.
14.2 Which (if any) of the simple extrapolation models presented in Section 14.1 do you think might be suitable for forecasting the GNP? The Consumer Price Index? A short- term interest rate? Annual production of wheat? Explain.
14.1 Go back to Example 14.1 and use the data for monthly department store sales to estimate a quadratic trend model. Use the estimated model to obtain an extrapolated value for sales for April 1974. Try to evaluate your model in comparison to the other four estimated in Example 14.1, and explain
13.3 The following equations describe a simple "cobweb" model of a competitive market: Demand: Supply: Q = b + bP-1 d
13.2 Consider the following simple multiplier-accelerator macroeconomic model: Ga- = b + b(C-G-1) Y = C++ G Note that investment is now a function of changes in consumption, rather than of changes in total GNP. (a) Determine the characteristic equation for this model, and find the associated
13.1 Show that if AA and AAS are both transient solutions to a model, i.e., both satisfy an equation such as (13.5), then the sum AA + AA must also be a solution.
Discuss some Research tools and sources of information
14 Collect foreign exchange spot rates and the corresponding forward rates for several currencies vis-à-vis the euro over a certain period. Build a panel data model to test the hypothesis that the forward premium is an unbiased predictor of future spot rate changes.
13 Repeat the above with the lagged inward FDI flows being included as an additional regressor. Apply pertinent approaches and procedures to estimate this dynamic panel data model.
12 Collect data on inward FDI at the country level for N host countries over a T year period and with K independent variables. Build and then estimate a panel data model (using EViews, LIMDEP, RATS or other packages). The dependent variable is inward FDI flows to the host country (with necessary
11 Repeat the above with the lagged debt-to-equity ratio or financial leverage being included as an additional regressor. The model becomes dynamic so pertinent approaches and procedures should be followed.
10 Collect data on variables related to the determination of corporate capital structure from FAME or Thomson ONE Banker and company annual reports.Construct a panel data set for N firms over a T year period with K independent variables. Build a panel data model where the dependent variable is the
9 Describe and discuss commonly applied approaches to estimating dynamic panel data models.
8 What is meant by dynamic panel data analysis? What kinds of issues may arise in performing dynamic panel data analysis?
7 What are featured by random parameter or random coefficient models in panel data modelling? Contrast random parameter models with random effects models.
6 What are fixed effects in panel data modelling? What are random effects in panel data modelling? Contrast them with each other with respect to the assumptions on the residual’s correlation structure.
5 Describe and contrast stacked and unstacked panel data structures in the organisation and presentation of panel data.
4 Why is it claimed that the use of panel data in finance and financial economics predated that in social-economic research, and the use is very extensive too?Provide a few of examples.
3 How do data samples used for testing PPP constitute panel data? What are the advantages of adopting panel data approaches to examining PPP?
2 Present and describe several commonly used panel data resulting from surveys in social-economic research, such as The National Longitudinal Surveys of Labor Market Experience of the US.
1 What is defined as panel data? How do cross-sectional data and longitudinal data/time series data form a panel data set?
12 Repeat the above case using Heckman’s two-stage procedure. Specifically, elaborate on the process in which a firm decides whether to engage in outward FDI or not and the process for the level of involvement of the firm, by choosing and justifying the set of independent variables for the two
11 Collect data on outward FDI at the firm level from various sources, e.g., company annual reports and relevant websites and databases. Implement a Tobit model by firstly estimating a probit model for corporate decisions to engage in outward FDI for the whole sample; and then secondly if a
10 Repeat the above case using Heckman’s two-stage procedure. Specifically elaborate on the decision process and the level of involvement process by choosing and justifying the set of independent variables for the two processes respectively.
9 Collect data on corporate use of derivatives in risk management from various sources, e.g., Thomson ONE Banker and company annual reports. Implement a Tobit model by firstly estimating a probit model for the whole sample for corporate decisions as whether to use derivatives in risk management;
8 Compare and contrast the Tobit model with the Heckman model with regard to their assumptions and estimation.
7 Present the Heckman model and illustrate the two-stage estimation procedure, paying attention to the decision process and the level of involvement process.
6 What is defined as an inverse Mill’s ratio? What is its links to the issues of bias in estimation of censored and truncated samples?
5 What is selection bias? How does selection bias arise? By what means can selection bias be detected and how can selection bias be corrected?
4 Describe and discuss the Tobit model with reference to its implementation and estimation, and the issues in its estimation.
3 Contrast censoring with truncation, and then contrast limited dependent variables associated with censoring and truncation with limited dependent variables associated with discrete choice.
2 Why and how do censoring and truncation arise? At what points do censoring and truncation usually come across?
1 What is featured by a censored sample? What is featured by a truncated sample?
10 Collect data on dual listings from various sources, e.g. Thomson ONE Banker, the websites of relevant stock exchanges and companies, and then run multinomial logistic regression for the choice of foreign stock exchanges for dual listings (using LIMDEP, GAUSS, RATS or other packages). The
9 Collect data from various sources, e.g. Acquisition Monthly, Thomson ONE Banker and company annual reports, and then estimate a probit model and a logistic regression model for the choice of payment methods in mergers and acquisitions (using LIMDEP, GAUSS, RATS or other packages). The dependent
8 What are marginal effects in discrete choice models? Why are the issues of marginal effects raised and addressed specifically for discrete choice models but not mentioned for linear regression and other linear models?
7 What are defined as ordered probit and ordered logit? What differentiates an ordered logit model from a multinomial logit model?
6 Present and describe the multinomial logit model and multinomial logistic regression, and further discuss their role in modelling discrete choice.
5 Contrast the probit model with the logit model, paying attention to their probability density functions.
4 Describe and discuss the logit model with regard to its functional form and in relation to logistic regression.
3 Describe and discuss the probit model with regard to its functional form.
2 What is defined as a limited dependent variable? What are features of limited dependent variables?
1 Describe binary choice and, in general, discrete choice, with a few examples in daily life, corporate and individual.
8 Collect data from DataStream and estimate phases and coherence for the following pairs of time series:(a) total returns of Tesco and Sainsbury’s,(b) total returns of Intel and Motorola.
7 Collect data from various sources and estimate phases and coherence for the following pairs of time series:(a) GDP of the US and the Canada,(b) GDP and retail sales of the UK.
6 Collect data from DataStream and estimate phases and coherence for the following pairs of time series:(a) the spot and forward foreign exchange rates of the UK£ vis-à-vis the US$,(b) the spot foreign exchange rates of the UK£ and Japanese yen vis-à-vis the US$.
5 Collect data from various sources and perform the Fourier transform for the following time series (using RATS, GAUSS or other packages):(a) GDP of selected countries,(b) total return series of selected companies,(c) foreign exchange rates of selected countries vis-à-vis the US$.
4 What are phases and coherence in spectral analysis? Contrast them with leads/lags and correlation in the time domain.
3 Describe the Fourier transform and the inverse Fourier transform.

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Financial Econometrics 1st Edition Peijie Wang - Solutions

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