Questions and Answers of Econometrics

3.1 In a population, mY = 75 and s2 Y = 45. Use the central limit theorem to answer the following questions:a. In a random sample of size n = 50, find Pr(Y 6 73).b. In a random sample of size n = 90,
3.8 Sketch a hypothetical scatterplot for a sample of size 10 for two random variables with a population correlation of (a) 1.0; (b) −1.0; (c) 0.9; (d) −0.5;(e) 0.0.
3.7 What is a scatterplot? What statistical features of a dataset can be represented using a scatterplot diagram?
3.6 Why does a confidence interval contain more information than the result of a single hypothesis test?
3.5 What is the difference between a null hypothesis and an alternative hypothesis? Among size, significance level, and power? Between a onesided alternative hypothesis and a two-sided alternative
3.4 Differentiate between standard error and standard deviation. How is the standard error of a sample mean calculated?
3.3 A population distribution has a mean of 15 and a variance of 10. Determine the mean and variance of Y from an i.i.d. sample from this population for(a) n = 50; (b) n = 500; and (c) n = 5000.
3.2 What is meant by the efficiency of an estimator? Which estimator is known as BLUE?
3.1 What is the difference between an unbiased estimator and a consistent estimator?
E2.1 On the text website, http://www.pearsonglobaleditions.com/Stock_Watson, you will find the spreadsheet Age_HourlyEarnings, which contains the joint distribution of age (Age) and average hourly
2.27 X and Z are two jointly distributed random variables. Suppose you know the value of Z, but not the value of X. Let X= E(X Z) denote a guess of the value of X using the information on Z, and
2.26 Suppose that Y1, Y2,c, Yn are random variables with a common mean mY, a common variance s2Y, and the same correlation r (so that the correlation between Yi and Yj is equal to r for all pairs i
2.25 (Review of summation notation) Let x1,c, xn denote a sequence of numbers, y1,c, yn denote another sequence of numbers, anda, b, and c denote three constants. Show thata. a ni = 1 axi = aa ni = 1
2.24 Suppose Yi is distributed i.i.d. N(0, s2) for i = 1, 2,c, n.a. Show that E(Y 2i > s2) = 1.b. Show that W = (1>s2)gni= 1Y 2 i is distributed x2 n.c. Show that E(W) = n. [Hint: Use your answer to
2.23 This exercise provides an example of a pair of random variables X and Y for which the conditional mean of Y given X depends on X but corr(X, Y) = 0. Let X and Z be two independently distributed
2.22 Suppose you have some money to invest—for simplicity, $1—and you are planning to put a fraction w into a stock market mutual fund and the rest, 1 - w, into a bond mutual fund. Suppose that
2.21 X is a random variable with moments E(X), E(X2), E(X3), and so forth.a. Show E(X - m)3 = E(X3) - 3[E(X2)][E(X)] + 2[E(X)]3.b. Show E(X - m)4 = E(X4) - 4[E(X)][E(X3)] + 6[E(X)]2[E(X2)] -3[E(X)]4.
2.20 Consider three random variables X, Y, and Z. Suppose that Y takes on k values y1,c, yk, that X takes on l values x1,c, xl, and that Z takes on m values z1,c, zm. The joint probability
2.19 Consider two random variables X and Y. Suppose that Y takes on k values y1,c, yk and that X takes on l values x1,c, xl.a. Show that Pr(Y = yj) = gli= 1Pr(Y = yj X = xi) Pr(X = xi). [Hint:Use
2.18 In any year, the weather may cause damages to a home. On a year-to-year basis, the damage is random. Let Y denote the dollar value of damages in any given year. Suppose that during 95% of the
2.17 Yi, i = 1,c, n, are i.i.d. Bernoulli random variables with p = 0.6. Let Y denote the sample mean.a. Use the central limit to compute approximations for i. Pr(Y 7 0.64) when n = 50.ii. Pr(Y 6
2.16 Y is distributed N(5, 100) and you want to calculate Pr(Y 6 3.6). Unfortunately, you do not have your textbook, and do not have access to a normal probability table like Appendix Table 1.
2.15 Suppose Yi, i = 1, 2,c, n, are i.i.d. random variables, each distributed N(10, 4).a. Compute Pr(9.6 … Y … 10.4) when (i) n = 20, (ii) n = 100, and(iii) n = 1000.b. Suppose c is a positive
2.14 In a population mY = 50 and J2 Y = 21. Use the central limit theorem to answer the following questions:a. In a random sample of size n = 50, find Pr(Y … 51).b. In a random sample of size n =
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,
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
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
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
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.
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
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
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:
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
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
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
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
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
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
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)
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.
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
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
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
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
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?
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.
18.1 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
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
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
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
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
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
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,
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
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)
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
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
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
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
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,
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
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.
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
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
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
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
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
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
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
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
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
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

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