Introduction To Econometrics 2nd Edition G. S. Maddala - Solutions

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4.2 Explain what is meant by the error term. What assumptions do we make about the error term when estimating an OLS regression?
4.1 What is a linear regression model? What is measured by the coefficients of a linear regression model—intercept b0 and slope b1? What is the ordinary least squares estimator?
E3.2 A consumer is given the chance to buy a baseball card for $1, but he declines the trade. If the consumer is now given the baseball card, will he be willing to sell it for $1? Standard consumer theory suggests yes, but behavioral economists have found that “ownership” tends to increase the
E3.1 On the text website, http://www.pearsonglobaleditions.com, you will find the data file CPS96_15, which contains an extended version of the data set used in Table 3.1 of the text for the years 1996 and 2015. It contains data on full-time workers, ages 25–34, with a high school diploma or a
3.22 Suppose Yi i.i.d.N1mY, s2 Y2 for i = 1,c, n. With s2 Y known, the t-statistic for testing H0: mY = 0 vs. H1: mY 7 0 is t = 1Y - 02 >SE1Y2, where SE1Y2 = sY>2n. Suppose sY = 10 and n = 100, so that SE1Y2 = 1. Using a test with a size of 5%, the null hypothesis is rejected if t 7 1.64.a.
3.21 Show that the pooled standard error 3SEpooled 1Ym - Yw24 given following Equation (3.23) equals the usual standard error for the difference in means in Equation (3.19) when the two group sizes are the same 1nm = nw2.
3.20 Suppose 1Xi, Yi2 are i.i.d. with finite fourth moments. Prove that the sample covariance is a consistent estimator of the population covariance; that is, sXY ¡P sXY, where sXY is defined in Equation (3.24). (Hint: Use the strategy of Appendix 3.3.)
3.19a. Y is an unbiased estimator of mY. Is Y2 an unbiased estimator of m2 Y?b. Y is a consistent estimator of mY. Is Y2 a consistent estimator of m2 Y?
3.18 This exercise shows that the sample variance is an unbiased estimator of the population variance when Y1,c, Yn are i.i.d. with mean mY and variance s2 Y.a. Use Equation (2.32) to show that E1Yi - Y2 2 = var1Yi 2 - 2cov1Yi, Y2 + var1Y2.b. Use Equation (2.34) to show that cov1Y, Yi 2 = s2 Y >
3.17 Read the box “Social Class or Education? Childhood Circumstances and Adult Earnings Revisited” in Section 3.5.a. Construct a 95% confidence interval for the difference in the household earnings of people whose father NS-SEC classification was higher between those with no educational
3.16 Assume that grades on a standardized test are known to have a mean of 500 for students in Europe. The test is administered to 600 randomly selected students in Ukraine; in this sample, the mean is 508, and the standard deviation (s) is 75.a. Construct a 95% confidence interval for the average
3.15 Ya and Yb are Bernoulli random variables from two different populations, denoted a andb. Suppose E(Ya) = pa and E(Yb) = pb. A random sample of size na is chosen from populationa, with a sample average denoted pna, and a random sample of size nb is chosen from populationb, with a sample average
3.14 Values of height in inches (X) and weight in pounds (Y) are recorded from a sample of 200 male college students. The resulting summary statistics are X = 71.2 in., Y = 164 lb, sX = 1.9 in., sY = 16.4 lb, sXY = 22.54 in. * lb, and rXY = 0.8. Convert these statistics to the metric system (meters
3.13 Data on fifth-grade test scores (reading and mathematics) for 400 school districts in Brussels yield average score Y = 712.1 and standard deviation sY = 23.2.a. Construct a 90% confidence interval for the mean test score in the population.b. When the districts were divided into districts with
3.12 To investigate possible gender discrimination in a British firm, a sample of 120 men and 150 women with similar job descriptions are selected at random. A summary of the resulting monthly salaries follows:Average Salary 1Y2 Standard Deviation 1sY2 n Men £8200 £450 120 Women £7900 £520
3.11 Consider the estimator Y, defined in Equation (3.1). Show that (a) E1Y2 = mY and (b) var1Y2 = 1.25s2 Y > n.
3.10 Suppose a new standardized test is given to 150 randomly selected third-grade students in Amsterdam. The sample average score Y on the test is 42 points, and the sample standard deviation, sY, is 6 points.a. The authors plan to administer the test to all third-grade students in Amsterdam.
3.9 Suppose that a plant manufactures integrated circuits with a mean life of 1000 hours and a standard deviation of 100 hours. An inventor claims to have developed an improved process that produces integrated circuits with a longer mean life and the same standard deviation. The plant manager
3.8 A new version of the SAT is given to 1500 randomly selected high school seniors. The sample mean test score is 1230, and the sample standard deviation is 145. Construct a 95% confidence interval for the population mean test score for high school seniors.
3.7 In a given population, 50% of the likely voters are women. A survey using a simple random sample of 1000 landline telephone numbers finds 55%women. Is there evidence that the survey is biased? Explain.
3.6 Let Y1c, Yn be i.i.d. draws from a distribution with mean m. A test of H0 : m = 10 vs. H1 : m 10 using the usual t-statistic yields a p-value of 0.07.a. Does the 90% confidence interval contain m = 10? Explain.b. Can you determine if m = 8 is contained in the 95% confidence interval? Explain.
3.5 A survey of 1000 registered voters is conducted, and the voters are asked to choose between candidate A and candidate B. Let p denote the fraction of voters in the population who prefer candidate A, and let pn denote the fraction of voters in the sample who prefer candidate A.a. You are
3.4 Using the data in Exercise 3.3:a. Construct a 95% confidence interval for p.b. Construct a 99% confidence interval for p.c. Why is the interval in (b) wider than the interval in (a)?d. Without doing any additional calculations, test the hypothesis H0 : p = 0.50 vs. H1 : p 0.50 at the 5%
3.3 In a poll of 500 likely voters, 270 responded that they would vote for the candidate from the democratic party, while 230 responded that they would vote for the candidate from the republican party. Let p denote the fraction of all likely voters who preferred the democratic candidate at the time
3.2 Let Y be a Bernoulli random variable with success probability Pr1Y = 12 = p, and let Y1,c, Yn be i.i.d. draws from this distribution. Let pn be the fraction of successes (1s) in this sample.a. Show that pn = Y.b. Show that pn is an unbiased estimator of p.c. Show that var1pn 2 = p11 - p2 >n.
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 Pr1Y 6 732.b. In a random sample of size n = 90, find Pr176 6 Y 6 772.c. In a random sample of size n = 120, find Pr1Y 7 692.
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; and (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 one-sided alternative hypothesis and a two-sided alternative hypothesis?
3.4 What is the difference between standard error and standard deviation? How is the standard error of the 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 = 5; (b) n = 500; and (c) n = 5000. Relate your answers to the law of large numbers.
3.2 What is meant by the efficiency of an estimator? Which estimator is known as BLUE?
3.1 Explain the difference between an unbiased estimator and a consistent estimator.
E2.1 On the text website, http://www.pearsonglobaleditions.com, you will find the spreadsheet Age_HourlyEarnings, which contains the joint distribution of age (Age) and average hourly earnings (AHE) for 25- to 34-year-old full-time workers in 2015 with an education level that exceeds a high school
2.28 Refer to Part B of Table 2.3 for the conditional distribution of the number of network failures M given network age A. Let Pr1A = 02 = 0.5; that is, you work in your room 50% of the time.a. Compute the probability of three network failures, Pr1M = 32.b. Use Bayes’ rule to compute Pr1A = 0 M
2.27 Consider the problem of predicting Y using another variable, X, so that the prediction of Y is some function of X, say g1X2. Suppose that the quality of the prediction is measured by the squared prediction error made on average over all predictions, that is, by E53Y - g1X2426. This exercise
2.26 Suppose that Y1, Y2,c, Yn are random variables with a common mean mY;a common variance s2 Y; and the same correlation r (so that the correlation between Yi and Yj is equal to r for all pairs i and j, where i j).a. Show that cov1Yi, Yj 2 = rs2 Y for i j.b. Suppose that n = 2. Show that E1Y2
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 xi,b. a ni = 1 1xi + yi2 = a ni = 1 xi + a ni = 1 yi,c. a ni = 1 a = n *a, andd. a ni = 1 1a + bxi +
2.24 Suppose Yi is distributed i.i.d. N10, s22 for i = 1, 2,c, n.a. Show that E1Y2 i > s22 = 1.b. Show that W = 11 > s22gn i = 1Y2 i is distributed x2 n.c. Show that E1W2 = n. [Hint: Use your answer to (a).]d. Show that V = Y1 C gn i = 2Y2 in - 1 is distributed tn - 1.
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 corr1X, Y2 = 0.Let X and Z be two independently distributed standard normal random variables, and let Y = X2 + Z.a. Show that E1YX2 = X2.b. Show that mY =
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 mutual fund. Suppose that $1 invested in a stock fund yields Rs after one year and that $1 invested in mutual fund yields. Rb. Suppose
2.21 X is a random variable with moments E1X2, E1X22, E1X32, and so forth.a. Show E1X - m2 3 = E1X32 - 33E1X2243E1X24 + 23E1X243.b. Show E1X - m24 = E1X42 - 43E1X243E1X324 + 63E1X2423E1X224 - 33E1X244.
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 distribution of X, Y, Z is Pr 1X = x, Y = y, Z = z2, and the conditional probability distribution of Y given
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 Pr1Y = yj2 = gl i = 1Pr1Y = yj X = xi2 Pr1X = xi2. [Hint:Use the definition of Pr1Y = yj X = xi2.]b. Use your answer to (a) to verify Equation (2.19).c.
2.18 In any year, the weather can inflict storm damage to a home. From year to year, the damage is random. Let Y denote the dollar value of damage in any given year. Suppose that in 95% of the years Y = $0, but in 5% of the years Y = $30,000.a. What are the mean and standard deviation of the damage
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 theorem to compute approximations for i. Pr1Y Ú 0.642 when n = 50.ii. Pr1Y … 0.562 when n = 200.b. How large would n need to be to ensure that Pr10.65 7 Y 7 0.552 =
2.16 Y is distributed N(10, 100) and you want to calculate Pr1Y … 5.82. Unfortunately, you do not have your textbook, and do not have access to a normal probability table like Appendix Table 1. However, you do have your computer and a computer program that can generate i.i.d. draws from the N(10,
2.15 Suppose Yi, I = 1, 2,c, n are i.i.d. random variables, each distributed N120, 42.a. Compute Pr119.6 … Y … 20.42 when (i) n = 25, (ii) n = 100, and(iii) n = 800.b. Suppose c is a positive number. Show that Pr120 - c … Y … 20 + c2 becomes close to 1.0 as n grows large.c. Use your answer
2.14 In a population, mY = 50 and s2 Y = 21. Use the central limit theorem to answer the following questions:a. In a random sample of size n = 50, find Pr1Y … 512.b. In a random sample of size n = 150, find Pr1Y 7 492.c. In a random sample of size n = 45, find Pr150.5 … Y … 512.
2.13 X is a Bernoulli random variable with Pr1X = 12 = 0.90; Y is distributed N(0, 4); W is distributed N(0, 16); and X, Y, and W are independent.Let S = XY + 11 - X2W. (That is, S = Y when X = 1, and S = W when X = 0.)a. Show that E1Y22 = 4 and E1W22 = 16.b. Show that E1Y32 = 0 and E1W32 = 0.
2.12 Compute the following probabilities:a. If Y is distributed t12, find Pr1Y … 1.362.b. If Y is distributed t30, find Pr1 -1.70 … Y … 1.702.c. If Y is distributed N(0, 1), find Pr1 -1.70 … Y … 1.702.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 1Y … 6.252.b. If Y is distributed x28, find Pr1Y … 15.512.c. If Y is distributed F8, , find Pr1Y … 1.942.d. Why are the answers to (b) and (c) the same?e. If Y is distributed x21, find Pr1Y … 0.52. (Hint: Use the
2.10 Compute the following probabilities:a. If Y is distributed N(4, 9), find Pr 1Y … 52.b. If Y is distributed N(5, 16), find Pr 1Y 7 22.c. If Y is distributed N(1, 4), find Pr 12 … Y … 52.d. If Y is distributed N(2, 1), find Pr 11 … Y … 42.
2.9 X and Y are discrete random variables with the following joint distribution:Exercises 95 Value of Y 2 4 6 8 10 Value of X 3 0.04 0.09 0.03 0.12 0.01 6 0.10 0.06 0.15 0.03 0.02 9 0.13 0.11 0.04 0.06 0.01 That is, Pr1X = 3, Y = 22 = 0.04, and so forth.a. Calculate the probability distribution,
2.8 The random variable Y has a mean of 4 and a variance of 19. Let Z = 31Y - 42.Find the mean and the variance of Z.
2.7 In a given population of two-earner 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 South Africa.a. Compute E(Y).b. The unemployment rate is the fraction of the labor force
2.5 In July, Lugano’s, a city in Switzerland, daily high temperature has a mean of 65oF and a standard deviation of 5oF. What are the mean, standard deviation, and variance in degrees Celsius?
2.4 Suppose X is a Bernoulli random variable with Pr1X = 12 = p.a. Show E1X42 = p.b. Show E1Xk2 = p for k 7 0.c. Suppose that p = 0.53. 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) s2 W and s2 V; and (c) sWV 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 coins are tossed.Assume the probability of a 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 = 90.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 probability distribution given in Figure 2.10a. You want to calculate Pr1Y … 0.22. Would it be reasonable to use normal approximation if n = 8? How about when n = 30 and n = 150? Explain.
2.5 Suppose that Y1,c, Yn are i.i.d. random variables with a N(2, 6) distribution.Sketch the probability density of Y when n = 2. Repeat this for n = 15 and n = 200. Describe how the densities differ. What is the relationship between your answers and the law of large numbers?
2.4 A math class has 100 students, and the mean student weight is 65 kg. A random sample of five students is selected from the class, and their average weight is calculated. Will the average weight of the students in the sample equal 65 kg? Why or why not? Use this example to explain why the sample
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 sex of the next person you meet, (b) the number of times a wireless network fails,(c) the time it takes to commute to school, and (d) whether it is raining or not.Explain why each can be thought of as random.
1.3 You are asked to study the causal effect of hours spent on employee training(measured in hours per worker per week) in a manufacturing plant on the productivity of its workers (output per worker per hour). Describe:a. an ideal randomized controlled experiment to measure this causal effect;b. an
1.2 Describe 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 Describe a hypothetical ideal randomized controlled experiment to study the effect of six hours of reading on the improvement of the vocabulary of high school students. Suggest some impediments to implementing this experiment in practice.
Illustrate the use of the AIC and BIC criteria in the choice between differ- ent ARMA models using one of these data sets.
Repeat Exercise 12 with the data in Table 13.4.
Repeat Exercise II with the data in Table 13.4.
Repeat Exercise 10 with the quarterly data on consumption and income in Table 13.4.
The seasonal effect in the two series in Table 13.7 can be taken into account by using seasonal dummies or by seasonal differencing as in the Box-Jen- kins approach. Which of these two methods is appropriate for these data sets?
In Exercise 10 compute the acf and plot the correlogram to determine whether the data are trend stationary or difference stationary. If they are difference stationary, determine the appropriate order of autoregression for the first differences of the two series using the methods in Table 13.1.
Table 13.7 gives data on monthly short-term interest rate (MRI) and the long-term interest rate (MR 240) for the period February 1950-December 1982. For each of the series estimate a regression on a time trend and sea- sonal dummies. Compute the R. Using the discussion of the different R measures
Suppose that the correlogram of a time series consisting of 100 observa- tions has 0.50. r = 0.63, r = -0.10. r = 0.08, r = -0.17, r = 0.13, 0.09, r = -0.05, r, 0.12, o -0.05. Suggest an ARMA model that would be appropriate. (Hint: The SE of each of the correlations 1/VN = 0.10. Values greater than
For a second-order AR process, show that the (theoretical) partial auto- correlation coefficient of order 2 is given by (pp)/(1 - p).
Consider the ARMA model X, 1.0x-0.5x+, -0.9,- + 0.2,-2 Expresse, as a function of X, and lagged values of X, by expanding &, (1 0.9L + 0.21) (11.0L + 0.5L)X, in powers of L.
Table 13.2 at the end of the chapter gives the Beveridge index, and Figure 13.2 gives the correlogram for the trend-free index (with trend eliminated by the ratio to moving-average method). Plot the correlograms for the raw index and the first differences of the series.
Which of the following AR(2) processes are stable? (a) X, 0.9X,, -0.2X-2+,. (b) X, (c) X, 0.8X +0.4X2 + ... 1.0X0.8X-2+.
Starting with the same initial value X, graph the difference stationary series X, X, 10+ E,..
Figure 13.1 shows the graph of the stationary series X, = 0.7X,- + E, with IN(5, 1). To generate the series we need the starting value X. We took ELX). We find it from the equation E(X) = 0.7E(X,-1) + E(E,) 0.7 +5. This gives 5/0.3 16.67. Using the same X, graph the nonstationary series X,
Explain how each of the following tests can be considered as a test for the coefficients of some extra variables added to the original equation. (a) Test for serial correlation. (b) Test for omitted variables. (c) Test for errors in variables. (d) Test for exogeneity. (e) J-test. (f) PSW
Comment on the following: In a study on the demand for money, an econ- omist regressed real cash balances on permanent income. To test whether the interest rate has been omitted, she regressed the residual from this regression on the interest rate and found that the coefficient was not signif-
Examine whether the following statements are true, false, or uncertain. Give a short explanation. If a statement is not true in general but is true under some conditions, state the conditions. (a) Since the least squares residuals are heteroskedastic and autocorre lated, even when the true errors
Explain the meaning of each of the following terms. (a) Least squares residual. (b) Predicted residual. (c) Studentized residual. (d) Recursive residual. (e) BLUS residual. (f) DFFITS. (g) Bounded influence estimation. (h) ARCH errors. (i) Post-data model construction. (i) Overparametrization with
Future prices have often been used as a proxy for expected prices in the cases of estimation of supply functions for agricultural commodities. How do you test whether the expectations implied in the future prices are ra- tional?
What is meant by weak tests for rationality? Are these tests really weak?
Explain tests for rationality based on (a) y- (b) var y, and var y,. (c) Tests for equality of coefficients.
Answer Exercise 4 with the Australian wine data discussed in Section 9.5 (data are in Table 9.2). Assume that the supply depends on expected price.
Estimate the hyperinflation model for Hungary and Germany using the data in Tables 10.1 and 10.2 and assuming the following. (a) Naive expectations Pr. = Pr. (b) Adaptive expectations. (c) Rational expectations.
In the demand and supply model for pork discussed in Section 9.3 (data are in Table 9.1), assume that supply depends on expected price, P. Estimate the model assuming the following. (a) Naive expectations P P. This is the Cobweb model. (b) Adaptive expectations. (c) Rational expectations.
Answer Exercise 2 if, instead of assuming adaptive expectations, we assume the following. (a) Naive expectations y, y-- (b) Rational expectations.
What are the problems one encounters in the OLS estimation under adaptive expectations in the following models? (a) Models of agricultural supply. (b) Models of hyperinflation. (c) Partial adjustment models. (d) Error correction models.
Explain the meaning of each of the following terms. (a) Adaptive expectations. (b) Regressive expectations. (c) Rational expectations.

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