Questions and Answers of Econometrics

Based on our discussion of individual and joint tests of hypothesis based, respectively, on the t and F tests, which of the following situations are likely?1. Reject the joint null on the basis of
Refer to Exercise 7.21.a. What are the real income and interest rate elasticities of real cash balances?b. Are the preceding elasticities statistically significant individually?c. Test the overall
From the data for 46 states in the United States for 1992, Baltagi obtained the following regression results:Log C = 4.30 – 1.24 log P + 0.17 log Y se = (0.91) (0.32)
From a sample of 209 firms, Wooldridge obtained the following regression results:where salary = salary of CEOsales = annual firm salesroe = return on equity in percentros = return on
Assuming that Y and X2, X3, . . . , Xkare jointly normally distributed and assuming that the null hypothesis is that the population partial correlations are individually equal to zero, R. A. Fisher
In studying the demand for farm tractors in the United States for the periods 1921€“1941 and 1948€“1957, Griliches€ obtained the following results:where Yt = value of
Consider the following wage-determination equation for the British economy* for the period 1950€“1969:whereW = wages and salaries per employeePF = prices of final output at factor costU =
A variation of the wage-determination equation given in Exercise 8.17 is as follows:whereW = wages and salaries per employeeV = unfilled job vacancies in Great Britain as a percentage of the total
For the demand for chicken function estimated in Eq. (8.6.24), is the estimated income elasticity equal to 1? Is the price elasticity equal to −1?
For the demand function in Eq. (8.6.24) how would you test the hypothesis that the income elasticity is equal in value but opposite in sign to the price elasticity of demand? Show the necessary
Table 5.6 gives data on GNP and four definitions of the money stock for the United States for 1970€“1983. Regressing GNP on the various definitions of money, we obtain the results shown in
State with reason whether the following statements are true, false, or uncertain. Be precise.a. The t test of significance discussed in this chapter requires that the sampling distributions of
Suppose the equation of an indifference curve between two goods isXiYi= Î²1+ Î²2XiHow would you estimate the parameters of this model? Apply the preceding model to the data in
To study the relationship between investment rate (investment expenditure as a ratio of the GDP) and savings rate (savings as a ratio of GDP), Martin Feldstein and Charles Horioka obtained data for a
The following table gives the variable definitions for various kinds of expenditures, total expenditure, income, age of household, and the number of children for a sample of 1,519 households drawn
Refer to the following table. Find out the rate of growth of expenditure on durable goods. What is the estimated semielasticity? Interpret your results. Would it make sense to run a double log
From the data given in the following table, find out the growth rate of expenditure on nondurable goods and compare your results with those obtained from Exercise 6.17. PCEXP Year or quarter
Consider the regression modelyi = β1 + β2xi + uiwhere yi = (Yi – Y̅ ) and xi = (Xi – X̅ ). In this case, the regression line must pass through the origin. True or false? Show your
Consider the log–linear model:ln Yi = β1 + β2 ln Xi + uiPlot Y on the vertical axis and X on the horizontal axis. Draw the curves showing the relationship between Y and X when β2 = 1, and when
The following table gives data for the U.K. on total consumer expenditure (in £ millions) and advertising expenditure (in £ millions) for 29 product categories.a. Considering the various
The following regression results were based on monthly data over the period January 1978 to December 1987:Ŷt = 0.00681 ................... + 0.75815Xtse = (0.02596) .................. (0.27009)t =
Consider the following models:Model I: Yi = β1 + β2Xi + uiModel II: Y*i α1 + α2X∗i + uiwhere Y* and X* are standardized variables. Show that α̂2 = β̂2(Sx/Sy ) and hence establish that
Consider the following regression model:1/Yi = β1 + β2 (1/Xi) + uiNeither Y nor X assumes zero value.a. Is this a linear regression model?b. How would you estimate this model?c. What is the
Consider the following models:ln Y∗i = α1 + α2 ln X∗i + u∗iln Yi = β1 + β2 ln Xi + uiwhere Y∗i = w1Yi and X∗i = w2Xi , the w’s being constants.a. Establish the relationships between
Between regressions (6.6.8) and (6.6.10), which model do you prefer? Why?
For the regression (6.6.8), test the hypothesis that the slope coefficient is not significantly different from 0.005.
Refer to Example 3.3 in Chapter 3 to complete the following:In Example 3.3The following table gives data on the number of cell phone subscribers and the number of personal computers (PCs), both per
Consider the following model:Yi = eβ1+β2Xi / (1 + eβ1+β2Xi)As it stands, is this a linear regression model? If not, what “trick,” if any, can you use to make it a linear regression model? How
Repeat Exercise 6.20 but refer to the demand for personal computers given in the following equation. Is there a difference between the estimated income elasticities for cell phones and personal
Graph the following models (for ease of exposition, we have omitted the observation subscript, i):a. Y = β1Xβ2, for β2 > 1, β2 = 1, 0 < β2 < 1, …b. Y = β1eβ2X, for β2 > 0 and
Refer to Problem 3.22.In exerciseTable 3.7 gives data on gold prices, the Consumer Price Index (CPI), and the New York Stock Exchange (NYSE) Index for the United States for the period 1974
Table 5.5 gives data on average public teacher pay (annual salary in dollars) and spending on public schools per pupil (dollars) in 1985 for 50 states and the District of Columbia.To find out if
Consider the following regression output:Ŷi = 0.2033 + 0.6560Xtse = (0.0976) (0.1961)r2 = 0.397 RSS = 0.0544 ESS = 0.0358where Y = labor force participation rate (LFPR) of women in 1972 and X =
The following table provides data on the lung cancer mortality index (100 = average) and the smoking index (100 = average) for 25 occupational groups.a. Plot the cancer mortality index against the
Following table gives annual data on the Consumer Price Index (CPI) and the Wholesale Price Index (WPI), also called Producer Price Index (PPI), for the U.S. economy for the period
R. A. Fisher has derived the sampling distribution of the correlation coefficient defined in Eq. (3.5.13).
Equation (5.3.5) can also be written asPr [β̂2 – 5α/2se(β̂2) < β2 < β̂2 + tα/2se (β̂2)] = 1 – αThat is, the weak inequality (≤) can be replaced by the strong inequality
Repeat the exercise in the preceding problem but let Y and X denote the male and female critical reading scores, respectively.Repeat exerciseRefer to the SAT data given in Exercise 2.16.In
Refer to the demand for cell phones regression given in Eq. (3.7.3).Eq (3.7.3)a. Is the estimated intercept coefficient significant at the 5 percent level of significance? What is the null hypothesis
Since 1986 the Economist has been publishing the Big Mac Index as a crude, and hilarious, measure of whether international currencies are at their €œcorrect€ exchange rate, as
Set up the ANOVA table in the manner of Table 5.4 for the regression model given in Eq. (3.7.2) and test the hypothesis that there is no relationship between food expenditure and total expenditure in
Suppose that the outcome of an experiment is classified as either a success or a failure. Letting X = 1 when the outcome is a success and X = 0 when it is a failure, the probability density, or mass,
A random variable X follows the exponential distribution if it has the following probability density function (PDF):f(X) = (1/θ)e-X/θ for X > 0= 0
By applying the second-order conditions for optimization (i.e., second-derivative test), show that the ML estimators of Î²1, Î²2, and Ïƒ2obtained by solving Eqs. (9),
€œIf two random variables are statistically independent, the coefficient of correlation between the two is zero. But the converse is not necessarily true; that is, zero correlation does not
Using the data given in Table 3.3, plot the number of cell phone subscribers against the number of personal computers in use. Is there any discernible relationship between the two? If so, how do you
Table 1.4 gives the foreign exchange rates for nine industrialized countries for the years 1985€“2006. Except for the United Kingdom, the exchange rate is defined as the units of foreign
Consider the following formulations of the two-variable PRF:Model I: Yi = β1 + β2Xi + uiModel II: Yi = α1 + α2(Xi – X̅ ) + uia. Find the estimators of β1 and α1. Are they identical?
Suppose you run the following regression:Yi = β̂1 + β̂2xi +ûiwhere, as usual, yi and xi are deviations from their respective mean values.What will be the value of β̂1? Why? Will β̂2 be the
Let r1= coefficient of correlation between n pairs of values (Yi, Xi) and r2= coefficient of correlation between n pairs of values (aXi+ b, cYi+ d), where a, b, c, and d are constants. Show that r1=
From a sample of 10 observations, the following results were obtained:ΣYi = 1,110 ΣXi = 1,700 ΣXiYi = 205,500ΣXi2 = 322,000
The following table gives data on gold prices, the Consumer Price Index (CPI), and the New York Stock Exchange (NYSE) Index for the United States for the period 1974 €“2006. The NYSE Index
If X1, X2, and X3 are uncorrelated variables each having the same standard deviation, show that the coefficient of correlation between X1 + X2 and X2 + X3 is equal to 1/2. Why is the correlation
Show that the estimates Î²Ì‚1= 1.572 and Î²Ì‚2= 1.357 used in the first experiment of Table 3.1 are in fact the OLS estimators.Table 3.1 Y2i (6) Y; X¢
According to Malinvaud (see footnote 11), the assumption that E(ui | Xi) = 0 is quite important. To see this, consider the PRF: Y = β1 + β2Xi + ui . Now consider two situations:(i) β1 = 0, β2 =
Table 3.8 gives data on gross domestic product (GDP) for the United States for the years 1959 2005.a. Plot the GDP data in current and constant (i.e., 2000) dollars against time.b. Letting Y denote
Consider the sample regressionYi = β̂1 + β̂2Xi + ûiImposing the restrictions (i) Σûi = 0 and (ii) Σ ûiXi = 0, obtain the estimators β̂1 and β̂2 and show that they are identical
In the regression Yi = β1 + β2Xi + ui suppose we multiply each X value by a constant, say, 2. Will it change the residuals and fitted values of Y? Explain. What if we add a constant value, say, 2,
Show that r2 defined in (3.5.5) ranges between 0 and 1. You may use the Cauchy–Schwarz inequality, which states that for any random variables X and Y the following relationship holds true:[E (XY)]2
Let β̂YX and β̂XY represent the slopes in the regression of Y on X and X on Y, respectively. Show thatβ̂YX β̂XY = r2Where r is the coefficient of correlation between X and Y.
Spearman’s rank correlation coefficient rs is defined as follows:rs = 1 – 6 Σd2 / n(n2 – 1)Where d = difference in the ranks assigned to the same individual or phenomenon and n = number of
Suppose in Exercise 3.6 that β̂YX β̂XY = 1. Does it matter then if we regress Y on X or X on Y? Explain carefully.
For the SAT example given in Exercise 2.16 do the following:In exerciseTable 2.9 gives data on mean Scholastic Aptitude Test (SAT) scores for collegebound seniors for 1972€“2007. These data
Show that Eq. (3.5.14) in fact measures the coefficient of determination.Apply the definition of r given in Eq. (3.5.13) and recall that Î£yiyÌ‚i =
Table 3.6 gives data on indexes of output per hour (X) and real compensation per hour (Y) for the business and nonfarm business sectors of the U.S. economy for 1960€“2005. The base year of
The relationship between nominal exchange rate and relative prices. From annual observations from 1985 to 2005, the following regression results were obtained, where Y = exchange rate of the
In Table 3.5, you are given the ranks of 10 students in midterm and final examinations in statistics. Compute Spearman€™s coefficient of rank correlation and interpret it. Student Rank E D
Regression without any regressor. Suppose you are given the model: Yi = β1 + ui.Use OLS to find the estimator of β1. What is its variance and the RSS? Does the estimated β1 make intuitive sense?
Explain with reason whether the following statements are true, false, or uncertain:a. Since the correlation between two variables, Y and X, can range from −1 to +1, this also means that cov (Y, X)
What is meant by an intrinsically linear regression model? If β2 in Exercise 2.7d were 0.8, would it be a linear or nonlinear regression model?
Are the following models linear regression models? Why or why not?a. Yi = eβ1 + β2Xi + uib. Yi = 1 / 1 + eβ1 + β2Xi + uic. ln Yi = β1 + β2 (1/Xi) + uid. Yi = β1 + (0.75-β1)e-β2(Xi-2) +uie.
Determine whether the following models are linear in the parameters, or the variables, or both. Which of these models are linear regression models? Descriptive Title Model Reciprocal Semilogarithmic
What do we mean by a linear regression model?
Why do we need regression analysis? Why not simply use the mean value of the regressand as its best value?
What is the role of the stochastic error term ui in regression analysis? What is the difference between the stochastic error term and the residual, ûi?
What is the difference between the population and sample regression functions? Is this a distinction without difference?
What is the conditional expectation function or the population regression function?
Table 2.10 presents data on mean SAT reasoning test scores classified by income for three kinds of tests: critical reading, mathematics, and writing. In Example 2.2, we presented Figure 2.7, which
Table 2.9 gives data on mean Scholastic Aptitude Test (SAT) scores for collegebound seniors for 1972€“2007. These data represent the critical reading and mathematics test scores for both
Table 2.8 gives data on expenditure on food and total expenditure, measured in rupees, for a sample of 55 rural households from India. (In early 2000, a U.S. dollar was about 40 Indian rupees.)a.
You are given the data in Table 2.7 for the United States for years 1980€“2006.a. Plot the male civilian labor force participation rate against male civilian unemployment rate. Eyeball a
Is the regression line shown in Figure I.3 of the Introduction the PRF or the SRF? Why? How would you interpret the scatterpoints around the regression line? Besides GDP, what other factors, or
What does the scattergram in Figure 2.10 reveal? On the basis of this diagram, would you argue that minimum wage laws are good for economic well being? Ratio of one year's salary at minimum wage to
You are given the scattergram in Figure 2.8 along with the regression line. What general conclusion do you draw from this diagram? Is the regression line sketched in the diagram a population
Consider the following nonstochastic models (i.e., models without the stochastic error term). Are they linear regression models? If not, is it possible, by suitable algebraic manipulations, to
The data presented in Table 1.6 were published in the March 1, 1984, issue of The Wall Street Journal. They relate to the advertising budget (in millions of dollars) of 21 firms for 1983 and millions
Controlled experiments in economics: On April 7, 2000, President Clinton signed into law a bill passed by both Houses of the U.S. Congress that lifted earnings limitations on Social Security
Suppose you were to develop an economic model of criminal activities, say, the hours spent in criminal activities (e.g., selling illegal drugs). What variables would you consider in developing such a
The data behind the M1 money supply in Figure 1.5 are given in Table 1.5. Can you give reasons why the money supply has been increasing over the time period shown in the table?Table 1.5 Seasonally
a. Using Table 1.3, plot the inflation rate of Canada, France, Germany, Italy, Japan, and the United Kingdom against the United States inflation rate.b. Comment generally about the behavior of the
Table 1.3 gives data on the Consumer Price Index (CPI) for seven industrialized countries with 1982€“1984 = 100 as the base of the index.a. From the given data, compute the inflation rate
Calculating VIFs typically involves running sets of auxiliary regressions, one regression for each independent variable in an equation. To get practice with this procedure, calculate the following:a.
Simultaneous equations make sense in cross-sectional as well as time-series applications. For example, James Ragan examined the effects of unemployment insurance (hereafter UI) eligibility standards
As an exercise to gain familiarity with the 2SLS program on your computer, take the data provided for the simple Keynesian model in Section 14.3, and:a. Estimate the investment function with OLS.b.
The word recursive is used to describe an equation that has an impact on a simultaneous system without any feedback from the system to the equation. Which of the equations in the following systems
In 2008, Goldman and Romley studied hospital demand by analyzing how 8,721 Medicare-covered pneumonia patients chose from among 117 hospitals in the greater Los Angeles area. The authors concluded
Because their college had just upgraded its residence halls, two seniors decided to build a model of the decision to live on campus. They collected data from 533 upper-class students (first-year
In 2001, Heo and Tan published an article in which they used the Granger causality model to test the relationship between economic growth and democracy. For years, political scientists have noted a
Some farmers were interested in predicting inches of growth of corn as a function of rainfall on a monthly basis, so they collected data from the growing season and estimated an equation of the
You€™ve been hired to determine the impact of advertising on gross sales revenue for €œFour Musketeers€ candy bars. Four Musketeers has the same price and more or less

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