Questions and Answers of Econometrics

Refer to the Longley data given in Section 10.10. Repeat the regression given in the table there by omitting the data for 1962; that is, run the regression for the period 1947–1961. Compare the two
Updated Longley data. We have extended the data given in Section 10.10 to include observations from 1959€“2005. The new data are in the following table. The data pertain to Y = number of
The following table gives data on median salaries of full professors in statistics in research universities in the United States for the academic year 2007.a. Plot median salaries against years in
You are given the following data:RSS1 based on the first 30 observations = 55, df = 25RSS2 based on the last 30 observations = 140, df = 25Carry out the Goldfeld–Quandt test of heteroscedasticity
The following table gives data on percent change per year for stock prices (Y) and consumer prices (X) for a cross section of 20 countries.a. Plot the data in a scattergram.b. Regress Y on X and
Table 11.10 from the website gives salary and related data on 447 executives of Fortune 500 companies. Data include salary = 1999 salary and bonuses; totcomp = 1999 CEO total compensation; tenure =
State with brief reason whether the following statements are true, false, or uncertain:a. In the presence of heteroscedasticity OLS estimators are biased as well as inefficient.b. If
In a regression of average wages (W, $) on the number of employees (N) for a random sample of 30 firms, the following regression results were obtained: Ŵ = 7.5 + 0.009N
a. Can you estimate the parameters of the modelsby the method of ordinary least squares? Why or why not?b. If not, can you suggest a method, informal or formal, of estimating the parameters of such
Prove that if wi = w, a constant, for each i, β∗2 and β̂2 as well as their variance are identical.
Refer to formulas (11.2.2) and (11.2.3). AssumeÏƒ2i = Ïƒ2kiwhere Ïƒ2 is a constant and where ki are known weights, not necessarily all equal.Using this assumption, show
In the modelYi = Î²2Xi + ui (There is no intercept)you are told that var (ui) = Ïƒ2X2i . Show that 2ΣΧΧ (ΣΧ) var (B2) =
Consider the three-variable linear regression model discussed in this chapter.a. Suppose you multiply all the X2 values by 2. What will be the effect of this rescaling, if any, on the estimates of
Determinants of price per ounce of cola. Cathy Schaefer, a student of mine, estimated the following regression from cross-sectional data of 77 observations:Pi = Î²0 + Î²1D1i +
Refer to the piecewise regression discussed in the text. Suppose there not only is a change in the slope coefficient at Xˆ—but also the regression line jumps, as shown in the following
In his study on the labor hours spent by the FDIC (Federal Deposit Insurance Corporation) on 91 bank examinations, R. J. Miller estimated the following function:WhereY = FDIC examiner labor hoursX1 =
Refer to the U.S. savings–income example discussed in Section 9.5.a. How would you obtain the standard errors of the regression coefficients given in Eqs. (9.5.5) and (9.5.6), which were obtained
Refer to regression (9.7.3). How would you test the hypothesis that the coefficients of D2 and D3 are the same? And that the coefficients of D2 and D4 are the same? If the coefficient of D3 is
To assess the effect of state right-to-work laws (which do not require membership in the union as a precondition of employment) on union membership, the following regression results were obtained,
In the following regression model:Yi = β1 + β2Di + uiY represents hourly wage in dollars and D is the dummy variable, taking a value of 1 for a college graduate and a value of 0 for a high-school
To study the rate of growth of population in Belize over the period 1970€“1992, Mukherjee et al. estimated the following models:where Pop = population in millions, t = trend variable, Dt =
Using the data given in the following, test the hypothesis that the error variances in the two subperiods 1958€“IV to 1966€“III and 1966€“IV to 1971€“II are
Using the methodology discussed in Chapter 8, compare the unrestricted and restricted regressions (9.7.3) and (9.7.4); that is, test for the validity of the imposed restrictions.
In the U.S. savings–income regression (9.5.4) discussed in the chapter, suppose that instead of using 1 and 0 values for the dummy variable you use Zi = a + bDi, where Di = 1 and 0, a = 2, and b =
Continuing with the savings–income regression (9.5.4), suppose you were to assign Di = 0 to observations in the second period and Di = 1 to observations in the first period. How would the results
Use the data given in the following table and consider the following model:ln Savingsi = Î²1 + Î²2 ln Incomei + Î²3 ln Di + uiwhere ln stands for natural log and
a. Show that if r1i = 0 for i = 2, 3, . . . , k then R1.23. . . k = 0b. What is the importance of this finding for the regression of variable X1(=Y) on X2, X3, . . . , Xk?
State with reason whether the following statements are true, false, or uncertain:a. Despite perfect multicollinearity, OLS estimators are BLUE.b. In cases of high multicollinearity, it is not
Consider the following modelYi = α1 + α2Di + βXi + uiwhere Y = annual salary of a college professorX = years of teaching experienceD = dummy for genderConsider three ways of defining the dummy
Refer to the quarterly appliance sales data given in the following table. Consider the following model:Salesi = Î±1 + Î±2D2i + Î±3D3i + Î±4D4i +
The following table 0gives data on quadrennial presidential elections in the United States from 1916 to 2004.*a. Using the data given in the following table, develop a suitable model to predict the
Refer to regression (9.6.4). Test the hypothesis that the rate of increase of average hourly earnings with respect to education differs by gender and race.
Refer to the regression (9.3.1). How would you modify the model to find out if there is any interaction between the gender and the region of residence dummies? Present the results based on this model
Stepwise regression. In deciding on the “best” set of explanatory variables for a regression model, researchers often follow the method of stepwise regression. In this method one proceeds either
Refer to Example 7.4. For this problem the correlation matrix is as follows:a. €œSince the zero-order correlations are very high, there must be serious multicollinearity.€
Refer to the illustrative example of Chapter 7 where we fitted the Cobb– Douglas production function to the manufacturing sector of all 50 states and the District of Columbia for 2005. The results
Suppose in the modelYi = β1 + β2X2i + β3X3i + uithat r23, the coefficient of correlation between X2 and X3, is zero. Therefore, someone suggests that you run the following regressions:Yi = α1 +
In data involving economic time series such as GNP, money supply, prices, income, unemployment, etc., multicollinearity is usually suspected. Why?
Consider the illustrative example of Section 10.6 (Example 10.1). How would you reconcile the difference in the marginal propensity to consume obtained from Eqs. (10.6.1) and (10.6.4)?
Consider the following model:Yt = β1 + β2Xt + β3Xt−1 + β4Xt−2 + β5Xt−3 + β6Xt−4 + utwhere Y = consumption, X = income, and t = time. The preceding model postulates that consumption
If the relation λ1X1i + λ2X2i + λ3X3i = 0 holds true for all values of λ1, λ2, and λ3, estimate r12.3, r13.2, and r23.1. Also find R21.23, R22.13, and R23.12. What is the degree of
In the model Yi = β1 + β2Di + ui , let Di = 0 for the first 40 observations and Di = 1 for the remaining 60 observations.You are told that ui has zero mean and a variance of 100. What are the mean
Refer to the child mortality example discussed in Chapter 8 (Example 8.1). The example there involved the regression of the child mortality (CM) rate on per capita GNP (PGNP) and female literacy rate
Consider the set of hypothetical data in the following table. Suppose you want to fit the modelYi = Î²1 + Î²2X2i + Î²3X3i + uito the data.a. Can you estimate the
In the k-variable linear regression model there are k normal equations to estimate the k unknowns. These normal equations are given in Appendix C. Assume that Xk is a perfect linear combination of
Refer to the U.S. savings–income regression discussed in the chapter. As analternative to Eq. (9.5.1), consider the following model:ln Yt = β1 + β2Dt + β3Xt + β4(Dt Xt ) + utwhere Y is savings
Refer to the Indian wage earners example (Section 9.12) and the data in Table 9.7.As a reminder, the variables are defined as follows:WI = weekly wage income in rupeesAge = age in yearsDsex = 1 for
From annual data for 1972–1979, William Nordhaus estimated the following model to explain the OPEC’s oil price behavior (standard errors in parentheses).Ŷt = 0.3x1t + 5.22x2tse = (0.03)
Consider the following regression results.* (The actual data are in the following table.)whereUN = unemployment rate, %V = job vacancy rate, %D = 1, for period beginning in 1966€“IV
Consider the following regression results (t ratios are in parentheses):where Y = wife€™s annual desired hours of work, calculated as usual hours of work per year plus weeks looking for
If you have monthly data over a number of years, how many dummy variables will you introduce to test the following hypotheses:a. All the 12 months of the year exhibit seasonal patterns.b. Only
Reconsider the savings–income regression in Section 8.7. Suppose we divide the sample into two periods as 1970–1982 and 1983–1995. Using the Chow test, decide if there is a structural change in
Return to Exercise 1.7, which gave data on advertising impressions retained and advertising expenditure for a sample of 21 firms. In Exercise 5.11 you were asked to plot these data and decide on an
Return to the child mortality example that we have discussed several times. In regression (7.6.2) we regressed child mortality (CM) on per capita GNP (PGNP) and female literacy rate (FLR). Now we
Refer to Example 8.3. Use the t test as shown in Eq. (8.6.4) to find out if there were constant returns to scale in the Mexican economy for the period of the study.In Example 8.3By way of
From annual data for the years 1968–1987, the following regression results were obtained:Ŷt = −859.92 + 0.6470X2t − 23.195X3t R2 = 0.9776 ……… (1)Ŷt = −261.09 + 0.2452X2t R2 = 0.9388
Critical values of R2 when true R2 0. Equation (8.4.11) gave the relationship between F and R2 under the hypothesis that all partial slope coefficients are simultaneously equal to zero (i.e., R2 =
Refer to Exercise 7.21c. Now that you have the necessary tools, which test(s) would you use to choose between the two models? Show the necessary computations. (the dependent variables in the two
Estimating the capital asset pricing model (CAPM). In Section 6.1 we considered briefly the well-known capital asset pricing model of modern portfolio theory. In empirical analysis, the CAPM is
Marc Nerlove has estimated the following cost function for electricity generation:Y = AXÎ² PÎ±1 PÎ±2 PÎ±3u
The demand for cable. The following table gives data used by a telephone cable manufacturer to predict sales to a major customer for the period 1968€“1983.The variables in the table are
Consider the Cobb–Douglas production functionY = β1Lβ2Kβ3 ......................(1)where Y = output, L = labor input, and K = capital input. Dividing (1) through by K, we get(Y/K) =
Energy prices and capital formation: United States, 1948€“1978. To test the hypothesis that a rise in the price of energy relative to output leads to a decline in the productivity of
The following is known as the transcendental production function (TPF), a generalization of the well-known Cobb€“Douglas production function:Yi = Î²1 LÎ²2
Refer to the U.S. defense budget outlay regression estimated in Exercise 7.18.a. Comment generally on the estimated regression results.b. Set up the ANOVA table and test the hypothesis that all the
Refer to Exercise 7.17 relating to wildcat activity.a. Is each of the estimated slope coefficients individually statistically significant at the 5 percent level?b. Would you reject the hypothesis
Refer to the demand for roses function of Exercise 7.16. Confining your considerations to the logarithmic specification,a. What is the estimated own-price elasticity of demand (i.e., elasticity with
Show that F tests of Eq. (8.4.18) and Eq. (8.6.10) are equivalent.
Suppose you want to study the behavior of sales of a product, say, automobiles over a number of years and suppose someone suggests you try the following models:Yt = β0 + β1tYt = α0 + α1t +
From the Phillips curve given in Eq. (6.7.3), is it possible to estimate the natural rate of unemployment? How?
Refer to the data in the following table. To find out if people who own PCs also own cell phones, run the following regression:CellPhonei= Î²1+ Î²2PCsi+ uia. Estimate the
Consider the following regression:SPIi = −17.8 + 33.2 Ginii se = (4.9) (11.8) r2= 0.16Where SPI = index of sociopolitical instability, average for
You are given the data in Table 6.7.**Fit the following model to these data and obtain the usual regression statistics and interpret the results:100 / (100 €“ Yi) = Î²1 +
Consider the data in the following table.Based on these data, estimate the following regressions:Yi = Î±1 + Î±2X2i + u1i Yi = Î»1 + Î»3X3i
From the following data estimate the partial regression coefficients, their standard errors, and the adjusted and unadjusted R2values: Ỹ = 367.693 X2 = 402.760 X3 = 8.0 E(X2i – X2)² = 84855.096
The demand for roses. The following table gives quarterly data on these variables:Y = quantity of roses sold, dozensX2= average wholesale price of roses, $/dozenX3= average wholesale price of
Wildcat activity. Wildcats are wells drilled to find and produce oil and/or gas in an improved area or to find a new reservoir in a field previously found to be productive of oil or gas or to extend
Show that Eq. (7.4.7) can also be expressed aswhere b23 is the slope coefficient in the regression of X2 on X3. Recall that b23 = Î£x2ix3i / Î£x23i. Ey,(x2 – b23x3i) E(x2;
In a multiple regression model you are told that the error term ui has the following probability distribution, namely, ui ∼ N(0, 4). How would you set up a Monte Carlo experiment to verify that the
Show that r212.3 = (R2 – r213 = (R2 – r213) / (1 – r213) and interpret the equation.
U.S. defense budget outlays, 1962€“1981. In order to explain the U.S. defense budget, you are asked to consider the following model:Yt = Î²1 + Î²2X2t +
If the relation α1X1 + α2X2 + α3X3 = 0 holds true for all values of X1, X2, and X3, find the values of the three partial correlation coefficients.
In a study of turnover in the labor market, James F. Ragan, Jr., obtained the following results for the U.S. economy for the period of 1950–I to 1979–IV.* (Figures in the parentheses are the
The demand for chicken in the United States, 1960€“1982. To study the per capita consumption of chicken in the United States, you are given the data in the following table,where Y =
Is it possible to obtain the following from a set of data?a. r23 = 0.9, r13 = −0.2, r12 = 0.8b. r12 = 0.6, r23 = −0.9, r31 = −0.5c. r21 = 0.01, r13 = 0.66, r23 = −0.7
Consider the following model:Yi = β1 + β2 Education i + β2 Years of experience + uiSuppose you leave out the years of experience variable. What kinds of problems or biases would you expect?
Show that β2 and β3 in Eq. (7.9.2) do, in fact, give output elasticities of labor and capital.
Consider the following demand function for money in the United States for the period 1980€“1998:Mt = Î²1 YÎ²2t rÎ²3t eutwhere M = real money demand, using
The following table gives data for the manufacturing sector of the Greek economy for the period 1961€“1987.a. See if the Cobb€“Douglas production function fits the data given in
The following table gives data for real consumption expenditure, real income, real wealth, and real interest rates for the U.S. for the years 1947€“2000.a. Given the data in the table,
In general R2≠ r212 + r213, but it is so only if r23 = 0. Comment and point out the significance of this finding.
Consider the following models.a. Will OLS estimates of Î±1 and Î²1 be the same? Why?b. Will OLS estimates of Î±3 and Î²3 be the same? Why?c. What is
Suppose you estimate the consumption functionYi = α1 + α2Xi + u1iand the savings functionZi = β1 + β2Xi + u2iwhere Y = consumption, Z = savings, X = income, and X = Y + Z, that is, income is
Suppose you express the Cobb€“Douglas model given in Eq. (7.9.1) as follows:Yi = Î²1XÎ²22i X Î²33i uiIf you take the log-transform of this model, you will
Regression through the origin. Consider the following regression through the origin:Yi = β̂2X2i + β̂3X3i + ûia. How would you go about estimating the unknowns?b. Will Σûi be zero for this
Refer to Exercise 7.24 and the data in the following table concerning four economic variables in the U.S. from 1947€“2000.a. Based on the regression of consumption expenditure on real
Suppose in the regressionln (Yi/X2i ) = α1 + α2 ln X2i + α3 ln X3i + uithe values of the regression coefficients and their standard errors are known. From this knowledge, how would you estimate
Refer to Section 8.8 and the data in the following table concerning disposable personal income and personal savings for the period 1970€“1995. In that section, the Chow test was introduced
Assume the following:Yi = β1 + β2X2i + β3X3i + β4X2i X3i + uiwhere Y is personal consumption expenditure, X2 is personal income, and X3 is personal wealth. The term (X2i X3i ) is known as the
You are given the following regression results:Ŷt = 16,899 − 2978.5X2t R2 = 0.6149 t

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