Questions and Answers of Econometrics

Consider the modelYi = β1 + β2X2i + ui…………….. (1)To find out whether this model is mis-specified because it omits the variable X3 from the model, you decide to regress the residuals
Consider the modelYi = β1 + β2X∗i + uiIn practice we measure X∗i by Xi such thata. Xi = X∗i + 5b. Xi = 3X∗ic. Xi = (X∗i + εi ), where εi is a purely random term with the usual
Refer to the regression Eqs. (13.3.1) and (13.3.2). In a manner similar to Eq. (13.3.3) show thatE(α̂1) = β1 + β3(X̅3 − b32X̅2)where b32 is the slope coefficient in the regression of the
Food expenditure in India. In the following table we have given data on expenditure on food and total expenditure for 55 families in India.a. Regress expenditure on food on total expenditure, and
Estimating ρ: The Cochrane–Orcutt (C–O) iterative procedure. As an illustration of this procedure, consider the two-variable model:Yt = β1 + β2Xt + ut …………………….. (1)and the
Return to the R&D example discussed in Section 11.7 (Exercise 11.10). Repeat the example using profits as the regressor. A priori, would you expect your results to be different from those using
Although log models as shown in Eq. (11.6.12) often reduce heteroscedasticity, one has to pay careful attention to the properties of the disturbance term of such models. For example, the modelYi =
For pedagogic purposes Hanushek and Jackson estimate the following model:Ct = Î²1 + Î²2GNPt + Î²3Dt + ui
For the data given in the following table, regress average compensation Y on average productivity X, treating employment size as the unit of observation. Interpret your results, and see if your
The following table gives data on the sales/cash ratio in U.S. manufacturing industries classified by the asset size of the establishment for the period 1971–I to 1973–IV. (The data are on a
Bartlett€™s homogeneity-of-variance test.* Suppose there are k independent sample variances s21, s22, . . . , s2kwith f1, f2, . . . , fkdf, each from populations which are normally
Consider the following regression-through-the origin model:Yi = βXi + ui, for i = 1, 2You are told that u1 ∼ N(0, σ2) and u2 ∼ N(0, 2σ2) and that they are
The following table gives data on 81 cars about MPG (average miles per gallons), HP (engine horsepower), VOL (cubic feet of cab space), SP (top speed, miles per hour), and WT (vehicle weight in 100
Repeat Exercise 11.16, but this time regress the logarithm of expenditure on food on the logarithm of total expenditure. If you observe heteroscedasticity in the linear model of Exercise 11.16 but
A shortcut to White’s test. As noted in the text, the White test can consume degrees of freedom if there are several regressors and if we introduce all the regressors, their squared terms, and
State whether the following statements are true or false. Briefly justify your answer.a. When autocorrelation is present, OLS estimators are biased as well as inefficient.b. The Durbin–Watson d
Given a sample of 50 observations and 4 explanatory variables, what can you say about autocorrelation if (a) d = 1.05? (b) d = 1.40? (c) d = 2.50? (d) d = 3.97?
In studying the movement in the production workers’ share in the value added (i.e., labor’s share), the following models were considered by Gujarati:*Model A: Yt = β0 + β1t + utModel B: Yt =
Detecting autocorrelation: von Neumann ratio test.* Assuming that the residual uÌ‚tare random drawings from normal distribution, von Neumann has shown that for large n, the ratiocalled
In a sequence of 17 residuals, 11 positive and 6 negative, the number of runs was 3. Is there evidence of autocorrelation? Would the answer change if there were 14 runs?
Theil€“Nagar Ï estimate based on d statistic. Theil and Nagar have suggested that, in small samples, instead of estimating Ï as (1 ˆ’ d/2), it should be
Refer to Exercise 7.19 about the demand function for chicken in the United States.a. Using the log–linear, or double-log, model, estimate the various auxiliary regressions. How many are there?b.
The following table gives data on imports, GDP, and the Consumer Price Index (CPI) for the United States over the period 1975€“2005. You are asked to consider the following model:ln
Klein and Goldberger attempted to fit the following regression model to the U.S. economy:Yi = Î²1 + Î²2X2i + Î²3X3i + Î²4X4i + uiwhere Y = consumption, X2
Critically evaluate the following statements:a. “In fact, multicollinearity is not a modeling error. It is a condition of deficient data.”b. “If it is not feasible to obtain more data, then one
From the annual data for the U.S. manufacturing sector for 1899€“1922, Dougherty obtained the following regression results:where Y = index of real output, K = index of real capital input, L
For the k-variable regression model, it can be shown that the variance of the kth (k = 2, 3, . . . , K) partial regression coefficient given in Eq. (7.5.6) can also be expressed aswhere
Verify that the standard errors of the sums of the slope coefficients estimated from Eqs. (10.5.6) and (10.5.7) are, respectively, 0.1549 and 0.1825.
Using Eqs. (7.4.12) and (7.4.15), show that when there is perfect collinearity, the variances of β̂2 and β̂3 are infinite.
Show that Eqs. (7.4.7) and (7.4.8) can also be expressed aswhere r23 is the coefficient of correlation between X2 and X3. (ΣΥ) (Σx)-(Σy) (Σ3) β- ΙΣ3/Σ-r3) (ΣΥκε) (Σx) -(Συ) (Σ) β
Consider the following model:GNPt = β1 + β2Mt + β3Mt−1 + β4(Mt −Mt−1) + utwhere GNPt = GNP at time t,Mt = money supply at time t, Mt−1 = money supply at time (t − 1), and (Mt −Mt−1)
Orthogonal explanatory variables. Suppose in the modelYi = β1 + β2X2i + β3X3i + · · ·+βk Xki + uiX2 to Xk are all uncorrelated. Such variables are called orthogonal variables. If this is the
Consider the following correlation matrix:Describe how you would find out from the correlation matrix whether (a) There is perfect collinearity, (b) There is less than perfect
Using matrix notation, it can be shownvar–cov (β̂) = σ2(X'X)−1What happens to this var–cov matrix:a. When there is perfect multicollinearity?b. When collinearity is high but not perfect?
In matrix notation it can be shown (see Appendix C) thatβ̂ = (X'X)−1X'ya. What happens to β̂ when there is perfect collinearity among the X’s?b. How would you know if perfect collinearity
Suppose all the zero-order correlation coefficients of X1(= Y), X2, . . . , Xk are equal to r.a. What is the value of R21.23 . . . k?b. What are the values of the first-order correlation coefficients?
Reestimate the model in Exercise 9.22 by adding the regressor, expenditure on durable goods.a. Is there a difference in the regression results you obtained in Exercise 9.22 and in this exercise? If
Consider the following model:Yi = β1 + β2Di + uiWhereDi = 0 for the first 20 observations and Di = 1 for the remaining 30 observations. You are also told that var (u2i) = 300.a. How would you
From data for 101 countries on per capita income in dollars (X) and life expectancy in years (Y) in the early 1970s, Sen and Srivastava obtained the following regression results: Ŷi = −2.40
To assess the effect of the Fed€™s policy of deregulating interest rates beginning in July 1979, Sidney Langer, a student of mine, estimated the following model for the quarterly period of
In regression (7.9.4), we presented the results of the Cobb–Douglas production function fitted to the manufacturing sector of all 50 states and Washington, DC, for 2005. On the basis of that
Establish statements (8.6.11) and (8.6.12).
Show that the F ratio of Eq. (8.4.16) is equal to the F ratio of Eq. (8.4.18). (ESS/TSS = R2.)
Estimating Qualcomm stock prices. As an example of the polynomial regression, consider data on the weekly stock prices of Qualcomm, Inc., a digital wireless telecommunications designer and
Repeat Exercise 3.25, replacing math scores for reading scores.
Given the assumptions in column 1 of the table, show that the assumptions in column 2 are equivalent to them.Assumptions of the Classical Model(1)……………………………....................
From the scattergram given in Figure 2.9, what general conclusions do you draw? What is the economic theory that underlies this scattergram? 10 11 12 6. Scarce land; Abundant land; less skilled
Estimating ρ: The Hildreth–Lu scanning or search procedure.* Since in the first order autoregressive schemeut = ρut−1 + εtρ is expected to lie between −1 and +1, Hildreth and Lu suggest a
The following table gives data on new passenger cars sold in the United States as a function of several variables.a. Develop a suitable linear or log€“linear model to estimate a demand
To assess the feasibility of a guaranteed annual wage (negative income tax), the Rand Corporation conducted a study to assess the response of labor supply (average hours of work) to increasing hourly
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

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