Questions and Answers of Econometrics

Compare and comment on the OLS and WLS regressions in Eqs. (15.7.3) and (15.7.1).
From data for 54 standard metropolitan statistical areas (SMSA), Demaris estimated the following logit model to explain high murder rate versus low murder rate:**ln Ôi = 1.1387 + 0.0014Pi +
In the probit regression given in the following table show that the intercept is equal to ˆ’Î¼x/Ïƒxand the slope is equal to 1/Ïƒx, where
Estimate the probabilities of owning a house at the various income levels underlying the regression (15.7.1). Plot them against income and comment on the resulting relationship.
In studying the purchase of durable goods Y (Y = 1 if purchased, Y = 0 if no purchase) as a function of several variables for a total of 762 households, Janet A. Fisherˆ—obtained the
For the home ownership data given in Table 15.1, the maximum likelihood estimates of the logit model are as follows:Comment on these results, bearing in mind that all values of income above 16
Refer to the data given in the following table. If YÌ‚iis negative, assume it to be equal to 0.01 and if it is greater than 1, assume it to be equal to 0.99. Recalculate the weights wiand
The following table gives data on real GDP, labor, and capital for Mexico for the period 1955€“1974. See if the multiplicative Cobb€“Douglas production function given in Eq.
Show that Î²ˆ—2of Eq. (11.3.8) can also be expressed asand var (Î²ˆ—2) given in Eq. (11.3.9) can also be expressed aswhere yˆ—i = Yi €“
Evaluate the following statement made by Henry Theil:Given the present state of the art, the most sensible procedure is to interpret confidence coefficients and significance limits liberally when
Commenting on the econometric methodology practiced in the 1950s and early 1960s, Blaug stated:. . . much of it [i.e., empirical research] is like playing tennis with the net down:instead of
According to Blaug, “There is no logic of proof but there is logic of disproof.” What does he mean by this?
Refer to the St. Louis model discussed in the text. Keeping in mind the problems associated with the nested F test, critically evaluate the results presented in regression (13.8.4).
Suppose the true model isYi = β1 + β2Xi + βX2i + β3X3i + uiBut you estimateYi = α1 + α2Xi + viIf you use observations of Y at X = −3, −2, −1, 0, 1, 2, 3, and estimate the “incorrect”
To see if the variable X2i belongs in the model Yi = β1 + β2Xi + ui , Ramsey’s RESET test would estimate the linear model, obtaining the estimated Yi values from this model [i.e., Ŷi = β̂1 +
Use the data for the demand for chicken given in Exercise 7.19. Suppose you are told that the true demand function isln Yt = β1 + β2 ln X2t + β3 ln X3t + β6 ln X6t + ut ……………… (1)but
Continue with Exercise 13.21. Strictly for pedagogical purposes, assume that model (2) is the true demand function.a. If we now estimate model (1), what type of specification error is committed in
Monte Carlo experiment.* Ten individuals had weekly permanent income as follows:$200, 220, 240, 260, 280, 300, 320, 340, 380, and 400. Permanent consumption (Y∗i) was related to permanent income
Continue with Exercise 13.25. Using the J test, how would you decide between the two models?Data from 13.25Refer to Exercise 8.26. With the definitions of the variables given there, consider the
Refer to Exercise 7.19, which is concerned with the demand for chicken in the United States. There you were given five models.a. What is the difference between model 1 and model 2? If model 2 is
Refer to the following table, which gives data on personal savings (Y) and personal disposable income (X) for the period 1970€“2005. Now consider the following models:How would you choose
Use the data in Exercise 13.28.To familiarize yourself with recursive least squares, estimate the savings functions for 1970€“1981, 1970€“1985, 1970€“1990, and
The data in the following table gives U.S. population, in millions of persons, for the period 1970€“2007. Fit the growth models given in Exercise 14.7 and decide which model gives a better
Continue with Exercise 13.29, but now use the updated data in Table 8.10.a. Suppose you estimate the savings function for 1970–1981. Using the parameters thus estimated and the personal disposable
Omission of a variable in the K-variable regression model. Refer to Eq. (13.3.3), which shows the bias in omitting the variable X3 from the model Yi = β1+β2X2i + β3X3i + ui . This can be
What is meant by intrinsically linear and intrinsically nonlinear regression models?
Since the error term in the Cobb–Douglas production function can be entered multiplicatively or additively, how would you decide between the two?
What is the difference between OLS and nonlinear least-squares (NLLS) estimation?
The relationship between pressure and temperature in saturated steam can be expressed as:*where Y = pressure and t = temperature. Using the method of nonlinear least squares (NLLS), obtain the normal
State whether the following statements are true or false.a. Statistical inference in NLLS regression cannot be made on the basis of the usual t, F, and χ2 tests even if the error term is assumed to
How would you linearize the CES production function discussed in the chapter? Show the necessary steps.
Models that describe the behavior of a variable over time are called growth models. Such models are used in a variety of fields, such as economics, biology, botany, ecology, and demography. Growth
The true model isY*i = Î²1 + Î²X*i + ui
Critically evaluate the following view expressed by Leamer:My interest in metastatistics [i.e., theory of inference actually drawn from data] stems from my observations of economists at work. The
Refer to Table 8.11, which gives data on personal savings (Y) and personal disposable income (X) for the period 1970â€“2005. Now consider the following models: Model A: Yt = Î±1 + Î±2Xt
Monte Carlo experiment.* Ten individuals had weekly permanent income as follows: $200, 220, 240, 260, 280, 300, 320, 340, 380, and 400. Permanent consumption (Y∗i) was related to permanent income
Continue with Exercise 13.21. Strictly for pedagogical purposes, assume that model (2) is the true demand function.a. If we now estimate model (1), what type of specification error is committed in
Use the data for the demand for chicken given in Exercise 7.19. Suppose you are told that the true demand function isln Yt = β1 + β2 ln X2t + β3 ln X3t + β6 ln X6t + ut
State with reason whether the following statements are true or false.†a. An observation can be influential but not an outlier.b. An observation can be an outlier but not influential.c. An
Refer to the St. Louis model discussed in the text. Keeping in mind the problems associated with the nested F test, critically evaluate the results presented in regression (13.8.4).
According to Blaug, “There is no logic of proof but there is logic of disproof.” What does he mean by this?
Evaluate the following statement made by Henry Theil:*Given the present state of the art, the most sensible procedure is to interpret confidence coefficients and significance limits liberally when
Refer to the demand function for chicken estimated in Eq. (8.6.23). Considering the attributes of a good model discussed in Section 13.1, could you say that this demand function is “correctly”
Suppose that the true model isYi = β1 Xi + ui ……………… (1)but instead of fitting this regression through the origin you routinely fit the usual intercept present model:Yi = α0 + α1Xi +
Continue with Exercise 13.2 but assume that it is model (2) that is the truth. Discuss the consequences of fitting the mis-specified model (1).In exercise 13.2Suppose that the true model isYi = β1
Suppose that the “true” model isYi = β1 + β2X2i + uibut we add an “irrelevant” variable X3 to the model (irrelevant in the sense that the true β3 coefficient attached to the variable X3 is
Consider the following “true” (Cobb–Douglas) production function:ln Yi = α0 + α1 ln L1i + α2 ln L2i + α3 ln Ki + uiwhereY = outputL1 = production laborL2 = nonproduction laborK = capitalBut
Refer to Eqs. (13.3.4) and (13.3.5). As you can see, Î±Ì‚2, although biased, has a smaller variance than Î²Ì‚2, which is unbiased. How would you decide
Show that β estimated from either Eq. (13.5.1) or Eq. (13.5.3) provides an unbiased estimate of true β.
Following Friedman€™s permanent income hypothesis, we may writeYˆ—i = Î± + Î²Xˆ—iwhere Yˆ—i = €œpermanent€
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

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