Questions and Answers of Econometrics

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
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=

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