Questions and Answers of Introductory Econometrics Modern

Consider the choice of values for aML and bML for the purpose of maximizing ln L(εâ•›1, εâ•›2, . . . , εn) in equation (15.55). Usually, this would require us to differentiate the
Consider the linear probability model.(a) When yi = 1, εi = 1 − (α + βxi). When yi = 0, εi = −(α + βxi). If p =P(yi = 1 | xi) and 1 − p = P(yi = 0 | xi), prove that E(εi) = p − (α +
Recall equation (4.37) for b.(a) Rewrite the numerator asWhy is this valid? (b) Consider the first term to the right of the equality in parta. Prove that(c) Consider the second term to the right of
Return to the probit results of table 15.3.(a) Interpret the likelihood ratio χ2 test. What should we conclude?(b) Construct the 95% confidence interval for the effect of years of schooling on the
Redraw figure 15.6 under the assumption that b is negative. Assume also that−(a + bxi) is positive but close to zero. In what direction does an increase in xi change the probability of observing an
Designate the probability that a newborn child will be a girl as π. The probability that a newborn child will be a boy is therefore 1 − π. Imagine that we have a sample that consists of a
Recall figures 15.4 and 15.5. Assume that εi and εj are both normally distributed with E(εi) = 0 and V(εi) = 36,000,000.(a) If we were to choose b = 6,000 and a = −28,000, what would P(yi <
Recall figure 15.3. Assume that εi is normally distributed with E(εi) = 0 and V(εi) = 36,000,000.(a) What is the critical value, call it Nâ•›.33, such that P(εi < Nâ•›.33) = .67?(b) If
Recall figure 15.1. Use the parameter values on which that figure is based.Assume that εi is normally distributed with E(εi) = 0 and V(εi) = 36,000,000.(a) What is the probability that εi is
Return to equation (14.34). Recall that both εi1 and εi2 have the properties of chapter 5, V(εi1) = V(εi2) and COV(εi1, εj2) = 0 for all i and j.(a) Use the rules of expectations to verify that
Return to equation (14.34).(a) Set p = 1 and verify that equation (14.34) reduces to equation (14.32) for the first period.(b) Set p = 0 and verify that equation (14.34) reduces to equation (14.32)
Return to the stratified regressions of table 14.4.(a) Compare the predicted earnings associated with different levels of schooling for men and women. Are there particular levels of education at
Return to the comparison between the regressions in tables 14.3 and 14.4.(a) Replicate the test in equation (14.29) for the regressions on the sample of men. Which is better, the unrestricted or
In our discussion of the dummy variables for schooling levels of table 14.4, we assert that the substantive results of a regression are the same, regardless of which dummy variable we omit. Table
In our discussion of tables 14.1 through 14.3, we note that the samples on which they are based include individuals who don’t work. We wonder whether the determinants of earnings are different
In our discussion of table 14.3, we note that there are provocative results regarding race and ethnicity. Table 14.6 presents another stratification of the pooled regression in table 14.3. In this
Consider the simplest version of equation (14.25). We have one dummy variable, x1i, and one other explanatory variable, x2i. Consequently, equationÂ€(14.25) reduces toa) In this specification,
Return to equation (14.25).(a) Set x1i = 0. Prove that, in this case, equation (14.25) reduces to equationÂ€(14.23).(b) Set x1i = 1. Prove that, in this case, equation (14.25) reduces to
Return to table 1.4.(a) Compare the regression in this table to that of figure 1.1. Explain why the regression of figure 1.1 is a restricted version of that in table 1.4. How many restrictions does
Return to exercise 12.10, where we prove that(a) Explain why all parts of that exercise, with the exception of partd, hold without modification when we have k rather than two explanatory
The intercept and slopes in equation (13.62) appear to differ substantially in magnitude from those in equation (13.63). Is this because the two regressions are contradictory or because some
The regression of equation (13.61), omitting the interaction between the dummy variable identifying women and the continuous variable for years of schooling, isExplain, with reference to section
Consider the sample regression of equation (13.58), where x1i is a dummy variable and x2i is a continuous variable:Imagine that x1i identifies women. Section 13.2 explains why we don’t want to add
Return to equations (13.59) and (13.60).(a) Test whether or not the three slopes in equation (13.59) are statistically significant. What do these tests indicate about the effect of being a black
Return to equation (13.49). Change x2i by Δxâ•›2. Follow the derivation in equations (13.50) through (13.52) to prove that = + 3**
Demonstrate that the population relationship of equation (11.1) forces the effects of race to be identical regardless of sex.(a) Subtract equation (13.42) from equation (13.44) to derive the expected
If we respecify the regression of equation (13.34) in the log-log form of equation(13.35), we obtain(a) Is the slope for the natural log of the CPI statistically significant? Why or why not?(b)
Consider the log-log specification of equation (13.38).(a) What is the expression for E(ln yi) in terms of α, β, and xi?(b) Change xi by Δx. Write the new expected value of the dependent variable
We return to the regression in equation (12.42) and respecify it in log-linear form:(a) Is the slope for the variable measuring the number of rooms statistically significant? Why or why not?(b)
Consider the log-linear specification of equation (13.35).(a) What is the expression for E(ln yi) in terms of α, β, and xi?(b) Change xi by Δx. Write the new expected value of the dependent
The regression of equation (11.41), with standard errors, is(a) Demonstrate, either with confidence intervals or two-tailed hypothesis tests, that b is statistically significant.(b) The quadratic
The regression of equation (11.43), with standard errors, is(a) Demonstrate, either with confidence intervals or two-tailed hypothesis tests, that b is statistically significant.(b) The quadratic
The quadratic version of the regression in equation (12.42) is(a) Interpret the signs and values of bâ•›1 and bâ•›2 following the analysis of section 13.3.(b) Check, either with confidence
Consider the regression in equation (13.23).(a) Demonstrate, either with confidence intervals or two-tailed hypothesis tests, thata, bâ•›1, and bâ•›2 are statistically significant.(b)
Return to the quadratic regression specification of equation (13.19). The predicted value of yi from this specification is(a) Change xi by Δx. Demonstrate that the new predicted value is(b) Subtract
Consider the regression in equation (13.20).(a) Demonstrate, either with confidence intervals or two-tailed hypothesis tests, thata, bâ•›1, and bâ•›2 are statistically significant.(b)
Imagine that we intend x2i to be a dummy variable in the populationÂ�relationship of equation (11.1).(a) Unfortunately, we make a mistake. We assign the value x2i = 2 to observations that have the
Section 13.2 alludes to two circumstances in which the regression calculations are not well defined.(a) Toward the end of section 13.2, we make the claim that the results pertaining to regression
Assume that εi has the properties of chapter 5.(a) Apply the rules of expectations from chapter 5 to equation (13.5) in order to derive equation (13.6).(b) Replace equation (13.1) in equation (13.7)
Consider the sample correlation between ˆx 1i, as given in equation (12.50), and ˆx 2i , as given in equation (12.51).(a) Demonstrate that(b) Use part a to prove equation (12.52). COV (2) = COV
Exercises 5.5 through 5.11 worked through the case of a population relationship that has no constant in the case with one explanatory variable.Let’s do the same thing here with two explanatory
Consider table 12.8.(a) Which estimate or estimates of b1 have confidence intervals that don’t contain the population value β1 = 4,000? How do we know?(b) Which estimate or estimates of b2
Imagine that we ran the following regression:(a) Recall our analysis beginning with equation (4.38). Replicate it to Â�demonstrate that, with this specification,(b) Demonstrate that, with this
In chapter 4, we observed that there isn’t any point to running a bivariate regression with only two observations. In exercise 4.4, we demonstrated that this regression would fit perfectly. A
Let’s revisit exercise 4.11 in the context of the regression of equation (11.12).(a) Prove that, when ˆyi is given by equation (11.13), its average is(b) The sample covariance between yi and ˆyi
Consider R2 for the regression of equation (11.12).(a) Verify that equation (4.48) is valid for the regression of equation (11.12).(b) Use equations (11.23) and (11.26) to prove that, when ˆyi is
Consider the information in equation (12.19). The sample for this regression has 1,000 observations.(a) Test the null hypothesis that the annual returns to a year of schooling are$4,000 against the
Consider the information in equation (12.18). The sample for this regression has 135 observations.(a) Create a confidence interval for the true effect of water availability on child mortality. Does
Consider the information in equation (12.17). The sample for this regression has 76 observations.(a) Create a confidence interval for the true effect of the Corruption Perceptions Index (CPI) on
Prove that the population V(b1) can be rewritten asWhat does this tell us about the factors that influence the precision of our estimate of β1? V(b)= 0 i=1
Prove that b1 is undefined if |CORR(x1i, x2i)| = 1.
Consider the relatively unlikely event that the correlation between x1i and x2i is zero.(a) Prove that, if this is true, the bias in equation (11.6) is zero.(b) Prove that, if this is true, the
Derive equation (12.14) from equation (12.12).(a) Transform the numerator of the term in equation (12.13) so that it equals the sample covariance of x1i and x2i.(b) Transform the denominator of the
Derive V(b2).(a) Derive an expression for b2 that is analogous to that for b1 in equationÂ€(12.1).(b) Reproduce the analysis in equations (12.2) through (12.6) in order to verify equation (12.8).
Demonstrate thata, as given by equation (11.28), is an unbiased estimator of α.(a) Expand the analysis of equations (5.39) and (5.40) to derive E(y) when yi is given by equation (11.1).(b) Prove
Replicate the analysis of section 11.5 to prove that bâ•›2 is an unbiased estimator of β2: E(bâ•›2) = β2.
Begin with the expression for bâ•›2 in equation (11.38). Using the auxiliary regressions in equations (11.2) and (11.11), reproduce the analysis of sectionÂ€11.4 to demonstrate that i=l i=l
Consider the intercept in equation (11.70).(a) What are the average values for the residuals of equations (11.68) and(11.69)?(b) According to equation (4.35), what is the formula for this
Consider the last three terms in equation (11.61).(a) Prove that the second and third terms to the right of the equality in equation (11.61) are equal.(b) Prove that the third and fourth terms to the
Derive equation (11.55).(a) Square and sum the residuals of equation (11.54) to obtain(b) Multiply the middle term byExplain why the result can be expressed as(c) Combine this new expression for the
Begin with the expression for bâ•›1 in equation (11.52).(a) Convert all of the summations in this expression into sample variances and covariances. Accomplish this by repeatedly multiplying the
Consider the omission of more than one explanatory variable. Imagine that the population relationship iswhere εi has all of the properties of chapter 5.(a) What is E(yi)?(b) We estimateWhat is(c)
Return to the solution for bâ•›2 in equation (11.38).(a) Rearrange equation (11.30) to solve for bâ•›1 in terms of bâ•›2.(b) Substitute the result of part a into equation (11.31) to
Let’s identify the bias that results if we omit x1i rather than x2i from theÂ�regression estimating equation (11.1). In other words, imagine that theÂ�population relationship is equation
Complete the derivations of equations (11.30) and (11.31).(a) Substitute equation (11.28) for a into the second normal equation, equation(11.22). Rearrange to obtain(b) Distribute the product and sum
Return to the first normal equation, equation (11.18). Mimic the derivation in equations (4.29) through (4.35) to solve for a.(a) Sum the individual terms separately and simplify to get(b) Divide
Return to the procedure in equations (4.22) through (4.28).(a) Derive equation (11.24) from equation (11.23).(b) Derive equation (11.27) from equation (11.26).
Examine the procedure in equations (4.13) through (4.15).(a) The term a is essentially inert throughout this procedure. Why?(b) Imagine applying this procedure to the sum of squared errors in
Examine the procedure in equations (4.9) through (4.12).(a) The term bxi is essentially inert throughout this procedure. Why?(b) Imagine applying this procedure to the sum of squared errors in
Follow the derivation in equations (4.40) through (4.42) to demonstrate that the term in equation (11.10) can be written as COV (x2) V(x)
Assume that the population relationship is given in equation (11.1) and that equation (5.5), E(εi) = 0, is true. Prove that E(yi) is as given in equation (11.5).
Derive equation (11.4) by following partsa, b, and c of exercise 5.12 and summing the result.
Consider the IV estimate of the effects of schooling on earnings in sectionÂ€10.9.(a) Examine the first-stage regression of equation (10.48). What does it imply about the comparison between the
Recall that the Staiger-Stock rule of thumb stipulates that the F-statistic for the first-stage regression should exceed 10 in order for bIV to be useful. We’re getting a little bit ahead of
Consider equation (10.34).(a) Offer an informal explanation of why the second term to the right of the equality in equation (10.34) might be written as(b) Demonstrate that, if the approximation in
Let’s derive the results in table 10.7.(a) Exercise 10.10a demonstrates that V(xi )=V(xi*) + V(νi ).Recall that V(xi*) = 8.5. Use the value for V(νi) to calculate V(xi) for each simulation in
Set CORR(zi, xi) = .5 in equation (10.45).(a) Demonstrate that this implies V(bIV) = 4 V(b).(b) Demonstrate that this implies SD(bIV) = 2 SD(b).(c) Demonstrate that the 95% confidence interval around
Equation (10.45) states that V(bIV) can be expressed asUse this expression to provide another explanation of why V(bIV) converges to zero as n converges to infinity. V(biv)= V(b) (CORR (z,, ;))
Construct the 95% confidence intervals around the values of bIV in table 10.5. How many of these confidence intervals contain the true valueβ = 4,000?
In table 10.5, examine the magnitude of the reduction in SD(bIV) that occurs with each tenfold increase in sample size.(a) Looking at the quantities in the expression for V(bIV) in equation (10.43),
Return to the calculation of the population value for SD(bIV) in table 10.5.(a) In the context of measurement error, equation (10.5) defines the observed value of the explanatory variable, xi, as xi
Recall the population expression for V(bIV) in equation (10.39):Complete the following derivation. Now that we’ve had nine chapters of practice at this sort of thing, we’re going to try it with a
Construct the 95% confidence intervals around the values of bIV in table 10.4.How many of these confidence intervals contain the true value β = 4,000?
Consider the demonstration that E(bIV) converges to β as the sample size increases, which begins with equations (10.33) and (10.34).(a) Simplify equation (10.33) to obtain equation (10.34). (b)
Equations (10.30) and (10.32) give bIV as(a) Follow equations (2.31) through (2.33) to transform the numerator into Σ − − i= n i i z z y y 1( )( ) and the denominator into Σ − − i= n i i z
Use the normal equations for the auxiliary regression of equation (10.21) and follow the development from equations (4.50) through (4.52) to prove equation (10.29), (2-7)=0. i=l
Use the result of exercise 3.7 and equation (3.10) to prove that, if the sample COV(εi, zi) = 0, then the sample COV(εi, c + dzi) = 0. Use this result to confirm that the sample COV(εi, ˆxi ) = 0.
Imagine that, in equation (10.9), xi was determined exactly by ai. Would the population relationship of equation (10.11) still be contaminated with endogeneity?Why or why not? In either case, give an
The discussion of equation (10.7) asserts that εi is uncorrelated with xi because it is uncorrelated with xi* and νi. Let’s prove that here.(a) Explain why the population covariance between two
Note 4 asserts that equation (10.5) can be understood as a special case of equation (5.1), where the constant is set to zero and the coefficient is set to one. Explain.
Consider again the population relationship of equation (5.1), yi =α +β xi +εi .Imagine that this relationship has second-order autocorrelation:ε γε ν i i i = + −2 , where νi is the shock
Return to the discussion of other forms of autocorrelation in section 9.9.(a) Why might the frequency of traffic accidents on different days display seventh-order autocorrelation?(b) Why might
Compare the GLS estimator of SD(a) in equation (9.51) to the OLS estimator calculated in equation (9.9).(a) Which is larger?(b) What must be true of the GLS estimator of SD(a) in comparison to the
Consider the GLS regression in equation (9.51).(a) Calculate the 95% confidence interval for α based on the information in equation (9.51) and the sample size of n = 1,000. What is the actual value
Consider again the population relationship of equation (5.1), y x i i i =α +β +ε .Assume that this relationship has fourth-order autocorrelation, as specified in equation (9.52):ε γε ν i i i =
Consider the relationship between εi and εi−2.(a) Follow equation (9.20) and the equation that succeeds it to demonstrate that(b) Using equations (9.10) and (9.12), replace εi with a function of
Figure 9.2 represents the disturbances generated by equation (9.10) from the shocks of figure 9.1 when ρ = .9. Figure 9.4 creates disturbances from the shocks of figure 9.1 using equation (9.10)
Consider the OLS regression in equation (9.9).(a) Calculate the 95% confidence interval for α based on the information in equation (9.9) and the sample size of n = 1,000. What is the actual value of
Section 9.4 asserts that the combination of the White heteroscedasticityconsistent variance estimator and the covariance estimator suggested in equation (9.7) must equal zero.(a) Explain why the sum
Suppose, again, we believe that larger values of the explanatory variables are associated with a larger range of values for the disturbance. Now suppose we believe that V(εi ) = xi2σ 2 .(a) In this

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