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About 833 results. 1294 free access solutions
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| m76571 | In Problem 4.11, the R-squared from estimating the model
log(salary) = (0 + (1logi (sales) + (2 log(mktval) + (3 profmarg
+ (4 ceoten + (5 fomten + u,
Using the data in CEOSAL2.RAW, was R2 = .353 (n = 177). When ceoten2 and comten2 are added, R2 = .375. Is there evidence of functional form misspecification in this model? |
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| m76572 | In Problem 4.2, we added the return on the firm s stock, ros, to a model explaining CEO salary; ros turned out to be insignificant. Now, define a dummy variable, rosneg, which is equal to one if ros < 0 and equal to zero if ros < 0. Use CEOSAL1 .RAW to estimate the model
log(salary) = (0 + (1 log(sales) + (2 roe +(3 rosneg + u.
Discuss the interpretation and statistical significance 3? |
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| m76585 | In Section 4.5, we used as an example testing the rationality of assessments of housing prices. There, we used a log-log model in price and assess [see equation (4.47)]. Here, we use a level-level formulation,
(i) In the simple regression model
price = (0 + (1 possess + u,
the assessment is rational if (1, = 1 and (0 = 0. The estimated equation is
First, test the hypothesis that H0: (0 = 0 against the two-sided alternative. Then, test H0: (1 = 1 against the two-sided alternative. What do you conclude?
(ii) To test the joint hypothesis that (0 = 0 and (1, = 1, we need the SSR in the restricted model. This amounts to computing
where n = 88, since the residuals in the restricted model are just pricei - assessi. (No estimation is needed for the restricted model because both parameters are specified under H0) This turns out to yield SSR = 209,448.99. Carry out the F test for the joint hypothesis.
(iii) Now, test H0: (2 = 0, (3 = 0, and (4 = 0 in the model
Price = (0 + (1 assess + (2 lotsize + (3 sqrft + (4 bdrms + u.
The R-squared from estimating this model using the same 88 houses is .829.
(iv) If the variance of price changes with assess, lotsize, sqrft, or bdrms, what can you say about the F test from part (iii)? |
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| m76594 | In testing for co integration between gfr and pe in Example 18.5, add t2 to equation (18.32) to obtain the OLS residuals. Include one lag in the augmented DF test. The 5% critical value for the test is -4.15. |
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| m76597 | In the example in equation (7.29), suppose that we define outlf to be one if the woman is out of the labor force, and zero otherwise.
(i) If we regress outlf on all of the independent variables in equation (7.29), what will happen to the intercept and slope estimates? (inlf = 1 - outlf. Plug this into the population equation inlf = (0 + (1 nwifeinc + (2 educ + ... and rearrange.)
(ii) What will happen to the standard errors on the intercept and slope estimates?
(iii) What will happen to the R-squared? |
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| m76601 | In the linear consumption function
The (estimated) marginal propensity to consume (MPC) out of income is simply the slope, , while the average propensity to consume (APC) is
Using observations for 100 families on annul income and consumption (both measured in dollars), the following equation is obtained:
(i) Interpret the intercept in this equation, and comment on its sign and magnitude.
(ii) What is the predicted consumption when family income is $30,000?
(iii) With inc on the x-axis, draw a graph of the estimated MPC and APC. |
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| m76602 | In the linear model given in equation (10.8), the explanatory variables xt = (xt1, ...., xtk) are said to be sequentially exogenous (sometimes called weakly exogenous) if
E(ut|xt, xt-1,_,, ...,x,) = 0, t = 1,2, .....,
So that the errors are unpredictable given current and all past values of the explanatory variables.
(i) Explain why sequential exogeneity is implied by strict exogeneity.
(ii) Explain why contemporaneous exogeneity is implied by sequential exogeneity.
(iii) Are the OLS estimators generally unbiased under the sequential exogeneity assumption? Explain.
(iv) Consider a model to explain the annual rate of HIV infections as a distributed lag of per capita condom usage for a state, region, or province:
E(HIVratet|pccont, pccontt-1,....,) = a0 + (0 pccnot + (1 pccon t-1.
+ (2 pccon t-2 + (3 pccon t-3.
Explain why this model satisfies the sequential exogeneity assumption. Does it seem likely that strict exogeneity holds too? |
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| m76603 | In the model (9.17), show that OLS consistently estimates a and ( if a1. is uncorrelated with xi. and bi. is uncorrelated with xi. and xi2, which are weaker assumptions than (9.19). [Write the equation as in (9.18) and recall from Chapter 5 that sufficient for consistency of OLS for the intercept and slope is E(ui) = 0 and Cov(xi, ui) = 0.] |
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| m76608 | In the simple linear regression model y = (0 + (1x + u. Suppose that E(u) ( 0. Letting a0 = E(u), show that the model can always be rewritten with the same slope, but a new intercept and error, where the new error has a zero expected value? |
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| m76609 | In the simple regression model (5.16), under the first four Gauss-Markov assumptions, we showed that estimators of the form (5.17) are consistent for the slope, (1 Given such an estimator, define an estimator of (0 by = - Show that plim = (0? |
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| m76610 | In the simple regression model under MLR. 1 through MLR.4, we argued that the slope estimator, is consistent for (1. Using = - , show that plim = (0. [You need to use the consistency of , and the law of large numbers, along with the fact that (0 = E(y) - (1E(x1).] |
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| m76623 | In this exercise, you are to compare OLS and LAD estimates of the effects of 401(k) plan eligibility on net financial assets. The model is
nettfa = (0 + (1inc + B2inc2 + (3age + (4age2 + (5male + (6e401k + u.
(i) Use the data in 401 KSUBS.RAW to estimate the equation by OLS and report the results in the usual form. Interpret the coefficient on e401k.
(ii) Use the OLS residuals to test for heteroskedasticity using the Breusch-Pagan test. Is u independent of the explanatory variables?
(iii) Estimate the equation by LAD and report the results in the same form as for OLS. Interpret the LAD estimate of (6.
(iv) Reconcile your findings from parts (i) and (iii). |
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| m76762 | Let d be a dummy (binary) variable and let z be a quantitative variable. Consider the model
y = (0 + (0d + (1z + (1d (z + u;
this is a general version of a model with an interaction between a dummy variable and a quantitative variable.
(i) Since it changes nothing important, set the error to zero, u = 0. Then, when d = 0 we can write the relationship between y and z as the function f0(z) = (0 +(1z. Write the same relationship when d = 1, where you should use/,(z) on the left-hand side to denote the linear function of z.
(ii) Assuming that (1 ( 0 (which means the two lines are not parallel), show that the value of z* such that f0 (z*) = f1 (z*) is z* = - (0/(1. This is the point at which the two lines intersect. Argue that z* is positive if and only if (0 and (1 have opposite signs.
(iii) Using the data in TWOYEAR.RAW, the following equation can be estimated:
where all coefficients and standard errors have been rounded to three decimal places. Using this equation, find the value of totcoll such that the predicted values of log(wage) are the same for men and women.
(iv) Based on the equation in part (iii), can women realistically get enough years of college so that their earnings catch up to those of men? Explain. |
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| m76764 | Let {ei: t = - 1, 0, 1, ...} be a sequence of independent, identically distributed random variables with mean zero and variance one. Define a stochastic process by
x, = et - (l/2)e1-1 + (l/2)e1-2, t = 1,2,....
(i) Find E(xt) and Var(xt). Do either of these depend on t?
(ii) Show that Corr(xt" xt+1) = -1/2 and Corr(xt, x1+2) = 1/3. (It is easiest to use the formula in Problem 11.1.)
(iii) What is Corr(x1, xl+h) for h > 2?
(iv) Is {xt} an asymptotically uncorrelated process? |
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| m76765 | Let gGDPt denote the annual percentage change in gross domestic product and let intt denote a short-term interest rate. Suppose that gGDPt is related to interest rates by
gGDPt = a0 + (0intt, + (1int t-1 + ut,
Where ut is uncorrelated with int1, intt-1, and all other past values of interest rates. Suppose that the Federal Reserve follows the policy rule:
intt, = (0 + (1 (gGDt-1 - 3) + vt,
Where (1, > 0. (When last year s GDP growth is above 3%, the Fed increases interest rates to prevent an "overheated" economy.) If vt is uncorrelated with all past values of intt and ut, argue that intt must be correlated with ut-1. (Lag the first equation for one time period and substitute for gGDPt-1 in the second equation.) Which Gauss-Markov assumption does this violate? |
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| m76766 | Let gGDPt denote the annual percentage change in gross domestic product and let intt denote a short term interest rate. Suppose that gGDPt is related to interest rates by
gGDPt = a0 + (0intt + (1 intt-1 + ut,
where ut is uncorrelated with intt, intt-1, and all other past values of interest rates. Suppose that the Federal Reserve follows the policy rule:
intt = (0 + (1 (gGDPt-1-3) + vt,
where (1 > 0. (When last year s GDP growth is above 3%, the Fed increases interest rates to prevent an "overheated" economy.) If v is uncorrelated with all past values of intt and ut, argue that intt must be correlated with ut-1 (Lag the first equation for one time period and substitute for gGDPt- t in the second equation.) Which Gauss-Markov assumption does this violate? |
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| m76767 | Let gMt be the annual growth in the money supply and let unem, be the unemployment rate. Assuming that unem, follows a stable AR(1) process, explain in detail how you would test whether gM Granger causes unem. |
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| m76768 | Let grad be a dummy variable for whether a student-athlete at a large university graduates in five years. Let hsGPA and SAT be high school grade point average and SAT score, respectively. Let study be the number of hours spent per week in an organized study hall. Suppose that, using data on 420 student-athletes, the following logit model is obtained:
P(grad = 1|hsGPA, SAT, study) = A(-1.17 + .24 hsGPA + .00058 SAT + .073 study),
where A(z) = exp(z)/[l + exp(z)] is the logit function. Holding hsGPA fixed at 3.0 and SAT fixed at 1,200, compute the estimated difference in the graduation probability for someone who spent 10 hours per week in study hall and someone who spent 5 hours per week. |
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| m76770 | Let hy6t denote the three-month holding yield (in percent) from buying a six-month T-bill at time (t - 1) and selling it at time t (three months hence) as a three month T-bill. Let hy3t-1 be the three month holding yield from buying a three month T-bill at time (t - 1). At time (t - 1), hy3t-1 is known, whereas hy6t is unknown because p3t (the price of three-month T-bills) is unknown at time (t - 1). The expectations hypothesis (EH) says that these two different three-month investments should be the same, on average. Mathematically, we can write this as a conditional expectation:
E(hy6t/It-1) = hy3t-1,
where It-1 denotes all observable information up through time r - 1. This suggests estimating the model
hy61, = (0 + (1hy3t-1 + ut,
and testing H0: (1 = 1. (We can also test H0: (0 = 0, but we often allow for a term premium for buying assets with different maturities, so that (0 ( 0.)
(i) Estimating the previous equation by OLS using the data in INTQRT.RAW (spaced every three months) gives
(.070) (.039)
n = 123, R2 = .866.
Do you reject H0: (1 = 1 against H0: (1 ( t 1 at the 1% significance level? Does the estimate seem practically different from one?
(ii) Another implication of the EH is that no other variables dated as t - 1 or earlier should help explain hy6t, once hy3t-1 has been controlled for. Including one lag of the spread between six-month and three-month T-bill rates gives
(.067) (.039) (.109)
n = 123, R2 = .885.
Now, is the coefficient on hy3t |
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| m76771 | Let inven, be the real value inventories in the United States during year t, let GDP, denote real gross domestic product, and let r3t denote the (ex post) real interest rate on three-month T-bills. The ex post real interest rate is (approximately) r3t = i3, - inft where i3t is the rate on three-month T-bills and inft is the annual inflation rate. The change in inventories, (invent, is the inventory investment for the year. The accelerator model of inventory investment is
(inven, = (0 + (1 GDPt + ut,
Where (1 > 0.
(i) Use the data in INVEN.RAW to estimate the accelerator model. Report the results in the usual form and interpret the equation. Is 1, statistically greater than zero?
(ii) If the real interest rate rises, then the opportunity cost of holding inventories rises, and so an increase in the real interest rate should decrease inventories. Add the real interest rate to the accelerator model and discuss the results.
(iii) Does the level of the real interest rate work better than the first difference, (r3t? |
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