Let Rur2 be the R-squared from this regression. To obtain the restricted R-squared, Rr2 , we need to reestimate the model reported in the problem but with the same observations used to estimate the unrestricted model. We would find the critical value from the F6, distribution. Therefore, the errors are serially uncorrelated. Especially after detrending there is little evidence of a unit root in log invpc. For log price , the first order autocorrelation is about. After detrending, the first order autocorrelation drops to.
We cannot confidently rule out a unit root in log price. Because differencing eliminates linear time trends, it is not surprising that the estimate on the trend is very small and very statistically insignificant.
Only if both parameters are zero does E returnt returnt-1 not depend on returnt The F statistic is about 2. The R- squared is about.
The time trend coefficient is very insignificant, so it is not needed in the equation. So the estimated LRP is now negative and significant, which is very different from the equation in levels, This is a good example of how differencing variables before including them in a regression can lead to very different conclusions than a regression in levels. The coefficient on gct-1 is also practically large, showing significant autocorrelation in consumption growth.
This regression basically shows that the change in prcfat cannot be explained by the change in unem or any of the policy variables. It does have some seasonality, which is why the R-squared is. Of course, this is not to say the levels regression is valid. But, as it turns out, we can reject a unit root in prcfat, and so we can at least justify using it in level form; see Computer Exercise Generally, the issue of whether to take first differences is very difficult, even for professional time series econometricians.
This is very little evidence against H0; the null is not rejected at any reasonable significance level. With multiple explanatory variables the formulas are more complicated but have similar features. There is no reason to worry about serial correlation in this example. But any kind of adjustment, either to obtain valid standard errors for OLS as in Section The t statistic is about 2. This means we should view the standard errors reported in equation The t statistic is well below one in absolute value, so there is no evidence of serial correlation in the accelerator model.
If we view the test of serial correlation as a test of dynamic misspecification, it reveals no dynamic misspecification in the accelerator model. The largest t statistic is on incum, which is estimated to have a large effect on the probability of winning.
But we must be careful here. So, for an incumbent Democrat running, we must add the coefficients on partyWH and incum together, and this nets out to about zero. The economic variables are less statistically significant than in equation The gnews interaction has a t statistic of about 1. Since the dependent variable is binary, this is a case where we must appeal to asymptotics.
Unfortunately, we have only 20 observations. The inflation variable has the expected sign but is not statistically significant. So 15 out of 20 elections through are correctly predicted. But, remember, we used data from these years to obtain the estimated equation. Because this is above. Therefore, there is little evidence of serial correlation in the errors. And, if anything, it is negative. In fact, all heteroskedasticity-robust standard errors are less than the usual OLS standard errors, making each variable more significant.
But we must remember that the standard errors in the LPM have only asymptotic justification. With only 20 observations it is not clear we should prefer the heteroskedasticity-robust standard errors to the usual ones. So there is no evidence that the average price of fish varies systematically within a week. Rough seas as measured by high waves would reduce the supply of fish shift the supply curve back , and this would result in a price increase.
One might argue that bad weather reduces the demand for fish at a market, too, but that would reduce price. If there are demand effects captured by the wave variables, they are being swamped by the supply effects. We can use the omitted variable bias table from Chapter 3, Table 3. Without wave2 and wave3, the coefficient on t seems to have a downward bias. Since we know the coefficients on wave2 and wave3 are positive, this means the wave variables are negatively correlated with t.
In other words, the seas were rougher, on average, at the beginning of the sample period. You can confirm this by regressing wave2 on t and wave3 on t. Further, the height of the waves is not influenced by past unexpected changes in log avgprc. Therefore, there is strong evidence of positive serial correlation.
The coefficient on wave3 drops by a relatively smaller amount, but its t statistic 1. The graph in part iii makes this clear, as does finding that the smallest variance estimate is 2. We should really compare adjusted R-squareds, because the ARCH 1 model contains only two total parameters. Therefore, after adjusting for the different df, the quadratic in return-1 fits better than the ARCH 1 model.
Therefore, an ARCH 2 model does not seem warranted. The adjusted R-squared is about. Therefore, there is very little evidence of first-order serial correlation. The variance of the error appears to be larger when the change in unemployment is larger. To account for the increase in average education levels, we obtain an additional effect: —. So the drop in average fertility if the average education level increased by 1. For example, in Example In Example Each person in the panel data set is exactly two years older on January 31, than on January 31, As we know, when we have an intercept in the model we cannot include an explanatory variable that is constant across i; this violates Assumption MLR.
Intuitively, since age changes by the same amount for everyone, we cannot distinguish the effect of age from the aggregate time effect. The increase from. But the very large. Prior to the policy change, the high earning group spent about By dropping highearn from the regression, we attribute to the policy change the difference between the two groups that would be observed without any intervention.
So we just use the usual F statistic for joint significance of the year dummies. The R-squared is about. This suggests that, at a minimum, we should compute heteroskedasticity-robust standard errors, t statistics, and F statistics. We could also use weighted least squares although the form of heteroskedasticity used here may not be sufficient; it does not depend on educ, age, and so on.
The F statistic for joint significance with 6 and 1, df is about 1. This is a bit misleading, however. The coefficients are large in magnitude as well. The coefficient on educ — which is for the base year, — is small and insignificant, suggesting little if any relationship between fertility and education in the early seventies. The estimates above are consistent with fertility becoming more linked to education as the years pass. The F statistic is insignificant because we are testing some insignificant coefficients along with some significant ones.
So the estimated effect is larger — the elasticity of price with respect to dist is. The coefficient on pctstu means that a one percentage point increase in pctstu increases rent by half a percent. The t statistic of five shows that, at least based on the usual analysis, pctstu is very statistically significant.
If ai is in the error term, the errors across the two time periods for each city are positively correlated, and this invalidates the usual OLS standard errors and t statistics.
Interestingly, the effect of pctstu is over twice as large as we estimated in the pooled OLS equation. Now, a one percentage point increase in pctstu is estimated to increase rental rates by about 1. Not surprisingly, we obtain a much less precise estimate when we difference although the OLS standard errors from part i are likely to be much too small because of the positive serial correlation in the errors within each city.
Note that serial correlation is no longer an issue because we have no time component in the first-differenced equation. On a four point scale, this a modest effect although it accumulates over four years of athletic eligibility. The fact that we are pooling across two semesters does not change that basic point. If we think harder, the direction of the bias is not clear, and this is where pooling across semesters plays a role. First, suppose we used only the fall term, when football is in season.
When we pool the two semesters we cannot, with a much more detailed analysis, determine which bias will dominate. The intercept in the first-differenced equation is the intercept for the spring. Interestingly, the in-season effect is larger now: term GPA is estimated to be about.
The t statistic is about —1. If some fraction of student-athletes take a lighter load during the season for those sports that have a true season , then term GPAs may tend to be higher, other things equal.
This would bias the results away from finding an effect of season on term GPA. The coefficients on the criminal justice variables change very modestly, and the statistical significance of each variable is also essentially unaffected. The contemporaneous spending variable, while still having a negative coefficient, is not at all statistically significant. Given the timing of the tests, a lagged effect is not surprising. In Michigan, the fourth grade math test is given in January, and so if preparation for the test begins a full year in advance, spending when the students are in third grade would at least partly matter.
The two-sided p-value is about. So we would be justified in dropping these variables, but they are not doing any harm. Therefore, there is no evidence that the return to education varied over this time period. Therefore, when we test all interaction terms as a group seven of them , we fail to reject the null that the union differential was constant over this period.
Most of the interactions are individually insignificant; in fact, only those for and are close. We can get joint insignificance by lumping several statistically insignificant variables in with one or two statistically significant ones.
But it is hard to ignore the practically large change from to There might be a problem in this example with the strict exogeneity assumption: perhaps union membership next year depends on unexpected wage changes this year. Because the variance is constant across t, by Problem This is what we wanted to show. When we specify an equation for each standardized final exam score, the errors in the different equations for the same student are certain to be correlated: students who have more unobserved ability tend to do better on all tests.
Unlike with a panel data set, where time is the natural ordering of the data within each cross-sectional unit, and the aggregate time effects apply to all units, intercepts for the different classes may not be needed. If all students took the same set of classes then this is similar to a panel data set, and we would want to put in different class intercepts. Thus, the different class intercepts based on arbitrarily ordering the classes for each student probably are not needed.
In other words, controlling for SATs and cumGPAs could be enough to obtain the ceteris paribus effect of class attendance. We could use fixed effects instead. Within each student we compute the demeaned data, where, for each student, the means are computed across classes.
We do not report an intercept because it gets removed by the time demeaning. The coefficient on y90t is identical to the intercept from the first difference estimation, and the slope coefficients and standard errors are identical to first differencing. We do not report an R-squared because none is comparable to the R-squared obtained from first differencing. Because there are three years, we would have a total of observations if each firm had data on all variables for all three years.
Instead, due to missing data, we can use only observations in the FE estimation. This is a practically large effect, and the t statistic is very large. It would if inertia played a role in training workers.
Therefore, if we omit binary indicators for occupation, the union wage differential may simply be picking up wage differences across occupations. Because some people change occupation over the period, we should include these in our analysis.
Of course the group we choose does not affect the estimated union wage differential. The fixed effect estimate on union, to four decimal places, is. Is this a large effect? Executions are relatively rare in most states, but murder rates are relatively low on average, too. For the unknown people whose lives might be saved via a deterrent effect, it would seem important. Somewhat surprisingly, this is well below the nonrobust standard error. See also Computer Exercise The next highest state was Virginia, with These are three-year totals.
The earlier finding of a deterrent effect is not robust to the time period used. Oddly, adding another year of data causes the standard error on the exec coefficient to markedly increase. The t statistic is 1. The F test for joint significant is 1.
Plus, when these variables are dropped from the regression, the coefficient on choice only falls to Therefore, it is essentially the same as the usual OLS standard error. This is not very surprising because at least of the observations can be assumed independent of one another.
The explanatory variables may adequately capture the within-family correlation. We have only 23 observations, and we are removing much of the variation in the explanatory variables except the gender variable by using within-family differences.
In the case of educ, the robust standard error is about. For married, the robust standard error is about. A similar change is evident for union from. For union, these are.
In both cases, the robust standard error is somewhat higher. Typically, the adjustment for FE has a smaller relative effect because FE removes the main source of positive serial correlation: the unobserved effect, ai. Remember, pooled OLS leaves ai in the error term. The usual standard errors for both pooled OLS and FE are invalid with serial correlation in the idiosyncratic errors, uit, but this correlation is usually of a smaller degree. And, in some applications, it is not unreasonable to think the uit have no serial correlation.
However, if we are being careful, we allow this possibility in computing our standard errors and test statistics. The error term u contains, among other things, family income, which has a positive effect on GPA and is also very likely to be correlated with PC ownership.
Therefore, family income certainly satisfies the second requirement for an instrumental variable: it is correlated with the endogenous explanatory variable [see But as we suggested in part i , faminc has a positive affect on GPA, so the first requirement for a good IV, If we had faminc we would include it as an explanatory variable in the equation; if it is the only important omitted variable correlated with PC, we could then estimate the expanded equation by OLS.
Some students who buy computers when given the grant would not have without the grant. Students who did not receive the grants might still own computers. Define a dummy variable, grant, equal to one if the student received a grant, and zero otherwise. Then, if grant was randomly assigned, it is uncorrelated with u. In particular, it is uncorrelated with family income and other socioeconomic factors in u. Further, grant should be correlated with PC: the probability of owning a PC should be significantly higher for student receiving grants.
Incidentally, if the university gave grant priority to low-income students, grant would be negatively correlated with u, and IV would be inconsistent. So the asymptotic bias is. This is a simple illustration of how a seemingly small correlation. Therefore, there is likely to be a self-selection problem: students that would do better anyway are also more likely to attend a choice school.
Since u1 does not contain income, random assignment of grants within income class means that grant designation is not correlated with unobservables such as student ability, motivation, and family support. In other words, after accounting for income, the grant amount must have some affect on choice. This seems reasonable, provided the grant amounts differ within each income class. This equation allows us to directly estimate the effect of increasing the grant amount on the test score, holding family income fixed.
From a policy perspective this is itself of some interest. If the initial equation is in levels or logs, xt and xt-1 are likely to be positively correlated. If the model is for first differences or percentage changes, there still may be positive or negative correlation between xt and xt Second, xt-1 will often be correlated with xt, and we can check this easily enough by running a regression of xt of xt This suggests estimating the equation by instrumental variables, where xt-1 is the IV for xt.
This is a reduced form simple regression equation. It shows that, controlling for no other factors, one more sibling in the family is associated with monthly salary that is about 2. The t statistic on sibs is about —4.
Of course sibs can be correlated with many things that should have a bearing on wage including, as we already saw, years of education. Note that brthord is missing for 83 observations. The equation predicts that every one-unit increase in brthord reduces predicted education by about.
In particular, the difference in predicted education for a first-born and fourth-born child is about. Dorf, Robert H. Byron Bird, Warren E. Larsen, Morris L.
Morton, D. Munson, Donald F. Young, Theodore H. Hassett, Donald G. Newnan, Ted G. Eschenbach, Jerome P. Kurose, Keith W. Hoffman James E. Smith and William A. Philips, J. Sze, Kwok K. McClellan, Ronald W. Lucas Jr. Ashcroft N.
Gere, Barry J. Crowe, Donald F. Roberson, Barbara C. Ehrhardt , Eugene F. Within U. Quantity: 1. Condition: Good. Light rubbing wear to cover, spine and page edges. Very minimal writing or notations in margins not affecting the text. Possible clean ex-library copy, with their stickers and or stamp s. Published by Brooks Cole, Published by HarperCollins. Condition: GOOD.
Spine creases, wear to binding and pages from reading. May contain limited notes, underlining or highlighting that does affect the text. Possible ex library copy, will have the markings and stickers associated from the library. Accessories such as CD, codes, toys, may not be included.
Published by Sharp Electronics Corporation. Possible clean ex-library copy, with their stickers and or stamp s. Published by Addison Wesley, Published by Prentice Hall, Condition: GOOD.
Spine creases, wear to binding and pages from reading. May contain limited notes, underlining or highlighting that does affect the text. Possible ex library copy, will have the markings and stickers associated from the library. Accessories such as CD, codes, toys, may not be included. Published by pearson, Published by McGraw Hill, New - Softcover Condition: Brand New. Condition: Brand New. Published by Prentice-Hall, Used Condition: Good.
Condition: Good. Former library book; may include library markings. Used book that is in clean, average condition without any missing pages. Published by MG-H, Used - Softcover Condition: Good. Goldstein, Larry J.
0コメント