To simplify the notation, after emphasizing the assumptions in the population model, and assuming random sampling, I condition only on the values of the explanatory variables in the sample. The number of volumes in the law library and the cost of tuition are both measures of school quality.
From equation (3.22) we have
This means that the simple regression estimator on average underestimates the effect of the training program. This is clearly a linear function of the yi: take the weights as wi = (zi −z )/szx.
The regression of educ on exper and tenure yields
4.7 (i) Although the standard error on hrsemp has not changed, the size of the coefficient has increased by half. Therefore, we can reject H0: β2 = 1 at the 5% level of significance (against the one-tailed alternative). v) The slope coefficient at inc in the simple regression is about .821, which is not very different from the .799 obtained in part (ii).
This would make little sense. Performance on math and science exams are measures of outputs of the educational process, and we would like to know how various educational inputs
Therefore, for predicting the price, the logo model is significantly better. ceteris paribus the effect of expendB on voteA is achieved by making changes and holding prtystrA, expendA and u fixed:. We can then obtain the t statistic we want as the coefficient on the black female dummy variable.
In Section 3.3 – in particular, in the discussion surrounding Table 3.2 – we discussed how to determine the direction of bias in the OLS estimators when an important variable (ability, in this
If n is the sample size, the df in the unrestricted model – the denominator df in the F distribution – is n – 8. When hsGPA2 is added to the regression, its coefficient is about 0.337 and its t statistic is about 1.56.
The estimated equation is
This means that one theoretical problem with the LPM – the ability to generate silly probability estimates – does not arise in this application. The R-squared in the weighted least squares estimate is larger than that from the OLS regression in part (i), but remember that they are not comparable. iv) With robust standard errors—that is, with standard errors robust to misspecifying the function h(x)—the equation holds.
The sample selection in this case is arguably endogenous. Because prospective students may look at campus crime as one factor in deciding where to attend college, colleges with high crime
This is a common finding in studies of school performance: family income (or related factors, such as living in poverty) are far more important in explaining student performance than per-pupil spending or other school characteristics. Note that for consistent OLS parameter estimation, we do not need e0 to be uncorrelated with turals*.). ii) The CEV assumptions are probably not true in this case. However, this is unlikely to be the case because tvhours* are directly dependent on the explanatory variables.
The coefficient for education becomes about .058 (se .006), so this is similar to the estimate obtained from IQ, although slightly larger and more precisely estimated.
With sales defined to be in billions of dollars, we obtain the following estimated equation using all companies in the sample
For testing βjc = βuniv, we can use the same trick as in Section 4.4 to obtain the standard error of the difference: replace univ with totcoll = jc + univ, and then the coefficient on jc is the difference in estimated returns, along with its standard error. The mean is much less sensitive to effects at the upper end of the distribution. It is important for students to learn that there are potential pitfalls inherent in using regression with time series data that are not present for cross-sectional applications.
It is also quite clear that in many applications there are likely to be some explanatory variables that are not strictly exogenous.
We follow the hint and write
Even series that appear to be roughly uncorrelated - such as stock returns - do not appear to be independently distributed, as you will see in Chapter 12 under dynamic forms of heteroskedasticity. We have to be careful when interpreting the results because we can simply find a. As discussed in Section 10.5, the usual R-squared can be misleading when the dependent variable is trending.
Write
This is another example of how the (marginal) significance of one variable (afdec6) can be masked by testing it together with two very insignificant variables.
The functional form was not specified, but a reasonable one is
Let post79 be a dummy variable equal to one for years after 1979, and zero otherwise
Adding log(prgnp) to equation (10.38) gives
The t-statistic on gyt-1 is only about 1.39, so it is not significant at the usual levels of significance. But the estimates are fairly close, given the size and marginal importance of the coefficient on inft-1. But maybe people became less careful when they were forced to wear seatbelts. v) The average prcfat is about 0.886, which means that on average just under one percent of all accidents are fatal.
This is one of those tough cases:. the correlation between unemt and unemt-1 is high but not particularly close to one. iv).
We can reason this from equation (12.4) because the usual OLS standard error is an
Because it is above 0.5, we would have predicted that Clinton would win the 1996 election, as he did. v) The regression of uˆt on uˆt−1 produces ρˆ ≈ -.164 with heteroscedasticity-robust standard error of about .195. But we must remember that the standard errors in the LPM have only asymptotic justification. The p-value is approximately .352, so there is little evidence of heteroskedasticity in the AR(1) model for gc.
However, the traditional assumption that the errors in the original equation are serially uncorrelated is not always good.
The first equation omits the 1981 year dummy variable, y81, and so does not allow any appreciation in nominal housing prices over the three year period in the absence of an incinerator
The effect of the law is measured by δ1, which is the change in the likelihood of a drunk driving arrest as a result of the new Florida law. So the estimated effect is larger – the price elasticity to distance is 0.062 after the combustion site has been chosen – but the t-statistic is only 1.24. The fixed effects estimates are unbiased estimators of the parameters on the time dummies in the original model.
Therefore, there is still evidence of a lagged spending effect after controlling for unobserved district effects. v) The change in the coefficient and significance on the lunch variable is the most dramatic.
Just use OLS on an expanded equation, where SAT and cumGPA are added as proxy variables for student ability and motivation; see Chapter 9
Although the difference in the estimates of the return to education is practically large, it is therefore not statistically significant. ii). The regression in part (iv) verifies that they are strongly positively correlated.) The more difficult issue is whether e401k can be taken as exogenous in the structural model. Therefore, there is strong evidence that p401k is endogenous in the structural equation (assuming, of course, that the IV, e401k, is exogenous). ii).
While significant in reduced form for education with other controls, the fact that it was insignificant in part (ii) is concerning.
We can easily see that the rank condition for identifying the second equation does not hold
Grnont is thus very significant in the reduced form for gcemt, and we can proceed with the IV estimation. IV). 16.16 (i) To estimate the demand equations, we need at least one exogenous variable that appears in the supply equation. ii) In order for wave2t and wave3t to be valid IVs for log(avgprct), we need two assumptions. There is indirect evidence for this in part three, as the two variables are jointly significant in the reduced form for log(avgprct).
Part (iii) provides evidence that there are day-of-the-week effects in the demand function.
For the immediate purpose of finding out the variables that determine whether accepted applicants choose to enroll, there is not a sample selection problem. The population of interest is
17.11 (i) The results for the Poisson regression model involving pcnv2, ptime862 and inc862 are given in the following table:. the standard error probabilities must be multiplied by σˆ, which is about 1.179. Therefore, MLE standard errors should increase by about 18%. iii) From table 17.3 we have the log-likelihood value for the limited model, Lr. The log-likelihood value for the unrestricted model is given in the table above as 17.12 (i) The results of the Poisson regression are given in the following table:. ii) Because the coefficient in black is so large, we get estimated proportionality.
Note also that black is highly statistically significant.). iii) From the above table, σˆ = .944, which shows that there is indeed underdispersion in the estimated model.
The results of an OLS regression using only the uncensored durations are given in the following table
Without an exclusion restriction in the log(wage) equation, ˆλ is almost a linear function of the other explanatory variables in the sample. However, there can be a . difference between eligibility and actual participation, since men can always refuse to participate if chosen.). v) The simple LPM results are. Participation in the job training program lowers the estimated probability of being unemployed in 1978 by 0.111, or 11.1 percentage points.
The results of the Poisson regression are given in the following table, along with the OLS estimates of a linear model for later reference.
Following the hint, we have
Clearly, we cannot think of the GDL model as a good approximation to the FDL model. Therefore, it is legitimate to treat gfrt as driftless if it is indeed a random walk. v) The prediction of gfrn+1 is simply gfrn, so the prediction error is simply Δgfrn+1 = gfrn+1 – gfrn. The standard error of the regression is slightly lower than that of the random walk model. vii).
The out-of-sample forecasting performance of the AR(2) model is worse than the no-drift random walk: the MAE for 1980 to 1984 is about 0.991 for the AR(2) model.
APPENDIX A
Since both variables are in proportional form, we can multiply the equation by 100 to convert each variable to proportional form, we can multiply the equation by 100 to convert each variable to. The first pound of fertilizer has the greatest effect, and each additional pound has a smaller effect than the previous pound.
APPENDIX B
The difference in expected GPAs is significant, but the difference in SAT scores is also quite large. ii) After the hint, we apply the law of repeated expectations.
APPENDIX C
In general, the mean of the ratios, Yi/Xi, is not the ratio of the means, W2 Y/. This non-equivalence is discussed a bit on page 676.). These are pretty similar ratings. value is below 0.05, we reject H0 against the one-sided alternative at the 5% level. iv). Therefore, H0 is rejected in favor of H1 at the 5% level, but not at the 1% level. iv) The p-value obtained from Stata is 0.029; this is half the p-value for the two-sided alternative.
This is well below the 5% critical value (based on the standard normal distribution, H1 is rejected at the 1% level. iv).
APPENDIX D
APPENDIX E
This follows directly from partitioned matrix multiplication in Appendix D. Write
