Economics 345 Applied Econometrics

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1 Economics 345 Applied Econometrics Problem Set 4--Solutions Prof: Martin Farnham Problem sets in this course are ungraded. An answer key will be posted on the course website within a few days of the release of each problem set. As noted in class, it is highly recommended that you make every effort to complete these problems before viewing the answer key. More Omitted Variables Bias 1) (this is a slight restatement of problem 3.8 from your text; I ve reworded it to make it clearer) Suppose that average worker productivity at manufacturing firms (avgprod) depends on two factors, average hours of training (avgtrain) and average worker ability (avgabil): avgprod = β 0 + β 1 avgtrain + β 2 avgabil + u assume that this equation satisfies the Gauss-Markov assumptions. Suppose that workers with lower ability tend to need more training. What, then, is the consequence of omitting avgabil from the RHS, for the estimate of the coefficient on avgtrain? See your notes on determining the sign of omitted variables bias, for help. If we omit avgabil from the RHS we mis-specify the model as avgprod = β 0 + β 1 avgtrain + u If avgabil affects avgprod and if avgabil and avgtrain are correlated, then omission of avgabil will lead to omitted variables bias. The sign of the bias is likely to be negative. This is because corr(avgtrain, avgabil) is negative (because lower ability workers require more training), and because avgabil positively affects avgprod (presumably higher ability workers tend to be more productive). Since the correlation between the omitted and included x variables is negative, and since the effect of the omitted variable on y is positive, we would expect the the overall sign of the bias to be negative. See your lecture notes for a table that lays out how to determine the sign of omitted variables bias. 2) Text Problem 3.9. This type of model is called a hedonic pricing model. It is a neat way to figure out the value of items that are not available on the market, and therefore that we cannot directly observe prices for. In this case, we re interested in figuring out how much people are

2 willing to pay for reductions in pollution. In other words, how much does cleaning up the air by X amount, increase people s willingness to pay for housing? If you ve ever wondered how policymakers try to figure out what it s worth to reduce pollution by a certain amount (for purposes of setting optimal abatement policy), this is one of the methods they use. i) If we believe people dislike nitrous oxide pollution (nox) then the coefficient on this variable should be negative. In other words, the higher the level of nox in the vicinity of a house, the less we should expect people to be willing to pay for the house. If we believe more rooms make a house more desirable, then the coefficient on rooms should be positive. That is, houses with more rooms should be more expensive (reflecting greater willingness to pay among potential bidders for the house). We can interpret beta1 as giving the percentage point decrease in willingness-to-pay for a house per percentage point increase in nox. In other words, beta1 is the elasticity of willingness-to-pay for housing with respect to nox levels. ii) nox and rooms may be negatively correlated because poor people have a higher tolerance for pollution (lower willingness-to-pay for clean air) and a higher tolerance for small homes (lower willingness-to-pay for extra rooms). So areas with high nox are likely to have homes with fewer than average rooms. Given that negative correlation between nox and rooms and the fact that rooms should have a positive effect on price, if we simply regress log(price) on log(nox) this will produce omitted variables bias of a negative sign. This is also referred to as causing downward bias of our estimate of beta1. iii) Given that we expected negative bias of our estimate of beta1 when excluding rooms from the regression, these results are consistent with our expectations. This doesn t guarantee that is closer to the true elasticity than , because it s possible that other relevant variables have been omitted, and that their omission is contributing to bias in the opposite direction (i.e. positive, or upward bias). By the way, just to be clear on the interpretation of the coefficient on nox, the estimate we obtain is This means that for each 1 percent increase in nox levels, the price of homes declines (on average) by about 0.7 percent. Variance of OLS Estimators 3) The goal of this problem is to get you to think about the math a bit, in particular with respect to the variance of the OLS slope estimators. Write out the variance of the OLS estimator for the slope coefficients for the general case of k right-hand-side (RHS) variables. In other words, the model we re picturing here is of the form:

3 y = β 0 + β 1 x β k x k + u Var( ˆβ j ) = σ 2 (x ij x j ) R j 2 ( ) = σ ( ) SST j 1 R j 2 Note: keep looking back at this formula as you ponder these answers a) Explain why, ceteris paribus, an increase in sample size will cause the variance of the jth (where j=1,2,,k) slope coefficient estimator to shrink. Increasing the sample size will cause the sum of squared deviations of x j to increase. This will increase the denominator, which will shrink the overall variance. Recall from the univariate case that increasing variation in an x variable increases our ability to identify the effect of that variable on y. It s like the case of a drug experiment on rats with cancer. Giving half the rats no treatment and half the rats a tiny amount of the drug will make it difficult to discern the average difference in outcomes between the treated rats and untreated rats. Giving the treatment group more of the drug will make for a starker comparison, and therefore a more precise estimate of the effect of the drug on outcomes. Better yet, if you have different levels of the drugs given to different rats in the treatment group, you maybe able to better map out the relationship between different levels of the drug and different outcomes. This can be useful for determining optimal dosage. b) Explain why, ceteris paribus, a decrease in the correlation between right-hand-side variables will cause the variance of the jth estimator to shrink. A decrease in the correlation between RHS variables will shrink the variance, because when there s less correlation between x j and the other RHS variables, the R-squared j will be smaller, and so 1 minus that R-squared will be bigger. If 1 minus the R-squared is bigger, then the denominator will be bigger, so the overall variance will be smaller. Think of the intuition like this. Suppose you conduct a series of controlled drug tests on rats with cancer. Each time you administer a drug to the treatment group of rats, you also give them a chance to exercise and you give them some extra nutritious food. You don t administer the exercise and food in a way that s perfectly collinear with the drug (imagine it s a slightly different amount of exercise and slightly different amount of food each time while the quantity of the drug treatment is the same in each test). This means getting the drug is highly (but not perfectly) collinear with getting exercise and the good food. Now suppose you sit down and run some regressions where you try to control for the amount of exercise and food the rats got, at the same time you try to measure for the effect of the drug. The problem is that exercise and good food tend to happen at the same time as treatment with the (potentially) cancer-curing drug. This makes it difficult to ascertain whether improvements in the rats health outcomes are due to the drug or due

4 to the exercise or due to the good food. When I say it becomes difficult to determine the effect of x (the drug) on y this is equivalent to saying the variance of the estimator of the effect of the drug goes up. If you were being more careful about the setup of the experiment, you would give identical exercise and food to rats with and without the drug (i.e. the treatment and control groups of rats). But in the real world, we can t perfectly control for this (even in the lab, rats who get the drug may choose to run around more because they feel better unless you tie them down, you may not be able to prevent them from exercising more). If you don t control for this, it will cause omitted variables bias. If you DO control for it, you avoid bias, but the variance of the slope estimator of interest will rise. c) Explain why, holding R-squared j and SST j constant but letting the sigma-squared vary, including more determinants of y on the RHS will lower the variance of the jth estimator. We discussed this in the univariate case. The more variables you include on the RHS, the more you re pulling unexplained variation out of u (your error term). Think of u as the linear combination of all factors that affect y that aren t included as RHS variables. Each time you pull one of these factors out and control for it, by including it as a RHS variable, you reduce the variance u. The variance of u is sigma-squared, so this shrinks when you include extra RHS variables (that have power to explain y). As sigma-squared shrinks, so does the variance of the jth estimator. d) Thinking back to (c), explain why, in general, we might expect including more RHS variables will increase the variance of the jth estimator. Under what specific conditions will the inclusion of more RHS variables lower the variance of the jth estimator? I m not looking for you to repeat the ceteris paribus condition in (c). Instead, discuss the relationship of the RHS variables with each other, in making your case. In (c) we made the ceteris paribus assumption that only sigma-squared would change when we included more RHS variables. In general, including more variables on the RHS will raise R-squared j, which will shrink the denominator and increase the variance of the jth estimator. So these effects will sometimes offset each other. Sometimes one effect will dominate, so we have to be careful to specify the conditions under which the variance will rise or fall. Including more RHS variables will increase the variance of the jth estimator when they tend to be highly correlated. If they re highly correlated, this will tend to raise the value of R-squared j more than it shrinks the value of sigma-squared. Including more RHS variables will decrease the variance of the jth estimator, when the added RHS variables are uncorrelated with the jth estimator, but have explanatory power for y. This is because their inclusion will lower sigma-squared without raising R- squared j.

5 Omitted Variables Bias AND Variance of the OLS Estimators Considered Together 4) Textbook problem i) I would expect them to be very different. This is because the omitted variables bias will tend to be large, in this case. Remember, the bias from omitting one RHS variable from a two-variable model (slightly different from this, but analogous) can be written as (see notes from chalkboard and page 96 of text) Bias( β 1 ) = β 2 δ1 where beta2 refers to the partial effect of x 2 on y, and delta 1 -tilde captures the degree of correlation between x 1 and x 2. Note that bias will be large when both these terms are large. It will be zero if either of these terms are zero. And as either term becomes very small (holding the other constant) the bias will shrink. ii) Again, think of the bias term above for guidance. If x 1 is almost uncorrelated with the other two potential RHS variables, then the bias from omitting them won t be great. So the two estimates will tend to be similar. iii) I would expect se( β 1 )to be smaller. This is because the R-squared j term is equal to zero in the univariate regression, but will be fairly large in the multivariate regression. Note that in this case, since the potential bias is small, it may be preferable to use the misspecified (univariate) model. This is an example of the bias-variance tradeoff that we have discussed so much. iv) In this case, I would expect se( ˆβ 1 )to be smaller. This is for reasons discussed above in problem 3.d. By including x 2 and x 3, you pull them out of u, and lower the variance of u (sigma-squared) and therefore the variance of the estimator of beta1. Recall that standard deviation is the square root of variance, and that standard error is an estimate (based on the random sample) of the standard deviation. So things that lower variance should be expected to lower se. EViews Problem In order for you to prepare for the lab exam at the end of term, it is important that you become comfortable enough with EViews to be able to do econometric analysis on your own (i.e. without asking your neighbor or your TA). Ways to prepare for that exam include reviewing the labs on your own time and working computer problems that I will include in the problem sets from here on in. Note that working these problems will also help prepare you for non-lab exams, as they focus on key econometric issues. 5) See Example 4.2 in the text. Work through this example, performing the analysis in EViews as you go (i.e. run the regression they run). The relevant data set is available in the folder where you ve found your lab datasets.

6 Example 4.2 is on page 133. a) Make sure you understand what the y variable is. Given your understanding of this variable, what do you expect the signs of the coefficients on totcomp, staff, and enroll to be? Explain in each case why you expect the sign you do. Since totcomp is meant to capture teacher quality, and because I would expect higher teacher quality to raise pass rates, I would expect this sign to be positive. Since staff captures overall teaching resources available to students, one would expect higher values would raise pass rates. Therefore this coefficient value would be expected to be positive. Enroll captures the number of students in the school, and hence the number of students over which available teaching resources must be divided. Since more enrollment means less resources per pupil, I would expect the sign of the coefficient on enroll to be negative. b) Do the coefficient estimates that you obtain match your expectations? Yes. c) Compare the coefficient on enroll with its standard error. At a glance, does it look like there is a statistically significant effect of enrollment on the percentage passing the math exam? Now formally test the hypothesis on the effect of enrollment, as set up in the example. Construct the t-statistic and carefully define the rejection region appropriate to testing this hypothesis at the 5% level. At a glance it doesn t look statistically significant, because the standard error is roughly the same magnitude of the coefficient estimate. At a glance, the t-stat for a null of zero will be approximately -1. This is unlikely to lie in the rejection region, except at very high significance levels. Formally, the hypothesis test is set up as follows: H 0 : β enroll = 0 H 1 : β enroll < 0 The t-statistic is / = Note that the t-stat given by EViews will be more correct, because it won t suffer from rounding error (or from as much as my calculation will suffer from). To find the rejection region, we need to remember this is a one-sided test with the alternative lying in the negative region. In other words, at the 5 percent significance level, we want to define any t-value lying at the fifth percentile or below as grounds for

7 rejection. Since the df are big in this case, we can use the standard normal distribution to find the fifth percentile value for the t-distribution corresponding to our null hypothesis. This looks to be at about We will reject the null if our t-stat lies to the left of this. Our t-stat is only about -0.9, so we fail to reject at the 5% significance level. d) Repeat the t-tests done in the example for totcomp and staff. For practice, perform the test on totcomp using a two-sided alternative (still at the 1% level). Carefully define the rejection region. Since they do the first 2 t-tests in the example, I won t repeat these. For the two-sided test on totcomp at the 1% level, we need to find the 0.5 th percentile and the 99.5 th percentile of the t-distribution. Again, with a large sample we can approximate this with the standard normal. The critical values are and (note: you don t need to look both these up. Once you find one, you can argue by the symmetry of the distribution that the other is just the negative of the first). So we will reject the null (with the 2-sided alternative at the 1% level) if our t-stat takes on a value that is less than or greater than Since the t-state for totcomp is 4.6, we reject the null of no effect. e) Construct a two-sided confidence interval for the estimate on totcomp, at the 99% confidence level (this should follow easily from (d)). The two-sided confidence interval is given by ˆβ totcomp ± c se( ˆβ totcomp ) = ± 2.575( ) So the confidence interval runs from about to about f) Reestimate the model, using the level-log form advocated in the example (for more on this form, see p49 of your text). Carefully interpret the coefficient on enroll. The coefficient estimate of on log(enroll) suggests that for a 1 percent increase in enrollment, the math pass rates are expected to fall by about units. Another way to put this is that for a 10 percent increase in enrollment the math pass rate declines by 0.13 percentage points (since the pass rate is measured in percentage units). g) The y variable in this case, is somewhat unusual. How is it different from other y variables we ve looked at? Can you see any potential problems? The y variable is constrained to lie between 0 and 100, since it is a percentage of total students. Usually, when we do OLS, we don t constrain the y variable in this way. One example of a problem that can arise is that for certain values of the x variables, we may obtain fitted values of y that lie outside the range. We may come across other examples of similarly constrained y variables later in the course.

8 h) Can you think of any forms of omitted variable bias that could be present in this model? What are other control variables that would probably be worth including on the RHS (if they were available)? Can we easily sign the bias, in this case? Let s consider bias of the coefficient on enroll. Sources of bias we want to consider are omitted variables that are correlated with enroll and that affect math pass rates. Consider something like the poverty rate in the district. High poverty rate areas tend to have quite a few children (poor families tend to have higher fertility rates than rich). And poor kids tend to have lower pass rates on standardized tests. So omitting the local poverty rate would potentially contribute to omitted variables bias on the coefficient for enroll. We can t easily sign the bias, unless we assume that enroll is uncorrelated with each of the other included RHS variables (and it almost certainly is correlated, at least with staff). Recall that once we have more than 2 included RHS variables, omitted variables bias becomes rather tricky to sign.