You have substantial latitude about what to emphasize in Chapter 1. I find it useful to talk about the economics of crime example (Example 1.1) and the wage example (Example 1.2) so that students see, at the outset, that econometrics is linked to economic reasoning, even if the economics is not complicated theory.
I like to familiarize students with the important data structures that empirical economists use, focusing primarily on cross-sectional and time series data sets, as these are what I cover in a first-semester course. It is probably a good idea to
mention the growing importance of data sets that have both a cross-sectional and time dimension.
I spend almost an entire lecture talking about the problems inherent in drawing causal inferences in the social sciences. I do this mostly through the agricultural yield, return to education, and crime examples. These examples also contrast
experimental and nonexperimental (observational) data. Students studying business and finance tend to find the term structure of interest rates example more relevant, although the issue there is testing the implication of a simple theory, as opposed to inferring causality. I have found that spending time talking about these examples, in place of a formal review of probability and statistics, is more successful (and more enjoyable for the students and me).
This is the chapter where I expect students to follow most, if not all, of the algebraic derivations. In class I like to derive at least the unbiasedness of the OLS slope coefficient, and usually I derive the variance. At a minimum, I talk about the factors affecting the variance. To simplify the notation, after I emphasize the assumptions in the population model, and assume random sampling, I just condition on the values of the explanatory variables in the sample. Technically, this is justified by random sampling because, for example, E(ui|x1,x2,…,xn) = E(ui|xi) by independent sampling. I find that students are able to focus on the key assumption SLR.4 and subsequently take my word about how conditioning on the independent variables in the sample is harmless. (If you prefer, the appendix to Chapter 3 does the
conditioning argument carefully.) Because statistical inference is no more difficult in multiple regression than in simple regression, I postpone inference until Chapter 4. (This reduces redundancy and allows you to focus on the interpretive differences between simple and multiple regression.)
You might notice how, compared with most other texts, I use relatively few assumptions to derive the unbiasedness of the OLS slope estimator, followed by the formula for its variance. This is because I do not introduce redundant or
unnecessary assumptions. For example, once SLR.4 is assumed, nothing further about the relationship between u and x is needed to obtain the unbiasedness of OLS under random sampling.
For undergraduates, I do not work through most of the derivations in this chapter, at least not in detail. Rather, I focus on interpreting the assumptions, which mostly concern the population. Other than random sampling, the only assumption that involves more than population considerations is the assumption about no perfect collinearity, where the possibility of perfect collinearity in the sample (even if it does not occur in the population) should be touched on. The more important issue is perfect collinearity in the population, but this is fairly easy to dispense with via examples. These come from my experiences with the kinds of model specification issues that beginners have trouble with.
The comparison of simple and multiple regression estimates – based on the particular sample at hand, as opposed to their statistical properties – usually makes a strong impression. Sometimes I do not bother with the “partialling out”
interpretation of multiple regression.
As far as statistical properties, notice how I treat the problem of including an irrelevant variable: no separate derivation is needed, as the result follows form Theorem 3.1.
I do like to derive the omitted variable bias in the simple case. This is not much more difficult than showing unbiasedness of OLS in the simple regression case under the first four Gauss-Markov assumptions. It is important to get the students thinking about this problem early on, and before too many additional (unnecessary) assumptions have been introduced.
I have intentionally kept the discussion of multicollinearity to a minimum. This partly indicates my bias, but it also reflects reality. It is, of course, very important for students to understand the potential consequences of having highly correlated independent variables. But this is often beyond our control, except that we can ask less of our multiple regression analysis. If two or more explanatory variables are highly correlated in the sample, we should not expect to precisely estimate their ceteris paribus effects in the population.
I find extensive treatments of multicollinearity, where one “tests” or somehow “solves” the multicollinearity problem, to be misleading, at best. Even the
organization of some texts gives the impression that imperfect multicollinearity is somehow a violation of the Gauss-Markov assumptions: they include
multicollinearity in a chapter or part of the book devoted to “violation of the basic assumptions,” or something like that. I have noticed that master’s students who have had some undergraduate econometrics are often confused on the multicollinearity issue. It is very important that students not confuse multicollinearity among the included explanatory variables in a regression model with the bias caused by omitting an important variable.
I do not prove the Gauss-Markov theorem. Instead, I emphasize its
implications. Sometimes, and certainly for advanced beginners, I put a special case of Problem 3.12 on a midterm exam, where I make a particular choice for the function g(x). Rather than have the students directly compare the variances, they should
appeal to the Gauss-Markov theorem for the superiority of OLS over any other linear, unbiased estimator.
At the start of this chapter is good time to remind students that a specific error distribution played no role in the results of Chapter 3. That is because only the first two moments were derived under the full set of Gauss-Markov assumptions. Nevertheless, normality is needed to obtain exact normal sampling distributions (conditional on the explanatory variables). I emphasize that the full set of CLM assumptions are used in this chapter, but that in Chapter 5 we relax the normality assumption and still perform approximately valid inference. One could argue that the classical linear model results could be skipped entirely, and that only large-sample analysis is needed. But, from a practical perspective, students still need to know where the t distribution comes from because virtually all regression packages report t statistics and obtain p-values off of the t distribution. I then find it very easy to
cover Chapter 5 quickly, by just saying we can drop normality and still use t statistics and the associated p-values as being approximately valid. Besides, occasionally students will have to analyze smaller data sets, especially if they do their own small surveys for a term project.
It is crucial to emphasize that we test hypotheses about unknown population parameters. I tell my students that they will be punished if they write something like
? = 0 on an exam or, even worse, H0: .632 = 0. H0:?1
One useful feature of Chapter 4 is its illustration of how to rewrite a population model so that it contains the parameter of interest in testing a single restriction. I find this is easier, both theoretically and practically, than computing variances that can, in some cases, depend on numerous covariance terms. The example of testing equality of the return to two- and four-year colleges illustrates the basic method, and shows that the respecified model can have a useful interpretation. Of course, some statistical packages now provide a standard error for linear combinations of estimates with a simple command, and that should be taught, too.
One can use an F test for single linear restrictions on multiple parameters, but this is less transparent than a t test and does not immediately produce the standard error needed for a confidence interval or for testing a one-sided alternative. The trick of rewriting the population model is useful in several instances, including obtaining confidence intervals for predictions in Chapter 6, as well as for obtaining confidence intervals for marginal effects in models with interactions (also in Chapter
The major league baseball player salary example illustrates the difference
between individual and joint significance when explanatory variables (rbisyr and hrunsyr in this case) are highly correlated. I tend to emphasize the R-squared form of the F statistic because, in practice, it is applicable a large percentage of the time, and it is much more readily computed. I do regret that this example is biased toward students in countries where baseball is played. Still, it is one of the better examples 3
of multicollinearity that I have come across, and students of all backgrounds seem to get the point.
Chapter 5 is short, but it is conceptually more difficult than the earlier chapters, primarily because it requires some knowledge of asymptotic properties of estimators. In class, I give a brief, heuristic description of consistency and asymptotic normality before stating the consistency and asymptotic normality of OLS. (Conveniently, the same assumptions that work for finite sample analysis work for asymptotic analysis.) More advanced students can follow the proof of consistency of the slope coefficient in the bivariate regression case. Section E.4 contains a full matrix treatment of asymptotic analysis appropriate for a master’s level course.
An explicit illustration of what happens to standard errors as the sample size grows emphasizes the importance of having a larger sample. I do not usually cover the LM statistic in a first-semester course, and I only briefly mention the asymptotic efficiency result. Without full use of matrix algebra combined with limit theorems for vectors and matrices, it is very difficult to prove asymptotic efficiency of OLS.
I think the conclusions of this chapter are important for students to know, even though they may not fully grasp the details. On exams I usually include true-false type questions, with explanation, to test the students’ understanding of asymptotics.
[For example: “In large samples we do not have to worry about omitted variable bias.” (False). Or “Even if the error term is not normally distributed, in large samples we can still compute approximately valid confidence intervals under the Gauss-Markov assumptions.” (True).]
I cover most of Chapter 6, but not all of the material in great detail. I use the example in Table 6.1 to quickly run through the effects of data scaling on the
important OLS statistics. (Students should already have a feel for the effects of data scaling on the coefficients, fitting values, and R-squared because it is covered in Chapter 2.) At most, I briefly mention beta coefficients; if students have a need for them, they can read this subsection.
The functional form material is important, and I spend some time on more complicated models involving logarithms, quadratics, and interactions. An
important point for models with quadratics, and especially interactions, is that we need to evaluate the partial effect at interesting values of the explanatory variables. Often, zero is not an interesting value for an explanatory variable and is well outside the range in the sample. Using the methods from Chapter 4, it is easy to obtain confidence intervals for the effects at interesting x values.
As far as goodness-of-fit, I only introduce the adjusted R-squared, as I think using a slew of goodness-of-fit measures to choose a model can be confusing to novices (and does not reflect empirical practice). It is important to discuss how, if we fixate on a high R-squared, we may wind up with a model that has no interesting ceteris paribus interpretation.
I often have students and colleagues ask if there is a simple way to predict y when log(y) has been used as the dependent variable, and to obtain a goodness-of-fit measure for the log(y) model that can be compared with the usual R-squared obtained when y is the dependent variable. The methods described in Section 6.4 are easy to implement and, unlike other approaches, do not require normality.
The section on prediction and residual analysis contains several important topics, including constructing prediction intervals. It is useful to see how much wider the prediction intervals are than the confidence interval for the conditional mean. I usually discuss some of the residual-analysis examples, as they have real-world applicability.
This is a fairly standard chapter on using qualitative information in regression analysis, although I try to emphasize examples with policy relevance (and only cross-sectional applications are included.).
In allowing for different slopes, it is important, as in Chapter 6, to appropriately interpret the parameters and to decide whether they are of direct interest. For
example, in the wage equation where the return to education is allowed to depend on gender, the coefficient on the female dummy variable is the wage differential between women and men at zero years of education. It is not surprising that we cannot estimate this very well, nor should we want to. In this particular example we would drop the interaction term because it is insignificant, but the issue of interpreting the parameters can arise in models where the interaction term is significant.
In discussing the Chow test, I think it is important to discuss testing for
differences in slope coefficients after allowing for an intercept difference. In many applications, a significant Chow statistic simply indicates intercept differences. (See the example in Section 7.4 on student-athlete GPAs in the text.) From a practical perspective, it is important to know whether the partial effects differ across groups or whether a constant differential is sufficient.
I admit that an unconventional feature of this chapter is its introduction of the linear probability model. I cover the LPM here for several reasons. First, the LPM is being used more and more because it is easier to interpret than probit or logit models. Plus, once the proper parameter scalings are done for probit and logit, the estimated effects are often similar to the LPM partial effects near the mean or median values of the explanatory variables. The theoretical drawbacks of the LPM are often of secondary importance in practice. Computer Exercise C7.9 is a good one to illustrate that, even with over 9,000 observations, the LPM can deliver fitted values strictly between zero and one for all observations.
If the LPM is not covered, many students will never know about using
econometrics to explain qualitative outcomes. This would be especially unfortunate for students who might need to read an article where an LPM is used, or who might want to estimate an LPM for a term paper or senior thesis. Once they are introduced to purpose and interpretation of the LPM, along with its shortcomings, they can tackle nonlinear models on their own or in a subsequent course.
A useful modification of the LPM estimated in equation (7.29) is to drop kidsge6 (because it is not significant) and then define two dummy variables, one for kidslt6 equal to one and the other for kidslt6 at least two. These can be included in place of kidslt6 (with no young children being the base group). This allows a diminishing marginal effect in an LPM. I was a bit surprised when a diminishing effect did not materialize.
This is a good place to remind students that homoskedasticity played no role in showing that OLS is unbiased for the parameters in the regression equation. In
addition, you probably should mention that there is nothing wrong with the R-squared or adjusted R-squared as goodness-of-fit measures. The key is that these are
estimates of the population R-squared, 1 – [Var(u)/Var(y)], where the variances are the unconditional variances in the population. The usual R-squared, and the adjusted version, consistently estimate the population R-squared whether or not Var(u|x) = Var(y|x) depends on x. Of course, heteroskedasticity causes the usual standard
errors, t statistics, and F statistics to be invalid, even in large samples, with or without normality.
By explicitly stating the homoskedasticity assumption as conditional on the explanatory variables that appear in the conditional mean, it is clear that only
heteroskedasticity that depends on the explanatory variables in the model affects the validity of standard errors and test statistics. The version of the Breusch-Pagan test in the text, and the White test, are ideally suited for detecting forms of
heteroskedasticity that invalidate inference obtained under homoskedasticity. If heteroskedasticity depends on an exogenous variable that does not also appear in the mean equation, this can be exploited in weighted least squares for efficiency, but only rarely is such a variable available. One case where such a variable is available is when an individual-level equation has been aggregated. I discuss this case in the text but I rarely have time to teach it.
As I mention in the text, other traditional tests for heteroskedasticity, such as the Park and Glejser tests, do not directly test what we want, or add too many
assumptions under the null. The Goldfeld-Quandt test only works when there is a natural way to order the data based on one independent variable. This is rare in practice, especially for cross-sectional applications.
Some argue that weighted least squares estimation is a relic, and is no longer necessary given the availability of heteroskedasticity-robust standard errors and test statistics. While I am sympathetic to this argument, it presumes that we do not care much about efficiency. Even in large samples, the OLS estimates may not be precise 6
enough to learn much about the population parameters. With substantial
heteroskedasticity we might do better with weighted least squares, even if the
weighting function is misspecified. As discussed in the text on pages 288-289, one can, and probably should, compute robust standard errors after weighted least squares. For asymptotic efficiency comparisons, these would be directly comparable to the heteroskedasiticity-robust standard errors for OLS.
Weighted least squares estimation of the LPM is a nice example of feasible GLS, at least when all fitted values are in the unit interval. Interestingly, in the LPM examples in the text and the LPM computer exercises, the heteroskedasticity-robust standard errors often differ by only small amounts from the usual standard errors. However, in a couple of cases the differences are notable, as in Computer Exercise C8.7.
The coverage of RESET in this chapter recognizes that it is a test for neglected nonlinearities, and it should not be expected to be more than that. (Formally, it can be shown that if an omitted variable has a conditional mean that is linear in the
included explanatory variables, RESET has no ability to detect the omitted variable. Interested readers may consult my chapter in Companion to Theoretical Econometrics, 2001, edited by Badi Baltagi.) I just teach students the F statistic version of the test. The Davidson-MacKinnon test can be useful for detecting functional form misspecification, especially when one has in mind a specific alternative, nonnested model. It has the advantage of always being a one degree of freedom test.
I think the proxy variable material is important, but the main points can be made with Examples 9.3 and 9.4. The first shows that controlling for IQ can substantially change the estimated return to education, and the omitted ability bias is in the expected direction. Interestingly, education and ability do not appear to have an interactive effect. Example 9.4 is a nice example of how controlling for a previous value of the dependent variable – something that is often possible with survey and nonsurvey data – can greatly affect a policy conclusion. Computer Exercise 9.3 is also a good illustration of this method.
I rarely get to teach the measurement error material, although the attenuation bias result for classical errors-in-variables is worth mentioning.
The result on exogenous sample selection is easy to discuss, with more details given in Chapter 17. The effects of outliers can be illustrated using the examples. I think the infant mortality example, Example 9.10, is useful for illustrating how a single influential observation can have a large effect on the OLS estimates.
With the growing importance of least absolute deviations, it makes sense to at least discuss the merits of LAD, at least in more advanced courses. Computer Exercise 9.9 is a good example to show how mean and median effects can be very different, even though there may not be “outliers” in the usual sense.
Because of its realism and its care in stating assumptions, this chapter puts a somewhat heavier burden on the instructor and student than traditional treatments of time series regression. Nevertheless, I think it is worth it. It is important that students learn that there are potential pitfalls inherent in using regression with time series data that are not present for cross-sectional applications. Trends, seasonality, and high persistence are ubiquitous in time series data. By this time, students should have a firm grasp of multiple regression mechanics and inference, and so you can focus on those features that make time series applications different from cross-sectional ones.
I think it is useful to discuss static and finite distributed lag models at the same time, as these at least have a shot at satisfying the Gauss-Markov assumptions. Many interesting examples have distributed lag dynamics. In discussing the time series versions of the CLM assumptions, I rely mostly on intuition. The notion of strict exogeneity is easy to discuss in terms of feedback. It is also pretty apparent that, in many applications, there are likely to be some explanatory variables that are not strictly exogenous. What the student should know is that, to conclude that OLS is unbiased – as opposed to consistent – we need to assume a very strong form of exogeneity of the regressors. Chapter 11 shows that only contemporaneous exogeneity is needed for consistency.
Although the text is careful in stating the assumptions, in class, after discussing strict exogeneity, I leave the conditioning on X implicit, especially when I discuss the no serial correlation assumption. As this is a new assumption I spend some time on it. (I also discuss why we did not need it for random sampling.)
Once the unbiasedness of OLS, the Gauss-Markov theorem, and the sampling distributions under the classical linear model assumptions have been covered – which can be done rather quickly – I focus on applications. Fortunately, the students already know about logarithms and dummy variables. I treat index numbers in this chapter because they arise in many time series examples.
A novel feature of the text is the discussion of how to compute goodness-of-fit measures with a trending or seasonal dependent variable. While detrending or deseasonalizing y is hardly perfect (and does not work with integrated processes), it is better than simply reporting the very high R-squareds that often come with time series regressions with trending variables.
Much of the material in this chapter is usually postponed, or not covered at all, in an introductory course. However, as Chapter 10 indicates, the set of time series applications that satisfy all of the classical linear model assumptions might be very small. In my experience, spurious time series regressions are the hallmark of many 8
student projects that use time series data. Therefore, students need to be alerted to the dangers of using highly persistent processes in time series regression equations. (Spurious regression problem and the notion of cointegration are covered in detail in Chapter 18.)
It is fairly easy to heuristically describe the difference between a weakly
dependent process and an integrated process. Using the MA(1) and the stable AR(1) examples is usually sufficient.
When the data are weakly dependent and the explanatory variables are
contemporaneously exogenous, OLS is consistent. This result has many applications, including the stable AR(1) regression model. When we add the appropriate
homoskedasticity and no serial correlation assumptions, the usual test statistics are asymptotically valid.
The random walk process is a good example of a unit root (highly persistent) process. In a one-semester course, the issue comes down to whether or not to first difference the data before specifying the linear model. While unit root tests are covered in Chapter 18, just computing the first-order autocorrelation is often
sufficient, perhaps after detrending. The examples in Section 11.3 illustrate how different first-difference results can be from estimating equations in levels.
Section 11.4 is novel in an introductory text, and simply points out that, if a model is dynamically complete in a well-defined sense, it should not have serial
correlation. Therefore, we need not worry about serial correlation when, say, we test the efficient market hypothesis. Section 11.5 further investigates the
homoskedasticity assumption, and, in a time series context, emphasizes that what is contained in the explanatory variables determines what kind of heteroskedasticity is ruled out by the usual OLS inference. These two sections could be skipped without loss of continuity.
Most of this chapter deals with serial correlation, but it also explicitly considers heteroskedasticity in time series regressions. The first section allows a review of what assumptions were needed to obtain both finite sample and asymptotic results. Just as with heteroskedasticity, serial correlation itself does not invalidate R-squared. In fact, if the data are stationary and weakly dependent, R-squared and adjusted R-squared consistently estimate the population R-squared (which is well-defined under stationarity).
Equation (12.4) is useful for explaining why the usual OLS standard errors are not generally valid with AR(1) serial correlation. It also provides a good starting point for discussing serial correlation-robust standard errors in Section 12.5. The subsection on serial correlation with lagged dependent variables is included to debunk the myth that OLS is always inconsistent with lagged dependent variables and serial correlation. I do not teach it to undergraduates, but I do to master’s students.
Section 12.2 is somewhat untraditional in that it begins with an asymptotic t test for AR(1) serial correlation (under strict exogeneity of the regressors). It may seem heretical not to give the Durbin-Watson statistic its usual prominence, but I do believe the DW test is less useful than the t test. With nonstrictly exogenous regressors I cover only the regression form of Durbin’s test, as the h statistic is asymptotically equivalent and not always computable.
Section 12.3, on GLS and FGLS estimation, is fairly standard, although I try to show how comparing OLS estimates and FGLS estimates is not so straightforward. Unfortunately, at the beginning level (and even beyond), it is difficult to choose a course of action when they are very different.
I do not usually cover Section 12.5 in a first-semester course, but, because some econometrics packages routinely compute fully robust standard errors, students can be pointed to Section 12.5 if they need to learn something about what the corrections do. I do cover Section 12.5 for a master’s level course in applied econometrics (after the first-semester course).
I also do not cover Section 12.6 in class; again, this is more to serve as a reference for more advanced students, particularly those with interests in finance. One important point is that ARCH is heteroskedasticity and not serial correlation, something that is confusing in many texts. If a model contains no serial correlation, the usual heteroskedasticity-robust statistics are valid. I have a brief subsection on correcting for a known form of heteroskedasticity and AR(1) errors in models with strictly exogenous regressors.
While this chapter falls under “Advanced Topics,” most of this chapter requires no more sophistication than the previous chapters. (In fact, I would argue that, with the possible exception of Section 13.5, this material is easier than some of the time series chapters.)
Pooling two or more independent cross sections is a straightforward extension of cross-sectional methods. Nothing new needs to be done in stating assumptions, except possibly mentioning that random sampling in each time period is sufficient. The practically important issue is allowing for different intercepts, and possibly different slopes, across time.
The natural experiment material and extensions of the difference-in-differences estimator is widely applicable and, with the aid of the examples, easy to understand. Two years of panel data are often available, in which case differencing across time is a simple way of removing g unobserved heterogeneity. If you have covered Chapter 9, you might compare this with a regression in levels using the second year of data, but where a lagged dependent variable is included. (The second approach only requires collecting information on the dependent variable in a previous year.) These often give similar answers. Two years of panel data, collected before and after a policy change, can be very powerful for policy analysis.
Having more than two periods of panel data causes slight complications in that the errors in the differenced equation may be serially correlated. (However, the traditional assumption that the errors in the original equation are serially uncorrelated is not always a good one. In other words, it is not always more appropriate to used fixed effects, as in Chapter 14, than first differencing.) With large N and relatively small T, a simple way to account for possible serial correlation after differencing is to compute standard errors that are robust to arbitrary serial correlation and hetero-skedasticity. Econometrics packages that do cluster analysis (such as Stata) often allow this by specifying each cross-sectional unit as its own cluster.
My preference is to view the fixed and random effects methods of estimation as applying to the same underlying unobserved effects model. The name “unobserved effect” is neutral to the issue of whether the time-constant effects should be treated as fixed parameters or random variables. With large N and relatively small T, it almost always makes sense to treat them as random variables, since we can just view the unobserved ai as being drawn from the population along with the observed variables. Especially for undergraduates and master’s students, it seems sensible to not raise the philosophical issues underlying the professional debate. In my mind, the key issue in most applications is whether the unobserved effect is correlated with the observed explanatory variables. The fixed effects transformation eliminates the unobserved effect entirely whereas the random effects transformation accounts for the serial correlation in the composite error via GLS. (Alternatively, the random effects transformation only eliminates a fraction of the unobserved effect.)
As a practical matter, the fixed effects and random effects estimates are closer when T is large or when the variance of the unobserved effect is large relative to the variance of the idiosyncratic error. I think Example 14.4 is representative of what often happens in applications that apply pooled OLS, random effects, and fixed effects, at least on the estimates of the marriage and union wage premiums. The random effects estimates are below pooled OLS and the fixed effects estimates are below the random effects estimates.
Choosing between the fixed effects transformation and first differencing is harder, although useful evidence can be obtained by testing for serial correlation in the first-difference estimation. If the AR(1) coefficient is significant and negative (say, less than ?.3, to pick a not quite arbitrary value), perhaps fixed effects is preferred.
Matched pairs samples have been profitably used in recent economic applications, and differencing or random effects methods can be applied. In an equation such as (14.12), there is probably no need to allow a different intercept for each sister provided that the labeling of sisters is random. The different intercepts might be needed if a certain feature of a sister that is not included in the observed controls is used to determine the ordering. A statistically significant intercept in the differenced equation would be evidence of this.
When I wrote the first edition, I took the novel approach of introducing instrumental variables as a way of solving the omitted variable (or unobserved heterogeneity) problem. Traditionally, a student’s first exposure to IV methods comes by way of simultaneous equations models. Occasionally, IV is first seen as a method to solve the measurement error problem. I have even seen texts where the first appearance of IV methods is to obtain a consistent estimator in an AR(1) model with AR(1) serial correlation.
The omitted variable problem is conceptually much easier than simultaneity, and stating the conditions needed for an IV to be valid in an omitted variable context is straightforward. Besides, most modern applications of IV have more of an
unobserved heterogeneity motivation. A leading example is estimating the return to education when unobserved ability is in the error term. We are not thinking that education and wages are jointly determined; for the vast majority of people, education is completed before we begin collecting information on wages or salaries. Similarly, in studying the effects of attending a certain type of school on student performance, the choice of school is made and then we observe performance on a test. Again, we are primarily concerned with unobserved factors that affect performance and may be correlated with school choice; it is not an issue of simultaneity.
The asymptotics underlying the simple IV estimator are no more difficult than for the OLS estimator in the bivariate regression model. Certainly consistency can be derived in class. It is also easy to demonstrate how, even just in terms of inconsistency, IV can be worse than OLS if the IV is not completely exogenous. At a minimum, it is important to always estimate the reduced form equation and test whether the IV is partially correlated with endogenous explanatory variable. The material on multicollinearity and 2SLS estimation is a direct extension of the OLS case. Using equation (15.43), it is easy to explain why multicollinearity is generally more of a problem with 2SLS estimation.
Another conceptually straightforward application of IV is to solve the
measurement error problem, although, because it requires two measures, it can be hard to implement in practice.
Testing for endogeneity and testing any overidentification restrictions is something that should be covered in second semester courses. The tests are fairly easy to motivate and are very easy to implement.
While I provide a treatment for time series applications in Section 15.7, I admit to having trouble finding compelling time series applications. These are likely to be found at a less aggregated level, where exogenous IVs have a chance of existing. (See also Chapter 16.)
I spend some time in Section 16.1 trying to distinguish between good and inappropriate uses of SEMs. Naturally, this is partly determined by my taste, and many applications fall into a gray area. But students who are going to learn about SEMS should know that just because two (or more) variables are jointly determined does not mean that it is appropriate to specify and estimate an SEM. I have seen many bad applications of SEMs where no equation in the system can stand on its own with an interesting ceteris paribus interpretation. In most cases, the researcher either wanted to estimate a tradeoff between two variables, controlling for other factors – in which case OLS is appropriate – or should have been estimating what is (often pejoratively) called the “reduced form.”
The identification of a two-equation SEM in Section 16.3 is fairly standard except that I emphasize that identification is a feature of the population. (The early work on SEMs also had this emphasis.) Given the treatment of 2SLS in Chapter 15, the rank condition is easy to state (and test).
Romer’s (1993) inflation and openness example is a nice example of using aggregate cross-sectional data. Purists may not like the labor supply example, but it has become common to view labor supply as being a two-tier decision. While there are different ways to model the two tiers, specifying a standard labor supply function conditional on working is not outside the realm of reasonable models.
Section 16.5 begins by expressing doubts of the usefulness of SEMs for aggregate models such as those that are specified based on standard macroeconomic models. Such models raise all kinds of thorny issues; these are ignored in virtually all texts, where such models are still used to illustrate SEM applications.
SEMs with panel data, which are covered in Section 16.6, are not covered in any other introductory text. Presumably, if you are teaching this material, it is to more advanced students in a second semester, perhaps even in a more applied course. Once students have seen first differencing or the within transformation, along with IV methods, they will find specifying and estimating models of the sort contained in Example 16.8 straightforward. Levitt’s example concerning prison populations is especially convincing because his instruments seem to be truly exogenous.
I emphasize to the students that, first and foremost, the reason we use the probit and logit models is to obtain more reasonable functional forms for the response probability. Once we move to a nonlinear model with a fully specified conditional distribution, it makes sense to use the efficient estimation procedure, maximum likelihood. It is important to spend some time on interpreting probit and logit estimates. In particular, the students should know the rules-of-thumb for comparing probit, logit, and LPM estimates. Beginners sometimes mistakenly think that,
because the probit and especially the logit estimates are much larger than the LPM estimates, the explanatory variables now have larger estimated effects on the response probabilities than in the LPM case. This may or may not be true.
I view the Tobit model, when properly applied, as improving functional form for corner solution outcomes. In most cases it is wrong to view a Tobit application as a data-censoring problem (unless there is true data censoring in collecting the data or because of institutional constraints). For example, in using survey data to estimate the demand for a new product, say a safer pesticide to be used in farming, some farmers will demand zero at the going price, while some will demand positive pounds per acre. There is no data censoring here; some farmers find it optimal to use none of the new pesticide. The Tobit model provides more realistic functional forms for E(y|x) and E(y|y > 0,x) than a linear model for y. With the Tobit model, students may be tempted to compare the Tobit estimates with those from the linear model and conclude that the Tobit estimates imply larger effects for the independent variables. But, as with probit and logit, the Tobit estimates must be scaled down to be comparable with OLS estimates in a linear model. [See Equation (17.27); for an example, see Computer Exercise C17.3.]
Poisson regression with an exponential conditional mean is used primarily to improve over a linear functional form for E(y|x). The parameters are easy to
interpret as semi-elasticities or elasticities. If the Poisson distributional assumption is correct, we can use the Poisson distribution compute probabilities, too. But over-dispersion is often present in count regression models, and standard errors and likelihood ratio statistics should be adjusted to reflect this. Some reviewers of the first edition complained about either the inclusion of this material or its location within the chapter. I think applications of count data models are on the rise: in microeconometric fields such as criminology, health economics, and industrial
organization, many interesting response variables come in the form of counts. One suggestion was that Poisson regression should not come between the Tobit model in Section 17.2 and Section 17.4, on censored and truncated regression. In fact, I put the Poisson regression model between these two topics on purpose: I hope it helps emphasize that the material in Section 17.2 is purely about functional form, as is Poisson regression. Sections 17.4 and 17.5 deal with underlying linear models, but where there is a data-observability problem.
Censored regression, truncated regression, and incidental truncation are used for missing data problems. Censored and truncated data sets usually result from sample design, as in duration analysis. Incidental truncation often arises from self-selection into a certain state, such as employment or participating in a training program. It is important to emphasize to students that the underlying models are classical linear models; if not for the missing data or sample selection problem, OLS would be the efficient estimation procedure.
Several of the topics in this chapter, including testing for unit roots and
cointegration, are now staples of applied time series analysis. Instructors who like their course to be more time series oriented might cover this chapter after Chapter 12, if time permits. Or, the chapter can be used as a reference for ambitious students who wish to be versed in recent time series developments.
The discussion of infinite distributed lag models, and in particular geometric DL and rational DL models, gives one particular interpretation of dynamic regression models. But one must emphasize that only under fairly restrictive assumptions on the serial correlation in the error of the infinite DL model does the dynamic regression consistently estimate the parameters in the lag distribution. Computer Exercise C18.1 provides a good illustration of how the GDL model, and a simple RDL model, can be too restrictive.
Example 18.5 tests for cointegration between the general fertility rate and the value of the personal exemption. There is not much evidence of cointegration, which sheds further doubt on the regressions in levels that were used in Chapter 10. The error correction model for holding yields in Example 18.7 is likely to be of interest to students in finance. As a class project, or a term project for a student, it would be interesting to update the data to see if the error correction model is stable over time.
The forecasting section is heavily oriented towards regression methods and, in particular, autoregressive models. These can be estimated using any econometrics package, and forecasts and mean absolute errors or root mean squared errors are easy to obtain. The interest rate data sets (for example, INTQRT.RAW) can be updated to do much more recent out-of-sample forecasting exercises.
This is a chapter that students should read if you have assigned them a term
paper. I used to allow students to choose their own topics, but this is difficult in a first-semester course, and places a heavy burden on instructors or teaching assistants, or both. I now assign a common topic and provide a data set with about six weeks left in the term. The data set is cross-sectional (because I teach time series at the end of the course), and I provide guidelines of the kinds of questions students should try to answer. (For example, I might ask them to answer the following questions: Is
there a marriage premium for NBA basketball players? If so, does it depend on race? Can the premium, if it exists, be explained by productivity differences?) The specifics are up to the students, and they are to craft a 10- to 15-page paper on their own. This gives them practice writing the results in a way that is easy-to-read, and forces them to interpret their findings. While leaving the topic to each student’s discretion is more interesting, I find that many students flounder with an open-ended assignment until it is too late. Naturally, for a second-semester course, or a senior seminar, students would be expected to design their own topic, collect their own data, and then write a more substantial term paper.