Testing

Abstract

This chapter treats hypothesis testing as an opportunity for the researcher to distinguish between three possible explanations for a set of empirical findings: random chance, the scientific hypothesis of primary interest, and alternative scientific hypotheses. The methods it offers to advance this goal involve refining the null hypothesis, while increasing the scrutiny of the primary scientific hypothesis of interest and the number of alternative scientific hypotheses that it must compete with. These methods are brought to life in applications to home sales in New England, multiproduct pricing in Major League Baseball, turnout in Congressional elections, and the link between abortion and crime.

Food for Thought

• 1. Chapter 6 discussed Gabaix’s (1999) explanation for Zipf’s Law.

  • (a) Articulate a parametric active null hypothesis for the city size distribution implied by Zipf’s law, and lay out a regression specification that would allow it to be tested.

  • (b) Do the same for a non-parametric active null.

  • (c) Articulate additional hypotheses that could be tested to examine Gabaix’s explanation for Zipf’s law more directly.

• 2. Hall’s (1978) classic paper on aggregate consumption argued that this variable should follow a random walk . Today this hypothesis could be tested using some sort of Dickey-Fuller test. Does this test treat Hall’s hypothesis as an active null , a refined passive null , or a traditional passive null? Do you think the data are sufficient to lend sufficient statistical power to such a test?

• 3. As mentioned in the chapter’s rational voter discussion, even the “improved” hypothesis tests that are used are not conclusive by themselves. One last piece of evidence is needed to round out support for the theory. How could it be obtained? (One answer to this question draws on a relatively obscure U.S . constitutional amendment.)

• 4. For years the rational voting literature was engrossed by the endogeneity of P and T, which arises because larger turnout in close elections will itself lower the probability that any given vote is decisive. General equilibrium models that accounted for this fact were deemed superior to partial equilibrium models that didn’t, though this “feedback effect” is inconsequential in practice.

  • (a) Show that this is true using a simple scale analysis, which incorporates the magnitudes in Fig. 9.4 and the fact that P is inversely proportional to the number of voters.

  • (b) What does this “inconsequentiality” imply for the magnitudes of the two terms on the left-hand side of Eq. 9.3? What does it imply for the predicted M-T relationship in the partial and general equilibrium models (as in Fig. 9.4)?

  • (c) Given your read of statistical significance in Fig. 9.4 and the conclusions arrived at above, sketch out a Venn diagram like Fig. 9.3, circumscribing the sets of estimates consistent with (not too unlikely under) the null hypothesis , the partial equilibrium model, and the general equilibrium model.

• 5. The hypothesis-testing technique known as “randomization inference” asks the researcher to impute the distribution of test statistics that would obtain if the treatment were randomly assigned across experimental units . In this book, instead, I have advocated creating an error structure that respects experimental content , which includes a random effect at the level of the experimental unit, when it differs from the unit of observation in the data. Compare these two approaches to hypothesis testing . How are they similar? How are they different? Which approach is more general?