## Choosing the More Likely Hypothesis

By Richard Startz, Department of Economics, University of California, Santa Barbara, USA, startz@ucsb.edu

Suggested Citation
Richard Startz (2014), "Choosing the More Likely Hypothesis", Foundations and Trends® in Econometrics: Vol. 7: No. 2, pp 119-189. http://dx.doi.org/10.1561/0800000028

Publication Date: 20 Nov 2014

Subjects

Keywords
C11 Bayes AnalysisC12 Hypothesis TestingC13 Estimation
Bayes TheoremHypothesis testingEconometric estimatorsParameter estimates

#### Journal details

 1. Introduction 2. Choosing Between Hypotheses 3. Bayes Theorem 4. A Simple Coin-Flipping Example 5. Regression Estimates 6. Diffuse Alternatives and the Lindley "Paradox" 7. Is the Stock Market Efficient? 8. Non-sharp Hypotheses 9. Bayes Theorem and Consistent Estimation 10. More General Bayesian Inference 11. The General Decision-theoretic Approach 12. A Practitioner's Guide to Choosing Between Hypotheses 13. Summary References

#### Abstract

Much of economists' statistical work centers on testing hypotheses in which parameter values are partitioned between a null hypothesis and an alternative hypothesis in order to distinguish two views about the world. Our traditional procedures are based on the probabilities of a test statistic under the null but ignore what the statistics say about the probability of the test statistic under the alternative. Traditional procedures are not intended to provide evidence for the relative probabilities of the null versus alternative hypotheses, but are regularly treated as if they do. Unfortunately, when used to distinguish two views of the world, traditional procedures can lead to wildly misleading inference. In order to correctly distinguish between two views of the world, one needs to report the probabilities of the hypotheses given parameter estimates rather than the probability of the parameter estimates given the hypotheses. This monograph shows why failing to consider the alternative hypothesis often leads to incorrect conclusions. I show that for most standard econometric estimators, it is not difficult to compute the proper probabilities using Bayes theorem. Simple formulas that require only information already available in standard estimation reports are provided. I emphasize that frequentist approaches for deciding between the null and alternative hypothesis are not free of priors. Rather, the usual procedures involve an implicit, unstated prior that is likely to be far from scientifically neutral.

DOI:10.1561/0800000028

#### Book details

ISBN: 978-1-60198-898-0
88 pp. \$65.00

ISBN: 978-1-60198-899-7
88 pp. \$120.00

1. Introduction
2. Choosing Between Hypotheses
3. Bayes Theorem
4. A Simple Coin-Flipping Example
5. Regression Estimates
6. Diffuse Alternatives and the Lindley "Paradox"
7. Is the Stock Market Efficient?
8. Non-sharp Hypotheses
9. Bayes Theorem and Consistent Estimation
10. More General Bayesian Inference
11. The General Decision-theoretic Approach
12. A Practitioner's Guide to Choosing Between Hypotheses
13. Summary
References

#### Choosing the More Likely Hypothesis

Much of economists' statistical work centers on testing hypotheses in which parameter values are partitioned between a null hypothesis and an alternative hypothesis in order to distinguish two views about the world. Our traditional procedures are based on the probabilities of a test statistic under the null but ignore what the statistics say about the probability of the test statistic under the alternative. Traditional procedures are not intended to provide evidence for the relative probabilities of the null versus alternative hypotheses, but are regularly treated as if they do. Unfortunately, when used to distinguish two views of the world, traditional procedures can lead to wildly misleading inference. In order to correctly distinguish between two views of the world, one needs to report the probabilities of the hypotheses given parameter estimates rather than the probability of the parameter estimates given the hypotheses.

Choosing the More Likely Hypothesis shows why failing to consider the alternative hypothesis often leads to incorrect conclusions. It shows that for most standard econometric estimators, it is not difficult to compute the proper probabilities using Bayes theorem. Simple formulas that require only readily available information in standard estimation reports are provided. The author emphasizes that frequentist approaches for deciding between the null and alternative hypothesis are not free of priors. Rather, the usual procedures involve an implicit, unstated prior that is likely to be far from scientifically neutral. 