VI. No explanatory variable is a perfect linear function of any other explan-
5.7 Summary and Exercises 5.8 Appendix: Econometric Lab #3
In this chapter, we return to the essence of econometrics—an effort to quan- tify economic relationships by analyzing sample data—and ask what conclu- sions we can draw from this quantification. Hypothesis testing goes beyond calculating estimates of the true population parameters to a much more complex set of questions. Hypothesis testing and statistical inference allow us to answer important questions about the real world from a sample. Is it likely that our result could have been obtained by chance? Would the results generated from our sample lead us to reject our original theories? How con- fident can we be that our estimate is close to the true value of the parameter?
This chapter starts with a brief introduction to the topic of hypothesis testing.
We then examine the t-test, typically used for hypothesis tests of individual regression coefficients. We next look at the confidence interval, a tool for evaluating the precision of our estimates, and we end the chapter by learn- ing how to use the F-test to determine whether whole groups of coefficients affect the dependent variable.
Hypothesis testing and the t-test should be familiar topics to readers with strong backgrounds in statistics, who are encouraged to skim this chapter and focus on only those applications that seem somewhat new. The development
M05_STUD2742_07_SE_C05.indd 115 1/6/16 5:08 PM
116 CHApTEr 5 HypotHesis testing and statistical inference
of hypothesis testing procedures is explained here in terms of the regression model, however, so parts of the chapter may be instructive even to those already skilled in statistics.
Our approach will be classical in nature, since we assume that the sample data are our best and only information about the population. An alternative, Bayesian statistics, uses a completely different definition of probability and does not use the sampling distribution concept.1
5.1 What Is Hypothesis Testing?
Hypothesis testing is used in a variety of settings. The Food and Drug Admin- istration (FDA), for example, tests new products before allowing their sale.
If the sample of people exposed to the new product shows some side effect significantly more frequently than would be expected to occur by chance, the FDA is likely to withhold approval of marketing that product. Similarly, economists have been statistically testing various relationships between consumption and income for almost a century; theories developed by John Maynard Keynes and Milton Friedman, among others, have been tested on mac- roeconomic and microeconomic data sets.
Although researchers are always interested in learning whether the theory in question is supported by estimates generated from a sample of real-world observations, it’s almost impossible to prove that a given hypothesis is correct.
All that can be done is to state that a particular sample conforms to a particu- lar hypothesis. Even though we cannot prove that a given theory is “correct”
using hypothesis testing, we often can reject a given hypothesis with a certain level of confidence. In such a case, the researcher concludes that it is very unlikely that the sample result would have been observed if the hypothesized theory were correct.
Classical Null and Alternative Hypotheses
The first step in hypothesis testing is to state the hypotheses to be tested.
This should be done before the equation is estimated because hypotheses
1. Bayesians, by being forced to state explicitly their prior expectations, tend to do most of their thinking before estimation, which is a good habit for a number of important reasons. For more on this approach, see Peter Kennedy, A Guide to Econometrics (Malden, MA: Blackwell, 2008), pp. 213–231. For more advanced coverage, see Tony Lancaster, An Introduction to Bayesian Econo- metrics (Oxford: Blackwell Publishing, 2004).
117 WHat is HypotHesis testing?
developed after estimation run the risk of being justifications of particular results rather than tests of the validity of those results.
The null hypothesis typically is a statement of the values that the researcher does not expect. The notation used to specify the null hypothesis is “H0:” followed by a statement of the range of values you do not expect. For example, if you expect a positive coefficient, then you don’t expect a zero or negative coefficient, and the null hypothesis is:
Null hypothesis H0: β… 0 (the values you do not expect)
The alternative hypothesis typically is a statement of the values that the researcher expects. The notation used to specify the alternative hypothesis is
“HA:” followed by a statement of the range of values you expect. To continue our previous example, if you expect a positive coefficient, then the alterna- tive hypothesis is:
Alternative hypothesis HA: β70 (the values you expect)
To test yourself, take a moment and think about what the null and alternative hypotheses will be if you expect a negative coefficient. That’s right, they’re:
H0: βÚ 0 HA: β60
The above hypotheses are for a one-sided test because the alternative hypotheses have values on only one side of the null hypothesis. Another approach is to use a two-sided test (or a two-tailed test) in which the alter- native hypothesis has values on both sides of the null hypothesis. For a two-sided test around zero, the null and alternative hypotheses are:
H0: β = 0 HA: β ≠0
We should note that there are a few rare cases in which we must violate our rule that the value you expect goes in the alternative hypothesis. Classical hypothesis testing requires that the null hypothesis contain the equal sign in some form (whether it be =, …, or Ú). This requirement means that researchers are forced to put the value they expect in the null hypothesis if their expectation includes an equal sign. This typically happens when the researcher specifies a particular value rather than a range. Luckily, such excep- tions are unusual in elementary applications.
With the exception of the unusual cases previously mentioned, economists always put what they expect in the alternative hypothesis. This allows us to make rather strong statements when we reject a null hypothesis. However, we
M05_STUD2742_07_SE_C05.indd 117 1/6/16 5:08 PM
118 CHApTEr 5 HypotHesis testing and statistical inference
can never say that we accept the null hypothesis; we must always say that we cannot reject the null hypothesis. As put by Jan Kmenta:
Just as a court pronounces a verdict as not guilty rather than innocent, so the conclusion of a statistical test is do not reject rather than accept.2