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Applied Econometrics
Problem Set 5--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.
Introduction: p-values for t-tests
You might wish to supplement what follows by browsing pages 133-135 of your text.
There are a couple reasons p-values can be useful to the econometrician (or someone
reading their research results). First, the p-value contains information about where in the
relevant t-distribution the t-statistic calculated for a particular null hypothesis lies. This
can eliminate the need to look up critical t-values for the chosen significance level of
your test. Second, if you just report that you reject the null at the 5% significance level
(for example), your reader doesn’t know whether this was a close call or not, unless they
take your t-stat and reference the t-table. If you give them a p-value, this tells them
whether your rejection of the null was a close call or whether it was a slam dunk (or
somewhere in between). I find the first reason particularly compelling.
Here’s an example. Suppose you’re testing the following hypothesis:
H 0 : β 1 = 0, against the alternative
H 1 : β 1 ≠ 0
If you’re performing this test at the 5% significance level, then you will reject the null if
your t-statistic lies at the 97.5 percentile or above, or if it lies at the 2.5 percentile or
below. In other words, the rejection regions sum up to an area of 0.05 under the t-
distribution, so it’s the top 0.025 and the bottom 0.025 of the distribution (because the
alternative is two-sided).
The p-value can be interpreted as telling us the lowest significance level at which the null
can be rejected, given the t-statistic we have calculated under the null. So if we just
happened to get a p-value in this case of 0.05, this would tell us that our t-statistic lies
either at the 2.5 percentile of the 97.5 percentile (and we could tell which, by looking at
whether the t-stat is positive or negative).
Suppose now that we obtained a p-value of 0.08. What would this tell us about where
our t-statistic lies in the relevant t-distribution? Well, for a two-sided alternative, if we’re
just on the border of rejection at the 8% significance level, this tells us our rejection
region would have to start at the 4 th percentile on the left hand side of the distribution,
economics 345 applied econometrics.pdf
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