Confidence Intervals Tobias Econ 472
The Design To illustrate the interpretation of confidence intervals in a regression context, we design the following experiment: Data sets of different sizes (either n=100 or n=2,500) are generated from the simple regression model: Tobias Econ 472
The Design, Continued We set: We draw the errors and the x’s independently from a N(0,1) distribution. The y’s are then calculated using our regression equation. Tobias Econ 472
The Design For each data set, we compute the OLS estimates. We then calculate a 95% confidence interval for the slope parameter as: where Tobias Econ 472
The Design, Continued In this design, we have imposed that the variance parameter is 1, and we assume that this is known. As discussed in class, the interpretation of confidence intervals is that, in repeated sampling from the population, the constructed intervals will cover or contain the true regression parameter a pre-specified percentage of the time. Tobias Econ 472
The Design, continued. So, if I continue to generate new data sets, run a regression and construct, say, a 95% confidence interval each time, then (approximately) 5 percent of the collected intervals should NOT contain the true parameter. Tobias Econ 472
The Design, Continued In the following 2 tables, we generate 1,000 data sets with n=100, and each time, we construct a confidence interval at the pre-specified level of significance. A sample of these intervals are reported in the following two tables. The last table performs the same exercise for 90% intervals using n=2,500 instead of n=100. Tobias Econ 472
A Sample of 95% Confidence Intervals Out of 1,000 trials, 53 of the intervals (5.3%) did not contain the true regression parameter. Tobias Econ 472
A Sample of 90% Confidence Intervals Of the 1,000 trials, 93 (9.3%) of the intervals did not contain the true parameter. These intervals are a bit tighter than those on the previous slide. Tobias Econ 472
90% Intervals with n=2,500 Of 1,000 trials, 116 (11.6%) fell outside the intervals. Importantly, note that these intervals are much tighter than those on the previous page, owing to the larger sample size. Tobias Econ 472
Hypothesis Testing Tobias Econ 472
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