1. Confidence Intervals
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Econ 472

2. 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:
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Econ 472

3. 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.
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Econ 472

4. The Design
For each data set, we compute the OLS estimates. We then calculate a 95% confidence interval for the slope parameter as:
where
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Econ 472

5. 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.
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Econ 472

6. 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.
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Econ 472

7. 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.
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Econ 472

8. A Sample of 95% Confidence Intervals
Out of 1,000 trials, 53 of the intervals (5.3%) did not contain the true regression parameter.
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Econ 472

9. 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.
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Econ 472

10. 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.
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Econ 472

11. Hypothesis Testing
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