January 31, 2012
The Distribution of Beta-Hat
We start from a previous equation and list some assumptions.
A little side trip shows that the assumption that the expected value of the errors equals zero is not important because the intercept will "soak up" the mean of the errors if it is not zero.
Our centerpiece proofs of the expected value and variance of the estimated beta:
It is important to understand that these conclusions are only valid if the assumptions hold. Stata will run the calculations in any case so, if the assumptions are not satisfied, the Stata results can be misleading.
We have seen a very similar analysis in ECON 400.
Posted by bparke at 07:52 PM | Comments (0)
aX+b Rules
We need some results from probability theory to support our analysis of regression models.
A numerical example "proof" of the expected value result:
A numerical example "proof" of the variance result:
These results are the basis for using the standard normal table for normal distributions.
Posted by bparke at 07:23 PM | Comments (0)
January 26, 2012
Left-Out Variables
We spent about half our time studying the 3-variable regression simulation on the EasyMetrics web site.
To understand our simulation results we made our first trip through one of the most important theoretical results in econometrics. The following algebra explains why leaving out a variable causes bias in the estimated coefficients.
Posted by bparke at 09:16 PM | Comments (0)
January 24, 2012
Why is the estimated slope stochastic?
We studied the EasyMetrics 2-variable regression simulation to see how the errors affect the estimated slope. The following algebra lays out a more mathematical explanation. The estimated coefficient is not equal to the true coefficient because of the sample covariance of the errors and the regressor.
The errors are part of the true model. They are unobservable because we do not know the true coefficients. We can use the estimated coefficients to calculate the residuals, which you can think of as "estimated errors."
The sample covariance of the residual and the regressor is zero by construction.
By the definition of the residual, which can be thought of a forecast error, the observed dependent variable is the sum of the predicted value from the estimated regression and the residual. We have just seen that the covariance of the regressor (and, hence, the prediction) and the error is zero by construction. This allows us to decompose the variance of the dependent variable into "explained" and "unexplained" variances. The R-squared is the explained variance as a percentage of the total variance.
Posted by bparke at 09:31 PM | Comments (0)
January 19, 2012
Probability and Statistics
We wrapped up by considering a move to 10,000 observations. The standard error of our estimated probability would then be 0.005, putting the critical values for the rejection region at 0.49 and 0.51. We would then reject the null if the sample percentage was 0.55.
Posted by bparke at 08:17 PM | Comments (0)
January 17, 2012
A First Look at Two-Variable Regressions
We spent a good bit of time experimenting with the EasyMetrics two-variable regression example.
We would like to acquire an intuitive understanding of how the errors pull the estimated regression line away from the true line that we would see if there were no errors.
The regression results have a counterpart in the basic example of estimating the mean of a sample of draws from a normal distribution.
A look at the future:
Posted by bparke at 08:17 PM | Comments (0)
January 12, 2012
What is a regression?
We worked on Help for New Students.
Posted by bparke at 10:12 PM | Comments (0)
January 10, 2012
Introduction
Welcome
You will need access to Stata. There are advantages to buying a copy. Stata GradPlan Look for Stata/IC. Small Stata will not allow you to do the exercises for this course.
It is cheaper to use the Virtual Computing Lab. UNC VCL. Stata should also be available on lab computers around campus.
The Wall Street Journal If money is tight, subscribe to the Wall Street Journal and use the VCL. Businesses hire people who major in economics and read the Wall Street Journal.
Follow economics at The Economics Roundtable. Veteran readers use an RSS reader. I recommend SharpReader (free).
The Econ Review has some interesting graphs.
Posted by bparke at 07:46 PM | Comments (0)