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Summary - GLM through Prob Models

Summary - GLM through Prob Models - Summary ResEc 702...

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Summary ResEc 702 – Econometrics I III. General Linear Model A. Introduction : Theory suggests the variables we should include in our GLM. In class, we reviewed results from a few microeconomic models. The consumer’s utility maximization problem was used because it also provides restrictions that might be incorporated into our modeling. What specific lessons can be learned from going through the process of specifying a theoretical model? B. Classical Regression Model : We revisited the same set of assumptions using matrix notation and added one additional assumption – no perfect multicollinearity. What, in general, do each of the assumptions mean for our statistical model? Why do we require these assumptions? C. OLS : Derived the OLS estimators using matrix notation. The process is the same. Discussed the final CRM assumption (no perfect multicollinearity) and its implications for estimation. D. Properties of b: Reviewed the properties of the OLS estimators. Derived the covariance matrix and considered variances for general linear model with two independent variables. The latter exercises, the Y = f(X1, X2) derivations using summation notation, allows us to see the effects of too much collinearity or zero collinearity. E. OLS estimator for 2 σ and Goodness of fit: 2 R F. Inference: We spent quite a bit of time on these topics including: Individual Parameter Inference –not much different from our approach in the Simple Linear Model. Tests of Significance for All Regressors – joint tests using, F-tests. Tests for Linear Combinations of Parameters – F-tests for Restrictions . This is perfectly general and allows us a test of the model (all parameters equal zero) and tests of sub-sets of parameters. As long as we can express our tests as R β = r , we can complete the test. Prediction Intervals – revisited the creation of confidence intervals for the expected value and an individual value. G. Specification Errors and Model Selection Tools: Discussed the two specification errors we might make and the consequences of the errors. First we considered the omission of relevant independent variables. We saw that the OLS estimators were biased, but they did have “less sampling variability.” But, as we discussed, the conclusion that sampling variability is less was made using the true value 2 . In the real world, we’ll be using a biased estimator for 2 , which will affect our estimates of sampling variability. (Plus, we will logically not explain as much variation in the dependent variable when we leave out important variables.) We then turned to the case of
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including irrelevant regressors. We found that b was unbiased, but inefficient. There are many model selection tools at our disposal. We talked about the use of theory and logic to properly specify the model.
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Summary - GLM through Prob Models - Summary ResEc 702...

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