1 / 14

Bivariate Regression Analysis

Bivariate Regression Analysis. Theoretical Models Basic Linear Models: Deterministic Version Basic Linear Models: Stochastic Version Statistical Assumptions Estimating Linear Models Residuals (and the Pursuit of Truth…) An Example. Theoretical Linear Models.

sean-fox
Download Presentation

Bivariate Regression Analysis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Bivariate Regression Analysis • Theoretical Models • Basic Linear Models: Deterministic Version • Basic Linear Models: Stochastic Version • Statistical Assumptions • Estimating Linear Models • Residuals (and the Pursuit of Truth…) • An Example

  2. Theoretical Linear Models • The basis of “causality” in models • Time ordering • Co-variation • Non-spuriousness • Examples • Fire Deaths f (# of fire trucks at the scene) • Job Retention f (current job satisfaction) • Income f (education)

  3. a b Deterministic Linear Models • Theoretical Model: • b0andb1are constant terms • b0 is the intercept • b1 is the slope • Xi is a predictor of Yi Yi b0 Xi

  4. Stochastic Linear Models • E[Yi] = b0+b1Xi • Variation in Y is caused by more than X: error (ei) • So:

  5. ei ei=0 X Assumptions Necessary for Estimating Linear Models 1. Errors have identical distributions Zero mean, same variance, across the range of X 2. Errors are independent of X and other ei 3. Errors are normally distributed

  6. Y X Normal, Independent & Identical ei Distributions (“Normal iid”) Problem: We don’t know: a) if error assumptions are true; b) values for b0 and b1 Solution: Estimate ‘em!

  7. Estimating Linear Models This is the formula for RESIDUALS -- which you will come to know and cherish.

  8. Residuals: Statistical Forensics • Residuals measure prediction error: • ei > 0 if Yi > Yi • ei < 0 if Yi < Yi Y ^ ^ X

  9. Stata and Regression: Predicting Incarceration with Average income • Stata dataset: Guns.dta • From “Data for empirical exercises” • What are your expectations? Why? • Stata command: • Regression: “regress incarc_rate avginc” • Output:

  10. In our data, some observed values are larger than would be predicted by average income alone Residual Analysis

  11. Normality of Residuals

  12. More on Normality: Q-Normal

  13. Distribution of Residuals by X

  14. BREAK TIME

More Related