1. Time Series Analysis – Chapter 2 Simple Regression Essentially, all models are wrong, but some are useful. - George Box Empirical Model-Building and Response Surfaces (1987), co-authored with Norman R. Draper, p. 424, ISBN 0471810339 George Box is the son-in-law of Sir Ronald Fisher.

2. Time Series Analysis – Chapter 2 Simple Regression Equation of a Line – Algebra Vs. Simple Regression – Statistics

3. Equation of a Line Example y = mx + b wage = 3.55educ – 33.8 y = wage in dollars per hour x = education in years completed Note: if I know how many years of education someone has completed I can predict their wage perfectly. Nothing else matters.

4. Simple Regression Example

7. Algebra vs. Statistics - Summary

10. StudentGPAACT 12.821 23.424 33.026 43.527 53.629 63.025 72.725 83.730

12. The Analysis of Variance Table Analysis of Variance Source DF SS MS F P Regression 1 0.59402 0.59402 8.20 0.029 Residual Error 6 0.43473 0.07245 Total 7 1.02875

13. ANOVA Models can be evaluated by examining variability. There are three types of variability that are quantified. Overall or total variability present in the data (SST) Variability explained by the regression model (SSR) Error variability that is unexplained (SSE) SST = SSR + SSE

14. ANOVA The larger the regression variability (SSR) is compared to the error variability (SSE) the more evidence there is that the model is explanatory. Analysis of Variance Source DF SS MS F P Regression 1 0.59402 0.59402 8.20 0.029 Residual Error 6 0.43473 0.07245 Total 7 1.02875

15. ANOVA – R 2 R 2 is the Coefficient of Determination R 2 = SSR/SST = 1 – SSE/SST TYPO on pg. 40!! R 2 is the percent of the variation in y (response variable) explained by x (explanatory variable). R-Sq = SSR/SST = 0.59402/ 1.02875 = 57.7%

16. ANOVA – r

17. ANOVA – R 2 vs. r R 2 always exists for simple regression and multiple regression and always has the same definition r only exists and makes sense for simple regression

18. Nobel Prize vs. # of McDonalds Explanatory variable is number of McDonalds a country has Response variable is number of Nobel Prizes that have been awarded that country.

19. Logs

20. Level – Level Model StudentGPAACT 12.821 23.424 33.026 43.527 53.629 63.025 72.725 83.730

21. Level – Log Model Dependent variable: y Independent variable: log(x) Not used in this chapter, discussed in future chapters.

22. Log – Level Model StudentGPAACTlog(GPA) 12.8211.02962 23.4241.22378 33.0261.09861 43.5271.25276 53.6291.28093 63.0251.09861 72.7250.99325 83.7301.30833

23. Log – Level Model

24. Level – Level Model

25. Log – Level Model

26. Is this still linear regression?

27. Log – Log Model StudentGPAACTlog(GPA)log(ACT) 12.8211.029623.04452 23.4241.223783.17805 33.0261.098613.25810 43.5271.252763.29584 53.6291.280933.36730 63.0251.098613.21888 72.7250.993253.21888 83.7301.308333.40120

28. Log – Log or Constant Elasticity Model

30. Simple Linear Regression Assumptions

31. SLR.2: The sample of size n used to estimate the model parameters is a random sample (sometimes called a simple random sample). What is the definition of a random sample?

32. Simple Linear Regression Assumptions SLR.3: The sample x values are not all the same value. OKNOT OK xy 3.424 3.026 3.527 3.629 3.025 2.725 xy 324 326 327 329 325

33. Simple Linear Regression Assumptions

35. Ordinary Least Squares Estimators

36. Ordinary Least Squares Minimize the sum of the squared residuals.

37. Ordinary Least Squares StudentGPAACTRESI1 12.8210.085714 23.4240.379121 33.026-0.225275 43.5270.172527 53.6290.068132 63.025-0.123077 72.725-0.423077 83.7300.065934

38. Ordinary Least Squares