Regression Analysis

1. Regression Analysis Gordon Stringer Gordon Stringer, UCCS

2. Regression Analysis • Regression Analysis: the study of the relationship between variables • Regression Analysis: one of the most commonly used tools for business analysis • Easy to use and applies to many situations Gordon Stringer, UCCS

3. Regression Analysis • Simple Regression: single explanatory variable • Multiple Regression: includes any number of explanatory variables. Gordon Stringer, UCCS

4. Regression Analysis • Dependant variable: the single variable being explained/ predicted by the regression model (response variable) • Independent variable: The explanatory variable(s) used to predict the dependant variable. (predictor variable) Gordon Stringer, UCCS

5. Regression Analysis • Linear Regression: straight-line relationship Form: y=mx+b • Non-linear: implies curved relationships, for example logarithmic relationships Gordon Stringer, UCCS

6. Data Types • Cross Sectional: data gathered from the same time period • Time Series: Involves data observed over equally spaced points in time. Gordon Stringer, UCCS

7. Graphing Relationships • Highlight your data, use chart wizard, choose XY (Scatter) to make a scatter plot Gordon Stringer, UCCS

8. Scatter Plot and Trend line • Click on a data point and add a trend line Gordon Stringer, UCCS

9. Scatter Plot and Trend line • Now you can see if there is a relationship between the variables. TREND uses the least squares method. Gordon Stringer, UCCS

10. Correlation • CORREL will calculate the correlation between the variables • =CORREL(array x, array y) or… • Tools>Data Analysis>Correlation Gordon Stringer, UCCS

11. Correlation • Correlation describes the strength of a linear relationship • It is described as between –1 and +1 • -1 strongest negative • +1 strongest positive • 0= no apparent relationship exists Gordon Stringer, UCCS

12. Simple Regression Model • Best fit using least squares method • Can use to explain or forecast Gordon Stringer, UCCS

13. Simple Regression Model • y = a + bx + e (Note: y = mx + b) • Coefficients: a and b • Variable a is the y intercept • Variable b is the slope of the line Gordon Stringer, UCCS

14. Simple Regression Model • Precision: accepted measure of accuracy is mean squared error • Average squared difference of actual and forecast Gordon Stringer, UCCS

15. Simple Regression Model • Average squared difference of actual and forecast • Squaring makes difference positive, and severity of large errors is emphasized Gordon Stringer, UCCS

16. Simple Regression Model • Error (residual) is difference of actual data point and the forecasted value of dependant variable y given the explanatory variable x. Error Gordon Stringer, UCCS

17. Simple Regression Model • Run the regression tool. • Tools>Data Analysis>Regression Gordon Stringer, UCCS

18. Simple Regression Model • Enter the variable data Gordon Stringer, UCCS

19. Simple Regression Model • Enter the variable data • y is dependent, x is independent Gordon Stringer, UCCS

20. Simple Regression Model • Check labels, if including column labels • Check Residuals, Confidence levels to displayed them in the output Gordon Stringer, UCCS

21. Simple Regression Model • The SUMMARY OUTPUT is displayed below Gordon Stringer, UCCS

22. Simple Regression Model • Multiple R is the correlation coefficient • =CORREL Gordon Stringer, UCCS

23. Simple Regression Model • R Square: Coefficient of Determination • =RSQ • Goodness of fit, or percentage of variation explained by the model Gordon Stringer, UCCS

24. Simple Regression Model • Adjusted R Square = 1- (Standard Error of Estimate)2 /(Standard Dev Y)2 Adjusts “R Square” downward to account for the number of independent variables used in the model. Gordon Stringer, UCCS

25. Simple Regression Model • Standard Error of the Estimate • Defines the uncertainty in estimating y with the regression model • =STEYX Gordon Stringer, UCCS

26. Simple Regression Model • Coefficients: • Slope • Standard Error • t-Stat, P-value Gordon Stringer, UCCS

27. Simple Regression Model • Coefficients: • Slope = 63.11 • Standard Error = 15.94 • t-Stat = 63.11/15.94 = 3.96; P-value = .0005 Gordon Stringer, UCCS

28. Simple Regression Model • y = mx + b • Y= a + bX + e • Ŷ = 56,104 + 63.11(Sq ft) + e • If X = 2,500 Square feet, then • $213,879 = 56,104 + 63.11(2,500) Gordon Stringer, UCCS

29. Simple Regression Model • Linearity • Independence • Homoscedasity • Normality Gordon Stringer, UCCS

30. Simple Regression Model • Linearity Gordon Stringer, UCCS

31. Simple Regression Model • Linearity Gordon Stringer, UCCS

32. Simple Regression Model • Independence: • Errors must not correlate • Trials must be independent Gordon Stringer, UCCS

33. Simple Regression Model • Homoscedasticity: • Constant variance • Scatter of errors does not change from trial to trial • Leads to misspecification of the uncertainty in the model, specifically with a forecast • Possible to underestimate the uncertainty • Try square root, logarithm, or reciprocal of y Gordon Stringer, UCCS

34. Simple Regression Model • Normality: • Errors should be normally distributed • Plot histogram of residuals Gordon Stringer, UCCS

35. Multiple Regression Model • Y = α + β1X1 + … + βkXk + ε • Bendrix Case Gordon Stringer, UCCS

36. Regression Modeling Philosophy • Nature of the relationships • Model Building Procedure • Determine dependent variable (y) • Determine potential independent variable (x) • Collect relevant data • Hypothesize the model form • Fitting the model • Diagnostic check: test for significance Gordon Stringer, UCCS