1. Gordon Stringer, UCCS1 Regression Analysis Gordon Stringer

2. Gordon Stringer, UCCS2 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

3. Gordon Stringer, UCCS3 Regression Analysis  Simple Regression: single explanatory variable  Multiple Regression: includes any number of explanatory variables.

4. Gordon Stringer, UCCS4 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)

5. Gordon Stringer, UCCS5 Regression Analysis  Linear Regression: straight-line relationship Form: y=mx+b  Non-linear: implies curved relationships, for example logarithmic relationships

6. Gordon Stringer, UCCS6 Data Types  Cross Sectional: data gathered from the same time period  Time Series: Involves data observed over equally spaced points in time.

7. Gordon Stringer, UCCS7 Graphing Relationships  Highlight your data, use chart wizard, choose XY (Scatter) to make a scatter plot

8. Gordon Stringer, UCCS8 Scatter Plot and Trend line  Click on a data point and add a trend line

9. Gordon Stringer, UCCS9 Scatter Plot and Trend line  Now you can see if there is a relationship between the variables. TREND uses the least squares method.

10. Gordon Stringer, UCCS10 Correlation  CORREL will calculate the correlation between the variables  =CORREL(array x, array y) or…  Tools>Data Analysis>Correlation

11. Gordon Stringer, UCCS11 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

12. Gordon Stringer, UCCS12 Simple Regression Model  Best fit using least squares method  Can use to explain or forecast

13. Gordon Stringer, UCCS13 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

14. Gordon Stringer, UCCS14 Simple Regression Model  Precision: accepted measure of accuracy is mean squared error  Average squared difference of actual and forecast

15. Gordon Stringer, UCCS15 Simple Regression Model  Average squared difference of actual and forecast  Squaring makes difference positive, and severity of large errors is emphasized

16. Gordon Stringer, UCCS16 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

17. Gordon Stringer, UCCS17 Simple Regression Model  Run the regression tool.  Tools>Data Analysis>Regression

18. Gordon Stringer, UCCS18 Simple Regression Model  Enter the variable data

19. Gordon Stringer, UCCS19 Simple Regression Model  Enter the variable data  y is dependent, x is independent

20. Gordon Stringer, UCCS20 Simple Regression Model  Check labels, if including column labels  Check Residuals, Confidence levels to displayed them in the output

21. Gordon Stringer, UCCS21 Simple Regression Model  The SUMMARY OUTPUT is displayed below

22. Gordon Stringer, UCCS22 Simple Regression Model  Multiple R is the correlation coefficient  =CORREL

23. Gordon Stringer, UCCS23 Simple Regression Model  R Square: Coefficient of Determination  =RSQ  Goodness of fit, or percentage of variation explained by the model

24. Gordon Stringer, UCCS24 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.

25. Gordon Stringer, UCCS25 Simple Regression Model  Standard Error of the Estimate  Defines the uncertainty in estimating y with the regression model  =STEYX

26. Gordon Stringer, UCCS26 Simple Regression Model  Coefficients: –Slope –Standard Error –t-Stat, P-value

27. Gordon Stringer, UCCS27 Simple Regression Model  Coefficients: –Slope = 63.11 –Standard Error = 15.94 –t-Stat = 63.11/15.94 = 3.96; P-value =.0005

28. Gordon Stringer, UCCS28 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)

29. Gordon Stringer, UCCS29 Simple Regression Model  Linearity  Independence  Homoscedasity  Normality

30. Gordon Stringer, UCCS30 Simple Regression Model  Linearity

31. Gordon Stringer, UCCS31 Simple Regression Model  Linearity

32. Gordon Stringer, UCCS32 Simple Regression Model  Independence: –Errors must not correlate –Trials must be independent

33. Gordon Stringer, UCCS33 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

34. Gordon Stringer, UCCS34 Simple Regression Model  Normality: Errors should be normally distributed Plot histogram of residuals

35. Gordon Stringer, UCCS35 Multiple Regression Model  Y = α + β 1 X 1 + … + β k X k + ε  Bendrix Case

36. Gordon Stringer, UCCS36 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