1. SESSION 49 - 52 Last Update 17 th June 2011 Regression

2. Lecturer:Florian Boehlandt University:University of Stellenbosch Business School Domain:http://www.hedge-fund- analysis.net/pages/vega.php

3. Learning Objectives 1.XY-Scatter Diagrams 2.Plotting the Regression Line 3.Coefficient Estimates 4.Pearson Coefficient of Correlation 5.Spearman Rank Correlation Coefficient

4. XY-Scatter Diagram To draw a scatter diagram we need data for two variables. In applications where one variable depends to some degree on the other variable, the dependent variable is labeled Y and the other, called the independent variable, X. The values for X and Y are combined into a single data point using the observations for X and Y as coordinates.

5. Example Temperature - Truck TempTrucks Obsxy 1112.5 2146.5 3208.5 42110.5 52311 62412 72613 82813.5 93015.5 103419

6. Regression Analysis Regression analysis is used to predict the value of one variable on the basis of the other variables. The first-order linear model describes the relationship between the dependent variable Y and the independent variable(s) X. The regression model with a as the y-intercept and m as the slope coefficient is of the form:

7. Example Temperature - Truck TempTrucks Obsxy 1112.5 2146.5 3208.5 42110.5 52311 62412 72613 82813.5 93015.5 103419 The estimators of the intercept a and slope coefficient b are based on drawing a straight line through the sample data:

8. Intercept and Slope The intercept a is the y-coordinate of the point where the linear function intersects the y-axis. The slope coefficient b is defined as the change in y for a unit change in x.

9. Fitted Line With Residuals The line drawn through the point is called the regression line.

10. Residuals Squared The regression or least square line represents a line that minimizes the sum of the squared differences between the points and the line.

11. Calculating Coefficients Raw Data (y-variable as dependent and x as independent variable): TempTrucks Obsxy 1112.5 2146.5 3208.5 42110.5 52311 62412 72613 82813.5 93015.5 103419

12. Solution TempTrucks Obsxyxyx^2 1112.527.5121 2146.591196 3208.5170400 42110.5220.5441 52311253529 62412288576 72613338676 82813.5378784 93015.5465900 1034196461156 Total23111228775779 Step1: Calculate the gradient (beta):

13. Solution TempTrucks Obsxyxyx^2 1112.527.5121 2146.591196 3208.5170400 42110.5220.5441 52311253529 62412288576 72613338676 82813.5378784 93015.5465900 1034196461156 Total23111228775779 Step 2: Calculate the intercept (alpha):

14. Interpreting the Coefficients The slope coefficient b may be interpreted as the change in the dependent variable y for a one unit change in x. In the previous example, a one unit change in temperature results in a b = 0.654 additional truckloads of cool drinks sold. The intercept a is the point at which the regression line and the y-axis intersect. If x = 0 lies far outside the range of sample values x, the interpretation of the intercept is not straight- forward. In the temperature-truck example, x = 0 lies outside the smallest and largest values for x in the sample. Interpreting the intercept for x would imply that at temperature of x = 0, the soft-drink sales decline to negative 3.914!

15. Point Prediction Upon obtaining the coefficient estimates we can predict the outcome for various x (point prediction) between the minimum and maximum sample observation using the regression function y = a + mx. For example: x = 16 degrees?y = 3.914 + 0.654*16y = 6.554 ≈ 7 truckloads X = 32 degrees?y = 3.914 + 0.654*32y = 17.023 ≈ 17 truckloads

16. Pearson Coefficient of Correlation The Pearson coefficient of correlation R may be used to test for linear association between variables. The coefficient is useful to determine whether or not a linear relationship exists between y and x. Note that variables may be positively or negatively correlated. R = 1 denotes perfect positive correlation, R = -1 signifies perfect negative correlation. R is defined for:

17. Type of Relationship DIRECT LINEAR RELATIONSHIP Small Dispersion Wide Dispersion INVERSE LINEAR RELATIONSHIP Small DispersionWide Dispersion NO LINEAR RELATIONSHIP Positive Linear Correlation exists 0 < r <+ 1 Negative Linear Correlation exists -1 < r < 0 No Correlation r = 0

18. Coefficient of Determination Squaring the Pearson coefficient of correlation delivers the coefficient of determination R 2 in regression. It may be interpreted as the proportion of variation in the dependent variable y that is explained by the variation in the explanatory variable x. R 2 is a measure of strength of the linear relationship between y and x.

19. Solution Step 3: Calculate R and R 2 TempTrucks Obsxyxyx^2y^2 1112.527.51216.25 2146.59119642.25 3208.517040072.25 42110.5220.5441110.25 52311253529121 62412288576144 72613338676169 82813.5378784182.25 93015.5465900240.25 1034196461156361 Total231112287757791448.5

20. Spearman Rank Correlation The standard coefficient of correlation allows for determining whether there is evidence of a linear relationship between two interval variables. In case where the variables are ordinal, or, if both variables are interval, the normality requirement may not be satisfied. A nonparametric test statistic called Spearman Rank Correlation Coefficient may be used under the circumstances.

21. Objective: Comparing 2 Variables Nominal Chi-Square test of a contingency table Nominal Analyzing the relationship between two variables Ordinal Data type? Spearman Rank Correlation Population Distribution? Error is normal or x and y bivariate normal x and y not bivariate normal Simple linear regression

22. Example Ranking Business Aspect Manag ementStaff Brand Equity11 Financial Controls23 Customer Service32 Planning Systems46 Research & Development54 Company Morale67 Productivity75 Below there is a list of organizational strengths that were independently ranked by management and staff and the managing director wished to know how closely correlated were the assessments:

23. Calculating R S Ranking Business AspectObs Manage mentStaffdd^2 Brand Equity11100 Financial Controls2231 Customer Service33211 Planning Systems446-24 Research & Development55411 Company Morale6671 Productivity77524 Total12