Regression Case

Your task is to apply the Market Model for each stock’s returns and the TSX Index returns. Do a complete regression analysis for each stock, following the analysis steps provided in the study guide and presented in the course materials. Write a report describing the analyses conducted. Your interpretations of the beta coefficient and coefficient of determination should be discussed in terms of the financial model.  In your conclusion, provide a summary table of all relevant parameters and discuss which stocks are most and least sensitive to changes in the index. Also explain which stocks’ risk could be diversified by creating an appropriate portfolio.  You do not need to develop a portfolio. 

Note that the last step "7 If the model fits the data, use the regression equation.” is not directly applicable other than your explanations of sensitivity and risk. This step really applies to using the model for prediction.

You should use the p-value criterion for any hypothesis tests conducted. Use only summary tables within the text of your report. If any data is within your text, it should be something you are discussing!! I do not need to see the raw data anywhere. You can include your Excel output in an Appendix if you like and refer to it there.

The intent of the case is for each student to work through several complete regression analyses (with help from your team for understanding :-)). However, when it comes to the final write up, if you organize it by regression steps (including all 9 stocks’ graphs together or a table in one place) rather than by stock, you can use one summary paragraph to explain the your analysis for each step.  This eliminates a lot of the repetition that results when organizing by individual stock. This is less writing for you and less reading for me ;-).  It makes my grading easier too and anything you can do to make your professor’s grading easier is always a good thing 

Procedure for Regression Diagnostics

 1.    Develop a model that has a theoretical basis.

 2.    Gather data for the variables in the model.

 3.    Draw scatter diagrams for all X,Y to determine whether a linear model appears to be appropriate and check for outliers.

 4.    Determine the regression equation and residuals.

 5.    Using the residuals, check the required conditions for the errors and re-check for outliers.

 a.    Is the error variable normally distributed with a mean of zero? Check for nonnormality

 Draw a histogram of the residuals and interpret the shape (Can use Standardized Residuals)

 b.    Is the error variance constant for all value of x? Check for Heteroscedasticity.

 If the variance of the errors is not the same for every value of x, then we have heteroscedasticity.  Check this by plotting the residuals against the predicted values of y (with Y on the x-axis) and looking for change in spread of the data points as you move across the x-axis. 

 c.    Are the errors independent? Check for Autocorrelation.

 Autocorrelation is when the error terms for different values of y are related to each other in some way.  This can be a problem for time-series data (and only needs to be checked for time-series data).  Plot the residuals against time periods and look for patterns – Do a line plot of the residuals

 d.    Check the existence of outliers and influential observations

 Look for any standardized residuals that are less than -2 or greater than +2

 6.    Assess the model fit.

 a.    Standard Error of the Estimate

 Compare to average Y

 b.    Coefficient of Determination (R2)

 Interpret it and evaluate it

 c.  Test Validity of Model

 ANOVA F test

 d.    Test significance of coefficients and interpret them

 t-test of beta

7.    If the model fits the data, use the regression equation.

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