Lecturer: Ing. Martina Hanová, PhD.
Regression analysis Regression analysis is a tool for analyzing relationships between financial variables: Identify the factors that are most responsible for a corporation's profits To estimate historical relationships among different financial assets Determine how much a change in interest rates will impact a portfolio of bonds Use this information to develop trading strategies and measure the risk contained in a portfolio
The value of a dependent variable is assumed to be related to the value of one or more independent variables. simple regression model multiple regression model.
assumptions of regression analysis is that the relationship between the dependent and independent variables is linear quickest ways to verify linearity is to graph the variables using a scatter plot.
to transform variables so that they do have a linear relationship.
The standard linear regression model may be estimated with a technique known as ordinary least squares. This results in formulas for the slope and intercept of the regression equation that "fit" the relationship between the independent variable (X) and dependent variable (Y) as closely as possible.
The coefficient of determination – R2 R2 can assume a value between 0 and 1 the closer R2 is to 1, the better the regression model explains the observed data. For a multiple regression model - the adjusted coefficient of determination
HYPOTHESIS TEST the null hypothesis is that all the slope coefficients of the model equal zero, with the alternative hypothesis that at least one of the slope coefficients is not equal to zero.
Each estimated coefficient in a regression equation must be tested to determine if it is statistically significant: The null hypothesis that's being tested is that the coefficient equals zero; if this hypothesis can't be rejected, the corresponding variable is not statistically significant.
Three of the most important violations that may be encountered: Autocorrelation, Heteroscedasticity Multicollinearity
RESIDUAL (Y i - Ŷ i ) Y i is the observed value of the dependent variable Ŷ i is the estimated value of the dependent variable Assumptions about the error term ε: E(ε) = 0. The variance of ε, denoted by σ 2, is the same for all values of X. The values of ε are independent. The error term ε has a normal distribution.
Much of residual analysis is based on an examination of graphical plots. A plot of the residuals against values of the independent variable X A plot of residuals against the predicted values of the dependent variable Ŷ A standardized residual plot A normal probability plot
Characteristics of a well-behaved residual vs. Ŷ : The residuals "bounce randomly" around the 0 line. The residuals roughly form a "horizontal band" around the 0 line. No one residual "stands out" from the basic random pattern of residuals.
An alternative to the residuals vs. fits plot is a "residuals vs. predictor plot."