定量投资分析课件 (3)

Assumptions of the Linear Regression Model
• The relationship between the dependent variable, Y, and the independent variable, X is linear in the parameters b0 and b1. • The independent variable, X , is not random. • The expected value of the error term is 0. • The variance of the error term is the same for all observations. • The error term, ε, is uncorrelated across observations. • The error term, ε, is normally distributed.
– We test the null hypothesis that all the slope coefficients in a regression are simultaneously equal to 0 by using an F-test.
ANOVA
• Analysis of variance (ANOVA) is a statistical procedure for dividing the total variability of a variable into components that can be attributed to different sources.
Explainedvariat ion R T ot alvariat ion Unexplaine d variat ion 1 T ot alvariat ion
2
1
2 ( Y Y ) i i 2 ( Y Y ) i i 1 i 1 n
n
^
Hypothesis Testing
Multiple Linear Regression Model
• Multiple linear regression allows us to determine the effect of more than one independent variable on a particular dependent variable.
Yi b0 b1X1i b 2 X 2i ... b k X ki i , for i 1, 2, . . ., n
• A slope coefficient, bj , measures how much the dependent variable, Y , changes when the independent variable, Xj , changes by one unit, holding all other independent variables constant. – In practice, software programs are used to estimate the multiple regression model.
RSS / 1 Mean regressionsum of squares F SSE /(n 2) Mean squared error
• where,
^ SSE Yi Yi i 1 n 2
RSS Yi Y i 1
Cov(X, Y) r s Xs Y
Correlations Among Stock Return Series
Testing the Significance of the Correlation Coefficient
• A t-test can be used to test the significance of the correlation coefficient.
– used to account for qualitative variables such male or female, month of the year effects, etc.
^
b1
ANOVA
• Analysis of variance (ANOVA) is a statistical procedure for dividing the total variability of a variable into components that can be attributed to different sources.
correlation coefficient
(Xi X)(Yi Y) Cov(X, Y) (n 1) i 1
n
2 ( X X ) i s2 X (n 1) i 1 n
sX s
2 X
• The sample correlation coefficient is,
Variables with Perfect Positive Correlation
Variables with Perfect Negative Correlation
Variables with a Correlation of 0
Variables with a Strong Non-Linear Association
Linear Regression Model
• Linear regression chooses the estimated or fitted parameters to minimize
Yi b0 b1 Xi i 1
n ^ ^
• Standard Error of the Estimate
t
r n2 1 r
2
Linear Regression
• Linear regression assumes a linear relationship between the dependent (Y) and the independent variables (X).
Yi b0 b1Xi i , i 1, . . . n
n ^
2
Limitations of Regression Analysis
• Regression relations can change over time, known as the issue of parameter instability. • Use of regression results specific to investment contexts is that public knowledge of regression relationships may negate their future usefulness. • If the regression assumptions are violated, hypothesis tests and predictions based on linear regression will not be valid.
Assumptions of the Multiple Linear Regression Model
• The relationship between the dependent variable, Y , and the independent variables, X1, X2, . . . , Xk, is linear. • The independent variables (X1, X2, . . . , Xk) are not random. Also, no exact linear relation exists between two or more of the independent variables. • The expected value of the error term, conditioned on the independent variables, is 0: E(| X1, X2, . . . , Xk) = 0. • The variance of the error term is the same for all observations • The error term is uncorrelated across observations • The error term is normally distributed.
n 1 2 R 1 1 R n k 1 s
• A dummy variable is qualitative variable that takes on a value of 1 if a particular condition is true and 0 if that condition is false.
• We can test to see if the slope coefficient is significant by using a t-test. ^
• We can also construct a confidence interval for b.
b1 b1 t s b1
b1 t c s ^
Topic 3: Regression
Correlation Analysis
• correlation analysis expresses the relationship between two data series using a single number. • The correlation coefficient is a measure of how closely related two data series are. • The correlation coefficient measures the linear association between two variables.