CFA Level 2 - Quantitative Methods

Linear Regression

regression equation : Y_i = b₀ + b₁X_i + ε_i, i = 1, …, n 

 (Yi−b‸0−b‸1Xi)2 
    means (dependent variable – predicted value of dependent variable)²


Standard Error of Estimates

ε‸i , the residual term in the regression.

SEE=⎛⎝⎜⎜⎜⎜⎜∑i=1n(Yi−b‸0−b‸1Xi)2n−2⎞⎠⎟⎟⎟⎟⎟1/2=⎛⎝⎜⎜⎜⎜⎜∑i=1n(ε‸i)2n−2⎞⎠⎟⎟⎟⎟⎟1/2
∑             i=1n(ε‸i)2


It is the sum of squared residuals


Coefficient of determination

If we call ∑i=1n(Yi−Y⎯⎯⎯)2 the total variation of Y and ∑i=1n(Yi−Y‸i)2 the unexplained variation from the regression,

then we can measure the explained variation from the regression using the following equation:

R2=ExplainedvariationTotalvariation=Totalvariation−UnexplainedvariationTotalvariation=1−UnexplainedvariationTotalvariation

Hypothesis testing

We will use a 95 percent confidence interval for our test, or we could say that the test has a significance level of 0.05.

The number of degrees of freedom equals the number of observations minus the number of parameters estimated

t-test of significance:
t=b‸1−b1sb‸1
The b1 is from the null hypothesis (hypothesized population value of the regression coefficient).
The b1bar is the estimated regression coefficient.
‸.
The t_c is critical t-value at the 0.05/0.01 significance level, from the book. The t is calculated. If t is greater than t_c, then we can reject the null hypothesis.
s is the standard error of the regression.

For example, the t-statistic is 2.50, and at the 0.05 significance level, t_c = 2.00; thus we reject the null hypothesis because t > t_c. This statement is equivalent to saying that we are 95 percent confident that the interval for the coefficient does not contain the value b1.

Often, financial analysts report the p-value or probability value for a particular hypothesis. The p-value is the smallest level of significance at which the null hypothesis can be rejected.

For example, if the p-value is 0.005, we can reject the hypothesis that the true parameter is equal to 0 at the 0.5 percent significance level (99.5 percent confidence).

Analysis of Variance (ANOVA)

Total SS = Regression SS + Residual SS

Regression SS:
∑i=1n(Yi−Y‸i)2  
Total SS:
 ∑i=1n(Yi−Y⎯⎯⎯)2 

F-test
The F-test for determining whether the slope coefficient equals to 0 is based on an F-statistic.
The F-statistic measures how well the regression equation explains the variation in the dependent variable. 
If the independent variable explains little of the variation in the dependent variable, the value of the F-statistic will be very small.

F statistic: Regression MSS /  Residual MSS
MSS: means sum of squares (SS divided by df)

F=RSS/1SSE/(n−2)=MeanregressionsumofsquaresMeansquarederror  

Confidence Interval of Regression Coefficient (Interval Prediction)
   CI = coeff + t_c * SE
t_c is from the confidence level. For large number of df:
90% confidence interval will have alpha value of 0.10 (two tail), and tc of 1.645
95% confidence interval will have alpha value of 0.05 (two tail), and tc of 1.96
99% confidence interval will have alpha value of 0.01 (two tail), and tc of 2.576

The end.