Simple Linear Regression Analysis -Investment Analysis

1.     Determining the Regression coefficient and Y-intercept Using the Method of Least Squares Regression

 

Fit a straight line through the scatter plot that will cause the sum of the squared residuals (measured in the Y-direction) to be minimized.

 

                      Y_i  =  b₀ + b₁X_i + e_I

 

                        b₁  = COV_YX/ (S_X²)

                        b_{0 =} Y - b₁ X

 

2.     The Standard Error of Estimate

 

The SEE is a measure of how imperfect the regression model is in predicting the dependent variable.

 

          SEE  = Sqrt(Sum of residuals squared/(n-2))

 

The interpretation of SEE is the similar to Std. Dev. Approximately 68% of the historical data points should lie within one standard deviation error of estimate on either side of the regression line (with errors measured in the vertical direction).

 

3.     Prediction Intervals on the Dependent Variable

 

Two sources of error are there on predicted values of dependent variables.

a.     From estimates of intercept and regression coefficients

b.     Variance above the estimate from the equation.

 

Hence variance associated with predicted value for given X value is:

 

S_t²  = (SEE)²[1 + (1/n) + ((X – X)²/(Sum of (x – x)²))]

 

Therefore, the following formula can be used to place a (1-  ) confidence interval aorund a predicted value of a dependent variable (   ) determined form a regression model. 

 

Y_i  =  b₀ + b₁X_i +(t _alpha/2,n-2* S_t)

 

In the graph, line of SEE of regression  around regression line will be a straight line. But the SEE of predicted values will be a parabolic line.

 

 

 

4.     Confidence Intervals on Regression Parameters

5.     Testing the Significance of Regression Parameters

 

A.   The Coefficient of Determination

B.   The Analysis of Variance

 

ANOVA      D of f          Sum of Squares   Mean Sum of Squares

 

Regression     1             Sum of (Y – Y)²                RSS/1

Residuals   n-2              Sum of (Y – Y)²             SSE/(n-2)

(Error)

 

Total          n-1              Sum of (Y – Y)²             TSS/(n-1)

1.     Regression Row

2.     Residuals Row

3.     Total Row

4.     Calculating R² from the Anova

 

R²      =   RSS/TSS

C.   The Global Test for the Significance of the Slope Coefficient

1.     F-tests

 

F(calc)  =  (RSS/1)/(SSE/(n-2))   = 

MSS of Regression/MSS of error

Original knol - http://knol.google.com/k/narayana-rao/simple-linear-regression-analysis/2utb2lsm2k7a/ 487