2.6_Linear+Regression+with+One+Regressor+一元线性回归

1-15FRM（Financial Risk Manager）金融风险管理师

Linear Regression with One Regressor

一元线性回归


Regression Analysis

A regression analysis has the goal of measuring how changes in one variable,

called a dependent or explained variable can be explained by changes in one or

more other variables called the independent or explanatory variables. The

regression analysis measures the relationship by estimating an equation (e.g.,

linear regression model). The parameters of the equation indicate the

relationship.

A scatter plot is a visual representation of the relationship between the

dependent variable and a given independent variable. It uses a standard two-

dimensional graph where the values of the dependent, or Y variable, are on the

vertical axis, and those of the independent, or X variable, are on the horizontal

axis.


Population Regression Function

Lockup(yrs) Average Return

5 10 14 14 15 12 13

6 17 12 15 16 10 14

7 16 19 19 13 13 16

8 15 20 19 15 16 17

9 21 20 16 20 18 19

10 20 17 21 23 19 20

Fugure 1:Hedge Fund Data

Returns(%) per year

Figure 2:Return Over Lockup Period


Population Regression Function

E (return | lockup period) = B0 + B1 × (lockup period)

Or more generally:

E (Yi | Xi) = B0 + B1 × (Xi)

Intercept Coefficient Slope Coefficient

0 1i i iY B B X

Error Term


Sample Regression Function and Ordinary least squares (OLS)

0 1i i iY b b X e

e2

Y

X

e1 e3

e4

Yi b b X ei i i 0 1

Y b b Xi i 0 1

22

i 0 1 iminimize e Y -(b b X )i


The Results of Ordinary least squares (OLS)


201405真题讲解


201405真题讲解

48. You are conducting an ordinary least squares regression of the returns on stocks Y

and X as Y=a + b × X +ε based on the past three year’s daily adjusted closing price

data. Prior to conducting the regression, you calculated the following information

from the data:

What is the slope of the resulting regression line?

A. 0.35

B. 0.45

C. 0.59

D. 0.77

Sample covariance 0.000181

Sample Variance of Stock X 0.000308

Sample Variance of Stock Y 0.000525

Sample mean return of stock X -0.03%

Sample mean return of Stock Y 0.03%

Notes 128页

0.000181

0.000.5877

0308


Example

Consider two stocks, A and B. Assume their annual returns are

jointly normally distributed, the marginal distribution of each stock

has mean 2% and standard deviation 10%, and the correlation is

0.9. What is the expected annual return of stock A if the annual

return of stock B is 3%?

A. 2%

B. 2.9%

C. 4.7%

D. 1.1%

Answer: B


Assumptions Underlying Linear Regression

Linear regression requires a number of assumptions. Most of the major assumptions pertain to she regression model’s residual term (i.e., error term). Three key assumptions are as follows:

1. The expected value of the error term, conditional on the independent variable, is zero ( )

2. All (X, Y) observations are independently and identically distributed (i.i.d.).

3. It is unlikely that large outliers will be observed in the data. Large outliers have the potential to create misleading regression results.

Additional assumptions include:

4. A linear relationship exists between the dependent and independent variable.

5. The model is correctly specified in that it includes the appropriate independent variable and does not omit variables.

6. The independent variable is uncorrelated with the error terms.

7. The variance of is constant for all Xi :

8. No serial correlation of the error terms exists

9. The error term is normally distributed

( | ) 0i iE X

i2Var( | )i iX

i i+j[i.e.,Corr(ε ,ε ) 0 for j=1,2,3...]


The Coefficient of Determination (R2)

_

( )iY Y ESS

Y

b0

__

( )iY Y TSS

( )i iY Y SSR

__

Y

0 1i iY b b X

X

2 1ESS SSR

RTSS TSS

2 2 2R R

__ __2 2 2

Total sum of squares = explained sum of squares + sum of squared residuals

( ) ( ) ( )

i iY Y Y Y Y Y

TSS ESS

SSR


The Standard Error of the Regression

The standard error of the regression (SER) measures the degree of variability

of the actual Y-values relative to the estimated Y-values from a regression

equation. The SER gauges the "fit" of the regression line. The smaller the

standard error, the better the fit.

The SER is the standard deviation of the error terms in the regression. As such,

SER is also referred to as the standard error of the residual, or the standard error

of estimate (SEE).

In some regressions, the relationship between the independent and dependent

variables is very strong (e.g., the relationship between 10-year Treasury bond

yields and mortgage rates). In other cases, the relationship is much weaker (e.g.,

the relationship between stock returns and inflation). SER will be low (relative to

total variability) if the relationship is very strong and high if the relationship is

weak.


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2.6_Linear+Regression+with+One+Regressor+一元线性回归