Lecture Applied econometrics course - Chapter 2: Multiple regression model

Lecture Applied econometrics course - Chapter 2: Multiple regression model has content: Why we need multiple regression model, estimation, R – Square, assumption, variance and standard error of parameters, the issues of multiple regression model, Illustration by Computer.

CHAPTER II MULTIPLE REGRESSION MODEL APPLIED ECONOMETRICS COURSE

 Variance and Standard Error of Parameters

 The issues of multiple regression model

 Illustration by Computer

NGUYEN BA TRUNG - 2016

I WHY WE NEED MULTIPLE REGRESSION MODEL?

 Reality and flexibility

 Avoid the biasedness caused by omitted variables

 Generally, multiple regression model is written by

 The terminology in (2.1) is similar as simple regression model

0 1 1

Y     X    X  u (2.1)

II ESTIMATION: OLS

 The matrix form of multiple regression model:

 Take derivative respect to parameters, we have:

II ESTIMATION: OLS

 The parameters are estimated in form:

Computer 1: WAGE1.wf

 Explain the meaning of education’s parameter?

 R2 is the non-decreasing function of the explained variables

 Therefore, R2 is not good indicator to measure the fit of your model

 You should use adjusted R-square instead:

Computer 2: WAGE1.wf

NGUYEN BA TRUNG - 2016

IV ASSUMPTION

 Assumption 1:

 Assump 2: X is the randomness

 Asssump 3: No perfect multi-collinearity

Under the assumption 1- 4, the estimators of OLS are unbiased:

Theorem 2: The unbiasedness of OLS

ˆ( j ) j

V VARIANCE AND STANDARD ERROR

 Assumption 5: Homoscedasticity

2

Var(u i)  

Under the assumption1- 5, the variance of estimator is unbiased:

Theorem 3: The unbiasedness of estimator’s variance

ˆ ( )

ˆ Var( )

Computer 3: WAGE1.wf

VI THE ISSUES OF MULTIPLE REGRESSION MODEL

6.1 The redundancy of explained variables

 Suppose that we have the true model as:

 This means that, X2i is a redundant variable in (2.8)

 Follow by the theorem 2, the estimators of (2.8) are still unbiased

(2.7)

(2.8)

VI THE ISSUES OF MULTIPLE REGRESSION MODEL

 But, the variance of estimator will decreases

 Lose the degree of freedoms

 Difficulty to get statistically significant of estimator

 Increase probability of multi-collinearity

6.2 The omitted variables

 The true model:

 The wrong model :

 We can show that:

 Where is the parameter obtained by regressing X1 on X2

 Thus, the omitted variable X2 will lead to biasedness:

Bias ( ) = 1  2 1 (2.12)

VI THE ISSUES OF MULTIPLE REGRESSION MODEL

VI THE ISSUES OF MULTIPLE REGRESSION MODEL

 The estimator of the model below will be positive or negative bias if omitted the povrate variable?

Solution: Proxy variable

Solution: Proxy variable

 Substitute in (2.13) by (2.14), and rearrange again: 𝒙𝟑∗

log(wage)     educ   exp er   ability  u

 If omitted the ability variable, the model will be positive or negative bias?

Example: Wage2.wf

 Assume that we have the true model as:

Log(wage) = β 0 + β 1 educ+ β 2 exper+ β 3 ability+u (true model)

Due to omitting the ability variable, we only estimate the following model:

Log(wage) = β 0 + β 1 educ+ β 2 exper +u (omitted model)

 whether β 1 will be biased or not? and why?

 Positive bias or negative bias? And why?

Example: Wage2.wf

APPLIED ECONOMETRICS COURSE

END OF THE CHAPTER II