Authors: Péter Elek, Anikó Bíró Supervised by: Péter Elek

ECONOMETRICS

ECONOMETRICS

Sponsored by a Grant TÁMOP-4.1.2-08/2/A/KMR-2009-0041 Course Material Developed by Department of Economics,

Faculty of Social Sciences, Eötvös Loránd University Budapest (ELTE) Department of Economics, Eötvös Loránd University Budapest

Institute of Economics, Hungarian Academy of Sciences Balassi Kiadó, Budapest

ECONOMETRICS

Authors: Péter Elek, Anikó Bíró Supervised by: Péter Elek

June 2010

ELTE Faculty of Social Sciences, Department of Economics

ECONOMETRICS

Week 5.

Multivariate regression II

Péter Elek, Anikó Bíró

Forecasting

Forecast error

Variance of forecast error (k regressors)

20 2

10 1

0

ˆ ˆ ˆ

ˆ x x

y      

0 20

2 2

10 1

1 0

0

ˆ ( ˆ ) ( ˆ )

ˆ y x x u

y              

ˆ ) ˆ ,

( )

( ) 1 (

1 ₀

1 1

0 2

m l

m m

k

l

k

m

l

l x x x Cov

n x

 





 

 



 

Estimating the variance of forecast error

Predicted expected value and its standard error =

= estimated constant and its standard error of auxiliary regression

u x

x x

x y

y

x x

x x

x x

y E y

x x

y

u x

x y









































) (

) (

) ,

| (

ˆ ˆ ˆ

ˆ

20 2

2 10

1 1

0

20 2

10 1

20 2

10 1

0

20 2

10 1 0

2 2 1

1























Sampling distribution of the coefficient estimates

If the assumptions are satisfied (normality and homoscedasticity, as well)

where RSS_i is the residual and TSS_i is the total sum of squares in the regression of x_i on the other explanatory variables, and R_i² is the

coefficient of determination in the same

regression (analogy with simple regression!)

 







 



 





2 2

2

, 1

~ ,

ˆ ~

i i

i

i N TSS R

N  RSS  



Omitting relevant variables I.

Simple regression

True model: y = β₁x₁ + β₂x₂ + u Estimated model: y = γ₁x₁ + u

Bias: Corr(x₁,x₂)>0 Corr(x₁,x₂)<0

β₂ >0 + –

β₂ <0 – +

2 12 1

1

2 1 1 2

1 2 1 2

2 1 1 1 1

ˆ ) ( ˆ













b E

x u x x

x x x

y x

i i

i i i

i i

i i i

i i

i i i













 

 

 

Omitting relevant variables II

k explanatory variables, k₁ + 1, …, k. omitted

u x

b x

b x

k i

b E

k jk

j j

j k

k j

ji i

i













 





1 1

1 1

1 1

...

,..., 1

, ˆ )

(

1







Omitting relevant variables, example

Wage tariff (2003)

Weak negative correlation between education and age

Partial effect of age is positive – if omitted, the estimated coefficient of education is slightly downward biased

Estimated equations

LOG(EARN) = 10.46 + 0.1547 EDUC9 + 0.0078 AGE LOG(EARN) = 10.79 + 0.1544 EDUC9

Irrelevant variables in the regression

True model: y = β₁x₁ + β₂x₂ + u

Estimated model: y = β₁x₁ + β₂x₂ + β₃x₃ + u, β₃ = 0 Does not affect unbiasedness

Variance increases:

RSS_i: from the regression of x_i on the other explanatory variables (additional regressor:

RSS_i decreases, except for these are uncorrelated)

i

i

RSS

Var

2

)

(   

t-test

“good” estimator of the variance of error term, therefore:

Two sided test: H₀: β_i = 0, H₁: β_i ≠ 0 One sided test: e.g. H₀: β_i = 0, H₁: β_i > 0 Confidence interval:

~

ˆ ) ( ˆ







k n i

i

i

t

SE 





ˆ ) ˆ (

i

i c SE 

  

~ 1 ˆ 1

2 2 1

2







  ^{ }

k n

k n

RSS

 



i

i RSS

SE(ˆ ) ˆ ² /

t-test, example

Credit approval, testing discrimination on settlement level:

approval_rate

= α + β

minority_rate

+ β

avg_inc

+ + β

avg_wealth

+ u

, i = 1…n

No difference according to minority ratio:

H

: β

= 0

Negative discrimination against minorities:

H

: β

< 0

Example, testing significance of a regressor

Do more experienced earn more, given education? (Wage tariff, 2003)

log(Earn_i) = α + β₁Educ + β₂ Exp_i + u_i H₀: β₂ = 0 H₁: β₂ > 0

Dependent Variable: LOG(EARN) Method: Least Squares

Included observations: 201971

Variable Coefficient Std. Error t-Statistic Prob.

C 10.556 0.004 2630.523 0.0000

EDUC9 0.164 0.001 320.482 0.0000

EXP 0.008 9.45E-05 79.859 0.0000

Example: relationship between earnings and years of education

log(Earn_i) = α + β₁Educ_y_i + u_i, Wage tariff 2003 (Univariate case: F-test is the square of t-test)

Analysis of variancie

Is the regression model useable?

Source of std. Dev.

Sum of squares

Degrees of freedom

Mean sum of squares

F

Explained (ESS)

R²S_yy k R²S_yy/k = MS₁ F =

MS₁/MS₂

Residual (RSS)

(1 – R²)S_yy n – k – 1 (1 – R²)S_yy/(n – k – 1) =

= MS₂

Total (TSS)

S_yy n – 1

F-test of the usability of the regression

H₀: β_i = 0 (i = 1,…,k) If H₀ satisfied

TSS ~ σ² Chi_n–1²

RSS ~ σ² Chi_n–k–1², ESS ~ σ² Chi_k² independent

Therefore:

So we reject H₀ if F > critical value of F_k,n–k–1 distribution

1 2 ,

2

) ~ 1 /(

) 1

(

/ )

1 /(

/



 



 



  F_{k n k}

k n R

k R k

n RSS

k F ESS

Seminar

Multivariate regression II

Practicing

Maddala: 4/1, 4/3, 4/4, 4/5, 4/6, 4/9, 4/10 Wooldridge: 4.12, 4.14, 4.17, 4.19, 6.15

Discussion

t- and F-tests

Forecasting with EViews

Data

Subsample of Wage tariff (see week 4)

Wooldridge housing price data (hprice.dta)