Econometrics – lecture 3 – multiple regression

The starting point for investigating the determination of Q is the intercept, 1... This causes the actual values of Q to deviate from the plane.. In this observation, u happens to have

Dr TU Thuy Anh Faculty of International Economics

MULTIPLE REGRESSION ANALYSIS: ESTIMATION

Chap 6 – S & W

K

L

1

Q = 1 + 2 L+ 3 K + u

The model has three dimensions, one each for Q, L, and K The starting point for investigating the determination of Q is the intercept, 1.

K

1

pure L

effect 1 + 2L

Q = 1 + 2L+ 3 K + u

pure K effect

L

1

1 + 3 K

Q

K

Q = 1 + 2 L+ 3 K + u

pure K effect

Pure L effect

1

1 + 3 K

1 + 2L + 3K

Q

K

1 + 2L

combined effect of L and K

Q = 1 + 2 L+ 3 K + u

1 + 2

pure K effect

pure L

effect

L

1

1 + 3K

1 + 2L + 3K

1 + 2L + 3 K + u

Q

K

1 + 2L

combined effect of L and K

u

Q = 1 + 2 L+ 3 K + u

1 + 2

The final element of the model is the disturbance term, u This causes the actual values of Q to deviate from the plane In this observation, u

happens to have a positive value

i i

i

Y  1  2 2  3 3 

i i

i b b X b X

i i

i i

i

i Y Y Y b b X b X

The regression coefficients are derived using the same least squares principle

used in simple regression analysis The fitted value of Y in observation i depends on our choice of b1, b2, and b3.

The residual e i in observation i is the difference between the actual and fitted values of Y.

We define RSS, the sum of the squares of the residuals, and choose b1, b2, and



 ei2 ( Yi b1 b2X2i b3 X3i )2

RSS

Model 3: OLS, using observations 1899-1922 (T = 24)

Dependent variable: q

coefficient std error t-ratio p-value

-const -4,85518 14,5403 -0,3339 0,7418

l 0,916609 0,149560 6,129 4,42e-06 ***

k 0,158596 0,0416823 3,805 0,0010 ***

Mean dependent var 165,9167 S.D dependent var 43,75318 Sum squared resid 2534,226 S.E of regression 10,98533 R-squared 0,942443 Adjusted R-squared 0,936961 F(2, 21) 171,9278 P-value(F) 9,57e-14

Log-likelihood -89,96960 Akaike criterion 185,9392 Schwarz criterion 189,4734 Hannan-Quinn 186,8768 rho 0,098491 Durbin-Watson 1,535082

A.1: The model is linear in parameters and correctly

specified.

A.2: There does not exist an exact linear relationship

among the regressors in the sample (No

multicolinearity).

A.3 The disturbance term has zero expectation

A.4 The disturbance term is homoskedastic

A.5 The values of the disturbance term have

independent distributions

A.6 The disturbance term has a normal distribution

u X

X

Y  1  2 2   k k 

Provided that the regression model assumptions are valid, the OLS

estimators in the multiple regression model are unbiased and efficient, as

 Example: X1 - 2 X2 =0

 Definition: X1 and X2 are perfectly multi-collinear if there exists b1, b2 such that:

 b1X1+b2X2 = a ( a: constant)

 at least one of (bi) is non-zero

 X1 and X2 are perfectly multi-collinear iff r(X1, X2) = +/- 1

 X1 and X2 are highly multi-collinear if r(X1, X2) is large

 ~X1 and X2 are highly multi-collinear if R2 of the model X1

on (a1, X2) is large

 Definition: X1, ,Xk are perfectly multi-colinear if there

exists b1, ,bk :

 b1X1+ +bkXk = a ( a: constant)

 at least one of (bi) is non-zero

 X1, ,Xk are highly multi-collinear R2 of the model (Xj on the rest and a1) is large.

 Assumption 6: no perfect multi-collinearity among X2, ,Xk



What would happen if you tried to run a regression when there is an exact linear relationship among the explanatory variables? The coefficient is not defined

u K

K L

Q   1   2   3   4 2 

Model 4: OLS, using observations 1899-1922 (T = 24)

Dependent variable: q

coefficient std error t-ratio p-value

-const -10,7774 16,4164 -0,6565 0,5190

l 0,822744 0,190860 4,311 0,0003 ***

k 0,312205 0,195927 1,593 0,1267

sq_k -0,000249224 0,000310481 -0,8027 0,4316

Mean dependent var 165,9167 S.D dependent var 43,75318 Sum squared resid 2455,130 S.E of regression 11,07955 R-squared 0,944239 Adjusted R-squared 0,935875 F(3, 20) 112,8920 P-value(F) 1,05e-12

Log-likelihood -89,58910 Akaike criterion 187,1782 Schwarz criterion 191,8904 Hannan-Quinn 188,4284 rho -0,083426 Durbin-Watson 1,737618