Econometrics – lecture 6 – extended forms of estimations

EXTENDED FORMS OF
REGRESSION MODELS
Dr. TU Thuy Anh
Faculty of International Economics

1


COBB-DOUGLAS FUNCTIONS
• Production function:Y = aKαLβ

=> lnY = a1+αln(K)+ βln(L)
=> lnY = a1+αln(K)+ βln(L) + u
• Example:
• Interpretation:
– α: when K increases by 1%, L unchanged then Y increases by

α%; the elasticity
– β: when L increases by 1%, K unchanged then Y increases by
α%
– a1: when K=L =1=> Y=exp(a1)
2


POLINOMIAL FUNCTION
 Cost function:
 TC = a1+a2Q?
 TC = a1+a2Q+a3 Q2?
 TC = a1+a2Q+a3 Q2 +a4Q3?
 TC = a1+a2Q+a3 Q2 +a4Q3+u

3


MODELS WITH DUMMY VARIABLES
 Returns and risk: returns = a1 + a2risk + u
 However:

2 different intercepts

4


MODELS WITH DUMMY VARIABLES
 How to solve the problem:
 run a regression for each group, or
 use one regression, but add “the state of the market” as another

independent variable *
 For the (*), need to “numerate” the “state of the market”: use

dummy variables
 =>returns = a1 + a2risk + a3D+ u (3)

5


DUMMY
Obs.

returns

risk

Bull/bear

D

1

11.5

3.0

Bull

1

2

11.7

3.5

Bear

0

3

12.0

4.0

Bear

0

..

6

..

30

11.8

2.6

Bull

1

31

12.0

3.5

Bear

0


THE INTERPRETATION OF a3
• Write (3) as:

returnsBull = a1 + a2risk + a3+ u (4)
returnsBear= a1 + a2risk +
u (5)
• a3: if risk = 0 then returns in the bull market are greater than

in the bear market by a3 unit
• Example:
– returns^ = 2 + 0.3risk + 0.1D?

• Note: we implicitly presume that a2 is the same in the two

markets

7


MODELS WITH DUMMY VARIABLES
 Q: Does the market respond the same to risk in two different

states?
 If not:

both slope and intercept are

different

8


MODELS WITH DUMMY VARIABLES
• returnsBull = a1 + a2risk + a3D+a4D*risk+ u
• exercise: interpret a4?
• Summary:
– qualitative variables: race, gender, area, joining trade

agreement,..may affect the dependent variable in a model
– => include dummies into the model to capture this effect
– the corresponding coefficients are used to compare between
groups
– we may include more than one dummies

9


MODELS WITH DUMMY VARIABLES
 returnsBull = a1 + a2risk + a3D+a4D*risk+ u
 exercise: interpret a4?
 Q: in what case we should use dummies?
 if a3 and a4 are not significant => should not
 How to test? F- test

10

HOW TO “DUMMIRIZE”
• If the qualitative variable consists of two categories (F/M;

bull/bear; post-graduates/undergraduates, etc.) => use 1
dummy: D = 1 if F; 0 if M
• If k≥ 2 categories: religions; race, ownership, ..=> use k-1
dummies
• Example: SOE, FDI and private enterprises
– D1 = 1 if SOE, 0 otherwise
– D2 = 1 if FDI, 0 otherwise
– => GDP = a1+a2D1+a3D2+a4K + a5L + u

11