Econometrics – lecture 6 – extended forms of estimations

TU Thuy Anh Faculty of International Economics 1 EXTENDED FORMS OF REGRESSION MODELS... MODELS WITH DUMMY VARIABLES Q: Does the market respond the same to risk in two different states?.

Dr TU Thuy Anh Faculty of International Economics

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EXTENDED FORMS OF REGRESSION MODELS

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%

– a : when K=L =1=> Y=exp(a )

POLINOMIAL FUNCTION

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 TC = a1+a2Q?

 TC = a1+a2Q+a3 Q2?

 TC = a1+a2Q+a3 Q2 +a4Q3?

 TC = a1+a2Q+a3 Q2 +a4Q3+u

MODELS WITH DUMMY VARIABLES

 Returns and risk: returns = a1 + a2risk + u

 However:

2 different intercepts

MODELS WITH DUMMY VARIABLES

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 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)

Obs returns risk Bull/bear

D

1

0

0

1

0

THE INTERPRETATION OF a3

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• 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

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

MODELS WITH DUMMY VARIABLES

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• 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

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

HOW TO “DUMMIRIZE”

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• 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