Econometrics – lecture 6 – extended forms of estimations

EXTENDED FORMS OF REGRESSION MODELS Dr TU Thuy Anh Faculty of International Economics 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) 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 MODELS WITH DUMMY VARIABLES  Returns and risk: returns = a1 + a2risk + u  However: different intercepts 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) DUMMY Obs returns risk Bull/bear D 11.5 3.0 Bull 11.7 3.5 Bear 12.0 4.0 Bear 30 11.8 2.6 Bull 31 12.0 3.5 Bear THE INTERPRETATION OF a3 • Write (3) as: returnsBull = a1 + a2risk + a3+ u (4) returnsBear= a1 + a2risk + u (5) • a3: if risk = then returns in the bull market are greater than in the bear market by a3 unit • Example: – returns^ = + 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 • 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 10 HOW TO “DUMMIRIZE” • If the qualitative variable consists of two categories (F/M; bull/bear; post-graduates/undergraduates, etc.) => use dummy: D = if F; if M • If k≥ categories: religions; race, ownership, => use k-1 dummies • Example: SOE, FDI and private enterprises – D1 = if SOE, otherwise – D2 = if FDI, otherwise – => GDP = a1+a2D1+a3D2+a4K + a5L + u 11 ... 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)... 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... ownership, => use k-1 dummies • Example: SOE, FDI and private enterprises – D1 = if SOE, otherwise – D2 = if FDI, otherwise – => GDP = a1+a2D1+a3D2+a4K + a5L + u 11