資料紹介
Econometrics Theory
1 Linear Regression Model
1.1 Basics and estimation
Let us conside a simple linear regression model below.
yt= b1+ b2xt+ ut; for t= 1;;n
This function is mapping fromxt toyt, that is,xt ! yt. Here we call each variables,yt as explained variable, or
dependent variable,xt as explanatory variable, or independent variable,b1,b2 as parameters andut as disturbance.
Orf(xt;yt)g is observed set,b1,b2 is unknown constant andutis un-observed variable.
As tofutg, we assume thatE(ut)= 0 and sometimes assume more. This assumption is valid because of the reason as
follows, that is, individualsin economic theory can be seen as economic man, or averageman, those who are rational,
hence in economic theory, we regard individuals as rational so if their characters are observed, consequentlyE(ut)=,
that is, the sum of their characters is trivial.
Then we need to estimate unknown parametersb1 and b2 by the observed setf(xt;yt)g and the assumption that
E(ut)= 0. For the modelyt= b1+ b2xt+ ut, we take expectaion butxtis random variable so that we take conditional
expectaion givenxt. Hence we obtain,
E(ytj xt) = E(b1+ b2xt+ utj xt)
= b1+ b2E(xtj xt)+ E(utj xt)
= b1+ b2xt+ E(utj xt)
Here assuming thatE(utj xt) = 0, we getE(ytj xt) = b1+ b2xt. NOTE thatE(utj xt) = 0 impliesE(ut) = 0 but not
vice versa. SinceEx[Eu(utj xt)] =Eu(ut), ifE(utj xt) = 0 then we haveE(ut) = 0. This logic is valid because of
the proof below, whereW1 andW2 are the support ofutandxtrespectively.
Eu(ut) =
Z
W1
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Econometrics Theory
1 Linear Regression Model
1.1 Basics and estimation
Let us conside a simple linear regression model below.
yt= b1+ b2xt+ ut; for t= 1;;n
This function is mapping fromxt toyt, that is,xt ! yt. Here we call each variables,yt as explained variable, or
dependent variable,xt as explanatory variable, or independent variable,b1,b2 as parameters andut as disturbance.
Orf(xt;yt)g is observed set,b1,b2 is unknown constant andutis un-observed variable.
As tofutg, we assume thatE(ut)= 0 ...
