Slide 6_2_distribution 最近の更新履歴 Keisuke Kawata's HP

Econometrics: Linear Regression

with one Regressor 2

Keisuke Kawata

Hiroshima University

䠄 _Review 䠅 Least Squares Assumptions

The least squires assumption 1. Your data is

2. The mean of is zero:

3. The conditional mean of u does not depend on : For any t,t_’,

• If the following least squares assumptions hold, OLS estimators � , � are –

– have

un iased a d onsistent esti ators.

the or al distri utio s u der the large sa ple size.

pure ra do sa pli g data.

� _� = .

� _{� �} = � _{� �}′ .

Random sampling and _� = assumptions

• The pure random sampling assumption is totally same requirement as the estimation for population mean.

• � _� = is not

If _{� �} =a, we should modify the population model as

�_� = �′ + � _� + _�′ where �′ =� + �, ^′ = − �.

Under the above specification, _{� �}^′ = must hold. The constant term can be interpreted as

Same conditional mean assumption

• In many case, � _{� �} = � _{� �}^{′ is} e.g.)

• The estimation about the effect of education year on income.

⇒Cognitive/non- og itive a ility a d pare t’s hara teristi s ay e differe e.

• The estimation about the effe t of pare t’s income on hild’s education outcomes.

⇒The parents location may be different.

Potential outcome and Same conditional mean

• � _{� �} = � _{� �}^′ is equivalent to _{� �� |�� = = � �� |�� =} Proof

• Because _{� = �� − � ��} ^⇒_{� � � = � �� �� = � − � ��}

• � _{� �} = = � _{� �} = ^⇒� �_� �_� = = � �_� �_� =

Graphical Intuition

• � _� = � _� ^{case ⇒}

�

�

� + �

Graphical Intuition

• � _{� �} > � _� ′_� ^if _� > _�′ ^⇒

�

� + �

Graphical Intuition

• � _{� �} < � _� ′_� ^if _� > _�′ ^⇒

�

�

� + �

Variance of OLS estimator

• Under Least Squares Assumptions, the estimators � , � have

• The distribution of � is � � , � _�1 ^where

� _{�1 = �} ^�� _��^{� − �� �}

�

where _�� is mean of _{�, and �� �} is the sample variance.

• The variance of estimator is small if –

Intuition

• Roughly speaking, we get the OLS estimator using the variation of explanation variable.

⇒ If there are no variation,

⇒ If the variation is not so large, the accuracy of OLS estimator is

Conclusion

• To get the estimator of linear population model, we often use the OLS estimation.

• The estimator of OLS is unbiased and consistent estimator if least squires assumptions hold.

• We can get accurate estimators if the sample size and the sample variance of explanation variable are large.