Econometrics: Linear Regression
with one Regressor 1
Keisuke Kawata
Hiroshima University
Comparing means from different populations
• In some cases, our interest treatment is not binary. e.g.,)
• The causal effects of education level(junior high, high, college) on wage.
⇒ If sample size in each education level groups, the estimation method for sub- sample means can directly apply.
䐟 Estimating the conditional mean income of sub-groups
� �� ℎ �ℎ , � �� ℎ �ℎ , �[ �� | �� � ] 䐠 Comparing these estimators.
Difference between college and high: � �� �� � − � �� ℎ �ℎ ,
� �� ℎ �ℎ − � � � ℎ �ℎ
Limitation
Our interest treatments are continuous variable
Using comparing means, can we get the estimator for continuous treatments? e.g., The diffe e e of hild edu atio le el a o di g to pa e t’s i o e.
⇒
• The number of potential value is so many.
⇒ The number of observation in each sub-group(e.g., 2000$, 2001$,2003$) is very s all at ost 1 .
⇒You cannot get No !!!!!
credible estimators of conditional means
Populatio odel
• In many cases, we use more general approach ⇒ Estimation for the parameters of
• If ou i te est is the effe t of ha gi g o e a ia le t e.g., pa e t’s i o e o another variable y (e.g., children education level), you should estimate the
following (linear) population model with one regressor:
he e i is a a e of o se atio , � ( ) and � ( ) are parameters, and u( ) is chapter the effects of other factors.
• Generally, � is called as , and � is
• Using the information of � and �, we try to estimate � , � , and �. population model.
� = � + � � + �
constant term coefficient error term
outcome treatment
Potential outcome and population model
• Let � and � denote population outcome.
• The real outcome of individual i, �, can be rewritten as
� = � � + − � �
= � − � � + �
= � � + � − � � + � − � �
�
:
Causal effect.Important note: We suppose that the causal effects are same among individuals
� � �
Interpretation of u
• We suppose that the value of outcome � is determined by treatments � and other factors (covariates) � .
e.g.) The population model about the relationship between education year and income
Education year Income
Birth place Pa e t’s so ial status
Cognitive/non-cognitive ability
u
Interpretation of u
• We suppose two sets �, �, � and �′, �′, �′ and can then write
� = � + � � + �
�′ = � + � �′ + �′
Combing above equations,
� − �′ = � � − �′ + � − �′
• If
� − �′
� − �′ = �
⇒ Gi e if the alue of is i eased i , the value of Y is increased in � .
� − �′ = ,
the effects of other factors, one
Graphical example
y
�
,
��
� + �
�What is good estimator?
y
t
• From the observed data �, � , we should get good estimators � , � , � of
� , � , � .
• There are many estimators.
Estimator 1
Estimator 2
Estimator 3
� � � ,
Ordinary Least Squares Estimator
Ordinary Least Squires Estimator (OLS Estimator):
• The estimators � , � a e dete i ed to i i ize the total s ua ed gap :
⇒ Graphically, � + � � is the a out dots.
�=
�
� − � − � �
ost fitted li e
OLS Estimator : Alternative interpretation
• The estimators � are determined by
� = � + � � + � ⇒ � = � − � − � �
• Using �, the total s ua ed gap a e e itte as
⇒ The OLS estimator is minimizing
�=
�
�
the squared error term.
The actual values of estimators (rewrite)
Using the concept of sample variance and covariance, the actual OLS estimators can be characterized by
• If t and y are positive (negative) correlated, the estimator of coefficient of t is
⇒ If the correlation is stronger, the absolute value of estimator is
• If � has no variation ( � = ), we cannot get
� = �=� � − �[ �] � − �[ �]
�=�
� − �[ �] =
��
� .
positi e egati e .
o e la ge esti ato .
The actual values of estimators
• The estimator of � can be characterized by
�[ �] = � + � �[ �] ⇒ � = �[ �] − � �[ �]
The property of estimators
• �, ��, � are random variables ⇒ The OLS estimators are
y
t
� + �
�a do a ia les.
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 ato s.
the o al dist i utio s u de the la ge sa ple size.
pu e a do sa pli g data.
� � = .
