Econometrics
F-test and Non-linear Population model
Keisuke Kawata
IDEC, Hiroshima University
Condition to identify all coefficients
• Assumption � � �, , … , � = � � �′, , … , � ensures just unbiasedness of ��.
• What’s assu ptio eeded to e sure u iased ess for all oeffi ie ts?
3’: � � �, … � = � � �′, ′ … ′� for any �, … � and �′, ′ … �′ .
Interpretation
• To u dersta d i terpretatio of o ditio 3’, e suppose the sample difference approach.
• Let think two binary variable X and T.
• The condition to obtain an unbiased estimator of the causal effect of T is
� � , �� = , � = = �[ � , |�� = , � = ] where � , is the potential outcome if � = and � =
• The condition to obtain an unbiased estimator of the causal effect of X is
⇒To obtain unbiased estimators of both X and T,
� � , �� = �, � = = � � , �� = �′, � = ′ , � �′ � ′
Hypothetical tests for a single coefficient
• Totally same as in the statistical tests in one regressor cases!!!!
Step 1. Under the null hypothesis �� = ��, , estimating the distribution of a OLS estimator
← From the central limit theorem, the distribution can be approximated as the Normal distribution.
← The variance of estimators can be estimated. Step 2. Calculating t-statistics.
Step 3. Calculating p-statistics← Interpretations of p-value is same as in single regressor cases.
Step 4. Constructing the confidence interval.
Hypothetical test of joint hypotheses
We try to test the following null hypothesis as
� : � = � , , … , �� = ��, , and the alternative hypothesis is then
test of Joint hypothesis
t-test?
• Can I use t statistics to test the joint hypothesis?
e.g.) If the p-value of null hypothesis � : � = and � : � = . are low, can we reject the null hypothesis � : � = and � = ?
⇒
← � and � may
Basic Ideas
� = and � = ⇒ Even if and are added in the regression,
F test
Step 1: We regress the population model with no restrictions as
� =
and calculate
Step 2: We regress the population model with restrictions � = � = as
and calculate
Step 3: We calculate actual difference of � between no restrictions and
restrictions .
Step 4: We calculate the probability that
⇒ If the probability is low ⇒ We can
Non-linear relationship
• Some time, economic theory and our experience predict non-linear relationship between outcome and treatment variables.
e.g.) The effect of class-size on average test score. Average score
Class size
The problem of miss-specification
• If we use a miss-specified population model ⇒ OLS estimators have
e.g.) If we assume the linear population model � = � + ���� + � even through actually � = � + � �� + � �� + �.
⇒Even if � � = � �′ , because � = � + � �� .
Estimation of non-linear population models
• Can we relax the linear assumption about population model?
⇒If the population model is additive separable, we can apply the OLS technique!!!
• If you would like to estimate the following population model
�[ �|�] = � + � � + � � + � � �
⇒ You can rewrite as
where � = � and � = � �.
⇒Totally equivalent to the OLS estimation with multi-repressors.
Quadratic population model
• Practically, there are two important non-linear specification; Quadratic and Log.
• To get i terpretatio a out the size effe ts of T, e ofte esti ate the quadratic model:
Note: We can include other explanation and control variables.
Interpretation: Quadratic population model
• The differentiation of E[Y|T] = � + � � + � T is
��[ |�]
�� =
⇒Marginal change of E[ |T] if T is marginally increased.
⇒If � ≠ , the marginal effect depends on the size of T. – � is positive ⇒ The effect of T is if T is large. – � is negative ⇒ The effect of T is if T is large.
Difficulty of specification
• How to determine the specification?⇒ There are no clear answer, you should determine the specification using the economic theory, experience and any special knowledge about your interest.
• We can suppose more high-order effects as
�[ �|�] = � + � � + � � + ⋯ + ����
⇒There exist costs to suppose high-order effects: Efficiency of the estimation goes down due to large number of coefficient.
• Some guide-line are existed: More high-order effect is supposed if the estimated coefficient of this effect is significant.
Practical matter
• If you would like to tackle miss-specification problem, you should use the non- parametric estimation.
⇒Ho e er…
In many case, assumption of linear relationship are still acceptable.
← This model can be interpreted as approximation of the true population mode.
• If you would like to find additional implication, you should try to estimate the quadratic population model.
Log population model
• We often estimate the population model including the log-value. Log-linear model:
Log-log model:
⇒ We can get OLS estimators using same method as in quadratic population model. Mathematical note: The differentiation of Z = � is
�
� =
Interpretation: Log-linear model
• The coefficient of log � = � + ���� + � means that
�� =
⇒ If T is i reased ith o e u it, changes
e.g.) If the OLS estimator of �� = . , we estimate that if T is increased, Y is increased with 10%.
Interpretation: Log-log model
• The coefficient of log � = � + �� log �� + � means that
�� =
⇒ �� is the of Y about T.
Mathematics note: the elasticity is defined by
∆
∆��
= ∆∆� � ⇒ ��� � � ∆� ⇒
⇒ If T is increased with 1%, Y is changed as
