Econometrics
Linear Regression with one Regressor 3
Keisuke Kawata
IDEC, Hiroshima University
Plan of talk
1. Hypothetical test and confidence interval 2. Binary explanation variable
3. Measure of fit
4. Assumption about the variance of error terms
Hypothetical test
• Hypothetical test is consisted by two hypothesis. Null Hypothesis:
Alternative Hypothesis:
⇒In many case, is important null hypothesis
←Only if the null can be rejected, we can argue the existence of the
• Actual process of hypothetical test is
1. Under the null hypothesis � = � , , estimating 2. Calculating
Significance level
• Same as the hypothetical test about the estimator of population mean, we often set 5% significance level (1% and 10% are also used).
⇒ If and only if p-value is less than 5%, we the null hypothesis.
• 5% significance level means that we incorrectly reject the true null hypothesis with 5%.
Notion: Even if the p-value is very high, we say � = � ,
Confidence intervals
• A confidence interval can be also constructed.
: a set of values that contains the true population coefficient with a certain prespecified probability ( ).
Confidence interval with 95% confidence level (95% confidence interval): contains the true � with .
• If and only if the p-statistics is lower than 5%, � , is the 95% confidence interval.
Question
• True/False question.
Suppose the pure random sampling data.
1. If the p-value of the null hypothesis � = is 0.99, we can say there are no causal relationship between the explanation and explained variables.
2. If there are some covariates, OLS estimators are not unbiased estimator.
3. If the sample size is totally large and least squares assumptions hold, the OLS estimator converge to the true(population) value.
4. Supposing that for any < ′, E[u| � = ]>E[u| � = ′]. The OLS estimator � is lower than the true value � .
Conditional mean and population model
• In the sample difference approach, the conditional means play an central role.
⇒The population model can be interpreted as the model of the conditional means.
• Sufficient condition of � � = , � � � = = �[ �| � = ′] is
� � � � , � � � = =
⇒ Population model can be rewritten as
�� = � + � � + � ⇒
←We assume the linear relationship between the conditional means and treatment.
� =
⇒� can be interpreted as the change of conditional means.
Population model in modern econometrics
• The population model can be interpreted as; Model of ��: �� = � + � � + �
Mode of conditional mean: �[��| � = ] = � + �
• In the modern empirical works in economics, we follow
⇒ To estimate average causal effect or difference, we would like to estimate the conditional means.
1st best estimator: ⇒Sometimes, we need to
estimate too many population values (conditional means).
Alternative estimator: ⇒By the assumption on the
fu tio al for , e a redu e the populatio alue hi h should e esti ated. the latter ie .
Regression with binary treatment
• The population model approach can apply for binary treatment variable. : takes one or zero to chapter the categorical information. e.g.)
– If you would like to estimate the effect of gender on income, the treatment is � = if male and � = if female
– If you would like to estimate the effect of nationality on income, the explanation variable is � = if native and � = if non-native.
• Dummy variable is sometimes called as indicator variable.
Graphical example
y
0 1 t
yi = � + � �
Notion: Interpretation
• Only interpretation of � is a bit different with continuous explanation variable cases.
Continuous case: � is the estimator of changing of y if t increases as one unit when the effect of other covariates (u) is constant.
Binary case: � is the estimator of of y between t=1 and t=0 groups when the effect of other factors (u) is totally equal between groups.
Measures of Fits
• You may have interest how well that the prediction � + � predicts the data.
⇒ Does the prediction account for much or for little of the variation in the explained variable?
• To answer such questions, we use a statistics � .
⇒
The �
• � : The fraction of the sample variance of � predicted by �.
⇒ Using OLS estimator � , � , we can calculate the predicted value � as
� =
From the definition of �, we can decompose the value of explained variable by the predicted value and others as
� =
• Formally, the � can be defined as the ratio of the explained sum of squires to the total sum of squires.
� + � �
� + �
�
• We define � as
� =
• The � range is between 0 and 1.
• � is low ⇒ The prediction power of X is
�=�
� −
�=�
� −
lo
• � , y itself, i ply that this regressio is either good or ad .
⇒Low � tell us only that influence outcomes.
⇔The estimator of � a d it’s p-value tell us the of the
Graphical example
Y = � + �
:SER is lower and � is high :SER is high and � is low
Graphical example
Y
:� is high, but p-value is high. : � is low, but p-value is low
Heteroskedasticity and Homoskedasticity
• To get the unbiased and consistent estimator, we only assume
• If we can set additionally assumption about , we can show more strong arguments and simplify the calculation about the variance of estimator.
4. homoskedasticity: the variance of the conditional distribution of u given t is constant for any t ⇒
⇒If above assumption does not hold, we say the error term is heteroskedastic.
� � � =
Graphical example: Homoskedasticity case
y y = � + �
Graphical example: Heteroskedasticity case
y
y = � + �
“upposi g that if t<t’, �� < �� | ′ .
Practical mean of Homoskedasticity
• In modern works, we often use OLS estimator Homoskedasticity assumption ← The problems coming from Heteroskedastic are . 1. OLS estimators are not efficient ⇒ If sample size is enough large, the loss of
efficiency is not big problem because the variance of estimators are
2. The form of estimated variance is complicated ⇒ The power of your PC is
• In many cases, the requirement of Homoskedasticity is
⇒ The distribution of error terms is Heteroskedastic.
Question
• True/False question.
Suppose the pure random sampling data.
1. Supposing � of your estimation is totally large. The large part of fluctuation of the explained variable cannot be explained by the explanation variable.
2. In above case, the explanation variable has only small causal effect on the explained variable.
3. If the error term is heteroskedastic, the OLS estimators are unbiased estimators.
