Consistency
Clive W. J. Granger: “If you can’t get it right
as n goes to infinity, you shouldn’t be in this
business.”
Consistency
What is consistency? If an estimator, 𝛽𝑗 , is
consistent, then the distribution of 𝛽𝑗
becomes more and more tightly distributed
around 𝛽𝑗 as the sample size grows. As 𝑛 →
∞ the distribution of 𝛽𝑗 collapses to the
single point 𝛽𝑗 .
Consistency
In the general model:
𝑦 = 𝛽𝑜 + 𝛽1 𝑥1 + 𝛽2 𝑥2 + ⋯ + 𝛽𝑘 𝑥𝑘 + 𝑢,
all OLS parameter estimators are consistent.
Example
Trying to estimate the mean or average human IQ:
• In Minitab take samples of 10, 100, and 1000.
• Have Minitab compute the sample mean, 𝑋,
𝑠
and the standard error of the sample mean, .
𝑛
• A 95% CI can be created.
• Data can be plotted.
Testing MLR Assumption 5
MLR Assumption 5: The error, u, has the same
variance given any values of the explanatory
variables. This is the assumption of
homoskedasticity.
Run a MLR analysis and graph the residuals vs.
each predictor variable and vs. fits and vs. order.
Testing MLR Assumption 5
Use SMOKE data set
Response: cigs
Predictors: educ, cigpric, age
In Minitab under Regression -> Regression click on
the Graphs box and at the bottom select the three
predictor variables to plot the residuals vs. these
variables.
Testing MLR Assumption 5 –
Examples of Heteroskedasticity
Testing MLR Assumption 5 –
Examples of Heteroskedasticity
Testing MLR Assumption 5 –
Examples of Heteroskedasticity
Testing MLR Assumption 6
MLR Assumption 6: The error, u, is independent of
the explanatory or predictor variables and is
normally distributed with mean zero and variance
𝜎 2.
Run a MLR analysis and make histogram of
residuals and normal probability plot of residuals.
Testing MLR Assumption 6 –
Examples of Normally Distributed
Residuals
Testing MLR Assumption 6 –
Examples of Normally Distributed
Residuals
Testing MLR Assumption 6 –
Examples of Non-Normally
Distributed Residuals
Testing MLR Assumption 6 –
Examples of Non-Normally
Distributed Residuals
Testing MLR Assumption 6 –
Examples of Normally Distributed
Residuals
Testing MLR Assumption 6 –
Examples of Normally Distributed
Residuals
Testing MLR Assumption 6 –
Examples of Non-Normally
Distributed Residuals
Testing MLR Assumption 6 –
Examples of Non-Normally
Distributed Residuals
Testing for Assumptions of
Normality and Homoskedasticity
Use the data set 4th Graders Feet
Regress foot length on childs age
Comment on normality and homoskedasticity
assumptions.
Testing for Assumptions of
Normality and Homoskedasticity
Use the data set TWOYEAR
Regress lwage on jc
Comment on normality and homoskedasticity
assumptions.
Testing for Assumptions of
Normality and Homoskedasticity
Use the data set SMOKE
Regress cigs on age
Comment on normality and homoskedasticity
assumptions.