Fixed regressor model
1. Linear regression model
is linear in the parameters.
2. Fixed X
X considered fixed in repeated samples
(non-random across samples). Note: A nonrandom variable (i.e. constant) always have
zero covariance with any o
Multiple-Choice and True/False Questions
The expressions are as follows
b0 = y b1 x
(x x)(yi y)
b1 = i=1
i=1 (xi x)
Note that we can replace the numerator in the e
Statistical properties of OLS1
Statistical properties of OLS estimators = Properties of the sampling distribution/probability distribution
of OLS estimators.
I. Important: estimators are random variables!
A random variable is a variable whose value is de
Hypothesis testing: Interpreting the p-value
The p-value: probability, under the null, of obtaining a test statistic that exceeds in absolute
value the value of the test statistic obtained from the sample ().
Decision rule: for a
Notes: Hypothesis Testing
Hypothesis about the true but unknown population parameter:
Is the obtained from the sample compatible with the stated hypothesis?
Is the obtained from the sample sufficiently close to the hypothesized value so that
Units of measurement and standardized variables (Section 6.2, 6.3 and 7.6)
Effects of rescaling
Consider the following regression:
Let and , and consider the regression using and :
Applying the OLS method to both regressions, we can show the follow
Predictions and Confidence Intervals
Problem: predicting the value of for an arbitrary value of , denoted .
There are two values we might be interested in predicting:
i) prediction of the conditional mean value of Y corresponding to .
ii) prediction of th
Review of Sections 3.2 to 3.4
Derivations and proofs
Practice doing the proofs! In the assignments and on the final exam, you will be asked to
write proofs. The review sheets I regularly post on cuLearn are a good way to check that you
are able to derive
Review of Section 3.1
Derivations and proofs
Lots of algebra! Some of the following results were obtained in class. Make sure that
you are able to derive them yourself (and check the checkboxes!). Try also to prove on
your own the remaining results mentio
Regression through the origin/ regression without a constant (Section 6.1 and
Applying the OLS method () we obtain the OLS estimator of . There is also a particular formula
for the variance of and for .
Lets compare the formulas obtained wit
Coefficient of determination
Recall that and (deviation form)
Note: variation means the sum of squares of the deviations of a variable from its mean
= Total Sum of Squares = the total variation of the values about their sample mean.
Matthieu Lajoie (100739404), Daanish Garda, Mathew Voyce
a. The coefficient of education of 0.070 suggests that for each additional year of education,
we can expect earnings to increase by 7%.
b. The 90% confidence interval for