Assumption MLR.3 Notes (No Perfect Collinearity)

1
Assumption MLR.3 Notes(No Perfect Collinearity)

• Perfect Collinearity can exist if
• One variable is a constant multiple of another
• Logs are used inappropriately
• One variable is a linear function of two or more
  other variables
• In general, all of these issues are easy to fix,
  once they are identified.

2
Assumption MLR.3 Notes(No Perfect Collinearity)

• One variable is a constant multiple of another
• -ie Assume that Joe only drinks coffee at work,
  and drinks exactly 3 cups of coffee every day he
  works. Therefore

-including both coffee and work in the regression
would cause perfect collinearity the regression
would fail
3
Assumption MLR.3 Notes(No Perfect Collinearity)
2) Logs are used inappropriately -Consider the
following equation and apply log rules
-a variable is included twice, causing an
inability to estimate B1 and B2 separately -note
that geek and geek2 could both have been used, as
they are not linearly related
4
Assumption MLR.3 Notes(No Perfect Collinearity)
3) One variable is a linear function of two or
more other variables -Consider a teenager who
spends all their income on movies and clothes
-if income and expenditures on both movies and
clothes are in the regression, perfect
collinearity exists and the regression fails
5
3.3 Fixing Multicollinearity

• Drop a variable from the model.
• If one variable is a multiple of another, it adds
  nothing to consider it twice
• Ditto for logs
• If the elements of a sum are in a regression, the
  sum itself is redundant. (Alternately one of the
  elements can be omitted).

6
3.3 Multicollinearity and N
-Assumption MLR.3 can also fail is N is too
small -in general, MLR.3 will always fail if
nltk1 (the number of parameters) -even if ngtk1,
MLR.3 may fail due to a bad sample Next we have
the most important assumption for proving OLSs
unbiasedness
7
Assumption MLR.4(Zero Conditional Mean)
The error u has an expected value of zero given
any values of the independent variables. In
other words,
8
Assumption MLR.4 Notes(Zero Conditional Mean)

• MLR.4 fails if the functional relationship is
  misspecified
• A variable is not included the correct way
• -ie consumption is included in the regression
  but not consumption2 and the true relationship is
  quadratic
• 2) A variable is included the incorrect way
• -ie log(consumption) is included in the
  regression but consumption is the true
  relationship
• -In these cases, the estimators are biased

9
Assumption MLR.4 Notes(Zero Conditional Mean)

• MLR.4 also fails if one omits an important factor
  correlated with any x
• -this can be due to ignorance or data
  restrictions
• MLR.4 also fails due to
• Measurement error (ch. 15)
• 2) An independent variable is jointly determined
  with y (ch. 16)

10
Assumption MLR.4 Notes(Zero Conditional Mean)
When MLR.4 holds, we have EXOGENOUS EXPLANATORY
VARIABLES When MLR.4 does not hold (xj is
correlated with u), xj is an ENDOGENOUS
EXPLANATORY VARIABLE
11
MLR.3 vs. MLR.4(Cage Match)
MRL.3 deals with relationships among independent
variables -if it fails, OLS cannot run MLR.4
deals with relationships between u and
independent variables -it is easier to
miss -it is more important
12
Theorem 3.1(Unbiasedness of OLS)
Under assumptions MLR.1 through MLR.4,
For any values of the population parameter Bj.
In other words, the OLS estimators are unbiased
estimators of the population parameters.
13
3.3 Is OLS valid?

• IF OLS runs (B estimates are found)
• -MLR.3 is satisfied
• IF the sample is random
• -MLR.2 is satisfied
• IF we have some reason to suspect a true
  relationship
• -MLR.1 is valid
• Therefore if we believe MLR.4 holds true
• -OLS is valid!

14
3.3 What is unbiasedness?

• Our estimates of Bhat are all numbers
• -numbers are fixed, and cannot be biased or
  unbiased
• MLR.1 through MLR.4 comments on the OLS
  PROCEEDURE
• -Is our assumptions hold true, our OLS
  PROCEEDURE is unbiased.
• In other words
• we have no reason to believe our estimate is
  more likely to be too big or more likely to be
  too small.

15
3.3 Irrelevant Variables in a Regression Model
Including independent variables that do not
actually affect y (irrelevant variables) is also
called OVERSPECIFYING THE MODEL -Consider the
model
-where x3 has no impact on y B30 -x3 may or may
not be correlated with x2 and x1 -in terms of
expectations
16
3.3 Irrelevant Variables in a Regression Model
From theorem 3.1, B1hat and B2hat are unbiased
since MLR.1 to MLR.4 still hold We furthermore
expect that
-even though B3hat may not be zero, it will
average out to zero across samples -Including
irrelevant variables doesnt affect OLS
unbiasedness, but we will see it affect OLS
variance