3.4 The Components of the OLS Variances: Multicollinearity

We see in (3.51) that the variance of Bjhat depends on three factors: σ2, SSTj and Rj

2:

1)The error variance, σ2

Larger error variance = Larger OLS variance-more “noise” in the equation makes it more difficult to accurately estimate partial effects of the variables-one can reduce the error variance by adding (valid) variables to the equation

3.4 The Components of the OLS Variances: Multicollinearity

2) The Total Sample Variation in xj, SSTj

Larger xj variance – Smaller OLSj variance-increasing sample size keeps increasing SSTj since 2)( jijj xxSST

-This still assumes that we have a random sample

3.4 The Components of the OLS Variances: Multicollinearity

3) Linear relationships among x variables: Rj2

Larger correlation in x’s – Bigger OLSj variance-Rj

2 is the most difficult component to understand - Rj

2 differs from the typical R2 in that it measures the goodness of fit of:

ikkiiij xxxx ...ˆ 22110

-Where xj itself is not considered an explanatory variable

3.4 The Components of the OLS Variances: Multicollinearity

3) Linear relationships among x variables: Rj2

-In general, Rj2 is the total variation in xj that is

explained by the other independent variables-If Rj

2=1, MLR.3 (and OLS) fails due to perfect multicollinearity (xj is a perfect linear combination of the other x’s)

Note that: 2

jR as )ˆ( jVar -High (but not perfect) correlation between independent variables is MULTICOLLINEARITY

3.4 Multicollinearity

-Note that an Rj2 close to 1 DOES NOT violate MLR. 3

-unfortunately, the “problem” of multicollinearity is hard to define

-No Rj2 is accepted as being too high

-A high Rj2 can always be offset by a high SSTj or a low σ2

-Ultimately, how big is Bjhat relative to its standard error?

3.4 Multicollinearity-Ceteris Paribus, it is best to have little correlation

between xj and all other independent variables

-Dropping independent variables will reduce multicollinearity-But if these variables are valid, we have created bias

-Multicollinearity can always be fought by collecting more data

-Sometimes multicollinearity is due to over specifying independent variables:

3.4 Multicollinearity Example-In a study of heart disease, our economic model is:

Heart disease=f(fast food, junk food, other)

-Unfortunately, Rfast food2 is high, showing a high correlation

between fast food and other x variables (especially junk food)-since fast food and junk food are so correlated, they should be

examined together; their separate effects are difficult to calculate

-Breaking up variables that can be added together can often cause Multicollinearity

3.4 Multicollineairity-it is important to note that multicollinearity

may not affect ALL OLS estimates-take the following equation:

uxxxy 3322110 -if x2 and x3 are correlated, Var(B2hat) and Var(B3hat) will be large (due to

multicollinearity)-HOWEVER, from (3.51), if x1 is fully uncorrelated with x2 and x3, R1

2=0 and

1

2

1)ˆ( SSTVar

3.4 Including Variables-Whether or not to include an independent variable is a balance between bias and variance:-take the following equation:

(A) ˆˆˆˆ 22110 xxy

-where both variables, x1 and x2, are included

-Compare to the following equation with x2 omitted:

(B) ~~~

110 xy If the true B2≠0 and x1 and x2 have ANY correlation, B1tilde is biased

-Focusing on bias, B1hat is preferred

3.4 Including Variables-Considering variance complicates things-From (3.51), we know that:

)(A' )1(

)ˆar(2

11

2

1 RSSTV

-Modifying a proof from chapter 2, we know that:

)(B' )~

ar(1

2

1 SSTV

-It is evident that unless x1 and x2 are uncorrelated in the sample, Var(B1tilde) is always smaller than Var(B1hat).

3.4 Including Variables-Obviously, if x1 and x2 aren’t correlated, we have no bias and no multicollinearity

-If x1 and x2 are correlated:

1) If B2≠0, B1tilde is biased, B1hat is unbiased

Var(B1tilde)< Var(B1hat)

2) If B2≠0, B1tilde is unbiased, B1hat is unbiased

Var(B1tilde)< Var(B1hat)

-Obviously in the second situation omit x2. If it has no real impact on y, adding it only causes multicollinearity and reduces OLS’s efficiency-Never include irrelevant variables

3.4 Including Variables-In the first case (B2≠0), leaving x2 out of the model results in a biased estimator of B1

-If the bias is small compared to the variance advantages, traditional econometricians have omitted x 2

-However, 2 points argue for including x2:

1) Bias doesn’t shrink with n, but variance does2) Error variance increases with omitted variables

3.4 Including Variables1) Sample size, bias and variance

-from discussion on (3.45), roughly bias doesn’t increase with sample size-from (3.51), increasing sample size increases SSTj and therefore decreases

variance:

2)( jijj xxSST

-One can avoid bias and fight multicollinearity by increasing sample size

3.4 Including Variables2) Error variance and omitted variables

-When x2 is omitted and B2≠0, (3.55) underestimates error

-Without including x2 in the model, x2’s variance is added to the error’s variance

-higher error variance increases Bjhat’s variance

3.4 Estimating σ2

-In order to obtain unbiased estimators of Var(Bjhat), we must first find an unbiased estimator of σ2.

-Since we know that σ2=E(u2), an unbiased estimator of σ2

would be:

22 1ˆ iun

-Unfortunately, this is not a true estimator as we do not observe the errors ui.

3.4 Estimating σ2

-We know that errors and residuals can be written as:

ikkiiii

ikkiiii

xxxyu

xxxyu

ˆ...ˆˆˆˆ

...

22110

22110

Therefore a natural estimate of σ2 would replace u with uhat-However, as seen in the bivariate case, this leads to bias, and we had

to divide by n-2 to become a consistent estimator

3.4 Estimating σ2

-To make our estimate of σ2 consistent, we divide by the degrees of freedom n-k-1:

(3.56) )1()1(

ˆˆ

2

2

kn

SSR

kn

ui

Where k is the number of independent variables-Notice in the bivariate case k=1 and the denominator is n-2. Also

note:

)parameters estimated of(number -

ns)observatio ofnumber (

)1(

kndf

3.4 Estimating σ2

-Technically, n-k-1 comes from the fact that E(SSR=(n-k-1)σ2

-Intuitively, from OLS’s first order conditions:

0ˆx and 0ˆ ij ii uu

There are therefore k+1 restrictions on OLS residuals (j=1,2,…k)-If we therefore have n-(k+1) residuals we can use these restrictions

to find the remaining residuals

Theorem 3.3(Unbiased Estimation of

σ2)Under the Gauss-Markov Assumptions

MLR. 1 through MLR. 5,

22 )ˆ( ENote: This proof requires matrix algebra

and is found in Appendix E

Theorem 3.3 Notes-the positive square root of σhat2, σhat, is called

the STANDARD ERROR OF THE REGRESSION (SER), or the STANDARD ERROR OF THE ESTIMATE

-SER is an estimator of the standard deviation of the error term

-when another independent variable is added to the equation, both SSR and the degrees of freedom fall-Therefore an additional variable may increase or decrease SER

Theorem 3.3 NotesIn order to construct confidence intervals

and perform hypothesis tests, we need the STANDARD DEVIATION OF BJHAT:

)1()ˆ(

2jj

jRSST

sd

Since σ is unknown, we replace it with its estimator,

σhat, to give us the STANDARD ERROR OF BJHAT:

(3.58) )1(

ˆ)ˆ(

2jj

jRSST

se

3.4 Standard Error Notes-since the standard error depends on σhat, it has a sampling distribution

-Furthermore, standard error comes from the variance formula, which relies on homoskedasticity (MLR.5)

-While heteroskedasticity doesn’t cause bias in Bjhat, it does affect its variance and therefore cause bias in its standard errors-Chapter 8 covers how to correct for heteroskedasticity

3.5 Efficiency of OLS - BLUE-MLR. 1 through MLR. 4 show that OLS is unbiased, but many unbiased estimators exist

-HOWEVER, using MLR.1 through MLR.5, OLS’s estimate B jhat of Bj is BLUE:

Best

Linear

Unbiased

Estimator

3.5 Efficiency of OLS - BLUEEstimator

-OLS is an estimator as “it is a rule that can be applied to any sample of data to produce an estimate”

Unbiased

-Since OLS’s estimate has the property

0,1,...kj )ˆ( jjE OLS is unbiased

3.5 Efficiency of OLS - BLUELinear

-OLS’s estimates are linear since Bjhat can be expressed as a linear function of the data on the dependent variable

(3.59) ˆiijj yw

Where wij is a function of independent variables

-This is evident from equation (3.22)

3.5 Efficiency of OLS - BLUEBest

-OLS is best since it has the smallest variance of all linear unbiased estimators

The Gauss-Markov theorem states that, given assumptions MLR. 1 through MLR.5, for any other estimator Bjtilde that is linear and unbiased:

)~

()ˆ( jj VarVar And this equality is generally strict

Theorem 3.4(Gauss-Markov Theorem)

Under the Assumptions MLR. 1 through MLR. 5,

k ˆ,...ˆ,ˆ10

Are respectively the best linear unbiased estimators (BLUE’s) of

k ,..., 10

Theorem 3.4 Notes-if our assumptions hold, no linear unbiased estimator will be a better choice than OLS-if we find any other unbiased linear estimator, its variance will be at least as big as OLS’s -If MLR.4 fails, OLS is biased and Theorem 3.4 fails-If MLR.5 (homoskedasticity) fails, OLS is not biased but no longer has the smallest

variance, it is LUE