week 5 class ec221

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Week 5 Class EC221

Will Matcham

[email protected]

OH: Thursday 2:303:30, 32LIF 1.30

Will Matcham (LSE) Week 5 Class EC221 October 2014 1 / 12

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Outline

Today is a demanding class...

RegisterHand back problem setsNext weeks class

1 Material

Variance and Covariance RulesPS3 Q4 Skipped Last WeekMultiple Linear Regression Model

2 Problem Set 4Q1 Not DoingQ2 Next WeekQ3Q4 ToughQ5 Tougher

3 Additional Class TopicWill Matcham (LSE) Week 5 Class EC221 October 2014 2 / 12

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Content RevisionVariance Rules

Denition of variance of random variable X , and covariance of tworandom variables X , Y .Scalar variance and covariance rules:

1 V (a + X ) = V (X )

2 Cov (a + X , b + Y ) = cov (X , Y )

3 V (aX ) = a V (X ) a

4 Cov (aX , bY ) = a Cov (X , Y ) b

5 V ( i X i ) = i j Cov (X i , X j ) = i V (X i ) + i j i = j

Cov (X i , X j )

6 Cov ( i X i , j Y j ) = i j cov (X i , Y j )Denition of variance of random vector X and covariance of randomvectors X, Y. What do these matrices look like?How do above properties generalise for random vectors (at least the

rst 4)?Will Matcham (LSE) Week 5 Class EC221 October 2014 3 / 12

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Problem Set 3Question 4

Something harder! The structure of how to approach it isstraightforward. The statistics of the steps in between are not

however.

When dealing with variances and covariances, please, try rst towork with the variance and covariance rules . Only introduce themas expectations later on if you are stuck. Expectations are nasty towork with!


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Multiple Linear Regression ModelHas much changed?

Not really! Our assumptions have mostly the same intuition, we are just using different notation.

Our assumptions:1 A1 The model is still linear in parameters but now we have k

parameters and not 2.2 A2 Perfect collinearity is about identication. Are the regressors

linearly independent so that the OLS can identify out each of theparameters. Solving systems and dummy variable trap.

3 A3 Exogeneity: our errors really are ignorance. Strong: implies errorshave zero mean and no covariance between regressors and errors.

4 A4 Again a convenience assumption for the moment. Demand a verysimple covariance structure between our errors. Total iid data. Willchange in LT but for the moment makes OLS the best.

Switching from scalar form and matrix form. In EC221 we mainly use

matrix form. Further econometric courses require mastery of both.Will Matcham (LSE) Week 5 Class EC221 October 2014 5 / 12

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Multiple Linear Regression ModelThe formula that you simply must remember

OLS derivation follows the same intuition but everything is now avector or a matrix.Thing to minimise:

S ( ) = y y 2X y + X X FOC:

2X y + 2 X X = 0 = = ( X X ) 1X y

Symmetry of X X really helps here.SOC: 2X X is pd so our objective function is strictly convex and wehave hit a strict global minimum at the solution.


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This is a slug! So much working!

Question really is to prove the point that we want to move to themultivariate model, get into linear algebra, and forget about thecalculus derivations.

Tip is to dening terms and putting in the averages at will. You

probably didnt get what is in the solutions, but you learn withexperience the tricks to use.


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This is the only question that is not just doing technical work, butrequires some econometric intuition and thought.

For the second part, which requires the thought, you need to thinkabout what a change of units is, and how we frame a change of unitsin the question we are given.

Fit of a regression is synonymous with looking at residuals.

Why is this important? Regressors are regressors, they explain thesame amount regardless of their unit. Otherwise you could hack your

regression and arbitrarily search for a matrix P to blow up theregressors with, to obtain the best t.

Think of regressors as information to explain the y . You cant gamethe regression process by changing your unit, and t the model better.

Only option: add (or take away) regressors, or change the model!Will Matcham (LSE) Week 5 Class EC221 October 2014 8 / 12

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I will present the crux of the argument and cover the details at theend.

Do not worry if you did not get this rst time. This does not meanyou are going to struggle with this course. Problem sets are a learningprocess, not a mock exam.


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As I mentioned in week 2 class, you need to know the propertiesabout idempotency, and understand how they help you.

Total instinct: if A is symmetric not only is is diagonalisable but it isorthogonally diagonalisable.


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Additional ContentThe details of Q4 and Q5

In Q4 we used that if u

N (0, V ), where V is symmetric and

positive denite, then V 1/ 2u N (0, I ).How is this matrix V 1/ 2 dened? We want it to be such thatV 1/ 2V 1/ 2 = V 1.Diagonalise V to get V = PDP 1. Taking inverses then gives

V 1

= PD 1

P 1

. Lets try V 1/ 2

= PD 1/ 2

P 1

.This works since

V 1/ 2V 1/ 2 = PD 1/ 2P 1PD 1/ 2P 1 = PD 1P 1 = V 1

Still one nal detail:

D 1/ 2 =1/ 1 . . . 0... . . . ...

0 . . . 1/ nThis is well dened:

V being pd ensures i > 0,

i .


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Additional ContentThe details of Q4 and Q5

In Q4 I dened V 1/ 2u =: q

N (0, I ). I then argued that each

q i N (0, 1).From this I concluded thatq q =

n

i =1

q 2i 2nIt is indeed a sum of n squared standard normal terms. What have Imissed?This crucial exception to the rule in statistics: a joint normal havingzero covariance implies that all random variables in the vector areindependent. We did this in PS2 Q2 for the bivariate case.This rule gives us the independence of the q i that we need:

q N (0, I ) = q i iidN (0, 1)


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