⇐40241/22) - MTWK 5 - VASSILIS Ps 4

RAWIR'S SPEAKING NOTES (UNOFFICIALtoNEENT)

QI① y,=p , aff a set

, for t -- I , . .IStep I : PROBLEM WITH MODEL①

Model④ is non -linear in parameters . We cannot directly uselinear regression methods to estimate this specification .

Step 2 : PROPOSED fourteen

Model①maybelinearsed using a log transformation :↳

y ye = Igp , + Palghat' f ,logMe + logE- , or

It = if , + fzkz.tt It + ME ,where

It log ye j Int := logEnt for h=2,3 ;{Me : -- log4 ; and if , := leg fi . -②

Step 3 : BRIEF ANALYSIS OF MODEL②

Model② is linear in parameters as far as if, if and §are concerned

.

It is a standard multiple linear regression modelwhere iff is regressed on a constant , It and size .

Of course the model remains non-linear with respect to fi .④ the economists anqoqbjswgfasmggeb.ee+help explain why

we may not{ care about f , in a ?

4) as Vassilis' solution explains , we retain consistency for¢ , := exp / .

i. e. non-linearitywrtf , may not be abig deal .We also will need to ensure that a log transformation is feasible .Assumption 1 : Yt 70 , nzt 70 , R3E> 0 , Et 70

r all t.

Step 4 : FURTHER ANALYSIS OF MODEL②" "

Proposition 1 : Say we had Vassilis ' At under Model① and

nzt and R3E were not constant over t .

Then I :-. fine %) is a full rank matrix(provided logf) does not introduce any linear dependence) .

× 11 a YPROPOSITION 2 : Say we had Vassilis

' A3Rfi and A4GMiidThen

, TIM ,and Effy't=rIT for a positive

constant scalar v.

Under Model②,. . .

byPropositions 1 and2 and Assumption1,the OL5 estimator for

0 := If, ,pa ,fg)'is the best linear unbiased estimator

due to the fans -Markov theorem. If ,additionally ,we

assume by- normality for 4. , then the 04 estimator of0 is the best unbiased estimator .

UNDERGRADQ2 Say y f. + Ifpnnht + Et , for t-f.li STYLE

Thenye . ,=p.

+ IFfikht - I + set - I , for t -- 2 ,- . . ,T .So the"FD model" is defined as

by; II.fondant + bet , for t -- 2 . . . . ,T ,where by =

yEye . ,and Aunt and AG are defined

analogously .

Say y= if + Xp + E ,where

%stfRADX is a Txk matrix ; STYLE

{iq.iq ' of us : Istria are equivalent)

Then the" FD Model" is defined as Ay where A is . . .

let's visualise together . . .DATA y i n , . .

-

[ IGNORE =) Y , I 211

[I I 0 . . .0] yz I K12

f. 0 -I 1 . . . 0 ] yz I K13

i. : : .. .

YT- I '

K1T-1YT/KIT

n⇒÷¥÷¥

So if we define a f-1) xT matrix

A := - I 1 0. . .

O O

O -I 1. . . 0 0

: : : : :

0 00 . - .- I 1

then the transformation Ay=Axp + A-E) yields a"FD model

"

as required .

Note.Ai :O

. Infact , Afi) :O for any tatter .

(b) Say the GM assumptions held in the"

levels model"

.

At : fi XI has rank ktlAL : y = 4×31 +

E,EIEI = 0

A3F : Li XI is fixed in repeated templesA-4am : E1E = FIT

- Thenthe FWL theorem and the Gfm theorem tell us

%,= 4'M X'Mig is the BLUE .

In other words, ttfvalp) - varffo.SI/70 ,

for tone WE § ,and where it

, f.1 denotes the smallest

eigenvalue .

But then we can easily compare

foffAxIAx)IIAx)IAy) = 4'A'A xIx' A'Ay ,

with %, because :

i )a og

is linear sink for B : 'A'AXP X ' A'A,

Irsas =By .④) ffs as is unabated since

F-(pas as) =p + EH'

A'AXTX ' A'A e) =pand so the Gm theorem tell us that

''ilvarlets..) -varlp.is/70 ,that is

, ④↳ is a " better"LUE than Paso↳ .

+ APPENDIX A : COMMON QUESTION FROM STUDENTSRAGVIR

,I DON'T UNDERSTAND WHY YOU ARE USING ALL THESE

FORMULAS WITH"

Mi"IN THEM

. HELP?"

Consider our model again : y = Z0 to where Z :-.fi X /,

and 0 :-. BYThe 04 estimator would be & : =#E)

'

e'y , a 4+4×1

vector. If we compare Vario) with Var(fans) ,

that would justbe silly because the former is CKH by htt) and the latter is

Ck by k) .We can't even compute Var@l-Varlfb.rs as) .

• If we compared Var (④xj'

X'y) with Var fools) , we

are indeed able to do so,but HxjX'y is Not the right

way to estimate f.You can't just ignore i.• The only correct way is to run a partitioned regression .

APPENDIX B :PROFESSOR RAGVIR'S EXTRA QUESTION

IMAGINE You ARE THE EC402TA .FOR 5 MINS

.

YOUR STUDENT ASKS You :

Clearly , pm as & foes are both as estimators .To see this , note :they both have the usual #

'II' I' 5 form ,

where forLEVELS : if :-. Miy ;

I := Mix ,and for

FD : if:-. Ay ; I := Ax .

That is, Pa,

= fmiximixjlm.tl'

Mig = ④ 'mixI'

X'Miy , and

pas ai x)' AXTYAX)

'

Ay = Y' A'AX5

'x' A' Ay .

The 61M theorem tells us that Old estimatorsare BLUE (under suitable assumptions) .

However,

here,we are using the GIM theorem to saythat

one as estimator is better than another?

Strange , isn't it ? Explain ,please !

Q3.

. ZE1R", twµz ,Vz)

- constant qE Rn

4)Define w :=q'Z .

Find EIW) & Varlw) .

F-4) = q' Eft) = q'µz .

Var 4) = of Var g=q' Vzq .

lifeii.

" since there exists a a¥%£.LY#efthEehY5IEisk aperfect linear relations p .

You don't need to read this (sketch of a)proof . Last year students

(wanted me to give extra intuition for Vassilis

' solution so I came upwith the reasoning below .

You can just stick to the official solution if you'resatisfied with that .

V

④) let's assume WL06 that there exists just one vector q gosit

.Var E) = 0

.

Then, recognise that

i. Var loft) = of Vzq = 0 iff Vzq=0 for qto ;2

.

Null(Ve) : = { qe Rn : Vzq = 0 } and soNullity(Vz) := dim{NullUHf- I ;

3. By the rank - nullity theorem ,

rank t NullityIvz) = n.

i. By 1,43 ,rank = n - I

.

APPENDIX C : VASSILIS' Famous ASSUMPTIONS

RAGVIR :

VASSILIS :

"

Assume Al,A2 ,A3Rfi , A4bM,A5N .

"

GREENE : f. . . has his own set of"

A"

s. . .]

OTHER : f. c. his/her own method . . .]

SUMMARY : We are all saying the something !