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⇐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 [email protected] 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 !