EC402 (4/22) - MT WK4 - VASSILIS PS3

2AWIR'S SPEAKING NOTES (UNOFFICIAL 6N

TODAY's CLASS TRY AT HOME / ASK IN D. It .= ←

QI Easy ,but conceptually important .Q2 (a) To DoQ2 4) To Do Q 341,434) , Q4-

Very good for gaining confidence with034) To Do algebra ; but no new tricksfor me to teach here .

Q5 MAYBE

Q6 To Do

0241 ye .- ftp.attst.t-f.itGiven @ =

'

Hy , find explicit expressions for §, ,fz .

- let X = fi al where µ .

. . .

,')'

and a -- la , , . - in i y=Iy , , . . . ,yH'

'

'

= if:(1¥ Ef£ ,where is =

iioia-iarii.HNg) i. f-µ: Its

⇒ I. =fIEaaI÷EE If ?Eµ IS

& fine ⇒KEY.EE# I

REWRITING IN A MORE FAMILIAR

WAYNote that T If,

In , the-D) =j. .=TIE

,hey+-TE51

ease do try to derive if not obvious . The only trick is to realise II.Zt =TE1

Clearly then ,T.IE?at-Ef=TfIIyE-TEYThus

, E-- EYE⇒of I II.He -E5and &, . . .

on next slide.

III.→ II. - I:c . E.ii.a.µwhat if we + f- TEJ in the numerator ?

i.e. Aff -- EIII e- this + Friis - III.Either= g-t.E.in?--iEf-n-IiIIn.ye-TayI:.pi---y-pI

Q2Ib) liven varlff.io I find explicit expressions for var ftp.varlfdand Gulf ,E) .

From partµ , wehave thatz

- I 2

• UH = fine . I:/ Is ,where D= i' in'n - innii

Tz

= II nE - But 0t= I1- E.a . i t.EE - II.⇒'

= E.IE/T-ie ,I- a i I E.int-M

(Explicit answers not written here but can be read off the above .)

Quai var III = E EgaI/ T II.H-E5

T.iq a)if E- a inE⇒⇒+ 2 II.

,atE - TE2]

I e-N' + To]

= of + FITIII.-It

Q3lb) see 02lb)

434) To show : EIEI If = Ef EYE) = 04.21 .

/

Why is this actually a really interesting result for us?

Slept : Define My :-. IT - xlx'xT'X'

, {÷'

X'yand E- y- Xp .

Then,I = Mxy = My E for model y =Xp +E .

> ifthese equalities are not obvious ,then do hr to derivethem to helpsolidify you? intuition .

Step2.Next, E if = EF44IM,e)title'M×sfstate > the

wh? EITRLE'MX4Y! false'M×3ftIfEfssYM×ftwo red ¥

?Tr { FIT My} = ETr{M×f = 8Tr{I , - xlx'x) if"

why?"s

explicitly in = o' fifty - trfxlx.xtxff-oftrhI-if.ir/HxlxiII/

your answer. whiff,{It} - Tr⇐+3) = EEK) .

Step3 : For us ,kid : . EfE' E) = ELI2) as required .

Q4(a) We have y= Xp + E = ynts .

Show that TSS -- Ess +Rss

.

Slept :y'y +Ellyn+E) = yniy + sty + y"Et Est .

Note that j'Etty? It ^ =dM× × pants?oif I

i.

y'

y = ydty + EE .

Step 2 : Note that in = I'M,E =µ if's = 0 if × contains a constant .

Further, if if = 0 ,

then i = i so if = 5 .

Step3 : y'y - T5 = y" is -TF2-1EI- 2

⇒ III. a of his'

= IIII - a II. joy + T5+

.

%

⇒ Infuse -IT = Eily, + IT IE ,or THEIRS

.

04lb) show :RE fcqyt.ge/f→ note typo in the question .

let y*:=ly - is) , and yd* ty - 5) , still assuming X contains a constant .

siepi.ie?--:Es--sj= I .

y*'y*

Slept . y4'yn*=f*'If - jI=J*'fy-E -j] - y*'fy±Ef .Steps . j*E=fxHxTx'y*Jr×e=y4 ' X' xlx'xYx')E=O.

Steps 42,3 ⇒ RZ = (5*15)'

= 6%1 ye.SI .

*hittin

05. y = Xp + E , where X is a Txk matrix . Suppose (WhoG) the last

regressor is scaled by d.i.e. Q:=XA where ⇐FIFI

,

'ISaywe use OLS : y = xp +E. Then we scale X by A. That is ,

we fit y=QIt nf .If we force 1=4 (which is another wayof saying we use as again) ,then clearly : xp = Qd = XA I ⇒ & = Atsonata nip ? It

i. %. =/& ; , j-

- I,. . .

,K- 1

Rift , f- K .

XK is measured originally in f- million ,and now in foods , i.e. A-1000 .

if its confusing , see this :

/salarykm) salary¥000)

Jeff Bezos 0.3 300

Ragvir 92 92,000

i. e. 1=1000 here

So the regression coefficientswill be sealed by a factor of0.001 relative to the original results .

The estimated marginal effect on y of"I unit

"

change in XK will bethink

big← f-µ when the"

I unit"is" 1 million

"

,but it will be f- Kk= FK4000144 when the " 1 unit" is just

" 1 thousand". ¥nk

SmallerII

" "

Qb.

A3Rmi : E XI - E - STRONG

ASRSM : Efsiirh.it/=0fnI?Ih.NTXkNTXIAssume:y=Xp+E,EIsf=0"WEAK

"

NTXI KX1

I E11µ. ??•

,

ii: t.i.me,In, EN1

y.no ?m Efaitsis) for all if u )" " IYNT ENT

ey .unobserved heterogeneity

(a) A3F ⇒ A3R#A3R¥ .

4) yi fait + sitF- Init sit -s¥%?j. o ⇒ n.sn/lmi,Asrsm .

(b) Say R=I and ni yit - I- To verify A3Rmi , we need to Condition on leads and lags as well asthe current value of regretsors . That is ,check E1Eit / f- . isit-2 , bit -1)Yi I]but this is impossible since realisation of snit affects

Y it for tsi .

If still confused , here's tome extra intuition"

We can't have it both ways"- either we treat sit as random

in which case we cannot claim all regressor valves / leads aregiven .

QR we treat sit as given so you can sensibly Condition ,but then why are we asking about Eff of a non - randomobject ?

Next consider AsRsm.

Say yie-pyie.it fit and of course Efait sie -If 0 .

Then,E fyit.IEif = F- f(fyit.at sit -I sit) f- 0 .

⇒ A3µi ,A3R④ .