Term 2 Lecture notes EC226 Econometrics Mastering 'Metrics - Score a first too

Wat Serial Correlation
Distribution
of Coefficient in
Dynamic Time Series Models .
(vs) I :




Estimation of time sevel serial condation
of form of linear dependence
:
Presence over
Ols model of Yo
some




Recap T, - time
for some series
, zz

The autocorrelation Pictoral representation of which
this lineor
dependency, is
:


Function (ACF)
of C againstj) form of
I plots values
measured in the a correlation between Ez and Exx




Moving
to
T S .




for different 12 .




correlation
model
1) O
zen I Vk
--

cor(zz
(2
Co
that is : cor r za
=
Et , -
n
-


li I
↳
-



O
-
-


+, -




v(zz)V(te -

k
I v(zi) -
y




It Bo B YE1 from lag of f=1 191 ju
-
+ +
Ef aise dependent variable where and l
=
, issues ·
, .




,




(i)
E(Et/yt 1) = 0 =
t(dely y ,
,
. .




+
ye 1 ,
ye
...

y 0
ez ,
2
+ n =
-o as h get
bigger ; fo =
)(z +, 7) :




↳




&
!
strict
enogeneity is
r possible
Consider A&F in P
if
types Models :




4
(V(((y 1) +
=
0 t 1) White Noise
Ptypes of Model

MA ARMA
Autoregresive (AR) AR ;
Wil Cor (Ez , Es (y) = 0 Ets
:
;

roite
/0 04 (MA)
White
proce
(iv) Et 14 + 3)
Honing Average
-
N

large
,




d
enogeneity
las we
4) Autoregrenie Moving average
(ARMA) .




Samlim
* ~
- As
enogeneity
·
strict isn't possible -o
we replace (i) / :
Autocorrelation Function & White Noise Process (vi)
Ii) assumption of temporaneom enogeneity : [(dily , Yo , Ys .
. . .




.y .)
) =0 White Noise Process :



in words -
expectation of er ror

term is unconditional/unrelated on all value
of Y that happened up model :
Ex
=

Ex
-
(E) = 0 ↳
(4 k ,
= 0 to
until the previous va l u e ·




VIEl :
83 EWN(0 04 ,




station see



Straitlas umption o wedevel
vie


if ze E
-
E(zy E(4y) 0 constant
·


= = =
Mear
all

-
came for
↳adchen E(zz) Elke
-
z+ M+ 4 M+
= =




Mean


② v(y)) 5y V(z) = constant
->
+ t nuance
·
= -




③ (yt ytn) Un ((z 2)
>
-
ou
,
=
,
= 0 to
4 *
whet rol voe
previous
.

some




>
-
graph indicates :
if the Mocen in
"shocked" today ,
100 % of the


(W NI shoch remains
today but in a l l
future perod
WEAA
-




-ACF
. ,




DEPENDENCY .
There is to the shoch whatsoever -




no
memory

condition :
Corlyt Yen) Un - 0 =
as h get bigges
↑
,




↳

↳
Lov
we
between
t a ke Gobs
observations
.
must
get smaller
,
the further it on

Each , dissipated
immediately is
- rent food .




creater similar condition to
sampling
a random
.

(1) Find Mat : 1 , N (p V (b , 1)
older 1 Model in which
cr of proces was
determined
by for.
val u e of
-of
,

: He
-



.
process



E AR (1) Model (vi) i
I'll
Hypother's Austing should also not i nv i l l e fitats ,
but the Xtat .




an add assumpt ou
·




the
-(i
ou
v(Ge & Could i e


*
ill
small a re


fol
a


PEz1
ols is bione
coefficien long is
large Et
conditions
of as the +
·
a re as
sanes him .




>
-
for I t to be stationor .




where it in a WN
process an d 10/11 (and have process in
stationary) -




Notes.
p
=

0 - Le derivation in Lecture


Note :
useful for proofs in to know it is a
purely random pocen & mated to all
including
past value of Ez

, continued
.




III
- -




Diagrammatically
-




·

p ,
10 ,
Gro

Diag i to



# Torammatical
goin decay zuo
9 0 20
·


. ·
, ,



f ·

if the proces is shocked today ,
100 % of the short is remembered

today
,
at period I
, of is remembered
,
ther
for every find pl ·
4. + & = complex roots
.

3
for
= 1 2 s
j , , ....,




O
Lautoregrenine
parameter
& o

back
low sucoil you agent shocked
①reces been GENERALIZATION AR(3) :


path Given joule
2 :



E to es.

of the path
.




out


= 47 ,
+
Pret +
-3 +
Et



& o
process
↳
autoregressive parameter/coefficient .

ARP - E =

4, e + -2 +... +
Ptp + -C.N
27




Defining the
lag operator 1 ,
s .
A (z =
E -,
and 1'z ,
=
zej we
in this c a re -y
talked written as :




can write th Model as : VIze =
Vo =
Divi+ UntPatz . . .

&POP Note-Make sure

what each
to understand

of the

letters
V .
=
4 , 80
+
Prk +
&K +... +
PUP-1
Mear




=
=

P(Ez +
Ex
=
12 (l PH) -
= 4
+
=
ze =
I- PLT'Et 82 0, 8 =

.
+
aro +
934 +... +
%000-m


(PL)" &L P2
+
PL+ in which
. . .




Now : =
1 + + case :
...,




024 03 )Et E 94 + En P E 928j2 + Pojp ja Pt
°
+ + =
+ + +
Vi
=
&Vie + . .

>
+ ...
- - .




-




this in a MAIO)

be solved back substitution or in the first part Yule-walker MOVING MALI) MA(L) MALG)
by MODELS
can like AVERAGE ,
:

, ,

EQ
. weighted a r.
of new. random shocks
.
(4) =
0


MA(1) "E+=
G 08 Et
-
this
&
+
,
in case :
v(Et) =




cor(42 4) 0
jf0
=




Auto Regressive (AR2) ; AR(3) ; AR(p) Models Ot(4+ ) + E(at)
,


2 -


(E(7t
=
E(04 + ,
+ Ex) =

,
=
0/


(V(zd) =
Vo
=
(1 + 8462
Ex in
stationary
-
18 ,
+
02/
ARLI -
zz =
P ,
z
+ + $277- +
Et (((zt =zi) ,
=
y
=
062


WiN is to be (4) (zz 2) 0
and the assumed
stationary Lov d
in zt
=
where
Et
=
a
proces process ,




%
E(e) E(zz j) V(ze V(zz j) deine
AY
ht to
yield = /184 44 Lives O
=
equations : So =

f
=
;
=
so and = :
e =
,
-




E
E(z) (1 4, q)t(t)
=
- =
0
joll
V (z) d =
.
=


difl Pek + s Wote : the MA(1) can be written as an
infinite AR frocess to knows as
atibility
i



Ywell
Cor ( +) ,
=
0 .
=

06 +
Put
.




Co(z + K Pik ,
z = =

Pik MA(2) >
-
En
= 0 4 , .,
+
02 & 2
+ Ex
,




((zz ,
7 z- 3)
=

Us
=


Pik +
P28 simlor
yules-walked equation -




Diagrammatically :



Scen
By for
3

Puls the shoch
on

P6 (1) 2
-
extension MA(z) remembers
periods
- =

i
-




.
.
. >
- MA : When shocked remembers
the shock for I period
. MA(4) remembers shorth for q peroch length of M .
Al -




/ +0
3-Wf =
Ivonance .
-C- Piet Pulju joz
.