代写代考 1 On the board Lecture 4

1 On the board Lecture 4
General, de-meaned, ARMA(1,1):
yt = 1yt1 1″t1 + “t: Letís get variance and autocovariances of this process:
Var(yt) = Var(1yt1 1″t1 +”t)

Copyright By PowCoder代写 加微信 powcoder

= 21V ar (yt) + 212 2112 + 2:
So: 0=Var(yt):
Need j1j < 1 for stationarity. First autocovariance: Cov (yt; yt1) = = = 1 = Letís get second covariance: Cov (yt; yt2) = = 2 = j = "1 = = "2= "t= Letís forecast 2 periods ahead: Et (yt+2) = = Et(yt+j) = y11y0+1"0 y2 1y1 + 1"1; yt 1yt1 + 1"t1: Et (1yt+1 1"t+1 + "t+2) 1Et (yt+1) ; 1Et (yt+j1) ; j > 1:
= 1+21 2112: 1  21
Cov (1yt1 1″t1 + “t; yt1) Cov (1yt1 1″t1; yt1)
1V ar (yt) 12;
Cov (1yt1 1″t1 + “t; yt2) 1Cov (yt1; yt2) ;
1 j1,j>1:
Forecasting with ARMA(1,1):
Et (yt+1) = Et (1yt 1″t + “t+1) = 1yt1″t:
But,howdoweget”t? Wegetthisbyassuming”0 =E(“)=0andy0 =E(y)= 0 (the latter is only true in this case as we assumed a demeaned ARMA(1,1).
Note that “t  N 0; 2
is the standard assumption, estimate process by max- imum likelihood.

1.1 Useful alternative present value equation
The present value formula says:
Pt = P1 Et (Dt+j) or Pt = P1 Et (Dt+j=Dt) :
j=1 1 + Et (Rt;t+j) Dt j=1 1 + Et (Rt;t+j)
Highly non-linear, hard to relate to linear prediction models. Campbell and Shiller introduce useful log-linear approximation that is accurate and allows us to operate in linear world.
Start with the deÖnition of gross returns:
Here, we have assumed that Dt > 0. Always true for the market, some Örms
have zero dividends. HW gives alternative model for this case. Take logs:
rt+1 = ln epdt+1 + 1
pdt +
dt+1;
where
dt+1 = ln (Dt+1=Dt), pdt = ln (Pt=Dt). Recall Örst-order Taylor ex-
pansion of f (x) around x0:
f (x)  f (x0 ) + f 0 (x0 ) (x x0 ) :
Dothisforlnepdt+1 +1
aroundEepdt
:
lnepdt+1 +1
 lnEepdt
+1
+ Eepdt
pdt+1 pd

Pt+1 + Dt+1 Pt
= Pt+1=Dt+1 + 1  Dt+1 :
where  is a constant term and  = E(epdt ) . If average P/D ratio in sample
= 0 + 1pdt+1;
0 1 E(epdt)+1
is 30, then 1 = 30=31 = 0:97. Now, we have a log-linear return equation: rt+1  0 + 1pdt+1 pdt +
dt+1:
This is due to Campbell and Shiller (1988). Derive and expression for pdt by iterating forward:
pdt  constant +
dt+1 rt+1 + 1pdt+1
= constant +
Pdt+1 rt+1 + 1 (
dt+2 rt+2) + 21pdt+2
= constant+1j1(
d r )+limpd : j=1 1 t+j t+j !1 1 t+
Note: lim!1 1 = 0. If pd ratio is stationary, lim!1 1pdt+ = 0. pd =constant+P1 j1(
d r ):
t 1 t+j t+j j=1

Investors trade in the market and use their expectations:
pd =constant+ P1 j1E (
d ) P1 j1E (r ):
Letís assume that xt = Et (rt+1) E (rt+1) is an AR(1) process. Then:
We can now evaluate:
t 1tt+j 1tt+j j=1 j=1
Et (xt+1) Et (xt+2) Et (xt+j)
= 1xt; = 21xt; = j1xt:
P1 j1E (x ) = P1 j1j1x 1 t t+j 1 1 t

程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com