1 On the board Lecture 3
AR(1) with intercept:
rt = 0 + 1rt 1 +”t; E(rt) = E( 0 + 1rt 1 +”t)
= 0+ 1E(rt); E(rt) = 0 :
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Recall AR(1) (unconditional mean zero so no intercept, for simplicity):
xt+1 = 1xt + “t+1: Forecasting at various horizons:
Stationarity:
Et (xt+1) = Et (xt+2) = = Et(xt+j) =
Et( 1xt+1+”t+2) 21xt;
Var(xt+1) = Var( 1xt)+Var(“t) = 21V ar (xt+1) + 2
2 Var(xt+1) = 1 2:
j = 1j 1;
Corr (xt+j ; xt) = 1Corr (xt+j 1; xt) :
Stationarity from autocorrelation function:
j 1j 1 = 0; (1 1B)j = 0;
where B is the backshift (or lag, L) operator. We need j 1j < 1: We could solve: 1 1z = 0:
Solution, z = 1= 1. So, z needs to be bigger than 1 in absolute value. Forecast error:
Autocorrelation of an AR(1):
V art (xt+1) =
= V art ("t+1) = 2;
Vart( 1xt+"t+1)
V art (xt+2) =
= 21V art (xt+1) + 2
V art ( 1 xt+1 + "t+2 ) = 1+ 212:
1.1 AR(2) process
AR(2) with unconditional mean of zero:
Forecasting:
Et (xt+3) Autocovariance function:
Et (xt+1) Et (xt+2)
= Et ( 1xt + 2xt 1 + "t+1) = 1xt + 2xt 1:
= Et ( 1xt+1 + 2xt + "t+2) = Et ( 1xt+1 + 2xt + "t+2) = 1Et (xt+1) + 2xt:
= 1 ( 1xt + 2xt 1) + 2xt = 21 + 2 xt + 1 2xt 1:
= Et ( 1xt+2 + 2xt+1 + "t+3) = 1Et (xt+2) + 2Et (xt+1) :
1xt 1 + 2xt 2 + "t:
E (xt j xt) = 1E (xt j xt 1) + 2E (xt j xt 2) + E (xt j "t) j = 1 j 1+ 2 j 2;forj>1:
The autocorrelation function simply divides this by variance:
j 1j 1 2j 2 = 0: 1 1B 2B2j = 0;
(1 !1B)(1 !2B)j = 0:
Use same trick as for AR(1) above. If j!1j < 1 and j!2j < 1 we have
To establish stationarity:
stationarity. Now, solve: Which gives:
1 1z 2z2 =0: q 2
z1;2 = 1 1 +4 2: 2 2
Ok, we have stationarity if z =! 1 > 1 and z =! 1 > 1. Note that z 11 22
could be complex. In this case, we have cyclical behavior.
1.2 Moving Average models
Consier special case of AR(1):
xt + 1xt 1 + 21xt 2 + 31xt 3 + ::: = “t:
Consider the same process, one lag back in time:
xt 1 + 1xt 2 + 21xt 3 + ::: = “t 1: Multiply the last equation by 1 and subtract latter from former:
xt + 1xt 1 + 21xt 2 + ::: 1xt 1 21xt 2 ::: = “t 1″t 1 xt = “t 1″t 1:
This is an MA(1) model. I can add an intercept:
rt = + “t 1″t 1: What¨s the unconditional average:
E (rt) = : What is its unconditional variance:
Var(rt) = Var(“t 1″t 1)
Autocovariances:
= 2 1+21: What about forecasting:
Et (rt+1 ) = Et (rt+2 ) =
Recall, an AR(1) can be written:
xt = P1 j1″t j j=0
Et(+”t+1 1″t) = 1″t:
Et ( + “t+2 1″t+1) = :
= E ((“t+1 1″t) (“t 1″t 1))
= E ((“t+2 1″t+1) (“t 1″t 1))
I.e., it¨s an MA(1).
A very common predictive model is the ARMA(1,1):
xt = 1xt 1 1″t 1 + “t:
Consider an ARIMA(1,1,0) for yt. Then the di¡ìerence yt = yt yt 1
follows an AR(1): So:
yt = + 1yt 1 + “t:
yt =+yt 1 + 1yt 1 +”t:
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