Correlogram, ARMA Models, ARMA Forecasts

1. Correlogram

1.1. definition

1.1.1. The correlation between ut and ut-1

1.1.1.1. CORR(ut, ut-1) = E(ut,ut-1) / E(ut2)

1.1.1.1.1. 自己前后一位的, 整体关系

1.1.1.1.2. Where we have assumed homoscedasticity so that var(ut) = var(ut-1)

1.2. estimate of the correlation

1.2.1. correlation coefficient

1.2.1.1. r1= Eut^ut-1^ / Eut^^2

1.2.1.2. rs= Eut^ut-s^ / Eut^^2

1.2.1.2.1. AR(1) ut = put-1 + et

1.3. The autocovariance/autocorrelation function:

1.3.1. E(yt - E(yt))(yt-s - E(yt-s)) = Rs , s = 0, 1, 2, ...

1.3.2. Js = Rs / R0, s =0, 1, 2, ...

1.3.3. a plot of Js against s = 0,1,2,.. will give the acf or corrlogram

1.4. If the correlogram declines exponentially it suggests that the errors follow an AR(1) process

1.5. Test Stetistics for autocorrelation

1.5.1. Q= T * E"k=1~m" Jk2 ~ X2m

1.5.1.1. Box- Pierce

1.5.2. Q*= T(T+2) E"k=1~m" Jk2 / (T-k) ~X2m

1.5.2.1. Ljung -Box

1.6. Partial Autocorrelation Function/ PACF

1.6.1. The pacf measures the correlation between an observation k periods ago and the current observation after controlling for correlations at intermediate lags

1.6.2. Partial correlation coefficient at lag 2 (J22)

1.6.2.1. J22= (J2-J1^2) / (1 - J1^2)

1.6.2.1.1. J1 and J2 are the autocorrelation coefficients at lags 1 and 2

1.6.3. PACF (or partial correlogram) plot of Js or ps (correlation between xt and xt-s) against s eliminating the effects of the intervening values of xt-1 through xt-s+1 (as such, for example, the partial autocorrelation between xt and xt-2 for an AR(1) is equal to zero)

1.7. Autoregressive (AR) and Moving Average (MA) Processes

1.7.1. yt = u + E"t=1~p" 9i * yt-i + ut

1.7.1.1. yt = u + E"t=1~p" 9i * L^i * yt + ut

1.7.1.1.1. yt = u + B(L)ut

1.7.2. therefore the autoregressive model

1.7.2.1. b(L)yt = u + B(L)ut

1.7.3. application: An ARMA Model for Cap Rates

1.7.3.1. NCREIF all property cap rates

1.7.3.2. Sample: 1Q1978 – 4Q2007

1.7.3.3. First differences are taken