Lag 1 the acf pacf always at lag 2 22 2 12

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Unformatted text preview: unction (denoted τkk) (cont’d) • The PACF is useful for telling the difference between an AR process and an ARMA process. • In the case of an AR(p), there are direct connections between yt and yt-s only for s≤ p. • So for an AR(p), the theoretical PACF will be zero after lag p. • In the case of an MA(q), this can be written as an AR(∞), so there are direct connections between yt and all its previous values. • For an MA(q), the theoretical PACF will be geometrically declining. 27 ARMA Processes • By combining the AR(p) and MA(q) models, we can obtain an ARMA(p,q) model: where and θ ( L) = 1 + θ1L + θ 2 L2 + ... + θ q Lq or with 28 The Invertibility Condition • Similar to the stationarity condition, we typically require the MA(q) part of the model to have roots of θ(z)=0 greater than one in absolute value. • The mean of an ARMA series is given by • The autocorrelation function for an ARMA process will display combinations of behaviour derived from the AR and MA parts, but for lags beyond q, the ACF will simply be identical to the individual AR(p) model. 29 Summary of the Behaviour of the ACF for AR and MA Processes An autoregressive process has • a geometrically decaying ACF • number of spikes of PACF = AR order A moving average process has • Number of spikes of ACF = MA order • a geometrically decaying PACF 30 Some sample ACF and PACF plots for standard processes The ACF and PACF are estimated using 100,000 simulated observations with disturbances drawn from a normal distribution. ACF and PACF for an MA(1) Model: yt = – 0.5ut-1 + ut 31 ACF and PACF for an MA(2) Model: yt = 0.5ut-1 - 0.25ut-2 + ut 32 ACF and PACF for a slowly decaying AR(1) Model: yt = 0.9yt-1 + ut 33 ACF and PACF for a more rapidly decaying AR(1) Model: yt = 0.5yt-1 + ut 34 ACF and PACF for a more rapidly decaying AR(1) Model with Negative Coefficient: yt = -0.5yt-1 + ut 35 ACF and PACF for a Non-stationary Model (i.e. a unit coefficient): yt = yt-1 + ut 36 ACF and PACF for an ARMA(1,1): yt = 0.5yt-1 + 0.5ut-1 + ut 37 Building ARMA Models - The Box Jenkins Approach • Box and Jenkins (1970) were the first to approach the task of estimating an ARMA model in a systematic manner. There are 3 steps to their approach: 1. Identification 2. Estimation 3. Model diagnostic checking Step 1: - Involves determining the order of the model. - Use of graphical procedures 38 Building ARMA Models - The Box Jenkins Approach (cont’d) Step 2: - Estimation of the parameters - Can be done using least squares or maximum likelihood depending on the model. Step 3: - Model checking Box and Jenkins suggest 2 methods: - deliberate overfitting - residual diagnostics 39 Some More Recent Developments in ARMA Modelling • • Identification would typically not be done using acf’s. We want to form a parsimonious model. • Reasons: - variance of estimators is inversely proportional to the number of degrees of freedom. - models which are profligate might be inclined to f...
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