6.6 ARIMA Model. Autoregressive Integrated Moving Average (ARIMA ) Model. ARIMA(p, d, q) Process

Dimensione: px

Iniziare la visualizzazioe della pagina:

Download "6.6 ARIMA Model. Autoregressive Integrated Moving Average (ARIMA ) Model. ARIMA(p, d, q) Process"

Viola Turco
5 anni fa
Visualizzazioni

1 ARIMA Model Autoregressive Integrated Moving Average (ARIMA ) Model ARIMA(p, d, q) Process φ(l) d y t = θ(l)ɛ t, where d y t = d 1 (1 L)y t = d 1 y t d 1 y t 1 = (1 L) d y t for d = 1, 2,, and 0 y t = y t. ARIMA(0,1,0) Model Consider the model: y t = y t y t 1 + ɛ t, ɛ t N(0, σ 2 ), y 0 = 0, which is rewritten as: y t = ɛ t + ɛ t ɛ 1. E(y t ) = 0, γ(0) = V(y t ) = σ 2 t, γ(τ) = Cov(y t, y t τ ) = E(y t y t τ ) = σ 2 (t τ),

2 129 which implies that γ(τ) is time-dependent. = y t is not stationary. ρ(τ) = Cov(y t, y t τ ) V(yt ) V(y t τ ) = t τ t t τ = t τ. t That is, ρ(τ) gradually decreases with slow speed. 6.7 SARIMA Model Seasonal ARIMA (SARIMA) Process: 1. SARIMA(p, d, q) φ(l) d s y t = θ(l)ɛ t, where s y t = (1 L s )y t = y t y t s.

3 130 s = 4 when y t denotes quarterly date and s = 12 when y t represents monthly data. 6.8 Optimal Prediction 1. AR(p) Process: y t = φ 1 y t φ p y t p + ɛ t (a) Define: E(y t+k Y t ) = y t+k t, where Y t denotes all the information available at time t. Taking the conditional expectation of y t+k = φ 1 y t+k 1 + +φ p y t+k p +ɛ t+k on both sides, y t+k t = φ 1 y t+k 1 t + + φ p y t+k p t, where y s t = y s for s t.

4 131 (b) Optimal prediction is given by solving the above differential equation. 2. MA(q) Process: y t = ɛ t + θ 1 ɛ t θ q ɛ t q (a) Let ˆɛ T, ˆɛ T 1,, ˆɛ 1 be the estimated errors. (b) y t+k = ɛ t+k + θ 1 ɛ t+k θ q ɛ t+k q (c) Therefore, y t+k t = ɛ t+k t + θ 1 ɛ t+k 1 t + + θ q ɛ t+k q t, where ɛ s t = 0 for s > t and ɛ s t = ˆɛ s for s t. 3. ARMA(p, q) Process: y t = φ 1 y t φ p y t p + ɛ t + θ 1 ɛ t θ q ɛ t q (a) y t+k = φ 1 y t+k φ p y t+k p + ɛ t+k + θ 1 ɛ t+k θ q ɛ t+k q

5 132 (b) Optimal prediction is: y t+k t = φ 1 y t+k 1 t + + φ p y t+k p t + ɛ t+k t + θ 1 ɛ t+k 1 t + + θ q ɛ t+k q t, where y s t = y s and ɛ s t = ˆɛ s for s t, and ɛ s t = 0 for s > t.

6 Identification ( ) 1. Based on AIC or SBIC given d, s, we obtain p, q. (a) AIC (Akaike s Information Criterion ) AIC = 2 log(likelihood) + 2k, where k = p + q, which is the number of parameters estimated. (b) SBIC (Shwarz s Bayesian Information Criterion) SBIC = 2 log(likelihood) + k log T, where T denotes the number of observations. 2. From the sample autocorrelation coefficient function ˆρ(k) and the partial autocorrelation coefficient function ˆφ k,k for k = 1, 2,, we obtain p, d, q, s.

7 134 AR(p) Process MA(q) Process Autocorrelation Function Gradually decreasing ρ(k) = 0, k = q + 1, q + 2, Partial Autocorrelation Function φ(k, k) = 0, Gradually decreasing k = p + 1, p + 2, (a) Compute s y t to remove seasonality. Compute the autocovariance functions of s y t. If the autocovariance functions have period s, we take (1 L s ), again. (b) Determine the order of difference. Compute the partial autocovariance functions every time. If the autocovariance functions decrease as τ is large, go to the next step. (c) Determine the order of AR terms (i.e., p).

8 135 Compute the partial autocovariance functions every time. The partial autocovariance functions are close to zero after some τ, go to the next step. (d) Determine the order of MA terms (i.e., q). Compute the autocovariance functions every time. If the autocovariance functions are randomly around zero, end of the procedure.

9 Example of SARIMA using Consumption Data Construct SARIMA model using monthly and seasonally unadjusted consumption expenditure data and STATA12. Estimation Period: Jan., 1970 Dec., 2012 (T = 516). gen time=_n. tsset time time variable: time, 1 to 516 delta: 1 unit. corrgram expend LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]

10 gen dexp=expend-l.expend (1 missing value generated). corrgram dexp LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]

11 138. gen sdex=dexp-l12.dexp (13 missing values generated). corrgram sdex LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor] arima sdex, ar(1,2) ma(1) (setting optimization to BHHH) Iteration 0: log likelihood = Iteration 1: log likelihood =

12 139 Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood = (switching optimization to BFGS) Iteration 5: log likelihood = Iteration 6: log likelihood = Iteration 7: log likelihood = Iteration 8: log likelihood = ARIMA regression Sample: Number of obs = 503 Wald chi2(3) = Log likelihood = Prob > chi2 = OPG sdex Coef. Std. Err. z P> z [95% Conf. Interval] sdex _cons ARMA ar L L ma L /sigma

13 140 Note: The test of the variance against zero is one sided, and the two-sided confidence interval is truncated at zero.. estat ic Model Obs ll(null) ll(model) df AIC BIC Note: N=Obs used in calculating BIC; see [R] BIC note. predict resid, r (13 missing values generated). corrgram resid LAG AC PAC Q Prob>Q [Autocorrelation] [Partial Autocor]

Documenti analoghi

Constant Propagation. A More Complex Semilattice A Nondistributive Framework

Constant Propagation. A More Complex Semilattice A Nondistributive Framework Constant Propagation A More Complex Semilattice A Nondistributive Framework 1 The Point Instead of doing constant folding by RD s, we can maintain information about what constant, if any, a variable has