Moving Average Model • Moving average model of order q (MA( q )): x t = w t + θ 1 w t − 1 + θ 2 w t − 2 + · · · + θ q w t − q where: – θ 1 , θ 2 , . . . , θ q are constants with θ q � = 0; – w t is Gaussian white noise wn(0 , σ 2 w ). • Note that w t is uncorrelated with x t − j , j = 1 , 2 , . . . . 1
• In operator form: x t = θ ( B ) w t , where the moving average operator θ ( B ) is θ ( B ) = 1 + θ 1 B + θ 2 B 2 + · · · + θ q B q . • Compare with the autoregressive model φ ( B ) x t = w t . • The moving average process is stationary for any values of θ 1 , θ 2 , . . . , θ q . 2
Moments • Mean: E ( x t ) = 0. • Autocovariances: � � γ ( h ) = cov x t + h , x t � � = E θ j w t + h − j θ k w t − k j k = σ 2 � θ k θ k + h w k = 0 if h > q. 3
• The MA( q ) model is characterized by γ ( q ) = σ 2 w θ q � = 0 γ ( h ) = 0 for h > q. • The contrast between the ACF of – a moving average model, which is zero except for a finite number of lags h – an autoregressive model, which goes to zero geometrically makes the sample ACF an important tool in deciding what model to fit. 4
Inversion • Example: MA(1) x t = w t + θw t − 1 = (1 + θB ) w t , so if | θ | < 1, w t = (1 + θB ) − 1 x t = π ( B ) x t , where ∞ ( − θ ) j B j . � π ( B ) = j =0 • So x t satisfies an infinite autoregression : ∞ − ( − θ ) j x t − j + w t , � x t = j =1 5
Autoregressive Moving Average Models • Combine! ARMA( p, q ): x t = φ 1 x t − 1 + φ 2 x t − 2 + · · · + φ p x t − p + w t + θ 1 w t − 1 + θ 2 w t − 2 + · · · + θ q w t − q . • In operator form: φ ( B ) x t = θ ( B ) w t . 6
Issues in ARMA Models Parameter redundancy: if φ ( z ) and θ ( z ) have any common fac- tors, they can be canceled out, so the model is the same as one with lower orders. We assume no redundancy. Causality: If φ ( z ) � = 0 for | z | ≤ 1, x t can be written in terms of present and past w s. We assume causality. Invertibility: If θ ( z ) � = 0 for | z | ≤ 1, w t can be written in terms of present and past x s, and x t can be written as an infinite autoregression. We assume invertibility. 7
Using proc arima • Example: fit an MA(1) model to the differences of the log varve thicknesses. • options linesize = 80; ods html file = ’../varve1.html’; data varve; infile ’../data/varve.dat’; input varve; lv = log(varve); dlv = dif(lv); run; 8
proc arima data = varve; title ’Fit an MA(1) model to differences of log varve’; identify var = dlv; estimate q = 1; run; • proc arima output
Using some proc arima options • Example: fit an IMA(1) model to the log varve thicknesses. • options linesize = 80; ods html file = ’varve2.html’; data varve; infile ’varve.dat’; input varve; lv = log(varve); run; 9
proc arima data = varve; title ’Fit an IMA(1, 1) model to log varve, using ML’; title2 ’Use minic option to identify a good model’; identify var = lv(1) minic; estimate q = 1 method = ml; estimate q = 2 method = ml; estimate p = 1 q = 1 method = ml; run; • proc arima output
Notes on the proc arima output • For the MA(1) model, the “Autocorrelation Check of Resid- uals” rejects the null hypothesis that the residuals are white noise. – If the series really had MA(1) structure, the residuals would be white noise. – So the MA(1) model is not a good fit for this series. • For both the MA(2) and the ARMA(1 , 1) models, the “Chi- Square” statistics are not significant, so these models both seem satisfactory. ARMA(1 , 1) has the better AIC and SBC. 10
Using R • Fit a given model and test the residuals as white noise: varve.ma1 = arima(diff(log(varve)), order = c(p = 0, d = 0, q = 1)); varve.ma1; Box.test(residuals(varve.ma1), lag = 6, type = "Ljung", fitdf = 1); • Note: the fitdf argument indicates that these are residuals from a fit with a single parameter. 11
• As in proc arima , differencing can be carried out within arima() : varve.ima1 = arima(log(varve), order = c(0, 1, 1)); varve.ima1; Box.test(residuals(varve.ima1), 6, "Ljung", 1); • But note that you cannot include the intercept, so the results are not identical. – Rerun the original analysis with no intercept: arima(diff(log(varve)), order = c(0, 0, 1), include.mean = FALSE); 12
• Make a table of AICs: AICtable = matrix(NA, 5, 5); dimnames(AICtable) = list(paste("p =", 0:4), paste("q =", 0:4)); for (p in 0:4) { for (q in 0:4) { varve.arma = arima(diff(log(varve)), order = c(p, 0, q)); AICtable[p+1, q+1] = AIC(varve.arma); } } AICtable; • Note: proc arima ’s MINIC option tabulates (an approximation to) BIC, not AIC. 13
• Make a table of BICs: BICtable = matrix(NA, 5, 5); dimnames(BICtable) = list(paste("p =", 0:4), paste("q =", 0:4)); for (p in 0:4) { for (q in 0:4) { varve.arma = arima(diff(log(varve)), order = c(p, 0, q)); BICtable[p+1, q+1] = AIC(varve.arma, k = log(length(varve) - 1)); } } BICtable; • Both tables suggest ARMA(1 , 1). 14
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