Seasonal ARIMA Models Many time series collected on a monthly or - - PowerPoint PPT Presentation

Seasonal ARIMA Models

• Many time series collected on a monthly or quarterly basis

have seasonal behavior.

• Similarly hourly data and daily behavior.
• E.g. Johnson & Johnson quarterly earnings; discussion typi-

cally focuses on comparison with: – previous quarter; – same quarter, previous year.

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• That is, we compare xt with xt−1 and xt−4.
• More generally, we compare xt with xt−1 and xt−s, where

– s = 4 for quarterly data, – s = 12 for monthly data, – s = 24 for daily effects in hourly data, – s = 168 for weekly effects in hourly data, etc.

• This suggests modeling xt in terms of xt−1 and xt−s.

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Pure Seasonal ARMA

• The pure seasonal ARMA model has the form

xt = Φ1xt−s + Φ2xt−2s + · · · + ΦPxt−Ps + wt + Θ1wt−s + Θ2wt−2s + · · · + ΘQwt−Qs.

• Notation: ARMA(P, Q)s.
• In operator form:

ΦP(Bs)xt = ΘQ(Bs)wt.

• ΦP(Bs) and ΘQ(Bs) are seasonal autoregressive and moving

average operators.

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Multiplicative Seasonal ARMA

• ACF of pure seasonal ARMA is nonzero only at lags s, 2s,

. . . ; most seasonal time series have other nonzero values.

• For such series, w(s)

t

= ΘQ(Bs)−1ΦP(Bs)xt is not white noise for any choice of ΦP and ΘQ.

• But suppose that for some ΦP and ΘQ, w(s)

t

is ARMA(p, q): φp(B)w(s)

t

= θq(B)wt, where {wt} is white noise.

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• Then xt satisfies

ΦP(Bs)φp(B)xt = ΘQ(Bs)θq(B)wt.

• This

is the Multiplicative Seasonal ARMA model ARMA(p, q) × (P, Q)s.

• The non-seasonal parts φp and θq control short-term corre-

lations (up to half a season, lag ≈ s/2), while the seasonal parts ΦP and ΘQ control the decay of the correlations over multiple seasons.

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Example: Johnson & Johnson earnings; R analysis par(mfrow = c(2, 1)) plot(log(jj)) jjl = lm(log(jj) ~ time(jj) + factor(cycle(jj))) summary(aov(jjl)) jjf = ts(fitted(jjl), start = start(jj), frequency = frequency(jj)) lines(jjf, col = 2, lty = 2) jjr = ts(residuals(jjl), start = start(jj), frequency = frequency(jj)) plot(jjr) acf(jjr) pacf(jjr)

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• PACF is simpler than ACF:

– ACF spikes at lags 4, 8, perhaps 12; of these, PACF spikes

• nly at lag 4;

– apart from lags 4, 8, . . . , PACF drops off faster.

• (P)ACF indicates neither simple ARMA nor simple ARMA4.
• PACF suggests ARMA(2, 0) × (1, 0)4:

jja = arima(jjr, order = c(2, 0, 0), seasonal = list(order = c(1, 0, 0), period = 4)) print(jja) tsdiag(jja)

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• Note: the original fit of the straight line and seasonal dum-

mies was by OLS; – possibly inefficient; – invalid inferences (standard errors, etc.).

• Solution: refit as part of the time series model.

x = model.matrix( ~ time(jj) + factor(cycle(jj))) jja = arima(log(jj), order = c(2, 0, 0), seasonal = list(order = c(1, 0, 0), period = 4), xreg = x, include.mean = FALSE) print(jja) tsdiag(jja)

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Notes:

• The time series being fitted is the original unadjusted log(jj).
• The regressors are specified as the matrix argument xreg.
• arima does not check for linear dependence, so we must either
• mit one dummy variable from xreg or use include.mean =

FALSE in arima.

• Regression parameter estimates are similar to OLS, but stan-

dard errors are roughly doubled.

• Using SAS: proc arima program and output.

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Multiplicative Seasonal ARIMA

• The seasonal difference operator is ∇s = 1 − Bs.
• Some series show slow decay of ACF only at lags s, 2s, . . . ,

which suggests seasonal differencing. – But note: seasonal means also give slow decay of ACF at those lags.

• The

Multiplicative Seasonal ARIMA model ARIMA(p, d, q) × (P, D, Q)s is ΦP(Bs)φp(B)∇D

s ∇dxt = ΘQ(Bs)θq(B)wt.

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