You are diagnosing a stationary daily production series using the Box-Jenkins approach. The sample ACF shows statistically significant spikes at lags 1 and 2 and is essentially zero for all higher lags. The sample PACF, in contrast, declines gradually without a sharp cut-off. Which model is the most parsimonious first candidate to fit to these data?
For an MA(q) process, the theoretical ACF cuts off after lag q, whereas the PACF tails off. Observing non-zero autocorrelations at lags 1 and 2 only, together with a tapering PACF, is therefore the textbook pattern for an MA(2) model.
AR(p) processes display the opposite behaviour (PACF cuts off, ACF tails off), so an AR(2) does not match the diagnostics.
ARMA(1,1) models generally produce ACF and PACF that both tail off rather than a sharp cut-off in the ACF.
A first-differenced random walk (ARIMA 0,1,0) is non-stationary before differencing and would show a very slow decay in the ACF, not an immediate cut-off. Hence the moving average model of order 2 is the most appropriate starting point.
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What is the ACF, and how does it help identify MA(q) processes?
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Why does the PACF tail off for an MA(q) process, and how is it different from AR(p)?
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How does the ARIMA model differ from stationary models like AR or MA?