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Change it on the Demand Analysis page.
ACF & PACF Analysis
Identify ARIMA(p,d,q) order parameters using Autocorrelation and Partial Autocorrelation Functions.
Automatically Recommended Order
ARIMA( 1, 0, 1 )
ACF lag cutoff: q = 1. PACF lag cutoff: p = 1. Order d = 0.
ACF - Autocorrelation Function
→ identifies q
ACF Definition
$$\rho(k) = \frac{\gamma(k)}{\gamma(0)} = \frac{\text{Cov}(Y_t, Y_{t-k})}{\text{Var}(Y_t)}$$
γ(k) = autocovariance at lag k
ρ(k) ∈ [−1, 1] = autocorrelation at lag k
Confidence bounds: $\pm \dfrac{1.96}{\sqrt{n}}$ (Bartlett's formula)
ρ(k) ∈ [−1, 1] = autocorrelation at lag k
Confidence bounds: $\pm \dfrac{1.96}{\sqrt{n}}$ (Bartlett's formula)
How to read ACF:
• MA(q): ACF cuts off abruptly after lag q (significant spikes only for lags 1 to q)
• Significant lags: None beyond lag 0
• Suggested q = 1
• MA(q): ACF cuts off abruptly after lag q (significant spikes only for lags 1 to q)
• Significant lags: None beyond lag 0
• Suggested q = 1
PACF - Partial Autocorrelation Function
→ identifies p
PACF Definition
$$\phi_{kk} = \text{Corr}(Y_t, Y_{t-k} \mid Y_{t-1}, \ldots, Y_{t-k+1})$$
PACF measures the direct relationship between $Y_t$ and $Y_{t-k}$ after removing the indirect effects of intermediate lags 1 through k-1.
Estimated using the Yule-Walker equations.
Estimated using the Yule-Walker equations.
How to read PACF:
• AR(p): PACF cuts off abruptly after lag p
• Significant lags: None beyond lag 0
• Suggested p = 1
• AR(p): PACF cuts off abruptly after lag p
• Significant lags: None beyond lag 0
• Suggested p = 1
ACF & PACF - Applied to Differenced Series (d=0)
ARIMA Model Selection Rules (Box-Jenkins Methodology)
| ACF Pattern | PACF Pattern | Model |
|---|---|---|
| Tails off slowly | Cuts off at lag p | AR(p) |
| Cuts off at lag q | Tails off slowly | MA(q) |
| Tails off slowly | Tails off slowly | ARMA(p,q) |
| Non-stationary pattern | Very slow decay | Difference (d↑) |
Information Criteria
$$\text{AIC} = -2\ln(\hat{L}) + 2k$$
$$\text{BIC} = -2\ln(\hat{L}) + k\ln(n)$$
where $\hat{L}$ = maximised likelihood, $k$ = parameters, $n$ = sample size.
Lower AIC/BIC = Better model fit. BIC penalises complexity more than AIC.
Lower AIC/BIC = Better model fit. BIC penalises complexity more than AIC.