Fit a seasonal ARIMA model to the classic Box-Jenkins airline passenger counts (monthly, 1949-1960) in your browser. Set the orders with the sliders, then press Update Model and the chart refits by conditional least squares: the dashed line is the forecast, and the band around it is a 95% prediction interval.
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Answer keysearches p, q ≤ 3 and P, Q ≤ 1 with your d, D and m
Current Model: SARIMA(1,1,0)(0,1,0)₁₂
📚 Understanding SARIMA Models
SARIMA(p,d,q)(P,D,Q)ₘ
SARIMA (Seasonal ARIMA) extends ARIMA by adding seasonal components to capture repeating patterns at fixed intervals (like monthly or quarterly seasonality).
Non-Seasonal Parameters (p,d,q)
p: AR order - captures short-term autocorrelation
d: Differencing degree - removes trend
q: MA order - models error terms
Seasonal Parameters (P,D,Q)ₘ
P: Seasonal AR - relates current to m periods ago
D: Seasonal differencing - removes seasonal trend
Q: Seasonal MA - seasonal error structure
m: Season length (12 for monthly, 4 for quarterly)
Airline Passenger Dataset
Period: Monthly, Jan 1949 - Dec 1960 (144 observations)
Default model: SARIMA(1,1,0)(0,1,0)₁₂, a parsimonious starting point. Press Find the Best Fit to rank the alternatives.
Features: Parsimonious model with differencing handling both trend and seasonality
Model Selection Tips
Use ACF/PACF plots to identify p and q
Apply differencing (d,D) until series is stationary
Set m to match your data frequency (12 for monthly)
Compare models using AIC, AICc, or BIC criteria
Educational Insight: The default model SARIMA(1,1,0)(0,1,0)₁₂ uses:
• d=1 to remove the trend through first differencing
• D=1 to remove seasonal pattern through seasonal differencing
• p=1 for short-term AR(1) autocorrelation
• P=0, Q=0 meaning seasonal differencing alone handles seasonality without additional AR/MA terms
• This is a parsimonious model that captures both trend and seasonality efficiently
⭐ What does "Find the Best Fit" do?
It fits every SARIMA(p, d, q)(P, D, Q)m with p and q from 0 to 3 and P and Q from 0 to 1, keeping the d, D and m you set, which is 64 candidate models. Each one is estimated by the same conditional least squares that runs when you press Update Model. The candidates are then ranked by AICc, the corrected Akaike information criterion, and the winner is applied to the sliders and the chart. The table lists the top six so you can see how close the race was.
The criterion. AICc = n·log(σ̂²) + 2k + 2k(k+1)/(n−k−1), where σ̂² is the residual variance and k counts the coefficients plus one for the variance. The first term rewards fit, the second charges for every extra parameter, and the third is a small-sample correction. Lower is better. Every model's σ̂² is computed on the same window of observations (the one the largest candidate needs after its lags), so the numbers are comparable.
Why d, D and m are not searched. Differencing changes the series being modelled, so residual variances of models with different d or D are not comparable and neither are their AICc values. In practice the differencing orders are chosen first, from plots and unit-root tests such as KPSS and OCSB, and only p, q, P, Q are ranked by an information criterion. That is exactly what R's auto.arima does. For this series d = 1 and D = 1 is the standard choice; try others and search again to see how the answer moves.
Dropped candidates. A fit whose AR part is non-stationary or whose MA part is non-invertible is discarded rather than ranked, again following auto.arima. The panel says how many were dropped.
Reading the table. Differences in AICc below about 2 are ties in practice, and among tied models the one with fewer parameters is preferred. With d = D = 1 the winner is a seven-coefficient SARIMA(3,1,3)(1,1,0)12 that beats SARIMA(2,1,1)(0,1,0)12, the model auto.arima returns for this series, by only about three points. Compare their forecast bands before deciding the extra four coefficients are worth having. The default SARIMA(1,1,0)(0,1,0)12 sits about five points behind with a single coefficient, which is why it makes a good classroom baseline.
The lesson the criterion cannot give you. The seasonal swings in this series grow with the level, and no ARIMA on the raw counts can reproduce that. On log(y), R's auto.arima returns the classic "airline model" SARIMA(0,1,1)(0,1,1)12, with just two coefficients. Model selection ranks the candidates you offer it; choosing the transformation is still your job.
Approximation. This tool uses conditional least squares while statsmodels and R use the exact Gaussian likelihood. On this series both give the same top five models by AICc and coefficients that agree to a few hundredths; rankings can differ only among near-ties.