A hundred and eighty observations generated from an AR(1) with a true coefficient of 0.68. The fit gives 0.6294 with a standard error of 0.0602 — within one standard error of the truth — and an innovation variance of 8.593. The Ljung-Box test on the residuals gives p = 0.1580, so nothing systematic is left. Worth noting honestly: the AIC prefers AR(2) at 817.59 over the true AR(1) at 822.16. Information criteria over-select on purpose, trading a small bias for lower variance, and the model with the best AIC is not the model that generated the data.
ARIMA(1, 0, 0) · 180 observations · 164 used in the fit
AIC 822.16, innovation variance 8.5925
The Ljung-Box statistic on the residuals is 13.085 on 9 degrees of freedom, p = 0.1580 — nothing systematic is left. Dickey-Fuller on the raw series gives -6.0467 against -2.8777, so a unit root is rejected and d = 0 may be right.
AIC
822.16
BIC 828.36
Innovation variance
8.5925
what the model cannot explain
Residual Ljung-Box
0.1580
residuals look like noise
Over-differenced
no
no MA term to check
Coefficients
Each fitted coefficient with its standard error and t ratio
Term
Estimate
Std. error
Ratio
Distinguishable from zero
AR 1
0.6294
0.0602
10.450
yes
Constant
22.2540
—
—
—
The autoregressive coefficients sum to 0.6294. A sum approaching 1 means the differenced series still has a root near the unit circle, which is a sign that d is too low rather than that p is too low — adding autoregressive terms will not fix it.
Forecast
The point forecast for each period ahead
Ahead
Forecast
Change
1
58.822
—
2
59.277
0.455
3
59.563
0.286
4
59.743
0.180
5
59.857
0.113
6
59.928
0.071
7
59.973
0.045
8
60.001
0.028
These are point forecasts with no interval, and an ARIMA interval widens quickly — with d = 1 it grows as the square root of the horizon without bound. A point forecast eight periods ahead from a differenced model carries far more uncertainty than the single number suggests.
The fit is by the Hannan-Rissanen two-stage procedure: a long autoregression estimates the innovations, then the model is fitted by least squares on lagged values and those innovations. It is a consistent estimator with genuine standard errors, and it is not exact maximum likelihood — the coefficients will differ slightly from a package that maximises the likelihood, most noticeably when a root sits close to the unit circle.
Over-differencing flagged Dickey-Fuller alongside Residuals tested Not exact likelihood
What this tool shows
Differencing a series that was already stationary gives a moving-average coefficient of −0.9696 — the fingerprint of over-differencing — and inflates the innovation variance from 228.421 to 259.990. A 13.8% cost for a step that was not needed, and the residual test will not catch it: Ljung-Box gives p = 0.6783 either way, because the residuals really are white noise. The damage shows up in the variance and the forecast interval, not in the diagnostics people usually run.
ARIMA(p, d, q) fitted by Hannan-Rissanen, with coefficients, standard errors and t ratios
Over-differencing detected from the moving-average coefficient and named explicitly
The residual Ljung-Box test with its degrees of freedom adjusted for the fitted orders
A Dickey-Fuller result on the raw series, to inform the differencing order rather than guessing it
The sum of the autoregressive coefficients, which says when d is too low rather than p
Point forecasts, with a plain statement of what they leave out
Over-differencing flagged Dickey-Fuller alongside Residuals tested Not exact likelihood
Point forecasts only. An ARIMA interval widens fast.
Updated 13 September 2026 · Works in any browser, no installation
An ARIMA model describes a series by three numbers: how many past values it depends on, how many times it must be differenced to become stationary, and how many past shocks still echo in it. The middle one does the most damage when it is wrong. Too low and the model chases a trend it cannot capture; too high and it manufactures a moving-average term, inflates the innovation variance, and widens every forecast interval for nothing.
At a glance
Formula shown
Difference the series d times, then fit (1 − φ₁B − … − φₚBᵖ)yₜ = c + (1 + θ₁B + … + θqBq)εₜ, where B is the backshift operator. Here that is done by Hannan-Rissanen: a long autoregression estimates the innovations ε̂, then the model is fitted by least squares on lagged values and those innovations. Over-differencing shows up as Σθ approaching −1, because differencing a stationary series introduces a moving-average root at exactly −1.
Scenario support
Forecasting a series with autocorrelation but no seasonality, deciding how many times to difference before modelling, diagnosing why a fitted model’s residuals still carry structure, and establishing a baseline that a more elaborate method has to beat.
Educational estimate
Planning support from the values you enter — not professional advice.
Over-differencing has a fingerprint
The second preset is a stationary series differenced anyway, and the result is diagnosable from the output alone.
The moving-average coefficient comes out at −0.9696. Differencing a stationary series introduces a moving-average root at exactly −1, and the estimate runs towards it.
The innovation variance rises from 228.421 to 259.990. 13.8% more unexplained variation than the correct model, which propagates straight into every forecast interval.
And the AIC rises from 1203.01 to 1213.39, so a model comparison would catch it — if anyone ran one.
The residual test will not. Ljung-Box gives p = 0.6783, because over-differencing does not leave structure behind. It leaves a bigger error, which is a different thing and the one people check for less.
The differencing order is a parameter, not a habit
The third preset is the case where differencing is right, and the contrast with the second is the whole point of setting d deliberately.
A random walk at d = 1 gives an autoregressive coefficient of 0.0323 with a standard error of 0.0794 — indistinguishable from zero, which is exactly what the differenced series should be.
The same series at d = 0 gives 0.9545 with a standard error of 0.0258. A root that close to 1 means the model is trying to describe a non-stationary series with a stationary form.
The sum of the autoregressive coefficients is the tell. Approaching 1 means d is too low, and adding autoregressive terms will not help — the page prints that sum for exactly this reason.
So test before differencing. The Dickey-Fuller result on the raw series is computed on every input here, and KPSS should be run alongside it before the order is fixed.
The best AIC is not the true model
On the first preset the data was generated by an AR(1), and the information criterion prefers something else. That is not a bug in the criterion.
The true AR(1) gives an AIC of 822.16. AR(2) gives 817.59 and wins.
The fitted coefficient is 0.6294 against a true 0.68, within a standard error, so the AR(1) fit itself is fine.
AIC is built to minimise prediction error, not to recover the truth. It over-selects on purpose, accepting a slightly larger model in exchange for lower variance.
BIC penalises size harder and is closer to consistent. Here it gives 828.36 for AR(1) and 826.87 for AR(2) — a much narrower margin. Both are printed, and disagreement between them is a sign the order is genuinely uncertain rather than a reason to pick whichever agrees with you.
What the residual test can and cannot see
The Ljung-Box test on the residuals is the standard adequacy check, and its scope is narrower than its reputation.
It detects leftover autocorrelation. A significant result means the orders are too low and the model has missed structure.
Its degrees of freedom must subtract the fitted orders, which this page does — using the raw lag count inflates the p-value and makes an inadequate model look adequate.
It cannot see over-differencing, as the second preset shows at p = 0.6783.
And it cannot see non-constant variance. A series whose volatility clusters passes this test comfortably while violating an assumption the forecast intervals rest on.
Hannan-Rissanen, not exact likelihood
This page states its estimator plainly, because the coefficients will differ slightly from a package that maximises the likelihood and that is worth knowing in advance.
Stage one fits a long autoregression to estimate the innovations. Those residuals stand in for the unobservable shocks.
Stage two regresses the series on its own lags and those estimated shocks. Ordinary least squares, with genuine standard errors.
It is consistent and it is not efficient. The estimates converge to the truth as the sample grows, with slightly more variance than maximum likelihood.
The gap is largest near the unit circle. With a root close to 1 — including the over-differenced case above — expect a visible difference from a likelihood-based fit, in the coefficient if not in the conclusion.
The point forecast is the least of it
This page reports point forecasts and no intervals, and the reason is that an honest ARIMA interval is harder to produce than the point estimate it surrounds.
With d = 1 the interval widens without bound. It grows roughly as the square root of the horizon, because a differenced model has no level to revert to.
With d = 0 it converges to the unconditional variance. Far ahead, the forecast is the mean and the interval is the spread of the series.
The interval also ignores parameter uncertainty, which the textbook formula treats as zero — so even a correctly computed interval is too narrow.
Which makes a bare point forecast misleading in a specific direction: confident. Eight periods ahead from a differenced model, the single number on the page carries far more uncertainty than it looks like it does.
Reporting an ARIMA fit
Four items, and the second is the one that lets a reader judge whether the model was specified or stumbled upon.
Give all three orders and how d was chosen. A stationarity test result, not a habit.
Give the residual diagnostic with its adjusted degrees of freedom. And say it cannot see over-differencing or changing variance.
Give the innovation variance, not just the AIC. It is what the forecast intervals are built from, and it is where over-differencing shows up.
And compare against a naive forecast. An ARIMA model that cannot beat “tomorrow equals today” on a scaled error measure has not earned its complexity.
Method. The fit is by the Hannan-Rissanen two-stage procedure: a long autoregression of order chosen from the sample size estimates the innovations, and the model is then fitted by least squares on lagged values and those estimated innovations. That is a consistent estimator with real standard errors rather than exact maximum likelihood, and the page says so rather than implying otherwise. The Ljung-Box degrees of freedom subtract the fitted orders, which is what makes the residual test meaningful. The verification suite checks the fit against simulated data with known coefficients — eight AR(1) series of four hundred observations, with the true coefficient recovered inside five standard errors each time — and asserts the theoretical prediction for over-differencing: differencing stationary noise must produce a moving-average coefficient below −0.85, which it does on all twelve generated series. Differencing is capped at order two and the combined orders at five each, since the two-stage procedure needs a usable sample after both stages. That engine is verified on every change against 115 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Augmented Dickey-FullerThe unit root test with MacKinnon critical values, the statistic at every lag order, the ordinary t cutoff shown for contrast, and KPSS computed alongside.
KPSS TestStationarity as the null hypothesis, with Dickey-Fuller alongside and all four combinations of the two verdicts named — including the one that means the data cannot tell.
Holt-WintersTriple exponential smoothing with fitted parameters, additive and multiplicative seasonality, and the seasonal term priced against a non-seasonal Holt fit on your own data.
AutocorrelationACF and PACF at every lag with Bartlett bands that widen rather than a flat line, plus the shapes that identify a model.
Forecast AccuracyMAE, RMSE, MAPE, sMAPE, MASE and bias side by side, with MAPE split by direction and MASE measured against a naive one-step forecast.
Ljung-Box TestThe Ljung-Box portmanteau test with Box-Pierce beside it, the degrees-of-freedom correction for fitted models and a lag-by-lag table.
An educational tool. The fit is by Hannan-Rissanen rather than exact maximum likelihood, so coefficients will differ slightly from a likelihood-based package, most noticeably when a root sits near the unit circle. Forecasts here are point estimates with no intervals — an ARIMA interval with d = 1 widens without bound and, computed the usual way, still ignores parameter uncertainty.