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The augmented Dickey-Fuller test, against the null of a unit root.

public static DickeyFullerResult AugmentedDickeyFuller(ReadOnlySpan<double> series, DickeyFullerOptions options = null)

Parameters — series are the observations, in time order. options sets the trend terms, the lag rule and its maximum, or null for the reference's defaults: a constant, Akaike's criterion, and Schwert's maximum lag.

Returns — DickeyFullerResult: the statistic, MacKinnon's p-value and critical values, and the lag the regression used.

Exceptions — ArgumentException when series carries a non-finite value, is constant, lies on one straight line — a deterministic trend, with no stochastic component to test — is too short for its trend terms and default lag, or builds a lagged design with no unique least-squares solution at the lag used or at any lag the search tries; or when options asks for a maximum lag above n/2 − terms − 1 or one that leaves the widest regression no degree of freedom.

Example — a drifting series of 24 points. The lag search keeps three lagged differences, so the regression fits 20 rows.

using Lodestar.Stats.TimeSeries;

double[] walk = [0.0, 1.2, 0.7, 2.1, 3.0, 2.4, 3.9, 5.1, 4.6, 6.0, 7.3, 6.8,
                 8.2, 9.5, 9.1, 10.4, 11.8, 11.2, 12.7, 14.0, 13.5, 14.9, 16.2, 15.8];

DickeyFullerResult result = Stationarity.AugmentedDickeyFuller(walk);

double statistic = Math.Round(result.Statistic, 4);             // => 1.9817
double p = Math.Round(result.PValue, 4);                        // => 0.9986
int usedLag = result.UsedLag;                                   // => 3
int rows = result.ObservationCount;                             // => 20
double fivePercent = Math.Round(result.CriticalValues[1], 4);   // => -3.0216
double akaike = Math.Round(result.InformationCriterion, 4);     // => -37.8399

Remarks — the regression is Δx[t] on the lagged level x[t−1], the lagged differences and the trend terms; the statistic is the lagged level's t statistic, and each fit is OrdinaryLeastSquares.Estimate from Lodestar.Stats.Regression, which stops short of the inference table a lag search never reads.

The lag search fits every candidate over the same rows, n − maxLag − 1 of them, so their criteria compare; only the chosen lag is refitted over its own, longer sample. That is the reference's rule, and fitting each candidate over its longest sample instead changes which lag wins on short series.

The p-value is MacKinnon's (1994) response surface — exactly 1 above the table and exactly 0 below it — and the critical values his 2010 ones, for the refitted row count.

using Lodestar.Stats.TimeSeries;

double[] walk = [0.0, 1.2, 0.7, 2.1, 3.0, 2.4, 3.9, 5.1, 4.6, 6.0, 7.3, 6.8,
                 8.2, 9.5, 9.1, 10.4, 11.8, 11.2, 12.7, 14.0, 13.5, 14.9, 16.2, 15.8];

DickeyFullerResult trend = Stationarity.AugmentedDickeyFuller(
    walk,
    new DickeyFullerOptions
    {
        Regression = TrendTerms.ConstantAndTrend,
        LagSelection = LagSelection.Fixed,
        MaxLag = 1,
    });

double statistic = Math.Round(trend.Statistic, 4);  // => -8.9132
double p = Math.Round(trend.PValue, 4);             // => 0

Applies to — net10.0, netstandard2.0.

See also — Stationarity.Kpss, DickeyFullerOptions, DickeyFullerResult, the Python equivalence table.