Stationarity: KPSS, Phillips-Perron, DF-GLS, Zivot-Andrews and variance ratio tests - library for MetaTrader 5

Stationarity.mqh has five tests that tell a random walk from a series that comes back to its mean, beyond the augmented Dickey-Fuller test: KPSS (whose null is the other way round: stationarity), Phillips-Perron, DF-GLS of Elliott, Rothenberg and Stock (more power than ADF near a unit root), Zivot-Andrews (a unit root against a stationary series with one structural break, with the date of the break) and the Lo-MacKinlay variance ratio. Every test gives its statistic, p-value, critical values at 1%, 5% and 10%, and the lags or bandwidth used. No DLL.

Usage

#include <Stationarity\Stationarity.mqh>

vector y;                                  // for example log prices
// ... fill y ...
SStationarityResult r;
if(CStationarity::KPSS(y, r, ST_TREND_CONSTANT))                 // null: stationary
   PrintFormat("KPSS %.3f, p-value %.3f, bandwidth %d", r.stat, r.pvalue, r.lags);
if(CStationarity::DFGLS(y, r, ST_TREND_LINEAR))                  // null: unit root
   PrintFormat("DF-GLS %.3f, p-value %.3f, %d lags", r.stat, r.pvalue, r.lags);
if(CStationarity::PhillipsPerron(y, r))
   PrintFormat("Phillips-Perron %.3f, p-value %.3f", r.stat, r.pvalue);
if(CStationarity::ZivotAndrews(y, r, ZA_BREAK_BOTH))             // unit root against one break
   PrintFormat("Zivot-Andrews %.3f, p-value %.3f, break at bar %d", r.stat, r.pvalue, (int)r.extra);
if(CStationarity::VarianceRatio(y, r, 10))                       // random walk?
   PrintFormat("VR(10) %.3f, p-value %.3f", r.extra, r.pvalue);

The tests

• KPSS, constant or trend: bandwidth chosen automatically (Hobijn, Franses and Ooms), or 12 (n/100)^(1/4), or given. p-values by interpolation in the simulated quantiles.
• PhillipsPerron, no constant, constant or trend: Z-tau or Z-rho, Newey-West long-run variance, MacKinnon p-values and critical values.
• DFGLS, constant or trend: the series is detrended by GLS with c = -7 or -13.5, the lag chosen by AIC, BIC or t-statistic on the OLS-detrended series.
• ZivotAndrews: break in the level, in the trend, or both, every date between the trimmed ends (15% by default); the regression at each date reuses the cross-products of the columns that do not move, so 2 800 bars take about 0.1 s.
• VarianceRatio: q-period against 1-period variance, overlapping or not, debiased, with the heteroskedasticity robust variance of Lo and MacKinlay. The ratio is in r.extra; below 1 means mean reversion, above 1 momentum.
• SelectLags (the lag search of the Dickey-Fuller regression) and the DickeyFullerP / DickeyFullerCrit functions for t and normalized bias (z) statistics are public.

The tables of p-values and critical values are in StationarityTables.mqh.

Example

The script StationarityReport.mq5 runs the tests on the log price and on the returns of a symbol and prints each result with its verdict at 5%, the date of the most likely break, and the variance ratios at 2 to 60 bars (picture 1). On EURUSD D1 the price behaves as a random walk and the returns are stationary, as expected; the most likely break is November 2022, just after the low (picture 2).

Checked

Against arch 8.0 (arch.unitroot) on six series (a random walk, a stationary AR(1), a trend-stationary series, a series with a break in its level, the log price of gold on H4 and of BTCUSD on D1), 204 tests with every trend, lag rule and option:

• the same lags, bandwidths, numbers of observations, break dates and critical values;
• statistics within 3e-11 and p-values within 2e-11.

Files

Stationarity.mqh and StationarityTables.mqh go to MQL5\Include\Stationarity, StationarityReport.mq5 to MQL5\Scripts\Stationarity.