ADF - Augmented Dickey-Fuller Test

// This Pine Script™ code is subject to the terms of the Mozilla Public License 2.0
// https://mozilla.org/MPL/2.0/
// © QuanTAlib

//@version=6
indicator("ADF - Augmented Dickey-Fuller Test", shorttitle="ADF", overlay=false, precision=4)

// ─── Inputs ────────────────────────────────────────────────────────────
int    period     = input.int(50, "Period", minval=20, maxval=500)
int    maxLag     = input.int(0,  "Max Lag (0=auto)", minval=0, maxval=10)
string regModel   = input.string("Constant", "Regression", options=["NoConstant", "Constant", "ConstantAndTrend"])

// ─── Helper: OLS solve 2×2 (constant + gamma, no lags for simplicity) ──
// Full ADF with Cholesky is too complex for Pine; this is a reference only.
// For production use, the C# implementation is authoritative.

adf_pvalue(float[] data, int p, int mLag, string reg) =>
    int n = array.size(data)
    if n < 20
        1.0
    else
        // Compute first differences
        int nd = n - 1
        float[] dy = array.new_float(nd, 0.0)
        for i = 0 to nd - 1
            array.set(dy, i, array.get(data, i + 1) - array.get(data, i))

        // Simple ADF: regress dy on y_{t-1} (constant model, no lags)
        // X = [1, y_{t-1}], Y = dy_t
        float sx = 0.0, sy = 0.0, sxx = 0.0, sxy = 0.0, syy = 0.0
        int startIdx = math.max(mLag, 0)
        int nObs = nd - startIdx
        if nObs < 3
            1.0
        else
            for t = startIdx to nd - 1
                float x = array.get(data, t)  // y_{t-1} in level
                float y = array.get(dy, t)
                sx += x
                sy += y
                sxx += x * x
                sxy += x * y
                syy += y * y

            // OLS for constant model: [alpha, gamma]
            float xbar = sx / nObs
            float ybar = sy / nObs
            float denom = sxx - sx * sx / nObs
            if math.abs(denom) < 1e-15
                1.0
            else
                float gamma = (sxy - sx * sy / nObs) / denom
                float alpha = ybar - gamma * xbar

                // RSS
                float rss = 0.0
                for t = startIdx to nd - 1
                    float x = array.get(data, t)
                    float y = array.get(dy, t)
                    float resid = y - alpha - gamma * x
                    rss += resid * resid

                float s2 = rss / (nObs - 2)
                if s2 <= 0
                    1.0
                else
                    float se = math.sqrt(s2 / denom)
                    if se <= 1e-15
                        1.0
                    else
                        float tStat = gamma / se

                        // MacKinnon (1994) p-value for N=1, regression-dependent
                        float tauStar = reg == "NoConstant" ? -1.04 : reg == "ConstantAndTrend" ? -2.89 : -1.61
                        float tauMin  = reg == "NoConstant" ? -19.04 : reg == "ConstantAndTrend" ? -16.18 : -18.83
                        float tauMax  = reg == "NoConstant" ? 1e308 : reg == "ConstantAndTrend" ? 0.70 : 2.74

                        if tStat < tauMin
                            0.0
                        else if tStat > tauMax
                            1.0
                        else
                            float poly = 0.0
                            if tStat <= tauStar
                                // Small-p coefficients (regression-dependent)
                                if reg == "NoConstant"
                                    poly := 0.6344 + 1.2378 * tStat + 0.032496 * tStat * tStat
                                else if reg == "ConstantAndTrend"
                                    poly := 3.2512 + 1.6047 * tStat + 0.049588 * tStat * tStat
                                else
                                    poly := 2.1659 + 1.4412 * tStat + 0.038269 * tStat * tStat
                            else
                                // Large-p coefficients (regression-dependent)
                                float t2 = tStat * tStat
                                float t3 = t2 * tStat
                                if reg == "NoConstant"
                                    poly := 0.4797 + 0.93557 * tStat - 0.06999 * t2 + 0.033066 * t3
                                else if reg == "ConstantAndTrend"
                                    poly := 2.5261 + 0.61654 * tStat - 0.37956 * t2 - 0.060285 * t3
                                else
                                    poly := 1.7339 + 0.93202 * tStat - 0.12745 * t2 - 0.010368 * t3

                            // Normal CDF via A&S 7.1.26 erf → Φ(x) = 0.5·(1 + erf(x/√2))
                            int    sgn  = poly < 0 ? -1 : 1
                            float  z    = math.abs(poly) * 0.70710678118654752
                            float  tt   = 1.0 / (1.0 + 0.3275911 * z)
                            float  tt2  = tt * tt
                            float  tt3  = tt2 * tt
                            float  tt4  = tt3 * tt
                            float  tt5  = tt4 * tt
                            float  erfcZ = (0.254829592 * tt - 0.284496736 * tt2 + 1.421413741 * tt3 - 1.453152027 * tt4 + 1.061405429 * tt5) * math.exp(-z * z)
                            float  erfZ  = 1.0 - erfcZ
                            float  result = 0.5 * (1.0 + sgn * erfZ)
                            result

// ─── Main ──────────────────────────────────────────────────────────────
float[] window = array.new_float(0)
for i = 0 to period - 1
    array.push(window, close[period - 1 - i])

float pval = adf_pvalue(window, period, maxLag, regModel)

// ─── Plot ──────────────────────────────────────────────────────────────
plot(pval, "ADF p-value", color=color.blue, linewidth=2)
hline(0.01, "1% significance", color=color.red, linestyle=hline.style_dotted)
hline(0.05, "5% significance", color=color.orange, linestyle=hline.style_dashed)
hline(0.10, "10% significance", color=color.gray, linestyle=hline.style_dotted)

bgcolor(pval < 0.05 ? color.new(color.green, 90) : na)