Granger 2003 - Spurious Regression Monte Carlo in Python

Summary

A single reproducible NumPy/SciPy script (seeded) that demonstrates every quantitative claim around Granger’s p. 363 account of spurious regression: (1) independent random walks regressed in levels reject a true null 67-90% of the time at nominal 5%, with $∣ t ∣$ growing like $T$ , $R^{2} \approx 0.24$ that never shrinks and DW $\to 0$ , while differencing restores 4-6% size; (2) a Granger-Newbold design with 1-5 random-walk regressors at $T = 50$ pushes the joint $F$ -test rejection rate from 67% to 98%; (3) Yule’s three shapes of the correlation distribution for $I (0)$ , $I (1)$ , $I (2)$ series; (4) the Engle-Granger residual test separates a spurious pair (4% rejection) from a cointegrated pair (100%); (5) the two-step ECM recovers adjustment speed $- 0.5$ and short-run effect $2.0$ . A companion plotting script draws a sample spurious pair, Yule’s three histograms and the $T$ growth of the spurious $t$ .

Overview

The lecture describes the Granger-Newbold result in words only (“a small simulation published in 1974”, p. 363). This note turns it — and the surrounding theory in Granger 2003 - Spurious Regression from Yule to Granger-Newbold and Granger 2003 - Cointegration and the Error-Correction Representation — into a simulation study, written in the aims / data-generating mechanisms / estimands / methods / performance-measures structure of Morris White Crowther 2019 - Simulation Studies (ADEMP) - Overview.

ADEMP element	This study
Aims	Show the size distortion of OLS inference with independent $I (1)$ series; show that differencing and cointegration-based methods fix it
Data-generating mechanisms	Gaussian random walks $x_{t} = \sum_{s \leq t} ε_{s}$ ; $T \in {10, 50, 100, 200, 500}$ ; independent pairs, $k = 1..5$ independent regressors, once/twice-cumulated noise, and a cointegrated pair $y_{t} = 1 + 2 x_{t} + u_{t}$ , $u_{t} = 0.5 u_{t - 1} + ε_{t}$
Estimands	slope $β$ (true 0 for spurious pairs, 2 for the cointegrated pair); ECM adjustment $γ = - 0.5$ and impact $δ = 2$
Methods	OLS in levels; OLS in first differences; joint $F$ -test; sample correlation; Engle-Granger residual ADF test; two-step ECM
Performance measures	rejection rate (empirical size/power), median $∣ t ∣$ , mean $R^{2}$ , median DW, $Pr (R^{2} > D W)$ , mean and SD of estimates

Replications: 5,000 (Exp. 1-2), 20,000 (Exp. 3), 2,000 (Exp. 4-5). Monte Carlo SE of a rejection rate $p$ is $p (1 - p) / R$ , e.g. $0.67 \cdot 0.33/5000 \approx 0.007$ ; all differences discussed below are many MC SEs apart.

Main Content

The script

Save as spurious_sim.py and run python spurious_sim.py (requires numpy, scipy). Output is printed as markdown table rows.

import numpy as np
import scipy.stats as st
 
def ols(y, X):
    """OLS with intercept already in X. Returns beta, t-stats, R2, DW."""
    beta, *_ = np.linalg.lstsq(X, y, rcond=None)
    resid = y - X @ beta
    n, k = X.shape
    s2 = resid @ resid / (n - k)
    se = np.sqrt(s2 * np.diag(np.linalg.inv(X.T @ X)))
    r2 = 1 - resid @ resid / np.sum((y - y.mean()) ** 2)
    dw = np.sum(np.diff(resid) ** 2) / (resid @ resid)
    return beta, beta / se, r2, dw, resid
 
def add_const(*cols):
    return np.column_stack([np.ones(len(cols[0]))] + list(cols))
 
# ---- Experiment 1: two independent random walks, levels vs differences ----
def exp1(n, reps=5000, seed=0):
    rng = np.random.default_rng(seed)
    rows = {"levels": [], "differences": []}
    for _ in range(reps):
        x = np.cumsum(rng.standard_normal(n))
        y = np.cumsum(rng.standard_normal(n))
        _, t, r2, dw, _ = ols(y, add_const(x))
        rows["levels"].append((t[1], r2, dw))
        _, t, r2, dw, _ = ols(np.diff(y), add_const(np.diff(x)))
        rows["differences"].append((t[1], r2, dw))
    for k, v in rows.items():
        t, r2, dw = map(np.array, zip(*v))
        print(f"| {n} | {k} | {np.mean(np.abs(t) > 1.96):.2f} | {np.median(np.abs(t)):.2f} | "
              f"{r2.mean():.2f} | {np.median(dw):.2f} | {np.mean(r2 > dw):.2f} |")
 
# ---- Experiment 2: Granger-Newbold style, k independent random-walk regressors, T=50 ----
def exp2(n=50, reps=5000, seed=1):
    rng = np.random.default_rng(seed)
    for k in range(1, 6):
        rej, r2s, dws = [], [], []
        for _ in range(reps):
            y = np.cumsum(rng.standard_normal(n))
            Xs = np.cumsum(rng.standard_normal((n, k)), axis=0)
            X = add_const(*Xs.T)
            beta, t, r2, dw, resid = ols(y, X)
            # joint F test of all slopes = 0
            rss1 = resid @ resid
            rss0 = np.sum((y - y.mean()) ** 2)
            F = ((rss0 - rss1) / k) / (rss1 / (n - k - 1))
            rej.append(F)
            r2s.append(r2); dws.append(dw)
        crit = st.f.ppf(0.95, k, n - k - 1)
        print(f"| {k} | {np.mean(np.array(rej) > crit):.2f} | {np.mean(r2s):.2f} | {np.median(dws):.2f} |")
 
# ---- Experiment 3: Yule-style sampling distribution of correlation ----
def exp3(n=10, reps=20000, seed=2):
    rng = np.random.default_rng(seed)
    for order, name in [(0, "white noise I(0)"), (1, "random walk I(1)"), (2, "twice-cumulated I(2)")]:
        rs = []
        for _ in range(reps):
            x = rng.standard_normal(n); y = rng.standard_normal(n)
            for _ in range(order):
                x = np.cumsum(x); y = np.cumsum(y)
            rs.append(np.corrcoef(x, y)[0, 1])
        rs = np.array(rs)
        print(f"| {name} | {n} | {np.mean(np.abs(rs) > 0.5):.2f} | {np.mean(np.abs(rs) > 0.9):.2f} | {np.std(rs):.2f} |")
 
# ---- Experiment 4: a genuinely cointegrated pair vs spurious pair (Engle-Granger step 2) ----
def adf_t(e, lags=1):
    """Dickey-Fuller t on residuals: de_t = rho e_{t-1} + sum gamma de_{t-j}; no constant."""
    de = np.diff(e)
    Y = de[lags:]
    cols = [e[lags:-1]] + [de[lags - j:-j] for j in range(1, lags + 1)]
    X = np.column_stack(cols)
    beta, *_ = np.linalg.lstsq(X, Y, rcond=None)
    r = Y - X @ beta
    s2 = r @ r / (len(Y) - X.shape[1])
    se = np.sqrt(s2 * np.linalg.inv(X.T @ X)[0, 0])
    return beta[0] / se
 
def exp4(n=200, reps=2000, seed=3, crit=-3.37):
    # crit: approx. 5% Engle-Granger critical value, 2 variables with constant (MacKinnon), T~200
    rng = np.random.default_rng(seed)
    res = {"independent RWs": [], "cointegrated (y = 2x + AR(1) error)": []}
    for _ in range(reps):
        x = np.cumsum(rng.standard_normal(n))
        y_ind = np.cumsum(rng.standard_normal(n))
        u = np.zeros(n)
        eps = rng.standard_normal(n)
        for t in range(1, n):
            u[t] = 0.5 * u[t - 1] + eps[t]
        y_coint = 1.0 + 2.0 * x + u
        for k, y in [("independent RWs", y_ind), ("cointegrated (y = 2x + AR(1) error)", y_coint)]:
            beta, t, r2, dw, e = ols(y, add_const(x))
            res[k].append((beta[1], r2, dw, adf_t(e) < crit))
    for k, v in res.items():
        b, r2, dw, rej = map(np.array, zip(*v))
        print(f"| {k} | {np.median(b):.2f} | {np.std(b):.3f} | {r2.mean():.2f} | {np.median(dw):.2f} | {rej.mean():.2f} |")
 
 
# ---- Experiment 5: two-step Engle-Granger ECM on the cointegrated pair ----
def exp5(n=200, reps=2000, seed=4):
    rng = np.random.default_rng(seed)
    out = []
    for _ in range(reps):
        x = np.cumsum(rng.standard_normal(n))
        u = np.zeros(n); eps = rng.standard_normal(n)
        for t in range(1, n):
            u[t] = 0.5 * u[t - 1] + eps[t]
        y = 1.0 + 2.0 * x + u
        b = ols(y, add_const(x))[0]
        z = y - b[0] - b[1] * x                       # step 1: equilibrium error
        dy, dx = np.diff(y), np.diff(x)
        beta, t, r2, dw, _ = ols(dy, add_const(z[:-1], dx))   # step 2: ECM
        out.append((beta[1], beta[2], t[1]))
    g, d, tg = map(np.array, zip(*out))
    print(f"| gamma (adjustment, true -0.5) | {g.mean():.3f} | {g.std():.3f} |")
    print(f"| delta (short-run, true 2.0) | {d.mean():.3f} | {d.std():.3f} |")
    print(f"| median t(gamma) | {np.median(tg):.2f} | |")
 
if __name__ == "__main__":
    print("Exp1"); [exp1(n) for n in (50, 100, 500)]
    print("Exp2"); exp2()
    print("Exp3"); exp3()
    print("Exp4"); exp4()
    print("Exp5"); exp5()

Experiment 1 — levels vs differences ( $T = 50, 100, 500$ ; 5,000 reps; seed 0)

Independent random walks, OLS of $y$ on $x$

$T$ Regression $Pr (∣ t ∣ > 1.96)$ median $∣ t ∣$ mean $R^{2}$ median DW $Pr (R^{2} > D W)$
50 levels 0.67 3.14 0.24 0.30 0.31
50 differences 0.06 0.69 0.02 2.00 0.00
100 levels 0.76 4.62 0.25 0.15 0.55
100 differences 0.05 0.69 0.01 2.00 0.00
500 levels 0.90 10.01 0.24 0.03 0.80
500 differences 0.04 0.67 0.00 2.00 0.00

Interpretation: matches Phillips’ asymptotics (theorem): size distortion grows with $T$ ; median $∣ t ∣$ scales like $T$ (ratio 10.01/4.62 = 2.17 vs $5 = 2.24$ ); $R^{2}$ converges to a non-degenerate distribution (mean $\approx 0.24$ ); DW $\to 0$ ; the $R^{2} > D W$ flag (rule of thumb) catches more spurious fits as $T$ grows. Differenced regressions have correct size (0.04-0.06), median $∣ t ∣ \approx 0.67$ (the $∣ N (0, 1) ∣$ median is 0.674) and DW $= 2$ .

$T$	Regression	$Pr (∣ t ∣ > 1.96)$	median $∣ t ∣$	mean $R^{2}$	median DW	$Pr (R^{2} > D W)$
50	levels	0.67	3.14	0.24	0.30	0.31
50	differences	0.06	0.69	0.02	2.00	0.00
100	levels	0.76	4.62	0.25	0.15	0.55
100	differences	0.05	0.69	0.01	2.00	0.00
500	levels	0.90	10.01	0.24	0.03	0.80
500	differences	0.04	0.67	0.00	2.00	0.00

Experiment 2 — Granger-Newbold design with $k$ regressors ( $T = 50$ ; 5,000 reps; seed 1)

One random walk on $k$ independent random walks; joint $F$ -test of all slopes $= 0$ at 5%

$k$ regressors rejection rate mean $R^{2}$ median DW
1 0.67 0.25 0.30
2 0.85 0.40 0.42
3 0.92 0.51 0.55
4 0.95 0.58 0.68
5 0.98 0.65 0.80

Interpretation: the qualitative Granger-Newbold (1974) finding — rejection of a true null in most replications with one regressor, approaching certainty with five — is reproduced; mean $R^{2}$ climbs to 0.65 from pure noise. DW rises with $k$ only because more regressors absorb more of the wandering; it stays far below 2 and the $R^{2} > D W$ pattern is typical.

$k$ regressors	rejection rate	mean $R^{2}$	median DW
1	0.67	0.25	0.30
2	0.85	0.40	0.42
3	0.92	0.51	0.55
4	0.95	0.58	0.68
5	0.98	0.65	0.80

Experiment 3 — Yule’s correlation shapes ( $T = 10$ ; 20,000 reps; seed 2)

Sampling distribution of $r (x, y)$ for independent series of three integration orders

Series type $T$ $Pr (∣ r ∣ > 0.5)$ $Pr (∣ r ∣ > 0.9)$ SD of $r$
white noise $I (0)$ 10 0.14 0.00 0.33
random walk $I (1)$ 10 0.43 0.02 0.52
twice-cumulated $I (2)$ 10 0.84 0.50 0.82

Interpretation: Yule’s (1926) three cases (described in Granger 2003 - Spurious Regression from Yule to Granger-Newbold): concentrated around 0 for random series; spread over most of $(- 1, 1)$ once cumulated; U-shaped with half the mass beyond $∣ r ∣ > 0.9$ when cumulated twice. Yule’s 0.95 church-marriage/mortality correlation is ordinary under the $I (2)$ -like row.

Series type	$T$	$Pr (∣ r ∣ > 0.5)$	$Pr (∣ r ∣ > 0.9)$	SD of $r$
white noise $I (0)$	10	0.14	0.00	0.33
random walk $I (1)$	10	0.43	0.02	0.52
twice-cumulated $I (2)$	10	0.84	0.50	0.82

Experiment 4 — Engle-Granger test, spurious vs cointegrated ( $T = 200$ ; 2,000 reps; seed 3)

Levels OLS plus ADF(1) test on residuals, 5% critical value $\approx - 3.37$

Critical value: $- 3.37$ is the finite-sample ( $T \approx 200$ ) Engle-Granger/MacKinnon 5% value for two variables with a constant. The asymptotic value, about $- 3.34$ , is the one quoted in the Engle-Granger algorithm.

Pair median $\hat{β}$ SD of $\hat{β}$ mean $R^{2}$ median DW EG rejects “no cointegration”
independent RWs (true $β = 0$ ) 0.02 0.621 0.24 0.07 0.04
cointegrated, $y = 1 + 2 x + u$ , $u \sim A R (1)$ , $ϕ = 0.5$ 2.00 0.032 0.98 1.04 1.00

Interpretation: $\hat{β}$ in the spurious case is centred near 0 but with SD 0.62 that does not shrink with $T$ (Phillips); in the cointegrated case it is superconsistent (SD 0.03). The residual unit-root test, with Engle-Granger rather than Dickey-Fuller critical values (approximately the MacKinnon value for two variables with a constant), holds its size (0.04) and has full power here. DW alone would not separate them perfectly at small $T$ , but the gap (0.07 vs 1.04) is large.

Pair	median $\hat{β}$	SD of $\hat{β}$	mean $R^{2}$	median DW	EG rejects “no cointegration”
independent RWs (true $β = 0$ )	0.02	0.621	0.24	0.07	0.04
cointegrated, $y = 1 + 2 x + u$ , $u \sim A R (1)$ , $ϕ = 0.5$	2.00	0.032	0.98	1.04	1.00

Experiment 5 — two-step ECM on the cointegrated pair ( $T = 200$ ; 2,000 reps; seed 4)

$Δ y_{t} = μ + γ \overset{z}{^}_{t - 1} + δ Δ x_{t} + ε_{t}$ with $\overset{z}{^}_{t} = y_{t} - \overset{α}{^} - \hat{β} x_{t}$

Parameter Mean estimate SD
$γ$ (adjustment, true $- 0.5$ ) $- 0.522$ 0.063
$δ$ (short-run, true $2.0$ ) 1.999 0.074
median $t (\overset{γ}{^})$ $- 8.25$ —

Interpretation: the Engle-Granger two-step recovers the error-correction dynamics with a small finite-sample bias in $γ$ (about $- 0.02$ , from using estimated $\overset{z}{^}$ ). This is Granger’s “Putting the analysis in the form of an error-correction model resolves many of the difficulties found with spurious regression” (p. 363) in numbers.

Parameter	Mean estimate	SD
$γ$ (adjustment, true $- 0.5$ )	$- 0.522$	0.063
$δ$ (short-run, true $2.0$ )	1.999	0.074
median $t (\overset{γ}{^})$	$- 8.25$	—

Figures — what Granger chose not to draw

Granger replaced diagrams with “internal visualization” (p. 360; see Granger 2003 - Econometrics, Integrated Series and the String-of-Pearls Picture). The three figures below make the simulation’s claims visible. A second, self-contained script produces them: save it as spurious_plots.py and run python spurious_plots.py <output_dir> (requires numpy, matplotlib; about 15 s). Figure 2 uses the same seed and generator as Experiment 3, so its numbers match that table exactly.

"""Figures for the spurious-regression note. Requires numpy, matplotlib.
Writes three PNGs into the directory given as argv[1] (default: current dir)."""
import sys, os
import numpy as np
import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt
 
OUT = sys.argv[1] if len(sys.argv) > 1 else "."
BLUE, ORANGE = "#2a78d6", "#eb6834"
INK, MUTED, SURFACE, GRID = "#0b0b0b", "#52514e", "#fcfcfb", "#e4e3df"
plt.rcParams.update({
    "figure.facecolor": SURFACE, "axes.facecolor": SURFACE,
    "axes.edgecolor": MUTED, "axes.labelcolor": INK, "text.color": INK,
    "xtick.color": MUTED, "ytick.color": MUTED, "axes.grid": True,
    "grid.color": GRID, "grid.linewidth": 0.8, "axes.spines.top": False,
    "axes.spines.right": False, "font.size": 10,
})
 
def ols_t(y, x):
    X = np.column_stack([np.ones_like(x), x])
    b, *_ = np.linalg.lstsq(X, y, rcond=None)
    e = y - X @ b
    s2 = e @ e / (len(y) - 2)
    se = np.sqrt(s2 * np.linalg.inv(X.T @ X)[1, 1])
    r2 = 1 - e @ e / np.sum((y - y.mean()) ** 2)
    return b, b[1] / se, r2
 
# Figure 1: one spurious pair (first draw from seed 0; not selected for drama
# beyond taking the first draw of the generator)
rng = np.random.default_rng(0)
T = 100
x = np.cumsum(rng.standard_normal(T)); y = np.cumsum(rng.standard_normal(T))
b, t, r2 = ols_t(y, x)
fig, ax = plt.subplots(1, 2, figsize=(10, 3.8))
ax[0].plot(x, color=BLUE, lw=2, label="x (random walk)")
ax[0].plot(y, color=ORANGE, lw=2, label="y (independent random walk)")
ax[0].set_xlabel("t"); ax[0].set_ylabel("level"); ax[0].legend(frameon=False)
ax[0].set_title("Two independent random walks, T = 100", loc="left")
ax[1].scatter(x, y, s=16, color=BLUE, edgecolor=SURFACE, linewidth=0.8)
xs = np.linspace(x.min(), x.max(), 2)
ax[1].plot(xs, b[0] + b[1] * xs, color=INK, lw=2)
ax[1].set_xlabel("x"); ax[1].set_ylabel("y")
ax[1].set_title(f"Levels OLS: slope {b[1]:.2f}, t = {t:.1f}, R² = {r2:.2f}", loc="left")
fig.tight_layout(); fig.savefig(os.path.join(OUT, "Granger 2003 - spurious pair.png"), dpi=150)
print(f"Fig1 slope={b[1]:.3f} t={t:.2f} R2={r2:.3f}")
 
# Figure 2: Yule's three shapes (same generator as Experiment 3: seed 2, 20,000 reps, T=10)
rng = np.random.default_rng(2)
n, reps = 10, 20000
fig, ax = plt.subplots(1, 3, figsize=(10, 3.2), sharey=True)
for i, (order, name) in enumerate([(0, "white noise I(0)"), (1, "random walk I(1)"),
                                   (2, "twice-cumulated I(2)")]):
    rs = np.empty(reps)
    for r in range(reps):
        a = rng.standard_normal(n); c = rng.standard_normal(n)
        for _ in range(order):
            a = np.cumsum(a); c = np.cumsum(c)
        rs[r] = np.corrcoef(a, c)[0, 1]
    ax[i].hist(rs, bins=np.linspace(-1, 1, 41), color=BLUE, edgecolor=SURFACE, linewidth=1)
    ax[i].set_title(f"{name}\nSD(r) = {rs.std():.2f}", loc="left")
    ax[i].set_xlabel("sample correlation r (T = 10)")
    print(f"Fig2 {name}: P(|r|>0.9)={np.mean(np.abs(rs)>0.9):.2f} SD={rs.std():.2f}")
ax[0].set_ylabel("count of 20,000 pairs")
fig.tight_layout(); fig.savefig(os.path.join(OUT, "Granger 2003 - Yule correlation shapes.png"), dpi=150)
 
# Figure 3: median |t| in levels vs differences against T, with sqrt(T) reference
rng = np.random.default_rng(5)
Ts = [25, 50, 100, 200, 400, 800]
lev, dif = [], []
for T in Ts:
    tl, td = [], []
    for _ in range(2000):
        x = np.cumsum(rng.standard_normal(T)); y = np.cumsum(rng.standard_normal(T))
        tl.append(abs(ols_t(y, x)[1])); td.append(abs(ols_t(np.diff(y), np.diff(x))[1]))
    lev.append(np.median(tl)); dif.append(np.median(td))
    print(f"Fig3 T={T}: median|t| levels={lev[-1]:.2f} differences={dif[-1]:.2f}")
fig, ax = plt.subplots(figsize=(6.5, 4))
ax.plot(Ts, lev, "o-", color=BLUE, lw=2, ms=8, label="levels (spurious)")
ax.plot(Ts, dif, "o-", color=ORANGE, lw=2, ms=8, label="first differences")
ref = lev[0] * np.sqrt(np.array(Ts) / Ts[0])
ax.plot(Ts, ref, "--", color=MUTED, lw=1.5, label="∝ √T reference")
ax.axhline(1.96, color=MUTED, lw=1, ls=":")
ax.text(Ts[-1], 1.75, "|t| = 1.96", color=MUTED, ha="right")
ax.set_xscale("log"); ax.set_yscale("log")
ax.set_xticks(Ts); ax.set_xticklabels([str(t) for t in Ts])
ax.set_xlabel("sample size T (log scale)"); ax.set_ylabel("median |t| (log scale)")
ax.set_title("Spurious t grows like √T; differenced t does not", loc="left")
ax.legend(frameon=False)
fig.tight_layout(); fig.savefig(os.path.join(OUT, "Granger 2003 - t growth with T.png"), dpi=150)

Printed output (NumPy 2.2.6, Matplotlib 3.10.3):

Fig1 slope=-0.270 t=-6.75 R2=0.318
Fig2 white noise I(0): P(|r|>0.9)=0.00 SD=0.33
Fig2 random walk I(1): P(|r|>0.9)=0.02 SD=0.52
Fig2 twice-cumulated I(2): P(|r|>0.9)=0.50 SD=0.82
Fig3 T=25: median|t| levels=2.20 differences=0.70
Fig3 T=50: median|t| levels=3.25 differences=0.67
Fig3 T=100: median|t| levels=4.42 differences=0.68
Fig3 T=200: median|t| levels=6.66 differences=0.70
Fig3 T=400: median|t| levels=9.16 differences=0.67
Fig3 T=800: median|t| levels=12.56 differences=0.69

Figure 1 — one spurious pair

Reading: the first draw from seed 0, not a selected example. Two independent random walks of length 100 wander apart: $x$ drifts up and $y$ drifts down. The levels regression reports slope $- 0.27$ , $t = - 6.8$ and $R^{2} = 0.32$ , a “highly significant” negative relationship between series that share nothing. The scatter shows the cause: consecutive points are joined in time, so the cloud is a few long excursions, not 100 independent observations. That is Granger’s string of pearls (p. 360).

Figure 2 — Yule's three correlation shapes ( $T = 10$ , 20,000 pairs each)

Reading: the histograms behind Experiment 3. White noise gives a hump centred on 0 (SD 0.33, close to $1/ 10 - 1$ ). Random walks give an almost flat distribution over $(- 1, 1)$ (SD 0.52). Twice-cumulated series give a U shape with half the mass beyond $∣ r ∣ > 0.9$ (SD 0.82). These are the bell, flat and U shapes Yule (1926) reported for random series, conjunct series with random differences, and conjunct series with conjunct differences; his own SDs were about 0.32 and 0.51-0.55 for the first two (Granger 2003 - Spurious Regression from Yule to Granger-Newbold).

Figure 3 — the spurious $t$ grows like $T$ (2,000 pairs per $T$ , seed 5)

Reading: on log-log axes the median levels $∣ t ∣$ follows the dashed $T$ reference line from 2.2 at $T = 25$ to 12.6 at $T = 800$ , as Phillips (1986) predicts. The differenced regression stays flat near 0.67-0.70, the median of $∣ N (0, 1) ∣$ . More data makes the spurious finding more convincing, which is why a larger sample is no protection.

Examples

Adapting the script as a persistence-matched null for a real analysis

Setup: an MMM or driver regression of a weekly KPI ( $T \approx 104$ ) on a candidate series returns $t = 5.1$ in levels.

Steps: (1) fit an AR(1) or ARIMA(0,1,0) to the candidate series; (2) generate 2,000 surrogate candidates from that fitted process, independent of the KPI; (3) rerun the same regression (same controls, same transformations) with each surrogate; (4) report where 5.1 falls in the surrogate distribution of $∣ t ∣$ . With random-walk-like candidates at $T \approx 100$ , the median surrogate $∣ t ∣$ is about 4.6 (Exp. 1), so $t = 5.1$ is unremarkable — evidence comparable to a coin flip, not to $p < 1 0^{- 6}$ .

Why it works: the null distribution inherits the persistence that invalidates the textbook $t$ distribution. The same logic (calibrate a procedure on data where the truth is known) underlies Designing Simulated-Data Experiments and Zhang Vaver 2017 - Aggregate Marketing System Simulator - Overview.

Connections

Theory being illustrated: Granger 2003 - Spurious Regression from Yule to Granger-Newbold, Granger 2003 - Cointegration and the Error-Correction Representation.
Simulation-study design and Monte Carlo error: Morris White Crowther 2019 - Simulation Studies (ADEMP) - Overview, Designing Simulated-Data Experiments.
Time-ordered validation as the out-of-sample counterpart: Bergmeir Hyndman Koo 2018 - CV for Time Series - Overview, Forecast Evaluation and Backtesting.
Marketing time-series modelling with unit roots: Single Marketing Time Series, Multivariate Persistence and Cointegration.

Second Brain

Explorer

Granger 2003 - Spurious Regression Monte Carlo in Python

Granger 2003 - Spurious Regression Monte Carlo in Python

Overview

Main Content

The script

Experiment 1 — levels vs differences ( $T = 50, 100, 500$ ; 5,000 reps; seed 0)

Experiment 2 — Granger-Newbold design with $k$ regressors ( $T = 50$ ; 5,000 reps; seed 1)

Experiment 3 — Yule’s correlation shapes ( $T = 10$ ; 20,000 reps; seed 2)

Experiment 4 — Engle-Granger test, spurious vs cointegrated ( $T = 200$ ; 2,000 reps; seed 3)

Experiment 5 — two-step ECM on the cointegrated pair ( $T = 200$ ; 2,000 reps; seed 4)

Figures — what Granger chose not to draw

Examples

Connections

See Also

Graph View

Table of Contents

Backlinks

Second Brain

Explorer

Granger 2003 - Spurious Regression Monte Carlo in Python

Granger 2003 - Spurious Regression Monte Carlo in Python

Overview

Main Content

The script

Experiment 1 — levels vs differences (T=50,100,500; 5,000 reps; seed 0)

Experiment 2 — Granger-Newbold design with k regressors (T=50; 5,000 reps; seed 1)

Experiment 3 — Yule’s correlation shapes (T=10; 20,000 reps; seed 2)

Experiment 4 — Engle-Granger test, spurious vs cointegrated (T=200; 2,000 reps; seed 3)

Experiment 5 — two-step ECM on the cointegrated pair (T=200; 2,000 reps; seed 4)

Figures — what Granger chose not to draw

Examples

Connections

See Also

Graph View

Table of Contents

Backlinks

Experiment 1 — levels vs differences ( $T = 50, 100, 500$ ; 5,000 reps; seed 0)

Experiment 2 — Granger-Newbold design with $k$ regressors ( $T = 50$ ; 5,000 reps; seed 1)

Experiment 3 — Yule’s correlation shapes ( $T = 10$ ; 20,000 reps; seed 2)

Experiment 4 — Engle-Granger test, spurious vs cointegrated ( $T = 200$ ; 2,000 reps; seed 3)

Experiment 5 — two-step ECM on the cointegrated pair ( $T = 200$ ; 2,000 reps; seed 4)