Replication — Hansen (1999): SETAR on the sunspot numbers¶

The SETAR model was invented for this series. Tong & Lim (1980, JRSS-B) introduced threshold autoregression with the annual Wolf sunspot numbers as the headline application, and the series has been the canonical SETAR test bed ever since. This page replicates the two-regime sunspot SETAR that tsecon.setar can express exactly: Hansen (1999, "Testing for linearity", Journal of Economic Surveys 13(5):551–576), who refits the series with a common AR order p = 11 in both regimes, one threshold, one delay — and reports a threshold of 7.4 (on the square-root scale), a delay of 2, and a bootstrap rejection of linearity with p ≈ 0.03.

.venv/bin/python docs/examples/replication_setar_sunspots.py

The data is the annual Wolf (Zürich) sunspot series, 1700–1988 — the sample of Tong (1990, Appendix 3) that Hansen used — committed at fixtures/sunspots_tong.csv (public observational data, vendored from statsmodels.datasets.sunspots with attribution), so this runs fully offline — tsecon ships no data loaders.

The result¶

quantity	Hansen (1999)	tsecon	verdict
sample	1700–1988 annual means	same (289 years; 278 usable after 11 lags)	exact
transform	y = 2(√(1+N) − 1)	same	exact
AR order p, both regimes	11 (with intercepts)	11	exact
trimming	≥ 10% of observations per regime	10% (identical fit at 15%: the split is interior)	match
delay d	2	2 (searched over {1, 2}; also wins over 1..11)	exact
threshold γ̂ (transformed)	7.4	7.4234	match at published precision
threshold (raw counts)	≈ 21	21.2	match
linearity: bootstrap p	≈ 0.03	0.022–0.032 across seeds (0.025 at the pinned seed)	match

Two of these rows deserve their honesty spelled out:

The threshold row. A SETAR threshold is identified only up to the gap between adjacent order statistics of the threshold variable — every γ in [7.4234, 7.4446) (raw counts [21.2, 21.3)) produces the identical fitted model, and implementations legitimately differ in which endpoint of that flat interval they print. Hansen's published two-digit 7.4 lies in exactly this gap, so "rounds to 7.4 and the gap is [21.2, 21.3)" is the strongest claim a replication can honestly pin — and it is what the regression test pins, alongside a 1e-10 pin on tsecon's own 7.4234 so the fit cannot drift while the round-to-published check still passes.

The p-value row. tsecon's setar_test implements the homoskedastic residual bootstrap for Hansen's F12 = n(S₀ − S₁)/S₁ (here 69.75 at d = 2, with S₀ = 1134.85 from the AR(11) and S₁ = 907.24 from the SETAR; at the rejected delay d = 1 the statistic drops to 40.09 and does not reject). The paper reports p ≈ 0.03 for the sunspot series and also computes heteroskedasticity-robust variants that moderate the evidence; the pinned claim is therefore the verdict — rejection at 5% — with the seeded p-value landing at 0.02–0.03, not bit-equality with any one of the paper's bootstrap schemes.

The exact spec, in tsecon¶

import numpy as np, csv

rows = [r for r in csv.reader(open("fixtures/sunspots_tong.csv"))
        if r and not r[0].startswith("#")][1:]
counts = np.array([float(r[1]) for r in rows])       # raw Wolf numbers
y = 2.0 * (np.sqrt(1.0 + counts) - 1.0)              # Ghaddar-Tong (1981)

r = tsecon.setar(y, p=11, delays=[1, 2], trim=0.10)  # Hansen's search space
r["delay"], r["threshold"]                           # 2, 7.4234
t = tsecon.setar_test(y, p=11, delay=2, trim=0.10, n_boot=199, seed=7)
t["stat"], t["p_value"]                              # 69.75, 0.025

delays=[1, 2] reproduces Hansen's joint search over the delay (tsDyn's replication of the same example searches thDelay = 0:1, which is d ∈ {1, 2} in the literature's convention — tsDyn counts the delay from the first lag). The choice is not load-bearing: an unrestricted search over d = 1..11 also lands on d = 2, and the test suite checks that too.

The script also prints tsecon's per-regime coefficients, standard errors, regime split (86 low / 192 high, 31%/69%) and per-regime variances (6.12 / 2.52 — the low regime is genuinely noisier), for side-by-side reading against the paper's Table 2. Those are printed, not pinned: the pinned quantities are the ones whose published values this page states above.

An information-criteria footnote, because it is honest and instructive: under the shared n·ln(SSR/n) + penalty convention, AIC prefers the SETAR(2) (378.8 vs 415.0) while BIC prefers the AR(11) (458.6 vs 469.5) — the threshold model carries 25 parameters and BIC charges for them. tsDyn's executed replication of the same example orders the criteria the same way. The reason the SETAR still wins the argument is the test, not the IC: the F12 bootstrap rejects linearity, which is precisely why Hansen frames the problem as testing rather than criterion-shopping.

What this is, and is not¶

This reproduces the sunspot SETAR(2) quantities Hansen (1999) reports — threshold, delay, and the linearity verdict — on the same data, transform, order, and trimming. It is a replication of the body-text estimates, not a bitwise port of the paper's tables: Table 2's per-regime least-squares coefficients are printed for comparison but not pinned, and the paper's several bootstrap/asymptotic p-value variants are represented here by the one scheme tsecon implements.

The deeper honesty note is about which published sunspot SETAR is being replicated. The most famous ones — Tong & Lim's (1980) SETAR(2; 3, 11) with d = 8, and Ghaddar & Tong's (1981) SETAR(2; 4, 12) — use regime-specific AR orders. tsecon.setar deliberately fits a common order p in both regimes (the Hansen 1997/1999 design its estimator and test are built on), so those models are expressible only as constrained versions of an order-11/12 fit — the constrained-vs-unconstrained relationship the literature itself notes. Rather than fudge a comparison across that gap, this page targets the published fit whose specification the shipped estimator expresses exactly and says so. The transcription golden (fixtures/setar.json) already pins the algorithm at 1e-10; what this page adds is the anchor to a published fit of real data.

Citation. Hansen, B. E. (1999), "Testing for Linearity," Journal of Economic Surveys 13(5):551–576. Tong, H. & Lim, K. S. (1980), "Threshold Autoregression, Limit Cycles and Cyclical Data," JRSS-B 42(3):245–292. Ghaddar, D. K. & Tong, H. (1981), "Data Transformation and Self-Exciting Threshold Autoregression," JRSS-C 30(3). Tong, H. (1990), Non-linear Time Series: A Dynamical System Approach, OUP. Sunspot data: public-domain Wolf/Zürich annual means (NOAA NGDC; the series is maintained today by SILSO, Royal Observatory of Belgium), vendored via statsmodels.