Replication — Bai & Perron (2003)¶

Bai & Perron (2003, J. Applied Econometrics 18(1)) is the canonical paper for dating multiple structural breaks, and bai_perron implements its machinery. The natural validation target is therefore the paper's own empirical application: a mean-shift model of the US ex-post real interest rate, quarterly 1961Q1–1986Q3 (T = 103) — the Garcia-Perron (1996) series.

The published answer: the level of the real rate breaks at 1972:3 and 1980:3 (the partition BP's BIC and LWZ criteria select), with the paper's own HAC-robust sequential procedure adding a third, less sharply dated break at 1966:4. The two-break segment means are 1.36, −1.80, 5.64: mildly positive real rates through 1972, negative real rates through the 1970s, and sharply positive rates after the Volcker disinflation. The 1980:3 break is dated to within a few quarters; the earlier breaks are not.

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

The data ships as RealInt in R's strucchange package (byte-identical to real in Perron's own mbreaks package) and is committed to the repository at fixtures/realint_bai_perron.csv with attribution, so this runs fully offline — tsecon ships no data loaders. The whole analysis is one call on an intercept-only design (mean shift = the constant is the only switching coefficient):

bp = tsecon.bai_perron(rate, np.ones((len(rate), 1)), max_breaks=5, trim=0.15)

The result¶

Break dates — exact match at every partition size

partition	published (BP 2003)	tsecon (dynamic program)
m = 1	1980:3	1980Q3
m = 2	1972:3, 1980:3	1972Q3, 1980Q3
m = 3	1966:4, 1972:3, 1980:3	1966Q4, 1972Q3, 1980Q3

The global SSR path (ssr_path = 1214.92, 645.00, 455.95, 445.18) matches R strucchange's reported RSS for the same partitions to every printed digit.

How many breaks — same procedure, one honest difference

criterion	published count	tsecon
sequential supF(l+1\|l), 5%, HAC-robust F (the paper's headline)	3	not implemented (classical F only)
sequential supF(l+1\|l), 5%, classical F	—	2
BIC, LWZ (BP's information criteria)	2	not implemented

bai_perron runs exactly the paper's sequential procedure at the published 5% critical values (8.58, 10.13, 11.14 for q = 1, trim 0.15 — reproduced verbatim), but computes classical supF statistics, where the paper's published statistics are HAC-robust. The two sequences on this data:

	supF(1\|0)	supF(2\|1)	supF(3\|2)	supF(4\|3)
paper (HAC, prewhitened)	57.91	33.93	14.73	0.03
tsecon (classical)	89.24	52.20	7.41	0.04
5% critical value	8.58	10.13	11.14	11.83

Both sequences scream "at least two breaks". They part company at supF(3|2): the HAC statistic clears the bar and BP's sequential procedure selects three breaks, while the classical statistic does not, so tsecon stops at two — the same count BP's own BIC and LWZ select (BP note the criteria are biased downward under serial correlation, which is why they prefer three). tsecon's classical sequence is not a private convention: Perron's mbreaks package with robust=0 produces 89.245, 52.204, 7.414 — matching tsecon to the third decimal — and R strucchange, also classical, selects the same two-break model by BIC. The third break tsecon would add is available and exact: break_dates_by_m[2] is 1966Q4, 1972Q3, 1980Q3.

Segment means — match to published rounding

regime	published mean (HAC se)	tsecon mean (classical OLS se)
1961Q1–1972Q3	1.36 (0.16)	1.3550 (0.188)
1972Q4–1980Q3	−1.80 (0.51)	−1.7961 (0.452)
1980Q4–1986Q3	5.64 (0.60)	5.6429 (0.566)

The point estimates are segment averages and agree with the published values to their printed rounding (and with strucchange's coefficients to 1e-6). The standard errors are different constructions — per-regime classical OLS here, HAC in the paper — and are reported side by side, not compared. For the three-break partition the segment means 1.8236, 0.8661, −1.7961, 5.6429 likewise match the published 1.82, 0.87, −1.80, 5.64.

Break-date confidence intervals — different estimator, stated plainly

break	tsecon 90%	tsecon 95%	published 95% (BP 2003)
1972Q3	1971Q2–1973Q4	1970Q4–1974Q2	1970:3–1972:4 (3-break model, heterogeneity-robust HAC)
1980Q3	1980Q1–1981Q1	1980Q1–1981Q1	(no independently verified published value)

These columns are not the same estimator, on three counts, all documented on the structural-breaks model card: tsecon ships the Bai (1997) homogeneous case with classical variance, the paper's CIs are the heterogeneity-robust variant with HAC variance; the paper's CIs condition on the three-break partition, tsecon reports CIs for its selected two-break model; and this object is famously fragile — the JAE replication study (Zeileis & Kleiber 2005) reports that some of BP's published CIs could not be reproduced in strucchange under any settings, with the discrepancy attributed to numerical problems in BP's original GAUSS code. Every implementation of this CI disagrees at the level of a few quarters on this series (strucchange's homogeneous 95% for 1972Q3: 1971Q2–1973Q4; mbreaks with the paper's settings: 1970Q1–1973Q2).

What does replicate — and it is the paper's substantive point about the CIs — is the asymmetry of precision: the 1980:3 Volcker break is pinned to a 5-quarter 95% window while the 1972:3 break needs 15 quarters, in every implementation including this one.

Cross-implementation validation¶

Because the published table mixes HAC statistics tsecon deliberately does not compute, the replication was additionally cross-checked, number by number, against the two maintained R implementations run on the identical committed series:

strucchange (breakpoints(RealInt ~ 1, h = 15), classical, the implementation validated against BP in the JAE replication section): same partitions for m = 1..3, same RSS path to every printed digit, same coefficients to 1e-6, same BIC-selected count (2).
mbreaks (Perron's own R port, mdl('rate', eps1 = 0.15)): with the paper's HAC settings it reproduces the published table (supF 57.91/33.93/ 14.73, sequential → 3, KT/BIC/LWZ → 2, means 1.355/−1.796/5.643); with robust = 0 its classical supF sequence matches tsecon's to the third decimal.

One geometry detail for anyone re-running this: with trim = 0.15 tsecon uses minimum segment length h = ceil(0.15 · 103) = 16 where the R packages use h = 15. This changes nothing through m = 3 (identical partitions and SSRs); at m = 4 and 5 — beyond anything selected — the admissible sets differ and so do the optimal partitions.

What this is, and is not¶

This reproduces Bai & Perron's break dates exactly at every partition size the paper reports, their segment means to published rounding, their published critical values verbatim, and their information-criteria break count via the same sequential procedure the paper uses — with the variance-estimator caveat above. It does not reproduce the paper's HAC-robust test statistics or its heterogeneity-robust break-date CIs, because tsecon does not implement those variants; where a published number is not comparable, the tables above say so rather than stretching a match.

Citation. Bai, J. & Perron, P. (2003), "Computation and Analysis of Multiple Structural Change Models," Journal of Applied Econometrics 18(1):1–22. Data: Garcia, R. & Perron, P. (1996), REStat 78:111–125, as distributed in R strucchange (Zeileis et al.) and mbreaks (Nguyen, Perron & Yamamoto). Validation study: Zeileis, A. & Kleiber, C. (2005), "Validating Multiple Structural Change Models — A Case Study," J. Applied Econometrics 20:685–690.