## _wp0875 — Size Tests—Weiner DGP with No Breaks

## Source details

**Canonical URL:** [_wp0875 — Size Tests—Weiner DGP with No Breaks](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp0875.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp0875.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp0875.pdf.json)

---

### Introduction
- Objective: extend Bai and Perron (1998, 2003a, 2003b — BP) methodology to improve small-sample performance (time series with as low as 50 observations).
- Main sources of BP test distortion identified:
  - 1) a small sample size,
  - 2) a small break size,
  - 3) a small segment size and breaks clustering,
  - 4) the use of heteroskedasticity and autocorrelation corrections.
- Key extensions proposed:
  - Use sample-specific (bootstrap) critical values estimated from a mimicking process under the null instead of BP’s asymptotic (Wiener) critical values.
  - Investigate misspecification of the DGP and treatment of serial correlation via parametric modeling versus HAC corrections.
  - Examine small-sample performance of BP’s sequential supF and sequential Dmax tests, with attention to segment-size effects.
- Target empirical context: sample size of 50 observations, with true number of breaks up to two.

### The BP methodology (summary)
- Two-part structure:
  - Identification of any number of breaks (global minimizing or sequential procedures).
  - Testing significance of identified breaks using asymptotic critical values (Wiener processes).
- Finite-sample complications:
  - Global minimization can create non-nested break sets across t and t+1 breaks, so SSR ratios depend on sample-specific parameters.
  - BP’s sequential remedy reduces segment size, worsening small-sample performance.
- Autocorrelation and heteroskedasticity in BP:
  - Corrections via Newey-West or including lagged dependent variables.
  - BP provide asymptotic critical values for a Wiener process and recommend larger minimal segment sizes for small samples and using corrections only when strongly warranted.

### A. The model and testing framework
- Pure structural change model (dropping xβ′): tt yx z u β δ ′′ = ++ (equation (1) in source); m is number of breaks; u can be non iid under the null.
- Break-location procedures:
  - Global minimization: choose break locations iT, 1,...,i m to minimize SSR via dynamic programming.
  - Sequential approach: find single-break minimizer, then additional breaks within partitions.
- Testing for existence of breaks:
  - sup F(0,m) compares SSE under null (no breaks) and alternative (k breaks).
  - Double-maximum tests (Dmax): maxDaFn = max1, , maxsup (0, ) n nm DaFn = = K — variants maxUD and maxWD depending on weights na.
  - F(1l+|l) (l versus l+1 breaks): perform l+1 tests of zero versus one break across partitions, reject if any partition rejects.
- Criteria for number of breaks:
  - Information criteria alternatives: BIC (Yao 1988); LWZ (Liu et al. 1997).
  - BP sequential approach is robust to heteroskedasticity and autocorrelation via embedded variance-covariance estimation; information criteria do not adjust for these.

### A modified BP methodology for small samples
- Two main extensions:
  1) Sample-specific critical values (bootstrap mimicking process)
     - Procedure:
       - Define a mimicking process with structural parameters estimated under the null.
       - Simulate with bootstrapped residuals to obtain Monte Carlo sample-specific critical values each time a test is run.
       - When testing 2 breaks versus 1 break, include the estimated single break in the mimicking process so critical values account for break size/location and segment lengths/variances.
     - Objectives attained:
       - 1) correction for small sample size,
       - 2) reduction in misspecification bias,
       - 3) control for segment-specific heteroskedasticity.
     - Advantages of bootstrapped segment-specific residuals:
       - Calculate sample-size-specific critical values;
       - Relax normality assumption of residuals;
       - Account for segmental heteroskedasticity.
  2) Treatment of serial correlation (two contrasting approaches)
     - Parametric approach:
       - Model serial correlation explicitly (e.g., AR(1)); BP allow adding 1t y − as regressor so errors become iid and breaks regression is no longer a pure structural model.
     - HAC robust errors approach:
       - When serial correlation is complex/unknown, model as a Wiener process and use an HAC estimate of the variance-covariance matrix.

### Methods comparison and small-sample considerations
- Sample-specific critical values generally produce better test results even with some misspecification.
- Parametric method:
  - Superior when order of serial correlation is known.
  - Potential problems: (i) overspecification; (ii) number of introduced parameters may be too large relative to sample size.
  - Estimation of at least two parameters (parametric) versus one (non-parametric robust) can reduce power in small samples.
  - Sequential procedures aggravate small-sample problems because statistics are estimated in partitions with fewer observations.
  - For AR(1) with a large β and at least one significant break, power erodes as β increases because F-statistics under the null are contaminated by fitted AR(1).

### Alternative statistics for finding multiple breaks
- Sequential UDmax:
  - UDmax(n) = max{supF(0,1), supF(0,2),..., supF(0,n)}.
  - Distribution unknown; critical values obtained by bootstrapping.
  - Captures more breaks when 1-break alternatives are rejected but 2-break alternatives accepted.
- Global UDmax (GUDMax):
  - Uses whole sample and embeds n-1 breaks at n-th step, generating critical values with n-1 breaks embedded.
  - Distribution depends on size of embedded breaks and sample-specific parameters; critical values estimated “on the fly”.
  - Computationally intensive; tends to be conservative in practice.
- Sequential procedures issue:
  - Globally SSR-minimizing n+1 breaks will not generally nest globally-SSR-minimizing n breaks, potentially leading to low power for sequential UDmax.

### Size tests — objectives and setup
- Purpose:
  - (i) Assess improvement in test size owing to sample-specific critical values.
  - (ii) Infer whether parametric or non-parametric modeling performs better when serial correlation is present.
- Experiments often use sample size 50 and true size set to 10 percent.

### D. Empirical results — Size tests (selected exact entries)
- Table 1: Size Tests—Weiner DGP With No Breaks (True size is 10 percent; 500 MC simulations; sample-specific CVs generated with 500 MC within each of the 500 runs)
  - Asymptotic 1/ vs Sample Specific:
    - SupF(0,1) 0.05 0.08
    - SupF(0,2) 0.05 0.09
    - SupF(0,3) 0.04 0.10
    - UDmax(3) 0.08 0.08
- Table 2: Size Tests with Autocorrelated DGP with No Breaks and Parametric Estimation (True size is 10 percent; 500 MC simulations)
  - Panel A (True model: yt = 0.9*yt-1 + ut, ut ~ N(0,1) and i.i.d.)
    - Asymptotic 1/ vs Sample Specific:
      - SupF(0,1) 0.26 0.15
      - SupF(0,2) 0.36 0.14
      - SupF(0,3) 0.41 0.15
      - UDmax(3) 0.44 0.14
  - Panel B (True model: yt = 0.5*yt-1 + ut, ut ~ N(0,1) and i.i.d.)
    - Asymptotic 1/ vs Sample Specific:
      - SupF(0,1) 0.08 0.10
      - SupF(0,2) 0.10 0.11
      - SupF(0,3) 0.09 0.09
      - UDmax(3) 0.15 0.10
- Table 3: Size Tests with Autocorrelated DGP with No Breaks and Standard Errors Robust to Serial Correlation (True size is 10 percent; 500 MC simulations)
  - Panel A (True model: yt = ut, ut = 0.9*ut-1 + et, et ~ N(0,1) and i.i.d.)
    - Asymptotic 1/ vs Sample Specific:
      - SupF(0,1) 0.56 0.59
      - SupF(0,2) 0.81 0.81
      - SupF(0,3) 0.89 0.89
      - UDmax(3) 0.87 0.85
  - Panel B (True model: yt = ut, ut = 0.5*ut-1 + et, et ~ N(0,1) and i.i.d.)
    - Asymptotic 1/ vs Sample Specific:
      - SupF(0,1) 0.20 0.26
      - SupF(0,2) 0.32 0.34
      - SupF(0,3) 0.34 0.34
      - UDmax(3) 0.41 0.34
- Table 4: Size Tests with DGP with No Breaks and Over- and Under-Specification of Degree of Autocorrelation (True size is 10 percent; 500 MC simulations)
  - Overspecification:
    - SupF(0,1) 0.06 0.09
    - SupF(0,2) 0.03 0.09
    - SupF(0,3) 0.04 0.09
    - UDmax(3) 0.08 0.08
  - Underspecification:
    - SupF(0,1) 0.90 0.94
    - SupF(0,2) 0.96 0.97
    - SupF(0,3) 0.96 0.98
    - UDmax(3) 0.97 0.97
  - Additional block:
    - SupF(0,1) 0.38 0.52
    - SupF(0,2) 0.55 0.66
    - SupF(0,3) 0.54 0.68
    - UDmax(3) 0.61 0.60

- Size-test conclusions:
  - Using asymptotic Wiener critical values can cause substantial size distortions in small samples and non-Wiener DGPs.
  - Sample-specific critical values generally improve size control.
  - When serial correlation is present:
    - Parametric estimation (correct AR(1)) with sample size 50 and theoretical size 10 percent:
      - Asymptotic actual size ranges: 8-15 percent for autoregressive parameter 0.5; 26-44 percent for 0.9.
      - Sample-specific actual sizes: 9-11 percent (β = 0.5) and 14-15 percent (β = 0.9).
    - Non-parametric robust-errors approaches perform poorly in small samples with high autocorrelation (e.g., UDmax(3) p-values: 0.87 percent asymptotic and 0.85 percent sample-specific when autocorrelation high).
  - If serial correlation is present and its order is known, parametric modeling is preferred for size considerations.

### E. Empirical results — Power tests (selected exact entries and conclusions)
- Simulation design:
  - Sample size 50; DGP either Wiener or AR(1) with autoregressive parameter 0.5 or 0.9.
  - Mimicking process: either Wiener with robust errors or AR(1).
  - Statistics compared:
    - (i) asymptotic sequential supF (Bai-Perron),
    - (ii) sample-specific sequential supF,
    - (iii) sequential UDmax,
    - (iv) global UDMax (GUDMax).
  - Break structures: symmetric U-form, asymmetric U-form, downhill; break sizes measured in multiples of standard deviations (e.g., {1,0,1}).
  - Loss function: L = (1/M) * sum_{i=1}^M |N - N̂_i|.
  - True size is 10 percent; 500 MC simulations; sample-specific CVs generated with 500 MC within each run.

- Panel A (True model: yt = ut, ut ~ N(0,1) and i.i.d.) — selected probabilities and loss values (columns correspond to BP asympt. 1/, S supF 2/, SUDmax 3/, AUDmax 4/ in two blocks)
  - Prob k=0 0.68 0.11 0.03 0.03 0.77 0.16 0.05 0.05
  - Prob k=1 0.12 0.10 0.10 0.34 0.14 0.12 0.16 0.41
  - Prob k=2 0.18 0.63 0.70 0.55 0.09 0.57 0.62 0.50
  - Prob k≥3 0.01 0.16 0.16 0.08 0.00 0.15 0.17 0.04
  - Loss value 5/ 1.51 0.49 0.33 0.48 1.68 0.58 0.43 0.55
  - Additional blocks in Panel A preserve exact numeric strings in source.

- Panel B (True model: yt = ut, ut = 0.5*ut-1 + et) — selected probabilities and loss values
  - Prob k=0 0.70 0.26 0.06 0.06 0.82 0.31 0.18 0.18
  - Prob k=1 0.23 0.19 0.15 0.58 0.14 0.34 0.41 0.49
  - Prob k=2 0.07 0.33 0.41 0.32 0.04 0.28 0.32 0.28
  - Prob k≥3 0.01 0.22 0.38 0.04 0.00 0.07 0.09 0.04
  - Loss value 5/ 1.63 0.94 0.66 0.75 1.78 1.03 0.86 0.91

- Panel C (True model: yt = ut, ut = 0.9*ut-1 + et) — selected probabilities and loss values
  - Prob k=0 0.39 0.11 0.01 0.01 0.59 0.20 0.08 0.08
  - Prob k=1 0.31 0.17 0.07 0.62 0.32 0.49 0.67 0.47
  - Prob k=2 0.21 0.28 0.22 0.30 0.08 0.22 0.21 0.22
  - Prob k≥3 0.09 0.44 0.70 0.07 0.00 0.08 0.04 0.23
  - Loss value 5/ 1.18 0.83 0.79 0.70 1.51 0.98 0.86 0.85

- Power-test conclusions (from simulation evidence):
  - Sample-specific critical value procedures broadly and considerably outperform asymptotic procedures on power.
  - For finding the correct number of breaks, Wiener modeling with robust errors deals with autocorrelation better than AR(1) modeling for power.
  - Among sample-specific approaches, sequential UDmax is best-performing in a majority of cases.
  - GUDmax performs best when DGP has high autocorrelation; it is substantially more conservative (tends to underestimate number of breaks) while sequential UDmax is more likely to overestimate.
  - Sequential UDmax and GUDmax maintain better power than BP asymptotic statistic even for large autocorrelation.
  - If high autocorrelation is suspected, non-parametric (robust errors) approach may yield better power.

### Appendix — Algorithms (size and power test sketches)
- Size test algorithm (selected steps):
  - 0. Select model parameters: generated process nature (Wiener or AR(1)), mimicking process nature, maximum number of breaks, significance level.
  - 1. Simulate N1 initial generated processes with no breaks.
  - 2. Use global minimizer to determine locations and sizes of potential breaks; obtain statistics.
  - 3. Estimate each generated process according to mimicking process under null of no breaks; extract residuals.
  - 4. Bootstrap to simulate N2 mimicking processes for each generated process using residuals and estimated parameters.
  - 5. Obtain statistics for mimicking processes, sort, and find critical values.
  - 6. Determine significance of original statistic.
  - 7–8. Count occurrences and compute test size as proportion identifying at least one break.
- Power test procedures (sequential UDmax and global UDmax) are described in step-by-step form (steps 0–13) mirroring the size test flow but embedding estimated breaks into mimicking processes and iterating across segments.
- Implementation note: Appendix I outlines algorithms in more detail. GAUSS programs are available from the authors on request.

### Summary and conclusions
- Focus: small samples (sample size 50) with up to two true breaks.
- Main innovations and findings:
  - Sample-specific critical values via mimicking-null Monte-Carlo simulation substantially improve both test size and power in small samples across nearly all tested DGP specifications; computationally intensive but effective.
  - Treatment of serial correlation:
    - For size, parametric AR(1) modeling is preferred when serial correlation order is known.
    - For power (finding correct number of breaks), Wiener modeling with robust errors performs better than AR(1).
  - Break-number selection alternatives:
    - Sequential UDmax (tests 0 breaks vs unknown number) — generally best-performing across many cases.
    - Global UDmax (GUDMax) — performs best under high autocorrelation; conservative tendency (underestimates number of breaks).
  - Both UDmax variants generally perform substantially better than BP asymptotic procedures in the small-sample experiments reported.

*Source: _wp0875 - 1. Size Tests—Weiner DGP with No Breaks ..........................................................................21*

### 1. Size Tests—Weiner DGP with No Breaks ..........................................................................21

### 1. Size Tests—Weiner DGP with No Breaks ..........................................................................21

### Introduction
- Objective: extend Bai and Perron (1998, 2003a, 2003b — BP) methodology to improve small-sample performance (time series with as low as 50 observations).
- Main sources of test distortion in BP (BP (2006) simulation):
  - 1) a small sample size,
  - 2) a small break size,
  - 3) a small segment size and breaks clustering,
  - 4) the use of heteroskedasticity and autocorrelation corrections.
- Key extensions proposed:
  - Use of sample-specific (bootstrap) critical values estimated from a mimicking process under the null instead of relying on BP’s asymptotic (Wiener) critical values.
  - Investigation of misspecification of the data generating process (DGP) and treatment of serial correlation via parametric modeling versus HAC corrections.
  - Examination of small-sample performance of BP’s sequential supF test and the sequential Dmax test, with special attention to segment-size effects.
- Targeted empirical context: sample size of 50 observations, with true number of breaks up to two (inspired by Berg, Ostry and Zettelmeyer, 2006).

### The BP methodology (summary)
- Two-part structure:
  - Identification of any number of breaks in a time series (global minimizing or sequential procedures).
  - Testing the statistical significance of identified breaks using asymptotic critical values (generated for Wiener processes).
- Finite-sample complication:
  - When breaks are found by global minimization, hypotheses for sequential testing can be non-nested (the set of t breaks may not be a subset of t+1 breaks), so SSR ratios do not asymptotically follow the usual F-distribution and depend on sample-specific parameters (e.g., break size).
  - BP’s solution: test for one break versus zero breaks within segments formed by existing breaks — this avoids non-nestedness but reduces segment size, worsening small-sample performance.
- BP’s accommodation of autocorrelation and heteroskedasticity:
  - BP allow corrections via Newey-West or including lagged dependent variables as regressors.
  - BP provide asymptotic critical values for a Wiener process and recommend larger minimal segment sizes for small samples and using autocorrelation/heteroskedasticity corrections only when strongly warranted.

A. The model
- Pure structural change model used in this paper (dropping xβ′ so all coefficients may change):
  - tt yx z u β δ ′′ = ++  (equation (1) in source)
  - m is number of breaks; t indexes observations; u can be non iid under the null.
- Break-location procedures:
  - Global minimization: choose break locations iT, 1,...,i m to minimize SSR via dynamic programming.
  - Sequential approach: find single-break minimizer, then find additional breaks within resulting partitions.
- Trade-offs:
  - Global minimization asymptotically selects the largest breaks but creates non-nestedness across numbers of breaks, complicating sequential inference.
  - Sequential methods may select smaller segments and can suffer when segment sizes are very small.

B. Testing for the existence of breaks
- BP generalize Andrews (1993) tests for multiple breaks and aim for robustness to serial correlation and heterogeneity under the null.

B.1. Zero versus a fixed number of breaks
- Test: F-ratio comparing SSE under null (no breaks) and alternative (k breaks) — denoted sup F(0, m) for global minimized breaks.
- Practical complication: estimation of variance-covariance matrix for δ in presence of autocorrelation and heteroskedasticity; BP propose a simpler asymptotically equivalent estimator, but this can distort size and power in small samples.

B.2. Zero versus an unknown number of breaks
- Double-maximum tests (Dmax) based on sup F(0, m):
  - maxDaFn = max1, , maxsup (0, ) n nm DaFn = = K  (equation (2) in source)
  - If weights na = 1 for all n, the statistic is maxUD; if na are chosen as functions of asymptotic critical values to equalize marginal p-values, the statistic is maxWD.
- Finite-sample distortions in variance-covariance estimation affect the size and power of maxUD and maxWD tests.

B.3. l versus 1l+ breaks
- F(1l+|l) compares SSE for l breaks versus l+1 breaks; equivalent to performing l+1 tests of zero breaks versus one break across partitions.
- BP propose testing for one break versus zero within each partition using sup F(0,1); rejection in at least one partition implies l+1 breaks are significant.
- Critical values under the null depend on sample-specific factors such as break size and residual properties.

B.4. Criteria for finding the number of breaks
- Alternatives: information criteria (BIC by Yao (1988); LWZ by Liu et al. (1997)) or sequence of sup F statistics as suggested by BP.
- BP approach is robust to heteroskedasticity and autocorrelation via variance-covariance estimation embedded in tests; information criteria based approaches do not adjust for these features.
- Sequential BP approach can misestimate the number of significant breaks when regimes switch up and down — supF sequence may lack power if one-break alternatives are badly misspecified; use of Dmax can reduce but not eliminate this problem.

### A modified BP methodology for small samples
- Two main proposed extensions:

1) Sample-specific critical values (bootstrap mimicking process)
- Procedure:
  - Define a “mimicking process” with structural parameters estimated from the data under the null.
  - Simulate this process with bootstrapped residuals to obtain Monte Carlo sample-specific critical values each time a test is run on a given time series.
  - Example: after rejecting 0 breaks and identifying the largest break, when testing 2 breaks versus 1 break the mimicking process includes the estimated single break so critical values account for size and location of that break and segment lengths and variances.
- Objectives attained:
  - 1) correction for the small sample size,
  - 2) reduction in misspecification bias,
  - 3) control for segment-specific heteroskedasticity.
- Advantages of bootstrapped segment-specific residuals:
  - Calculate sample-size-specific critical values;
  - Relax normality assumption of residuals;
  - Account for segmental heteroskedasticity.

2) Treatment of serial correlation (two contrasting approaches)
- Parametric approach:
  - If serial correlation is suspected and its order is guesstimated, model the process explicitly (e.g., AR(1)); BP allows adding 1t y − as a regressor so errors become iid and breaks regression is no longer a pure structural model.
- HAC robust errors approach:
  - When nature of serial correlation is complex or unknown, model the process as a Wiener process and use an HAC estimate of the variance-covariance matrix.

### Implications and practical considerations (from provided text)
- Using asymptotic critical values generated for Wiener processes can cause considerable size and power distortions in small samples and for non-Wiener DGPs.
- Sequential supF testing performance can be poor when segment size becomes small; Dmax variants may perform better in some cases but identification problems can persist with multiple breaks.
- Small-sample focus: sample sizes as low as 50 observations and up to two true breaks are emphasized for method development and simulation evaluation.

*Source: _wp0875 - 1. Size Tests—Weiner DGP with No Breaks ..........................................................................21*

### Appendix I outlines the algorithms in more detail. GAUSS programs are available from the authors on request.

### _wp0875 - Appendix I outlines the algorithms in more detail. GAUSS programs are available from the authors on request.

### Methods comparison and small-sample considerations
- Risk of misspecifying the underlying process exists for both asymptotic and sample-specific critical values; sample-specific critical values produce better test results even with some misspecification.
- Parametric method advantages and disadvantages:
  - Should be superior when the order of serial correlation is known. (Footnote 5: potential problems with parametric modeling are: (i) overspecification; (ii) number of introduced parameters may be too large relative to the sample size.)
  - Estimation of at least two parameters (parametric) versus one (second method) can reduce power in small samples owing to poorly estimated model parameters.
  - Sequential procedures aggravate the small-sample problem because statistics are estimated in partitions with fewer observations.
  - For an AR(1) process with a large β and at least one significant break, power erodes as β increases because F-statistics under the null are contaminated by the fitted AR(1) process. (Footnote 6 provides an illustrative algebraic relation.)

### Alternative statistics for finding multiple breaks
- Sequential UDmax:
  - Tests null of 0 breaks against alternative of an unknown number of breaks to avoid power distortions.
  - UDmax(n) = max{supF(0,1), supF(0,2),..., supF(0,n)}.
  - Distribution is unknown because UDmax(n) can be any supF(0,i), i=1,n; critical values are obtained by bootstrapping.
  - Should capture more breaks when 1-break alternatives are rejected although 2-break alternatives would be accepted.
- Global UDmax (GUDMax):
  - Uses the whole sample: at the n-th step embed n-1 breaks and generate critical values for UDmax statistics with n-1 breaks embedded; test null of these specific n-1 breaks against alternative of at least n breaks.
  - Distribution depends on the size of each of the n-1 breaks, standard error of the series, and other sample-specific parameters; analytic distribution unknown.
  - Proposes estimating critical values “on the fly” for each application despite computational cost.
- Sequential procedures: globally SSR-minimizing n+1 breaks will not generally nest globally-SSR-minimizing n breaks, which may lead to low power for sequential UDmax.

### Size tests (focus on null of zero breaks; sample size 50 in many experiments)
- Purpose:
  - (i) Assess magnitude of improvement in test size owing to sample-specific critical values.
  - (ii) Infer which modeling—parametric or non-parametric—performs better when serial correlation is present.
- Wiener DGP, no serial correlation assumed in modeling:
  - Table 1 examples for sample size of 50:
    - Asymptotic actual size: 4-5 percent for sup F’s; 8 percent for UDmax(3).
    - Sample-specific procedure: 8-10 percent for sup F’s; 8 percent for UDmax.
  - Conclusion: sample-specific procedure outperforms asymptotic in this setting.
- Case A: Serial correlation present
  - Parametric estimation (correctly modeling AR(1)):
    - For sample size of 50 with serial correlation and theoretical size 10 percent:
      - Asymptotic procedure actual size range: 8-15 percent for autoregressive parameter of 0.5; 26-44 percent for 0.9.
      - Sample-specific procedure actual sizes: 9-11 percent (β = 0.5) and 14-15 percent (β = 0.9).
  - Non-parametric with robust error:
    - Both asymptotic and sample-specific procedures perform poorly in small samples with high autocorrelation:
      - UDmax(3) p-values: 0.87 percent (asymptotic) and 0.85 percent (sample-specific) when autocorrelation coefficient is high.
      - For moderate autocorrelation (0.5), sample-specific outperforms asymptotic: UDmax p-values 0.41 percent (sample-specific) and 0.34 percent (asymptotic) as reported (BP 2003b, Table 1, DGP-4, section 4).
    - Conclusion: If serial correlation is present, parametric modeling is preferred for size considerations because robust error correction does not sufficiently improve test size.
- Case B: Serial correlation absent (overspecifying AR(1) when none exists)
  - With sample-specific critical values, overspecification via AR(1) modeling does not lead to a problem (p-values 8-9 percent).
  - Asymptotic results are inferior (p-values 3-8 percent).
  - Note: unnecessary White heteroskedasticity correction substantially worsens BP asymptotic size results but has no important effect on size of sample-specific tests.

### Power tests (sample size 50; DGP either Wiener or AR(1) with autoregressive parameter 0.5 or 0.9)
- Mimicking process for modeling: either Wiener with robust errors or AR(1).
- Four statistics reported:
  - (i) asymptotic sequential supF (Bai-Perron statistic),
  - (ii) sample-specific sequential supF,
  - (iii) sequential UDmax,
  - (iv) global UDMax (GUDMax).
- Break embedding and structures:
  - Two breaks embedded; three break-structure cases considered:
    - (1) symmetric U-form: downbreak followed by equally large upbreak,
    - (2) asymmetric U-form,
    - (3) downhill: two consecutive downbreaks.
  - For cases 1 and 3: two sizes of breaks considered: medium and large.
  - Break locations divide the sample into three approximately equal subsamples; standard deviation constant across segments.
  - Break sizes measured in multiples of standard deviations (e.g., {1,0,1} denotes two breaks of size 1 standard deviation forming symmetric U-shape).
- Loss function for performance:
  - L = (1/M) * sum_{i=1}^M |N - N̂_i| where N is true number of breaks, N̂_i is estimated number in run i; equivalent formulation provided.
- Simulation results (Table 5) — key conclusions:
  - Sample-specific critical value procedures broadly and considerably outperform asymptotic procedures.
  - Contrary to size results, Wiener modeling with robust errors deals with autocorrelation better than AR(1) modeling for power: if goal is finding right number of breaks, robust errors are generally better.
  - Among sample-specific approaches, sequential UDmax is the best-performing statistic in a majority of cases.
  - GUDmax performs best when DGP has high autocorrelation; it is substantially more conservative (errors are mostly underestimation), while sequential UDmax is more likely to overestimate.
  - Sequential UDmax and GUDmax maintain better power than BP asymptotic statistic even for large autocorrelation.
  - If high autocorrelation is suspected, non-parametric approach may yield better power.

### Summary and conclusions
- Focus: small samples (sample size 50) with up to two true breaks.
- Main innovations and findings:
  - Sample-specific critical values: derive critical values by mimicking the null hypothesis and estimating critical values via Monte-Carlo simulations rather than tabulating asymptotic critical values.
    - Computationally intensive but produces substantially superior results for both test size and power in small samples across nearly all tested DGP specifications.
  - Serial correlation treatment:
    - Using robust errors handles autocorrelation better than parametric AR(1) modeling when goal is finding correct number of breaks.
  - Break number selection methods:
    - Two proposed alternatives to the BP sequential Sup F test:
      - Sequential UDmax (similar to BP but tests 0 breaks vs unknown number of breaks) — generally best-performing across many cases.
      - Global UDmax (GUDMax) — performs best under high autocorrelation; conservative tendency (underestimates number of breaks).
    - Both alternatives generally perform substantially better than BP asymptotic proposal.

### Appendix — Algorithms (size tests, sketch)
- Size tests algorithm steps (selected):
  - 0. Select model parameters: nature of generated process (Wiener or AR(1)), nature of mimicking process (Wiener, Wiener with robust option, or AR(1)), maximum number of breaks, significance level.
  - 1. Simulate N1 initial generated processes with no breaks.
  - 2. Use the global minimizer procedure to determine locations and sizes of potential breaks; obtain statistics for generated processes.
- Note: Appendix I contains further algorithmic detail; GAUSS programs available from the authors on request.

*Source: _wp0875 - Appendix I outlines the algorithms in more detail. GAUSS programs are available from the authors on request.*

### 3. Estimate each of the generated processes according to the nature of the mimicking

### 3. Estimate each of the generated processes according to the nature of the mimicking process, under the null of no breaks for the whole series, extract the residuals.

### A. Size test procedure (overview)
- Step 3: Estimate each generated process according to the nature of the mimicking process, under the null of no breaks for the whole series, extract the residuals.
- Step 4: Use the bootstrap to simulate N2 mimicking processes for each of the generated processes (use the residuals and the estimated parameters from step 3).
- Step 5: Obtain the statistics for the mimicking processes, sort the statistics, and find the critical values, in accordance with the significance level.
- Step 6: Determine if the statistic of the generated process is significant.
- Step 7: If significant, count this case as at least one break is found; if not significant, count this case as no breaks are found.
- Step 8: The size of the test is the proportion of the times the procedure identifies at least one break.

### B. Power tests for the sequential UDmax statistic (procedure)
- Step 0: Select the model parameters. Important choices: nature of the generated process (Wiener or AR(1)), nature of the mimicking process (Wiener with the robust option or AR(1)), maximum number of breaks, significance level, structure of the breaks to be embedded.
- Step 1: Simulate N1 initial processes (generated processes) with the break structure defined in step 0.
- Step 2: Use the global minimizer procedure to determine location and size of potential breaks. Obtain the statistic for the generated processes.
- Step 3: Estimate each generated process according to the mimicking process under the null of no breaks for the whole series, extract residuals.
- Step 4: Use the bootstrap to simulate N2 mimicking processes for each generated process (use residuals and estimated parameters from step 3).
- Step 5: Obtain statistics for mimicking processes, sort them, and find critical values consistent with the significance level.
- Step 6: Determine if the statistic for the generated process is significant.
- Step 7: If not significant, stop and count as no breaks found; if significant, continue.
- Step 8: Using location of the 1st break from step 2, estimate the generated process on each of the two segments according to the mimicking process under the null of no breaks on each segment; extract segment-specific residuals.
- Step 9: Repeat steps 4–6 on each of the two segments.
- Step 10: If one rejects the null on at least one segment, assume there are two breaks (three segments).
- Step 11: Continue the hypothesis-testing process on segments until reaching maximum number of breaks or failing to reject the null on every segment.
- Step 12: For the generated processes, count times one finds 1 break, 2 breaks, ..., N breaks.
- Step 13: The power of the test is the proportion of times the procedure identifies the number of breaks correctly.

### C. Power tests, the global UDmax (procedure)
- Step 0: Select model parameters (same important choices as in B).
- Step 1: Simulate N1 initial processes with the break structure defined in step 0.
- Step 2: Use global minimizers procedure to determine location and size of potential breaks. Obtain statistics for generated processes.
- Step 3: Estimate each generated process according to the mimicking process under the null of no breaks for the whole series, extract residuals.
- Step 4: Use the bootstrap to simulate N2 mimicking processes for each generated process (use residuals and estimated parameters from step 3).
- Step 5: Obtain statistics for mimicking processes, sort them, and find critical values.
- Step 6: Determine if the statistic of the generated process is significant.
- Step 7: If not significant, stop and count as no breaks found; if significant, continue.
- Step 8: Using the location and size of the 1st break from step 2, construct N2 new mimicking processes based on the segment-specific bootstrapped residuals and the embedded break.
- Step 9: Repeat steps 5–6 for the new mimicking series. (The statistic for the generated series is still based on the whole sample.)
- Step 10: If one rejects the null using the new critical values, assume there are two breaks.
- Step 11: Embed the second break in yet other N2 new mimicking processes. Continue hypothesis testing based on series with embedded breaks until reaching the maximum number of breaks or failing to reject the null.
- Step 12: For the generated processes, count how many times one finds 1 break, 2 breaks, ..., N breaks.
- Step 13: The power of the test is the proportion of the times the procedure identifies the number of breaks correctly.

### D. Empirical results — Size tests (selected table entries preserved exactly)
- Table 1: Size Tests—Weiner DGP With No Breaks
  - Asymptotic 1/ vs Sample Specific:
    - SupF(0,1) 0.05 0.08
    - SupF(0,2) 0.05 0.09
    - SupF(0,3) 0.04 0.10
    - UDmax(3) 0.08 0.08
  - Notes: True size is 10 percent. 500 Monte Carlo (MC) simulations were performed. Sample specific critical values were generated for the DGP process (50 observations), with 500 MC simulation runs within each of the 500 MC runs performed.

- Table 2: Size Tests with Autocorrelated DGP with No Breaks and Parametric Estimation
  - Panel A (True model: yt = 0.9*yt-1 + ut, ut ~ N(0,1) and i.i.d.)
    - Asymptotic 1/ vs Sample Specific:
      - SupF(0,1) 0.26 0.15
      - SupF(0,2) 0.36 0.14
      - SupF(0,3) 0.41 0.15
      - UDmax(3) 0.44 0.14
  - Panel B (True model: yt = 0.5*yt-1 + ut, ut ~ N(0,1) and i.i.d.)
    - Asymptotic 1/ vs Sample Specific:
      - SupF(0,1) 0.08 0.10
      - SupF(0,2) 0.10 0.11
      - SupF(0,3) 0.09 0.09
      - UDmax(3) 0.15 0.10
  - Notes: True size is 10 percent. 500 Monte Carlo (MC) simulations were performed. Sample specific critical values were generated for the DGP process (50 observations), with 500 MC simulation runs within each of the 500 MC runs performed.

- Table 3: Size Tests with Autocorrelated DGP with No Breaks and Standard Errors Robust to Serial Correlation
  - Panel A (True model: yt = ut, ut = 0.9*ut-1 + et, et ~ N(0,1) and i.i.d.)
    - Asymptotic 1/ vs Sample Specific:
      - SupF(0,1) 0.56 0.59
      - SupF(0,2) 0.81 0.81
      - SupF(0,3) 0.89 0.89
      - UDmax(3) 0.87 0.85
  - Panel B (True model: yt = ut, ut = 0.5*ut-1 + et, et ~ N(0,1) and i.i.d.)
    - Asymptotic 1/ vs Sample Specific:
      - SupF(0,1) 0.20 0.26
      - SupF(0,2) 0.32 0.34
      - SupF(0,3) 0.34 0.34
      - UDmax(3) 0.41 0.34
  - Notes: True size is 10 percent. 500 Monte Carlo (MC) simulations were performed. Sample specific critical values were generated for the DGP process (50 observations), with 500 MC simulation runs within each of the 500 MC runs performed.

- Table 4: Size Tests with DGP with No Breaks and Over- and Under-Specification of Degree of Autocorrelation
  - Case of overspecification (Asymptotic 1/ vs Sample Specific):
    - SupF(0,1) 0.06 0.09
    - SupF(0,2) 0.03 0.09
    - SupF(0,3) 0.04 0.09
    - UDmax(3) 0.08 0.08
  - Case of underspecification (Asymptotic 1/ vs Sample Specific):
    - SupF(0,1) 0.90 0.94
    - SupF(0,2) 0.96 0.97
    - SupF(0,3) 0.96 0.98
    - UDmax(3) 0.97 0.97
  - Additional block (Asymptotic 1/ vs Sample Specific):
    - SupF(0,1) 0.38 0.52
    - SupF(0,2) 0.55 0.66
    - SupF(0,3) 0.54 0.68
    - UDmax(3) 0.61 0.60
  - Notes: True size is 10 percent. 500 Monte Carlo (MC) simulations were performed. Sample specific critical values were generated for the DGP process (50 observations), with 500 MC simulation runs within each of the 500 MC runs performed.

### E. Empirical results — Power tests (selected entries from Table 5; Panel A, B, C preserved)
- General notes for Table 5:
  - 1/ Sequential testing using asymptotic critical values (Bai and Perron (1998)).
  - 2/ Sequential sup F testing using sample specific critical values.
  - 3/ Sequential UDmax testing using sample specific critical values.
  - 4/ The adjusted global UDmax methodology.
  - 5/ Using the loss function defined for power tests (page 14).
  - True size is 10 percent. 500 Monte Carlo (MC) simulations were performed. Sample specific critical values were generated for the DGP process (50 observations), with 500 MC simulation runs within each of the 500 MC runs performed.
  - Both true underlying process and the simulated model have two breaks. First one located at observation number 17 and second at observation number 34. Within each break interval, time series dynamics are governed by a Wiener process and the averages are a multiple of the unitary standard deviation as indicated in the first column of the table.

- Panel A: True model: yt = ut, ut ~ N(0,1) and i.i.d. (Averages in s.d.)
  - Selected probabilities and loss values (columns correspond to BP asypt. 1/, S supF 2/, SUDmax 3/, AUDmax 4/ in two blocks shown)
    - Prob k=0 0.68 0.11 0.03 0.03 0.77 0.16 0.05 0.05
    - Prob k=1 0.12 0.10 0.10 0.34 0.14 0.12 0.16 0.41
    - Prob k=2 0.18 0.63 0.70 0.55 0.09 0.57 0.62 0.50
    - Prob k≥3 0.01 0.16 0.16 0.08 0.00 0.15 0.17 0.04
    - Loss value 5/ 1.51 0.49 0.33 0.48 1.68 0.58 0.43 0.55
  - Additional blocks in Panel A (examples):
    - Prob k=0 0.73 0.05 0.00 0.00 0.55 0.00 0.00 0.00
    - Prob k=1 0.00 0.00 0.00 0.03 0.00 0.02 0.03 0.05
    - Prob k=2 0.24 0.78 0.82 0.83 0.43 0.83 0.81 0.87
    - Prob k≥3 0.03 0.17 0.18 0.14 0.02 0.15 0.16 0.08
    - Loss value 5/ 1.48 0.28 0.18 0.17 1.12 0.17 0.19 0.13
  - Further selected rows preserve exact numeric strings as in source for other averaged s.d. blocks.

- Panel B: True model: yt = ut, ut = 0.5*ut-1 + et, et ~ N(0,1) and i.i.d. (Averages in s.d.)
  - Selected probabilities and loss values (first block examples):
    - Prob k=0 0.70 0.26 0.06 0.06 0.82 0.31 0.18 0.18
    - Prob k=1 0.23 0.19 0.15 0.58 0.14 0.34 0.41 0.49
    - Prob k=2 0.07 0.33 0.41 0.32 0.04 0.28 0.32 0.28
    - Prob k≥3 0.01 0.22 0.38 0.04 0.00 0.07 0.09 0.04
    - Loss value 5/ 1.63 0.94 0.66 0.75 1.78 1.03 0.86 0.91
  - Additional blocks and loss values preserved exactly as in source.

- Panel C: True model: yt = ut, ut = 0.9*ut-1 + et, et ~ N(0,1) and i.i.d. (Averages in s.d.)
  - Selected probabilities and loss values (examples):
    - Prob k=0 0.39 0.11 0.01 0.01 0.59 0.20 0.08 0.08
    - Prob k=1 0.31 0.17 0.07 0.62 0.32 0.49 0.67 0.47
    - Prob k=2 0.21 0.28 0.22 0.30 0.08 0.22 0.21 0.22
    - Prob k≥3 0.09 0.44 0.70 0.07 0.00 0.08 0.04 0.23
    - Loss value 5/ 1.18 0.83 0.79 0.70 1.51 0.98 0.86 0.85
  - Additional rows and blocks are preserved exactly as in the source tables.

*Source: _wp0875 - 3. Estimate each of the generated processes according to the nature of the mimicking process (extracted from supplied IMF PDF).*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp0875.pdf_