# Minimum LM Unit Root Test Cuts False Break Signals to 3.9%

Cite as: Gorak, R. (2026). Minimum LM Unit Root Test Cuts False Break Signals to 3.9%. Tradicted. https://www.tradicted.com/research/lee-minimum-2013/
Paper: Junsoo Lee and Mark C. Strazicich — *Minimum LM unit root test with one structural break*
Published in: Economics Bulletin (2013)
Original: http://econ.appstate.edu/RePEc/pdf/wp0417.pdf

Key finding: The minimum LM unit root test's null rejection rate stays at 3.9% for a break of δ3 = 10, versus 50.6% for the Zivot-Andrews test (Lee and Strazicich, 2013).

---
The minimum LM unit root test endogenously locates a single structural break in a series' intercept and trend. It then tests whether the series is trend stationary or contains a unit root. Researchers use it to check whether a shock permanently shifted a series' level. Lee and Strazicich present the test in "Minimum LM Unit Root Test with One Structural Break" (2013). They ran 5,000 Monte Carlo replications at T = 100. Across break sizes of δ3 = 0 to 10, the null rejection rate stayed near 3.9%–5.7%. Over the same range, the Zivot-Andrews test's rejection rate rose to 50.6%.

## What the Study Found

With no break present, the LM test rejects a true unit root null 5.7% of the time, near the nominal 5% level. At the largest break tested, δ3 = 10, the LM null rejection rate stays at 3.9% versus 50.6% for the Zivot-Andrews (ZA) test. Under the alternative with no break, the LM test correctly rejects a false null 71.0% of the time versus 38.9% for the ZA test. At δ3 = 10 the LM test's power reaches 45.4%, compared with 98.7% for the ZA test. The LM test also locates the true break correctly 48.0% of the time under the null and 89.8% under the alternative at δ3 = 10.

> "In contrast to similar ADF-type endogenous break tests, the one-break minimum LM unit root test tends to estimate the break point correctly and is free of size distortions and spurious rejections in the presence of a unit root with break."
>
> Lee and Strazicich (2013), *Minimum LM Unit Root Test with One Structural Break*, p. 10.

## Methodology

Lee and Strazicich (2013) test the one-break minimum LM unit root test using Monte Carlo simulations rather than an empirical dataset. Each simulation generates 5,000 replications of pseudo-iid N(0,1) random numbers using the Gauss RNDNS procedure. Every replication uses a sample size of T = 100 observations with a true structural break placed at TB = 50. The test statistic is corrected for autocorrelation using the Ng and Perron (1995) general-to-specific lag selection procedure.

## Key Statistics

| Metric | Finding | Context |
|---|---|---|
| Null rejection rate, no break | 5.7% | LM test, δ3 = 0, β = 1 |
| Null rejection rate, δ3 = 10 | 3.9% | LM test vs. 50.6% for the ZA test |
| Power under alternative, no break | 71.0% | LM test, β = 0.8, δ3 = 0 |
| Power under alternative, δ3 = 10 | 45.4% | LM test vs. 98.7% (spurious) for the ZA test |
| 5% critical value, Model A | -3.566 | T = 100 |
| Minimum LM test statistic | Inf τ̃(λ̃) = Infλ τ̃(λ) | break selected at the most negative LM t-statistic (Eq. 4) |

## Minimum LM vs Zivot-Andrews Unit Root Test

| Measure | Minimum LM Test | Zivot-Andrews (ZA) Test |
|---|---|---|
| Null rejection rate, no break (δ3 = 0) | 5.7% | 6.0% |
| Null rejection rate, largest break (δ3 = 10) | 3.9% | 50.6% |
| Power under alternative, no break (δ3 = 0) | 71.0% | 38.9% |
| Power under alternative, largest break (δ3 = 10) | 45.4% | 98.7% (inflated by spurious rejections) |

## Why This Matters

Unit root tests are a standard tool for testing whether a price or macro series follows a random walk or reverts to trend. That question is central to tests of weak-form market efficiency. A test that spuriously rejects the unit root null under a break can mislead researchers. They may conclude a series is predictable when it is actually a random walk with a permanent shift. The minimum LM test's null distribution does not depend on a break's size or location. Researchers therefore do not need new critical values when a break is suspected. A competing endogenous-break test can look more powerful under the alternative. That apparent power is often just the same size distortion inflating its rejections under the null. This lets applied work test for mean reversion around known shocks without inflating false signals of predictability.

## FAQ

### What does the minimum LM unit root test find about false break signals?

The minimum LM unit root test keeps its null rejection rate near 3.9%–5.7% across break sizes from δ3 = 0 to δ3 = 10. That stays close to the nominal 5% significance level. Lee and Strazicich (2013) show the Zivot-Andrews test's rejection rate climbs to 50.6% at the same break size.

### Does the minimum LM test need new critical values when a break occurs?

The minimum LM test's null rejection rate stays near 3.9% even at the largest break tested, δ3 = 10. Lee and Strazicich (2013) show this invariance means critical values do not need to be resimulated for a suspected break. The Zivot-Andrews test's distribution, by contrast, shifts with the break's size and location.

### How accurately does the test locate the true structural break?

The minimum LM test locates the true break correctly 48.0% of the time under the null at δ3 = 10. It locates the break correctly 89.8% of the time under the alternative at the same break size. Lee and Strazicich (2013) find break-point accuracy improves as breaks grow larger.

### What is a structural break in a unit root test?

The minimum LM test's 5% critical value is -3.566 for Model A. Model A defines a structural break as a one-time shift in a series' intercept under the alternative hypothesis. Lee and Strazicich (2013) compare the test statistic against this value to judge whether a series is trend stationary despite the break.

## Source

Lee, Junsoo and Mark C. Strazicich. (2013). Minimum LM Unit Root Test with One Structural Break. Economics Bulletin.

[Read the full paper →](http://econ.appstate.edu/RePEc/pdf/wp0417.pdf)