# Structural Break Tests Robust to Regression Misspecification

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Unconditional Mean and Variance Break Tests

**Assumption**

**1.**

- (i)
- $\mathbf{E}\left({u}_{t}\right)=0$ and $\mathbf{AVar}\left(\right)open="("\; close=")">{T}^{-1/2}{\sum}_{t=1}^{\left[T\lambda \right]}{u}_{t}$;
- (ii)
- for some $d>4$, ${\mathrm{sup}}_{t}\mathbf{E}{\left|{u}_{t}\right|}^{d}<\infty $ and $\left\{{u}_{t}\right\}$ is ${\mathcal{L}}_{2}$-near epoch dependent of size ${c}_{m}=O\left({m}^{-1}\right)$ on $\left\{{g}_{t}\right\}$, i.e., ${\left(\right)}_{{u}_{t}}2$ with ${c}_{m}=O\left({m}^{-1}\right)$ where ${\mathcal{G}}_{t-m}^{t+m}=\sigma ({g}_{t-m},\dots ,{g}_{t+m})$, and $\left\{{g}_{t}\right\}$ is either ϕ-mixing of size ${m}^{-d/\left(\right)open="("\; close=")">2(d-1)}$ or α-mixing of size ${m}^{-d/(d-2)}$.7

**Assumption**

**2.**

- (i)
- $\mathbf{AVar}\left(\right)open="("\; close=")">{T}^{-1/2}{\sum}_{t=1}^{\left[T\lambda \right]}|{y}_{t}-\overline{y}|$, for some ${\mathbf{v}}_{a}>0$; and
- (ii)
- $\mathbf{AVar}\left(\right)open="("\; close=")">{T}^{-1/2}{\sum}_{t=1}^{\left[T\lambda \right]}{({y}_{t}-\overline{y})}^{2}$, for some ${\mathbf{v}}_{s}>0$.

**Theorem**

**1.**

## 3. Conditional Mean and Variance Break Tests

#### 3.1. Correct Specification

**Assumption**

**3.**

- (i)
- $\mathbf{E}\left({x}_{t}{\u03f5}_{t}\right)=0$, $\mathbf{AVar}({T}^{-1/2}{\sum}_{t=1}^{\left[T\lambda \right]}{x}_{t}{\u03f5}_{t})=\lambda \mathbf{V}$ and ${T}^{-1}{\sum}_{t=1}^{\left[T\lambda \right]}$${x}_{t}{x}_{t}^{\prime}$$\stackrel{P}{\u27f6}\lambda Q$, two positive definite (pd) matrices of constants; and
- (ii)
- for some $d>4$, ${\mathrm{sup}}_{t}{\parallel {x}_{t}{\u03f5}_{t}\parallel}_{d}<\infty $ and $\left\{{x}_{t}{\u03f5}_{t}\right\}$ is ${\mathcal{L}}_{2}$-near epoch dependent of size ${c}_{m}=O\left({m}^{-1}\right)$ on $\left\{{h}_{t}\right\}$, and $\left\{{h}_{t}\right\}$ is either ϕ-mixing of size ${m}^{-d/\left(\right)open="("\; close=")">2(d-1)}$ or α-mixing of size ${m}^{-d/(d-2)}$.

**Theorem**

**2.**

#### 3.2. Dynamic Misspecification

**Assumption**

**4.**

- (i)
- $\mathbf{E}\left({w}_{t}\right)=0$, $\mathbf{AVar}({T}^{-1/2}{\sum}_{t=1}^{\left[T\lambda \right]}{w}_{t})=\lambda \mathbf{H}$, a pd matrix of constants;
- (ii)
- for some $d>4$, ${\mathrm{sup}}_{t}{\parallel {w}_{t}\parallel}_{d}<\infty $ and $\left\{{w}_{t}\right\}$ is ${\mathcal{L}}_{2}$-near epoch dependent of size ${d}_{m}=O\left({m}^{-1}\right)$ on either an ϕ-mixing process of size ${m}^{-d/\left(\right)open="("\; close=")">2(d-1)}$ or an α-mixing process of size ${m}^{-d/(d-2)}$;
- (iii)
- ${Q}_{\left(12\right)}\ne {O}_{{p}_{1}\times {p}_{2}}$, where ${O}_{{p}_{1}\times {p}_{2}}$ is the ${p}_{1}\times {p}_{2}$ null matrix; and
- (iv)
- ${T}^{-1}{\sum}_{1\lambda}{w}_{t}{w}_{t}^{\prime}\stackrel{P}{\u27f6}\lambda \mathsf{\Omega}$, a pd matrix of constants.

**Theorem**

**3.**

- (i)
- If $C{M}_{T}^{*}$ is constructed under heteroskedasticity,$$C{M}_{T}^{*}\Rightarrow \underset{\lambda}{\mathrm{sup}}\frac{{\mathcal{B}}_{s}^{{*}^{\prime}}\left(\lambda \right)\phantom{\rule{4pt}{0ex}}A\phantom{\rule{4pt}{0ex}}{\mathcal{B}}_{s}^{*}\left(\lambda \right)}{\lambda (1-\lambda )}.$$
- (ii)
- Let $\nu ={\sigma}_{\u03f5}^{2}+{\theta}_{\left(2\right)}^{\prime}[{Q}_{\left(2\right)}-{Q}_{\left(12\right)}^{\prime}{Q}_{\left(1\right)}^{-1}{Q}_{\left(12\right)}]{\theta}_{\left(2\right)}$. If $C{M}_{T}^{*}$ is constructed under homoskedasticity, then the result in (i) holds, with $A={\nu}^{-1}{\mathbf{H}}^{*\mathbf{1}/{\mathbf{2}}^{\prime}}[\left(\xi {\xi}^{\prime}\right)\otimes {Q}_{\left(1\right)}^{-1}]{\mathbf{H}}^{*\mathbf{1}/\mathbf{2}}$.

## 4. Simulation Results

#### 4.1. Correct Specification and Various Misspecifications

#### 4.2. Dynamic Misspecification

## 5. Empirical Illustrations

#### 5.1. Unemployment Rate

#### 5.2. Industrial Production Growth

#### 5.3. Interest Rates

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proofs of Theorems 1–3

**Proof**

**(Proof**

**of**

**Theorem**

**1.)**

**Proof**

**(Proof**

**of**

**Theorem**

**2.)**

**Proof**

**(Proof**

**of**

**Theorem**

**3.)**

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1 | There are a few notable exceptions. For example, Rapach and Wohar (2005) directly tested and found multiple unconditional mean shifts in international interest rates and inflation using the Bai and Perron (1998) tests. Elliott and Müller (2007) and Eo and Morley (2015) considered in their simulations a break in the unconditional mean (and variance) of a time series. Their focus was on constructing confidence sets with good coverage for small and large breaks, by inverting structural break tests. Recently, Müller and Watson (2017) proposed new methods for detecting low-frequency mean or trend changes. Our paper is different as it highlights the properties of the existing sup Wald test for a break in the unconditional and conditional mean and variance of a time series. |

2 | Non-stationary processes with a trend break and unit root errors, whose first-differences exhibit mean shifts with stationary errors, have been analyzed in many papers. However, as Vogelsang (1998, 1999) showed, to recover monotonic power, testing the first-differenced series for a mean shift is better than testing the level series for a trend shift. |

3 | The only case where our test has comparatively low power to the conditional mean test is in a correctly specified dynamic model with an intercept very close to zero. This case is further discussed in Section 3. |

4 | |

5 | Even though UM tests are not routinely used, they are a special case of the HAC-adjusted conditional break-point test in, e.g., Bai and Perron (1998), when the only regressor is an intercept. In addition, a CUSUM (cumulative sum) variant of this test for iid data is in Pitarakis (2004). As shown in Appendix A, proof of Theorem 1, for unconditional break tests, there is an explicit asymptotic relationship between the CUSUM test and the sup Wald test. However, as the Appendix shows, the conclusion of the two tests based on asymptotic critical values is in general different. Since there is strong evidence for the non-monotonic power of the CUSUM test (see, e.g., Vogelsang 1999), the paper focuses on the sup Wald test instead. |

6 | Note that a break in the expected absolute value of a demeaned series is not the same as a variance break only under certain conditions. |

7 | Here, ${\parallel \xb7\parallel}_{2}={(\mathbf{E}\parallel \xb7\parallel}^{2}{)}^{1/2}$ stands for the ${\mathcal{L}}_{2}$-norm, and $|\xb7|$ stands for the Euclidean norm. |

8 | For a proof that the most common GARCH model, GARCH(1,1), is near-epoch dependent and therefore fits our assumptions, see (Hansen 1991). |

9 | Simulation evidence for this statement is available from the authors upon request. |

10 | |

11 | The weights mentioned in Newey and West (1994) are set equal to one as usual for scalar cases. |

12 | Here, “⇒” indicates weak convergence in the Skorohod metric. |

13 | The simulation section shows that the CM tests are severely oversized with dynamic misspecification. |

14 | We extend the $\mathbf{vec}(A,B)$ notation to denote stacking in a vector all columns of A, then all columns of B, one by one, in order, even when $A,B$ do not have the same number of rows, and we let ${\mathbf{vec}}^{\prime}(A,B)={\left[\mathbf{vec}(A,B)\right]}^{\prime}$. |

15 | Overspecifying the number of lags or regressors is not a problem, as the coefficients on the additional regressors or lags will converge to zero. |

16 | This was also mentioned in (Chong 2003), in the comments after their Theorem 3. |

17 | A formal proof of this statement can be found in (Hall et al. 2012, Supplemental Appendix, page 23.) |

18 | The unconditional mean and variance sup Wald tests require a long-run variance estimator. We report the Newey and West (1994) HAC estimator with the data dependent bandwidth therein and the Bartlett kernel, as explained in detail in Section 2. The Andrews (1991) fixed bandwidth HAC estimator leads to slightly worse performance across all tests and designs; results are available upon request from the authors. |

19 | For a static DGP with i.i.d. errors (DGP3, detailed below), we report the power of the tests based on asymptotic critical values rather than size-adjusted powers, because the size distortions are minor. |

20 | The size-adjusted powers are computed as follows: for a DGP under the alternative of one break, we take the parameter values after the break, and use these parameter values for generating the DGP under the null, which will have the same sample size as the DGP under the alternative. We simulate this null DGP and take the 95% quantile of the empirical distribution of a test statistic as the empirical critical value to be used. We then simulate the corresponding DGP under the alternative, and calculate the empirical rejection frequency using the corresponding empirical critical values. Note that, by construction, all size-adjusted power plots start at 5%, which is the corresponding empirical rejection frequency for a DGP under the null of no break using its simulated empirical 95% critical values. |

21 | Only for the static model in DGP3, we used 2000 simulations because they were sufficient to get accurate Monte Carlo results. |

22 | Note that, for DGP1, the unconditional mean of ${y}_{t}$ is equal to ${\alpha}_{t}/(1-{\beta}_{t})$. If ${\alpha}_{t}/(1-{\beta}_{t})$ is close to zero regardless of t, the UM test will, by design, have little power for a break in the slope ${\beta}_{t}$. Therefore, if a slope break is the only break of interest, it should be tested directly via the CM test for partial structural change in slopes. |

23 | Note that we do not plot size-adjusted powers for this DGP, and we only do so in general for correctly specified models. |

24 | The results are very similar for the UA and CA tests and they are available upon request. |

25 | Other types of model misspecifications may also affect the size and power of the (CM and CV) structural break tests. Analyzing them is beyond the scope of this paper, but further results regarding these misspecifications can be found in (Chong 2003; Pitarakis 2004), among others. |

26 | For one test in Table 10, the critical values are not available because this test entails 26 parameters, while critical values are available, to our knowledge, only for maximum 20 parameters. However, from (Andrews 2003), it is evident that the critical values are strictly increasing in the number of parameters, so it is reasonable to assume that the critical values for 26 parameters should be above the critical values for 20 parameters. |

27 | The means are: $6.11$ for the unemployment rate, $17.73$ for the interest rate, and $0.002$ for the industrial production growth. For the power of the UM test, the means themselves are not important as long as they are non-zero, and as long as the t-statistics for these means (also known as signal-noise ratios) reject the null hypothesis of a zero mean. |

28 | Note that, in this case, the AIC and BIC with a maximum number of 30 lags picks the same number of lags as when a maximum of 12 lags is imposed. |

29 | In addition, note that the test statistic $\mathrm{sup}}_{\lambda \in [\u03f5,1-\u03f5]}{\sqrt{UM}}_{T$ is known in statistics as a “weighted version” of the CUSUM test—see (Aue and Horvath 2012, p. 5). |

**Figure 1.**DGP1: Size-adjusted power for a correctly specified AR(1) model with iid errors. Note: $Wald$ is the CM test, and $Wal{d}_{U}$ is the UM test.

**Figure 2.**DGP3: Power for a correctly specified static model with iid errors. Note: $Wald$ is the CM test, and $Wal{d}_{U}$ is the UM test.

**Figure 3.**DGP4: Size-adjusted power for a correctly specified static model with AR(1) errors. Note: $Wald$ is the CM test, and $Wal{d}_{U}$ is the UM test.

**Figure 4.**DGP1 (

**left**); and DGP4 (

**right**) with size-adjusted power for a correctly specified model: with an AR(1) lag and iid errors (

**left**); or with static regressors and AR(1) errors (

**right**). Note: $Wald$ is the CV test, and $Wal{d}_{U}$ is the UV test.

**Figure 5.**Empirical distribution of the number of lags selected by AIC and BIC. DGP2: AR(4) model: ${y}_{t}=1+\mathbf{0.1}\phantom{\rule{4pt}{0ex}}{y}_{t-1}+0.2\phantom{\rule{4pt}{0ex}}{y}_{t-2}+0.15\phantom{\rule{4pt}{0ex}}{y}_{t-3}+0.075\phantom{\rule{4pt}{0ex}}{y}_{t-4}+{\u03f5}_{t}$.

**Figure 6.**Empirical distribution of the number of lags selected by AIC and BIC. DGP2: AR(4) model: ${y}_{t}=1+\mathbf{0.2}\phantom{\rule{4pt}{0ex}}{y}_{t-1}+0.2\phantom{\rule{4pt}{0ex}}{y}_{t-2}+0.15\phantom{\rule{4pt}{0ex}}{y}_{t-3}+0.075\phantom{\rule{4pt}{0ex}}{y}_{t-4}+{\u03f5}_{t}$.

**Figure 7.**Empirical distribution of the number of lags selected by AIC and BIC. DGP2: AR(4) model: ${y}_{t}=1+\mathbf{0.3}\phantom{\rule{4pt}{0ex}}{y}_{t-1}+0.2\phantom{\rule{4pt}{0ex}}{y}_{t-2}+0.15\phantom{\rule{4pt}{0ex}}{y}_{t-3}+0.075\phantom{\rule{4pt}{0ex}}{y}_{t-4}+{\u03f5}_{t}$.

**Figure 8.**Empirical distribution of the number of lags selected by AIC and BIC. DGP2: AR(4) model: ${y}_{t}=1+0.1\phantom{\rule{4pt}{0ex}}{y}_{t-1}+\mathbf{0.175}\phantom{\rule{4pt}{0ex}}{y}_{t-4}+{\u03f5}_{t}$.

**Figure 9.**Empirical distribution of the number of lags selected by AIC and BIC. DGP2: AR(4) model: ${y}_{t}=1+0.1\phantom{\rule{4pt}{0ex}}{y}_{t-1}+\mathbf{0.275}\phantom{\rule{4pt}{0ex}}{y}_{t-4}+{\u03f5}_{t}$.

**Figure 10.**Empirical distribution of the number of lags selected by AIC and BIC. DGP2: AR(4) model: ${y}_{t}=1+0.1\phantom{\rule{4pt}{0ex}}{y}_{t-1}+\mathbf{0.375}\phantom{\rule{4pt}{0ex}}{y}_{t-4}+{\u03f5}_{t}$.

DGP | Trim | Model | ${\mathit{UM}}_{\mathit{T}}^{*}$ | ${\mathit{CM}}_{\mathit{T}}^{*}$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}$ | $\mathit{\beta}$ | T = 100 | 200 | 500 | 1000 | 100 | 200 | 500 | 1000 | ||

DGP1—AR(1) model, iid errors | 15% | 1 | 0.1 | 0.034 | 0.040 | 0.047 | 0.052 | 0.067 | 0.056 | 0.050 | 0.050 |

0.2 | 0.039 | 0.047 | 0.054 | 0.056 | 0.078 | 0.060 | 0.050 | 0.051 | |||

0.3 | 0.040 | 0.048 | 0.053 | 0.057 | 0.082 | 0.063 | 0.056 | 0.052 | |||

0.4 | 0.044 | 0.055 | 0.056 | 0.056 | 0.091 | 0.064 | 0.058 | 0.048 | |||

0.5 | 0.052 | 0.052 | 0.062 | 0.060 | 0.107 | 0.068 | 0.063 | 0.050 | |||

0.6 | 0.061 | 0.058 | 0.070 | 0.065 | 0.120 | 0.079 | 0.055 | 0.058 | |||

0.7 | 0.082 | 0.072 | 0.075 | 0.069 | 0.141 | 0.092 | 0.062 | 0.058 | |||

DGP3—static model, iid errors | 15% | 1 | 0.1 | 0.067 | 0.052 | 0.060 | 0.050 | 0.075 | 0.063 | 0.059 | 0.041 |

0.2 | 0.073 | 0.053 | 0.049 | 0.043 | 0.086 | 0.052 | 0.054 | 0.047 | |||

0.3 | 0.067 | 0.056 | 0.054 | 0.052 | 0.083 | 0.062 | 0.049 | 0.049 | |||

0.4 | 0.084 | 0.055 | 0.049 | 0.052 | 0.091 | 0.058 | 0.042 | 0.052 | |||

0.5 | 0.061 | 0.053 | 0.058 | 0.051 | 0.070 | 0.056 | 0.061 | 0.051 | |||

0.6 | 0.058 | 0.058 | 0.046 | 0.054 | 0.080 | 0.056 | 0.044 | 0.052 | |||

0.7 | 0.073 | 0.056 | 0.056 | 0.055 | 0.081 | 0.059 | 0.054 | 0.048 | |||

0.8 | 0.063 | 0.060 | 0.047 | 0.048 | 0.079 | 0.059 | 0.051 | 0.048 | |||

0.9 | 0.069 | 0.062 | 0.055 | 0.050 | 0.092 | 0.065 | 0.050 | 0.052 | |||

DGP 4—static model, AR(1) errors | 15% | 1 | 0.1 | 0.063 | 0.055 | 0.067 | 0.064 | 0.115 | 0.107 | 0.090 | 0.098 |

0.3 | 0.065 | 0.061 | 0.066 | 0.062 | 0.113 | 0.111 | 0.090 | 0.100 | |||

0.5 | 0.063 | 0.060 | 0.063 | 0.059 | 0.122 | 0.109 | 0.091 | 0.106 | |||

0.7 | 0.060 | 0.061 | 0.070 | 0.060 | 0.111 | 0.113 | 0.104 | 0.109 | |||

0.9 | 0.064 | 0.061 | 0.065 | 0.065 | 0.109 | 0.106 | 0.086 | 0.099 |

DGP | Estimated Model | Trim | Model | ${\mathit{UM}}_{\mathit{T}}^{*}$ | ${\mathit{CM}}_{\mathit{T}}^{*}$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}$ | $\mathit{\beta}$ | T = 100 | 200 | 500 | 1000 | 100 | 200 | 500 | 1000 | |||

DGP2—AR(4), iid errors | AR(1) | 15% | 1 | 0.1 | 0.157 | 0.114 | 0.114 | 0.088 | 0.427 | 0.463 | 0.486 | 0.508 |

0.2 | 0.191 | 0.131 | 0.132 | 0.102 | 0.493 | 0.509 | 0.531 | 0.560 | ||||

0.3 | 0.231 | 0.171 | 0.174 | 0.130 | 0.554 | 0.579 | 0.606 | 0.616 | ||||

DGP2—AR(4), iid errors | AR(2) | 15% | 1 | 0.1 | 0.157 | 0.113 | 0.114 | 0.088 | 0.419 | 0.371 | 0.336 | 0.332 |

0.2 | 0.190 | 0.131 | 0.131 | 0.103 | 0.455 | 0.378 | 0.344 | 0.336 | ||||

0.3 | 0.230 | 0.170 | 0.173 | 0.129 | 0.496 | 0.421 | 0.374 | 0.358 | ||||

DGP3—static model, iid errors | AR(1) | 15% | 1 | 0.1 | 0.067 | 0.052 | 0.060 | 0.050 | 0.067 | 0.054 | 0.053 | 0.046 |

0.2 | 0.073 | 0.053 | 0.049 | 0.043 | 0.067 | 0.054 | 0.047 | 0.049 | ||||

0.3 | 0.067 | 0.056 | 0.054 | 0.052 | 0.070 | 0.056 | 0.048 | 0.056 | ||||

0.4 | 0.084 | 0.055 | 0.049 | 0.052 | 0.070 | 0.053 | 0.048 | 0.052 | ||||

0.5 | 0.061 | 0.053 | 0.058 | 0.051 | 0.065 | 0.055 | 0.051 | 0.051 | ||||

0.6 | 0.058 | 0.058 | 0.046 | 0.054 | 0.074 | 0.053 | 0.049 | 0.051 | ||||

0.7 | 0.073 | 0.056 | 0.056 | 0.055 | 0.067 | 0.055 | 0.049 | 0.048 | ||||

0.8 | 0.063 | 0.060 | 0.047 | 0.048 | 0.073 | 0.054 | 0.050 | 0.051 | ||||

0.9 | 0.069 | 0.062 | 0.055 | 0.050 | 0.065 | 0.053 | 0.053 | 0.052 | ||||

DGP4—static model, AR(1) errors | ${X}_{t}^{2}$ instead of ${X}_{t}$ | 15% | 1 | 0.1 | 0.063 | 0.055 | 0.067 | 0.064 | 0.207 | 0.165 | 0.111 | 0.121 |

0.3 | 0.065 | 0.061 | 0.066 | 0.062 | 0.204 | 0.158 | 0.120 | 0.125 | ||||

0.5 | 0.063 | 0.060 | 0.063 | 0.059 | 0.211 | 0.162 | 0.121 | 0.131 | ||||

0.7 | 0.060 | 0.061 | 0.070 | 0.060 | 0.209 | 0.160 | 0.127 | 0.128 | ||||

0.9 | 0.064 | 0.061 | 0.065 | 0.065 | 0.216 | 0.173 | 0.137 | 0.143 |

DGP | Trim | Model | ${\mathbf{UV}}_{\mathit{T}}^{*}$ | ${\mathbf{CV}}_{\mathit{T}}^{*}$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}$ | $\mathit{\sigma}$ | T = 100 | 200 | 500 | 1000 | 100 | 200 | 500 | 1000 | ||

DGP1—AR(1) model, iid errors | 15% | 1 | 1 | 0.042 | 0.052 | 0.051 | 0.059 | 0.0753 | 0.0570 | 0.050 | 0.048 |

1.2 | 0.042 | 0.047 | 0.051 | 0.050 | 0.0730 | 0.0533 | 0.050 | 0.048 | |||

1.4 | 0.039 | 0.046 | 0.050 | 0.056 | 0.0754 | 0.0584 | 0.050 | 0.052 | |||

1.6 | 0.040 | 0.043 | 0.052 | 0.054 | 0.0745 | 0.0517 | 0.053 | 0.047 | |||

1.8 | 0.040 | 0.045 | 0.052 | 0.055 | 0.0735 | 0.0543 | 0.053 | 0.051 | |||

2.0 | 0.040 | 0.047 | 0.052 | 0.054 | 0.0775 | 0.0572 | 0.053 | 0.052 | |||

2.2 | 0.041 | 0.044 | 0.054 | 0.057 | 0.0768 | 0.0592 | 0.050 | 0.051 | |||

2.4 | 0.043 | 0.049 | 0.054 | 0.055 | 0.0774 | 0.0595 | 0.051 | 0.048 | |||

2.6 | 0.043 | 0.047 | 0.052 | 0.052 | 0.0738 | 0.0581 | 0.049 | 0.048 | |||

DGP4—static model, AR(1) errors | 15% | 1 | 1 | 0.041 | 0.051 | 0.052 | 0.057 | 0.076 | 0.067 | 0.059 | 0.060 |

1.2 | 0.041 | 0.049 | 0.053 | 0.055 | 0.075 | 0.067 | 0.060 | 0.058 | |||

1.4 | 0.042 | 0.048 | 0.053 | 0.058 | 0.076 | 0.066 | 0.060 | 0.060 | |||

1.6 | 0.039 | 0.047 | 0.056 | 0.054 | 0.071 | 0.064 | 0.063 | 0.056 | |||

1.8 | 0.041 | 0.048 | 0.056 | 0.053 | 0.077 | 0.066 | 0.063 | 0.057 | |||

2.0 | 0.043 | 0.048 | 0.055 | 0.057 | 0.076 | 0.062 | 0.063 | 0.061 | |||

2.2 | 0.039 | 0.049 | 0.058 | 0.052 | 0.073 | 0.064 | 0.063 | 0.056 | |||

2.4 | 0.042 | 0.053 | 0.056 | 0.054 | 0.075 | 0.072 | 0.061 | 0.057 | |||

2.6 | 0.045 | 0.048 | 0.053 | 0.057 | 0.072 | 0.061 | 0.058 | 0.060 |

DGP | Estimated Model | Trim | Model | ${\mathbf{UV}}_{\mathit{T}}^{*}$ | ${\mathbf{CV}}_{\mathit{T}}^{*}$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\alpha}$ | $\mathit{\sigma}$ | T = 100 | 200 | 500 | 1000 | 100 | 200 | 500 | 1000 | |||

DGP2—AR(4) model, iid errors | AR(1) | 15% | 1 | 1 | 0.057 | 0.064 | 0.075 | 0.063 | 0.110 | 0.107 | 0.124 | 0.121 |

1.2 | 0.061 | 0.063 | 0.072 | 0.065 | 0.112 | 0.102 | 0.118 | 0.129 | ||||

1.4 | 0.059 | 0.068 | 0.072 | 0.065 | 0.116 | 0.112 | 0.119 | 0.129 | ||||

1.6 | 0.062 | 0.061 | 0.072 | 0.063 | 0.116 | 0.100 | 0.118 | 0.125 | ||||

1.8 | 0.055 | 0.056 | 0.073 | 0.067 | 0.113 | 0.104 | 0.120 | 0.125 | ||||

2 | 0.057 | 0.064 | 0.068 | 0.062 | 0.119 | 0.106 | 0.115 | 0.123 | ||||

2.2 | 0.060 | 0.063 | 0.077 | 0.064 | 0.118 | 0.099 | 0.119 | 0.122 | ||||

2.4 | 0.057 | 0.063 | 0.075 | 0.067 | 0.108 | 0.108 | 0.121 | 0.125 | ||||

2.6 | 0.062 | 0.065 | 0.074 | 0.065 | 0.111 | 0.113 | 0.116 | 0.125 | ||||

DGP1—AR(1) model, iid errors | AR(4) | 15% | 1 | 1 | 0.042 | 0.052 | 0.051 | 0.059 | 0.075 | 0.054 | 0.047 | 0.047 |

1.2 | 0.042 | 0.047 | 0.051 | 0.050 | 0.075 | 0.052 | 0.048 | 0.046 | ||||

1.4 | 0.039 | 0.046 | 0.050 | 0.056 | 0.070 | 0.054 | 0.049 | 0.050 | ||||

1.6 | 0.040 | 0.043 | 0.052 | 0.054 | 0.079 | 0.052 | 0.049 | 0.046 | ||||

1.8 | 0.040 | 0.045 | 0.052 | 0.055 | 0.076 | 0.051 | 0.049 | 0.050 | ||||

2 | 0.040 | 0.047 | 0.052 | 0.054 | 0.075 | 0.056 | 0.050 | 0.050 | ||||

2.2 | 0.041 | 0.044 | 0.054 | 0.057 | 0.073 | 0.051 | 0.047 | 0.049 | ||||

2.4 | 0.043 | 0.049 | 0.054 | 0.055 | 0.074 | 0.053 | 0.048 | 0.046 | ||||

2.6 | 0.043 | 0.047 | 0.052 | 0.052 | 0.076 | 0.055 | 0.047 | 0.046 | ||||

DGP3—static model, iid errors | AR(1) | 15% | 1 | 1 | 0.030 | 0.033 | 0.041 | 0.047 | 0.077 | 0.057 | 0.051 | 0.051 |

1.2 | 0.030 | 0.032 | 0.040 | 0.048 | 0.076 | 0.058 | 0.048 | 0.053 | ||||

1.4 | 0.029 | 0.032 | 0.043 | 0.047 | 0.075 | 0.055 | 0.055 | 0.051 | ||||

1.6 | 0.028 | 0.032 | 0.041 | 0.044 | 0.072 | 0.054 | 0.049 | 0.050 | ||||

1.8 | 0.028 | 0.033 | 0.039 | 0.049 | 0.071 | 0.054 | 0.047 | 0.053 | ||||

2 | 0.029 | 0.037 | 0.040 | 0.048 | 0.074 | 0.061 | 0.049 | 0.054 | ||||

2.2 | 0.029 | 0.033 | 0.040 | 0.043 | 0.080 | 0.057 | 0.049 | 0.045 | ||||

2.4 | 0.026 | 0.036 | 0.039 | 0.044 | 0.069 | 0.058 | 0.048 | 0.049 | ||||

2.6 | 0.029 | 0.032 | 0.045 | 0.045 | 0.074 | 0.055 | 0.056 | 0.050 | ||||

DGP4—static model, AR(1) errors | ${X}_{t}^{2}$ instead of ${X}_{t}$ | 15% | 1 | 1 | 0.041 | 0.051 | 0.052 | 0.057 | 0.075 | 0.064 | 0.059 | 0.058 |

1.2 | 0.041 | 0.049 | 0.053 | 0.055 | 0.074 | 0.063 | 0.058 | 0.055 | ||||

1.4 | 0.042 | 0.048 | 0.053 | 0.058 | 0.073 | 0.064 | 0.058 | 0.058 | ||||

1.6 | 0.039 | 0.047 | 0.056 | 0.054 | 0.071 | 0.063 | 0.063 | 0.057 | ||||

1.8 | 0.041 | 0.048 | 0.056 | 0.053 | 0.070 | 0.064 | 0.060 | 0.059 | ||||

2 | 0.043 | 0.048 | 0.055 | 0.057 | 0.076 | 0.065 | 0.058 | 0.059 | ||||

2.2 | 0.039 | 0.049 | 0.058 | 0.052 | 0.069 | 0.064 | 0.063 | 0.054 | ||||

2.4 | 0.042 | 0.053 | 0.056 | 0.054 | 0.075 | 0.067 | 0.060 | 0.052 | ||||

2.6 | 0.045 | 0.048 | 0.053 | 0.057 | 0.075 | 0.060 | 0.057 | 0.058 |

DGP 2: AR(4) Model: ${\mathit{y}}_{\mathit{t}}=1+\mathit{\beta}\phantom{\rule{4pt}{0ex}}{\mathit{y}}_{\mathit{t}-1}+0.2\phantom{\rule{4pt}{0ex}}{\mathit{y}}_{\mathit{t}-2}+0.15\phantom{\rule{4pt}{0ex}}{\mathit{y}}_{\mathit{t}-3}+0.75\phantom{\rule{4pt}{0ex}}{\mathit{y}}_{\mathit{t}-4}+{\mathit{\u03f5}}_{\mathit{t}}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{k}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |||

${\mathit{UM}}_{\mathbf{T}}^{*}$ | ${\mathit{CM}}_{\mathbf{T}}^{*}$ | |||||||||||

$T=100$ | $\beta =0.1$ | 0.153 | 0.563 | 0.468 | 0.309 | 0.246 | 0.292 | 0.346 | 0.447 | 0.536 | 0.661 | |

$\beta =0.2$ | 0.178 | 0.714 | 0.515 | 0.332 | 0.271 | 0.323 | 0.377 | 0.465 | 0.558 | 0.676 | ||

$\beta =0.3$ | 0.221 | 0.849 | 0.577 | 0.373 | 0.300 | 0.354 | 0.407 | 0.500 | 0.592 | 0.699 | ||

$T=200$ | $\beta =0.1$ | 0.110 | 0.612 | 0.464 | 0.243 | 0.151 | 0.140 | 0.154 | 0.169 | 0.184 | 0.215 | |

$\beta =0.2$ | 0.134 | 0.774 | 0.527 | 0.265 | 0.168 | 0.149 | 0.160 | 0.180 | 0.200 | 0.230 | ||

$\beta =0.3$ | 0.170 | 0.893 | 0.585 | 0.292 | 0.182 | 0.165 | 0.178 | 0.198 | 0.217 | 0.245 | ||

$T=500$ | $\beta =0.1$ | 0.121 | 0.660 | 0.492 | 0.210 | 0.109 | 0.082 | 0.083 | 0.088 | 0.092 | 0.095 | |

$\beta =0.2$ | 0.131 | 0.813 | 0.545 | 0.214 | 0.105 | 0.077 | 0.076 | 0.082 | 0.085 | 0.087 | ||

$\beta =0.3$ | 0.170 | 0.924 | 0.606 | 0.243 | 0.119 | 0.086 | 0.086 | 0.094 | 0.095 | 0.101 | ||

$T=1000$ | $\beta =0.1$ | 0.087 | 0.682 | 0.501 | 0.190 | 0.097 | 0.064 | 0.061 | 0.067 | 0.067 | 0.066 | |

$\beta =0.2$ | 0.099 | 0.838 | 0.569 | 0.203 | 0.097 | 0.065 | 0.065 | 0.069 | 0.069 | 0.070 | ||

$\beta =0.3$ | 0.119 | 0.944 | 0.614 | 0.217 | 0.096 | 0.066 | 0.063 | 0.064 | 0.065 | 0.065 |

DGP 2: AR(4) Model: ${\mathit{y}}_{\mathit{t}}=1+0.1\phantom{\rule{4pt}{0ex}}{\mathit{y}}_{\mathit{t}-1}+{\mathit{\gamma}}_{3}\phantom{\rule{4pt}{0ex}}{\mathit{y}}_{\mathit{t}-4}+{\mathit{\u03f5}}_{\mathit{t}}$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathbf{k}$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |||

${\mathit{UM}}_{\mathbf{T}}^{*}$ | ${\mathit{CM}}_{\mathbf{T}}^{*}$ | |||||||||||

$T=100$ | ${\gamma}_{3}=0.175$ | 0.084 | 0.233 | 0.157 | 0.235 | 0.275 | 0.254 | 0.306 | 0.399 | 0.498 | 0.624 | |

${\gamma}_{3}=0.275$ | 0.143 | 0.315 | 0.219 | 0.335 | 0.383 | 0.272 | 0.327 | 0.421 | 0.513 | 0.637 | ||

${\gamma}_{3}=0.375$ | 0.214 | 0.405 | 0.305 | 0.454 | 0.516 | 0.295 | 0.355 | 0.455 | 0.542 | 0.656 | ||

$T=200$ | ${\gamma}_{3}=0.175$ | 0.106 | 0.239 | 0.139 | 0.191 | 0.203 | 0.122 | 0.127 | 0.152 | 0.164 | 0.191 | |

${\gamma}_{3}=0.275$ | 0.168 | 0.342 | 0.221 | 0.315 | 0.327 | 0.128 | 0.137 | 0.163 | 0.183 | 0.208 | ||

${\gamma}_{3}=0.375$ | 0.242 | 0.455 | 0.326 | 0.453 | 0.479 | 0.138 | 0.148 | 0.172 | 0.194 | 0.226 | ||

$T=500$ | ${\gamma}_{3}=0.175$ | 0.138 | 0.262 | 0.149 | 0.193 | 0.187 | 0.076 | 0.081 | 0.088 | 0.091 | 0.096 | |

${\gamma}_{3}=0.275$ | 0.191 | 0.365 | 0.221 | 0.300 | 0.293 | 0.071 | 0.072 | 0.075 | 0.079 | 0.084 | ||

${\gamma}_{3}=0.375$ | 0.279 | 0.509 | 0.351 | 0.469 | 0.467 | 0.077 | 0.078 | 0.086 | 0.084 | 0.091 | ||

$T=1000$ | ${\gamma}_{3}=0.175$ | 0.072 | 0.266 | 0.138 | 0.171 | 0.164 | 0.060 | 0.063 | 0.062 | 0.062 | 0.062 | |

${\gamma}_{3}=0.275$ | 0.077 | 0.385 | 0.224 | 0.312 | 0.303 | 0.063 | 0.062 | 0.063 | 0.065 | 0.069 | ||

${\gamma}_{3}=0.375$ | 0.092 | 0.539 | 0.357 | 0.475 | 0.460 | 0.063 | 0.063 | 0.063 | 0.066 | 0.068 |

Moments/Models | Trimming | Statistic Value | Critical Value | Break Fraction | Break Date |
---|---|---|---|---|---|

Unconditional Mean sup Wald tests: | |||||

$E\left({y}_{t}\right)$ | 10% | 9.585 * | 9.11 | 0.892 | Oct-08 |

5% | 9.585 | 9.71 | 0.892 | - | |

Conditional Mean sup Wald tests: | |||||

AR(1) | 10% | 11.408 | 12.17 | 0.419 | - |

5% | 15.494 * | 12.80 | 0.929 | Nov-10 | |

AR(4) | 10% | 14.736 | 18.86 | 0.139 | - |

5% | 38.485 * | 19.57 | 0.911 | Dec-09 | |

AR(5)—BIC | 10% | 18.205 | 20.81 | 0.121 | - |

5% | 36.777 * | 21.53 | 0.928 | Nov-10 | |

AR(6)—AIC | 10% | 25.214 * | 22.62 | 0.880 | Apr-08 |

5% | 31.279 * | 23.41 | 0.928 | Nov-10 | |

AR(12) | 10% | 30.343 | 32.76 | 0.879 | - |

5% | 56.090 * | 33.63 | 0.949 | Jan-12 | |

DL(1) with macro factors—BIC | 10% | 13.935 | 16.91 | 0.451 | - |

5% | 18.612 * | 17.54 | 0.949 | Mar-09 | |

DL(7) with macro factors—AIC | 10% | 19.665 | 37.43 | 0.619 | - |

5% | 25.653 | 38.35 | 0.934 | - | |

DL(12) with uncertainty factors—AIC/BIC | 10% | 150.259 * | 32.76 | 0.173 | Mar-70 |

5% | 1826 * | 33.63 | 0.941 | Jan-09 |

Moments | Test | Trimming | Statistic Value | Critical Value | Break Fraction | Break Date |
---|---|---|---|---|---|---|

Macro Factor | ||||||

E(${y}_{t}$) | $U{M}_{T}^{*}$ | 10% | 2.646 | 9.11 | 0.281 | - |

5% | 2.646 | 9.71 | 0.281 | - | ||

Macro Uncertainty Factor | ||||||

E(${y}_{t}$) | $U{M}_{T}^{*}$ | 10% | 5.214 | 9.11 | 0.898 | - |

5% | 6.295 | 9.71 | 0.921 | - | ||

Macro Factor | ||||||

$Var$(${y}_{t}$) | $U{V}_{T}^{*}$ | 10% | 4.277 | 9.11 | 0.898 | - |

5% | 10.181 * | 9.71 | 0.935 | Jul-08 | ||

Macro Uncertainty Factor | ||||||

$Var$(${y}_{t}$) | $U{V}_{T}^{*}$ | 10% | 4.969 | 9.11 | 0.898 | - |

5% | 8.317 | 9.71 | 0.928 | - |

Moments/Models | Statistic Value | Critical Value | Break Fraction | Break Date |
---|---|---|---|---|

Unconditional Variance sup Wald tests: | ||||

$Var\left({y}_{t}\right)$ | 5.011 | 9.11 | 0.447 | - |

Conditional Variance sup Wald tests: | ||||

AR(1) | 15.278 * | 9.11 | 0.476 | Feb-86 |

AR(4) | 11.948 * | 9.11 | 0.471 | Feb-86 |

AR(5)—BIC | 13.024 * | 9.11 | 0.469 | Feb-86 |

AR(6)—AIC | 17.660 * | 9.11 | 0.468 | Feb-86 |

AR(12) | 14.753 * | 9.11 | 0.457 | Feb-86 |

DL(1) with macro factors—BIC | 15.098 * | 9.11 | 0.275 | May-74 |

DL(7) with macro factors—AIC | 15.044 * | 9.11 | 0.259 | Apr-74 |

DL(12) with uncertainty factors—AIC/BIC | 6.375 | 9.11 | 0.415 | - |

Moments/Models | Trimming | Statistic Value | Critical Value | Break Fraction | Break Date |
---|---|---|---|---|---|

Unconditional Mean sup Wald tests: | |||||

$E\left({y}_{t}\right)$ | 10% | 3.191 | 9.11 | 0.252 | - |

5% | 3.191 | 9.71 | 0.252 | - | |

Conditional Mean sup Wald tests: | |||||

AR(1) | 10% | 13.949 * | 12.17 | 0.401 | Jan-82 |

5% | 28.948 * | 12.80 | 0.933 | Feb-11 | |

AR(3)—BIC | 10% | 19.147 * | 16.91 | 0.399 | Jan-82 |

5% | 22.813 * | 17.54 | 0.922 | Jul-10 | |

AR(4) | 10% | 25.432 * | 18.86 | 0.398 | Jan-82 |

5% | 25.432 * | 19.57 | 0.398 | Jan-82 | |

AR(5)—AIC | 10% | 28.486 * | 20.81 | 0.397 | Jan-82 |

5% | 28.486 * | 21.53 | 0.397 | Jan-82 | |

AR(12) | 10% | 35.846 * | 32.76 | 0.392 | Feb-82 |

5% | 37.517 * | 33.63 | 0.924 | Sep-10 | |

DL(12) with macro factors—AIC/BIC | 10% | 32.600 | >43.47 | 0.102 | - |

5% | 36.215 | >44.46 | 0.060 | - | |

DL(1) with uncertainty factors—BIC | 10% | 11.632 | 12.17 | 0.539 | - |

5% | 26.729 * | 12.80 | 0.948 | May-09 | |

DL(5) with uncertainty factors—AIC | 10% | 17.761 | 20.81 | 0.254 | - |

5% | 20.115 | 21.53 | 0.935 | - |

Moments/Models | Statistic Value | Critical Value | Break Fraction | Break Date |
---|---|---|---|---|

Unconditional Variance sup Wald tests: | ||||

$Var\left({y}_{t}\right)$ | 5.535 | 9.11 | 0.437 | - |

Conditional Variance sup Wald tests: | ||||

AR(1) | 10.164 * | 9.11 | 0.437 | Jan-84 |

AR(3)—BIC | 12.535 * | 9.11 | 0.433 | Jan-84 |

AR(4) | 13.302 * | 9.11 | 0.413 | Jan-83 |

AR(5)—AIC | 12.168 * | 9.11 | 0.411 | Jan-83 |

AR(12) | 10.495 * | 9.11 | 0.398 | Jan-83 |

DL(12) with macro factors—AIC/BIC | 8.267 | 9.11 | 0.442 | - |

DL(1) with uncertainty factors—BIC | 13.318 * | 9.11 | 0.455 | Jan-84 |

DL(5) with uncertainty factors—AIC | 10.938 * | 9.11 | 0.448 | Jan-84 |

Moments/Models | Trimming | Statistic Value | Critical Value | Break Fraction | Break Date |
---|---|---|---|---|---|

Unconditional Mean sup Wald tests: | |||||

$E\left({y}_{t}\right)$ | 10% | 15.797 * | 9.11 | 0.752 | Apr-01 |

5% | 15.797* | 9.71 | 0.752 | Apr-01 | |

Conditional Mean sup Wald tests: | |||||

AR(1) | 10% | 57.077 * | 12.17 | 0.893 | Dec-08 |

5% | 57.077 * | 12.80 | 0.893 | Dec-08 | |

AR(4) | 10% | 54.459 * | 18.86 | 0.893 | Dec-08 |

5% | 54.459 * | 19.57 | 0.893 | Dec-08 | |

AR(9)—BIC | 10% | 125.086 * | 27.77 | 0.892 | Dec-08 |

5% | 125.086 * | 28.64 | 0.892 | Dec-08 | |

AR(12)—AIC | 10% | 129.647 * | 32.76 | 0.891 | Dec-08 |

5% | 129.647 * | 33.63 | 0.891 | Dec-08 | |

DL(1) with macro factors—AIC/BIC | 10% | 8.081 | 16.91 | 0.409 | - |

5% | 25.995 * | 17.54 | 0.934 | Jun-08 | |

DL(1) with uncertainty factors—AIC/BIC | 10% | 850.945 * | 12.17 | 0.791 | Apr-01 |

5% | 1840 * | 12.80 | 0.942 | Jan-09 |

Moments/Models | Statistic Value | Critical Value | Break Fraction | Break Date |
---|---|---|---|---|

Unconditional Variance sup Wald tests: | ||||

$Var\left({y}_{t}\right)$ | 4.852 | 9.11 | 0.409 | - |

Conditional Variance sup Wald tests: | ||||

AR(1) | 21.591 * | 9.11 | 0.410 | Sep-82 |

AR(4) | 24.721 * | 9.11 | 0.407 | Oct-82 |

AR(9)—BIC | 20.799 * | 9.11 | 0.394 | Aug-82 |

AR(12)—AIC | 23.787 * | 9.11 | 0.388 | Aug-82 |

DL(1) with macro factors—AIC/BIC | 12.637 * | 9.11 | 0.433 | Aug-82 |

DL(1) with uncertainty factors—AIC/BIC | 9.678 * | 9.11 | 0.428 | Aug-82 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Abi Morshed, A.; Andreou, E.; Boldea, O.
Structural Break Tests Robust to Regression Misspecification. *Econometrics* **2018**, *6*, 27.
https://doi.org/10.3390/econometrics6020027

**AMA Style**

Abi Morshed A, Andreou E, Boldea O.
Structural Break Tests Robust to Regression Misspecification. *Econometrics*. 2018; 6(2):27.
https://doi.org/10.3390/econometrics6020027

**Chicago/Turabian Style**

Abi Morshed, Alaa, Elena Andreou, and Otilia Boldea.
2018. "Structural Break Tests Robust to Regression Misspecification" *Econometrics* 6, no. 2: 27.
https://doi.org/10.3390/econometrics6020027