# A Joint Chow Test for Structural Instability

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Test Statistics

## 3. Model and Assumptions

**Assumption 1.**$\left|\mathrm{eigen}\right(\mathbf{D}\left)\right|=1$ and $\mathrm{rank}({D}_{1},\dots ,{D}_{dim\mathbf{D}})=dim\mathbf{D}.$

**Assumption 2.**${\xi}_{t}$ is a martingale difference sequence with respect to the natural filtration ${\mathcal{F}}_{t}$, so $\mathsf{E}\left({\xi}_{t}\right|{\mathcal{F}}_{t-1})=0$. The initial values ${X}_{0},\dots {X}_{1-k}$ are ${\mathcal{F}}_{0}$-measurable and:

**Assumption 3.**The explosive roots of $\mathbf{B}$ have geometric multiplicity of unity. That is, for all complex λ with $\left|\lambda \right|>1$, $\mathrm{rank}(\mathbf{B}-\lambda {\mathbf{I}}_{pk})\ge pk-1$.

**Assumption 4.**Let ${\mathcal{G}}_{t}$ be the sigma field over ${\mathcal{F}}_{t}$ and ${Z}_{t}$. Then, $({\epsilon}_{t},{\mathcal{G}}_{t})$ is a martingale difference sequence, i.e., $\mathsf{E}\left({\epsilon}_{t}\right|{\mathcal{G}}_{t-1})=0$.

**Assumption 5.**${\epsilon}_{t}\stackrel{\mathrm{iid}}{\sim}\mathrm{N}(0,{\sigma}^{2})$.

## 4. Main Results

**Theorem 6.**Under Assumptions 1, 2, 3 and 4,

**Lemma 7.**Suppose ${\mathsf{C}}_{1,t}^{2}-{({q}_{t}/\sigma )}^{2}\stackrel{\mathrm{as}}{\to}0\phantom{\rule{4.pt}{0ex}}as\phantom{\rule{4.pt}{0ex}}t\to \infty $. Then:

**Lemma 8.**Let $\left\{{X}_{n}\right\}$ be independent Gaussian random variables with mean zero and variance one. Let ${Z}_{n}={max}_{1\le j\le n}\left|{X}_{j}\right|$. Then, ${a}_{n}({Z}_{n}-{b}_{n})$ converges in distribution to Λ, where ${a}_{n}={(2logn)}^{1/2}$ and ${b}_{n}={(2logn)}^{1/2}-{(8logn)}^{-1/2}(loglogn+log\pi )$.

**Lemma 9.**Under Assumption 5,

**Theorem 10.**Under Assumptions 1, 2, 3, 4 and 5, with some $g\left(T\right)\to \infty $,

**Corollary 11.**Under the same assumptions, $2\xb7exp(-{\mathsf{SC}}_{T}^{2})\sim {\chi}_{2}^{2}$. A test based on this result should reject for small values of the statistic.

## 5. Finite-Sample Corrections

## 6. Simulation Study

#### 6.1. Autoregressive Data-Generating Process

**Table 1.**Simulated rejection frequency for ${\mathsf{SC}}^{\mathsf{2}}$ and ${\mathsf{SC}}^{*\mathsf{2}}$ under the Gaussian autoregression in Equation (29). 200,000 repetitions, $\mathsf{MCSE}\le 0.1$.

T | Autoregressive Coefficient (α) | ||||||||
---|---|---|---|---|---|---|---|---|---|

−1.03 | −1.00 | −0.50 | 0.00 | 0.50 | 0.90 | 1.00 | 1.03 | ||

5% Nominal Size | |||||||||

Intercept included in model (M1) | |||||||||

25 | ${\mathsf{SC}}^{\mathsf{2}}$ | 14.52 | 14.44 | 13.92 | 14.40 | 15.82 | 19.28 | 19.86 | 20.21 |

${\mathsf{SC}}^{\mathsf{2}*}$ | 5.30 | 5.26 | 5.01 | 5.24 | 5.78 | 7.28 | 7.59 | 7.75 | |

50 | ${\mathsf{SC}}^{\mathsf{2}}$ | 12.80 | 12.72 | 12.32 | 12.60 | 13.50 | 16.13 | 16.97 | 17.43 |

${\mathsf{SC}}^{\mathsf{2}*}$ | 5.17 | 5.15 | 4.92 | 5.05 | 5.38 | 6.52 | 7.00 | 7.27 | |

100 | ${\mathsf{SC}}^{\mathsf{2}}$ | 10.43 | 10.41 | 10.15 | 10.36 | 10.73 | 12.34 | 13.27 | 13.85 |

${\mathsf{SC}}^{\mathsf{2}*}$ | 5.09 | 5.05 | 4.95 | 5.00 | 5.08 | 5.82 | 6.38 | 6.74 | |

No intercept included in model (M2) | |||||||||

25 | ${\mathsf{SC}}^{\mathsf{2}}$ | 15.28 | 15.23 | 14.51 | 14.49 | 14.62 | 15.11 | 15.33 | 15.39 |

${\mathsf{SC}}^{\mathsf{2}*}$ | 5.49 | 5.46 | 5.20 | 5.19 | 5.23 | 5.44 | 5.49 | 5.53 | |

50 | ${\mathsf{SC}}^{\mathsf{2}}$ | 13.17 | 13.12 | 12.72 | 12.72 | 12.71 | 13.01 | 13.20 | 13.21 |

${\mathsf{SC}}^{\mathsf{2}*}$ | 5.33 | 5.29 | 5.06 | 5.03 | 5.09 | 5.20 | 5.26 | 5.27 | |

100 | ${\mathsf{SC}}^{\mathsf{2}}$ | 10.64 | 10.60 | 10.27 | 10.31 | 10.34 | 10.42 | 10.62 | 10.63 |

${\mathsf{SC}}^{\mathsf{2}*}$ | 5.17 | 5.12 | 5.02 | 4.99 | 5.01 | 5.03 | 5.13 | 5.16 | |

1% Nominal Size | |||||||||

No intercept included in model (M1) | |||||||||

25 | ${\mathsf{SC}}^{\mathsf{2}}$ | 6.30 | 6.29 | 5.98 | 6.24 | 7.06 | 8.87 | 9.16 | 9.33 |

${\mathsf{SC}}^{\mathsf{2}*}$ | 1.09 | 1.10 | 1.04 | 1.07 | 1.24 | 1.64 | 1.75 | 1.79 | |

50 | ${\mathsf{SC}}^{\mathsf{2}}$ | 4.77 | 4.77 | 4.56 | 4.70 | 5.16 | 6.44 | 6.83 | 7.04 |

${\mathsf{SC}}^{\mathsf{2}*}$ | 1.08 | 1.08 | 1.02 | 1.06 | 1.15 | 1.45 | 1.57 | 1.60 | |

100 | ${\mathsf{SC}}^{\mathsf{2}}$ | 3.36 | 3.36 | 3.24 | 3.31 | 3.47 | 4.22 | 4.59 | 4.74 |

${\mathsf{SC}}^{\mathsf{2}*}$ | 1.05 | 1.05 | 1.02 | 1.02 | 1.04 | 1.21 | 1.36 | 1.43 | |

No intercept included in model (M2) | |||||||||

25 | ${\mathsf{SC}}^{\mathsf{2}}$ | 6.71 | 6.69 | 6.25 | 6.22 | 6.34 | 6.59 | 6.71 | 6.73 |

${\mathsf{SC}}^{\mathsf{2}*}$ | 1.16 | 1.15 | 1.06 | 1.04 | 1.06 | 1.12 | 1.18 | 1.19 | |

50 | ${\mathsf{SC}}^{\mathsf{2}}$ | 4.98 | 4.97 | 4.70 | 4.69 | 4.78 | 4.90 | 4.95 | 4.95 |

${\mathsf{SC}}^{\mathsf{2}*}$ | 1.12 | 1.11 | 1.04 | 1.05 | 1.05 | 1.07 | 1.10 | 1.09 | |

100 | ${\mathsf{SC}}^{\mathsf{2}}$ | 3.41 | 3.37 | 3.24 | 3.24 | 3.25 | 3.33 | 3.39 | 3.40 |

${\mathsf{SC}}^{\mathsf{2}*}$ | 1.06 | 1.05 | 1.02 | 1.01 | 1.00 | 1.00 | 1.03 | 1.04 |

**Table 2.**Simulated rejection frequency for ${\mathsf{SC}}^{*\mathsf{2}}$ and $\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}$, possibly combined with normality test Φ, under autoregression in Equation (29) with various error distributions. 50,000 repetitions, $\mathsf{MCSE}\le 0.25$.

T | Error Distribution | |||||
---|---|---|---|---|---|---|

Φ | ${\mathit{t}}_{\mathbf{50}}$ | ${\mathit{t}}_{\mathbf{10}}$ | ${\mathit{t}}_{\mathbf{5}}$ | ${\mathit{\chi}}_{\mathbf{3}\mathbf{,}\mathbf{cent}\mathbf{.}}^{\mathbf{2}}$ | ||

Unconditional tests | ||||||

50 | ${\mathsf{SC}}^{*2}$ | 5.0 | 6.6 | 15.0 | 28.8 | 40.9 |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}$ | 6.0 | 6.0 | 6.0 | 5.9 | 6.3 | |

100 | ${\mathsf{SC}}^{*2}$ | 5.0 | 7.4 | 22.2 | 45.0 | 59.9 |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}$ | 4.9 | 5.0 | 4.9 | 4.7 | 4.9 | |

Joint tests | ||||||

50 | Φ | 4.9 | 6.7 | 19.0 | 41.0 | 95.1 |

${\mathsf{SC}}^{*2}|\Phi $ | 3.4 | 3.9 | 6.2 | 8.0 | * 7.4 | |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}|\Phi $ | 6.0 | 6.0 | 6.1 | 6.0 | * 6.7 | |

${\mathsf{SC}}^{*2}+\Phi $ | 8.1 | 10.4 | 24.0 | 45.8 | * 95.5 | |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}+\Phi $ | 10.6 | 12.3 | 23.9 | 44.6 | * 95.4 | |

100 | Φ | 4.8 | 7.6 | 28.7 | 63.0 | 100.0 |

${\mathsf{SC}}^{*2}|\Phi $ | 3.3 | 4.2 | 7.2 | 8.6 | ||

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}|\Phi $ | 5.0 | 5.0 | 5.0 | 4.9 | ||

${\mathsf{SC}}^{*2}+\Phi $ | 8.0 | 11.5 | 33.9 | 66.2 | * 100.0 | |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}+\Phi $ | 9.5 | 12.2 | 32.3 | 64.8 | * 100.0 |

**Table 3.**Simulated rejection frequency for ${\mathsf{SC}}^{*\mathsf{2}}$ and $\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}$, possibly combined with normality test Φ, under the process in Equation (30) with a break of magnitude γ at time τ. 50,000 repetitions, $\mathsf{MCSE}\le 0.5$.

T | Break Timing (τ) | |||||||
---|---|---|---|---|---|---|---|---|

0.5T | T-2 | T-1 | ||||||

Post-Break Constant (γ) | ||||||||

0.0 | 2.0 | 4.0 | 2.0 | 4.0 | 2.0 | 4.0 | ||

Unconditional tests | ||||||||

25 | ${\mathsf{SC}}^{*2}$ | 5.5 | 13.7 | 51.2 | 21.0 | 78.7 | 16.0 | 70.3 |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}$ | 10.3 | 90.4 | 99.9 | 19.9 | 44.8 | 14.1 | 28.1 | |

50 | ${\mathsf{SC}}^{*2}$ | 5.2 | 17.1 | 67.6 | 18.6 | 83.2 | 13.3 | 69.9 |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}$ | 6.0 | 99.8 | 100.0 | 10.2 | 34.5 | 7.1 | 11.5 | |

100 | ${\mathsf{SC}}^{*2}$ | 5.1 | 20.0 | 77.9 | 16.1 | 84.2 | 11.5 | 67.6 |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}$ | 4.9 | 100.0 | 100.0 | 7.2 | 31.2 | 5.4 | 7.4 | |

Joint tests | ||||||||

25 | Φ | 5.1 | 4.3 | 15.6 | 9.6 | 37.9 | 9.9 | 53.9 |

${\mathsf{SC}}^{*2}|\Phi $ | 3.8 | 12.5 | 45.3 | 14.8 | 66.7 | 9.4 | 38.3 | |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}|\Phi $ | 10.2 | 90.4 | 99.9 | 19.6 | 52.1 | 13.1 | 22.6 | |

${\mathsf{SC}}^{*2}+\Phi $ | 8.7 | 16.2 | 53.8 | 22.9 | 79.4 | 18.4 | 71.5 | |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}+\Phi $ | 14.8 | 90.8 | 99.9 | 27.3 | 70.3 | 21.7 | 64.3 | |

50 | Φ | 4.8 | 4.4 | 17.0 | 11.1 | 61.9 | 9.5 | 58.6 |

${\mathsf{SC}}^{*2}|\Phi $ | 3.5 | 15.7 | 62.6 | 11.3 | 58.6 | 7.2 | 31.1 | |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}|\Phi $ | 6.0 | 99.8 | 100.0 | 9.8 | 33.9 | 6.9 | 9.0 | |

${\mathsf{SC}}^{*2}+\Phi $ | 8.2 | 19.4 | 69.0 | 21.1 | 84.2 | 16.1 | 71.5 | |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}+\Phi $ | 10.6 | 99.9 | 100.0 | 19.8 | 74.8 | 15.8 | 62.3 | |

100 | Φ | 4.7 | 4.2 | 15.6 | 11.2 | 74.7 | 8.5 | 57.4 |

${\mathsf{SC}}^{*2}|\Phi $ | 3.4 | 18.7 | 74.7 | 8.9 | 43.5 | 6.4 | 28.4 | |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}|\Phi $ | 4.9 | 100.0 | 100.0 | 6.8 | 19.3 | 5.4 | 6.3 | |

${\mathsf{SC}}^{*2}+\Phi $ | 8.0 | 22.1 | 78.7 | 19.1 | 85.7 | 14.4 | 69.5 | |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}+\Phi $ | 9.4 | 100.0 | 100.0 | 17.2 | 79.6 | 13.5 | 60.0 |

**Table 4.**Simulated rejection frequency for ${\mathsf{SC}}^{*\mathsf{2}}$ and $\mathsf{sup}\phantom{\rule{4pt}{0ex}}\mathsf{F}$, possibly combined with normality test Φ, under the process in Equation (31) with a break of magnitude δ. 50,000 repetitions, $\mathsf{MCSE}\le 0.5$.

α | T | Outlier Magnitude (δ) | ||||||
---|---|---|---|---|---|---|---|---|

0.0 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | |||

Unconditional tests | ||||||||

0.5 | 50 | ${\mathsf{SC}}^{*2}$ | 5.5 | 5.8 | 10.6 | 28.7 | 58.7 | 84.9 |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}{\mathsf{t}}^{\mathsf{2}}$ | 4.7 | 5.0 | 10.5 | 31.8 | 66.2 | 90.7 | ||

100 | ${\mathsf{SC}}^{*2}$ | 5.2 | 5.5 | 9.8 | 28.1 | 61.1 | 87.9 | |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}{\mathsf{t}}^{\mathsf{2}}$ | 4.8 | 5.1 | 9.8 | 29.9 | 65.1 | 90.8 | ||

0.9 | 50 | ${\mathsf{SC}}^{*2}$ | 6.6 | 7.0 | 12.0 | 30.2 | 59.7 | 85.3 |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}{\mathsf{t}}^{\mathsf{2}}$ | 4.7 | 4.9 | 10.3 | 31.3 | 65.5 | 90.3 | ||

100 | ${\mathsf{SC}}^{*2}$ | 5.9 | 6.3 | 10.8 | 28.9 | 61.4 | 87.9 | |

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}{\mathsf{t}}^{\mathsf{2}}$ | 4.9 | 5.1 | 9.8 | 29.7 | 64.9 | 90.6 | ||

Joint tests | ||||||||

0.5 | 50 | Φ | 4.9 | 5.0 | 9.6 | 27.7 | 59.3 | 86.6 |

${\mathsf{SC}}^{*2}|\Phi $ | 3.8 | 4.0 | 5.8 | 10.8 | 18.0 | 25.6 | ||

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}{\mathsf{t}}^{\mathsf{2}}|\Phi $ | 2.2 | 2.4 | 4.0 | 10.3 | 22.7 | 36.8 | ||

${\mathsf{SC}}^{*2}+\Phi $ | 8.5 | 8.8 | 14.8 | 35.5 | 66.6 | 90.0 | ||

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}{\mathsf{t}}^{\mathsf{2}}+\Phi $ | 7.0 | 7.3 | 13.2 | 35.2 | 68.5 | 91.5 | ||

100 | Φ | 4.8 | 5.0 | 8.7 | 25.0 | 57.1 | 86.0 | |

${\mathsf{SC}}^{*2}|\Phi $ | 3.5 | 3.7 | 5.2 | 11.0 | 20.9 | 32.0 | ||

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}{\mathsf{t}}^{\mathsf{2}}|\Phi $ | 2.7 | 2.8 | 4.4 | 10.9 | 24.3 | 40.0 | ||

${\mathsf{SC}}^{*2}+\Phi $ | 8.1 | 8.5 | 13.4 | 33.2 | 66.1 | 90.5 | ||

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}{\mathsf{t}}^{\mathsf{2}}+\Phi $ | 7.3 | 7.7 | 12.7 | 33.2 | 67.5 | 91.6 | ||

0.9 | 50 | Φ | 4.9 | 5.1 | 9.4 | 27.1 | 58.6 | 85.9 |

${\mathsf{SC}}^{*2}|\Phi $ | 4.9 | 5.2 | 7.3 | 13.1 | 21.6 | 30.6 | ||

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}{\mathsf{t}}^{\mathsf{2}}|\Phi $ | 2.2 | 2.3 | 4.0 | 10.3 | 22.6 | 37.5 | ||

${\mathsf{SC}}^{*2}+\Phi $ | 9.5 | 10.0 | 16.0 | 36.6 | 67.5 | 90.2 | ||

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}{\mathsf{t}}^{\mathsf{2}}+\Phi $ | 6.9 | 7.2 | 13.0 | 34.6 | 68.0 | 91.2 | ||

100 | Φ | 4.7 | 5.0 | 8.6 | 24.7 | 56.8 | 85.8 | |

${\mathsf{SC}}^{*2}|\Phi $ | 4.2 | 4.4 | 6.2 | 12.2 | 22.2 | 33.6 | ||

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}{\mathsf{t}}^{\mathsf{2}}|\Phi $ | 2.7 | 2.8 | 4.4 | 10.9 | 24.2 | 39.5 | ||

${\mathsf{SC}}^{*2}+\Phi $ | 8.7 | 9.2 | 14.3 | 33.8 | 66.4 | 90.6 | ||

$\mathsf{sup}\phantom{\rule{4pt}{0ex}}{\mathsf{t}}^{\mathsf{2}}+\Phi $ | 7.3 | 7.6 | 12.6 | 32.9 | 67.2 | 91.4 |

#### 6.2. Autoregressive Distributed Lag Data-Generating Process

T | Autoregressive Coefficient (ψ) | ||||
---|---|---|---|---|---|

0.25 | 1.00 | ||||

50 | ${\mathsf{SC}}^{2}$ | 15.0 | 16.3 | ||

${\mathsf{SC}}^{*2}$ | 6.1 | 6.7 | |||

100 | ${\mathsf{SC}}^{2}$ | 11.6 | 12.2 | ||

${\mathsf{SC}}^{*2}$ | 5.4 | 5.8 |

ψ | Break Timing(τ) | |||||||
---|---|---|---|---|---|---|---|---|

0.5T | T-2 | T-1 | ||||||

Post-Break Constant (ν) | ||||||||

0.0 | 2.0 | 4.0 | 2.0 | 4.0 | 2.0 | 4.0 | ||

0.25 | ${\mathsf{SC}}^{*2}$ | 6.1 | 16.3 | 64.6 | 19.7 | 83.2 | 13.8 | 68.2 |

Φ | 4.9 | 6.8 | 34.1 | 10.9 | 61.8 | 9.1 | 54.9 | |

${\mathsf{SC}}^{*2}|\Phi $ | 4.5 | 13.4 | 51.2 | 12.7 | 58.8 | 8.4 | 33.1 | |

${\mathsf{SC}}^{*2}+\Phi $ | 9.2 | 19.3 | 67.8 | 22.2 | 84.3 | 16.7 | 69.8 | |

1.00 | ${\mathsf{SC}}^{*2}$ | 6.7 | 16.0 | 62.0 | 19.4 | 81.2 | 14.2 | 67.2 |

Φ | 5.0 | 6.0 | 25.1 | 9.7 | 54.5 | 8.7 | 51.8 | |

${\mathsf{SC}}^{*2}|\Phi $ | 5.3 | 13.7 | 52.6 | 13.5 | 60.8 | 9.1 | 35.2 | |

${\mathsf{SC}}^{*2}+\Phi $ | 10.0 | 18.9 | 64.5 | 21.9 | 82.2 | 17.0 | 68.8 |

ψ | Outlier Magnitude (ν) | ||||||
---|---|---|---|---|---|---|---|

0.0 | 1.0 | 2.0 | 3.0 | 4.0 | 5.0 | ||

0.25 | ${\mathsf{SC}}^{*2}$ | 6.1 | 6.3 | 10.4 | 25.5 | 53.0 | 79.1 |

Φ | 4.9 | 5.2 | 9.1 | 25.4 | 55.4 | 83.1 | |

${\mathsf{SC}}^{*2}|\Phi $ | 4.5 | 4.6 | 6.3 | 10.8 | 17.5 | 24.8 | |

${\mathsf{SC}}^{*2}+\Phi $ | 9.2 | 9.6 | 14.8 | 33.5 | 63.2 | 87.3 | |

1.00 | ${\mathsf{SC}}^{*2}$ | 6.7 | 7.1 | 11.0 | 25.4 | 51.5 | 77.4 |

Φ | 5.0 | 5.3 | 9.3 | 25.9 | 55.9 | 83.5 | |

${\mathsf{SC}}^{*2}|\Phi $ | 5.3 | 5.5 | 7.1 | 11.7 | 18.5 | 26.0 | |

${\mathsf{SC}}^{*2}+\Phi $ | 10.0 | 10.5 | 15.7 | 34.6 | 64.1 | 87.8 |

## 7. Empirical Illustration

**Figure 1.**U.K. GDP series. (

**a**) Series in logs; (

**b**) scaled residuals; (

**c**) pointwise one-step Chow tests; the horizontal line is the 1% critical value; (

**d**) simultaneous Chow test; the horizontal lines are the 1% (top) and 5% (bottom) critical values.

## 8. Conclusions

## Supplementary Materials

Supplementary File 1## Acknowledgements

## Author Contributions

## A. Proofs

#### A.1. Notation

#### A.2. Three-Way Process Decomposition

#### A.3. Preliminary Asymptotic Results

**Lemma 12.**Suppose Assumptions 1, 2 and 3 hold with $\alpha >4$ only. Then, for all $\beta >1/\alpha $ and $\zeta <1/8$,

- (i)
- ${C}_{RW}\stackrel{\mathrm{a}.\mathrm{s}.}{=}\mathrm{o}\left({t}^{-\zeta /2}\right)$,
- (ii)
- ${C}_{\xi S}\stackrel{\mathrm{a}.\mathrm{s}.}{=}\mathrm{o}\left({t}^{\beta -1/2}\right)$,
- (iii)
- ${S}_{RR\xb7W}^{-1}\stackrel{\mathrm{a}.\mathrm{s}.}{=}{S}_{RR}^{-1/2}\xb7\{1+\mathrm{o}\left(1\right)\}\xb7{S}_{RR}^{-1/2}$,
- (iv)
- ${S}_{{\xi}_{2}{\xi}_{2}\xb7S}^{-1}\stackrel{\mathrm{a}.\mathrm{s}.}{=}{S}_{{\xi}_{2}{\xi}_{2}}^{-1/2}\xb7\{1+\mathrm{o}\left(1\right)\}\xb7{S}_{{\xi}_{2}{\xi}_{2}}^{-1/2}$,
- (v)
- ${S}_{RR}^{-1/2}{R}_{t-1}\stackrel{\mathrm{a}.\mathrm{s}.}{=}\mathrm{o}\left({t}^{-\zeta /2}\right)$,
- (vi)
- ${S}_{WW}^{-1/2}{W}_{t-1}\stackrel{\mathrm{a}.\mathrm{s}.}{=}\mathrm{O}\left(1\right)$,
- (vii)
- ${S}_{RR}^{-1/2}{\left(R\right|W)}_{t}\stackrel{\mathrm{a}.\mathrm{s}.}{=}\mathrm{o}\left({t}^{-\zeta /2}\right)$, and
- (viii)
- ${S}_{{\xi}_{2}{\xi}_{2}}^{-1/2}{\left({\xi}_{2}\right|S)}_{t}\stackrel{\mathrm{a}.\mathrm{s}.}{=}\mathrm{o}\left({t}^{\beta -1/2}\right)$.

**Proof.**Result (i) is proven by decomposing the correlation to apply results from [25], so that:

**Lemma 13.**Under Assumptions 1, 2 and 3 with $\alpha >4$ and with $\beta >1/\alpha $,

- (i)
- ${\sum}_{s=1}^{t-1}{\epsilon}_{s}{S}_{s-1}^{\prime}{S}_{SS}^{-1/2}\stackrel{\mathrm{a}.\mathrm{s}.}{=}\mathrm{o}\left({t}^{\beta}\right)$,
- (ii)
- ${\sum}_{s=1}^{t-1}{\epsilon}_{s}{R}_{s-1}^{\prime}{S}_{RR}^{-1/2}\stackrel{\mathrm{a}.\mathrm{s}.}{=}\mathrm{O}\left[{(logt)}^{1/2}\right]$,
- (iii)
- ${\sum}_{s=1}^{t-1}{\epsilon}_{s}{W}_{s-1}^{\prime}{S}_{WW}^{-1/2}\stackrel{\mathrm{a}.\mathrm{s}.}{=}\mathrm{o}\left({t}^{\beta}\right)$,
- (iv)
- ${\sum}_{s=1}^{t-1}{\epsilon}_{s}{\left(R\right|W)}_{s}^{\prime}{S}_{RR}^{-1/2}\stackrel{\mathrm{a}.\mathrm{s}.}{=}\mathrm{O}\left[{(logt)}^{1/2}\right]+\mathrm{o}\left({t}^{\beta -1/16}\right)$.

**Proof.**For (i), (ii) and (iii), use [25], Theorem 2.4. For (iv), write:

**Lemma 14.**Under Assumptions 1, 2, 3 and 4,

- (i)
- ${\sum}_{s=1}^{t-1}{\epsilon}_{s}{\xi}_{2,s}^{\prime}{S}_{{\xi}_{2}{\xi}_{2}}^{-1/2}\stackrel{\mathrm{a}.\mathrm{s}.}{=}\mathrm{O}\left[{(logt)}^{1/2}\right]$,
- (ii)
- ${\sum}_{s=1}^{t-1}{\epsilon}_{s}{\left({\xi}_{2}\right|S)}_{s}^{\prime}{S}_{{\xi}_{2}{\xi}_{2}}^{-1/2}\stackrel{\mathrm{a}.\mathrm{s}.}{=}\mathrm{o}\left({t}^{2\beta -1/2}\right)+\mathrm{O}\left[{(logt)}^{1/2}\right]$, the latter term dominating when $\alpha >4$.

**Proof.**For (i), use [46], Lemma 1(iii), and [44], Corollary 1(iii). For (ii), write:

#### A.4. Proof of Theorem 6

**Lemma 15.**Under Assumptions 1, 2 and 3,

**Proof.**Divide the statistic into two parts using that:

**Lemma 16.**Under Assumptions 1, 2, 3 and 4:

**Proof.**Once again, we take the proof in two steps, using that:

**Proof of Theorem 6.**We aim to show that:

#### A.5. Proof of Lemma 7

**Proof.**Theorem 6 shows that ${\mathsf{C}}_{1,t}^{2}-{q}_{t}^{2}$ vanishes almost surely. Egorov’s theorem ([47], Theorem 18.4) then shows that ${\mathsf{C}}_{1,t}^{2}-{q}_{t}^{2}$ vanishes uniformly on a set with large probability. That is,

#### A.6. Proof of Lemma 8 (Correction to Lemma 1 of [30])

**Proof.**The first part of Deo’s lemma, determining the domain of attraction as Λ, is correct. The second part, determining the norming sequences, is in error. Deo cites ([48] p. 374) for this calculation. There, Cramér calculates the norming sequences for a sequence of independent standard normal random variables (with a right tail differing from the density of interest in only a constant factor). We follow the slightly more direct approach of [49], Theorem 1.5.3.

#### A.7. Proof of Lemma 9

**Proof.**Consider the normalised linear process:

#### A.8. Proof of Theorem 10

**Proof.**By a property of inequalities, we can establish a lower bound on the supremum statistic,

## Conflicts of Interest

## References

- G.C. Chow. “Tests of equality between sets of coefficients in two linear regressions.” Econometrica 28 (1960): 591–605. [Google Scholar] [CrossRef]
- T. Kimura. “The impact of financial uncertainties on money demand in Europe.” In Monetary Analysis: Tools and Applications. Edited by H.-J. Klöckers and C. Willeke. Frankfurt am Main, Germany: European Central Bank, 2001, pp. 97–116. [Google Scholar]
- D.F. Hendry. “Using PC-GIVE in econometrics teaching.” Oxf. Bull. Econ. Stat. 48 (1986): 87–98. [Google Scholar] [CrossRef]
- Quantitative Micro Software (QMS). EViews 7 User’s Guide II. Irvine, CA, US: QMS, 2009. [Google Scholar]
- R.L. Brown, and J. Durbin. “Methods of investigating whether a regression relationship is constant over time.” In Selected Statistical Papers, European Meeting. Mathematical Centre Tracts, 26; Amsterdam, the Netherlands: Mathematisch Centrum, 1968. [Google Scholar]
- R.L. Brown, J. Durbin, and J.M. Evans. “Techniques for testing the constancy of regression relationships over time.” J. R. Stat. Soc. Ser. B 37 (1975): 149–192. [Google Scholar]
- W. Krämer, W. Ploberger, and R. Alt. “Testing for structural change in dynamic models.” Econometrica 56 (1988): 1355–1369. [Google Scholar] [CrossRef]
- W. Ploberger, and W. Krämer. “On studentizing a test for structural change.” Econ. Lett. 20 (1986): 341–344. [Google Scholar] [CrossRef]
- B. Nielsen, and J.S. Sohkanen. “Asymptotic behavior of the CUSUM of squares test under stochastic and deterministic time trends.” Econom. Theory 27 (2011): 913–927. [Google Scholar] [CrossRef]
- J.S. Galpin, and D.M. Hawkins. “The use of recursive residuals in checking model fit in linear regression.” Am. Stat. 38 (1984): 94–105. [Google Scholar]
- J.-M. Dufour. “Recursive stability analysis of linear regression relationships: An exploratory methodology.” J. Econom. 19 (1982): 31–76. [Google Scholar] [CrossRef]
- P. Perron. “Dealing with structural breaks.” In Palgrave Handbook of Econometrics. Edited by T.C. Mills and K. Patterson. Basingstoke, UK: Palgrave Macmillan, 2006, pp. 278–352. [Google Scholar]
- D.W.K. Andrews. “Tests for parameter instability and structural change with unknown change point.” Econometrica 61 (1993): 821–856. [Google Scholar] [CrossRef]
- R.E. Quandt. “Tests of the hypothesis that a linear regression system obeys two separate regimes.” J. Am. Stat. Assoc. 55 (1960): 324–330. [Google Scholar] [CrossRef]
- D.W.K. Andrews. “End-of-sample instability tests.” Econometrica 71 (2003): 1661–1694. [Google Scholar] [CrossRef]
- K.S. Srikantan. “Testing for the single outlier in a regression model.” Sankhya Indian J. Stat. Ser. A 23 (1961): 251–260. [Google Scholar]
- I. Chang, G.C. Tiao, and C. Chen. “Estimation of time series parameters in the presence of outliers.” Technometrics 31 (1988): 193–204. [Google Scholar] [CrossRef]
- C. Chen, and L.M. Liu. “Forecasting time series with outliers.” J. Forecast. 12 (1993): 13–35. [Google Scholar] [CrossRef]
- A.J. Fox. “Outliers in time series.” J. R. Stat. Soc. Ser. B 34 (1972): 350–363. [Google Scholar]
- V. Barnett, and T. Lewis. Outliers in Statistical Data, 3rd ed. New York, NY, USA: Wiley, 1994. [Google Scholar]
- T.L. Lai, and C.-Z. Wei. “Extended least squares and their applications to adaptive control and prediction in linear systems.” Autom. Control 31 (1986): 898–906. [Google Scholar]
- M. Duflo. Random Iterative Models. Berlin, Germany: Springer-Verlag, 1997. [Google Scholar]
- D.F. Hendry. Dynamic Econometrics. Oxford, UK: Oxford University Press, 1995. [Google Scholar]
- S. Johansen. “A Bartlett correction factor for tests on the cointegrating relations.” Econom. Theory 16 (2000): 740–778. [Google Scholar] [CrossRef]
- B. Nielsen. “Strong consistency results for least squares estimators in general vector autoregressions with deterministic terms.” Econom. Theory 21 (2005): 534–561. [Google Scholar] [CrossRef]
- D. Bauer. “Almost sure bounds on the estimation error for OLS estimators when the regressors include certain MFI(1) processes.” Econom. Theory 25 (2009): 571–582. [Google Scholar] [CrossRef]
- J.H. Stock, and M.W. Watson. “Has the business cycle changed and why? ” NBER Macroecon. Annu. 17 (2002): 159–230. [Google Scholar]
- B. Nielsen. Singular Vector Autoregressions with Deterministic Terms: Strong Consistency and Lag Order Determination. Nuffield College Discussion Paper; Oxford, UK: Nuffield College, 2008. [Google Scholar]
- E. Engler, and B. Nielsen. “The empirical process of autoregressive residuals.” Econom. J. 12 (2009): 367–381. [Google Scholar] [CrossRef]
- C.M. Deo. “Some limit theorems for maxima of absolute values of Gaussian sequences.” Sankhyā Indian J. Stat. Ser. A 34 (1972): 289–292. [Google Scholar]
- J.A. Doornik. Object-Oriented Matrix Programming Using Ox, 3rd ed. London, UK: Timberlake Consultants Press, 2007. [Google Scholar]
- J.A. Doornik, and H. Hansen. “An omnibus test for univariate and multivariate normality.” Oxf. Bull. Econ. Stat. 70 (2008): 927–939. [Google Scholar] [CrossRef]
- D.W.K. Andrews. Tests for Parameter Instability and Structural Change with Unknown Change Point. Cowles Foundation Discussion Papers; New Haven, CT, USA: Cowles Foundation for Research in Economics, Yale University, 1990, Volume 943. [Google Scholar]
- R.D. Cook, and S. Weisberg. Residuals and Influence in Regression. New York, NY, USA: Chapman and Hall, 1982. [Google Scholar]
- L. Kilian, and U. Demiroglu. “Residual based tests for normality in autoregressions: Asymptotic theory and simulations.” J. Econ. Bus. Control 18 (2000): 40–50. [Google Scholar]
- B. Nielsen. “Order determination in general vector autoregressions.” In Time Series and Related Topics: In Memory of Ching-Zong Wei. IMS Lecture Notes and Monograph Series; Edited by H.-C. Ho, C.-K. Ing and T.L. Lai. Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006, Volume 52, pp. 93–112. [Google Scholar]
- J.A. Doornik, and D.F. Hendry. Empirical Econometric Modelling—PcGive 14. London, UK: Timberlake Consultants, 2013, Volume 1. [Google Scholar]
- D.F. Hendry, and B. Nielsen. Econometric Modelling. Princeton, NJ, USA: Princeton University Press, 2007. [Google Scholar]
- R. Lynch, and C. Richardson. “Discussion.” J. Off. Stat. 20 (2004): 623–629. [Google Scholar]
- K.D. Patterson, and S.M. Heravi. “Revisions to official data on U.S. GNP: A multivariate assessment of different vintages (with discussion).” J. Off. Stat. 20 (2004): 573–602. [Google Scholar]
- S. Johansen, and B. Nielsen. Outlier Detection Algorithms for Least Squares Time Series. Nuffield College Discussion Paper 2014-W04; Oxford, UK: Nuffield College, 2014. [Google Scholar]
- P. Burridge, and A.M.R. Taylor. “Additive outlier detection via extreme-value theory.” J. Time Ser. Anal. 27 (2006): 685–701. [Google Scholar] [CrossRef]
- I. Weissman. “Estimation of parameters and larger quantiles based on the k largest observations.” J. Am. Stat. Assoc. 73 (1978): 812–815. [Google Scholar] [CrossRef]
- T.L. Lai, and C.-Z. Wei. “Asymptotic properties of multivariate weighted sums with applications to stochastic regression in linear dynamic systems.” In Multivariate Analysis VI. Edited by P.R. Krishnaiah. Amsterdam, the Netherlands: North Holland, 1985, pp. 375–393. [Google Scholar]
- S.R. Searle. Matrix Algebra Useful for Statistics. New York, NY, USA: John Wiley and Sons, 1982. [Google Scholar]
- T.L. Lai, and C.-Z. Wei. “Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems.” Ann. Stat. 10 (1982): 154–166. [Google Scholar] [CrossRef]
- J. Davidson. Stochastic Limit Theory. Oxford, UK: Oxford University Press, 1994. [Google Scholar]
- H. Cramér. Mathematical Methods in Statistics. Princeton, NJ, USA: Princeton University Press, 1946. [Google Scholar]
- M.R. Leadbetter, G. Lindgren, and H. Rootzén. Extremes and Related Properties of Random Sequences and Processes. New York, NY, USA: Springer-Verlag, 1982. [Google Scholar]
- P. Embrechts, C. Klüppelberg, and T. Mikosch. Modelling Extremal Events for Insurance and Finance. Berlin, Germany: Springer, 1997. [Google Scholar]
- M.R. Leadbetter, and H. Rootzen. “Extremal theory for stochastic processes.” Ann. Probab. 16 (1988): 431–478. [Google Scholar] [CrossRef]

^{1}ABMI series from Office of National Statistics, seasonally adjusted, 2010 prices, release 20 December 2013.

© 2015 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 license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nielsen, B.; Whitby, A.
A Joint Chow Test for Structural Instability. *Econometrics* **2015**, *3*, 156-186.
https://doi.org/10.3390/econometrics3010156

**AMA Style**

Nielsen B, Whitby A.
A Joint Chow Test for Structural Instability. *Econometrics*. 2015; 3(1):156-186.
https://doi.org/10.3390/econometrics3010156

**Chicago/Turabian Style**

Nielsen, Bent, and Andrew Whitby.
2015. "A Joint Chow Test for Structural Instability" *Econometrics* 3, no. 1: 156-186.
https://doi.org/10.3390/econometrics3010156