# Asymptotic Theory for Cointegration Analysis When the Cointegration Rank Is Deficient

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model without Deterministic Terms

#### 2.1. Model and Hypotheses

#### 2.2. Granger-Johansen Representation

**I(1)**

**Condition.**

#### 2.3. Test Statistics

#### 2.4. Asymptotic Theory for the Rank Test

**Theorem**

**1.**

#### 2.5. Asymptotic Theory for the Test on the Cointegrating Vectors

**Theorem**

**2.**

**Theorem**

**3.**

#### 2.6. The Case of Nearly Deficient Rank

**Theorem**

**4**

**.**Consider the data generating process (13). Let ${B}_{u}$ be a bivariate standard Brownian motion on $[0,1]$ and let ${J}_{u}$ be the bivariate Ornstein-Uhlenbeck process given by

**Theorem**

**5.**

## 3. The Model with a Constant

#### 3.1. Model and Hypotheses

#### 3.2. Granger-Johansen Representation

#### 3.3. Test Statistics

#### 3.4. Asymptotic Theory for the Rank Tests

**Theorem**

**6.**

**Theorem**

**7.**

**Theorem**

**8.**

#### 3.5. Asymptotic Theory for the Test on the Cointegrating Vectors

**Theorem**

**9.**

**Theorem**

**10.**

## 4. Applications of Results

#### 4.1. Finite Sample Theory

#### 4.2. Identification Robust Inference

**Theorem**

**11.**

**Theorem**

**12.**

#### 4.3. Empirical Illustration

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proofs

**Proof**

**of**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**5.**

**Proof**

**of**

**Theorem**

**6.**

**Proof**

**of**

**Theorem**

**7.**

**Proof**

**of**

**Theorem**

**8.**

**Proof**

**of**

**Theorem**

**9.**

**Proof**

**of**

**Theorem**

**10.**

**Proof**

**of**

**Theorem**

**11.**

**Proof**

**of**

**Theorem**

**12.**

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**Table 1.**Quantiles, mean, and variance of $LR\{{\mathsf{H}}_{z}(r)|{\mathsf{H}}_{z}(p)\}$, where the data generating process has rank $s=\mathrm{rank}\phantom{\rule{0.166667em}{0ex}}\Pi \le r$.

$\mathit{r}-\mathit{s}$ | $\mathit{p}-\mathit{r}$ | 50% | 80% | 85% | 90% | 95% | 97.5% | 99% | Mean | Var |
---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 0.60 | 1.88 | — | 2.98 | 4.13 | 5.32 | 6.94 | 1.14 | 2.22 |

2 | 5.48 | 8.48 | 9.31 | 10.44 | 12.30 | 14.07 | 16.34 | 6.09 | 10.61 | |

3 | 14.39 | 18.94 | 20.13 | 21.70 | 24.22 | 26.54 | 29.37 | 15.02 | 25.13 | |

4 | 27.29 | 33.35 | 34.88 | 36.91 | 40.04 | 42.93 | 46.45 | 27.93 | 45.66 | |

1 | 1 | 0.36 | 1.13 | 1.38 | 1.74 | 2.35 | 2.98 | 3.81 | 0.67 | 0.70 |

2 | 4.27 | 6.25 | 6.78 | 7.50 | 8.65 | 9.76 | 11.14 | 4.61 | 4.66 | |

3 | 11.92 | 15.20 | 16.04 | 17.14 | 18.88 | 20.50 | 22.48 | 12.31 | 13.22 | |

4 | 23.47 | 28.09 | 29.25 | 30.76 | 33.10 | 35.21 | 37.83 | 23.89 | 26.96 | |

2 | 1 | 0.30 | 0.97 | 1.18 | 1.48 | 1.98 | 2.47 | 3.11 | 0.56 | 0.48 |

2 | 3.93 | 5.57 | 6.01 | 6.59 | 7.51 | 8.38 | 9.46 | 4.18 | 3.24 | |

3 | 11.04 | 13.82 | 14.53 | 15.46 | 16.91 | 18.24 | 19.87 | 11.34 | 9.63 | |

4 | 21.84 | 25.83 | 26.82 | 28.11 | 30.09 | 31.91 | 34.13 | 22.18 | 20.21 |

**Table 2.**Quantiles, mean, and variance of $LR\{{\mathsf{H}}_{z,\beta}(r)|{\mathsf{H}}_{z}(r)\}$, where the data generating process has rank $s=\mathrm{rank}\phantom{\rule{0.166667em}{0ex}}\Pi \le r$.

p | r | s | 50% | 80% | 85% | 90% | 95% | 97.5% | 99% | Mean | Var |
---|---|---|---|---|---|---|---|---|---|---|---|

2 | 1 | 1 | 0.45 | 1.64 | 2.07 | 2.71 | 3.84 | 5.02 | 6.63 | 1 | 2 |

0 | 2.62 | 5.44 | 6.22 | 7.30 | 9.05 | 10.75 | 12.96 | 3.31 | 8.71 | ||

3 | 2 | 2 | 1.39 | 3.22 | 3.79 | 4.61 | 5.99 | 7.38 | 9.21 | 2 | 4 |

0 | 5.80 | 9.42 | 10.40 | 11.71 | 13.82 | 15.77 | 18.27 | 6.42 | 15.53 | ||

3 | 1 | 1 | 1.39 | 3.22 | 3.79 | 4.61 | 5.99 | 7.38 | 9.21 | 2 | 4 |

0 | 6.79 | 10.58 | 11.57 | 12.89 | 15.02 | 17.00 | 19.49 | 7.33 | 17.52 |

**Table 3.**Quantiles, mean, and variance of $LR\{{\mathsf{H}}_{z,\beta}(r)|{\mathsf{H}}_{z}(p)\}$, where the data generating process has rank $s=\mathrm{rank}\phantom{\rule{0.166667em}{0ex}}\Pi \le r$.

p | r | s | 50% | 80% | 85% | 90% | 95% | 97.5% | 99% | Mean | Var |
---|---|---|---|---|---|---|---|---|---|---|---|

2 | 1 | 1 | 1.54 | 3.43 | 4.01 | 4.83 | 6.22 | 7.62 | 9.47 | 2.15 | 4.23 |

0 | 3.35 | 6.11 | 6.89 | 7.95 | 9.70 | 11.38 | 13.57 | 3.98 | 8.82 | ||

3 | 2 | 2 | 2.52 | 4.85 | 5.53 | 6.48 | 8.07 | 9.60 | 11.62 | 3.15 | 6.26 |

0 | 6.36 | 9.96 | 10.92 | 12.22 | 14.32 | 16.29 | 18.79 | 6.98 | 15.35 | ||

3 | 1 | 1 | 7.50 | 11.03 | 11.98 | 13.27 | 15.34 | 17.30 | 19.81 | 8.13 | 14.73 |

0 | 11.33 | 15.73 | 16.88 | 18.41 | 20.83 | 23.09 | 25.91 | 11.96 | 23.31 |

**Table 4.**Quantiles, mean, and variance of $LR\{{\mathsf{H}}_{c\ell}(r)|{\mathsf{H}}_{c\ell}(p)\}$, where the data generating process satisfies ${\mathsf{H}}_{c\ell}^{\circ}(s)={\mathsf{H}}_{c\ell}(s)\backslash {\mathsf{H}}_{c}(s)$ with $s\le r$.

$\mathit{r}-\mathit{s}$ | $\mathit{p}-\mathit{r}$ | 50% | 80% | 85% | 90% | 95% | 97.5% | 99% | Mean | Var |
---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 0.45 | 1.64 | 2.07 | 2.71 | 3.84 | 5.02 | 6.63 | 1 | 2 |

2 | 7.61 | 11.09 | 12.04 | 13.30 | 15.35 | 17.27 | 19.74 | 8.24 | 14.29 | |

3 | 18.66 | 23.72 | 25.03 | 26.76 | 29.47 | 31.95 | 34.99 | 19.29 | 31.38 | |

4 | 33.52 | 40.07 | 41.71 | 43.86 | 47.22 | 50.21 | 53.94 | 34.15 | 53.86 | |

1 | 1 | 0.38 | 1.33 | 1.66 | 2.13 | 2.93 | 3.72 | 4.74 | 0.79 | 1.08 |

2 | 6.01 | 8.34 | 8.96 | 9.78 | 11.10 | 12.34 | 13.87 | 6.37 | 6.53 | |

3 | 15.49 | 19.14 | 20.08 | 21.30 | 23.21 | 24.99 | 27.14 | 15.88 | 16.73 | |

4 | 28.82 | 33.82 | 35.07 | 36.70 | 39.20 | 41.50 | 44.27 | 29.24 | 31.96 | |

2 | 1 | 0.34 | 1.19 | 1.47 | 1.87 | 2.55 | 3.19 | 4.00 | 0.69 | 0.79 |

2 | 5.43 | 7.34 | 7.84 | 8.51 | 9.57 | 10.56 | 11.81 | 5.70 | 4.46 | |

3 | 14.17 | 17.26 | 18.04 | 19.05 | 20.64 | 22.09 | 23.86 | 14.48 | 12.00 | |

4 | 26.62 | 30.92 | 31.98 | 33.38 | 35.52 | 37.46 | 39.79 | 26.95 | 23.82 |

**Table 5.**Quantiles, mean, and variance of $LR\{{\mathsf{H}}_{c\ell}(r)|{\mathsf{H}}_{c\ell}(p)\}$, where the data generating process satisfies ${\mathsf{H}}_{c}^{\circ}(s)={\mathsf{H}}_{c}(s)\backslash {\mathsf{H}}_{c\ell}(s-1)$ with $s\le r$.

$\mathit{r}-\mathit{s}$ | $\mathit{p}-\mathit{r}$ | 50% | 80% | 85% | 90% | 95% | 97.5% | 99% | Mean | Var |
---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2.45 | 4.90 | 5.60 | 6.56 | 8.15 | 9.72 | 11.71 | 3.04 | 6.95 |

2 | 9.39 | 13.36 | 14.41 | 15.80 | 18.03 | 20.14 | 22.80 | 10.03 | 18.66 | |

3 | 20.30 | 25.70 | 27.09 | 28.89 | 31.75 | 34.37 | 37.61 | 20.95 | 35.73 | |

4 | 35.19 | 42.01 | 43.71 | 45.94 | 49.38 | 52.52 | 56.31 | 35.84 | 58.26 | |

1 | 1 | 1.51 | 3.12 | 3.55 | 4.12 | 5.04 | 5.92 | 7.03 | 1.87 | 2.72 |

2 | 7.21 | 9.95 | 10.66 | 11.61 | 13.09 | 14.47 | 16.21 | 7.60 | 8.95 | |

3 | 16.78 | 20.75 | 21.75 | 23.08 | 25.13 | 26.98 | 29.32 | 17.20 | 19.57 | |

4 | 30.25 | 35.49 | 36.81 | 38.51 | 41.15 | 43.56 | 46.46 | 30.69 | 35.22 | |

2 | 1 | 1.16 | 2.54 | 2.89 | 3.36 | 4.09 | 4.76 | 5.62 | 1.48 | 1.81 |

2 | 6.38 | 8.66 | 9.25 | 10.03 | 11.26 | 12.40 | 13.80 | 6.69 | 6.23 | |

3 | 15.27 | 18.64 | 19.49 | 20.61 | 22.35 | 23.94 | 25.88 | 15.61 | 14.27 | |

4 | 28.00 | 32.45 | 33.58 | 35.05 | 37.32 | 39.37 | 41.85 | 28.26 | 26.55 |

**Table 6.**Quantiles, mean, and variance of $LR\{{\mathsf{H}}_{c}(r)|{\mathsf{H}}_{c}(p)\}$, where the data generating process satisfies ${\mathsf{H}}_{c}^{\circ}(s)={\mathsf{H}}_{c}(s)\backslash {\mathsf{H}}_{c\ell}(s-1)$ with $s\le r$.

$\mathit{r}-\mathit{s}$ | $\mathit{p}-\mathit{r}$ | 50% | 80% | 85% | 90% | 95% | 97.5% | 99% | Mean | Var |
---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 3.44 | 5.86 | 6.56 | 7.52 | 9.13 | 10.69 | 12.74 | 4.04 | 6.89 |

2 | 11.40 | 15.43 | 16.49 | 17.91 | 20.18 | 22.33 | 25.03 | 12.02 | 19.50 | |

3 | 23.31 | 28.86 | 30.28 | 32.15 | 35.06 | 37.74 | 41.04 | 23.95 | 38.13 | |

4 | 39.20 | 46.23 | 47.99 | 50.28 | 53.82 | 57.05 | 61.01 | 39.84 | 62.48 | |

1 | 1 | 2.74 | 4.27 | 4.70 | 5.27 | 6.21 | 7.10 | 8.25 | 3.05 | 2.75 |

2 | 9.47 | 12.30 | 13.04 | 14.01 | 15.54 | 16.96 | 18.74 | 9.84 | 9.81 | |

3 | 20.04 | 24.19 | 25.25 | 26.63 | 28.76 | 30.71 | 33.13 | 20.45 | 21.78 | |

4 | 34.51 | 40.03 | 41.40 | 43.17 | 45.93 | 48.43 | 51.41 | 34.95 | 39.09 | |

2 | 1 | 2.62 | 3.89 | 4.22 | 4.68 | 5.41 | 6.10 | 6.96 | 2.84 | 1.87 |

2 | 8.86 | 11.26 | 11.87 | 12.67 | 13.93 | 15.10 | 16.54 | 9.14 | 7.06 | |

3 | 18.77 | 22.37 | 23.27 | 24.43 | 26.23 | 27.88 | 29.91 | 19.09 | 16.34 | |

4 | 32.40 | 37.23 | 38.43 | 39.98 | 42.35 | 44.52 | 47.08 | 32.76 | 30.09 |

**Table 7.**Quantiles, mean, and variance of $LR\{{\mathsf{H}}_{c,\beta}(r)|{\mathsf{H}}_{c}(r)\}$, where the data generating process satisfies ${\mathsf{H}}_{c,\beta}^{\circ}(s)$.

p | r | s | 50% | 80% | 85% | 90% | 95% | 97.5% | 99% | Mean | Var |
---|---|---|---|---|---|---|---|---|---|---|---|

2 | 1 | 1 | 1.39 | 3.22 | 3.79 | 4.61 | 5.99 | 7.38 | 9.21 | 2 | 4 |

0 | 6.34 | 9.84 | 10.78 | 12.02 | 14.05 | 15.96 | 18.41 | 6.87 | 15.09 | ||

3 | 2 | 2 | 3.36 | 5.99 | 6.75 | 7.78 | 9.49 | 11.14 | 13.28 | 4 | 8 |

0 | 12.45 | 17.48 | 18.79 | 20.53 | 23.26 | 25.76 | 28.91 | 13.12 | 30.71 | ||

3 | 1 | 1 | 2.37 | 4.64 | 5.32 | 6.25 | 7.82 | 9.35 | 11.35 | 3 | 6 |

0 | 10.60 | 14.82 | 15.92 | 17.36 | 19.66 | 21.79 | 24.48 | 11.07 | 22.93 |

**Table 8.**Quantiles, mean, and variance of $LR\{{\mathsf{H}}_{c,\beta}(r)|{\mathsf{H}}_{c}(p)\}$, where the data generating process satisfies ${\mathsf{H}}_{c,\beta}^{\circ}(s)$.

p | r | s | 50% | 80% | 85% | 90% | 95% | 97.5% | 99% | Mean | Var |
---|---|---|---|---|---|---|---|---|---|---|---|

2 | 1 | 1 | 5.44 | 8.50 | 9.34 | 10.50 | 12.38 | 14.17 | 16.50 | 6.07 | 10.98 |

0 | 9.32 | 13.37 | 14.44 | 15.88 | 18.18 | 20.31 | 22.98 | 9.94 | 19.72 | ||

3 | 2 | 2 | 7.44 | 11.02 | 11.99 | 13.29 | 15.37 | 17.37 | 19.88 | 8.09 | 15.09 |

0 | 15.37 | 20.48 | 21.80 | 23.54 | 26.26 | 28.78 | 31.88 | 15.99 | 32.22 | ||

3 | 1 | 1 | 14.46 | 19.08 | 20.28 | 21.88 | 24.39 | 26.74 | 29.64 | 15.10 | 25.77 |

0 | 20.35 | 25.89 | 27.31 | 29.15 | 32.04 | 34.72 | 38.02 | 20.96 | 38.07 |

Test | ${\mathit{b}}_{12,\mathit{t}}$ | ${\mathit{b}}_{24,\mathit{t}}$ | Test | System |
---|---|---|---|---|

${\chi}_{normality}^{2}\left(2\right)$ | $\underset{[0.15]}{3.8}$ | $\underset{[0.13]}{4.1}$ | ${\chi}_{normality}^{2}\left(4\right)$ | $\underset{[0.36]}{4.3}$ |

${F}_{ar,1-7}\left(7,144\right)$ | $\underset{[0.11]}{1.7}$ | $\underset{[0.45]}{1.0}$ | ${F}_{ar,1-7}\left(28,272\right)$ | $\underset{[0.24]}{1.2}$ |

${F}_{arch,1-7}\left(7,147\right)$ | $\underset{[0.09]}{1.8}$ | $\underset{[0.41]}{1.0}$ |

Hypothesis | r | Likelihood | $\mathit{LR}$ | p-Value | |
---|---|---|---|---|---|

$\mathit{s}=\mathit{r}$ | ${\mathsf{H}}_{\mathit{c}}(0)$ | ||||

${\mathsf{H}}_{c\ell}(2)={\mathsf{H}}_{c}(2)$ | 2 | 134.63 | |||

${\mathsf{H}}_{c\ell}(1)$ | 1 | 133.71 | 1.8 | 0.18 | 0.39 |

${\mathsf{H}}_{c}(1)$ | 1 | 133.71 | 1.8 | 0.80 | 0.75 |

${\mathsf{H}}_{c\ell}(0)$ | 0 | 129.70 | 9.8 | 0.30 | 0.46 |

${\mathsf{H}}_{c}(0)$ | 0 | 129.21 | 10.8 | 0.57 | 0.57 |

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**MDPI and ACS Style**

Bernstein, D.H.; Nielsen, B. Asymptotic Theory for Cointegration Analysis When the Cointegration Rank Is Deficient. *Econometrics* **2019**, *7*, 6.
https://doi.org/10.3390/econometrics7010006

**AMA Style**

Bernstein DH, Nielsen B. Asymptotic Theory for Cointegration Analysis When the Cointegration Rank Is Deficient. *Econometrics*. 2019; 7(1):6.
https://doi.org/10.3390/econometrics7010006

**Chicago/Turabian Style**

Bernstein, David H., and Bent Nielsen. 2019. "Asymptotic Theory for Cointegration Analysis When the Cointegration Rank Is Deficient" *Econometrics* 7, no. 1: 6.
https://doi.org/10.3390/econometrics7010006