# Partial Cointegrated Vector Autoregressive Models with Structural Breaks in Deterministic Terms

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Models and Representations

#### 2.1. Previous Models

#### 2.2. The Partial Model with Structural Breaks

#### 2.3. Representations

**Assumption**

**1.**

**Theorem**

**1.**

#### 2.4. The Partial Model with Shifts in The Level

## 3. Testing for Cointegrating Rank in the Partial Models

#### 3.1. Rank Test Statistic

#### 3.2. Asymptotic Distribution of the Test Statistic

**Assumption**

**2.**

- (i)
- $\mathsf{E}({\epsilon}_{t}{\epsilon}_{t}^{\prime})=\mathsf{\Omega}$;
- (ii)
- ${T}^{-1}{\sum}_{t=1}^{T}\mathsf{E}({\epsilon}_{t}{\epsilon}_{t}^{\prime}|{\mathcal{F}}_{t-1})\stackrel{\mathsf{P}}{\to}\mathsf{\Omega}$;
- (iii)
- either of the following boundedness conditions
- (a)
- ${sup}_{t\in \mathbb{N}}\mathsf{E}\{{\epsilon}_{t}^{\prime}{\epsilon}_{t}{1}_{({\epsilon}_{t}^{\prime}{\epsilon}_{t}>a)}|{\mathcal{F}}_{t-1}\}\stackrel{\mathsf{P}}{\to}0$ as $a\to \infty $;
- (b)
- ${sup}_{t\in \mathbb{N}}\mathsf{E}{|{\epsilon}_{t}|}^{4}<\infty .$

**Assumption**

**3.**

- (i)
- $\mathsf{Var}({\epsilon}_{t}|{\mathcal{F}}_{t-1})=\mathsf{\Omega}$$a.s.$, where Ω is positive definite;
- (ii)
- ${sup}_{t\in \mathbb{N}}\mathsf{E}(|{\epsilon}_{t}{|}^{2+\xi}|{\mathcal{F}}_{t-1})<\infty $$a.s.$ for some $\xi >0.$

**Lemma**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4**

**.**Let ${B}^{[1]},\dots ,{B}^{[q]}$ be independent $(p-r)$-dimensional standard Brownian motions and define

#### 3.3. Asymptotic Distribution for the Broken Constant Case

**Theorem**

**5.**

## 4. Approximations of the Asymptotic Distributions

#### 4.1. Derivation of Response Surface

#### 4.2. Implementation of Response Surface

## 5. Empirical Illustration

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Tables for Response Surfaces

$log{\mathit{\lambda}}_{\mathit{p}-\mathit{r}}$ | $log{\mathit{\delta}}_{\mathit{p}-\mathit{r}}$ | $\mathsf{Cov}({\mathsf{T}}_{1},{\mathsf{T}}_{2})$ | |||
---|---|---|---|---|---|

const. | 4.14 | const. | 0.5987 | const. | −1.298 |

${(p-r)}^{-1}$ | −6.301 | $p-r$ | −0.0538 | ${1}_{(2)}$ | 0.03616 |

${(p-r)}^{-2}$ | 5.8842 | a | −1.039 | ${1}_{(4)}$ | −0.027 |

${(p-r)}^{-3}$ | −2.32576 | b | −0.39 | ${(p-r)}^{-3}$ | −2.022 |

$p-r$ | 0.17 | ${(p-r)}^{2}$ | 0.00686 | a | −8.689 |

a | 2.6165 | ${a}^{2}$ | 5.547 | b | 2.225 |

b | 2.5245 | $ab$ | 2.331 | ${a}^{2}$ | 59.77 |

$(p-r)a$ | −0.0572 | ${b}^{2}$ | 1.841 | $ab$ | 24.31 |

$(p-r)b$ | −0.0971 | ${(p-r)}^{3}$ | −0.00033 | ${b}^{2}$ | −5.156 |

${a}^{2}$ | −7.550 | ${a}^{3}$ | −10.42 | ${a}^{3}$ | −133.5 |

$ab$ | −5.323 | ${ab}^{2}$ | −4.325 | ${ab}^{2}$ | −59.05 |

${b}^{2}$ | −7.412 | ${b}^{3}$ | −2.553 | ${a(p-r)}^{-1}$ | −29.55 |

${(p-r)}^{3}$ | −0.000124 | ${a(p-r)}^{-1}$ | 9.905 | ${b(p-r)}^{-1}$ | −66.58 |

$(p-r)ab$ | 0.161 | ${b(p-r)}^{-1}$ | 1.862 | ${b}^{2}{(p-r)}^{-1}$ | 255.3 |

${(p-r)b}^{2}$ | 0.179 | ${a}^{2}{(p-r)}^{-1}$ | −61.09 | ${a}^{3}{(p-r)}^{-1}$ | 280.5 |

${a}^{3}$ | 10.40 | ${ab(p-r)}^{-1}$ | −17.09 | ${ab}^{2}{(p-r)}^{-1}$ | 155.3 |

${ab}^{2}$ | 6.096 | ${b}^{2}{(p-r)}^{-1}$ | −11.48 | ${b}^{3}{(p-r)}^{-1}$ | −240 |

${b}^{3}$ | 5.851 | ${a}^{3}{(p-r)}^{-1}$ | 117.68 | ${a(p-r)}^{-2}$ | 21.32 |

${a(p-r)}^{-1}$ | −8.860 | ${ab}^{2}{(p-r)}^{-1}$ | 35.19 | ${b(p-r)}^{-2}$ | 71.68 |

${b(p-r)}^{-1}$ | −4.948 | ${b}^{3}{(p-r)}^{-1}$ | 18.6 | ${b}^{2}{(p-r)}^{-2}$ | −305.7 |

${a}^{2}{(p-r)}^{-1}$ | 46.15 | ${a(p-r)}^{-2}$ | −8.836 | ${a}^{2}{b(p-r)}^{-2}$ | −321.1 |

${ab(p-r)}^{-1}$ | 31.85 | ${b(p-r)}^{-2}$ | 1.033 | ${b}^{3}{(p-r)}^{-2}$ | 332.1 |

${b}^{2}{(p-r)}^{-1}$ | 26.12 | ${a}^{2}{(p-r)}^{-2}$ | 66.94 | ${(p-r)1}_{(3)}$ | 0.038 |

${a}^{3}{(p-r)}^{-1}$ | −86.58 | ${ab(p-r)}^{-2}$ | 10.84 | ${b}^{2}{1}_{(3)}$ | −0.184 |

${ab}^{2}{(p-r)}^{-1}$ | −50.50 | ${a}^{3}{(p-r)}^{-2}$ | −140.88 | ||

${b}^{3}{(p-r)}^{-1}$ | −28.78 | ${ab}^{2}{(p-r)}^{-2}$ | −30.16 | ||

${a(p-r)}^{-2}$ | 5.296 | ${b}^{3}{(p-r)}^{-2}$ | −10.05 | ||

${b(p-r)}^{-2}$ | 2.386 | ${a1}_{(1)}$ | 2.107 | ||

${a}^{2}{(p-r)}^{-2}$ | −29.03 | ${b1}_{(1)}$ | −1.029 | ||

${ab(p-r)}^{-2}$ | −19.46 | ${a}^{2}{1}_{(1)}$ | −20.63 | ||

${b}^{2}{(p-r)}^{-2}$ | −13.42 | ${b}^{2}{1}_{(1)}$ | 3.511 | ||

${a}^{3}{(p-r)}^{-2}$ | 62.00 | ${a}^{3}{1}_{(1)}$ | 45.85 | ||

${a}^{2}{b(p-r)}^{-2}$ | −5.880 | ${ab}^{2}{1}_{(1)}$ | 4.267 | ||

${ab}^{2}{(p-r)}^{-2}$ | 34.59 | ${(p-r)b}^{2}{1}_{(2)}$ | 0.062 | ||

${b}^{3}{(p-r)}^{-2}$ | 15.93 |

$log{\mathit{\lambda}}_{\mathit{p}-\mathit{r}}$ | $log{\mathit{\delta}}_{\mathit{p}-\mathit{r}}$ | $\mathsf{Cov}({\mathsf{T}}_{1},{\mathsf{T}}_{2})$ | |||
---|---|---|---|---|---|

const. | 4.95486 | const. | 0.4472 | const. | −1.531 |

${(p-r)}^{-1}$ | −9.263 | ${(p-r)}^{-2}$ | 1.17564 | ${(p-r)}^{-1}$ | 0.9029 |

${(p-r)}^{-2}$ | 9.162 | ${(p-r)}^{-3}$ | −1.5294 | a | 4.164 |

${(p-r)}^{-3}$ | −3.662 | b | 0.8286 | ${(p-r)}^{2}$ | 0.01579 |

a | 3.05 | $(p-r)b$ | −0.0646 | $(p-r)b$ | 0.3388 |

b | 0.3315 | $ab$ | 1.75 | $ab$ | −27.16 |

${(p-r)}^{2}$ | 0.01738 | ${(p-r)b}^{2}$ | 0.04051 | ${b}^{2}$ | −14.15 |

$(p-r)a$ | −0.128 | ${a}^{3}$ | −2.084 | ${(p-r)}^{3}$ | −0.0013 |

${a}^{2}$ | −14.61 | ${ab}^{2}$ | −3.698 | ${(p-r)}^{2}b$ | −0.0167 |

$ab$ | −4.14 | ${b}^{3}$ | −0.788 | ${a}^{3}$ | −19.65 |

${b}^{2}$ | −2.419 | ${a(p-r)}^{-1}$ | −4.819 | ${a}^{2}b$ | 14.03 |

${(p-r)}^{3}$ | −0.00084 | ${b(p-r)}^{-1}$ | −3.897 | ${ab}^{2}$ | 42.2 |

${(p-r)a}^{2}$ | 0.3264 | ${a}^{2}{(p-r)}^{-1}$ | 30.49 | ${b}^{3}$ | 17.43 |

$(p-r)ab$ | 0.1302 | ${ab(p-r)}^{-1}$ | −5.108 | ${a(p-r)}^{-1}$ | −77.72 |

${(p-r)b}^{2}$ | 0.0266 | ${b}^{2}{(p-r)}^{-1}$ | 2.273 | ${b(p-r)}^{-1}$ | −20.52 |

${a}^{3}$ | 21.56 | ${a}^{3}{(p-r)}^{-1}$ | −40.9 | ${a}^{2}{(p-r)}^{-1}$ | 278.7 |

${ab}^{2}$ | 5.56 | ${ab}^{2}{(p-r)}^{-1}$ | 13.37 | ${ab(p-r)}^{-1}$ | 313.6 |

${b}^{3}$ | 3.03 | ${a(p-r)}^{-2}$ | 16 | ${b}^{2}{(p-r)}^{-1}$ | 169.1 |

${a(p-r)}^{-1}$ | −5.742 | ${b(p-r)}^{-2}$ | 3.795 | ${a}^{3}{(p-r)}^{-1}$ | −461.7 |

${b(p-r)}^{-1}$ | 3.339 | ${a}^{2}{(p-r)}^{-2}$ | −110.5 | ${ab}^{2}{(p-r)}^{-1}$ | −562.9 |

${a}^{2}{(p-r)}^{-1}$ | 44.2 | ${a}^{3}{(p-r)}^{-2}$ | 184.8 | ${b}^{3}{(p-r)}^{-1}$ | −221.2 |

${ab(p-r)}^{-1}$ | 9.66 | ${ab}^{2}{(p-r)}^{-2}$ | −4.478 | ${a(p-r)}^{-2}$ | 81.64 |

${b}^{2}{(p-r)}^{-1}$ | −4.44 | ${(p-r)1}_{(1)}$ | 0.5014 | ${a}^{2}{(p-r)}^{-2}$ | −315 |

${a}^{3}{(p-r)}^{-1}$ | −81.67 | ${a1}_{(1)}$ | −9.833 | ${ab(p-r)}^{-2}$ | −384.8 |

${ab}^{2}{(p-r)}^{-1}$ | −15.2 | ${a}^{2}{1}_{(1)}$ | 73.02 | ${b}^{2}{(p-r)}^{-2}$ | −114.6 |

${a(p-r)}^{-2}$ | 2.41 | ${b}^{2}{1}_{(1)}$ | −5.835 | ${a}^{3}{(p-r)}^{-2}$ | 804 |

${b(p-r)}^{-2}$ | −3.44 | ${a}^{3}{1}_{(1)}$ | −130.2 | ${a}^{2}{b(p-r)}^{-2}$ | −290 |

${a}^{2}{(p-r)}^{-2}$ | −24.23 | ${b}^{3}{1}_{(1)}$ | 4.743 | ${ab}^{2}{(p-r)}^{-2}$ | 860.7 |

${b}^{2}{(p-r)}^{-2}$ | 9.6 | ${(p-r)}^{2}{a1}_{(2)}$ | −0.2472 | ${b}^{3}{(p-r)}^{-2}$ | 205.2 |

${a}^{3}{(p-r)}^{-2}$ | 47.34 | ${(p-r)}^{2}{b1}_{(2)}$ | 0.06919 | ${b}^{2}{1}_{(2)}$ | 0.18 |

${b}^{3}{(p-r)}^{-2}$ | −7.22 | ${(p-r)a}^{2}{1}_{(2)}$ | 3.765 | ${(p-r)}^{3}{1}_{(2)}$ | −0.00017 |

${(p-r)b}^{2}{1}_{(2)}$ | −0.884 | ${(p-r)a1}_{(3)}$ | 1.337 | ||

${a}^{3}{1}_{(2)}$ | −14.06 | ${(p-r)b1}_{(3)}$ | −0.0215 | ||

${b}^{3}{1}_{(2)}$ | 1.944 | ${(p-r)}^{2}{a1}_{(3)}$ | −0.408 |

## Appendix B. Proof of the Granger–Johansen Representation

**Proof**

**of**

**Theorem**

**1.**

## Appendix C. Proofs of Asymptotic Results

#### Appendix C.1. A High Level Assumption

**Assumption**

**A1.**

**Lemma**

**A1.**

- (a)
- ${T}^{-1}{\sum}_{t=1}^{T}{\epsilon}_{t}{\epsilon}_{t}^{\prime}\stackrel{\mathsf{P}}{\to}\mathsf{\Omega}$;
- (b)
- ${T}^{-1}{\sum}_{t=1}^{T}\mathsf{E}\{{\epsilon}_{t}^{\prime}{\epsilon}_{t}{1}_{({\epsilon}_{t}^{\prime}{\epsilon}_{t}>\delta T)}|{\mathcal{F}}_{t-1}\}\stackrel{\mathsf{P}}{\to}0$ for all $\delta >0$;
- (c)
- ${T}^{-1}{\sum}_{t=1}^{T}\mathsf{E}\{{\epsilon}_{t}^{\prime}{\epsilon}_{t}{1}_{({\epsilon}_{t}^{\prime}{\epsilon}_{t}>\delta T)}\}\to 0$ for all $\delta >0$;
- (d)
- ${max}_{1\le t\le T}|{\epsilon}_{t}^{2}|/T\stackrel{\mathsf{P}}{\to}0.$

**Proof**

**of**

**Lemma**

**A1.**

**Lemma**

**A2.**

**Proof**

**of**

**Lemma**

**A2.**

**Proof**

**of**

**Lemma**

**1.**

**Remark**

**A1.**

#### Appendix C.2. Several Lemmas for the Partial Systems

**Lemma**

**A3.**

**Proof**

**of**

**Lemma**

**A3.**

**Lemma**

**A4**

**.**Suppose that Assumptions 1 and A1 are satisfied under ${\alpha}_{z}=0$. Then,

**Lemma**

**A5.**

**Proof**

**of**

**Lemma**

**A5.**

**Lemma**

**A6.**

**Proof**

**of**

**Lemma**

**A6.**

#### Appendix C.3. Proofs of the Theorems in Section 3

**Proof**

**of**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**3.**

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**Figure 1.**Data. (

**a**) $t{b}_{t}$ is the trade balance between the UK and Germany; (

**b**) $dul{c}_{t}$ is the unit labour cost differential between the UK and Germany; (

**c**) ${y}_{t}$ and ${y}_{t}^{*}$ are the logs of the UK and German gross domestic products, respectively; (

**d**) $pp{p}_{t}$ is the terms of trade.

$\mathit{p}-\mathit{r}$ | $\mathit{m}-\mathit{r}$ | a | b | ${\mathit{q}}_{95}$ | ${\mathit{q}}_{95}^{*}$ | $\left(\right)open="|"\; close="|">{\mathit{q}}_{95}/{\mathit{q}}_{95}^{*}-1$ | $\mathbf{\Delta}{\mathit{p}}^{\mathit{app}}$ |
---|---|---|---|---|---|---|---|

2 | 1 | $0.0$ | $0.0$ | $15.45$ | $15.33$ | $0.0078$ | $0.0058$ |

2 | 1 | $0.0$ | $0.3$ | $21.25$ | $21.25$ | $0.0000$ | $0.0000$ |

2 | 1 | $0.1$ | $0.4$ | $25.63$ | $25.76$ | $0.0050$ | $-0.0065$ |

2 | 1 | $0.2$ | $0.3$ | $27.23$ | $27.11$ | $0.0044$ | $0.0055$ |

2 | 1 | $0.3$ | $0.3$ | $27.74$ | $27.62$ | $0.0043$ | $0.0056$ |

4 | 3 | $0.0$ | $0.0$ | $50.29$ | $50.08$ | $0.0042$ | $0.0100$ |

4 | 3 | $0.0$ | $0.3$ | $65.09$ | $64.97$ | $0.0018$ | $0.0057$ |

4 | 3 | $0.1$ | $0.4$ | $77.01$ | $76.84$ | $0.0022$ | $0.0082$ |

4 | 3 | $0.2$ | $0.3$ | $80.25$ | $80.11$ | $0.0017$ | $0.0068$ |

4 | 3 | $0.3$ | $0.3$ | $81.92$ | $81.84$ | $0.0010$ | $0.0039$ |

5 | 3 | $0.0$ | $0.0$ | $57.35$ | $57.32$ | $0.0005$ | $0.0015$ |

5 | 3 | $0.0$ | $0.3$ | $72.27$ | $72.03$ | $0.0033$ | $0.0112$ |

5 | 3 | $0.1$ | $0.4$ | $84.00$ | $83.98$ | $0.0002$ | $0.0010$ |

5 | 3 | $0.2$ | $0.3$ | $87.23$ | $87.10$ | $0.0015$ | $0.0063$ |

5 | 3 | $0.3$ | $0.3$ | $88.44$ | $88.47$ | $0.0003$ | $-0.0015$ |

7 | 4 | $0.0$ | $0.0$ | $91.64$ | $91.79$ | $0.0016$ | $-0.0077$ |

7 | 4 | $0.0$ | $0.3$ | $110.97$ | $110.81$ | $0.0014$ | $0.0076$ |

7 | 4 | $0.1$ | $0.4$ | $126.33$ | $126.34$ | $0.0001$ | $-0.0005$ |

7 | 4 | $0.2$ | $0.3$ | $130.53$ | $130.07$ | $0.0035$ | $0.0215$ |

7 | 4 | $0.3$ | $0.3$ | $131.26$ | $131.45$ | $0.0014$ | $-0.0097$ |

$\mathit{p}-\mathit{r}$ | $\mathit{m}-\mathit{r}$ | a | b | ${\mathit{q}}_{95}$ | ${\mathit{q}}_{95}^{*}$ | $\left(\right)open="|"\; close="|">{\mathit{q}}_{95}/{\mathit{q}}_{95}^{*}-1$ | $\mathbf{\Delta}{\mathit{p}}^{\mathit{app}}$ |
---|---|---|---|---|---|---|---|

2 | 1 | $0.0$ | $0.0$ | $12.21$ | $12.28$ | $0.0057$ | $-0.0036$ |

2 | 1 | $0.0$ | $0.3$ | $15.51$ | $15.55$ | $0.0026$ | $-0.0020$ |

2 | 1 | $0.1$ | $0.4$ | $18.24$ | $18.35$ | $0.0060$ | $-0.0055$ |

2 | 1 | $0.2$ | $0.3$ | $18.71$ | $18.75$ | $0.0021$ | $-0.0020$ |

2 | 1 | $0.3$ | $0.3$ | $18.81$ | $18.87$ | $0.0032$ | $-0.0030$ |

4 | 3 | $0.0$ | $0.0$ | $42.76$ | $42.60$ | $0.0038$ | $0.0077$ |

4 | 3 | $0.0$ | $0.3$ | $50.66$ | $50.71$ | $0.0010$ | $-0.0024$ |

4 | 3 | $0.1$ | $0.4$ | $57.40$ | $57.67$ | $0.0047$ | $-0.0141$ |

4 | 3 | $0.2$ | $0.3$ | $58.63$ | $58.69$ | $0.0010$ | $-0.0029$ |

4 | 3 | $0.3$ | $0.3$ | $58.83$ | $58.90$ | $0.0012$ | $-0.0035$ |

5 | 3 | $0.0$ | $0.0$ | $50.06$ | $49.96$ | $0.0020$ | $0.0049$ |

5 | 3 | $0.0$ | $0.3$ | $57.88$ | $57.73$ | $0.0026$ | $0.0073$ |

5 | 3 | $0.1$ | $0.4$ | $64.64$ | $64.73$ | $0.0014$ | $-0.0046$ |

5 | 3 | $0.2$ | $0.3$ | $65.66$ | $65.50$ | $0.0024$ | $0.0078$ |

5 | 3 | $0.3$ | $0.3$ | $65.62$ | $65.66$ | $0.0006$ | $-0.0020$ |

7 | 4 | $0.0$ | $0.0$ | $82.47$ | $82.35$ | $0.0015$ | $0.0059$ |

7 | 4 | $0.0$ | $0.3$ | $92.22$ | $92.26$ | $0.0004$ | $-0.0020$ |

7 | 4 | $0.1$ | $0.4$ | $101.46$ | $101.56$ | $0.0010$ | $-0.0050$ |

7 | 4 | $0.2$ | $0.3$ | $102.01$ | $102.04$ | $0.0003$ | $-0.0015$ |

7 | 4 | $0.3$ | $0.3$ | $101.81$ | $102.16$ | $0.0034$ | $-0.0182$ |

Single-Eq. Tests | ${\mathit{tb}}_{\mathit{t}}$ | ${\mathit{dulc}}_{\mathit{t}}$ | Vector Tests | |
---|---|---|---|---|

F${}_{AR5}$(5,66) | $0.946[0.457]$ | $0.777[0.570]$ | F${}_{AR5}$(20,120) | $0.542[0.943]$ |

F${}_{ARCH4}$(4,84) | $0.469[0.758]$ | $0.511[0.728]$ | F${}_{HET}$(93,162) | $0.838[0.825]$ |

F${}_{HET}$(31,56) | $0.726[0.831]$ | $1.016[0.468]$ | ${\chi}_{ND}^{2}$(4) | $2.233[0.693]$ |

${\chi}_{ND}^{2}$(2) | $1.341[0.512]$ | $0.426[0.808]$ |

$\mathit{r}=0$ | $\mathit{r}\le 1$ | |
---|---|---|

$PLR\{{H}_{\ell}(r)\left(\right)open="|"\; close>{H}_{\ell}(2)$ | $56.610{[0.014]}^{*}$ | $21.964[0.148]$ |

95% limit quantiles | $50.864$ | $26.334$ |

${\mathit{y}}_{\mathit{t}}$ | ${\mathit{y}}_{\mathit{t}}^{*}$ | ${\mathit{ppp}}_{\mathit{t}}$ |
---|---|---|

$0.004[0.951]$ | $1.183[0.277]$ | $1.954[0.162]$ |

© 2019 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**

Kurita, T.; Nielsen, B.
Partial Cointegrated Vector Autoregressive Models with Structural Breaks in Deterministic Terms. *Econometrics* **2019**, *7*, 42.
https://doi.org/10.3390/econometrics7040042

**AMA Style**

Kurita T, Nielsen B.
Partial Cointegrated Vector Autoregressive Models with Structural Breaks in Deterministic Terms. *Econometrics*. 2019; 7(4):42.
https://doi.org/10.3390/econometrics7040042

**Chicago/Turabian Style**

Kurita, Takamitsu, and Bent Nielsen.
2019. "Partial Cointegrated Vector Autoregressive Models with Structural Breaks in Deterministic Terms" *Econometrics* 7, no. 4: 42.
https://doi.org/10.3390/econometrics7040042