# Improved Inference on Cointegrating Vectors in the Presence of a near Unit Root Using Adjusted Quantiles

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## Abstract

**:**

## 1. Introduction

## 2. The Vector Autoregressive Model with near Unit Roots

#### 2.1. The Model

**Assumption**

**1.**

**Lemma**

**1.**

**Theorem**

**2.**

#### 2.2. Asymptotic Distributions

**Theorem**

**3.**

**Corollary**

**1.**

## 3. Critical Value Adjustment for Test on $\beta $ in the CVAR with near Unit Roots

#### 3.1. Bonferroni Bounds

#### 3.2. Adjusted Bonferroni Bounds

#### 3.3. The Simulation Study of Elliott (1998)

#### 3.4. Results with Bonferroni Quantiles and Adjusted Bonferroni Quantiles for ${Q}_{\beta}$

#### 3.5. A Few Examples of Other DGPs

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Proofs

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**3.**

**Proof**

**of**

**Corollary**

**1.**

## Appendix B. Figures

**Figure A1.**Top panel: Rejection frequency of the test ${Q}_{\beta}$ for $\gamma =0$ using the ${\chi}^{2}{\left(1\right)}_{0.90}$ quantile as a function of c and $\rho $. Bottom panel: Rejection frequency of the 5% test ${Q}_{r}$ for $r=1$ using Table 15.1 in Johansen (1996) as a function of c. $N=1000$ simulations of $T=100$ observations from the DGP (15) and (16).

**Figure A2.**Quantiles and fitted values in the distributions of $\widehat{c}$ and ${Q}_{\beta}$ as a function of c for different values of $\rho $; $N=1000$ simulations of $T=100$ observations from the DGP (15) and (16).

**Figure A3.**Rejection frequency of the test ${Q}_{\beta}$ for $\gamma =0$ using the ${\chi}^{2}{\left(1\right)}_{0.90}$ quantile (Unadjusted), the Bonferroni quantile in (20) for $\xi =95\%$ and $\eta =5\%$ (Bonf) and the adjusted Bonferroni quantile in (21) for $\eta =5\%$ (Adjusted Bonf) as a function of c for different values of $\rho $; $N=1000$ simulations of $T=100$ observations from the DGP (15) and (16).

**Figure A4.**Rejection frequency of the test ${Q}_{\beta}$ for $\gamma =0$ using the ${\chi}^{2}{\left(1\right)}_{0.90}$ quantile (c = 0, chisq) and the Bonferroni quantile in (20) for $\xi =95\%$ and $\eta =5\%,$ as a function of $\gamma $ for different values of c and $\rho $; $N=1000$ simulations of $T=100$ observations from the DGP (15) and (16).

**Figure A5.**Rejection frequency of the test ${Q}_{\beta}$ for $\gamma =0$ using the ${\chi}^{2}{\left(1\right)}_{0.90}$ quantile (c=0, chisq) and the adjusted Bonferroni quantile in (21) for $\eta =5\%$ as a function of $\gamma $ for different values of c and $\rho $; $N=1000$ simulations of $T=100$ observations from the DGP (15) and (16).

**Figure A6.**Rejection frequency of the test ${Q}_{\beta}$ for $\gamma =0$ using the ${\chi}^{2}{\left(1\right)}_{0.90}$ quantile (Unadjusted), the Bonferroni quantile in (20) for $\xi =95\%$ and $\eta =5\%$ (Bonf) and the adjusted Bonferroni quantile in (21) for $\eta =5\%$ (Adjusted Bonf) as a function of c; $N=1000$ simulations of $T=100$ observations from the DGPs in Table 1.

**Figure A7.**Rejection frequency of the test ${Q}_{\beta}$ for $\gamma =0$ using the ${\chi}^{2}{\left(1\right)}_{0.90}$ quantile (c = 0, chisq) and the Bonferroni quantile in (20) for $\xi =95\%$ and $\eta =5\%,$ as a function of $\gamma $ for different values of c; $N=1000$ simulations of $T=100$ observations from the DGPs in Table 1.

**Figure A8.**Rejection frequency of the test ${Q}_{\beta}$ for $\gamma =0$ using the ${\chi}^{2}{\left(1\right)}_{0.90}$ quantile (c = 0, chisq) and the adjusted Bonferroni quantile in (21) for $\eta =5\%$ as a function of $\gamma $ for different values of c; $N=1000$ simulations of $T=100$ observations from the DGPs in Table 1.

## References

- Cavanagh, Christopher L., Graham Elliott, and James H. Stock. 1995. Inference in Models with Nearly Integrated Regressors. Econometric Theory 11: 1131–47. [Google Scholar] [CrossRef]
- Di Iorio, Francesca, Stefano Fachin, and Riccardo Lucchetti. 2016. Can you do the wrong thing and still be right? Hypothesis testing in I(2) and near-I(2) cointegrated VARs. Applied Economics 48: 3665–78. [Google Scholar] [CrossRef]
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- Johansen, Søren. 1996. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press. [Google Scholar]
- McCloskey, Adam. 2017. Bonferroni-based size-correction for nonstandard testing problems. Journal of Econometrics. in press. Available online: http://www.sciencedirect.com/science/article/pii/S0304407617300556 (accessed on 12 June 2017). [CrossRef]
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**Table 1.**The matrix $\Pi $ for four different DGPs given by $\alpha =-\beta ={(-1,1)}^{\prime}/2$ which are the basis for the simulations of rejection probabilities for the adjusted test for $\beta ={(1,-1)}^{\prime}/2$. The positions of $c/T$ give the different ${\alpha}_{1}$ and ${\beta}_{1}$.

Four DGPs Allowing for near Unit Roots, $\Omega ={\mathit{I}}_{2}$ | |||
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1: | $\left(\right)$ | 2: | $\left(\right)$ |

3: | $\left(\right)$ | 4: | $\left(\right)$ |

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## Share and Cite

**MDPI and ACS Style**

Franchi, M.; Johansen, S.
Improved Inference on Cointegrating Vectors in the Presence of a near Unit Root Using Adjusted Quantiles. *Econometrics* **2017**, *5*, 25.
https://doi.org/10.3390/econometrics5020025

**AMA Style**

Franchi M, Johansen S.
Improved Inference on Cointegrating Vectors in the Presence of a near Unit Root Using Adjusted Quantiles. *Econometrics*. 2017; 5(2):25.
https://doi.org/10.3390/econometrics5020025

**Chicago/Turabian Style**

Franchi, Massimo, and Søren Johansen.
2017. "Improved Inference on Cointegrating Vectors in the Presence of a near Unit Root Using Adjusted Quantiles" *Econometrics* 5, no. 2: 25.
https://doi.org/10.3390/econometrics5020025