If the above assumptions hold, the estimates of the regression coefficients will converge to their true values. Let *D*_{T} be a diagonal matrix. The presence of *D*_{T} is due to the different convergence rates of the coefficients associated with the deterministic trend and integrated regressors. When the regression model contains an intercept such that *d*_{t}=1, we have $${D}_{T}=diag\mathrm{(}{T}^{\frac{1}{2}}\mathrm{,}\text{\hspace{0.17em}}T{I}_{k}\mathrm{}\mathrm{)}\mathrm{.}$$ In the case of a linear trend such that *d*_{t}=(1, *t*)′, we have *diag*(1, *T*^{–1})*d*_{[Tr]}⇒*B*_{d}(*r*)=(1, *r*)′. Since the convergence rate of the coefficient on the trending variable is *T*, the diagonal matrix becomes $${D}_{T}=diag\mathrm{(}{T}^{\frac{1}{2}}\mathrm{,}\text{\hspace{0.17em}}T{I}_{k+1}\mathrm{}\mathrm{)}\mathrm{.}$$ From now on the integral “∫” denotes integration from 0 to 1 and the argument of the Brownian motions are “*r*,” 0≤*r*≤1, unless otherwise specified. Similar to Theorem 1 in Xiao (2009), under the null hypothesis of cointegration denoted *H*_{0}, the limit distribution of the coefficient estimator is given by the following representation:

**Theorem 1** *Under H*_{0} *and Assumptions* 1–3, $$\widehat{\theta}\mathrm{(}\tau \mathrm{)}$$ *is a consistent estimator of θ*(*τ*) *and*

$${D}_{T}\mathrm{(}\widehat{\theta}\mathrm{(}\tau \mathrm{)}-\theta \mathrm{(}\tau \mathrm{)}\mathrm{)}\Rightarrow {\left[f\mathrm{(}{F}^{-1}\mathrm{(}\tau \mathrm{)}\mathrm{)}{\displaystyle \int}{B}_{z}{{B}^{\prime}}_{z}\right]}^{-1}\left[{\displaystyle \int}{B}_{z}d{B}_{\psi}+{\overline{\Delta}}_{x\psi}\right]\mathrm{,}\text{\hspace{1em}(8)}$$(8)

*where* $${B}_{z}=\mathrm{(}{{B}^{\prime}}_{d}\mathrm{,}\text{\hspace{0.17em}}{{B}^{\prime}}_{x}{\mathrm{)}}^{\prime},\text{\hspace{1em}}{\overline{\Delta}}_{x\psi}=\mathrm{(}0,\text{\hspace{0.17em}}{{\Delta}^{\prime}}_{x\psi}{\mathrm{)}}^{\prime},$$ *and* $${\Delta}_{x\psi}={\displaystyle {\sum}_{t=0}^{\infty}}E\mathrm{(}{\xi}_{2t}{\psi}_{\tau}\mathrm{(}{u}_{0}\mathrm{(}\tau \mathrm{)}\mathrm{)}\mathrm{)}$$ *is the one-sided long run covariance between ξ*_{2t} *and ψ*_{τ}(*u*_{t}(*τ*)).

The asymptotic representation is composed of integrals of Brownian motions and a bias term. In particular, the asymptotic distribution of the cointegrating coefficient estimator $$\widehat{\beta}\mathrm{(}\tau \mathrm{)}$$ is as follows:

$$T\mathrm{(}\widehat{\beta}\mathrm{(}\tau \mathrm{)}-\beta \mathrm{(}\tau \mathrm{)}\mathrm{)}\Rightarrow {\left[f\mathrm{(}{F}^{-1}\mathrm{(}\tau \mathrm{)}\mathrm{)}{\displaystyle \int}{\underset{\_}{B}}_{xd}{\underset{\_}{B}}_{xd}^{\prime}\right]}^{-1}\left[{\displaystyle \int}{\underset{\_}{B}}_{xd}d{B}_{\psi}+{\Delta}_{x\psi}\right]\mathrm{,}\text{\hspace{1em}(9)}$$(9)

where $${\underset{\_}{B}}_{xd}={B}_{x}-\mathrm{(}{\displaystyle \int}{B}_{x}{d}^{\prime}\mathrm{)}{\mathrm{(}{\displaystyle \int}d{{B}^{\prime}}_{d}\mathrm{)}}^{-1}{B}_{d}$$ is a *k*-dimensional demeaned or detrended Brownian motion. In the presence of serial cross correlation, the limit distribution of $$\widehat{\beta}\mathrm{(}\tau \mathrm{)}$$ has a second-order bias term Δ_{xψ}. Also, the above limit distribution depends on the nuisance parameter Ω, since the distribution of $$\int}{\underset{\_}{B}}_{xd}d{B}_{\psi$$ depends on the correlation between *B*_{x} and *B*_{ψ}, which is unknown in general. In order to develop useful inference procedures, the bias and nuisance parameter need to be removed. Moreover, to make the model parsimonious, instead of including leads and lags of the nonstationary regressors as in the quantile-varying cointegration model of equation (9) from Section 3.1 of Xiao (2009), this paper exploits the Phillips-Hansen type fully modified estimator to correct the serial correlation and endogeneity. As suggested by Phillips and Hansen (1990) and Phillips and Loretan (1991), the fully modified estimator displays superior properties than those of the usual estimator^{3}.

Following Phillips and Hansen (1990), define $${\psi}_{\tau}^{+}\mathrm{(}{u}_{t}\mathrm{(}\tau \mathrm{)}\mathrm{)}={\psi}_{\tau}\mathrm{(}{u}_{t}\mathrm{(}\tau \mathrm{)}\mathrm{)}-{\Omega}_{\psi x}{\Omega}_{xx}^{-1}{\xi}_{2t}$$ so that $${\psi}_{\tau}^{+}\mathrm{(}{u}_{t}\mathrm{(}\tau \mathrm{)}\mathrm{)}$$ is uncorrelated with ξ_{2t} and has variance $${\omega}_{\psi \mathrm{.}x}^{2}={\omega}_{\psi}^{2}-{\Omega}_{\psi x}{\Omega}_{xx}^{-1}{\Omega}_{x\psi}.$$ Then we have

$$\begin{array}{c}{T}^{-1}{\displaystyle \sum _{t=1}^{T}}{x}_{t}{\psi}_{\tau}^{+}\mathrm{(}{u}_{t}\mathrm{(}\tau \mathrm{)}\mathrm{)}\Rightarrow {\displaystyle \int}{B}_{x}d{B}_{\psi \mathrm{.}x}+{\Delta}_{x\psi}^{+}\\ ={\omega}_{\psi \mathrm{.}x}{\displaystyle \int}{B}_{x}dW+{\Delta}_{x\psi}^{+}\mathrm{,}\end{array}$$

where $${B}_{\psi \mathrm{.}x}={B}_{\psi}-{\Omega}_{\psi x}{\Omega}_{xx}^{-1}{B}_{x}$$ is a Brownian motion independent of *B*_{x}, *W*=*B*_{ψ.x}/*ω*_{ψ.x} is a standard Brownian motion, and $${\Delta}_{x\psi}^{+}={\Delta}_{x\psi}-{\Omega}_{\psi x}{\Omega}_{xx}^{-1}{\Delta}_{xx}$$ is the one-sided long run covariance between *ξ*_{2t} and $${\psi}_{\tau}^{+}\mathrm{(}{u}_{t}\mathrm{(}\tau \mathrm{)}\mathrm{)}\mathrm{.}$$

The long run variance $${\omega}_{\psi}^{2}$$ can be estimated by the following kernel estimator:

$${\widehat{\omega}}_{\psi}^{2}={\displaystyle \sum _{h=-M}^{M}}K\mathrm{(}\frac{h}{M}\mathrm{)}{\widehat{\Gamma}}_{\psi \psi}\mathrm{(}h\mathrm{}\mathrm{)}\mathrm{,}$$

where $${\widehat{\Gamma}}_{\psi \psi}\mathrm{(}h\mathrm{)}={T}^{-1}{\displaystyle {\sum}_{t=1}^{T-h}}{\psi}_{\tau}^{2}\mathrm{(}{\widehat{u}}_{t+h}\mathrm{(}\tau \mathrm{)}\mathrm{)}$$ with quantile regression residual $${\widehat{u}}_{t}\mathrm{(}\tau \mathrm{)}={y}_{t}-{\widehat{\theta}}^{\prime}\mathrm{(}\tau \mathrm{)}{z}_{t},K\mathrm{(}\cdot \mathrm{)}$$ is a kernel function defined on [–1, 1] with *K*(0)=1, and *M* is the bandwidth that *M*→∞ and *M*/*T*→0. The other elements in the long run variance Ω_{τ} and the one-sided long run covariance can be estimated in the similar fashion. Let $${\widehat{\Delta}}_{x\psi}^{+}={\widehat{\Delta}}_{x\psi}-{\widehat{\Omega}}_{\psi x}{\widehat{\Omega}}_{xx}^{-1}{\widehat{\Delta}}_{xx},$$ where $${\widehat{\Omega}}_{\psi x}={{\widehat{\Omega}}^{\prime}}_{x\psi},\text{\hspace{0.17em}}{\widehat{\Omega}}_{xx},\text{\hspace{0.17em}}{\widehat{\Delta}}_{x\psi},$$ and $${\widehat{\Delta}}_{xx}$$ are the kernel estimators:

$$\begin{array}{c}{\widehat{\Omega}}_{x\psi}={\displaystyle \sum _{h=-M}^{M}}K\mathrm{(}\frac{h}{M}\mathrm{)}{\widehat{\Gamma}}_{x\psi}\mathrm{(}h\mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{1em}}{\widehat{\Delta}}_{x\psi}={\displaystyle \sum _{h=0}^{M}}K\mathrm{(}\frac{h}{M}\mathrm{)}{\widehat{\Gamma}}_{x\psi}\mathrm{(}h\mathrm{}\mathrm{)}\mathrm{,}\\ {\widehat{\Omega}}_{xx}={\displaystyle \sum _{h=-M}^{M}}K\mathrm{(}\frac{h}{M}\mathrm{)}{\widehat{\Gamma}}_{xx}\mathrm{(}h\mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{1em}}{\widehat{\Delta}}_{xx}={\displaystyle \sum _{h=0}^{M}}K\mathrm{(}\frac{h}{M}\mathrm{)}{\widehat{\Gamma}}_{xx}\mathrm{(}h\mathrm{}\mathrm{)}\mathrm{,}\end{array}$$

where $${\widehat{\Gamma}}_{x\psi}\mathrm{(}h\mathrm{)}={T}^{-1}{\displaystyle {\sum}_{t=1}^{T-h}}\Delta {x}_{t}{\psi}_{\tau}\mathrm{(}{\widehat{u}}_{t+h}\mathrm{(}\tau \mathrm{)}\mathrm{)}$$ and $${\widehat{\Gamma}}_{xx}\mathrm{(}h\mathrm{)}={T}^{-1}{\displaystyle {\sum}_{t=1}^{T-h}}\Delta {x}_{t}\Delta {{x}^{\prime}}_{t+h}.$$ Under the null of cointegration $${\widehat{\omega}}_{\psi}^{2},\text{\hspace{0.17em}}{\widehat{\Omega}}_{x\psi},\text{\hspace{0.17em}}{\widehat{\Delta}}_{x\psi},\text{\hspace{0.17em}}{\widehat{\Omega}}_{xx},$$ and $${\widehat{\Delta}}_{xx}$$ are consistent estimators of $${\omega}_{\psi}^{2},\text{\hspace{0.17em}}{\Omega}_{x\psi},\text{\hspace{0.17em}}{\Delta}_{x\psi},\text{\hspace{0.17em}}{\Omega}_{xx},$$ and Δ_{xx}^{4}. As suggested by Andrews (1991) and Xiao and Phillips (2002), the Bartlett kernel and a plug-in bandwidth are adopted such that

$$K\mathrm{(}u\mathrm{)}=\{\begin{array}{ll}1-\mathrm{|}u\mathrm{|}\hfill & \text{for\hspace{0.17em}|}u\text{|}\le \text{1}\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$

and $$M=O\mathrm{(}{T}^{\frac{1}{3}}\mathrm{}\mathrm{)}\mathrm{.}$$

The regression coefficient estimator after the modification, $${\widehat{\theta}}^{+}\mathrm{(}\tau \mathrm{)}=\mathrm{(}{\widehat{\alpha}}^{\prime}\mathrm{(}\tau \mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{0.17em}}{{\widehat{\beta}}^{\prime}}^{+}\mathrm{(}\tau \mathrm{)}{\mathrm{)}}^{\prime},$$ is

$${\widehat{\theta}}^{+}\mathrm{(}\tau \mathrm{)}=\widehat{\theta}\mathrm{(}\tau \mathrm{)}-{\left[\widehat{f\mathrm{(}{F}^{-1}\mathrm{(}\tau \mathrm{)}\mathrm{)}}{\displaystyle \sum _{t=1}^{T}}{z}_{t}{{z}^{\prime}}_{t}\right]}^{-1}\left[{\displaystyle \sum _{t=1}^{T}}{z}_{t}{\widehat{\Omega}}_{\psi x}{\widehat{\Omega}}_{xx}^{-1}\Delta {x}_{t}+T{\overline{\widehat{\Delta}}}_{x\psi}^{+}\right]\mathrm{,}\text{\hspace{1em}(10)}$$(10)

where $${\overline{\widehat{\Delta}}}_{x\psi}^{+}=\mathrm{(}\mathrm{0,}\text{\hspace{0.17em}}{{\widehat{\Delta}}^{\prime}}_{x\psi}^{+}{\mathrm{)}}^{\prime}.$$ In particular, the fully modified estimator of the coefficients associated with the I(1) regressors is given by

$${\widehat{\beta}}^{+}\mathrm{(}\tau \mathrm{)}=\widehat{\beta}\mathrm{(}\tau \mathrm{)}-{\left[\widehat{f\mathrm{(}{F}^{-1}\mathrm{(}\tau \mathrm{)}\mathrm{)}}{\displaystyle \sum _{t=1}^{T}}{\underset{\_}{x}}_{t}^{d}{\underset{\_}{x}}^{\prime}{}_{t}^{d}\right]}^{-1}\left[{\displaystyle \sum _{t=1}^{T}}{\underset{\_}{x}}_{t}^{d}{\widehat{\Omega}}_{\psi x}{\widehat{\Omega}}_{xx}^{-1}\Delta {x}_{t}+T{\widehat{\Delta}}_{x\psi}^{+}\right]\mathrm{,}\text{\hspace{1em}(11)}$$(11)

where $${\underset{\_}{x}}_{t}^{d}$$ denotes the demeaned or detrended regressors and $$\widehat{f\mathrm{(}{F}^{-1}\mathrm{(}\tau \mathrm{)}\mathrm{)}}$$ is a nonparametric consistent estimator of the density function *f*(*F*^{–1}(*τ*)). The density function can be estimated using the Gaussian kernel and Silverman’s “rule-of-thumb” bandwidth.

Consequently, the fully modified estimator$${\widehat{\beta}}^{+}\mathrm{(}\tau \mathrm{)}$$follows a mixed normal distribution in the limit such that

$$\begin{array}{c}T\mathrm{(}{\widehat{\beta}}^{+}\mathrm{(}\tau \mathrm{)}-\beta \mathrm{(}\tau \mathrm{)}\mathrm{)}\Rightarrow {\left[f\mathrm{(}{F}^{-1}\mathrm{(}\tau \mathrm{)}\mathrm{)}{\displaystyle \int}{\underset{\_}{B}}_{xd}{\underset{\_}{B}}^{\prime}{}_{xd}\right]}^{-1}{\omega}_{\psi \mathrm{.}x}{\displaystyle \int}{\underset{\_}{B}}_{xd}dW\\ ~MN\mathrm{(}\mathrm{0,}\text{\hspace{0.17em}}\frac{{\omega}_{\psi \mathrm{.}x}^{2}}{f{\mathrm{(}{F}^{-1}\mathrm{(}\tau \mathrm{)}\mathrm{)}}^{2}}{\left[{\displaystyle \int}{\underset{\_}{B}}_{xd}{\underset{\_}{B}}^{\prime}{}_{xd}\right]}^{-1}\mathrm{)}\mathrm{,}\end{array}$$

where $${\omega}_{\psi \mathrm{.}x}^{2}$$ is estimated by $${\widehat{\omega}}_{\psi \mathrm{.}x}^{2}={\widehat{\omega}}_{\psi}^{2}-{\widehat{\Omega}}_{\psi x}{\widehat{\Omega}}_{xx}^{-1}{\widehat{\Omega}}_{x\psi},$$ which is calculated by the nonparametric kernel method. Therefore, the asymptotic distribution of $${\widehat{\theta}}^{+}\mathrm{(}\tau \mathrm{)}$$ is as follows:

**Theorem 2** *Under H*_{0} *and Assumptions* 1–3, $${\widehat{\theta}}^{+}\mathrm{(}\tau \mathrm{)}$$ *is a consistent estimator of θ*(*τ*) *and*

$${D}_{T}\mathrm{(}{\widehat{\theta}}^{+}\mathrm{(}\tau \mathrm{)}-\theta \mathrm{(}\tau \mathrm{)}\mathrm{)}\Rightarrow {\left[f\mathrm{(}{F}^{-1}\mathrm{(}\tau \mathrm{)}\mathrm{)}{\displaystyle \int}{B}_{z}{{B}^{\prime}}_{z}\right]}^{-1}{\omega}_{\psi \mathrm{.}x}{\displaystyle \int}{B}_{z}dW\text{\hspace{1em}(12)}$$(12)

$$\sim MN\mathrm{(}\mathrm{0,}\text{\hspace{0.17em}}\frac{{\omega}_{\psi \mathrm{.}x}^{2}}{f{\mathrm{(}{F}^{-1}\mathrm{(}\tau \mathrm{)}\mathrm{)}}^{2}}{\left[{\displaystyle \int}{B}_{z}{{B}^{\prime}}_{z}\right]}^{-1}\mathrm{)}\mathrm{,}\text{\hspace{1em}(13)}$$(13)

*where* $${B}_{z}=\mathrm{(}{{B}^{\prime}}_{d}\mathrm{,}\text{\hspace{0.17em}}{{B}^{\prime}}_{x}{\mathrm{)}}^{\prime}$$ *and* $${\omega}_{\psi \mathrm{.}x}^{2}={\omega}_{\psi}^{2}-{\Omega}_{\psi x}{\Omega}_{xx}^{-1}{\Omega}_{x\psi}.$$

In the linear regression case, the cumulated sum of the residuals, which is $${T}^{-\text{\hspace{0.17em}}\frac{1}{2}}{\displaystyle {\sum}_{t=1}^{n}}{\widehat{u}}_{t},\text{\hspace{0.17em}}n=\mathrm{1,}\text{\hspace{0.17em}}\dots \mathrm{,}\text{\hspace{0.17em}}T,$$ converges under the null of cointegration and diverges to infinity under the alternative. Analogously, in the quantile regression model, the cumulated sum of $${\psi}_{\tau}\mathrm{(}{\widehat{u}}_{t}\mathrm{(}\tau \mathrm{)}\mathrm{)}$$ satisfies the following conditions^{5}:

$$\underset{n=\mathrm{1,}\text{\hspace{0.17em}}\dots \text{\hspace{0.17em}}\mathrm{,}T}{\text{max}}{T}^{-\frac{1}{2}}\left|{\displaystyle \sum _{t=1}^{n}}{\psi}_{\tau}\mathrm{(}{\widehat{u}}_{t}\mathrm{(}\tau \mathrm{)}\mathrm{)}\right|=\mathrm{(}\begin{array}{ll}{O}_{p}\mathrm{(}1\mathrm{)}\hfill & \text{under\hspace{0.17em}}{H}_{\text{0}}\hfill \\ {O}_{p}\mathrm{(}T\mathrm{)}\hfill & \text{under\hspace{0.17em}}{H}_{\text{1}}\mathrm{.}\hfill \end{array}\text{\hspace{1em}(14)}$$(14)

Under *H*_{0}, the cumulated sum of the fully modified residuals converges to a functional of Brownian motions. For the quantile regression model, the residuals from the fully modified quantile regression is calculated as $${\widehat{u}}_{t}^{+}\mathrm{(}\tau \mathrm{)}={y}_{t}^{+}-{{z}^{\prime}}_{t}{\widehat{\theta}}^{+}\mathrm{(}\tau \mathrm{)},$$ where $${y}_{t}^{+}={y}_{t}-{\widehat{\Omega}}_{\psi x}{\widehat{\Omega}}_{xx}^{-1}\Delta {x}_{t}.$$ The CUSUM test statistic is given by

$$C{S}_{T}\mathrm{(}\tau \mathrm{)}=\underset{n=\mathrm{1,}\dots \mathrm{,}T}{\text{max}}\frac{1}{{\widehat{\omega}}_{\psi \mathrm{.}x}\sqrt{T}}\left|{\displaystyle \sum _{t=1}^{n}}{\psi}_{\tau}\mathrm{(}{\widehat{u}}_{t}^{+}\mathrm{(}\tau \mathrm{)}\mathrm{)}\right|\mathrm{.}\text{\hspace{1em}(15)}$$(15)

In particular,

$${T}^{-\frac{1}{2}}{\displaystyle \sum _{t=1}^{\mathrm{[}Tr\mathrm{]}}}{\psi}_{\tau}\mathrm{(}{\widehat{u}}_{t}^{+}\mathrm{(}\tau \mathrm{)}\mathrm{)}\Rightarrow {B}_{\psi \mathrm{.}x}-\left[{\displaystyle \int}d{B}_{\psi \mathrm{.}x}{{B}^{\prime}}_{z}\right]{\left[{\displaystyle \int}{B}_{z}{{B}^{\prime}}_{z}\right]}^{-1}{\displaystyle {\int}_{0}^{r}}{B}_{z}\text{\hspace{1em}(16)}$$(16)

$$={\omega}_{\psi \mathrm{.}x}\left\{{W}_{1}-\left[{\displaystyle \int}d{W}_{1}{S}^{\prime}\right]{\left[{\displaystyle \int}S{S}^{\prime}\right]}^{-1}{\displaystyle {\int}_{0}^{r}}S\right\}\mathrm{,}\text{\hspace{1em}(17)}$$(17)

where $$S=\mathrm{(}{{B}^{\prime}}_{d}\mathrm{,}\text{\hspace{0.17em}}{{W}^{\prime}}_{2}{\mathrm{)}}^{\prime}$$ and *W*_{1} and *W*_{2} are one and *k*-dimensional standard Brownian motions that are independent of each other.

Define

$$\underset{\_}{W}\mathrm{(}r\mathrm{)}={W}_{1}-\left[{\displaystyle \int}d{W}_{1}{S}^{\prime}\right]{\left[{\displaystyle \int}S{S}^{\prime}\right]}^{-1}{\displaystyle {\int}_{0}^{r}}S\mathrm{.}$$

Then, for a certain quantile *τ*, the asymptotic representation of the CUSUM test statistic is as follows:

**Theorem 3** *Under H*_{0} *and Assumptions* 1–3,

$$C{S}_{T}\mathrm{(}\tau \mathrm{)}\Rightarrow \underset{0\le r\le 1}{\text{sup}}\mathrm{|}\underset{\_}{W}\mathrm{(}r\mathrm{)}|\mathrm{.}\text{\hspace{1em}(18)}$$(18)

For each quantile level *τ*, the asymptotic distribution is the same as that from the linear case. Simulated critical values are tabulated in and from Hao and Inder (1996) and from Xiao and Phillips (2002). Hao and Inder (1996) also consider the CUSUM statistic, however, for testing the null hypothesis of parameter constancy. Under the null of no structural change, the test statistic from their model has the same asymptotic distribution as the cumulative sum statistic here. The limit distributions are different under the alternatives. The empirical size and power properties of the CUSUM test for cointegration are evaluated via simulation by Xiao and Phillips (2002). In the case of quantile regression, the finite sample properties of the test are similar to those from the fully modified OLS regression.

Table 1 Cointegration among Treasury bill rates (yields: 4, 13, 26 weeks).

Table 2 Cointegration among financial commercial paper rates (yields: 30, 60, 90 days).

Under *H*_{1}, the cumulated sum of the fully modified residuals diverges to infinity. Note that the fully modified residuals of the quantile regression model can be rewritten as:

$${\widehat{u}}_{t}^{+}\mathrm{(}\tau \mathrm{)}={\widehat{u}}_{t}\mathrm{(}\tau \mathrm{)}-{\widehat{\Omega}}_{\psi x}{\widehat{\Omega}}_{xx}^{-1}\Delta {x}_{t}+{{z}^{\prime}}_{t}{\left[\widehat{f\mathrm{(}{F}^{-1}\mathrm{(}\tau \mathrm{)}\mathrm{)}}{\displaystyle \sum _{t=1}^{T}}{z}_{t}{{z}^{\prime}}_{t}\right]}^{-1}\left[{\displaystyle \sum _{t=1}^{T}}{z}_{t}{\widehat{\Omega}}_{\psi x}{\widehat{\Omega}}_{xx}^{-1}\Delta {x}_{t}+T{\overline{\widehat{\Delta}}}_{x\psi}^{+}\right]\mathrm{.}$$

As shown in Lemma 1 of Xiao and Phillips (2002), under *H*_{1}, as *T*→∞, *M*→∞, and *M*/*T*→0,

$${\widehat{\Omega}}_{\psi x}={{\widehat{\Omega}}^{\prime}}_{x\psi}={O}_{p}\mathrm{(}M\mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{1em}}{\widehat{\Delta}}_{x\psi}={O}_{p}\mathrm{(}M\mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{1em}}{\widehat{\omega}}_{\psi \mathrm{.}x}^{2}={O}_{p}\mathrm{(}TM\mathrm{}\mathrm{)}\mathrm{.}$$

Furthermore, we have

$${\widehat{\Omega}}_{\psi x}{\widehat{\Omega}}_{xx}^{-1}{x}_{\mathrm{[}Tr\mathrm{]}}={O}_{p}\mathrm{(}{T}^{\frac{1}{2}}M\mathrm{)}\text{\hspace{0.17em}\hspace{0.17em}and}\text{\hspace{0.17em}}{\displaystyle \sum _{t=1}^{\mathrm{[}Tr\mathrm{]}}}{{z}^{\prime}}_{t}{\left[\widehat{f\mathrm{(}{F}^{-1}\mathrm{(}\tau \mathrm{)}\mathrm{)}}{\displaystyle \sum _{t=1}^{T}}{z}_{t}{{z}^{\prime}}_{t}\right]}^{-1}\left[{\displaystyle \sum _{t=1}^{T}}{z}_{t}{\widehat{\Omega}}_{\psi x}{\widehat{\Omega}}_{xx}^{-1}\Delta {x}_{t}+T{\overline{\widehat{\Delta}}}_{x\psi}^{+}\right]={O}_{p}\mathrm{(}{T}^{\frac{1}{2}}M\mathrm{}\mathrm{)}\mathrm{.}$$

The following theorem presents consistency of the test:

**Theorem 4** *Under H*_{1} *and Assumptions* 1–3*, as T*→∞, *Pr*(*CS*_{T}(*τ*)>*A*_{T})→*1 for any nonstochastic sequence* $${A}_{T}=o\mathrm{(}{T}^{\frac{1}{2}}{M}^{-\frac{1}{2}}\mathrm{}\mathrm{)}\mathrm{.}$$

Therefore, under the alternative of no cointegration the test statistic diverges and the divergence rate depends on the bandwidth.

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