Testing for and dating structural break in smooth time-varying cointegration parameters, with an application to retail gasoline price and crude oil price long-run relationship

Abstract

This paper outlines a methodology to test for structural break in a smooth time-varying cointegration model. We show how such a problem can be brought down to the standard procedure proposed by Hansen (J Bus Econ Stat 10:321–335, 1992). As an application, we investigate the long-run relationship between the crude oil price and the gasoline retail price for Switzerland.

Appendix

The limiting distribution of $\text {CS}_T$ under time-varying cointegration can be obtained using Lemma 2 in Bierens and Martins (2010) and Theorem 1 of Xiao and Phillips (2002).

Lemma A.1

Under the assumption of $\Delta x_{t}$ being a strictly stationary zero-mean $p$-variate process such that $\Delta x_{t}=e_{t}$ with $e_{t}$ i.i.d $(0,\varOmega )$, defining $Z_{t}$ as in equation (7) and denoting $\varOmega ^{\frac{1}{2}}$ as the Cholesky factorization of $\varOmega$, i.e. $\varOmega =\varOmega ^{\frac{1}{2}}\varOmega ^{\frac{1}{2}\prime }$, using Lemma 2 of Bierens and Martins (2010), it comes the following limiting distributions:

$$\begin{aligned}&\frac{1}{T} \sum _{t=1}^{T} Z_{t}^{\prime }e_{t} \Longrightarrow \varOmega ^{\frac{1}{2}} \int _{0}^{1} S^{(m)\prime }(r) dW_2(r),\\&\frac{1}{T^{2}} \sum _{t=1}^{T} Z_{t}^{\prime }Z_{t} \Longrightarrow \varOmega ^{\frac{1}{2}} \left( \int _{0}^{1} S^{(m)\prime }(r) S^{(m)}(r)\text {d}r \right) \varOmega ^{\frac{1}{2} \prime }, \end{aligned}$$

where $S^{(m)}(r)$ is defined as:

$$\begin{aligned} S^{(m)}(r)^{\prime }=\left( W_2(r)^{\prime }, \sqrt{2}\cos (\pi r)W_2(r)^{\prime },\ldots , \sqrt{2}\cos (m\pi r)W_2(r)^{\prime }\right) , \end{aligned}$$

where $W_2(r)$ is a $(p-1)$ vector Brownian motion.

Note that the above results are immediate if $e_{t}$ is serially correlated such as $e_{t}=\Phi (L)v_t=\sum _{s=0}^{\infty }\Phi _se_{t-s}$, and we assume that $\sum _{s=0}^{\infty }s|\phi ^{(s)}_{ij}|<\infty$ for each i,j=1,...,n for $\phi ^{(s)}_{ij}$, the row i, column j element of $\Phi _s$ and that $(v_t)$ is an i.i.d. sequence with mean 0, variance $\varOmega _v$ with the factorization $\varOmega _v=\varOmega _v^{\frac{1}{2}}\varOmega _v^{\frac{1}{2}\prime }$, finite fourth moments. We have in this case:

$$\begin{aligned} \frac{1}{T} \sum _{t=1}^{T} Z_{t}^{\prime }e_{t} \Longrightarrow \varLambda ^{\frac{1}{2}} \left( \int _{0}^{1} S^{(m)\prime }(r) dW_2(r)\right) \varLambda ^{\frac{1}{2}\prime }+\sum _{k=1}^\infty \varOmega _k; \end{aligned}$$

and

$$\begin{aligned} \frac{1}{T^{2}} \sum _{t=1}^{T} Z_{t}^{\prime }Z_{t} \Longrightarrow \varLambda ^{\frac{1}{2}} \left( \int _{0}^{1} S^{(m)\prime }(r) S^{(m)}(r)\text {d}r \right) \varLambda ^{\frac{1}{2} \prime }, \end{aligned}$$

where $\varOmega _s=\mathbb {E}(e_te_{t-s}^\prime )=\sum _{k=0}^\infty \Phi _{s+k}\varOmega _v\Phi _{k}^\prime$ for $s=0,1,2,\ldots$, and where $\varLambda ^{\frac{1}{2}}=\Phi (1)\varOmega _{v}^{\frac{1}{2}}$ with $\Phi (1)$ nonsingular.

Using the previous results and following Phillips and Hansen (1990) and Phillips (1995) to establish the limiting distribution of $1/\sqrt{T}\left| \sum _{t=1}^{[Tr]}\hat{u}^+_t\right|$, with $r\in [0,1]$, the convergence (12) can be stated, which is similar to the standard time-invariant case presented in Theorem 1 of Xiao and Phillips (2002). $\square$