Testing for Periodic Integration with a Changing Mean

Abstract

In this paper we extend the test of periodic integration proposed by Boswijk and Franses (J Time Ser Anal 17:221–245, 1996) allowing for a change in the mean. We provide the asymptotic distribution and show that is the square of the distribution obtained by Perron and Vogelsang (J Bus Econ Stat 10:467–470, 1992a, J Bus Econ Stat 10:301–320, 1992b). In a Monte-Carlo experiment we show a good behaviour of the test in terms of size and power. Finally we have illustrated the use of the test in an empirical application to the case of external imbalances in the eurozone.

Appendices

Appendix 1: Proof

Proof

First note that from (5) it is possible to write:

$$\begin{aligned} \mathbf {y}_{\tau }-\mathbf {y}_{\tau -1}=\left( \mathbf {\Theta }_{0}+\mathbf { \Theta }_{1}L\right) \mathbf {\Psi }(L)^{-1}\mathbf {e}_{\tau }=\mathbf {C} \left( L\right) \mathbf {u}_{\tau } \end{aligned}$$

(23)

with $\mathbf {u}_{\tau }=\mathbf {\Psi }(L)^{-1}\mathbf {e}_{\tau }$, them we have that:

$$\begin{aligned} \mathbf {y}_{\tau }= & {} \mathbf {y}_{0}+\mathbf {C}\left( 1\right) \mathop {\displaystyle \sum }\limits _{j=1}^{\tau }\mathbf {u}_{j}+O_{p}\left( 1\right) \\= & {} \mathbf {y}_{0}+\mathbf {ab}^{\prime }\mathop {\displaystyle \sum }\limits _{j=1}^{\tau }\mathbf {u} _{j}+O_{p}\left( 1\right) . \end{aligned}$$

Replace $\mathbf {u}_{\tau }$ by $\left( \mathbf {\gamma }D\left( {\textit{NB}}\right) _{\tau }+\mathbf {u}_{\tau }\right)$, hence we have:

$$\begin{aligned} \mathbf {y}_{\tau }=\mathbf {y}_{0}+\mathbf {ab}^{\prime }\mathop {\displaystyle \sum }\limits _{j=1}^{\tau }\mathbf {u}_{j}+\mathbf {ab}^{\prime }\mathbf {\gamma } {\textit{DU}}_{\tau }+O_{p}\left( 1\right) \end{aligned}$$

As in Perron and Vogelsang (1992a) it is possible to write for $\tilde{y} _{s\tau }=y_{s\tau }-\hat{\mu }_{s}-\hat{\gamma }_{s}^{*}{\textit{DU}}_{s\tau }$, where $\bar{y}_{s}^{a}=N_{b}^{-1}\sum _{\tau =1}^{N_{b}}y_{s\tau }=\lambda ^{-1}N^{-1}\sum _{\tau =1}^{N_{b}}y_{s\tau }$ and $\bar{y}_{s}^{b}=\left( N-N_{b}\right) ^{-1}\sum _{\tau =N_{b}+1}^{N}y_{s\tau }=\left( 1-\lambda \right) ^{-1}N^{-1}\sum _{\tau =N_{b}+1}^{N}y_{s\tau }$:

$$\begin{aligned}&\displaystyle \tilde{y}_{s\tau } =y_{s\tau }-\bar{y}_{s}^{a}=\mathbf {a}_{s}S_{\tau }- \mathbf {a}_{s}\bar{S}_{a}\quad if\quad \tau \le N_{B} \nonumber \\&\displaystyle \tilde{y}_{s\tau } =y_{s\tau }-\bar{y}_{s}^{b}=\mathbf {a}_{s}S_{\tau }- \mathbf {a}_{s}\bar{S}_{b}-\mathbf {a}_{s}\mathbf {b}^{\prime }\mathbf {\gamma } \left( 1-\lambda ^{\prime }\right) /\left( 1-\lambda \right) \quad if\quad N_{B}\le \tau \le N_{B}^{\prime } \nonumber \\&\displaystyle \tilde{y}_{s\tau } =y_{s\tau }-\bar{y}_{s}^{b}=\mathbf {a}_{s}S_{\tau }- \mathbf {a}_{s}\bar{S}_{b}+\mathbf {a}_{s}\mathbf {b}^{\prime }\mathbf {\gamma }- \mathbf {a}_{s}\mathbf {b}^{\prime }\mathbf {\gamma }\left( 1-\lambda ^{\prime }\right) /\left( 1-\lambda \right) \quad if\quad N_{B}^{\prime }\le \tau \le N\nonumber \\ \end{aligned}$$

(24)

with $S_{\tau }=\mathbf {b}^{\prime }\mathop {\displaystyle \sum }\nolimits _{j=1}^{\tau }\mathbf {u} _{j}$, $\bar{S}_{a}=N_{b}^{-1}\sum _{\tau =1}^{N_{b}}S_{\tau }=\lambda ^{-1}N^{-1}\sum _{\tau =1}^{N_{b}}S_{\tau }$ and $\bar{S}_{b}=\left( N-N_{b}\right) ^{-1}\sum _{\tau =N_{b}+1}^{N}S_{\tau }=\left( 1-\lambda \right) ^{-1}N^{-1}\sum _{\tau =N_{b}+1}^{N}S_{\tau }$. Additionally we define $\tilde{y}_{s\tau }^{*}$ ,without serial correlation, as the residuals from a projection of $\tilde{y}_{s\tau }$ on $D\left( {\textit{NB}}\right) _{s,\tau }$ and ,in the presence of serial correlation, as the the residuals from a projection of $\tilde{y}_{s\tau }$ on $D\left( {\textit{NB}}\right) _{s,\tau }$ and its $p-1$ lags. Assume for simplicity the absence of serial correlation, hence:

$$\begin{aligned}&\displaystyle \tilde{y}_{s\tau }^{*} =\mathbf {a}_{s}S_{\tau }-\mathbf {a}_{s}\bar{S} _{a}\quad if\quad \tau \le N_{B} \nonumber \\&\displaystyle \tilde{y}_{s\tau }^{*} =0\quad if\quad \tau =N_{B}+1 \nonumber \\&\displaystyle \tilde{y}_{s\tau }^{*} =\mathbf {a}_{s}S_{\tau }-\mathbf {a}_{s}\bar{S} _{b}-\mathbf {a}_{s}\mathbf {b}^{\prime }\mathbf {\gamma }\left( 1-\lambda ^{\prime }\right) {/}\left( 1-\lambda \right) \quad if\quad N_{B}+1\le \tau \le N_{B}^{\prime } \nonumber \\&\displaystyle \tilde{y}_{s\tau }^{*} =\mathbf {a}_{s}S_{\tau }-\mathbf {a}_{s}\bar{S} _{b}+\mathbf {a}_{s}\mathbf {b}^{\prime }\mathbf {\gamma }-\mathbf {a}_{s} \mathbf {b}^{\prime }\mathbf {\gamma }\left( 1-\lambda ^{\prime }\right) {/}\left( 1-\lambda \right) \quad if\quad N_{B}^{\prime }\le \tau \le N\nonumber \\ \end{aligned}$$

(25)

Following the lines of the proof of Theorem 1 in Boswijk and Franses (1996) it is convenient to write (10)/(11) using conventional time subscripts and seasonal dummy variable notation ($D_{st}$ taking the value unity when observation t falls in season s and zero otherwise). Employing this notation yields the representation (see Boswijk and Franses 1996, p. 238):

$$\begin{aligned} \tilde{y}_{t}^{*}=\pi _{1}D_{1t}\tilde{y}_{t-1}^{*}+\sum _{s=1}^{4}\varphi _{s}D_{st}\tilde{y}_{t-1}^{*}+\sum _{s=1}^{4}\sum _{j=1}^{p-1}\psi _{js}\left( D_{st}\tilde{y}_{t-j}^{*}-\varphi _{s-j}D_{st}\tilde{y}_{t-j-1}^{*}\right) +\varepsilon _{t} \end{aligned}$$

(26)

where the restrictions $\varphi _{1}\varphi _{2}\varphi _{3}\varphi _{4}=1$ is imposed. Note that since the deterministic terms enter unrestrictedly then $\tilde{y}_{t}^{*}$ are the residuals as defined in (24)/(25). Let $\theta =\left[ \theta _{1},\theta _{2}^{\prime },\theta _{3}^{\prime }\right] ^{\prime }$ denote the full parameter vector with $\theta _{1}=\pi _{1}$, $\theta _{2}^{\prime }=\left[ \varphi _{2},\varphi _{3},\varphi _{4}\right]$ and $\theta _{3}^{\prime }=\left[ \psi _{11},\ldots ,\psi _{1,p-1},\ldots ,\psi _{41},\ldots ,\psi _{4,p-1}\right]$. Under the null hypothesis $\pi _{1}=0$, hence this parameter is associated with the unit root while, $\varphi _{2}$, $\varphi _{3}$ and $\varphi _{4}$ are cointegration parameters (with $\varphi _{1}$ defined from the periodic unit root restriction as $\varphi _{1}=\left( \varphi _{2}\varphi _{3}\varphi _{4}\right) ^{-1}$), and $\theta _{3}$ collects the parameters associated with the stationary regressors in (26). Let $z_{t}=\left[ z_{t}^{1},z_{t}^{2\prime },z_{t}^{3\prime }\right] ^{\prime }$ be defined conformably with $\theta$ as $z_{t}=\partial \tilde{y}_{t}/\partial \theta$ , and hence

(27)

Note that for $z_{t}^{1}$ we have that

$$\begin{aligned} \sigma ^{-2}N^{-1}\mathop {\displaystyle \sum }\limits _{t=1}^{T}z_{t}^{1}\varepsilon _{t}= & {} \sigma ^{-2}N^{-1}\mathop {\displaystyle \sum }\limits _{t=1}^{T}D_{1t}\tilde{y}_{t-1}\varepsilon _{t}=\sigma ^{-2}N^{-1}\mathop {\displaystyle \sum }\limits _{\tau =1}^{N}\tilde{y}_{4,\tau -1}\varepsilon _{1\tau } \nonumber \\= & {} \sigma ^{-2}N^{-1}\sum \mathbf {a}_{4}S_{\tau -1}\varepsilon _{1\tau }-\sigma ^{-2}N^{-1}\mathbf {a}_{4}\bar{S}_{a}\sum _{\tau =1}^{N_{b}}\varepsilon _{1\tau } \nonumber \\&-\,\sigma ^{-2}N^{-1}\mathbf {a}_{4}\bar{S}_{a}\sum _{\tau =N_{b}+1}^{N}\varepsilon _{1\tau }+O_{p}\left( 1\right) \end{aligned}$$

(28)

and

$$\begin{aligned} \sigma ^{-2}N^{-2}\mathop {\displaystyle \sum }\limits _{t=1}^{T}\left( z_{t}^{1}\right) ^{2}= & {} \sigma ^{-2}N^{-2}\mathop {\displaystyle \sum }\limits _{t=1}^{T}\left( D_{1t}\tilde{y} _{t-1}\right) ^{2}=\sigma ^{-2}N^{-2}\mathop {\displaystyle \sum }\limits _{\tau =1}^{N}\left( \tilde{y}_{4,\tau -1}\right) ^{2} \nonumber \\= & {} \sigma ^{-2}N^{-2}\sum \left( \mathbf {a}_{4}S_{\tau -1}\right) ^{2}\nonumber \\&+\,\sigma ^{-2}N^{-1}\lambda \left( \mathbf {a}_{4}\bar{S}_{a}\right) ^{2}+\sigma ^{-2}N^{-1}\left( 1-\lambda \right) \left( \mathbf {a}_{4}\bar{S} _{b}\right) ^{2}\nonumber \\&-\,2\sigma ^{-2}N^{-2}\mathbf {a}_{4}\bar{S}_{a}\sum _{\tau =1}^{N_{b}}\left( \mathbf {a}_{4}S_{\tau -1}\right) \nonumber \\&-\,2\sigma ^{-2}N^{-2}\mathbf {a}_{4}\bar{S} _{b}\sum _{\tau =N_{b}+1}^{N}\left( \mathbf {a}_{4}S_{\tau -1}\right) +o_{p}\left( 1\right) . \end{aligned}$$

(29)

From lemma 1 in Boswijk and Franses (1996) it is possible to establish:

$$\begin{aligned} \sigma ^{-2}N^{-1}\sum \mathbf {a}_{s}S_{\tau -1}\varepsilon _{1\tau }\Rightarrow & {} \sigma ^{-2}\omega a_{s}\mathop {\displaystyle \int }\nolimits _{0}^{1}w\left( r\right) dE_{1}\left( r\right) \nonumber \\ \sigma ^{-1}N^{-3/2}\sum \mathbf {a}_{s}S_{\tau -1}\Rightarrow & {} \sigma ^{-1}\omega a_{s}\mathop {\displaystyle \int }\nolimits _{0}^{1}w\left( r\right) dr \nonumber \\ \sigma ^{-1}N^{-1/2}\sum \varepsilon _{1\tau }\Rightarrow & {} \sigma ^{-1}E_{1}\left( 1\right) \nonumber \\ \sigma ^{-1}N^{-3/2}\sum _{N_{B}}\mathbf {a}_{s}S_{\tau -1}\Rightarrow & {} \sigma ^{-1}\omega a_{4}\mathop {\displaystyle \int }\nolimits _{\lambda }^{1}w\left( r\right) dr \nonumber \\ \sigma ^{-1}N^{-1/2}\sum _{N_{B}}\varepsilon _{1\tau }\Rightarrow & {} \sigma ^{-1}\left( E_{1}\left( 1\right) -E_{1}\left( \lambda \right) \right) \nonumber \\ \sigma ^{-2}N^{-2}\sum \left( \mathbf {a}_{s}S_{\tau -1}\right) ^{2}\Rightarrow & {} \sigma ^{-2}\omega ^{2}a_{s}^{2}\mathop {\displaystyle \int }\nolimits _{0}^{1}\left[ w\left( r\right) \right] ^{2}dr \nonumber \\ \sigma ^{-1}N^{-1/2}\mathbf {a}_{s}\bar{S}_{a}= & {} \sigma ^{-1}\lambda ^{-1}N^{-3/2}\sum _{\tau =1}^{N_{b}}\mathbf {a}_{s}S_{\tau }\Rightarrow \sigma ^{-1}\omega a_{s}\lambda ^{-1}\mathop {\displaystyle \int }\nolimits _{0}^{\lambda }w\left( r\right) dr \nonumber \\ \sigma ^{-1}N^{-1/2}\mathbf {a}_{s}\bar{S}_{b}= & {} \sigma ^{-1}\left( 1-\lambda \right) ^{-1}N^{-3/2}\nonumber \\&\quad \times \sum _{\tau =N_{b}+1}^{N}\mathbf {a}_{s}S_{\tau }\Rightarrow \sigma ^{-1}\omega a_{s}\left( 1-\lambda \right) ^{-1}\mathop {\displaystyle \int }\nolimits _{\lambda }^{1}w\left( r\right) dr\nonumber \\ \end{aligned}$$

(30)

Hence we have that:

$$\begin{aligned} \sigma ^{-2}N^{-1}\mathop {\displaystyle \sum }\limits _{t=1}^{T}z_{t}^{1}\varepsilon _{t}\Rightarrow & {} \sigma ^{-2}\omega a_{4}\left[ \mathop {\displaystyle \int }\nolimits _{0}^{1}w\left( r\right) dE_{1}\left( r\right) -\lambda ^{-1}\left[ \mathop {\displaystyle \int }\nolimits _{0}^{\lambda }w\left( r\right) dr\right] E_{1}\left( \lambda \right) \right. \nonumber \\&-\left. \left( 1-\lambda \right) ^{-1}\left[ \mathop {\displaystyle \int }\nolimits _{\lambda }^{1}w\left( r\right) dr\right] \left( E_{1}\left( 1\right) -E_{1}\left( \lambda \right) \right) \right] \nonumber \\= & {} \sigma ^{-2}\omega a_{4}\left[ NU\left( E_{1}\right) ,\left( \lambda \right) \right] \nonumber \\&where {:} \nonumber \\ \left[ NU\left( E_{1}\right) ,\left( \lambda \right) \right]= & {} \mathop {\displaystyle \int }\nolimits _{0}^{1}w\left( r\right) dE_{1}\left( r\right) -\lambda ^{-1}\left[ \mathop {\displaystyle \int }\nolimits _{0}^{\lambda }w\left( r\right) dr\right] E_{1}\left( \lambda \right) \nonumber \\&-\left( 1-\lambda \right) ^{-1}\left[ \mathop {\displaystyle \int }\nolimits _{\lambda }^{1}w\left( r\right) dr\right] \left( E_{1}\left( 1\right) -E_{1}\left( \lambda \right) \right) \end{aligned}$$

(31)

and:

(32)

Note also that form (27) and following the lines of (28)–(33) it is possible to establish:

(33)

and $\left[ {\textit{DE}}\left( \lambda \right) \right]$ defined in (33). Under the periodic unit root null hypothesis the ${\textit{PAR}}(p-1)$ regressors $D_{st}y_{t-j}-\varphi _{s-j}D_{st}y_{t-j-1}$ collected in the vector $z_{t}^{3}$ are stationary with

$$\begin{aligned} \sigma ^{-2}N^{-1}\sum z_{t}^{3}\varepsilon _{\tau }\Rightarrow & {} N\left( 0,V_{3}\right) \nonumber \\ \sigma ^{-2}N^{-2}\sum z_{t}^{3}z_{t}^{3\prime }\rightarrow & {} V_{3}. \end{aligned}$$

(34)

As a consequence of the different rates of convergence for the parameter estimates corresponding to the nonstationary PI regressors and those for the stationary ${\textit{PAR}}(p-1)$ component in the augmented regression (10) or (11), we have that:

$$\begin{aligned} N^{-2}\sum z_{t}^{3}z_{t}^{2\prime }= & {} O_{p}\left( 1\right) \\ N^{-2}\sum z_{t}^{3}z_{t}^{1}= & {} O_{p}\left( 1\right) . \end{aligned}$$

Following Boswijk and Franses (1996), we establish the distribution of ${\textit{LR}}_{io}\left( \lambda \right)$ using

$$\begin{aligned} {\textit{LR}}_{io}\left( \lambda \right) =\frac{\left( N\hat{\theta }_{1}\right) ^{2}}{ \left( Y_{N}^{-1}Q_{\theta }Y_{N}^{-1}\right) ^{11}}+o_{p}\left( 1\right) . \end{aligned}$$

(35)

where $Y_{N}=diag\left[ N\times I_{4},N^{1/2}\times I_{4\left( p-1\right) } \right]$, $\left( Y_{N}^{-1}Q_{\theta }Y_{N}^{-1}\right) ^{11}$ is the first element of the principal diagonal of the inverse matrix $\left( Y_{N}^{-1}Q_{\theta }Y_{N}^{-1}\right) ^{-1}$, $N\hat{\theta }_{1}$ is the first element of $\left( Y_{N}^{-1}Q_{\theta }Y_{N}^{-1}\right) ^{-1}Y_{N}^{-1}q_{\theta }$, and $q_{\theta }$ and $Q_{\theta }$ are the score and negative of the Hessian matrix, respectively, formulated in terms of $\theta$. Note that, as in Boswijk and Franses (1996),

$$\begin{aligned} \left( Y_{N}^{-1}Q_{\theta }Y_{N}^{-1}\right) ^{-1}Y_{N}^{-1}q_{\theta }=\left( \sigma ^{-2}Y_{N}^{-1}\mathop {\displaystyle \sum }z_{t}z_{t}^{\prime }Y_{N}^{-1}\right) ^{-1}\sigma ^{-2}Y_{N}^{-1}\mathop {\displaystyle \sum }z_{t}\varepsilon _{t}. \end{aligned}$$

From (31), (33)–(34) it is easy to see that

(36)

Therefore,

$$\begin{aligned} \left( Y_{N}^{-1}Q_{\theta }Y_{N}^{-1}\right) ^{-1}Y_{N}^{-1}q_{\theta }\Rightarrow \left[ \begin{array}{c} \left[ {\textit{DE}}\left( \lambda \right) \right] ^{-1}\sigma ^{-1}\left( K^{\prime }K\right) ^{-1}K^{\prime }\left[ NU\left( E\right) ,\left( \lambda \right) \right] \\ N\left( 0,V_{3}^{-1}\right) \end{array} \right] . \end{aligned}$$

(37)

Note that $\left[ {\textit{DE}}\left( \lambda \right) \right]$ is a scalar and also that for $\sigma ^{-1}\left( K^{\prime }K\right) ^{-1}K^{\prime }\left[ NU\left( E\right) ,\left( \lambda \right) \right]$ it is possible to write:

(38)

Now, partitioning $K=\left[ K_{1}\vdots K_{2}\right]$ to focus on the first element of $\left( Y_{N}^{-1}Q_{\theta }Y_{N}^{-1}\right) ^{-1}Y_{N}^{-1}q_{\theta }$, namely $N\hat{\theta }_{1}$, (37) and (38) implies

$$\begin{aligned} N\hat{\theta }_{1}\Rightarrow & {} \left[ {\textit{DE}}\left( \lambda \right) \right] ^{-1}\left\{ \mathop {\displaystyle \int }\nolimits _{0}^{1}w\left( r\right) dS_{1}\left( r\right) \right. \\&-\lambda ^{-1}\left[ \mathop {\displaystyle \int }\nolimits _{0}^{\lambda }w\left( r\right) dr\right] S_{1}\left( \lambda \right) \\&\left. -\left( 1-\lambda \right) ^{-1}\left[ \mathop {\displaystyle \int }\nolimits _{\lambda }^{1}w\left( r\right) dr\right] \left( S_{1}\left( 1\right) -S_{1}\left( \lambda \right) \right) \right\} \\ where&:&\\ S_{1}\left( r\right)= & {} \sigma ^{-1}\left( K_{1}^{\prime }M_{2}K_{1}\right) ^{-1}K_{1}^{\prime }M_{2}E\left( r\right) \\ M_{2}= & {} I-K_{2}\left( K_{2}^{\prime }K_{2}\right) ^{-1}K_{2}^{\prime }. \end{aligned}$$

In Boswijk and Franses (1996) it is shown that $S_{1}\left( r\right) =\left( K_{1}^{\prime }M_{2}K_{1}\right) ^{-1/2}w\left( r\right)$ hence we have:

$$\begin{aligned} N\hat{\theta }_{1}\Rightarrow & {} \left( K_{1}^{\prime }M_{2}K_{1}\right) ^{-1/2}\left[ {\textit{DE}}\right] ^{-1}\left\{ \mathop {\displaystyle \int }\nolimits _{0}^{1}w\left( r\right) dw\left( r\right) \right. \nonumber \\&-\lambda ^{-1}\left[ \mathop {\displaystyle \int }\nolimits _{0}^{\lambda }w\left( r\right) dr\right] w\left( \lambda \right) \nonumber \\&\left. -\left( 1-\lambda \right) ^{-1}\left[ \mathop {\displaystyle \int }\nolimits _{\lambda }^{1}w\left( r\right) dr\right] \left( w\left( 1\right) -w\left( \lambda \right) \right) \right\} \nonumber \\= & {} \left( K_{1}^{\prime }M_{2}K_{1}\right) ^{-1/2}\left[ {\textit{DE}}\left( \lambda \right) \right] ^{-1}\left[ NU\left( \lambda \right) \right] \nonumber \\ with&:&\nonumber \\ \left[ NU\left( \lambda \right) \right]= & {} \mathop {\displaystyle \int }\nolimits _{0}^{1}w\left( r\right) dw\left( r\right) -\lambda ^{-1}\left[ \mathop {\displaystyle \int }\nolimits _{0}^{\lambda }w\left( r\right) dr\right] w\left( \lambda \right) \nonumber \\&-\left( 1-\lambda \right) ^{-1}\left[ \mathop {\displaystyle \int }\nolimits _{\lambda }^{1}w\left( r\right) dr\right] \left( w\left( 1\right) -w\left( \lambda \right) \right) \end{aligned}$$

(39)

note also that:

$$\begin{aligned} \left( Y_{N}^{-1}Q_{\theta }Y_{N}^{-1}\right) ^{11}\Rightarrow \left( K^{\prime }K\right) ^{11}\left[ {\textit{DE}}\right] ^{-1}=\left( K_{1}^{\prime }M_{2}K_{1}\right) ^{-1}\left[ {\textit{DE}}\left( \lambda \right) \right] ^{-1}. \end{aligned}$$

(40)

Then finally substituting (39) and (40) into (3) the required result is easily obtained. $\square$

Appendix 2: Tables

See Tables 1, 2, 3, 4, 5, 6, 7 and 8.

Table 1 Empirical quantiles of ${\textit{LR}}_{io}$

Table 2 Empirical size and power of ${\textit{LR}}_{io}$ for (15) with i)

Table 3 Empirical size and power of ${\textit{LR}}_{io}$ for (15) with ii)

Table 4 Empirical size and power of ${\textit{LR}}_{io}$ for (15) with iii)

Table 5 Empirical size and power of ${\textit{LR}}_{io}$ for (17) with i)

Table 6 Empirical size and power of ${\textit{LR}}_{io}$ for (17) with ii)

Table 7 Empirical size and power of ${\textit{LR}}_{io}$ for (17) with iii)

Table 8 Results

Appendix 3: Figures

See Figs. 1, 2, 3, 4 and 5.

Fig. 1

Evolution of exp/gdp and imp/gdp for Spain

Fig. 2

Evolution of exp/gdp and imp/gdp for Italy

Fig. 3

Evolution of exp/gdp and imp/gdp for Finland

Fig. 4

Evolution of exp/gdp and imp/gdp for The Netherlands

Fig. 5

Evolution of exp/gdp and imp/gdp for France

Appendix 4: Empirical Literature

Authors	Countries analyzed	Period	Variables	Techniques
Arize (2002)	50, all continents	Quart., 73-98	Nom. X/GDP, M/GDP dom. curr.	Johansen, SW, Hansen
Fountas and Wu (1999)	US	Quart., 67-94	X, M, real, nominal, relative	EG, structural breaks
Greenidge et al. (2012)	Barbados	Annu. 60-2006	Real X/GDP, M/GDP	ERS, NP, KPSS, Johansen, DOLS
Hamori (2009)	G-7 countries	Annu, 60-2005	X and M, mill. US $, trade bal	panel u. roots, coint., IPS, Pedroni
Herzer and Nowak-Lehmann (2006)	Chile	Annu, 75-2004	Real X and M domest. currency	Gregory-Hansen, DOLS, ECM
Holmes et al. (2011)	India	Annu. 50-2003	X/GDP, M/GDP	Johansen, Saikkonen and Lütkepohl,
				Breitung, Breitung and Taylor
Husted (1992)	US	Quart. 67-89	Nom., real, differenced ratios	EG, ADF, Perron-breaks
Irandoust and Sjoo (2000)	Sweden	Quart. 80-95	Nom., real, X, M/GDP dom. curr.	VECM, Johansen, stability tests
Irandoust and Ericsson (2004)	Fr, G, I, Sw, UK, US	Quart. 71-97	Real, log, seasonally adj.	VECM, Johansen, stability tests
Nag and Mukherjee (2012)	India	Annu. 50-2008	Real X and M (inclusive interest)	Lee and Strazicich, Gregory-Hansen
Narayan and Narayan (2005)	22 least developed	Annu. 60-2000	Nominal X and M	bounds ARDL, ECM, Hansen, DOLS