Savings and investments in the OECD: a panel cointegration study with a new bootstrap test

Abstract

In this paper we test for the existence of a stable long-run savings–investments relationship in 18 OECD economies over the period 1970–2007. Although individual modelling provides only very weak support to the hypothesis of a link between savings and investments, this cannot be ruled out as individual time series tests may have low power. We thus construct a new bootstrap test for panel cointegration robust to short- and long-run dependence across units. This test provides evidence of a long-run savings–investments relationship in most of the countries, with USA the most notable exception. However, the elasticities generally smaller than 1 suggest that market imperfections mostly cause only partial home biases.

Appendix

1.1 Asympotic validity of the RSB no cointegration test

The panel cointegration statistics proposed in this paper are akin to the group mean tests by Pedroni (1999, 2004). We will thus analogously investigate the asymptotic properties of the statistics by means of sequential limit arguments with T assumed to grow large prior to N. A detailed discussion of the implication of this type of asymptotics can be found in Pedroni (2004). For convenience, let us first recall the general set-up already introduced in Sect. 3.2. Let $z_{it}=({y} _{it},x_{it})^{\prime }$, where $t=1,\ldots , T$ and $i = 1,\ldots , N$ index, respectively, the time periods and the cross section units, be integrated bivariate processes such that $x_{it}=x_{it-1}+u_{it}^{x},y_{it}=\beta _{i}x_{it}+u_{it}^{y}$, where $u_{it}^{x}\sim N(0,\sigma _{xi}^{2})$ and $u_{it}^{y}$ may be either stationary (under cointegration, in which case $\beta _{i}$ is the long-run elasticity of $y_{it}$ to $x_{it}$) or not (in which case $\beta _{i}$ is the short-run elasticity). Define further for a given unit $i$ the AR(1)-filtered process $v_{it}$ mapping the null hypothesis of no cointegration through the parameter $\rho _{i}:$

$$\begin{aligned} v_{it}:=u_{it}^{y}-\rho _{i}u_{it-1}^{y}. \end{aligned}$$

(14)

To establish the asymptotic validity of our panel bootstrap procedure we first of all need to prove, for a given unit $i,$ that the results presented in PPP for unit root tests also hold when the object of interest is a vector of estimated cointegrating residuals. Recalling that in our case the unit root test is applied to the second-step residuals $\widetilde{u}_{it}$, defined in Eq. (7), we need to show that the empirical coefficient statistic:

$$\begin{aligned} T(\widetilde{\rho }_{i}-1)=\frac{T^{-1}\sum _{t=2}^{T}\widetilde{u}_{it-1}\widetilde{\nu }_{it}}{T^{-2}\sum _{t=2}^{T}\widetilde{u}_{it-1}^{2}} \end{aligned}$$

(15)

where $\widetilde{\nu }_{it}=\widetilde{u}_{it}^{y}-\widetilde{\rho }_{i} \widetilde{u}_{it-1}^{y}$, and the bootstrap coefficient statistic:

$$\begin{aligned} T(\widetilde{\rho }_{i}^{*}-1)=\frac{T^{-1}\sum _{t=2}^{T} \widetilde{u}_{it-1}^{*}\widetilde{v}_{it}^{*}}{T^{-2}\sum _{t=2}^{T}\widetilde{u}_{it-1}^{*^{2}}} \end{aligned}$$

(16)

where $\widetilde{u}_{it}^{*}$ and $\widetilde{v}_{it}^{*}$ are the analogues of $\widetilde{u}_{it}$ and $\widetilde{\nu }_{it}$ in the bootstrap world, have the same limiting distributions.

To this end, we need to assume the error processes and the estimators involved in (15) and (16) to satisfy some conditions. More precisely, we assume for each unit $i$:

• A1 The vector $\eta _{it}^{\prime }=(u_{it}^{x},v_{it})$ is an independent component stationary ergodic process with zero mean and finite variance (see Phillips and Ouliaris 1990, henceforth PO, condition C1 and Eq. 3). Hence, $\eta _{it}=\sum _{j=-\infty }^{\infty }A_{ij}\xi _{it-j}$ with $\sum _{j=-\infty }^{\infty }\left\| A_{ij}\right\| <\infty$ and $A_{i}(1)=\sum _{j=-\infty }^{\infty }A_{ij},$ where the $\xi _{it-j}^{\prime }s$ are i.i.d. $(0,\Sigma _{i} )$ with $\Sigma _{i}$ positive definite. All stationary ARMA processes satisfy these conditions.

• A2 For the vector process $\eta _{it}$ it holds that $S_{iT}(r)=T^{-1/2}\sum _{t=1}^{[Tr]}\eta _{it}\rightarrow B_{i}(r)$, $r \in [0,1],$ where $B_{i}(r)$ is a bivariate Brownian motion with covariance matrix

  $$\begin{aligned} \Omega _{i} =\underset{T\rightarrow \infty }{\lim }T^{-1}E\left[ \left( \sum _{t=1}^{T}\eta _{it}\right) ^{\prime }\left( \sum _{t=1}^{T}\eta _{it}\right) \right] . \end{aligned}$$

• A3 For $v_{it}$ and $u_{it}^{y}$ conditions (i)–(v) in PPP hold, so that:

  • (i) $\mathrm I\!E \left| v_{it}\right| ^{6+\delta }<\infty$;

  • (ii) $\mathrm I\!E \left| u_{it}^{y}\right| ^{6+\delta }<\infty ;$

  • (iii) $(\alpha$-mixing): $\sum _{k}k^{2}[\alpha _{v_{i}}(k)]^{\frac{\delta }{6+\delta }}<\infty ;$

  • (iv) if $\rho _{i}=1,$ $\sum _{k}k^{2}[\alpha _{u_{i}^{y}}(k)]^{\frac{\delta }{6+\delta }}<\infty ;$

  • (v) the spectral density of $v_{it}$, $f_{v_{i}},$ satisfies $f_{v_{i}}(0)>0.$

  Conditions (i)–(iv) are needed to ensure the asymptotic validity of the Stationary Bootstrap (Politis and Romano 1994), while condition (v) is introduced, as in PPP, to exclude the degenerate case $var(\sqrt{T}\textit{m}(v_{i}))=0,$ where $\textit{m}(v_{i})=T^{-1}\sum _{t=1}^{T}v_{it}$.

• A4 The estimator $\widehat{\rho }_{i}$ of the AR(1) coefficient in Eq. (14) satisfies:

  $$\begin{aligned} \widehat{\rho }_{i}=\rho _{i}+\left\{ \begin{array}{l} o_{p}(1) \\ O_{p}(T^{-1}) \end{array} \right. \begin{array}{l} \text{ if } \rho _{i}<1 \text{(cointegration) } \\ \text{ if } \rho _{i}=1 \text{(no } \text{ cointegration) } \end{array} \end{aligned}$$

  The estimator $\widehat{\beta }_{i}^{d}$ of the cointegrating coefficient satisfies:

  $$\begin{aligned} \widehat{\beta }_{i}^{d}=\beta _{i}+\left\{ \begin{array}{l} O_{p}(T^{-1}) \\ O_{p}(T^{-\frac{1}{2}}) \end{array} \right. \begin{array}{l} \text{ if } \rho _{i}<1 \text{(cointegration) } \\ \text{ if } \rho _{i}=1 \text{(no } \text{ cointegration) } \end{array} \end{aligned}$$

  Assumptions A1 ensure that the Stationary Bootstrap can be applied to $\Delta x_{it}=u_{it}^{x}$. Assumptions A1–A2 ensure that the no cointegration statistic $T(\widetilde{\rho }_{i}-1)$ has the same limiting distribution of the coefficient statistic for a unit root (Hansen 1990, Theorem 2). Finally, Assumptions A3–A5 allow us to state Lemma 1 below, which extends to the estimated residuals $\left\{ \widehat{ \widehat{u}}_{it}^{d}\right\}$ point ($i$) of PPP’s Lemma 3. Building on Lemma 1, we shall first state Proposition 1, our counterpart of PPP’s Theorem 1, and finally, in Propositions 2 and 3, state the asymptotic validity of the bootstrap cointegration and panel cointegration tests.

Propositions 1 and 2 below are on the asymptotic behaviour as $T\rightarrow \infty$ for a given unit $i$; we will introduce the panel dimension in Proposition 3.

Lemma 1

Let $\overline{u}_{it}^{d}=\widehat{\widehat{u}}_{it}^{d}-\frac{1}{T-1} \sum _{\tau =2}^{T}\widehat{\widehat{u}}_{i\tau }^{d}$ and $\overline{v}_{it}=v_{it}-\frac{1}{T-1}\sum _{\tau = 2}^{T}v_{i\tau }$ . For a given unit i, define $u_{it}^{d*}$ and $v_{it}^{*},$ obtained applying the Stationary Bootstrap, respectively, to $\overline{u} _{it}^{d}$ and $\overline{v}_{it},$ and $\mathbb E ^{*}(\cdot )$ the expectation in the bootstrap world. Recall that $\theta _{T}$ is the coefficient of the geometric distribution employed in the resampling algorithm. Then, under Assumptions A3–A5, if $\theta _{T}\rightarrow 0$ and $\sqrt{T}\theta _{T}\rightarrow \infty$:

$$\begin{aligned} \mathbb E ^{*}[T^{-1/2}\sum _{m=1}^{K_{\lfloor Tr\rfloor }}\sum _{s=1}^{L_{m}}\overline{u}_{i\varsigma _{m+s}}^{d}-T^{-1/2}\sum _{m=1}^{K_{\lfloor Tr\rfloor }}\sum _{s=1}^{L_{m}}v_{i\varsigma _{m+s}}]^{2}\rightarrow 0. \end{aligned}$$

Proof

First of all, recall that the residual $\widehat{\widehat{u}} _{it}^{d}$ from Eq. (6) can be rewritten as

$$\begin{aligned} \widehat{\widehat{u}}_{it}^{d}&= \widetilde{u}_{it}-\widehat{\rho }_{i} \widetilde{u}_{it-1} \\&= (y_{it}-\widehat{\beta }_{i}^{d}x_{it})-\widehat{\rho }_{i}(y_{it-1}- \widehat{\beta }_{i}^{d}x_{it-1}). \end{aligned}$$

Expanding $y_{it}$ as $y_{it}=\beta _{i}x_{it}+u_{it}^{y}$ and in turn $u_{it}^{y}$ as $u_{it}^{y}$ $=\rho _{i}u_{it-1}^{y}+v_{it}$ we obtain:

$$\begin{aligned} \widehat{\widehat{u}}_{it}^{d}&= u_{it}^{y}-\hat{\rho }_{i}u_{it-1}^{y}+( \widehat{\beta }_{i}^{d}-\beta _{i})x_{it}-\hat{\rho }_{i}(\widehat{\beta } _{i}^{d}-\beta _{i})x_{it-1} \\&= v_{it}+(\rho _{i}-\hat{\rho }_{i})u_{it-1}^{y}+(\widehat{\beta } _{i}^{d}-\beta _{i})x_{it}-\hat{\rho }_{i}(\widehat{\beta }_{i}^{d}-\beta _{i})x_{it-1}. \end{aligned}$$

so that the centred residual $\overline{u}_{it}^{d}$ turns out to be the sum of the unobservable noise $v_{it}$ (centred on the mean) and three other terms:

$$\begin{aligned} \overline{u}_{it}^{d}&= \rho _{i}u_{it-1}^{y}+v_{it}-(\widehat{\beta } _{i}^{d}-\beta _{i})\left( x_{it}-\frac{1}{T-1}\sum _{\tau =2}^{T}x_{i\tau }\right) \\&\quad +\hat{\rho }_{i}(\widehat{\beta }_{i}^{d}-\beta _{i})\left( x_{it-1}-\frac{1}{T-1} \sum _{\tau =2}^{T}x_{i\tau -1}\right) \\&\quad -\hat{\rho }_{i}u_{it-1}^{y}+\hat{\rho }_{i}\frac{1}{T-1}\sum _{\tau =2}^{T}u_{it-1}^{y}-\frac{1}{T-1}\sum _{\tau =2}^{T}(\rho _{i}u_{it-1}^{y}+v_{it}) \\&= (v_{it}-\frac{1}{T-1}\sum _{\tau =2}^{T}v_{it})-(\widehat{\rho } _{i}-\rho _{i})\left( u_{it-1}^{y}-\frac{1}{T-1}\sum _{\tau =2}^{T}u_{i\tau -1}^{y}\right) \\&\quad -(\widehat{\beta }_{i}^{d}-\beta _{i})\left( x_{it}-\frac{1}{T-1} \sum _{\tau =2}^{T}x_{i\tau }\right) \!+\!\hat{\rho }_{i}(\widehat{\beta } _{i}^{d}-\beta _{i})\left( x_{it-1}\!-\!\frac{1}{T-1}\sum _{\tau =2}^{T}x_{i\tau -1}\right) \end{aligned}$$

Taking into account the block structure (cf. steps 2.1–2.4), the (normalised) sum of these residuals can be written as:

$$\begin{aligned} T^{-1/2}\sum _{m=1}^{K_{\lfloor Tr\rfloor }}\sum _{s=1}^{L_{m}}\overline{u} _{it}^{d}&= T^{-1/2}\sum _{m=1}^{K_{\lfloor Tr\rfloor }}\sum _{s=1}^{L_{m}}\left( v_{i\varsigma _{m+s}}-\frac{1}{T-1}\sum _{\tau =2}^{T}v_{i\tau }\right) \\&\quad -T^{-1/2}\sum _{m=1}^{K_{\lfloor Tr\rfloor }}\sum _{s=1}^{L_{m}}(\rho _{i}- \hat{\rho }_{i})\left( u_{i\varsigma _{m+s-1}}^{y}-\frac{1}{T-1}\sum _{\tau =2}^{T}u_{i\tau -1}^{y}\right) \\&\quad -T^{-1/2}\sum _{m=1}^{K_{\lfloor Tr\rfloor }}\sum _{s=1}^{L_{m}}(\widehat{\beta }_{i}^{d}-\beta _{i})\left( x_{i\varsigma _{m+s}}-\frac{1}{T-1} \sum _{\tau =2}^{T}x_{i\tau }\right) \\&\quad +T^{-1/2}\sum _{m=1}^{K_{\lfloor Tr\rfloor }}\sum _{s=1}^{L_{m}}\hat{\rho } _{i}(\widehat{\beta }_{i}^{d}-\beta _{i})\left( x_{i\varsigma _{m+s-1}}-\frac{1}{T-1}\sum _{\tau =2}^{T}x_{i\tau -1}\right) \\&= A+B+C+D. \end{aligned}$$

To prove the Lemma we need to show that $B,C$ and $D$ converge to zero uniformly in $r$. Let us examine $C$ first. Following PPP (proof of Eq. (19), p. 622) we need to show that $\mathbb E ^{*}(C^{2})\rightarrow 0,$ or, equivalently (PPP, Eq. (26), p. 623) that

$$\begin{aligned} \mathbb E ^{*}\left[ (\widehat{\beta }_{i}^{d}-\beta _{i})\sum _{m=1}^{K_{\lfloor Tr\rfloor }}\sum _{s=1}^{L_{m}}\left( x_{i\varsigma _{m+s}}-\frac{1}{T-1}\sum _{\tau =2}^{T}x_{i\tau }\right) \right] ^{2}=O_{p}(\theta _{T}^{-1}T^{-\frac{3}{2}}). \end{aligned}$$

First of all define $\overline{x}_{it}=x_{it}-T^{-1}\sum _{\tau =2}^{T}x_{i\tau }$ and $V_{im}^{*}=\sum _{s=1}^{L_{m}}\overline{x}_{i\varsigma _{m+s-1}}$; note that $\mathbb E ^{*}(V_{im}^{*})=0$ (PPP, Eq. (24), p. 622). Next, recall that $x_{it}$ is the cumulated sum of the stationary process $u_{it}^{x}.$ Then,

$$\begin{aligned} \mathbb E ^{*}(x_{i\varsigma _{m+s-1}}^{2})=\frac{1}{T-1} \sum _{t=2}^{T}\left( \sum _{\tau =1}^{t}u_{i\tau }^{x}\right) ^{2}=O_{p}(T) \end{aligned}$$

so that $\mathbb E ^{*}\left( \overline{x}_{i\varsigma _{m+s-1}}^{2}\right) =Var^{*}(\overline{x}_{i\varsigma _{m+s-1}})\le \mathbb E ^{*}(x_{i\varsigma _{m+s-1}}^{2})=O_{p}(T)$. From this result and PPP Eq. (30) we obtain

$$\begin{aligned} \mathbb E ^{*}(V_{im}^{*2})=\mathbb E ^{*}\left( \sum _{s=1}^{L_{m}} \overline{x}_{i\varsigma _{m+s-1}}\right) ^{2}=O_{p}(\theta _{T}^{-2}T). \end{aligned}$$

Since $\mathbb E ^{*}(V_{im}^{*})=0$, $\mathbb E (\sum _{m=1}^{K_{\lfloor Tr\rfloor }}V_{im}^{*})^{2}=\mathbb E ^{*}(K_{\lfloor Tr\rfloor })\mathbb E ^{*}(V_{im}^{*})^{2}=O_{p}(\theta _{T}^{-1}T^{2})$ (PPP, p. 624). Finally, since Assumption A5 ensures that under the null hypothesis of no cointegration $(\widehat{\beta } _{i}^{d}-\beta _{i})=O_{p}(T^{-\frac{1}{2}}),$ it follows that

$$\begin{aligned} \mathbb E ^{*}\left[ (\widehat{\beta }_{i}^{d}-\beta _{i})\sum _{m=1}^{K_{\lfloor Tr\rfloor }}V_{im}^{*}\right] ^{2}=O_{p}(\theta _{T}^{-1}T^{-\frac{3}{2}}). \end{aligned}$$

uniformly in $r$. By Slutsky’s theorem the same proof can be applied to $D$, so that

$$\begin{aligned} \mathbb E ^{*}\left[ \hat{\rho }_{i}(\widehat{\beta }_{i}^{d}-\beta _{i})\sum _{m=1}^{K_{\lfloor Tr\rfloor }}V_{im}^{*}\right] ^{2}=O_{p}(\theta _{T}^{-1}T^{-\frac{3}{2}}), \end{aligned}$$

uniformly in $r$. Finally, recall that under the null hypothesis $\rho _{i}=1$ and the $u_{it}^{y\prime }s$ are the cumulated sum of the stationary process $v_{it},$ while under Assumption A4, $(\widehat{\rho }_{i}-\rho _{i})=O_{p}(T^{-1}).$ Then the same arguments used above apply to $B$, so that

$$\begin{aligned} \mathbb E ^{*}\left[ (\rho _{i}-\hat{\rho }_{i})\sum _{m=1}^{K_{\lfloor Tr\rfloor }}\sum _{s=1}^{L_{m}}\left( u_{i\varsigma _{m+s-1}}^{y}-\frac{1}{T-1} \sum _{\tau =2}^{T}u_{i\tau -1}^{y}\right) \right] ^{2}=O_{p}(\theta _{T}^{-1}T^{-1}) \end{aligned}$$

uniformly in $r.$ This completes the proof. $\square$

Convergence in mean square of the estimated residuals to the unobservable noise allows us to exploit all the results contained in PPP’s Lemmas 3 and 4. Of particular importance is the convergence in probability in the bootstrap world (denoted by $\overset{p^{*}}{\rightarrow }$) of the variance of $u_{it}^{d*},$ $\widehat{\sigma }_{iT}^{*2}=Var^{*}(T^{-1}\sum _{t=1}^{T}u_{it}^{d*}),$ to the variance of $v_{it}^{*},$ $\sigma _{iT}^{*2}=Var^{*}(T^{-1}\sum _{t=1}^{T}v_{it}^{*})$, and to the long-run variance of $v_{it},\sigma _{i\infty }^{2}=2\pi f_{v_{i}}(0)$:

$$\begin{aligned}&\widehat{\sigma }_{iT}^{*2}-\sigma _{iT}^{*2}\overset{p^{*}}{\rightarrow }0 \nonumber \\&\widehat{\sigma }_{iT}^{*2}-\sigma _{i\infty }^{2}\overset{p^{*}}{ \rightarrow }0. \end{aligned}$$

We can now state Proposition 1.

Proposition 1

Define the partial sum process $S_{iT}^{*}(r),$ $r\in [0,1],$ where $S_{iT}^{*}(r)=\frac{1}{\sqrt{T}\widehat{\sigma } _{ir}^{*}}\sum _{t=1}^{[Tr]}u_{it}^{d*}.$ Under A3–A5, if $\theta _{T}\rightarrow 0$ and $\sqrt{T}\theta _{T}\rightarrow \infty ,$ when $T\rightarrow \infty$ it holds that $S_{iT}^{*}(\cdot ) \overset{d^{*}}{\rightarrow }W_{i}$ in probability, where $\overset{d^{*} }{\rightarrow }$ denotes convergence in distribution in the bootstrap world and $W_{i}$ is a standard Wiener process.

Proof

First of all, define the process $R_{iT}(r)$, partial sum of the noises $\overline{v}_{it}$:

$$\begin{aligned} R_{iT}(r)=\frac{1}{\sqrt{T}\sigma _{iT}^{*}}\sum _{m=1}^{k_{\lfloor Tr\rfloor }}\sum _{s=1}^{L_{m}}\overline{v}_{i\varsigma _{m+s}} \end{aligned}$$

By (33) in PPP, $R_{iT}(r)\overset{d^{*}}{\rightarrow }W$. Further, from our Lemma 1 and (17) above and PPP’s Lemma 3 it holds that $S_{iT}^{*}(r)-R_{iT}(r)\overset{p^{*}}{\rightarrow }0$ uniform in $r$. It then follows that $S_{iT}^{*}\overset{d^{*}}{\rightarrow } W_{i}$ and the proof is complete. $\square$

Proposition 1 allows us to extend (either directly or by straightforward application of the continuous mapping theorem) Lemma 5 in PPP (which includes a set of convergence results entirely analogous to those of Lemma 1 in Phillips 1986) to quantities derived from the estimated cointegrating residuals.

Recalling that $\sigma _{i\infty }^{2}$ is the asymptotic variance of $v_{it},$ and defining $\sigma _{i} ^{2}$ as the asymptotic variance of its cumulated sums (see e.g. Phillips 1986, p. 314), in our case we have:

• (a) $T^{-2}\sum _{t=2}^{T}\widetilde{u}_{it-1}^{*2} \overset{d^{*}}{\rightarrow }{\sigma _{i}^{2}} \int _{0}^{1}W_{i}^{2}(r)\text{ d }r.$

• (b) $T^{-1}\sum _{t=2}^{T}\widetilde{u}_{it-1}^{*} \widetilde{v}_{it}^{*}\overset{d^{*}}{\rightarrow }\frac{1}{2} (\sigma _{i} ^{2}W_{i}^{2}(1)-\sigma _{i\infty }^{2}).$

• (c) $T^{-3/2}\sum _{t=2}^{T}\widetilde{u}_{it-1}^{*} \overset{d^{*}}{\rightarrow }\sigma _{i} \int _{0}^{1}W_{i}(r)\text{ d }r.$

• (d) $T^{-1/2}\sum _{t=1}^{T}\widetilde{v}_{it}^{*}\overset{d^{*}}{\rightarrow }\sigma _{i\infty }W_{i}(1).$

The proofs are based on Proposition 1, consistency of the two-step estimator $\beta _{i}^{d*}$ for the bootstrap population parameter $\widehat{\beta }_{i}^{d},$ and standard arguments. For instance, for result (a):

$$\begin{aligned} T^{-2}\sum _{t=2}^{T}\widetilde{u}_{it-1}^{*2}&= T^{-2}\sum _{t=2}^{T}(y_{it-1}^{*}-\beta _{i}^{d*}x_{it-1}^{*})^{2} \\&= T^{-2}\sum _{t=2}^{T}\left[ \left( \sum _{s=1}^{t-1}\Delta y_{is}^{*}\right) -\beta _{i}^{d*}\left( \sum _{s=1}^{t-1}\Delta x_{is-1}^{*}\right) \right] ^{2} \\&= T^{-2}\sum _{t=2}^{T}\left[ \left( \sum _{s=1}^{t-1}(\widehat{\beta } _{i}^{d}\Delta x_{is}^{*}+u_{is}^{d*}\right) -\beta _{i}^{d*}\left( \sum _{s=1}^{t-1}\Delta x_{is-1}^{*}\right) \right] ^{2} \\&= T^{-2}\sum _{t=2}^{T}\left[ (\widehat{\beta }_{i}^{d}-\beta _{i}^{d*})\sum _{s=1}^{t-1}\Delta x_{is}^{*}+\sum _{s=1}^{t-1}u_{is}^{d*}\right] ^{2}. \end{aligned}$$

Since $\beta _{i}^{d*}\overset{p}{\rightarrow }\widehat{\beta }_{i}^{d}$ the limit depends only on $(\sum _{s=1}^{t-1}u_{is}^{d*})^{2},$ which by Proposition 1 and continuos mapping theorem is known to converge to $\sigma _{i} ^{2}\int _{0}^{1}W_{i}^{2}(r)\text{ d }r$. We can now state the following proposition, which generalises PPP’s Theorem 2 to tests of no cointegration.

Proposition 2

For a given unit i, assume that $\eta _{it}^{\prime }=(u_{it}^{x},v_{it})$ satisfies assumptions A1–A3. Then

$$\begin{aligned} \underset{c\in \mathcal R }{\sup }\left| P^{*}(T(\widetilde{\rho } _{i}^{*}-1)\le c|z_{1}\ldots z_{T})-P_{0}(T(\widetilde{\rho } _{i}-1)\le c)\right| \overset{p}{\longrightarrow }0 \end{aligned}$$

where $P^{*}$ is the bootstrap distribution and $P_{0}$ is the probability measure obtained under the true null hypothesis of no cointegration.

Proof

Consider (15) and (16). By Proposition 1 and the extension of PPP’s Lemma 5 presented in (a)–(d) above, all partial sums appearing in both statistics have the same limit distribution. Hence, by the continuous mapping theorem $T(\widetilde{\rho }_{i}-1)$ and $T(\widetilde{ \rho }_{i}^{*}-1)$ have the same limiting distribution also. $\square$

Proposition 2 ensures that a test based on the empirical bootstrap distribution will be asymptotically valid, as this distribution will be close to the true null distribution. The final step is stating the asymptotic validity under independence of a no cointegration panel test computed as the mean of the individual tests.

Proposition 3

Assume that $E(\eta _{it}\eta _{js}^{\prime })=0$ for each $i\ne j$ and each $s,t$. Then:

$$\begin{aligned} \underset{c\in \mathcal R }{\sup }\left| P^{*}(N^{-1}\sum _{i=1}^{N}T(\widetilde{\rho }_{i}^{*}-1)\le c|z_{i1}\ldots z_{iT})-P_{0}(N^{-1}\sum _{i=1}^{N}T(\widetilde{\rho } _{i}-1)\le c)\right| \overset{p}{\longrightarrow }0 \end{aligned}$$

where $P^{*}$ is the bootstrap distribution and $P_{0}$ is the probability measure obtained under the null hypothesis of no cointegration.

Proof

From Proposition 2 and the continuous mapping theorem. $\square$

PPP point out that their Theorem 1 is general enough to expect tests based on other statistics, such as the Dickey–Fuller test, to be asymptotically valid. Since Proposition 1 extends PPP’s Theorem 1 to cointegration residuals we can then expect a no cointegration panel test constructed as the mean of the individual HEG tests to be asymptotically valid also.

1.2 Data source and definitions

The data, in national currencies at current prices, have been downloaded from the OECD.stat database. Definitions are as follows: investment Gross capital formation (transaction code: P5S1); savings Net savings (transaction code B8NS1) plus Consumption of fixed capital (transaction code K1S1); gross domestic product transaction code B1_GS1.