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A pairwise-based approach to examining the Feldstein–Horioka condition of international capital mobility

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Abstract

We propose a pairwise procedure to test the Feldstein–Horioka condition of capital mobility. In contrast to the existing approach, we explicitly examine the relationship between domestic investment and foreign savings rather than domestic savings. In terms of addressing the Feldstein–Horioka puzzle, our results based on a panel of OECD and emerging market economies initially suggest that the depth and extent of capital mobility remain generally limited and that mobility has increased over the past 20 years. However, in contrast to existing studies, we find that capital mobility between Euro and EU pairs is more extensive than between pairs that involve other countries. If our sample is expanded to include emerging markets, we find that capital mobility has also increased though is weaker than for OECD economies. We provide additional insight in terms of consistency between our assessment of capital mobility based on the Feldstein–Horioka condition (a quantity approach) and a price approach based on real interest rate differentials.

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Notes

  1. We also include the possibility \(\alpha _{1,ij}<0\) as part of the case where capital is perfectly mobile. Since Eq. (5) can be rewritten as \(I_{i,t} = -\alpha _{0,ij} + (1-\alpha _{1,ij}) S_{j,t} + \alpha _{1,ij} S_{i,t} - u_{ij,t}\), \(\alpha _{1,ij}<0\) implies that \(I_{i,t}\) is very sensitive to \(S_{j,t}\) because \((1-\alpha _{1,ij})>1\).

  2. The cases where \(i=j\) are of no particular interest as they result in a regression of \((S_{i,t} - I_{i,t})\) against an intercept.

  3. Feldstein and Horioka (1980) do not include in their estimations France, Iceland, Luxembourg, Mexico, Norway, Portugal, Spain, Switzerland and Turkey. In addition, West Germany in the original FH sample is replaced by Germany.

  4. Chile, Israel and Korea entered the OECD in more recent years, and for this reason, they are not included in our group of 25 countries. For these three countries, the source of the data is also the World Bank.

  5. The results of the CD test are based on computer code developed by the authors using the Rats 8.2 computer econometric software.

  6. Given the extension of the time period, it is also worth considering the effect of possible structural breaks. For this, we implemented the more recent Hadri and Rao (2008) test, which tests the null of joint stationarity allowing for the possibility of one-time endogenously determined structural breaks as well as cross section dependence. Results not reported here indicate that the structural breaks identified by the Hadri and Rao testing procedure are not sufficient to alter our conclusion regarding the stationarity of the investment and savings series over the period of analysis.

  7. Ericsson et al. (1994) and Juselius (1999) argue that the order of integration is not a property inherent to a time series. That is, a time series may be integrated of order one during a specific sample period and integrated of order zero during another one.

  8. If we instead consider an EU-only sub-sample of 15 countries by adding Denmark, Sweden and the UK to the 12 Euro members, then 85 out of 210 (40 %) of the EU-only pairs are characterised by \(\hat{\alpha }_{1,ij}<0.5\). While the extent of capital mobility is slightly lower than in the Euro-only case, it is nonetheless greater than for the rest of the sample.

  9. When the indicator variable is defined as taking the value of one when the null of capital mobility is not rejected at the 5 % (1 %) significance level, the estimated slope coefficient in Eq. (6) remains negative although the resulting model is somewhat inferior and therefore not reported here.

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Correspondence to Mark J. Holmes.

Additional information

We thank Nataly Obando for help in obtaining some of the data. We also thank participants at the 7th FIW Research Conference on International Economics held in Vienna, and economics seminar participants at Massey University. An anonymous referee provided valuable comments that helped improve our paper. The usual disclaimer applies.

Appendix: Bootstrapping the empirical distribution of \(\overline{\hat{\alpha }}_{1,ij}\)

Appendix: Bootstrapping the empirical distribution of \(\overline{\hat{\alpha }}_{1,ij}\)

To bootstrap \(\overline{\hat{\alpha }}_{1,ij}\), we follow Pesaran et al. (2009) who consider the following model set-up:

$$\begin{aligned} y_{it}&= \varvec{\delta }_{i}^{'}\mathbf d _{t} + \varvec{\gamma }_{i}^{'}\mathbf f _{t} + \varepsilon _{it} \end{aligned}$$
(11)
$$\begin{aligned} \Delta \varepsilon _{it}&= \eta _{i} + \lambda _{i} \varepsilon _{i,t-1} + \sum _{l=1}^{p_i} \psi _{il} \Delta \varepsilon _{i,t-l} + \upsilon _{it}\end{aligned}$$
(12)
$$\begin{aligned} \Delta f_{st}&= \varvec{\mu }_{i}^{'}\mathbf d _{t} + \phi _{s}f_{s,t-1} + \sum _{l=1}^{p_s} \xi _{sl} \Delta f_{s,t-l} + e_{st} \end{aligned}$$
(13)

where \(y_{it}\) is investment (saving) as a proportion of GDP in country \(i\) at time \(t\), \(s=1,2,\ldots ,m\) is the number of common factors, \(\mathbf d _{t}=(1,t)^{'}\) is a vector of deterministic components that includes intercept and trend, \(\mathbf f _{t}\) is a \((m \times 1)\) vector of unobserved common factors, with elements denoted \(f_{st}\), and \(\varepsilon _{it}\) denotes the corresponding idiosyncratic elements. The unobserved common factors \(f_{st}\) and (or) the idiosyncratic elements \(\varepsilon _{it}\) may be \(I(0)\) or \(I(1)\).

In line with Pesaran et al. (2009), we use the cross-sectional average of \(y_{it}\), denoted \(\bar{y_{t}}=N^{-1}\sum _{i=1}^{N} y_{it}\), as an estimate of the common factor that may induce cross-sectional dependence across countries. To account for cross-sectional dependence, investment (saving) in each country is regressed on \(\bar{y_{t}}\):

$$\begin{aligned} y_{it}= \hat{\delta }_{1i} + \hat{\delta }_{2i} t + \hat{\gamma }_{1i} \bar{y}_{t} + \hat{\varepsilon }_{it}. \end{aligned}$$
(14)

It should be noted that in these factor equations, the trend term is included if it is statistically significant at the 5 % level.

The next step is to examine the time-series properties of \(\bar{y}_{t}\), which may be \(I(0)\) or \(I(1)\). This involves estimating the ADF(\(p\)) regression:

$$\begin{aligned} \Delta \bar{y}_{t}=\hat{\mu } + \hat{\phi }\bar{y}_{t-1} + \sum _{l=1}^{p} \hat{b}_l \Delta \bar{y}_{t-l} + \hat{e}_{t}, \end{aligned}$$
(15)

which may also include trend if it is statistically significant, and where the number of lags \(p\) may be determined, e.g. using the Akaike information criterion (AIC). To illustrate the implementation of the bootstrap, let us consider the case in which \(\bar{y}_{t}\) does not have a unit root, nor a deterministic trend (which appear to be the features that best characterise the resulting cross-sectional means of the investment and saving series). In this case, the bootstrap samples of \(\bar{y}_{t}\), denoted \(\bar{y}_{t}^{(b)}\), can be computed using the following generating mechanism:

$$\begin{aligned} \bar{y}_{t}^{(b)}=\hat{\mu } + \big (1+\hat{\phi }\big )\bar{y}_{t-1}^{(b)} + \sum _{l=1}^{p} \hat{b_l} \Delta \bar{y}_{t-l}^{(b)} + \hat{e}_{t}^{(b)} \end{aligned}$$
(16)

where bootstrap residuals \(\hat{e}_{t}^{(b)}\) are generated by randomly drawing with replacement from the set of estimated and centred residuals \(\hat{e}_{t}\) in Eq. (15) and where the first \((p+1)\) values of \(\bar{y}_{t}\) are used to initialise the process \(\bar{y}_{t}^{(b)}\).

In turn, the bootstrap samples of \(y_{it}\), denoted \(y_{it}^{(b)}\), are generated as:

$$\begin{aligned} y_{it}^{(b)}= \hat{\delta }_{1i} + \hat{\delta }_{2i} t + \hat{\gamma }_{1i} \bar{y}_{t}^{(b)} + \hat{\varepsilon }_{it}^{(b)}, \end{aligned}$$
(17)

where \(\hat{\delta }_{1i}\), \(\hat{\delta }_{2i}\) and \(\hat{\gamma }_{1i}\) are the OLS estimates of \(\delta _{1i}\), \(\delta _{2i}\) and \(\gamma _{1i}\) in Eq. (14), respectively, and

$$\begin{aligned} \varepsilon _{it}^{(b)} = \hat{\eta }_{i} + \big (1+\hat{\lambda }_{i}\big ) \varepsilon _{i,t-1}^{(b)} + \sum _{l=1}^{p_i} \hat{\psi }_{il} \Delta \varepsilon _{i,t-l}^{(b)} + \upsilon _{it}^{(b)}, \end{aligned}$$
(18)

where bootstrap residuals \(\upsilon _{it}^{(b)}\) are generated by randomly drawing with replacement from the set of estimated residuals \(\upsilon _{it}\) in Eq. (12) and where the first \((p+1)\) values of \(\hat{\varepsilon }_{it}\) are used to initialise the process \(\hat{\varepsilon }_{it}^{(b)}\). The AIC is used to determine the optimal number of lags.

The bootstrap procedure described above is applied to obtain the bootstrap samples of investment and saving (both as a share of GDP) by setting \(y_{it}=I_{it}\) and \(y_{it}=S_{it}\), respectively. Letting \(I_{it}^{(b)}\) and \(S_{it}^{(b)}\) be the respective bootstrap samples of \(I_{it}\) and \(S_{it}\), Eq. (5) is estimated \(b=1,\ldots ,B\) times to derive the empirical distribution of \(\overline{\hat{\alpha }}_{1,ij}\).

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Holmes, M.J., Otero, J. A pairwise-based approach to examining the Feldstein–Horioka condition of international capital mobility. Empir Econ 50, 279–297 (2016). https://doi.org/10.1007/s00181-015-0937-8

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