Intercountry feedback and spillover effects within the international supply and use framework: a Bayesian perspective^*

* This article is a revised version of the paper that won the International Input–Output Association's 2012 Wassily Leontief Memorial Prize in Bratislava.

ABSTRACT

This paper proposes a new framework for the estimation of product-level global and interregional feedback and spillover (FS) factor multipliers. The framework is directly based on interregional supply and use tables (SUTs) that could be rectangular and gives a possibility of taking account of the inherent input–output data uncertainty problems. A Bayesian econometric approach is applied to the framework using the first version of international SUTs in the World Input–Output Database. The obtained estimates of the global and intercountry FS output effects are discussed and presented at the world, country and product levels for the period of 1995–2009.

KEYWORDS:

• Global factor effects
• interregional feedback and spillover factor effects
• supply and use tables
• Bayesian techniques

JEL CLASSIFICATION CODES:

• R11
• R15
• C11

Acknowledgements

The author is grateful to the editors Bart Los and Michael Lahr for their patience and understanding. The author also thanks Bart Los, Michael Lahr and Ronald Miller for their extensive useful comments, and Bob Dröge for his assistance in working with the high performance computing facility of Millipede cluster offered by the Center for High Performance Computing and Visualisation of the University of Groningen.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

* This article is a revised version of the paper that won the International Input–Output Association's 2012 Wassily Leontief Memorial Prize in Bratislava.

1 Throughout the paper, the term ‘interregional’ is used for ‘intercountry’ whenever the focus is on intercountry effects. However, it is also a more general concept since various effects among different regions within a single country (or province) are also termed interregional effects.

2 The interregional spillover and feedback effects in the literature are also called open loop and closed loop effects, respectively. Open loop because the corresponding impact goes from one region into another region, and closed loop because the corresponding effect originates and via other regions ends (i.e. closes its impact loop) in the same region.

3 For a recent survey of the literature on diverse methods dealing with IO uncertainty issues, see Temursho (2017).

4 Also many IO analyses require the linking of an IO table to additional data sets such as international trade, employment and environmental statistics which are typically collected at an industry base. Linking these to symmetric IO tables that are either from the product-by-product or industry-by-industry type is problematic. Instead, SUTs being industry-by-product provide a natural link to the additional data sources.

5 This issue arises in e.g. Rueda-Cantuche (2011) where products in economy-wide SUTs are first disaggregated and only then OLS is used to estimate IO multipliers. This approach has a number of drawbacks that are discussed in detail in Temurshoev (2012).

6 For an extensive discussion and comparison of the frequentist versus Bayesian statistical approaches, see e.g. Poirier (1995).

7 Matrices are given in bold capital letters, vectors in bold lower case letters and scalars in italicized lower case letters. Vectors are columns by definition, thus row vectors are obtained by transposition indicated by a prime. $\mathbf{0}$ is a null array of appropriate dimension.

8 A unitary increase in final demand for region r's product j in our two-region framework is equivalent to setting all the elements of the final demand vector to zeros except its jth element for region r that is set to 1.

9 When the focus are regions within one country, ${\mathit{\theta }}_r$ indicates the national effects. Note that the change in final demand may also include a unitary change of exports of final goods from region r to the other region.

10 Note that by construction, this vector is non-negative.

11 If one added $\pm (\mathbf{I}-{\mathbf{A}}^{\prime })\tilde{\mathit{\theta }}$ to the right-hand side of the difference between Equations 4(4) $\begin{array}{rl}\mathit{\mu }& =(\mathbf{I}-{\mathbf{A}}^{\prime })\mathit{\theta },\end{array}$(4) and 5(5) $\begin{array}{rl}\mathit{\mu }& =(\mathbf{I}-{\tilde{\mathbf{A}}}^{\prime })\tilde{\mathit{\theta }}.\end{array}$(5) , instead of Equation 6(6) $\mathbf{0}=({\tilde{\mathbf{A}}}^{\prime }-{\mathbf{A}}^{\prime })\mathit{\theta }+(\mathbf{I}-{\tilde{\mathbf{A}}}^{\prime })(\mathit{\theta }-\tilde{\mathit{\theta }}).$(6) one would obtain $\mathbf{0}=({\tilde{\mathbf{A}}}^{\prime }-{\mathbf{A}}^{\prime })\tilde{\mathit{\theta }}+(\mathbf{I}-{\mathbf{A}}^{\prime })(\mathit{\theta }-\tilde{\mathit{\theta }})$. This together with Equation 5(5) $\begin{array}{rl}\mathit{\mu }& =(\mathbf{I}-{\tilde{\mathbf{A}}}^{\prime })\tilde{\mathit{\theta }}.\end{array}$(5) defines the alternative to Equation 7(7) $\left[\begin{array}{c}\mathit{\mu }\\ \mathbf{0}\end{array}\right]=\left[\begin{array}{cc}\mathbf{I}-{\mathbf{A}}^{\prime }& \mathbf{0}\\ {\tilde{\mathbf{A}}}^{\prime }-{\mathbf{A}}^{\prime }& \mathbf{I}-{\tilde{\mathbf{A}}}^{\prime }\end{array}\right]\left[\begin{array}{c}\mathit{\theta }\\ \mathit{\theta }-\tilde{\mathit{\theta }}\end{array}\right].$(7) regression-type system as $\left[\begin{array}{c}\mu \\ 0\end{array}\right]=\left[\begin{array}{cc}\mathrm{I}-{\tilde{\mathrm{A}}}^{\mathrm{\prime }}& 0\\ {\tilde{\mathrm{A}}}^{\mathrm{\prime }}-{\mathrm{A}}^{\mathrm{\prime }}& \mathrm{I}-{\mathrm{A}}^{\mathrm{\prime }}\end{array}\right]\left[\begin{array}{c}\tilde{\theta }\\ \theta -\tilde{\theta }\end{array}\right].$

12 Using the theory of partitioned matrices, it is easy to show that the inverse of interest has the following form: ${\left[\begin{array}{cc}\mathbf{I}-{\mathbf{A}}^{\prime }& \mathbf{0}\\ {\tilde{\mathbf{A}}}^{\prime }-{\mathbf{A}}^{\prime }& \mathbf{I}-{\tilde{\mathbf{A}}}^{\prime }\end{array}\right]}^{-1}=\left[\begin{array}{cc}{\mathbf{L}}^{\prime }& \mathbf{0}\\ -{\tilde{\mathbf{L}}}^{\prime }({\tilde{\mathbf{A}}}^{\prime }-{\mathbf{A}}^{\prime }){\mathbf{L}}^{\prime }& {\tilde{\mathbf{L}}}^{\prime }\end{array}\right].$

13 Besides the Use-regionalized system, as applied to $\mathbf{U}$ in Equation 8(8) $\mathbf{U}=\left[\begin{array}{cc}{\mathbf{U}}^{11}& {\mathbf{U}}^{12}\\ {\mathbf{U}}^{21}& {\mathbf{U}}^{22}\end{array}\right],\mathbf{U}=\left[\begin{array}{cc}{\mathbf{U}}^{11}& \mathbf{0}\\ \mathbf{0}& {\mathbf{U}}^{22}\end{array}\right]\mathrm{and}\mathbf{V}=\tilde{\mathbf{V}}=\left[\begin{array}{cc}{\mathbf{V}}^1& \mathbf{0}\\ \mathbf{0}& {\mathbf{V}}^2\end{array}\right].$(8) , a Make-regionalized SUTs framework is another option that instead adds destination-specific information to the supply table accounts, as discussed in Jackson and Schwarm (2011). However, the authors rightly mention (p. 195) that the preference for the Use-regionalized SUTs framework ‘is based on the foundation of production behaviour consistent with the demand-driven IO model rather than market share behaviour, which appears to be more consistent with a supply-driven IO model’. For further details on regionalization issues, see also Jackson (1998) and Lahr (2001).

14 Similar transformations show that the alternative system given in footnote 11 in terms of SUTs under product technology assumption is $\left[\begin{array}{c}\mathbf{e}\\ \mathbf{0}\end{array}\right]=\left[\begin{array}{cc}\mathbf{V}-{\tilde{\mathbf{U}}}^{\prime }& \mathbf{0}\\ {\tilde{\mathbf{U}}}^{\prime }-{\mathbf{U}}^{\prime }& \mathbf{V}-{\mathbf{U}}^{\prime }\end{array}\right]\left[\begin{array}{c}\tilde{\mathit{\theta }}\\ \mathit{\theta }-\tilde{\mathit{\theta }}\end{array}\right].$

15 Notice that both Equations 7(7) $\left[\begin{array}{c}\mathit{\mu }\\ \mathbf{0}\end{array}\right]=\left[\begin{array}{cc}\mathbf{I}-{\mathbf{A}}^{\prime }& \mathbf{0}\\ {\tilde{\mathbf{A}}}^{\prime }-{\mathbf{A}}^{\prime }& \mathbf{I}-{\tilde{\mathbf{A}}}^{\prime }\end{array}\right]\left[\begin{array}{c}\mathit{\theta }\\ \mathit{\theta }-\tilde{\mathit{\theta }}\end{array}\right].$(7) and 11(11) $\left[\begin{array}{c}\mathbf{e}\\ \mathbf{0}\end{array}\right]=\left[\begin{array}{cc}\mathbf{V}-{\mathbf{U}}^{\prime }& \mathbf{0}\\ {\tilde{\mathbf{U}}}^{\prime }-{\mathbf{U}}^{\prime }& \mathbf{V}-{\tilde{\mathbf{U}}}^{\prime }\end{array}\right]\left[\begin{array}{c}\mathit{\theta }\\ \mathit{\theta }-\tilde{\mathit{\theta }}\end{array}\right].$(11) allow for different number of products (and industries) across regions, i.e. one region can have more or less products (industries) than the other.

16 To be more precise, assume that all regions have n industries and p products, where p>n. In the two-region world setting, the number of parameters and equations in 11(11) $\left[\begin{array}{c}\mathbf{e}\\ \mathbf{0}\end{array}\right]=\left[\begin{array}{cc}\mathbf{V}-{\mathbf{U}}^{\prime }& \mathbf{0}\\ {\tilde{\mathbf{U}}}^{\prime }-{\mathbf{U}}^{\prime }& \mathbf{V}-{\tilde{\mathbf{U}}}^{\prime }\end{array}\right]\left[\begin{array}{c}\mathit{\theta }\\ \mathit{\theta }-\tilde{\mathit{\theta }}\end{array}\right].$(11) will be $4\times p$ and $4\times n$, respectively.

17 In general, when $\mathbf{e}$ is defined such that all its components are nullified except those for region r, the entries in $\mathit{\theta }-\tilde{\mathit{\theta }}$ have the following interpretation: those corresponding to region r are the interregional feedback effects for region r and those corresponding to region s ($\ne r$) represent the interregional spillover effects from region s into region r. That is, the last quantify region r's factor production due to unitary increase in final demand for region s ($\ne r$) products.

18 Also as rightly indicated by Bart Los, by aggregating products at a certain point one would arrive at a situation in which the supply table is (essentially) diagonal, which makes the use of SUTs entirely irrelevant as SUTs were designed to improve upon the original Leontief framework in accounting for secondary products.

19 For detailed discussion of the approach used in this paper, see Koop, (2003, chapter 6).

20 To be more precise, the likelihood is $p(\mathbf{y},\mathbf{X}|\mathit{\beta },h,\mathit{\lambda },\mathit{\eta })$. However, the second assumption implies that the likelihood can be written as $p(\mathbf{y},\mathbf{X}|\mathit{\beta },h,\mathit{\lambda },\mathit{\eta })=p\left(\mathbf{X}|\mathit{\eta }\right)p(\mathbf{y}|\mathbf{X},\mathit{\beta },h,\mathit{\lambda })$, hence without loss of information one can simply work with the likelihood function conditional on $\mathbf{X}$, $p(\mathbf{y}|\mathbf{X},\mathit{\beta },h,\mathit{\lambda })$. For notational convenience, the dependence on $\mathbf{X}$ is suppressed throughout the paper.

21 For SUTs settings of the basic transformation models, see Temurshoev (2012).

22 All the MATLAB files used for this purpose are available as online supplemental material. These were adopted from the relevant programs of Gary Koop's Bayesian Econometrics (Koop, 2003) and James LeSage's Econometrics Toolbox (LeSage, 1999). The large scale of the dataset requires heavy computing performance, hence I made use of the advanced high performance computing facility of Millipede cluster offered by the Center for High Performance Computing and Visualisation of the University of Groningen.

23 The following interesting observation (conjecture) can be made from Table 1: the industry technology estimate of the global output multiplier of roughly 2.52 is a weighted average of the corresponding product technology estimates of 2.05 and 3.17, where the weights are shares of products' contribution to the total of net outputs produced within the industry of interest. These last shares are reported in Table 1 as well. Thus $0.6145\times 2.0513+0.3855\times 3.1702\approx 2.5$ is quite close to the corresponding reported multiplier. The Leontief-inverse-based estimate of the FS output effect for Motor vehicles, trailers and semi-trailers is 0.4988, which is practically identical to 0.4965 that is reported for Other transport equipment. The corresponding Bayesian estimate for the first product is 0.1463 (reflecting its fewer international linkages). Hence, the industry technology estimate of the FS effect maybe approximated as$0.6145\times 0.1463+0.3855\times 0.9593\approx 0.46$.

24 See e.g. Bems et al. (2011) for the study of the effect of the 2008–2009 Great Recession on cross-border trade in intermediates.

25 Here first I compute the country-specific averages of the posterior means and of their lower and upper 90% HPDIs in 1995 and 2009, and then the growth rates for these three bounds separately. Thus the graph shows the evolution of the country-specific averages of these three statistics.

26 It must be noted that with Donald Trump winning the 2016 presidential election in the US and his intended protectionist (trade) policies, this trend could very well be weakened in the near future or even reversed, given the global importance of the US economy.