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On approximating the accelerator part in dynamic input–output models

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Abstract

We release the limitations of previous studies and instead of setting the crucial parameters of the dynamic endogenous input–output model with layers of techniques on an arbitrary basis we propose a new optimization-based approach to approximating of the elements of capital matrices on the basis of recent historical data. Using recent IO data we first formally prove that in comparison to arbitrarily adjusted dynamic IO models the new theoretical approach allows one to obtain a significantly better fit to the historical data in the short-run. This result has also some implications for the long-run analyses as it suggests that using the new approach for typical empirical applications of dynamic IO models with respect to modelling future behavior of economies seems relatively much more reasonable. Having this remark in mind, in the empirical part of the paper we use the new methodological approach in a particular case study. In an illustrative empirical application we try to forecast the possible evolution of sectoral classification in the Polish economy over the next 40 years.

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Fig. 1

Source: Own elaboration based on WIOD November 2016 Release

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Notes

  1. See e.g. Pfajfar and Dolinar (2000), Gurgul and Majdosz (2005), Gurgul and Lach (2015, 2016), among others.

  2. See e.g. Chenery and Watanabe (1958), Hewings and Romanos (1981), Hewings (1982), Defourny and Thorbecke (1984), Ćmiel and Gurgul (2002), Gurgul and Majdosz (2005), Gurgul and Lach (2015, 2016, 2017), among others.

  3. In general, static IO models allow solely (ex-post) analysis of historical data, while dynamic IO models allow also forecasting and simulating future behavior of an economy under various scenarios.

  4. Nowadays the forward linkages are often derived from output coefficients and not input coefficients (see e.g. Gurgul and Lach (2017) for most recent discussion on the topic and comparisons of widely used key-sector linkages). However, the new methodology of solving the dynamic endogenous input–output models with layers of techniques presented in this paper gives one opportunity to either forecast evolution of traditional measures of Rasmussen (1956), which is done in the illustrative empirical part of the paper, or forecast time paths of other widely used key-sector linkages recently reviewed by Temurshoev and Oosterhaven (2014).

  5. The introduction of depreciation rate, obsolescence rate and idle rate of capital extends the traditional open dynamic input–output model of Leontief. In general, the need for this type of extensions follows from the well-known problem of negative outputs that may occur in the solution of traditional (non-extended) dynamic input–output model. For more details on this issue see Leontief (1953), Duchin and Szyld (1985), ten Raa (1986) and Gurgul and Lach (2016).

  6. Comp. Griliches (1957), Grübler (1998), Köhler (2003), Pan (2006), Gurgul and Lach (2016), among others.

  7. See Gurgul and Lach (2016) for an extensive discussion on the advantages and disadvantages of using the logistic-curve-based model of technology change.

  8. A detailed description of the construction of matrices \(\mathbf{P}_t \) in (8) is presented in Gurgul and Lach (2016).

  9. \(\mathbf{X}_{j,OLD}^t \) stands for sector j’s total output by the old technical processes while \(\mathbf{X}_{j,NEW}^t \) denotes sector j’s total output by the new technical processes. Obviously \(X_j^t =X_{j,OLD}^t +X_{j,NEW}^t \).

  10. Henceforward \(\Delta \mathbf{X}^{0}=[\Delta X_1^0 ,\Delta X_2^0 ,\ldots ,\Delta X_n^0 ]\) and \(\mathbf{Y}_{INV}^0 =[Y_{1,INV}^0 ,Y_{2,INV}^0 ,\ldots ,Y_{n,INV}^0 ]\).

  11. Formula (13) ensures that \(\mathbf{B}_0 \Delta \mathbf{X}^{0}=\mathbf{Y}_{INV}^0 \), i.e. that the initial investment term in the dynamic IO model equals the initial gross fixed capital formation - the approximation of investments chosen by Gurgul and Lach (2016).

  12. It is important to underline that nowadays researchers usually have access to even highly disaggregated sector-specific data on investment published by central statistical offices. For example, in case of Poland such data has been published annually since 2004 by Central Statistical Office of Poland in reports titled Fixed Assets in National Economy (respective data may be accessed via the page: http://stat.gov.pl/en/topics/industry-construction-fixed-assets/fixed-assets). However, at the moment the intersectoral dependences are still not mapped out in the reports. This is why previous research (e.g. Gurgul and Lach (2016), among others) used arbitral approximations of the unknown structure of (square) capital matrix in dynamic IO model.

  13. This kind of data is available both in national statistical offices as well as in international agendas, like World Input–Output Database. The recently updated World Input–Output Database (WIOD November 2016 Release) consists of a series of detailed and reliable databases and covers 28 EU countries and 15 other major countries in the world for the period 2000–2014. For more details on the most recent WIOD database see Timmer et al. (2016).

  14. This way we assume that \({\bar{\mathbf{B }}}_0 \) is constant over the estimation period. This assumption reflects the well-known conviction that input–output coefficients are quite stable in the short-term (Pan 2006).

  15. As stressed by Pan (2006) the definition of output in sector j produced with old technology yields \(X_{j,OLD}^t =(1-d_{t-1}^j -\theta _t^j )X_j^{t-1} \).

  16. The chosen objective function has a clear economic background and was used in previous studies that were focused on solving dynamic endogenous IO models by minimizing the value of future idle capacities for all industries (Duchin and Szyld 1985; Edler and Rybakova 1992; Gurgul and Lach 2016, among others).

  17. We follow the arguments of Pan (2006) and assume that the input–output coefficients should change in accordance with technological progress in the long run. Taking these remarks into account, we decided to focus on a probable classification of the sectors of the Polish economy over the next four decades.

  18. Thus, such an analysis provides the opportunity to gain some information on the possible structural change of an economy as a consequence of the process of economic transformation.

  19. The development of The World Input Output Database is a long-awaited solution to the problem of the scarcity of international IO data. The latter seems to constitute necessary conditions for further growth of detailed multinational IO studies dealing with both analysis of historical data in modified and extended static IO models (see e.g. Luptáčik and Böhm 2010; Kovačić et al. 2015) as well as on scenario analysis in generalized dynamic IO models (comp. e.g. Dobos and Floriska 2008; Dobos and Tallos 2013; Gurgul and Lach 2016).

  20. The IO data on 2014 is the most recent available in WIOD 2016 Release.

  21. The sector-specific life spans (in years) were as follows: 30 (Construction), 10 (Manufacturing), 1 (Agriculture and Mining), 0.5 (remaining groups of sectors).

  22. To find more robust evidence supporting the hypothesis of significant improvement in short-run measures of goodness-of-fit of the new approach we solved the Optimization problem no 1 for two additional sets of IO data. We assumed the same estimation period and the same aggregation scheme as in the benchmark case but instead of the Polish data we used the WIOD data on two other countries. To minimize the risk of obtaining country-dependent results we chose two economies which significantly vary in terms of their levels of development, i.e. Germany and China. In both cases we compared the final values of the objective functions in Optimizationproblemno 1 with the initial values obtained for Brody’s (1966) and Gurgul and Lach’s (2016) specifications of capital matrices. In case of Germany the final value of the objective function was 5.7 (Brody’s 1966 specification) or 2.1 (Gurgul and Lach’s 2016 specification) times smaller than the starting value. In case of China it turned out that the final value of the objective function was 14.1 (Brody’s 1966 specification) or 9.9 (Gurgul and Lach’s 2016 specification) times smaller than the starting value.

  23. To model the sector-group-specific logistic curves we used the values of technical parameters suggested by Gurgul and Lach (2016). Similarly to Gurgul and Lach (2016), in the case of a lack of the required R&D expenditure data in the Eurostat database we used adequate data taken from the reports Science and technology in Poland published annually by the Central Statistical Office in Poland.

  24. Note that model (5) is nonlinear since it contains both technical coefficients and outputs as endogenous variables.

  25. The GAMS software is a powerful optimization tool with respect to both the inbuilt as well as user-modified solvers (see e.g. Ghezavati et al. 2017). Because of the specificity of GAMS summation and product notation the implementation of the model using the GAMS programing syntax is straightforward thanks to the product-based formulas (10) and (11).

  26. http://www.neos-server.org.

  27. A forecast time span of 40 years covers an entire Kondratieff cycle and thus a change in the list of key sectors is quite likely. Therefore, even before running the optimization procedures one may expect the results of our simulation to show some changes in the sectorial classification of the economy under study. However, an interesting and non-trivial question in this context is what this change will exactly likely look like. Giving (at least preliminary) answer to this important research question is one of the main goals of the simulation study.

  28. In Table 2 we present the detailed statistics obtained only for the dynamic model in which the initial capital matrix was approximated using the new approach described in Sect. 3.2, since—as we already mentioned—in this case the final objective function measuring the magnitude of error term in Optimizationproblemno 1 was several times smaller than the final objective obtained for the starting value based on Gurgul and Lach (2016) or Brody’s (1966) approximations. To see how the results differ due to the choice of the parameter estimation in the two last columns of Table 2 we additionally present the final sectorial classification obtained in the models in which the initial capital matrices were chosen on an arbitrary basis using Gurgul and Lach (2016) or Brody’s (1966) approach [these column names are “2054 (G–L)” and “2054 (B)”, respectively].

  29. Following Gurgul and Lach (2016) in the benchmark case we assumed a 5% nominal annual growth rate of non-investment final demand in all aggregated sectors of the Polish economy in the period forecasted.

  30. Note that Gurgul and Lach (2016) also used a different IO table for Poland (they used the input–output table derived from WIOD 2013 Release, i.e. a database with different sectorial divisions in comparison to its 2016 Release). The authors also used disaggregated IO tables.

  31. For the purpose of clarity of presentation in Fig. 1 we use tags for every second simulated data point. The shading area indicates the values of linkages which are below 1.

  32. The detailed results of all stages of sensitivity analysis are available from the authors upon request.

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Acknowledgements

We would like to thank the Editor of this journal, prof. Ulrike Leopold-Wildburger, and two anonymous Referees for valuable comments on earlier versions of the paper. Financial support for this paper from the National Science Centre of Poland (Research Grant No. DEC-2015/19/B/HS4/00088) is gratefully acknowledged.

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Gurgul, H., Lach, Ł. On approximating the accelerator part in dynamic input–output models. Cent Eur J Oper Res 27, 219–239 (2019). https://doi.org/10.1007/s10100-017-0502-y

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