Abstract
This paper analyzes the evolution of income elasticities of imports and verifies the validity original Thirlwall’s Law for eight Latin American countries overs the period 1961-2018. Given the econometric issues inherent to classical time-varying parameter regressions, a Bayesian estimation procedure is implemented in order to provide more robust parameter estimates. A stochastic volatility specification is also included to take into account the potential presence of conditional heteroskedasticity. While some Latin American economies showed a rather stable behavior, the income elasticities of imports for Brazil, Colombia and Mexico presented an upward trend from 1961 to 2018. Regarding the validity of original Thirlwall’s Law, there were evidences of the balance-of-payments (BOP) posing a restriction to the growth performance in all of the eight selected Latin American countries.
Keywords:
Import Elasticity; Thirlwall’s Law; Balance-of-Payments-Constrained Growth; Latin America; Bayesian Estimation; Time-Varying Parameter
Resumo
O objetivo desse estudo é analisar a evolução das elasticidades de importação e verificar a validade da versão original da Lei de Thirwall para oito países da América Latina durante o período de 1961 a 2018. Dadas as questões econométricas inerentes a regressões com parâmetros variantes no tempo, será utilizada uma abordagem bayesiana para a estimação. A fim de levar em consideração a presença de heterocedasticidade condicional, é incluída a especificação com volatilidade estocástica no modelo. Enquanto algumas economias da América Latina mostraram um comportamento estável, as elasticidades-renda de importação para o Brasil, Colômbia e México apresentaram uma tendência de elevação no período analisado. Em relação à validade da Lei de Thirwall, há evidências de que o balanço de pagamentos (BOP) coloca uma restrição à performance de crescimento em todos os oito países selecionados da América Latina.
Palavras-Chave:
Elasticidade de importação; Lei de Thirwall; crescimento restrito pelo balanço de pagamentos; América Latina; Estimação Bayesiana; Parâmetros tempo-variante
1. Introduction
As the world economy, Latin America has been the stage of both prosperity and recession periods. According to Lopes e Carvalho(2008), economic growth has been poor in Latin America for about three decades. From its economic collapse in the 1980s and, consequently, the later commitment to Neoliberal macroeconomic reforms, the extensive historical disequilibria in the balance-of-payments (BOP) faced by many developing countries have reinforced the need of evaluating the argument of whether international trade is de facto a constraint to economic performance.
Since the seminal paper by Thirlwall (1979Thirlwall, A. P. (1979), ‘Balance of Payments Constraint as an Explanation of International Growth Rate Differences’, Banca Nazionale del Lavoro Quarterly Review 32(128), 45-53.), which introduced the BOP-constrained growth framework, there has been a vast debate on its potential empirical applicability to economies throughout the world. While growth models of Neoclassical tradition rely on supply side issues such as factor accumulation, productivity increments or technological progress as driving forces of economic growth (Solow, 1956Solow, R. (1956), ‘A Contribution to the Theory of Economic Growth’, The Quarterly Journal of Economics 70(1), 65-94., 1957; Swan, 1956Swan, T. W. (1956), ‘Economic Growth and Capital Accumulation’, Economic Record 32(2), 334-361.; Romer, 1986Romer, D. (1986), ‘Increasing Returns and Long-Run Growth’, Journal of Political Economy 94(5), 1002-1037.; Lucas, 1988Lucas, R. E. (1988), ‘On the Mechanics of Economic Development’, Journal of Monetary Economics 22(1), 3-42.; Mankiw et al., 1992Mankiw, G., Romer, D. and Weil, D. (1992), ‘A Contribution to the Empirics of Economic Growth’, The Quarterly Journal of Economics 107(2), 407-437.; Aghion and Howitt, 1992Aghion, P. and Howitt, P. (1992), ‘A Model of Growth Through Creative Destruction’, Econometrica 60(2), 323- 351.), the former Post Keynesian approach postulates that output growth is demand-determined, with foreign exchange dynamics being crucial on explaining its sustainability. By embodying the assumption of trade deficits not being automatically corrected via adjustment of the real exchange rate, the presence of continuously increasing current account deficits would then hamper the growth of domestic output given the lack of also ever-growing financing sources. Hence, the insufficiency of foreign currency from trade imbalances would then set an upper limit to the growth rate of aggregate demand and, thus, to economic growth (Thirlwall and Hussain, 1982; McCombie, 1997McCombie, J. S. L. (1997), ‘On the Empirics of Balance-of-Payments-Constrained Growth’, Journal of Post Keynesian Economics 19(3), 345-375.).
Moreover, one should notice that the BOP-constrained growth approach has been criticized by disregarding the biases and inconsistent parameter estimates due to structural breaks. Since the study of Bairam (1993Bairam, E. (1993), ‘Static versus Dynamic Specifications and the Harrod Foreign Trade Multiplier’, Applied Economics 25(6), 739-742.), cointegration analysis has been the predominant econometric method adopted by the recent literature to evaluate the potential long-run relationship between economic growth and the BOP dynamics. However, given that this latter procedure requires a large sample size and that macroeconomic relations are likely to change over time, structural break problems have compromised the reliability of cointegration-based evidences (Bagnai, 2010Bagnai, A. (2010), ‘Structural Changes, Cointegration and the Empirics of Thirlwall’s Law’, Applied Economics 42(10), 1315-1329.).
Despite some recent attempts on dealing with these issues (Hieke, 1997Hieke, H. (1997), ‘Balance-of-Payments-Constrained Growth: A Reconsideration of the Evidences for the U.S. Economy’, Journal of Post Keynesian Economics 19(3), 313-325.; Atesoglu, 1997Atesoglu, H. S. (1997), ‘Balance-of-Payments-Constrained Growth Model and Its Implications for the United States’, Journal of Post Keynesian Economics 19(3), 327-335.; Bairam and Ng, 2001Bairam, E. and Ng, L. (2001), ‘Thirlwall’s Law and the Stability of Exports and Import Income Functions’, International Review of Applied Economics 15(3), 287-303.; Bértola et al., 2002Bértola, L., Higachi, H. and Porcile, G. (2002), ‘Balance-of-payments-constrained growth in Brazil: a test of Thirlwall’s Law, 1890-1973’, Journal of Post Keynesian Economics 25(1), 123-140.; Holland et al., 2004Holland, M., Vieira, F. and Canuto, O. (2004), ‘Economic growth and the balance-of-payments constraint in Latin America’, Investigación Económica LXIII, 45-74.; Pacheco-López and Thirlwall, 2006Pacheco-López, P. and Thirlwall, A. P. (2006), ‘Trade liberalization, the income elasticity of demand for imports, and growth in Latin America’, Journal of Post Keynesian Economics 29(1), 41-61.; Cimoli et al., 2010Cimoli, M., Porcile, G. and Rovira, S. (2010), ‘Structural change and the BOP-constraint: why did Latin America fail to converge?’, Cambridge Journal of Economics 34(2), 389-411.; Bagnai, 2010Bagnai, A. (2010), ‘Structural Changes, Cointegration and the Empirics of Thirlwall’s Law’, Applied Economics 42(10), 1315-1329.), the Post Keynesian empirical literature has yet to tackle parametric instability considering the whole sample period. According to McCombie (1997McCombie, J. S. L. (1997), ‘On the Empirics of Balance-of-Payments-Constrained Growth’, Journal of Post Keynesian Economics 19(3), 345-375.), the parameters associated to trade equations should evolve slowly over time. In general, classical recursive and rolling regression methods often incur statistical issues such as ad hoc window selection, loss of inference efficacy (mainly due to degrees of freedom restrictions) and biased parameter estimates. Therefore, the application of nonlinear Bayesian estimation methods is an appealing alternative. By amending the univariate regression framework to allow for parameters following a first-order random walk process, it is possible to capture smooth parametric shifts, thus being a more suitable procedure for describing time-varying economic dynamics.
This paper focuses on the evolution of income elasticities of imports in Latin America between 1961 and 2018 and verifies whether the BOP was indeed a constraint to the region’s economic growth. To this end, we first estimate time-varying parameter (TVP) univariate regressions with stochastic volatility for the Latin economies’ import functions, following closely the methodology procedure proposed by Nakajima (2011Nakajima, J. (2011), ‘Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Method-ology and Empirical Applications’, Monetary and Economic Studies 29, 107-142.) and Primiceri (2005Primiceri, G. (2005), ‘Time-Varying Structural Vector Autoregressions and Monetary Policy’, Review of Econo-metric Studies 72(3), 821-852.). The time-varying parameter specification provides time-varying income elasticity of imports estimates while the stochastic volatility takes into account the potential conditional heteroskedasticity that might arise from potential stochastic shocks over the volatility of the considered macroeconomic aggregates (Hamilton, 2010Hamilton, J. (2010), Macroeconomics and ARCH, in T. Bollerslev, J. R. Russell and M. Watson, eds, ‘Festschrift in Honor of Robert F. Engle’, Oxford University Press, pp. 79-96.). Based on these elasticity estimates, we then check the validity of Thirlwall’s Law for the region.
As to the contribution to the debate, to the best of our knowledge, this paper is the first application of a Bayesian approach to deal with potential parametric instability while estimating the income elasticities of imports in Latin America. Therefore, the novelty presented in our research is to verify the original Thirwall’s Law using a time varying parameter model and Bayesian inference. This would allow for more robust estimates and a better description of these elasticities’ movements over time. In fact, the results show that the volatility of imports has, in general, decreased from 1961 to 2018 in Latin America. Also, from the estimates for the time-varying income elasticities of imports, it is possible to divide the Latin American economies into two different groups, with the first one presenting a rather stable behavior over time, while the second one showed a clear upward trend. This increase is a repercussion of the region’s commitment to Neoliberal trade reforms. Regarding the validity of Thirlwall’s Law, we found favorable evidence for all the countries analyzed.
Besides this introduction and the concluding remarks, the paper is organized in four sections. First, Section 2 focuses on a historical overview of the trade developments in Latin America. Section 3 introduces the Post Keynesian theory of BOP-constrained growth. Section 4 describes the model structure and the estimation procedure of the TVP regression framework. Last, but not least, Section 5 presents the estimates for the time-varying income elasticities of imports, discussing their repercussions for the validity of Thirlwall’s Law in its seminal version and whether the selected economies were on a catching-up or falling behind trajectory in the period.
2. Trade Developments and Economic Growth in Latin America
From the 1950s, the development strategy in Latin America, mainly based on the adoption of import substitution policies, was responsible for reducing the openness and efficiency of the regional economies, increasing their external vulnerability, due to higher dependency on a narrow range of export products, and consequently hampering their ability to absorb external shocks (Rajagopal, 2005Rajagopal, D. (2005), ‘Where did the trade liberalization drive Latin American economy: a cross section analysis’, Applied Econometrics and International Development 6(2), 89-108.). Following the onset of the the oil crises in the 1970s, which considerably affected the developing economies that relied on the product, the external indebtedness faced in the 1980s produced large disparities of GDP per capita (Bleaney, 1999Bleaney, M. (1999), ‘Trade reform, macroeconomic performance and export growth in ten latin american coun-tries, 1979-95’, The Journal of International Trade & Economic Development: An International and Com-parative Review 8(1), 89-105.). In particular, the subsequent emergence of trade imbalances was a crucial element in deepening the economic growth slowdown and in the later commitment to Neoliberal trade reforms.
Recently, empirical research has provided evidences of a direct correlation between the Latin America’s external vulnerability and its rather volatile economic growth (Garret and Lange, 1986Garret, G. and Lange, P. (1986), ‘Performance in a hostile world: Economic growth in capitalist democracies, 1974-1982’, World Politics 38(4), 517-545.; Guillaumont, 2001Guillaumont, P. (2001), ‘On the economic vulnerability of low-income countries’, Working Paper 2001.02. CERDI.; Cordina, 2004Cordina, G. (2004), ‘Economic vulnerability and economic growth: some results from a Neo-Classical growth modelling approach’, Journal of Economic Development 29(2), 21-39.; Loayza and Raddatz, 2007Loayza, N. V. and Raddatz, C. (2007), ‘The Structural Determinants of External Vulnerability’, The World Bank Economic Review 21(3), 359-387.; Briguglio et al., 2008Briguglio, L., Cordina, G., Farrugia, N. and Vella, S. (2008), ‘Economic vulnerability and resilience - con-cepts and measurements’, WIDER Working Paper 2008.55. United Nations University - World Institute for Development Economics Research, Helsinki.; Seth and Ragab, 2012Seth, A. and Ragab, A. (2012), ‘Macroeconomic vulnerability in developing countries: approaches and issues’, Working Paper 94. Poverty Practice, Bureau for Development Policy, UNDP, Brasilia.). Indeed, as shown in Table 1, the period 1974-1986 presented a pronounced degree of GDP growth volatility in comparison to the period 1986-2007. In addition, the increase on the coefficient of variation in the last decade could be explained by the adverse effects of the 2008 financial crisis. Hence, these results suggest that the stability of Latin American growth rates has been intrinsically related to external financing fluctuations and commodity price cycles.
Despite the existence of distinct patterns, the overall Latin American trade volume has presented an upward trend since the mid 1970s, with an average growth rate of nearly 3.9% per year (Table 1). More specifically, the percentage of trade growth in the Latin American economies slightly increased from an average of 4.73% during 1961-1980 to an average rate of 5.02% during 1980-2000. On the other hand, the coefficient of variation declined sharply, posting a contraction from 2.29 to 0.78. Yet, in regard to the last decade, Latin American trade grew at a slower pace. With less than half the average rate observed historically - roughly 2% per year -, the growth of trade followed the economic downturn in the aftermath of recent global crisis. According to the International Monetary Fund (2016), the recent slowdown in global economic activity, mostly in investment, is responsible for up to three-fourths of the trade deceleration. Furthermore, the end of the 1986-2007 boom also exposed many primary-exporting countries in the region to sizable terms of trade shocks, with the correction in commodity prices resulting in an acute decrease in the value of Latin America’s trade.
In terms of development strategy, the shift from the import substitution paradigm toward the adoption of the export-led growth framework in the late 1970s placed trade integration as the cornerstone of the Latin American industrialization process. Trade had become the catalyst of economic growth. The structural adjustment under the neoclassical prescription essentially relied on liberalization reforms, financial deregulation and dismantling of state intervention. While Latin America benefited from increased trade openness in the mid 1980s and early 1990s, its level remained rather stable thereafter. For instance, the region’s degree of openness increased from 21% in 1983 to 45% in 2007 (Figure 1). This result suggests that trade reforms implemented in Latin America broadened the integration of the region in the international market. Also, from visual inspection of Figure 1, one can conclude that the degree of openness seems to have mirrored the behavior of Latin America’s GDP, particularly after the early 1980s. However, in comparison to other emerging and developed economies, the relatively low average trade integration in Latin America could be partially explained by unrelated trade factors, such as geography, high transport costs and the weak quality of institutions (De Ferranti et al., 2003De Ferranti, D., Lederman, D., Perry, G. and Suescún, R. (2003), ‘Trade for Development in Latin America and the Caribbean’, Trade Note 2003.10. World Bank, Washington, D.C.).
Even though several theoretical and empirical studies have propelled the positive impacts of openness on economic performance (Dollar, 1992Dollar, D. (1992), ‘Outward-oriented developing economies really do grow more rapidly: evidence from 95 LDCs, 1976-1985’, Economic Development and Cultural Change 40(3), 523-544.; Ben-David, 1993Ben-David, D. (1993), ‘Equalizing Exchange: Trade Liberalization and Income Converge’, Quarterly Journal of Economics 108(3), 653-679.; Krugman, 1994Krugman, P. (1994), ‘The Myth of Asia’s Miracle’, Foreign Affairs 73(6), 62-78.; Sachs et al., 1995Sachs, J. D., Warner, A., Aslund, A. and Fischer, S. (1995), ‘Economic Reform and the Process of Global Integration’, Brookings Papers on Economic Activity 26(1), 1-118.; Edwards, 1998Edwards, S. (1998), ‘Openness, Productivity and Growth: What do We Really Know?’, The Economic Journal 108(447), 383-398.; Frankel and Romer, 1999Frankel, J. A. and Romer, D. (1999), ‘Does trade cause growth?’, American Economic Review 89(3), 379-399.; Lucas, 2009Lucas, R. E. (2009), ‘Trade and the Diffusion of the Industrial Revolution’, American Economic Journal: Macroeconomics 1(1), 1-25.; Kim et al., 2012Kim, D., Lin, S. and Suen, Y. (2012), ‘Dynamic Effects of Financial Openness on Economic Growth and Macroeconomic Uncertainty’, Emerging Markets Finance & Trade 48(1), 25-54.), the debate has yet to be settled. From a structural Keynesian viewpoint, the export-led development would be ineffective in generating stable economic linkages and promoting genuine prosperity (Palley, 2002Palley, T. I. (2002), ‘A new development paradigm: domestic demand-led growth’, Discussion Paper - September.Foreign Policy in Focus (FPIF), Washington, D.C., 2004). Moreover, the restructuring of the productive system toward an export-oriented industrialization would be more subject to adverse external shocks in contrast to development strategies reliant on domestic demand (Foxley, 2009Foxley, A. (2009), Recovery: The Global Financial Crisis and Middle-Income Countries, Carnegie Endowment for International Peace, Washington, D.C.). Therefore, despite the upward path of Latin America’s trade performance engendered by its fairly commitment to the export-led growth paradigm, the resulting increase in its external vulnerability poses an additional predicament to the region. Consequently, in light of this pronounced sensitivity to international conditions, the recent end of the high commodity prices cycle pari passu the Chinese economic slowdown are likely to restrain even further its already stagnant economic performance. Ergo, as discussed by Ocampo (2013Ocampo, J. A. (2013), The history and challenges of Latin American development, United Nations Publication, Santiago.), rethinking Latin America’s development patterns and strategies is of major importance in order to promote social and economic welfare.
3. The Balance-of-Payments-Constrained Growth Model Revisited
Unless a country can finance ever-growing current account deficits, the economic performance of a given economy cannot grow faster than the rate consistent with BOP equilibrium (Thirlwall, 1979Thirlwall, A. P. (1979), ‘Balance of Payments Constraint as an Explanation of International Growth Rate Differences’, Banca Nazionale del Lavoro Quarterly Review 32(128), 45-53.). In the presence of recurrent trade imbalances and the relative scarcity of financing sources, the resulting insufficiency of foreign currency would then set an upper limit to the growth rate of aggregate demand and, thus, to economic growth.
Let the current account equilibrium condition be given by
where is the domestic price of exports; is exports; is the foreign price of imports; is imports; and is the nominal exchange rate measured as the domestic price of foreign currency. The export and import demand functions are defined as
and
where and are constants; is world income; is domestic income; is the price elasticity of exports; is the income elasticity of exports; is the price elasticity of imports; and is the income elasticity of imports. Note that this seminal model structure formalized by [71] specified elasticities as constant quantities over time.
After transforming equations (1) to (3) into their respective growth rates and solving the subsequent system of linear equations for the domestic income growth rate (), it is possible to define the economic growth compatible with BOP equilibrium (), that is,
with lower-case letters corresponding to the growth rates of the variables. The time subscripts were dropped for notational convenience. From the relation depicted by equation 4, an improvement in the real terms of trade, ceteris paribus, would increase the growth compatible with BOP equilibrium if the sum of the price elasticities of exports and imports is less than . Yet, in the presence of the Marshall-Lerner condition, , the deterioration in competitiveness due to an improvement in the real terms of trade would then decrease the growth compatible with BOP equilibrium. Also, only persistent depreciations of the exchange rate (i.e. ) would lead to a permanently higher growth trajectory for a given country. Last, but not least, there exists an inverse relation between the income elasticity of imports and the growth compatible with BOP equilibrium.
Following McCombie and Thirwall (1994McCombie, J. S. L. and Thirwall, A. P. (1994), Economic Growth and the Balance-of-Payments Constraint, 1 edn, St. Martin’s Press, New York.), if relative prices in international trade are neutral in the long-run (i.e. ), thus not presenting significant effects on the growth compatible with BOP equilibrium path of a given country, and if capital flows cannot finance current account deficits, equation 4 reduces to
This last equation has come to be known as Thirlwall’s Law. However, as discussed by Thirlwall (2011Thirlwall, A. P. (2011), ‘Balance of payments constrained growth models: history and overview’, PSL Quarterly Review 64(259), 307-351.) himself, this relationship and its implications were built on the previous theoretical macroeconomic developments of Harrod’s foreign trade multiplier (Harrod, 1933Harrod, R. (1933), International Economics, 1 edn, Cambridge University Press, CambriHicks, J. R. (1950), A Contribution to the Theory of the Trade Cycle, 1 edn, Clarendon Press, Oxford.), Hicks’ super-multiplier (Hicks, 1950) and the Prebisch-Singer hypothesis (Prebisch, 1959Prebisch, R. (1959), ‘Commercial Policy in the Underdeveloped Countries’, American Economic Review 49(2), 251-273., 1962; Singer, 1950).
Several empirical studies have been conducted in order to validate the applicability of Thirlwall’s Law for both developed and emerging economies (Thirlwall and Hussain, 1982Thirlwall, A. P. and Hussain, M. N. (1982), ‘The Balance of Payments Constraint, Capital Flows and Growth Rate Differences between Developing Countries’, Oxford Economic Papers 34(3), 498-510.; Bairam, 1988Bairam, E. (1988), ‘Balance of payments, the Harrod trade multiplier and economic growth: the European and North American experience 1970-1985’, Applied Economics 20(12), 1635-1642.; Atesoglu, 1993Atesoglu, H. S. (1993), ‘Balance of payments constrained growth: evidence from the United States’, Journal of Post Keynesian Economics 15(4), 507-514., 1997; Mc-Combie, 1997; Hieke, 1997Hieke, H. (1997), ‘Balance-of-Payments-Constrained Growth: A Reconsideration of the Evidences for the U.S. Economy’, Journal of Post Keynesian Economics 19(3), 313-325.; Alonso, 1999Alonso, J. A. (1999), ‘Growth and the external constraint: lessons from the Spanish case’, Applied Economics 31(2), 245-253.; Moreno-Brid, 1998Moreno-Brid, J. C. (1998), ‘On Capital Flows and the Balance of Payments Constrained Growth Model’, Journal of Post Keynesian Economics 21(2), 283-289., 1999; Bértola et al., 2002Bértola, L., Higachi, H. and Porcile, G. (2002), ‘Balance-of-payments-constrained growth in Brazil: a test of Thirlwall’s Law, 1890-1973’, Journal of Post Keynesian Economics 25(1), 123-140.; Razmi, 2005Razmi, A. (2005), ‘Balance-of-Payments-Constrained Growth Model: The Case of India’, Journal of Post Keynesian Economics 27(4), 655-687.; Kvedaras, 2005Kvedaras, V. (2005), ‘Explanation of economic growth differences in the CEE countries: importance of Bop constraint’, Baltic Journal of Economics 5(2), 48-65.; Garcimartín et al., 2008Garcimartín, C., Rivas, L. A. and De Sarralde, S. D. (2008), ‘Accounting for Irish growth: a balance-of-payments-constraint approach’, Journal of Post Keynesian Economics 30(3), 409-433.; Britto and McCombie, 2009Britto, G. and McCombie, J. S. L. (2009), ‘Thirlwall’s Law and the Long-Term Equilibrium Growth Rate: An Application to Brazil’, Journal of Post Keynesian Economics 32(1), 115-136.; Felipe et al., 2010Felipe, J., McCombie, J. S. L. and Naqvi, K. (2010), ‘Is Pakistan’s growth rate balance of payments constrained? Policies and implications for development and growth’, Oxford Development Studies 38(4), 477-496.; Bagnai, 2010Bagnai, A. (2010), ‘Structural Changes, Cointegration and the Empirics of Thirlwall’s Law’, Applied Economics 42(10), 1315-1329.). In general, the obtained results have confirmed the validity of the BOP-constrained growth framework.
For the specific case of Latin America, López and Cruz (2000López, J. and Cruz, A. (2000), ‘Thirlwall’s Law and Beyond: The Latin American Experience’, Journal of Post Keynesian Economics 22(3), 477-495.) incorporated the real exchange rate in the model in order to analyze its effect on the long-term economic growth of Argentina, Brazil, Colombia and Mexico. Their results supported the idea that, in the presence of the Marshall-Lerner condition, faster rate of output growth could be achieved by adopting competitive exchange rate levels.
Additionally, with a sample of ten Latin American economies, Holland et al. (2004Holland, M., Vieira, F. and Canuto, O. (2004), ‘Economic growth and the balance-of-payments constraint in Latin America’, Investigación Económica LXIII, 45-74.) tested the adherence of the predicted growth rate compatible with BOP equilibrium to the actual long-term growth rates. Despite the presence of some deviation, the results corroborated the argument of the BOP posing a restriction to the region’s long-term economic growth.
Pacheco-López and Thirlwall (2006Pacheco-López, P. and Thirlwall, A. P. (2006), ‘Trade liberalization, the income elasticity of demand for imports, and growth in Latin America’, Journal of Post Keynesian Economics 29(1), 41-61.) investigated whether the adoption of trade liberalization by the Latin American economies had increased their income elasticity of imports. The authors implemented rolling regres-sion models in order to capture a potential upward trend over time. Besides proving the validity of Thirlwall’s Law, their results also confirmed this trend for the country sample.
Similarly, using rolling regressions with panel data for a broad sample of developed and emerging economies, Cimoli et al. (2010Cimoli, M., Porcile, G. and Rovira, S. (2010), ‘Structural change and the BOP-constraint: why did Latin America fail to converge?’, Cambridge Journal of Economics 34(2), 389-411.) also showed that the Latin American income elasticities of imports have increased since 1970. Along with those results, the authors provided further evidences that the decrease in the income gap of some developing countries was intrinsically related to the transformation of their economic structures.
Moreover, based on a multi-sectoral perspective (see Araujo and Lima (2007Araujo, R. A. and Lima, G. T. (2007), ‘A structural economic dynamics approach to balance-of-payments-constrained growth’, Cambridge Journal of Economics 31(5), 755-774. URL: https://doi.org/10.1093/cje/bem006
https://doi.org/10.1093/cje/bem006...
) from a theoretical perspective), Gouvea and Lima (2010Gouvea, R. R. and Lima, G. (2010), ‘Structural change, balance of payments constraint and economic growth: evidence from the multi-sectoral Thirlwall’s Law’, Journal of Post Keynesian Economics 33(1), 169-204.) evaluated the implications of changes in the sectoral composition of exports and imports on the external constraint for a sample of Latin American and Asian countries. Both the original and the multisectoral versions of Thirlwall’s law were validated for Latin America.
4. The Time-Varying Parameter (TVP) Regression Model
A basic TVP regression model can be defined as
with and . Note that is a scalar of response; and are and vectors of covariates, respectively; is a vector of time-invariant coefficients; and is a vector of time-varying coefficients. In this paper, the regression model is assumed to have only time-varying coefficients (i.e. ).
As to the evolution process of , the parameters are assumed to follow a driftless first-order random walk specification, given by
with and . We assume that and . This specification, in contrast to Markov-switching models, allows for both temporary and permanent shifts instead of discrete breaks, thus being a more suitable characterization of potential changes in private sector behavior or the learning dynamics of economic agents. More specifically, the time-varying coefficient is imposed as to capture a possible nonlinearity, such as a gradual change or a structural break (Primiceri, 2005Primiceri, G. (2005), ‘Time-Varying Structural Vector Autoregressions and Monetary Policy’, Review of Econo-metric Studies 72(3), 821-852.; Nakajima, 2011Nakajima, J. (2011), ‘Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Method-ology and Empirical Applications’, Monetary and Economic Studies 29, 107-142.).
The stochastic volatility specification is included by imposing a log-volatility process for the time-varying variance , that is,
with ; ; and . Note that is the stochastic volatility. Moreover, we assume that ; ; ; and the initial condition is set based on the stationary distribution as .
Consider and as state variables. Then, the TVP regression model consists of a nonlinear non-Gaussian state space representation. Primiceri (2005Primiceri, G. (2005), ‘Time-Varying Structural Vector Autoregressions and Monetary Policy’, Review of Econo-metric Studies 72(3), 821-852.) argues that the Maximum Likelihood (ML) estimation cannot provide reliable estimates for this class of models due to intractability of the likelihood function. Therefore, the estimation procedure is mainly based on a Bayesian approach using the Markov Chain Monte Carlo (MCMC) method for the numerical evaluation of the posterior of the parameters.
By splitting up the original problem into a number of smaller steps, the Bayesian inference is able to deal with high-dimensional parameter space and potential nonlinearities in the likelihood function. Under the assumption of a certain prior probability density, , the MCMC algorithm is able to generate the joint posterior distribution of the parameters, , given by
where is the data; is the marginal distribution (also known as the normalizing constant); and is the likelihood function. In other words, the Bayesian inference consists of updating the prior information on by observing . However, since the posterior distribution usually does not have a closed form due to the intractability of the likelihood function and to the impossibility of an analytical assessment for the marginal distribution, the MCMC sampling methods are fairly robust algorithms capable of recursively sampling from the posterior distribution without computing them (Nakajima, 2011Nakajima, J. (2011), ‘Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Method-ology and Empirical Applications’, Monetary and Economic Studies 29, 107-142.).
Regarding the TVP regression model, after defining the prior density , one can obtain the joint posterior distribution , that is,
In this paper, the implementation of the MCMC algorithm to explore this posterior distribution follows the procedure developed by Nakajima (2011Nakajima, J. (2011), ‘Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Method-ology and Empirical Applications’, Monetary and Economic Studies 29, 107-142.), which is presented in the appendix.
5. Time-Varying Income Elasticity of Imports and Economic Growth in Latin America
Following the import function defined by Thirlwall (1979Thirlwall, A. P. (1979), ‘Balance of Payments Constraint as an Explanation of International Growth Rate Differences’, Banca Nazionale del Lavoro Quarterly Review 32(128), 45-53.), the TVP regression model is applied in order to obtain time-varying income elasticities of imports for eight Latin American countries1, namely, Bolivia, Brazil, Chile, Colombia, Ecuador, Mexico, Peru and Uruguay. In 2015, these countries represented together nearly 81% of Latin America GDP (at constant 2010 prices, in USD) (ECLAC, 2017). Hence, the time-varying econometric model for each country is defined as
where is the import growth rate; is the domestic output growth rate; is the real effective exchange rate (REER) growth rate; and . This is a linear in growth rates model, whose parameters are interpreted as elasticities1 1 In a study on measurement issues in estimating elasticity values, Casler (2015) found that the linear in growth rate regression specification provides more accurate estimates than the commonly used linear, logarithmic, logarithmic-linear and linear-logarithmic specifications. Furthermore, the results obtained by the author also advocates in favor of the natural logarithmic growth rate method in comparison to the standard base, the reverse base and the geometric-mean growth rate methods. . As discussed in Section 4, the time-varying parameters and follow a driftless random walk process, that is, , with and . Also, the time-varying variance is defined as a log-volatility process, so that , where and . Note that is the stochastic volatility.
Annual data on Gross Domestic Product (GDP), imports and exports of goods and services were obtained from the World Development Indicators (WDI) database, maintained by the World Bank (2020). These data were extracted at constant 2010 prices, in USD. As to the annual REER index, the data were collected from the database developed by Darvas (2012Darvas, Z. (2012), ‘Real effective exchange rates for 178 countries: a new database’, Working Paper 2012/06. Bruegel Institute, Belgium.), with its base changed to 2010 Last update: February 2020. The growth rates were constructed as the natural logarithm of the ratio of the period t and t − 1 values. The growth rate dataset spans from 1961 to 2018.
With respect to the choice of prior distributions, this paper follows closely the default values proposed in Nakajima (2011Nakajima, J. (2011), ‘Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Method-ology and Empirical Applications’, Monetary and Economic Studies 29, 107-142.):
These could be considered rather diffuse priors, which contribute to mitigate potential identification issues. Moreover, in order to compute the posterior estimates, the TVP regression model for each country is estimated by drawing 100,000 samples after the initial 10,000 samples were discarded in the burn-in period.
Autocorrelation and convergence are common issues associated to MCMC sampling methods. However, as shown from figures A.1 to A.7 in Appendix B, the sampling method efficiently produces uncorrelated samples, since the sample paths look stable and the sample autocorrelations drop stably for all countries. Also, from tables B.1 to B.8 in Appendix C, the convergence diagnosis (CD) and the inefficiency factors (IF) of Geweke (1992Geweke, J. (1992), Evaluating the accuracy of sampling-based approaches to the calculation of posterior mo-ments, in J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds, ‘Bayesian Statistics’, Oxford University Press, pp. 169-188.) confirm that, for the eight considered countries, the null hypothesis of convergence to the posterior distribution is not rejected for the parameters and state variables at the 10% significance level and that the sampling method is efficient, since the IF values are rather low.
Figure 2 plots the posterior estimates for the stochastic volatility of the Latin American countries. In general, the volatility of imports has decreased from red1961 to 2018 for all eight countries. More specifically, the estimates for Brazil, Chile, Ecuador, Peru and Uruguay show that the variance of their imports decreased after the mid 1980s, however slightly increasing after 2008. These evidences are in accordance with the stabilization of international trade after the 1970s and early 1980s oil crises and the uncertainty engendered by the recent financial crisis, respectively. On the other hand, regarding Bolivia and Mexico, the instability of their imports from the 1970s to the early 1980s is relatively higher than the other considered Latin American economies. Yet, specially in Mexico, the stochastic volatility of imports has steadily decreased after the mid 1980s. In the case of Colombia, the increase on the variance of imports in the 1990s is in line with the Colombian trade liberalization policies enacted during the period.
As the variance of imports depicts a reasonably pronounced time-varying behavior from 1961 to 2018, one should also expect time-variation on the estimates for the Latin American income elasticities of imports. In fact, as shown in Figure 11, the estimates for all eight countries presented some degree of time-variation. With exception of Chile, it is possible to combine the considered Latin American economies into two different groups based on the evolution pattern of their income elasticities of imports. The first group, consisting of Bolivia, Ecuador, Peru and Uruguay, showed rather stable estimates over time, with small changes in the early 1980s and mid 1990s. These results are close to those found by Pacheco-López and Thirlwall (2006Pacheco-López, P. and Thirlwall, A. P. (2006), ‘Trade liberalization, the income elasticity of demand for imports, and growth in Latin America’, Journal of Post Keynesian Economics 29(1), 41-61.). As to the second group, consisting of Brazil, Colombia and Mexico, there has been an upward trend in their estimates. Following the tradition set out by Thirlwall (1979), these evidences would suggest that the rate of economic growth compatible with the BOP equilibrium has decreased for these countries. For instance, the recent studies by Pacheco-López and Thirlwall (2006) and Cimoli et al. (2010Cimoli, M., Porcile, G. and Rovira, S. (2010), ‘Structural change and the BOP-constraint: why did Latin America fail to converge?’, Cambridge Journal of Economics 34(2), 389-411.) support these results, with the authors arguing that the increase in the income elasticities of imports in Latin America is highly correlated to the region’s commitment to Neoliberal trade liberalization policies, mainly in the form of tariff reductions, import quotas and licenses. Finally, in the case of Chile, despite its estimates being relatively volatile through the sample period, the country has recently shown income elasticity of imports levels similar to those from the beginning of the sample.
Even though these evidences are in favor of the Post Keynesian theory of BOP-constrained growth, the validity of the seminal model developed by Thirlwall (1979Thirlwall, A. P. (1979), ‘Balance of Payments Constraint as an Explanation of International Growth Rate Differences’, Banca Nazionale del Lavoro Quarterly Review 32(128), 45-53.) has yet to be properly tested. Following the strategy suggested by McCombie (1997McCombie, J. S. L. (1997), ‘On the Empirics of Balance-of-Payments-Constrained Growth’, Journal of Post Keynesian Economics 19(3), 345-375.), the objective is to statistically test whether the observed growth rate () and the theoretical rate of economic growth with BOP equilibrium () are different. If so, the result would then advocate against the validity of Thirlwall’s Law. Note that is computed based on equation 5. The method proposed by McCombie (1997) consist of the OLS (ordinary least square) specification to be tested for each country:
where is a white noise disturbance.
The Thirlwall’s Law is validated if the parameter is not statistically different from the unity and the parameter is not statistically different from zero. This is equivalent to:
Thus, according to Felipe et al. (2019Felipe, J., Lanzafame, M. and Estrada, G. (2019), ‘Is indonesia’s growth rate balance-of-payments- constrained? a time-varying estimation approach’, Review of Keynesian Economics 7(4), 557-553.) to test the Thirlwall’s Law, we can perform a unit root test in the series . The unit root test will allow assessing whether the the series are stationary, that is, mean-reverting processes. In addition, we need to verify that the variable is a zero-mean process. This can be tested by modelling an autoregressive process for the and test if the intercept is equal to zero. In summary, the approach proposed by Felipe et al. (2019) consists of testing the two conditions below:
-
mean-reverting process by unit root test. For the Thirlwall’s Law to be supported by the data, the null hypothesis of unit root should be rejected at the usual significance levels.
-
zero-mean process by modelling as autoregressive process and test if intercept is equal to zero. For the Thirlwall’s Law to be supported by the data, the null hypothesis that intercept is equal to zero should not be rejected at the usual significance levels.
In relation to the unit root tests, we will apply the standard Augmented Dickey-Fuller (ADF), Phillips-Perron and a robust ADF version to structural breaks. (Hamilton (2010Hamilton, J. (2010), Macroeconomics and ARCH, in T. Bollerslev, J. R. Russell and M. Watson, eds, ‘Festschrift in Honor of Robert F. Engle’, Oxford University Press, pp. 79-96.)).
In order to verify zero mean process, the following autoregressive process will be estimated (Felipe et al.(2019Felipe, J., Lanzafame, M. and Estrada, G. (2019), ‘Is indonesia’s growth rate balance-of-payments- constrained? a time-varying estimation approach’, Review of Keynesian Economics 7(4), 557-553.)):
Table 2 presents the estimation results. For all countries and unit root tests performed, the null hypothesis of unit root was rejected at the 1% significance level. On the other hand, apart Colombia and Mexico, the null hypothesis of θ being statistically equal to the zero is not rejected at the usual significance level. Note that, in the case of Mexico and Colombia, the null hypothesis is not rejected at the 5% significance level. Hence, the obtained results confirm the validity of Thirlwall’s Law for all of the country sample, which implies that the BOP has indeed posed a restriction to their economic performance from 1961 to 2018. Therefore, the results corroborate the conclusion of Holland et al. (2004Holland, M., Vieira, F. and Canuto, O. (2004), ‘Economic growth and the balance-of-payments constraint in Latin America’, Investigación Económica LXIII, 45-74.): no single economy is immune from its external sector constraint". For the case of Brazil, the evidence is in line with that found by JAYME JR (2003Jayme Jr, F. G. (2003), ‘Balance-of-payments-constrained economic growth in Brazil’, Brazilian Journal of Political Economy 23, 63 - 86. URL: http://www.scielo.br/scielo.php?script=sciarttextpid = S0101 − 31572003000100063nrm = iso
http://www.scielo.br/scielo.php?script=s...
) that tested Thirlwal’s Law using annual data from 1955 to 1998 and cointegration techniques and more recently by Ribeiro (2019Ribeiro, R. (2019), Balance-of-Payments Constrained Growth Model: The Brazilian Case over the Period 1980-2011, pp. 265-288.) from 1980 to 2010.
To sum up, the evidence obtained supports that for all countries are zero-mean stationary processes. Thus, the actual growth in Latin America tends to be equal to the BoPC growth rate on average and the short-term divergences between the two rates do not last in the long run.
One should note that the present approach to the BOP-constrained growth model has only considered the seminal model of Thirlwall (1979Thirlwall, A. P. (1979), ‘Balance of Payments Constraint as an Explanation of International Growth Rate Differences’, Banca Nazionale del Lavoro Quarterly Review 32(128), 45-53.). Therefore, these results should be considered as a preliminary investigation for the selected Latin American economies. Yet, the novelty of these results is associated to the use of a robust Bayesian econometric framework to estimate time-varying income elasticities of imports, controlling for stochastic volatility.
Furthermore, given the long-term nature of the theoretical rates of economic growth with BOP equilibrium, these estimates are also considered a measure for evaluating whether an economy has been in a catching-up trajectory relative to the world economy (Nassif et al., 2015Nassif, A., Feijó, C. A. and Araújo, E. (2015), ‘Structural change and economic development: is Brazil catching up or falling behind?’, Cambridge Journal of Economics 39(5), 1307-1332., 2016). Table 3 reports the average BOP-constrained growth rate for the selected Latin American economies as well as the average world growth rate. The World GDP data (at constant 2010 prices, in USD) were retrieved from the WDI database (World Bank, 2020).
With the exception of Peru, there are no evidences of a catching-up process by all other countries from 1961 to 2018. On the other hand, there are evidences of a catching-up process by Brazil and Mexico from 1970-1985. Yet, between 1985 and 2000, the estimates show that the Bolivian economy embarked on a trajectory of catching up, which continued during the period 2000-2015. When considering the last eighteen years (2000-2018), this catching-up process is only observed in Bolivia, Peru and Uruguay. Following Ocampo (2013Ocampo, J. A. (2013), The history and challenges of Latin American development, United Nations Publication, Santiago.), this cyclical process of catching-up and falling behind could be described as one of truncated convergence. Given the international trade slowdown and the recent economic crises, the observed trajectory shift by the Latin American countries could be the result of the fragility of their economies to external imbalances. However, one should note that the average BOP-constrained growth rate for Brazil in the period has relatively decreased in comparison to its past economic performance, indicating a trajectory of falling behind, which is in accordance with evidence found by Nassif et al. (2015Nassif, A., Feijó, C. A. and Araújo, E. (2015), ‘Structural change and economic development: is Brazil catching up or falling behind?’, Cambridge Journal of Economics 39(5), 1307-1332.) and Nassif et al. (2016). According to Ribeiro (2019Ribeiro, R. (2019), Balance-of-Payments Constrained Growth Model: The Brazilian Case over the Period 1980-2011, pp. 265-288.), the slow growth of labour productivity and lack of technological innovations and structural changes may have imposed severe supply restrictions to the growth of Brazilian economy, despite of the commodities boom in 2000s. Last, but not least, the results also highlight the fact that Uruguay remained on a falling behind trajectory until the 2000’ and in the last eighteen years embarked on a trajectory of catching up.
6. Concluding Remarks
Economic growth in Latin America has been an intriguing phenomenon, driving a vast variety of theoretical and empirical research. Given the recurrence of trade imbalances during its economic history, the Post Keynesian theory of BOP-constrained growth has emerged as a robust theoretical alternative regarding the potential limits of the region’s long-term economic performance. Several studies have tried to provide empirical evidences supporting these ideas (see e.g. Thirlwall and Hussain (1982Thirlwall, A. P. and Hussain, M. N. (1982), ‘The Balance of Payments Constraint, Capital Flows and Growth Rate Differences between Developing Countries’, Oxford Economic Papers 34(3), 498-510.), McCombie (1997McCombie, J. S. L. (1997), ‘On the Empirics of Balance-of-Payments-Constrained Growth’, Journal of Post Keynesian Economics 19(3), 345-375.), Holland et al. (2004Holland, M., Vieira, F. and Canuto, O. (2004), ‘Economic growth and the balance-of-payments constraint in Latin America’, Investigación Económica LXIII, 45-74.), Pacheco-López and Thirlwall (2006Pacheco-López, P. and Thirlwall, A. P. (2006), ‘Trade liberalization, the income elasticity of demand for imports, and growth in Latin America’, Journal of Post Keynesian Economics 29(1), 41-61.) and Cimoli et al. (2010Cimoli, M., Porcile, G. and Rovira, S. (2010), ‘Structural change and the BOP-constraint: why did Latin America fail to converge?’, Cambridge Journal of Economics 34(2), 389-411.)).
In general, however, the Post Keynesian research has overlooked econometric issues related to structural breaks and parameter instability over the sample period, compromising the reliability of the estimates due to biases and inconsistency. Despite some attempts on dealing with those issues (see e.g. Atesoglu (1997Atesoglu, H. S. (1997), ‘Balance-of-Payments-Constrained Growth Model and Its Implications for the United States’, Journal of Post Keynesian Economics 19(3), 327-335.), Bairam and Ng (2001Bairam, E. and Ng, L. (2001), ‘Thirlwall’s Law and the Stability of Exports and Import Income Functions’, International Review of Applied Economics 15(3), 287-303.), Bagnai (2010Bagnai, A. (2010), ‘Structural Changes, Cointegration and the Empirics of Thirlwall’s Law’, Applied Economics 42(10), 1315-1329.)), the estimation of time-varying parameters considering the whole sample is a subject that has yet to be properly explored. This paper intended to contribute to the debate on this mat-ter. Combining nonlinear Bayesian techniques and classical OLS estimations, TVP regressions with stochastic volatility for the import functions of eight Latin American economies were implemented and the respective validity of Thirlwall’s Law was tested.
For some Latin American economies, namely, Brazil, Colombia and Mexico, the obtained results showed that there has been an upward trend in their estimates of the income elasticity of imports from 1970 to 2015. On the other, for the second group of countries, consisting of Bolivia, Ecuador, Peru and Uruguay, a rather stable behavior was observed. In the particular case of Chile, even though is estimates were relatively volatile over time, the country has recently presented income elasticity of import demand levels similar to those from beginning of the sample. Moreover, the BOP-constrained model developed by Thirlwall (1979Thirlwall, A. P. (1979), ‘Balance of Payments Constraint as an Explanation of International Growth Rate Differences’, Banca Nazionale del Lavoro Quarterly Review 32(128), 45-53.) was validated for the period. This corroborates the idea of trade imbalances effectively limiting the Latin American economic performance for the last 50 years. As to a potential catching-up process in the region, the estimates for the theoretical rate of economic growth with BOP equilibrium not supported the idea of an overall catching-up trajectory for the region. Yet, one should note that the Bolivian economy has reversed its falling behind trajectory, while the Brazilian growth has relatively decreased in comparison to its past performance. On a side note, the results also highlighted the persistence of a falling behind trajectory by the Uruguayan economy for the whole period. Uruguay remained on a falling behind trajectory until the 2000’ and in the last eighteen years embarked on a trajectory of catching up.
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This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior Brasil (CAPES) - Finance Code 001. All errors remain ours.
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I thank the financial support from the National Council for Scientific and Technological Development - CNPq - Brazil.
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Would like to thank CNPq for financial support.
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JEL Classification
C11; E12; F43; O47; O54
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1
In a study on measurement issues in estimating elasticity values, Casler (2015) found that the linear in growth rate regression specification provides more accurate estimates than the commonly used linear, logarithmic, logarithmic-linear and linear-logarithmic specifications. Furthermore, the results obtained by the author also advocates in favor of the natural logarithmic growth rate method in comparison to the standard base, the reverse base and the geometric-mean growth rate methods.
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2
The objective of the accept-reject (AR) sampling, also known as the von Neumann sampling, is to sample from a target density , with being an unnormalized target density and an unknown normalizing constant. However, this target distribution might be impossible to simulate by an inverse transformation. Hence, suppose it is possible to sample from the density and that there exists a constant c for which for all . In order to obtain a draw from , the AR sampling proceeds to draw a candidate from and from the uniform distribution . If , is accepted. Otherwise, the process is repeated.
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3
Given the impossibility of direct sampling from the posterior distribution, the Metropolis-Hastings (MH) algorithm is a MCMC method implemented in order to simulate samples by combining the use of the full joint density function and a proposal distribution. This proposal distribution is usually chosen as to correspond as close as possible to the target conditional posterior distribution. More specifically, define as the probability density function of the proposal given the current point and as the acceptance rate from the current point to the proposal , which is given bywhere is the target posterior distribution and is a candidate from the proposal . The MH algorithm consists of choosing an arbitrary starting point and setting . Then, it generates a candidate and evaluates this candidate according to the probability . If is accepted, it sets ; otherwise, it sets . Finally, we set and repeat the process. According to [50], this iteration process produces the sample from the target conditional posterior distribution.
Appendices
A. MCMC algorithm by Nakajima (2011Nakajima, J. (2011), ‘Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Method-ology and Empirical Applications’, Monetary and Economic Studies 29, 107-142.)In this paper, the implementation of the MCMC algorithm to explore this posterior distribution follows the procedure developed by Nakajima (2011Nakajima, J. (2011), ‘Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Method-ology and Empirical Applications’, Monetary and Economic Studies 29, 107-142.), which is constructed as:
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Initialize , and ;
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Sample | , , , ;
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Sample | , , , , ;
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Sample | ;
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Sample | , , , , , ;
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Sample | , ;
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Sample | , ;
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Sample | , , , ;
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Go back to (2).
The steps of each sampling process are discussed below:
Sample The prior for is specified as , with the conditional posterior density given by
where
and , for . As shown by [50], note that the conditional posterior density is proportional to the kernel of the normal distribution whose mean and variance are and , respectively. Finally, a sample is drawn as | , , , .
Sample By considering as the state variable, the model given by equations (6) and (7) consists of a linear Gaussian state space model. As in Nakajima (2011Nakajima, J. (2011), ‘Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Method-ology and Empirical Applications’, Monetary and Economic Studies 29, 107-142.), is sampled simultaneously from the conditional posterior distribution through the algorithm of simulation smoother on the state space model
where , and . Then, the simulation smoother algorithm mainly draws , with , for , and consisting of all model parameters. In order to initialize the Kalman filter approach, we set and . Then, the Kalman filter is recursively run:
for . Subsequently, with and , the simulation smoother algorithm is performed, such that:
for . Regarding the initial state, is drawn as , with . After drawing , is computed by using the state equation (16), replacing by .
One should note that, for the TVP regression model, we have:
where is a zero vector and is a identity matrix.
Sample For the conditional posterior density of , the prior distribution for is specified as , where is the inverse-Wishart distribution. The conditional posterior distribution for is given by
where and . Equation (17) forms the kernel of the inverse-Wishart distribution, which depends only on . Last, but not least, we draw the sample as .
Sample h First, note that equations (6) and (8) consist of a nonlinear and non-Gaussian state space model. Following Nakajima (2011Nakajima, J. (2011), ‘Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Method-ology and Empirical Applications’, Monetary and Economic Studies 29, 107-142.), a multi-move sampler algorithm is implemented to draw samples from the exact posterior distribution of the original model. Consider the model as
with and . Also, let the stochastic volatility follow a driftless random walk process, that is,
where , and . Note that the disturbances and are specified as
with .
In order to sample a block (where , and ) from its joint posterior density, one should consider the draw of
where
and
One should note that . The posterior draw of is obtained by running the state equation along with the draw of given .
In order to sample from the density (20), Nakajima (2011Nakajima, J. (2011), ‘Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Method-ology and Empirical Applications’, Monetary and Economic Studies 29, 107-142.) advocates in favor of combining the classic accept-reject (AR) sampling2 2 The objective of the accept-reject (AR) sampling, also known as the von Neumann sampling, is to sample from a target density π(x)=f(x)/K, with f(x) being an unnormalized target density and K an unknown normalizing constant. However, this target distribution might be impossible to simulate by an inverse transformation. Hence, suppose it is possible to sample from the density h(x) and that there exists a constant c for which f(x)≤ch(x) for all x. In order to obtain a draw from π(x), the AR sampling proceeds to draw a candidate z from h and u from the uniform distribution U(0,1). If u≤f(z)/ch(z), z is accepted. Otherwise, the process is repeated. and the Metropolis-Hastings (MH) algorithm3 3 Given the impossibility of direct sampling from the posterior distribution, the Metropolis-Hastings (MH) algorithm is a MCMC method implemented in order to simulate samples by combining the use of the full joint density function and a proposal distribution. This proposal distribution is usually chosen as to correspond as close as possible to the target conditional posterior distribution. More specifically, define q(θ*|θ(i)) as the probability density function of the proposal given the current point θ(i) and α(θ0,θ*) as the acceptance rate from the current point θ0 to the proposal θ*, which is given by α ( θ 0 , θ * ) = min { 1, p ( θ * | y ) q ( θ 0 | θ * ) p ( θ 0 | y ) q ( θ * | θ 0 ) } where p(θ|y) is the target posterior distribution and θ* is a candidate from the proposal q(θ*|θ(i)). The MH algorithm consists of choosing an arbitrary starting point θ(0) and setting i=0. Then, it generates a candidate θ* and evaluates this candidate according to the probability α(θ(i),θ*). If θ* is accepted, it sets θ(i+1)=θ*; otherwise, it sets θ(i+1)=θ(i). Finally, we set i=i+1 and repeat the process. According to [50], this iteration process produces the sample from the target conditional posterior distribution. with a proposal distribution given by
where
and
for and (when ). One should note that
and
which are obtained from a second-order Taylor expansion of
around a certain point , namely,
For (when ), we have
and
Nakajima (2011Nakajima, J. (2011), ‘Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Method-ology and Empirical Applications’, Monetary and Economic Studies 29, 107-142.) argues that this proposal density has been chosen given its direct correspondence to the state space model
with , when , and . Given , a candidate point of for the AR-MH algorithm is drawn by implementing the simulation smoother procedure over the state space model characterized by (26).
Then, one must find as near as possible to the mode of the posterior density. This is performed as to guarantee efficiency during the sampling process. For this, [50] proposes the following steps:
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Initialize ;
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Compute and by equations (22), (23) and (25);
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Run the moment smoother using the current and on (26) and obtain for ;
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Replace by ;
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Go back to (2).
These steps are looped several times until reach near the mode of the posterior density. In addition, one should note that is the product in the simulation smoother as with . Following [50], is divided into blocks, that is, for with and , then sampling each block recursively. The author also highlights that the determination of is done by implementing the stochastic knots method, which proposes , for , with being a random sample from the uniform distribution . The is randomly chosen for each iteration of the MCMC sampling in order to guarantee a flexible draw of .
Sample First, the prior distribution for is defined as and it is assumed that in order to satisfy the restriction . Hence, the conditional posterior distribution of is
As discussed by [50], by omitting the term , the posterior distribution in (27) would then correspond to a kernel of the normal distribution, which allows the implementation of the MH algorithm.
Regarding the TVP regression model, in order to sample , a candidate sample is drawn as , where denotes the truncated normal distribution over the domain , and . Note that this proposal density is similar to the target conditional posterior distribution and is also truncated for the same domain of the target. As to the acceptance rate for the candidate from the current point , namely , it is given by
where is the probability density function of the proposal and is the old sample (current point) drawn in the previous iteration. One should note that the acceptance rate corresponds to the ratio of the omitted terms from the conditional posterior distribution of . As to the acceptance step, the candidate is accepted if , where is a random number drawn from the uniform distribution .
Sample The prior distribution for is set as , with referring to the inverse gamma distribution. Hence, the conditional posterior distribution of is
Since the posterior distribution (29) corresponds to the kernel of the inverse gamma distribution, the samples are simply drawn as , with and .
Sample As shown by [50], the process of sampling is similar to the sampling of . The prior distribution of is defined as , with its conditional posterior distribution given by
where and .
B. Estimation Results - Sample Autocorrelation
C. Estimation Results - MCMC Convergence
Editor Responsável:
Publication Dates
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Publication in this collection
18 Oct 2021 -
Date of issue
Jul-Sep 2021
History
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Received
13 Nov 2019 -
Accepted
05 Apr 2021