A BVAR Note on the J-Curve and the Marshall-Lerner Condition for Brazil

In the present work, the hypotheses of the J-curve and the Marshall-Lerner condition for Brazil from January 2003 to December 2019 were tested. The impulse-response function (IRF) and the variance decomposition (VD) of a Bayesian vector autoregressive model (Minnesota priors) served as instruments for the empirical verification of the above-mentioned hypotheses. The Bai and Perron (1998, 2003) structural break test was carried out, which identified two breaks and, consequently, three subsamples, from January 2003 to October 2007; December 2007 to June 2015; and July 2015 to December 2019. The results showed that the estimated BVAR empirically supports the hypotheses in question. In the short term, it is observed that a real depreciation of the Brazilian currency results, in the first five months, in a deficit in the trade balance. However, as of the fourth month, the result of the trade balance becomes positive, and it remains like that for longer than ten months. This means that one cannot reject the J-curve hypothesis. For a forecast horizon of 36 months, it was found that the Marshall-Lerner condition should not be rejected either. In other words, a currency devaluation causes an increase in the trade balance for longer than three years.


Introduction
This paper aims to analyze, through a Bayesian vector autoregressive (BVAR) model, the dynamic interaction between the exchange rate and the trade balance.The theoretical basis of these co-movements for an open economy can be found in Ivrendi and Guloglu (2010) and Rocha, Magalhã es and Brilhante (2022).The focus of this article, however, will be to analyze whether the empirical evidence corroborate the hypotheses of the J-curve and the Marshall-Lerner condition in Brazil, from January 2003 to December 2019.
The J-curve (Note 1) is generated, in a simplified way, by a currency depreciation that results, in the short term, in a trade deficit and, in the long term, in a trade surplus.According to Ramos Filho and Ferreira (2016), this phenomenon is explained by the relative rigidity in terms of exported and imported quantum, as a consequence of foreign exchange contracts (Note 2).The Marshall-Lerner condition, in its turn, establishes that an improvement in the trade balance will occur in response to a currency depreciation, if the volume of exports and imports is elastic in relation to the real exchange rate (Sonaglio, Scalco, & Campos, 2010;Arruda & Martins, 2020).
This study provides a contribution to the specialized literature because, as far as we know, the BVAR models have been little used to analyze the aforementioned hypotheses, especially for Brazil.the predictions about model variables, impulse-response functions and variance decomposition (Doan, Litterman, & Sims, 1983;Litterman, 1986;Banbura, Giannone, & Reichlin, 2010); (ii) Bayesian VAR eliminates the problem of the order of integration of model variables and also decreases the relevance of sample size (Sims & Uhlig, 1991).
There are several studies for the Brazilian economy, outside the scope of Bayesian analysis, that carried out empirical verifications of the hypotheses of the J-curve and the Marshall-Lerner condition.Arruda and Martins (2020) analyzed the impacts of a currency depreciation on total net exports of basic and industrialized goods in a panel for the Brazilian states in the period from January 1999 to December 2015 (Note 3).Using Panel Vector Autoregressive (PVAR) models, their results indicate the occurrence of the J-curve for the industrialized goods.Finally, using Panel Dynamic Ordinary Least Squares (PDOLS) estimators, the authors identified empirical evidence that validate the existence of the Marshall-Lerner condition, since the response of Brazilian states' net exports to a currency depreciation was positive.Març al et al. (2009) test the hypotheses of the J-curve and the stability of the relationship between the real exchange rate and the trade balance from the first trimester of 1980 to the fourth trimester of 2004 for Brazil.The results show that there is stability between the exchange rate and the trade balance, but they do not present favorable evidence of the existence of the J-curve.On the other hand, Neves and Lé lis (2007) estimate price and income elasticities of exports between 1980 and 2004 with a Panel Data approach and conclude that all Brazilian states present inelastic export demand in relation to price and income, except for Sã o Paulo, which has a significant participation of high value-added products.
Ramos Filho and Ferreira (2016) test the J-curve hypothesis for selected sectors of the Brazilian manufacturing industry, with annual data from 1996 to 2012.The authors used an ARDL model with cointegration (Pesaran et al., 2001) which presented results that only four sectors, inherent to those of high and low technological intensity, incompletely exhibit the J-curve effect.Thus, the authors conclude that the occurrence of the J-curve is not associated with the level of technological intensity of the sectors of the economy.
Finally, it is worth mentioning that there are several studies conducted for different countries that test the hypotheses of the J-curve and the Marshall-Lerner condition.Bahmani-Oskooee and Fariditavana (2016), for example, test the J-curve hypothesis for Canada, China, the United States and Japan using ARDL and NARDL models, with quarterly data from 1973 to 2014: the J-curve hypothesis is empirically supported for the United States via ARDL, while for China, via NARDL.Turkay (2014) uses the VEC approach to test the Marshall-Lerner condition hypothesis in Turkey's trade balance compared to the rest of the world between 1980 and 2012.The results obtained validate the hypothesis in question.Lastly, Nusair (2017) tests the J-curve hypothesis for 16 Eastern European countries, with quarterly data from 1994 to 2005.The author uses ARDL (linear) and NARDL (nonlinear) models and concludes that the nonlinear model is the most suitable to test the existence of the J-curve, given that it had no empirical support in Eastern European countries when using linear model, however, when using The ARDL and NARDL models have significant importance in the frequentist analysis of the J curve.These models were not used in this work, as the objective was a Bayesian analysis.However, such models can be used in future studies on this topic.
Besides this introduction, the following section discusses the Bayesian VAR methodology.In the third section, the analysis of the results is presented, followed by the final conclusions.

Econometric Methodology
A variety of Bayesian priors (Note 4) were developed to be used in vector autoregressive models, such as Litterman/Minnesota, Wishart-Normal, Sims-Zha Normal Wishat, Sims-Zha Norma Flat, and others.We opted for the Litterman/Minnesota method.In this method,  prior is usually distributed and conditional to matrix Σ  .Therefore, the Bayesian method can find posteriors for several different types of priors via simulation.However, for simplicity, assume that the prior of parameter  of the above regression model is normally distributed, that is: So the posteriors will also be normally distributed: mode and estimated mean will be given by a matrix of the weighted average of ordinary least squares estimates and the priors established in the research, whose posterior for  is given by the following expression: From equation (3), one can observe that the  estimator depends on the variance-covariance matrix of random errors, ∑  .It is necessary to make an estimate for this matrix.This can be done using Ordinary Least Squares (OLS) information, as follows (Note 5): It is essential, however, to establish some hypotheses on this variance-covariance matrix of random errors and on how it is estimated.There are three possibilities: (i) to use the residual variance estimates of an adjusted AR(1) model for each series; (ii) to replace Σ  by its estimate, Σ ̂, in which the diagonal elements of this matrix,   2 , correspond to the OLS estimates of the error variances of a VAR.In the present work, this procedure was used to obtain the estimated variance-covariance matrix of random errors; or (iii) to use the Σ  estimates of a complete VAR model (Note 6).
Once established how matrix Σ  will be estimated, then  priors should be calibrated:  1 ,  1 ,  2 , and  3 is a set of hyperparameters. 1 is  prior mean.In some cases, one can wish this prior to be equal to 1 or very close to 1, to capture the persistence in I(1) economic and financial time series.However, if the VAR series are in difference or in growth rate, then a choice of  1 = 0 would be more appropriate. 1 is the global adherence over the variance (first lag) and it controls the global adherence of  prior. 1 should be close to zero if there is more certainty about the prior, that is, when  1 = 0.1 is established, the prior information is allowed to dominate the sample information.In this situation, the prior is relatively strong (Note 7). 2 represents the relative adherence of other variables' variance.In other words,  2 controls the importance of the lag of variable  in the -th BVAR equation, with  ≠ , and it is called cross-variable weights.If cross-lags play a relevant role in each equation of the model, then  2 ≅ 1, otherwise  2 ≅ 0 (Note 8). 3 >0 represents the relative adherence of the variance of the lags and, as a result, the decline rate of these lags.If  3 =1, then there is a linear decline in lags (Moreira et al., 2015).After calibration of BVAR priors, the quality of the priors will be verified via robustness analysis.

Dataset
In this work, the variables have monthly periodicity, from January 2003 to December 2019 (Note 9).The model consists of four variables: one external variable (Source: FRED): i) U.S. imports, such as proxy for the income of the rest of the world (WI) (Note 10); and three domestic variables (Source: IPEADATA): ii) index of economic activity in Brazil/IBC-Br, such as domestic income proxy (GDPBR); iii) exports and imports (US$ FOB) that enable the construction of the trade balance, given by the ratio exports/imports (XM); iv) and real effective exchange rate (XR) (Notes 11 and 12).In a simplified way, the variables that make up the BVAR are shown in vector   , represented by equation 5.
= (WI  , GDPBR  , XR  , XM  ) (5) The second variable of Figure 1, presented in panel (b), is the IBC-Br and it captures well the dynamics of the Brazilian economy.The Brazilian growth in the first decade of the millennium, as well as the 2014-16 internal crisis, can be visually verified.The third variable, panel (c), in its turn, is an index for the real effective exchange rate for Brazil's main trading partners.Finally, the last variable, in panel (d), is the log-difference between Brazilian exports and imports.Given the subprime crisis, which began in 2008, and the Brazilian recession, from 2015 to 2016, it was decided to carry out the following structural break test by Bai andPerron (1998, 2003).
The variables will be used in level, although the four variables that make up the model are integrated of order one, (1), and that there is cointegration.This seeks to avoid imposing possibly incorrect restrictions on the model, according to Sims and Ulig (1991) and Stock and Watson (1990).In other words, even with (1) variables, the residuals continue to be stationary, given the inclusion of variables in level differences in the model (Hamilton, 1994).The model will be adjusted using the natural logarithm of the variables and the ordering of the variables follows the same as the one exposed in equation 5.
Figure 1 shows the model's endogenous variables in logarithm from January 2003 to December 2019.The first, panel (a), refers to the proxy to capture the world income, as already highlighted.One can notice that, from January 2003 to July 2008, the variable showed significant growth, reaching a global maximum in the last month.However, with the subprime financial crisis, in less than a year, the variable went from its global maximum to a figure close to its global minimum, in May 2009.
The second variable, presented in panel (b), is the IBC-Br and it captures well the dynamics of the Brazilian economy.The Brazilian growth in the first decade of the millennium, as well as the 2014-16 internal crisis, can be visually verified.The third variable, panel (c), in its turn, is an index for the real effective exchange rate for Brazil's main trading partners.Finally, the last variable, in panel (d), is the log-difference between Brazilian exports and imports.Given the subprime crisis, which began in 2008, and the Brazilian recession, from 2015 to 2016, it was decided to carry out the following structural break test by Bai andPerron (1998, 2003).

Estimation and Results
Bayesian VAR estimation process, for the three subsamples, was performed following the conventional routine of multivariate time series studies.The unit root test, KPSS, was performed, and it showed that all variables are non-stationary (Note 13); the Johansen test for cointegration was also performed and the conclusion drawn from it was that there is a cointegrating vector in the model (Note 14) (see Tables A2 and A5 in the Appendix, respectively); besides, the optimal number of lags used in the model was determined taking into account the information criteria of Schwartz, Akaike and Hannan-Quin and the elimination of the residual autocorrelation problem, which established that a BVAR with four lags would be the most suitable (Note 15).
The BVARs were estimated based on equation ( 5), for all subsamples.The following priors were used:  1 = 1, for the series are integrated of order one, I(1);  1 = 5, given that, in the present work, both the priors' information and the sample information are of vital importance in explaining the co-movements of the variables that make up the behavior of Brazil's trade balance;  2 = 0.99, for it is considered that cross-lags have a relevant role in each equation of the model; and  3 =1, since a linear decline in the lags of the variables that make up the model is allowed (Moreira et al., 2015).
Only for first subsample, 2003m01-2007m10, the J-curve hypothesis and the Marshall-Lerner condition are verified empirically, as will be seen later.For the other subsamples, 2007m12-2015m06; and 2015m07-2019m12, there is no empirical evidence to corroborate such assumptions (see Appendix Figures : A2 and A3).In other words, the period of economic instability that began with the subprime crisis, 2008, and intensified with the Brazilian recession, 2015-2016, significantly compromised the dynamic structure of the variables that make up the Bayesian VAR.
Since the adjusted BVAR, for the first subsample, did not present autocorrelated residuals (see Appendix Tables: A3 and A4), and it is stationary (Table A1) and robust (Note 16), then it can be used to analyze the co-movements of the variables that compose it via impulse-response function (IRF) and variance decomposition (VD) of forecast errors.
The impulse-response function and the variance decomposition of the estimated model showed that the co-movements between the variables are, in general, in accordance with the conventional economic theory.It is worth mentioning that even with I(1) and cointegrated variables, there are no spurious relations between the variables that make up the BVAR.
By using variables in level, however, the possibility of cointegration between the variables is implicitly allowed (Peersman & Smet, 2001), since the purpose of the analysis using autoregressive vector, VAR, SVAR or BVAR, is to determine the co-movements between the variables and not the estimated parameters.Therefore, the cointegration structure established between the variables is not an obstacle for the analysis of IRFs and VD of an estimated autoregressive vector (Sims, Stock, & Watson, 1990).

Impulse-Response Functions
In Figure 2, the accumulated impulse-response function (Note 17), via generalized method (Generalized IRF) (Note 18), shows the response of the trade balance (XM) to a standard deviation innovation in the real exchange rate (XR).The aim is to understand the short-term forecast (panel a) and the forecast of 36 moths (panel b)dynamic behavior of the trade balance, given a currency devaluation.
In the short term, panel (a), it can be observed that a real depreciation of the Brazilian currency results, in the first e third months, in a deficit in the trade balance.However, as of the fourth month, the result of the trade balance becomes positive and it remains like that for longer than ten months.This means that the existence of evidence for the J-curve hypothesis cannot be rejected.Note.The graphs show the median (black line) together with a 90% credible interval (grey shaded area).

Variance Decomposition
Table 1 shows the forecast error variance decomposition (VD) for four different forecast horizons.That is, for two, four, twelve and eighteen months.Each column shows, for the response variable -trade balance, the forecast error proportion that is explained by the structural shocks of each of the four variables that make up the model, duly listed on the left side of the table.Therefore, for each forecast horizon, the sum of the entries in each line is equivalent to 100%.The trade balance forecast errors, in the first four months, are explained by their own shocks, on average 89%.However, in the medium term or 18 months, its forecast errors are explained by its own shocks, around 69%, by world income shocks, 4%, by Brazilian GDP shocks, 19%, and by the real exchange rate, 7%.It can be observed that the real exchange rate, over time, significantly expands its participation in explaining the trade balance forecast errors, 0.78 to 7.19.

Robustness Analysis
The robustness analysis was performed by including the commodity price variable (PCOMM) in the estimated BVAR (Note 20).In summary, the model was estimated assuming that the commodity price variable is the most exogenous of the variables that make up the BVAR, resulting in the following ordering of the autoregressive vector: i) PCOMM, ii) WI, iii) GDPBR, iv) XR and v) XM.It is worth mentioning that the variable that captures commodity prices is the Consumer Price Index, which considers all commodities around the world and is made available by the U.S. Bureau of Labor Statistics (Note 21).
Four lags and the same priors established in the model were used.Finally, it was analyzed whether there are significant changes in the dynamics of impulse-response functions (generalized) for BVARs with and without commodity prices.

Conclusion
In this work, the hypotheses of the J-curve and the Marshall-Lerner condition were tested through the impulse-response function (IRF) and the variance decomposition (VD) of a Bayesian vector autoregressive model (Minnesota priors).The results showed that the BVAR estimated for Brazil empirically supports the hypotheses in question from January 2003 to December 2019.A depreciation of the Brazilian real results, in the first three months, in a deficit in Brazil's trade balance.However, from the fourth month onwards, the trade balance changes its behavior, remaining positive for longer than a year.The present study contributes to the specialized literature by using Bayesian VAR model, since it has been little used in studies that test the aforementioned hypotheses in Brazil.
The robustness analysis was performed by including the commodity price variable (PCOMM) in the estimated BVAR.It can be observed that the inclusion of commodity prices in the BVAR did not result in significant changes in the (generalized) IRF, which captures the real exchange rate shocks on the trade balance.Thus, it is concluded that the estimated BVAR model is robust and the results presented in the study in question are relevant for a good understanding of the dynamic relationship between trade balance and real exchange rate.
The study's findings offer significant insights for both practitioners and researchers.For policymakers and economists, understanding the time lag effects of currency devaluation on trade balance is crucial, as the validation of the J-curve hypothesis and the Marshall-Lerner condition underscores the importance of considering short-term adversities against long-term gains.Businesses engaged in international trade can also leverage these insights for strategic planning and operational adjustments.However, researchers should interpret these findings within the context of the study's limitations, including its focus on Brazil during a specific period, which might not encapsulate the dynamics in different temporal or economic settings.Additionally, the influence of unforeseen external shocks and the exclusion of micro-level and sector-specific factors could affect the generalizability and applicability of the results.
Future research could extend the BVAR model application to different economies or specific sectors to understand the broader applicability of the J-curve and the Marshall-Lerner conditions.Incorporating micro-level data could unveil nuanced sector-specific dynamics, offering a more granular understanding of currency depreciation effects.Moreover, longitudinal studies examining the impact of significant global economic shifts could shed light on the long-term implications of such events on trade balance and exchange rate dynamics.
This study contributes significantly to the field by applying the BVAR model to analyze complex economic phenomena in an emerging economy context.While it provides a robust framework for understanding the dynamic interaction between exchange rates and trade balance, acknowledging its limitations and the need for further research is imperative.Future studies should aim to broaden the scope of analysis, incorporate a diverse range of variables, and consider the evolving global economic landscape to enhance the robustness and applicability of the findings.
Further studies may consider employing the Bayesian time-varying parameter vector autoregressive (Bayesian TVP-VAR) methodology.This approach is advantageous as it captures the dynamics of trade balance responses to exchange rate shocks, taking into account specific timeframes within the sample, such as periods of economic boom and downturn.This could potentially be a significant extension of the current research.

Notes
Note 1.An aggregate analysis of the J-curve in accordance with the works of Neves andLé lis (2007), Març al et al. (2009) and Fligenspan (2009) will be carried out.Based on the results of the estimated BVAR, it can be observed that the problem of the aggregation bias has no influence on the quality of the model, since IRF showed a statistically significant relation of a real exchange rate devaluation on the trade balance.
Note 2. For further explanation on the J-curve, see Krugma and Obtsfeld (2000) and Kulkarni and Clarke (2009).
Note 3. From the perspective of regional economic models, according to Rickman (2010).
Note 4. The Bayesian estimation method, by treating the parameters as random variables, imposes important probability distributions on priors of a VAR model complete set of coefficients, in order to obtain parsimony in the number of parameters to be estimated.Thusly, when using a BVAR, the estimators are more robust and, consequently, there is an estimated model with a better predictive power.However, a proper choice of priors should impose some structure on the VAR that reflects the nature and process of generating data.This article closely follows the work on Bayesian VAR by Ouliaris et al. (2018).
Note 5. Another way to get information about matrix Σ  is to produce Bayesian estimates of it, which will require establishing priors on the variance matrix in question.
Note 6.However, this possibility is not highly recommended, since the estimated matrix may be singular.That is, there may not be enough information when the number of variables and lags in the VAR are too large.
Note 7. If  1 ≥ 10 is established, the prior is said to be uninformative/uncertain and the generated estimates will be close to the estimated coefficients of an unrestricted VAR.For more details, see Ouliaris et al. (2018).
Note 9.The analysis ends in December 2019, due to the shock of COVID-19.
Note 10.Imports of U.S. goods from the rest of the world in millions of dollars.The variable is made available by FRED already deseasonalized.Later, it was deflated by the U.S. Consumer Price Index.The US GDP was not used as a proxy for the rest of the world's income, as it is composed of domestic and foreign absorption.Therefore, American imports would be more relevant to the Brazilian trade balance.It should be noted that among the most important countries for Brazilian foreign trade, the United States presents the most reliable statistics.Data from the Chinese economy could also be used in the analysis, but we did not use them, given the lack of credibility of the statistics (HOLZ, 2014).
Note 11.Real Effective Exchange Rate -National Consumer Price Index (NCPI) -Exports: index (2010 average = 100).The real effective exchange rate is a weighted arithmetic mean of Brazil's bilateral real exchange rates relative to 23 trading partners selected.This series was calculated using national CPI as domestic price index.
Note 12.The seasonal adjustment in the trade balance and in the IBC-Br was carried out through X-13/ARIMA-SEATS.
Note 13.In unit root and cointegration tests a significance level of 5% is used.
Note 14.According to Álvarez and Ballabriga (1994), Litterman/Minnesota priors have a good performance in the presence of cointegration.The authors showed, in a Monte Carlo experiment, that the addition of long-term constraints to the priors does not improve, in non-asymptotic samples, the performance, predictive analysis, IRFs and VD of a Bayesian VAR.Using variables in level, however, the possibility of cointegration between the variables is implicitly allowed (Peersman & Smet, 2001).The purpose of the analysis using autoregressive vector, VAR, SVAR or BVAR, is to determine the co-movements between the variables and not the estimated parameters.Therefore, the cointegration structure established between the variables is not an obstacle for the analysis of IRFs and VD of an estimated autoregressive vector (Sims, Stock, & Watson, 1990).
Note 15.The results of the tests of unit root, cointegration, residual autocorrelation and Bayesian VAR stability can be consulted in the Appendix of this work.
Note 16.The robustness test will be presented in section 3.3.This test consists of introducing the price of commodities in the BVAR and analyzing IRFs' stability or lack thereof.If the introduction of this variable results in significantly different IRFs when compared to the IRFs of the original model, then the VAR is not robust.
Note 17.The non-cumulative IRF is available with the authors.It presents a similar dynamic to the accumulated IRF.The latter was the one chose to be presented in the work, as it provides a better idea of the J-curve.
Note 18. Generalized IRFs method is robust to changes in the order variables enter the BVAR (Pesaran & Shin, 1998).
Note 19.The stability of the model is measured by the BVAR stability test via Roots of the characteristic Polynomial, Table A1.
Note 20.For a robustness analysis in an autoregressive vector, see Luporini ( 2008).
Note 21.The same variable was incorporated into a SVAR model by Rocha, Magalhã es, and Brilhante (2022), who sought to investigate, among other things, the extent of the influence of commodity prices on credit cycles in Brazil.The study points out that commodity price index and U.S. GDP are relevant to consistently capture the dynamic interaction between credit and important domestic variables in Brazil, namely, GDP, inflation, interest rate and exchange rate.

Appendix A
From Tables A3 and A4, see that the model is well specified, that is, it does not suffer autocorrelation with respect to residuals.

Figure 1 .
Figure 1.Model variables Note.Except for the exchange rate, all other variables are deseasonalized.
Over the 36-month forecast period, panel (b), it can be verified that the Marshall-Lerner condition should not be rejected either.In other words, a currency devaluation causes an increase in the trade balance for longer than 36 months.Therefore, the estimated BVAR empirically supports the existence of the J-curve and the Marshall-Lerner condition for Brazil from January 2003 to October 2007 (Note 19).Therefore, the results obtained in the BVAR model are similar to the works ofArruda and  Martins (2020), Ramos Filho andFerreira (2016),Nusair (2017), Bahmani-Oskooee and Fariditavana (2016),Turkay (2014) and Bahmani-Oskooee,Goswami and Talukdar (2008).

Figure 2 .
Figure 2. Trade balance responses to an innovation of one standard deviation in the real exchange rate for 10 and 36 months using Generalized Factors

Figure 3
Figure3compares the model after the inclusion of commodity prices with the original model.It can be verified that the inclusion of commodity prices in the BVAR did not result in significant changes in the (generalized) IRF, which captures real exchange rate shocks on the Brazilian trade balance.Therefore, it is concluded that the model is robust and the results presented in the study are relevant for the empirical verification of the hypotheses of the J-curve and the Marshall-Lerner condition in Brazil.

Figure 3 .
Figure 3.Comparison of the trade balance responses to an innovation of one standard deviation in the real exchange rate without (original model) and with the inclusion of commodity prices Note.The graph shows the medians (black and blue line) together with both 90% credible intervals (grey and light blue shaded areas).

Figure A1 .
Figure A1.Trade balance responses to an innovation of one standard deviation in the real exchange rate for 10 months (2007m12 -2015m06) using Generalized Factors Note.The graph shows the median (black line) together with a 90% credible interval (grey shaded area).

Figure A2 .
Figure A2.Trade balance responses to an innovation of one standard deviation in the real exchange rate for 10 months (2015m7 -2019m12) using Generalized Factors

Table 1 .
Variance decompositionproportion of variance precision error per variable

Table A1 .
BVAR stability test -Roots of Characteristic Polynomial Note.No root lies outside the unit circle., which implies the stability of the model.Lag specification = 4. Endogenous variables: ln(GDPBR), ln(XR), ln(XM) and ln(WI).

Table A2 .
KPSS test for unit roots Null Hypotesis: the column variable is stationary.Bandwidth: 11 (Newey-West automatic) using Bartlett kernel.* indicates that the constant and linear trend were used to compute the test.On the other hand, ** indicates that just the constant was used to run the test.

Table A4 .
Likelihood ratio testresiduals Null hypothesis: No serial correlation at lags 1 to h. *Edgeworth expansion corrected likelihood ratio statistic.

Table A5 .
Trace and maximum eigenvalue testsNote.the optimal specification of the Johansen test was established based on the Akaike criterion.This specification is composed of intercept and trend in the error correction vector (1st part of the table) and no intercept in the VAR (2nd part of the table).Trace and maximum eigenvalue tests show, at a significance level of 5%, that there is only one cointegration equation in the model.