Impulse-Response Functions Analysis : An application to the Exchange Rate Pass- Through in Mexico

1. Sylvia Beatriz Guillermo Peon Facultad de Economia. Benemerita Universidad Autonoma de Puebla Martin Alberto Rodriguez Brindis Escuela de Economia y Negocios Universidad Anahuac Campus Oaxaca Impulse-Response FunctionsAnalysis: Anapplicationtothe Exchange Rate Pass-Through in Mexico

2. Impulse-Response FunctionsAnalysis: AnApplicationtothe Exchange Rate Pass-Through in Mexico Outline • Exchange Rate Pass-Through Definition • Time Series Frameworks and Estimation strategies for Impulse-Response Functions • SVAR model Estimation • VEC model Estimation using Stata VEC command • VEC model Estimation using a Two-Stage Procedure

3. Impulse-Response FunctionsAnalysis: AnApplicationtothe Exchange Rate Pass-Through in Mexico Definition • The Exchange Rate Pass-Through (ERPT) can be understood as the degree to which exchange ratechanges are passed on into domestic prices along the distribution chain. • Exchange rateshocks may affect prices at different stages both directly as well as indirectly. • Theconventional transmission mechanism of the exchange rate works in two stages: • Stage 1: the exchange rate changes have a direct effect on import prices • Stage 2: the mechanism works through its impact on producer prices and consumer prices

4. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico Two Estimation Approaches for IRFs • Our work presents an analysis of the ERPT mechanism for the Mexican economy after the formal adoption of inflation targeting (Jan 2001), using impulse-response functions (IRFs) as a tool to estimate the degree and timing of the effect of exchange rate depreciation changes on domestic prices • The analysis is carried out using two time series frameworks. • Recursive SVAR model: unlike the traditional VAR model, allows us to impose restrictions on the contemporaneous and lagged matrices of coefficients in order to improve estimation results. • VEC model: considers the possibility of valid cointegratingrelationships among the variables and allows us to incorporate the deviations from the long run equilibrium) as explanatory variables when modeling the short run behavior of the variables.

5. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico Data Set We use monthly observations of the following variables: • Oil price index (oilp) • Global Indicator of Economic Activity for Mexico (igae) • Nominal Exchange Rate Pesos/ USD (ex_rate) • Import price Index (impi) • Producer price Index (ppi) • Consumer price Index (cpi) • Nominal Interest Rate (i_rate) Sources: IFS, INEGI and Banxico Period of Analysis: 2001m1 to 2013m2 All series are I (1) except the Interest Rate, which is I (0): used ADFT All series are expressed in natural logs except the Interest Rate

6. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico SVAR Model • Given that we have monthly observations, we use the twelve-seasonal difference (or difference of order twelve) of each I(1) variable; that is, for the k-thI (1) variable in the system (Stata Seasonal Difference Operator S12.y): • The structural form model can be expressed as:

7. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico SVAR Model

8. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico SVAR Model • Normalization restrictions together with a Wold causal ordering (recursive structure), provide the K(K+1)/2 necessary restrictions to uniquely identify the structural shocks and impulse-responses (Just-identified SVAR). Thus, we can define A as a lower triangular matrix: • This set of restrictions also ensure just-identified IRFs which are qualitatively the same as the orthogonalizedIRFs based on a Cholesky decomposition of the variance-covariance matrix of the reduced form VAR disturbances. • However, we use an SVAR model in our study (Stata SVAR command) because it allows us to place some additional short run constraints –in addition to the traditional recursive structure− to help us improve the estimation of the structural impulse-response functions (IRFs). In other words, we estimate an overidentified SVAR model to analyze the structural IRFs.

9. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico SVAR Model • The lag order of the model is 2 and it was chosen according to Akaike Information Criterion (AIC) and Final Prediction Error (FPE) criterion (Lütkepohl (2005, pp 152) • In small samples, AIC and FPE may have better properties (choose the correct order more often). • Models based on these criteria may produce superior forecasts, because AIC and FPE are designed for minimizing the forecast error variance, in small as well as large samples. • Before placing any constraints (on matrix A and/or on the underlying VAR), we tested for residual autocorrelation using LM test (Stata command: varlmar) Note: when used after SVAR with constraints valmar shows only zeros and ones for the chi2-statistic and p-values respectively

10. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico SVAR Model Restrictions on the underlying VAR parameters • The aim of this estimation stage is to specify an underlying VAR model containing all necessary right-hand side variables and as parsimonious as possible; a model which could also help us to improve the accuracy of the implied impulse-responses • We used sequential elimination of regressors procedure suggested in Brüggemann et al (2003) and Lütkepohl (2005). • The procedure involves testing zero restrictions on individual coefficients (to eliminate lags of variables of the underlying VAR) in each of the seven equations. • At each step of the procedure a single regressor was sequentially eliminated in one equation if its corresponding P-value was higher than 0.1 • 66 insignificant regressor were eliminated in the model. • Checking Model Stability Note:varstable command may not be used after fitting An overidentified SVAR model.

11. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico SVAR Model • Over identifying restrictions on the matrix of contemporaneous effects A were determined following a procedure similar to the sequential elimination of regressors.Theseadditional zero restrictions correspond to setting Note: Using stata SVAR, this implies a definition of matrix A in the following way: matrix A = (1,0,0,0,0,0,0\0,1,0,0,0,0,0\.,0,1,0,0,0,0\.,0,.,1,0,0,0\.,0,.,.,1,0,0\.,.,.,.,.,1,0\.,0,.,0,0,.,1)

12. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico SVAR Model: Results Note: Stata does not compute cumulative Structural IRF´s Statadoes not compute bootstrap standard errors for overidentified structural VAR models. However, the structural IRF´s and forecast-error variance decompositions were estimated using the small-sample correction for the maximum likelihood estimator of the underlying VAR disturbances variance-covariance matrix (see Stata Time-Series Reference Manual).

13. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico Cumulative Responses to a one-percent change in Exchange Rate depreciation SVAR Model: Results Note: Statadoes not compute cumulative Structural IRF´s Because structural shocks are standardized to one-percent shock, the vertical axis in the figures indicates the estimated percentage point change in the respective response variable due to a one-percent shock, after s periods.

14. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico SVAR Model : Results • Cumulative Pass-Through Elasticity The CPTE at period s is the ratio of the cumulative response of the corresponding price index inflation to the cumulative response of the exchange rate depreciation, both evaluated s periods after the exchange rate shock (Capistrán, et al 2011). • Pass-through degree to import prices is the highest and it occurs immediately, with an impact elasticity (at s = 0) very close to one. It remains quite high (0.903), implying an almost complete ERPT at this stage of the distribution chain. • The impact effect of the pass-through on producer prices is about 0.11 which increases to 0.17 after nine months and decreases thereafter. By month 18 after the shock, 13.3 percent of the exchange rate depreciation is passed on into producer prices and it stays the same afterwards. • The CPTE of consumer prices is zero on impact and one month after the exchange rate shock. It barely increases to 0.026 after four months and because inflation responses to the exchange rate depreciation are zero thereafter, the elasticity of consumer prices ends up being 0.015 implying that only 1.5 percent of the exchange rate depreciation is passed on into consumer prices.

15. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model • The VEC approach uses the Cholesky decomposition of the residual variance–covariance matrix by imposing some necessary restrictions so that causal interpretation of the simple IRFs is possible. If cointegration exists, estimation of the IRFs provides a tool to identify when the effect of a shock to the exchange rate is transitory and when it is permanent.

16. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model • Exploring graphically some possible cointegratingrelations • Starting the estimation process by selecting the lag-order

17. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model Case 2: No linear trends in the differenced data (trends in levels are linear but NOT quadratic) and linear trend in cointegrating equations (cointegrating equations are trend stationary) Case 4: NO linear trends in the levels of the data and cointegrating equations stationary around a constant mean. • vecrankstata command to determine the number of cointegrating equations

18. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model • Unrestricted Estimation (no contraints on alfa and beta) was carried out with rtrend and rconstant • Stability and autocorrelation tests were also performed: • Model versions are stable: there are only K-r = 7-3 = 4 unit moduli and the remaining are less than 1. However, the estimated model with no linear trend in the levels of the variables, shows one additional moduli of 0.97, indicating that the rtrend model is better. • Found evidence of autocorrelation for lag orders 1 and 2. So we included one more lag in the estimation process.

19. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model : Results • The estimated cointegrating equations are the following:

20. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model: Results • Estimation with overidentifying restrictions on beta (cointegrating parameters) and restrictions on alfa (adjustment parameters) was carried out. However, STATA estimation results indicate that beta is underidentified. • We used Statadforceoption to get the beta and alfa parameter estimates when they are not identified. • LM test for identifying restrictions report chi2( 8) = 12.26 Prob > chi2 = 0.140 so restrictions are valid. • Stability test shows that the restricted model is stable, and veclmar command cannot be used in this case because it requires that the parameters in the cointegrating equations be exactly identified or overidentified. • Orthogonal Impulse-functions are estimated for both unrestricted and restricted models.

21. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model: Results * DEFINING CONSTRAINTS ON * COINTEGRATING PARAMETERS * bconstraints constraint 10 [_ce1]ppi = 1 constraint 11 [_ce1]cpi = 0 constraint 12 [_ce1]ex_rate = 0 constraint 13 [_ce1]oilp = 0 constraint 20 [_ce2]ppi = 0 constraint 21 [_ce2]cpi = 1 constraint 22 [_ce2]ex_rate = 0 constraint 30 [_ce3]ppi = 0 constraint 31 [_ce3]cpi = 0 constraint 32 [_ce3]ex_rate = 1 * DEFINING CONSTRAINTS ON * ADJUSTMENT PARAMETERS * aconstraints constraint103 [D_ppi]L1._ce3 = 0 constraint 201 [D_cpi]L1._ce1 = 0 constraint 402 [D_impi]L1._ce3 = 0 constraint 501 [D_oilp]L1._ce1 = 0 constraint 502 [D_oilp]L1._ce2 = 0 constraint701 [D_i_rate]L1._ce1 = 0 constraint703 [D_i_rate]L1._ce2 = 0

22. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model: Results

23. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model: Results Note: Statadoesnot compute Std. ErrorsforOIRFsestimatedwith VECM

24. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model: Results Note: PPI Response isverysensitivetoconstraints

25. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico • Some inconveniences found (for our particular study) when estimating the model using Stata VEC command: • If the order of the variables is important (as in our model) there could be a conflict with Johansen normalization restrictions used by Stataveccommand. Keeping the recursive (Wold causal) order imposed in the SVAR model (if possible) becomes very difficult because it implies to place several restrictions on the beta coefficients which easily lead to convergence NOT achieved when maximizing the log-likelihood function. • Stata PDF documentation files specify as technical note: • vecuses a switching algorithm developed by Boswijk (1995) to maximize the log- likelihoodwhen constraints are placed on the parameters. The starting values affect both the ability ofthe algorithm to find a maximum and its speed in finding that maximum. • Specifying starting values for BETA is complicated. VEC Model: Results

26. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model: Results • Some inconveniences found (for our particular study) when estimating the model using Stata VEC command: • IRF´s results are very sensitive to constraints on cointegrating parameters. • IRF´s are now estimated based on shocks to the levels of the variables, so these IRFs are not directly comparable with the ones obtained with the SVAR model. The analysis must be different under the two approaches. • The variables in differences are the simple first differences (not the seasonal differences that we used with the SVAR model). First differences represent (in our model) monthly growth rates, and we used annual growth rates with the SVAR. • We cannot asses the statistical significance of the IRF´s because standard errors are not computed when using vec command.

27. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model: TwoStageProcedure • As alternative estimation approach we use a two-stage procedure. We also follow Stata Time Series Manual two-step estimation method suggested when using veclmar command for VECM when the parameters of the cointegrating vectors (beta) are exactly identified or overidentified. • This method requires to have explored reasonable cointegrating relations, and requires to perform the corresponding stationarity tests to verify that each cointegrating relation is I(0).

28. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model: TwoStageProcedure • Advantages: • WekeepWold causal ordering of variables • Can imposeconstraintsoncontemporaneous and underlying VAR parameters and onadjustmentparametersin ordertoimproveestimationprecision . • Can estimateStructuralIRF´swithcorrespondingStd. Errors

29. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model: TwoStageProcedure: Results

30. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model: TwoStageProcedure

31. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico VEC Model: TwoStageProcedure

32. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico Conclusions • The SVAR model over-estimates the size and persistence (except for consumer price inflation) of responses to a one-percent Exchange Rate Depreciation shock. • However, the SVAR model under-estimates the CPT Elasticities. In other words, the estimated percentage of the exchange rate depreciation that is passed on into prices along the distribution chain is higher under the SVEC estimation approach. • The difference on CPTE between the two approaches is more evident for the consumer price index. Ten months after the shock, the SVEC and SVAR models estimate that 10% and 1.7% of the exchange rate depreciation is passed on into consumer prices respectively. This implies that taking into account deviations from the long-run equilibrium relationships in our ERPT analysis is important.

33. Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico References • Bruggemann, Ralf, Krolzig Hans-Martin and Lütkepohl, Helmut (2003). Comparison of Model Reduction Methods for VAR processes. Economics Papers 2003-W13, Economics Group, Nuffield College, University of Oxford. • Capistrán, Carlos, Ibarra-Ramirez, Raúl and Ramos-Francia, Manuel. (2011). El Traspaso de Movimientos del Tipo de Cambio a los Precios: Un Análsis para la Economía Mexicana. Banco de México. Documentos de Investigación. WorkingPaper No. 2011-12. • Lütkepohl, Helmut. (2005). New Introduction to Multiple Time Series Analysis. Springer-Verlag.