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

Impulse-Response Functions Analysis: An application to the Exchange Rate Pass-Through in MexicoSylvia Beatriz Guillermo PeonFacultad de Economia. Benemerita Universidad Autonoma de PueblaMartin Alberto Rodriguez BrindisEscuela de Economia y Negocios Universidad Anahuac Campus Oaxaca

OutlineImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoExchange 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

DefinitionImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoThe Exchange Rate Pass-Through (ERPT) can be understood as the degree to which exchange rate changes are passed on into domestic prices along the distribution chain. Exchange rate shocks may affect prices at different stages both directly as well as indirectly.The conventional 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

Two Estimation Approaches for IRFsOur 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 pricesThe 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 cointegrating relationships 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.

Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico4Data SetWe 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 BanxicoPeriod of Analysis: 2001m1 to 2013m2All series are I (1) except the Interest Rate, which is I (0): used ADFTAll series are expressed in natural logs except the Interest Rate

Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoSVAR ModelGiven 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-th I (1) variable in the system (Stata Seasonal Difference Operator S12.y):

The structural form model can be expressed as:

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

SVAR ModelImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico

SVAR ModelNormalization 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 orthogonalized IRFs 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.


SVAR ModelThe lag order of the model is 2 and it was chosen according to Akaike Information Criterion (AIC) and Final Prediction Error (FPE) criterion (Ltkepohl (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 constraintsvalmar shows only zeros and ones for thechi2-statistic and p-values respectively


SVAR ModelImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoRestrictions on the underlying VAR parametersThe 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 Brggemann et al (2003) and Ltkepohl (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 0.166 insignificant regressor were eliminated in the model.

Checking Model Stability

Note: varstable command may not be used after fittingAn overidentified SVAR model.

SVAR ModelImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoOver identifying restrictions on the matrix of contemporaneous effects A were determined following a procedure similar to the sequential elimination of regressors.These additional 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)

SVAR Model: ResultsImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoNote:Stata does not compute cumulative Structural IRFsStata does not compute bootstrap standard errors for overidentified structural VAR models. However, the structural IRFs 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).

SVAR Model: ResultsImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoNote:Stata does not compute cumulative Structural IRFsBecause 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.

Cumulative Responses to a one-percent change in Exchange Rate depreciation

SVAR Model : ResultsImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoCumulative Pass-Through ElasticityThe 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 (Capistrn, 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.

VEC ModelImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoThe VEC approach uses the Cholesky decomposition of the residual variancecovariance 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.

VEC ModelImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoExploring graphically some possible cointegrating relations

Starting the estimation process by selecting the lag-order

VEC ModelImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexicovecrank stata command to determine the number of cointegrating equations

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.

17VEC ModelImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoUnrestricted Estimation (no contraints on alfa and beta) was carried out with rtrend and rconstantStability 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.

VEC Model : ResultsImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoThe estimated cointegrating equations are the following:

VEC Model: ResultsImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoEstimation 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 Stata dforce option 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.

VEC Model: ResultsImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico * DEFINING CONSTRAINTS ON * COINTEGRATING PARAMETERS

* bconstraints constraint 10 [_ce1]ppi = 1constraint 11 [_ce1]cpi = 0 constraint 12 [_ce1]ex_rate = 0constraint 13 [_ce1]oilp = 0

constraint 20 [_ce2]ppi = 0constraint 21 [_ce2]cpi = 1constraint 22 [_ce2]ex_rate = 0

constraint 30 [_ce3]ppi = 0constraint 31 [_ce3]cpi = 0constraint 32 [_ce3]ex_rate = 1

* DEFINING CONSTRAINTS ON * ADJUSTMENT PARAMETERS

* aconstraints

constraint 103 [D_ppi]L1._ce3 = 0constraint 201 [D_cpi]L1._ce1 = 0constraint 402 [D_impi]L1._ce3 = 0constraint 501 [D_oilp]L1._ce1 = 0constraint 502 [D_oilp]L1._ce2 = 0constraint 701 [D_i_rate]L1._ce1 = 0constraint 703 [D_i_rate]L1._ce2 = 0VEC Model: ResultsImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico

VEC Model: ResultsImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoNote: Stata does not compute Std. Errors for OIRFs estimated with VECM

VEC Model: ResultsImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoNote: PPI Response is very sensitive to constraints

VEC Model: ResultsImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoSome 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 Stata vec command. 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: vec uses a switching algorithm developed by Boswijk (1995) to maximize the log-likelihood when constraints are placed on the parameters. The starting values affect both the ability of the algorithm to find a maximum and its speed in finding that maximum.Specifying starting values for BETA is complicated.

VEC Model: ResultsImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoSome inconveniences found (for our particular study) when estimating the model using Stata VEC command:

IRFs results are very sensitive to constraints on cointegrating parameters.

IRFs 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 IRFs because standard errors are not computed when using vec command. VEC Model: Two Stage ProcedureAs 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).


VEC Model: Two Stage ProcedureImpulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico

Advantages: We keep Wold causal ordering of variablesCan impose constraints on contemporaneous and underlying VAR parameters and on adjustment parameters in order to improve estimation precision .Can estimate Structural IRFs with corresponding Std. ErrorsVEC Model: Two Stage Procedure: ResultsImpulse-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.

Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in Mexico32References Impulse-Response Functions Analysis: An Application to the Exchange Rate Pass-Through in MexicoBruggemann, Ralf, Krolzig Hans-Martin and Ltkepohl, Helmut (2003). Comparison of Model Reduction Methods for VAR processes. Economics Papers 2003-W13, Economics Group, Nuffield College, University of Oxford.

Capistrn, Carlos, Ibarra-Ramirez, Ral and Ramos-Francia, Manuel. (2011). El Traspaso de Movimientos del Tipo de Cambio a los Precios: Un Anlsis para la Economa Mexicana. Banco de Mxico. Documentos de Investigacin. Working Paper No. 2011-12.

Ltkepohl, Helmut. (2005). New Introduction to Multiple Time Series Analysis. Springer-Verlag.

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