Asymmetric Exchange Rate Pass-Through: Evidence, Inflation Dynamics and Policy Implications for Brazil (1999–2016)

Abstract

We investigate the existence of an asymmetry in the exchange rate pass-through to the Brazilian consumer price index (CPI). Using a decomposition of the exchange rate series, into appreciations and depreciations of the Brazilian currency during the 1999–2016 period, we estimate Structural Vector Auto-regression (SVAR) models with different identifying restrictions. The results are robust and indicate a relevant asymmetric behavior of the exchange rate pass-through. Estimates indicate a pass-through of 16% in case of depreciation and of 5.8% in case of appreciation of Brazilian Real (BRL) against the US Dollar. Accordingly, the inflationary effect resulted from a (systematic) depreciation is only partially compensated by a deflationary effect of an (systematic) appreciation of the same magnitude, generating an inflationary bias that may cast doubts on inflation control strategies based solely on inflation targeting. Results provide a case against excess exchange volatility and capital mobility. A stable exchange rate favors price stability.

Appendix

1.1 SVAR Model

Consider a K-dimensional time series $Y_{t} = \left( {y_{1} ,y_{2} , y_{3} ,y_{4} } \right)^{'}$ where Y _t is a SVAR of finite order p of the structural form

$$AY_{t} = v_{0} + B_{1} Y_{t - 1} + \cdots + B_{p} Y_{t - p} + Bu_{t}$$

(1)

where A is a K × K matrix that defines the causal interrelationships among the contemporaneous variables, and u _t denotes a mean zero uncorrelated error term (also referred to as structural innovation or structural shock) with a variance-covariance matrix $E\left( {u_{t} ,u_{t}^{\prime } } \right) =\Sigma _{u}$. Because structural shocks are by definition uncorrelated, $\Sigma _{u}$ is a diagonal matrix (Kilian 2011).

Equation (1) cannot be estimated by ordinary least squares (OLS) since the variables have contemporaneous effects on each other. OLS estimates would suffer from simultaneous equation bias since the regressors and error terms would be correlated (Enders 2014).

In order to allow estimation, it is necessary to derive its reduced form representation. Premultiplication on both sides of Eq. (1) by A ⁻¹ allows the reduced form (2) to be obtained.

$$Y_{t} = c_{0} +\Phi _{1} Y_{t - 1} +\Phi _{2} Y_{t - 2} + \cdots +\Phi _{p} Y_{t - p} + e_{t}$$

(2)

Where $c_{0} = A^{ - 1} v_{0} ;\quad\Phi _{i} = A^{ - 1} B_{i} ;\quad Ae_{t} = Bu_{t}$.

Standard OLS method obtains consistent estimates of the reduced form (2) parameters $\Phi _{i}$, the reduced form errors e _t and their covariance matrix $E\left( {e_{t} e_{t}^{\prime } } \right) =\Sigma _{e}$ (Lütkepohl 2005).

However, the reduced form errors are correlated. Only in the special case where there are no contemporaneous effects among variables (i.e., matrix A elements, $a_{ij} (i \ne j)$, equals zero) the shocks will be uncorrelated.

It is possible, however, to recover the structural VAR coefficients and analyze how Y _t respond to structural shocks in u _t , from the estimates of the model in reduced form since, by construction, $Ae_{t} = Bu_{t}$. Hence, the variance of e _t is

$$\Sigma _{e} = A^{ - 1} B\Sigma _{u} B^{\prime } A^{{ - 1^{\prime } }}$$

(3)

$\Sigma _{e}$ can be consistently estimated from the reduced form by OLS, and the system can be solved for the unknown parameters provided that the number of unknown parameters do not exceed the number of equations. This involves imposing restrictions on matrix A. Usually, the most common approach is to impose a _ij  = 0 restrictions (Kilian 2011).²⁰

The assumption a _ij  = 0 means that yj does not have a contemporaneous effect on yi. The imposition of different restrictions will result in different impulse response functions depending on the correlation between errors in the reduced form. Only if all reduced form errors are uncorrelated impulse response functions will be the same regardless of the restrictions imposed.

1.1.1 Asymmetry

One possible approach to investigate the existence of asymmetric effects of x _t on y _t is to decompose the variable x _t into two new series: $x_{t}^{ + }$ of its positive variations and $x_{t}^{ - }$ of the negative variations.

Based on Schorderet (2004) and Granger and Yoon (2002) method, a time series can be decomposed as follows:

$$x_{t} = x_{0} + x_{t}^{ + } + x_{t}^{ - }$$

(4)

Where

$$x_{t}^{ + } = \sum\limits_{i = 1}^{t} {\theta_{i} } (\Delta x_{i} );\quad \left\{ {\begin{array}{*{20}l} {\theta_{i} = 1 \,{\text{se}}\,\Delta x_{i} > 0,} \hfill \\ {0,\,{\text{otherwise}}. } \hfill \\ \end{array} } \right.$$

(5)

$$x_{t}^{ - } = \sum\limits_{i = 1}^{t} {\theta_{i}^{*} } (\Delta x_{i} );\quad \left\{ {\begin{array}{*{20}l} { \theta_{i}^{*} = 1\,{\text{se}}\,\Delta x_{i} < 0, } \hfill \\ {0,\,{\text{otherwise}}. } \hfill \\ \end{array} } \right.$$

(6)

Such as that, x _t value, for all t, is equal to its initial value (x ₀) plus the sum of all its positive and negative variations up to t.

In this way, we have first difference of $x_{t}^{ + }$ and $x_{t}^{ - }$ series:

$$dx_{t}^{ + } = \theta_{i} (\Delta x_{i} );\quad \left\{ {\begin{array}{*{20}l} { \theta_{i} = 1 \,{\text{se}} \,\Delta x_{i} > 0, } \hfill \\ {0,\, {\text{otherwise}}. } \hfill \\ \end{array} } \right.$$

(7)

$$dx_{t}^{ - } = \theta_{i}^{*} (\Delta x_{i} );\quad \left\{ {\begin{array}{*{20}l} { \theta_{i}^{*} = 1\,{\text{se}}\,\Delta x_{i} < 0, } \hfill \\ {0,\,{\text{otherwise}}. } \hfill \\ \end{array} } \right.$$

(8)

The decomposition in form (5) and (6) is known in literature as decomposition by cumulative variations whereas the form in (7) and (8) is known as period-to-period variations.

Series decomposed by cumulative variations have unit root and cointegrate and are used in the estimation of error correction models (ECM) and its multivariate form vector error correction (VEC).

For purpose of this chapter, that estimates a model with the stationary variables in first difference, the most adequate decomposition, that was used, is the period-to-period decomposition.

1.1.2 Exchange Rate Pass-Through

The exchange rate pass-through can be calculated from impulse response functions estimated by the SVAR model. This method was used by McCarthy (2007) to calculate the exchange rate pass-through for several industrialized countries and by Belaisch (2003) and Araújo and Modenesi (2010) for Brazil.

$$R_{t,t + j} = \left( {\frac{{\Sigma {\kern 1pt} \Delta {\text{CPI}}_{t,t + j} }}{{\Sigma {\kern 1pt} \Delta {\text{ER}}_{t,t + j} }}} \right)\, \cdot \,100$$

(9)

Where ΔCPI is consumer price index variations and ΔER exchange rate variations.

1.1.3 Wald Coefficient Restriction Test

Wald coefficient restriction tests were performed in asymmetric models to test the hypothesis that the coefficients relative to exchange rate positive variations are statistically different from the negative variations.

The test was performed under two different null hypothesis specifications:

H ₀ (A):

Null hypothesis that the sum of the coefficients of $x^{ + }$ lags is equal to the sum of the coefficients of $x^{ - }$ lags.

H ₀ (B):

Null hypothesis that the lag coefficient i of y ⁺ is equal to the lag coefficient i of $y^{ - }$ for all lags.

To illustrate, generically, a two lags model, SVAR (2), and three variables, $Y_{t} = \left( {y_{1}^{ + } ,y_{1}^{ - } ,y_{2} } \right)^{'}$, where $y_{1}^{ + }$ and $y_{1}^{ - }$ are period-to-period decompositions of $y_{1}$

In reduced form:

$$Y_{t} = c_{0} +\Phi _{1} Y_{t - 1} +\Phi _{2} Y_{t - 2} + e_{t}$$

(10)

Can be written in the form

$$\left\{ {\begin{array}{*{20}l} {y_{{1_{t} }}^{ + } = c_{1} y_{{1_{t - 1} }}^{ + } + c_{2} y_{{1_{t - 2} }}^{ + } + c_{3} y_{{1_{t - 1} }}^{ - } + c_{4} y_{{1_{t - 2} }}^{ - } + c_{5} y_{{2_{t - 1} }} + c_{6} \Delta y_{{2_{t - 2} }} + c_{7} } \hfill \\ {y_{{1_{t} }}^{ - } = c_{8} y_{{1_{t - 1} }}^{ + } + c_{9} y_{{1_{t - 2} }}^{ + } + c_{10} y_{{1_{t - 1} }}^{ - } + c_{11} y_{{1_{t - 2} }}^{ - } + c_{12} y_{{2_{t - 1} }} + c_{13} \Delta y_{{2_{t - 2} }} + c_{14} } \hfill \\ {y_{{2_{t} }} = c_{15} y_{{1_{t - 1} }}^{ + } + c_{16} y_{{1_{t - 2} }}^{ + } + c_{17} y_{{1_{t - 1} }}^{ - } + c_{18} y_{{1_{t - 2} }}^{ - } + c_{19} y_{{2_{t - 1} }} + c_{20} \Delta y_{{2_{t - 2} }} + c_{21} } \hfill \\ \end{array} } \right.$$

The Wald test was then calculated under H ₀ with two different specifications

H ₀ (A):

$$H_{0} :c_{1} + c_{2} = c_{3} + c_{4} \,{\text{and}}\, c_{8} + c_{9} = c_{10} + c_{11} \, {\text{and}}\, c_{15} + c_{16} = c_{17} + c_{18 }$$

H ₀ (B):

$$H_{0} :c_{1} = c_{3} \,{\text{and}}\, c_{2} = c_{4} \,e\, c_{8} = c_{10} \,{\text{and}}\, c_{9} = c_{11} \,{\text{and}}\, c_{15} = c_{17} \, e\, c_{16} = c_{18 }$$

Under H ₀, the Wald statistic is asymptotically distributed as a χ ²(q), where q is the number of linear restrictions.

Wald test statistics reject the null hypothesis that coefficients are equal at conventional significance levels. Under the null hypothesis A, the test statistic was 8.38 (0.08 p-value). Under the null hypothesis B, test statistic was 16.57 (0.03 p-value). Therefore, results indicate the existence of asymmetry, that is, that exchange rate devaluations have different effects on inflation than exchange rate appreciations.