Estimating Impulse-Response Functions for Macroeconomic Models using Directional Quantiles

Gabriel Montes-Rojas

doi:10.1515/jtse-2021-0002

Article

Estimating Impulse-Response Functions for Macroeconomic Models using Directional Quantiles

Gabriel Montes-Rojas

Published/Copyright: December 17, 2021

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Journal of Time Series Econometrics Volume 14 Issue 2

Abstract

A multivariate vector autoregressive model is used to construct the distribution of the impulse-response functions of macroeconomics shocks. In particular, the paper studies the distribution of the short-, medium-, and long-term effects after a shock. Structural and reduced form quantile vector autoregressive models are developed where heterogeneity in conditional effects can be evaluated through multivariate quantile processes. The distribution of the responses can then be obtained by using uniformly distributed random vectors. An empirical example of exchange rate pass-through in Argentina is presented.

Keywords: impulse-response functions; vector autoregressive models; multivariate quantiles; pass-through

JEL Classification: C13; C14; C22

Corresponding author: Gabriel Montes-Rojas, Instituto Interdisciplinario de Economía Política de Buenos Aires (IIEP-BAIRES-UBA), Facultad de Ciencias Económicas, Universidad de Buenos Aires, Av. Córdoba 2122 2do piso, C1120AAQ, Ciudad Autónoma de Buenos Aires, Argentina, E-mail: gabriel.montes@fce.uba.ar

Appendix: A Univariate Example

Consider a location-scale univariate time-series model,

y t = b y t − 1 + a + ( 1 + d y t − 1 ) ε t ,

where ε t ∼ iid 0 , σ ε 2 with distribution function F _ɛ and quantile function Q _ɛ. Assume that 0 < b < 1 and d ≥ 0.

For this model,

E ( y t | y t − 1 ) = b y t − 1 + a ,

but

Q y t ( τ | y t − 1 ) = ( b + d Q ε ( τ ) ) y t − 1 + a + Q ε ( τ ) , τ ∈ ( 0,1 ) .

The conditional quantile y _t|y _t−1 depends on the quantiles of the shocks, and thus,

∂ E ( y t | y t − 1 ) ∂ y t − 1 = b ,

but

∂ Q y t ( τ | y t − 1 ) ∂ y t − 1 = ( b + d Q ε ( τ ) ) .

The model with d = 0 is the location-shift model, but d ≠ 0 is the more general location-scale-shift model.

In order to simulate a shock, suppose we start from a stationary state, y ̄ = b y ̄ + a = a 1 − b , and then we impose a shock at h = 0 of magnitude ɛ ₀ = 1. Then we have y 0 s = b y ̄ + a + ( 1 + d y ̄ ) 1 , and y h s = b y h − 1 s + a + 1 + d y h − 1 s ε h , for h > 0. Note that the initial response to the shock, y 0 s , depends here on the initial conditions, y ̄ . Compare this with the path without shock, y 0 0 = b y ̄ + a = y ̄ , and y h 0 = b y h − 1 0 + a + 1 + d y h − 1 0 ε h , for h > 0.

However, the quantiles η ∈ (0, 1) of the responses will depend on the specific distribution of ɛ. The η index here does not correspond to the quantile index of y _t|y _t−1, for which we use τ. Using simple calculations reveals Q ₁(τ|y ₀) = by ₀ + a + (1 + dy ₀)Q _ɛ(τ), where Q ₁(.|.) is the 1 period ahead quantile forecast. Consider now the quantile impulse response function, Q I R F 1 ( τ , 1 | y ̄ ) = Q 1 τ | y 0 s − Q 1 ( τ | y ̄ ) . Then in order to obtain Q Q I R F 1 ( η , 1 | y ̄ ) , the one-period ahead η quantile of the impulse response function, we would need to compute the distribution of Q I R F 1 ( u , 1 | y ̄ ) = Q 1 u | y 0 s − Q 1 ( u | y ̄ ) with u ∼ U(0, 1) and obtain the η-quantile. Note that this can be written as I R F 1 ( 1 | y ̄ ) + g 1 ( u , 1 , y ̄ ) where g ₁(.) is a function of the distribution of the shocks, the initial shock and initial conditions. Then Q I R F 1 ( u , 1 | y ̄ ) = I R F 1 ( 1 , y ̄ ) + d y 0 s Q ε ( u ) − d y ̄ Q ε ( u ) . The latest expression corresponds to the difference of the quantiles, and this is the parameter estimated by the QIRF analysis above. This is in general different from the quantile of the difference that, in this case, with y 0 s − y ̄ ≥ 0 , is Q y 1 s − y 1 0 ( η | y ̄ ) = I R F 1 ( 1 , y ̄ ) + d Q ε ( η ) ( y 0 s − y ̄ ) . Note that in the latter, the quantiles of ɛ correspond to those of y 1 s − y 1 0 . However, the quantiles of QIRF₁ are not indexed in general by those of ɛ. In other words, in general Q Q I R F 1 ( η , 1 | y ̄ ) ≠ Q 1 η | y 0 s − Q 1 ( η | y ̄ ) .

For h = 2, y 2 = b y 1 + a + ( 1 + d y 1 ) ε 2 = b 2 y 0 + a ( 1 + b ) + b ( 1 + d y 0 ) ε 1 + ( 1 + d ( a + b y 0 ) ) ε 2 + d ( 1 + d y 0 ) ε 1 ε 2 . Note that for h = 2, the quantiles would depend on the joint distribution of (ɛ ₁, ɛ ₂) which is non-standard even if they are independent shocks. Note that in the case that d = 0, i.e. location-shift model, the quantile function of bɛ ₁ + ɛ ₂ is a simple function (e.g. if ɛ _i ∼ iid N(0, 1), i = 1, 2 then bɛ ₁ + ɛ ₂ ∼ N(0, 1 + b ²)). When d ≠ 0, the nonlinear term ɛ ₁ ɛ ₂, however, produces a non-standard distribution.

Let Q I R F 2 ( ( τ 1 , τ 2 ) , 1 | y ̄ ) = Q 2 ( ( τ 1 , τ 2 ) | y 0 s ) − Q 2 ( ( τ 1 , τ 2 ) | y ̄ ) , where Q ₂(.|.) is the two-periods ahead forecast. Then, the distribution of this can be obtained by evaluating Q I R F 2 ( ( u 1 , u 2 ) , 1 | y ̄ ) , u _i ∼ iid U(0, 1), i = 1, 2. Note that Q I R F 2 ( ( u 1 , u 2 ) , 1 | y ̄ ) = I R F 2 ( 1 , y ̄ ) + g 2 ( ( u 1 , u 2 ) , 1 , y ̄ ) , where g ₂(.) is a function of the distribution of the shocks, the initial shock and initial conditions.

A generalization of the above results in the random variable Q I R F h ( u ( h ) , 1 | y ̄ ) = Q h ( u ( h ) | y 0 s ) − Q h ( u ( h ) | y ̄ ) = I R F h ( 1 , y ̄ ) + g h ( u ( h ) , 1 , y ̄ ) , where u ^(h) = (u ₁, …, u _h) is a sequence of independent uniform random variables and g _h(.) is a function of those and the shocks.

The RVARQ model is used to estimate the quantile functions semi-parametrically, in this case, using univariate QR models iteratively. An alternative simulation procedure would estimate a , b , d , σ ε 2 and then it would require to use a distribution of estimated ɛ to compute the quantiles and the corresponding quantile paths. This would require a double estimation procedure, as the parameters and the implied distribution of the shocks are related to each other. The RVARQ model does not need this as it estimates the quantiles directly.

References

Arcones, M., and B. Yu. 1995. “Central Limit Theorems for Empirical and U-Processes of Stationary Mixing Sequences.” Journal of Theoretical Probability 7: 47–71.10.1007/BF02213360Search in Google Scholar

Aron, J., R. Macdonald, and J. Muellbauer. 2014. “Exchange Rate Pass-Through in Developing and Emerging Markets: A Survey of Conceptual, Methodological and Policy Issues, and Selected Empirical Findings.” Journal of Development Studies 50 (1): 101–43. https://doi.org/10.1080/00220388.2013.847180.Search in Google Scholar

Bernard, C., and C. Czado. 2015. “Conditional Quantiles and Tail Dependence.” Journal of Multivariate Analysis 138: 104–26. https://doi.org/10.1016/j.jmva.2015.01.011.Search in Google Scholar

Carlier, G., V. Chernozhukov, and A. Galichon. 2016. “Vector Quantile Regression.” Annals of Statistics 44: 1165–92. https://doi.org/10.1214/15-aos1401.Search in Google Scholar

Caselli, F. G., and A. Roitman. 2016. “Non-linear Exchange Rate Pass-Through in Emerging Markets.” In IMF Working Paper WP/16/1.10.5089/9781513578262.001Search in Google Scholar

Ca‘Zorzi, M., E. Hahn, and M. Sánchez. 2007. “Exchange Rate Pass-Through in Emerging Markets.” IUP Journal of Monetary Economics 4: 84–102.10.2139/ssrn.970654Search in Google Scholar

Chavleishvili, S., and S. Manganelli. 2019. “Forecasting and Stress Testing with Quantile Vector Autoregression.” In Working Paper. Also available at http://www.simonemanganelli.org/Simone/Research_files/QVAR%20January%202\\,019.pdf.10.2139/ssrn.3489065Search in Google Scholar

Chernozhukov, V., and C. Hansen. 2005. “An IV Model of Quantile Treatment Effects.” Econometrica 73: 245–61. https://doi.org/10.1111/j.1468-0262.2005.00570.x.Search in Google Scholar

Chernozhukov, V., and C. Hansen. 2006. “Instrumental Quantile Regression Inference for Structural and Treatment Effects Models.” Journal of Econometrics 132: 491–525. https://doi.org/10.1016/j.jeconom.2005.02.009.Search in Google Scholar

Chernozhukov, V., and C. Hansen. 2008. “Instrumental Variable Quantile Regression: A Robust Inference Approach.” Journal of Econometrics 142: 379–98. https://doi.org/10.1016/j.jeconom.2007.06.005.Search in Google Scholar

Choudhri, E., and D. Hakura. 2006. “Exchange Rate Pass-Through to Domestic Prices: Does the Inflationary Environment Matter?” Journal of International Money and Finance 25: 614–39. https://doi.org/10.1016/j.jimonfin.2005.11.009.Search in Google Scholar

Christiano, L., M. Eichenbaum, and C. Evans. 1996. “The Effects of Monetary Policy Shocks: Evidence from the Flow of Funds.” The Review of Economics and Statistics 78: 16–34. https://doi.org/10.2307/2109845.Search in Google Scholar

Edwards, S. 2006. “The Relationship between Exchange Rates and Inflation Targeting Revisited.” In NBER Working Paper 12163.10.3386/w12163Search in Google Scholar

Engle, R. F., and S. Manganelli. 2004. “CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles.” Journal of Business & Economic Statistics 22: 367–81. https://doi.org/10.1198/073500104000000370.Search in Google Scholar

Escanciano, J. C., and C. Velasco. 2010. “Specification Tests of Parametric Dynamic Conditional Quantiles.” Journal of Econometrics 159: 209–21. https://doi.org/10.1016/j.jeconom.2010.06.003.Search in Google Scholar

Fan, J., and Q. Yao. 2005. Nonlinear Time Series: Nonparametric and Parametric Methods. Berlin: Springer.Search in Google Scholar

Firpo, S. 2007. “Efficient Semiparametric Estimation of Quantile Treatment Effects.” Econometrica 75 (1): 259–76. https://doi.org/10.1111/j.1468-0262.2007.00738.x.Search in Google Scholar

Galvao, A., and L. Wang. 2015. “Uniformly Semiparametric Efficient Estimation of Treatment Effects with a Continuous Treatment.” Journal of the American Statistical Association 110: 1528–42. https://doi.org/10.1080/01621459.2014.978005.Search in Google Scholar

Galvao, A., K. Kato, G. Montes-Rojas, and J. Olmo. 2014. “Testing Linearity against Threshold Effects: Uniform Inference in Quantile Regression.” Annals of the Institute of Mathematical Statistics 66 (2): 413–39. https://doi.org/10.1007/s10463-013-0418-9.Search in Google Scholar

Hallin, M., D. Paindaveine, and M. Šiman. 2010. “Multivariate Quantiles and Multiple-Output Regression Quantiles: From L1 Optimization to Halfspace Depth.” Annals of Statistics 38 (2): 635–69. https://doi.org/10.1214/09-aos723.Search in Google Scholar

Han, H., O. Linton, T. Oka, and Y.-J. Whang. 2016. “The Cross-Quantilogram: Measuring Quantile Dependence and Testing Directional Predictability between Time Series.” Journal of Econometrics 193 (1): 251–70. https://doi.org/10.1016/j.jeconom.2016.03.001.Search in Google Scholar

Inoue, A., and L. Kilian. 2013. “Inference on Impulse Response Functions in Structural VAR Models.” Journal of Econometrics 177: 1–13. https://doi.org/10.1016/j.jeconom.2013.02.009.Search in Google Scholar

Inoue, A., and L. Kilian. 2016. “Joint Confidence Sets for Structural Impulse Responses.” Journal of Econometrics 192: 421–32. https://doi.org/10.1016/j.jeconom.2016.02.008.Search in Google Scholar

Kato, K. 2009. “Asymptotics for Argmin Processes: Convexity Arguments.” Journal of Multivariate Analysis 100: 1816–29. https://doi.org/10.1016/j.jmva.2009.02.008.Search in Google Scholar

Kilian, L. 2009. “Not all Prices are Alike: Disentangling Demand and Supply Shocks in the Curde Oil Market.” The American Economic Review 99: 1053–69. https://doi.org/10.1257/aer.99.3.1053.Search in Google Scholar

Kilian, L., and S. Manganelli. 2007. “Quantifying the Risk of Deflation.” Journal of Money, Credit, and Banking 121: 1047–72. https://doi.org/10.1111/j.0022-2879.2007.00036.x.Search in Google Scholar

Koenker, R., and G. Basset. 1978. “Regression Quantiles.” Econometrica 46 (1): 33–50. https://doi.org/10.2307/1913643.Search in Google Scholar

Koenker, R., and Z. Xiao. 2006. “Quantile Autoregression.” Journal of the American Statistical Association 101: 980–90. https://doi.org/10.1198/016214506000000672.Search in Google Scholar

Ma, L., and R. Koenker. 2006. “Quantile Regression Methods for Recursive Structural Equation Models.” Journal of Econometrics 134 (2): 471–506. https://doi.org/10.1016/j.jeconom.2005.07.003.Search in Google Scholar

McCarthy, J. 2007. “Pass-Through of Exchange Rates and Import Prices to Domestic Inflation in Some Industrialized Economies.” Eastern Economic Journal 33 (4): 511–37. https://doi.org/10.1057/eej.2007.38.Search in Google Scholar

Menon, J. 1995. “Exchange Rate Pass-Through.” Journal of Economic Surveys 9: 197–231. https://doi.org/10.1111/j.1467-6419.1995.tb00114.x.Search in Google Scholar

Montes-Rojas, G. 2017. “Reduced Form Vector Directional Quantiles.” Journal of Multivariate Analysis 158: 20–30. https://doi.org/10.1016/j.jmva.2017.03.007.Search in Google Scholar

Montes-Rojas, G. 2019. “Multivariate Quantile Impulse Response Functions.” Journal of Time Series Analysis 40: 739–52.10.1111/jtsa.12452Search in Google Scholar

Paindaveine, D., and M. Šiman. 2011. “On Directional Multiple-Output Quantile Regression.” Journal of Multivariate Analysis 102: 193–212. https://doi.org/10.1016/j.jmva.2010.08.004.Search in Google Scholar

Paindaveine, D., and M. Šiman. 2012. “Computing Multiple-Output Regression Quantile Regions.” Computational Statistics & Data Analysis 56: 840–53. https://doi.org/10.1016/j.csda.2010.11.014.Search in Google Scholar

Qu, Z. 2008. “Testing for Structural Change in Quantile Regression.” Journal of Econometrics 146: 170–84. https://doi.org/10.1016/j.jeconom.2008.08.006.Search in Google Scholar

Ramey, V. 2016. “Macroeconomic Shocks and Their Propagation.” In Handbook of Macroeconomics, Vol. 2, edited by J. B. Taylor, and H. Uhlig, 71–162. North-Holland, Amsterdam: Elsevier.10.3386/w21978Search in Google Scholar

Ruzicka, J. 2021. Quantile Structural Vector Autoregression. University Carlos III, Madrid: Mimeo.Search in Google Scholar

Stock, J., and M. Watson. 2016. “Dynamic Factor Models, Factor-Augmented Vector Autoregressions, and Structural Vector Autoregression in Macroeconomics.” In Handbook of Macroeconomics, Vol. 2, edited by J. B. Taylor, and H. Uhlig, 415–525. North-Holland, Amsterdam: Elsevier.10.1016/bs.hesmac.2016.04.002Search in Google Scholar

Taylor, J. 2000. “Low Inflation, Pass-Through and the Pricing Power of Firms.” European Economic Review 44: 1389–408. https://doi.org/10.1016/s0014-2921(00)00037-4.Search in Google Scholar

Wei, Y. 2008. “An Approach to Multivariate Covariate-dependent Quantile Contours with Application to Bivariate Conditional Growth Charts.” Journal of the American Statistical Association 103 (481): 397–409. https://doi.org/10.1198/016214507000001472.Search in Google Scholar

White, H., T.-H. Kim, and S. Manganelli. 2015. “VAR for VaR: Measuring Tail Dependence Using Multivariate Regression Quantiles.” Journal of Econometrics 187: 169–88. https://doi.org/10.1016/j.jeconom.2015.02.004.Search in Google Scholar

Received: 2021-01-11

Revised: 2021-09-07

Accepted: 2021-11-24

Published Online: 2021-12-17

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/jtse-2021-0002

Keywords for this article

impulse-response functions; vector autoregressive models; multivariate quantiles; pass-through