RESEARCH ARTICLE
Forecasting inflation time series using score-driven dynamic models and combination methods: The case of Brazil
Carlos Henrique Dias Cordeiro de Castro,
Fernando Antonio Lucena Aiube,
Corresponding Author
Carlos Henrique Dias Cordeiro de Castro
UERJ – Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil
Correspondence
Carlos Henrique Dias Cordeiro de Castro, UERJ – Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil.
Email: [email protected]
Search for more papers by this authorFernando Antonio Lucena Aiube
UERJ – Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil
Search for more papers by this authorCarlos Henrique Dias Cordeiro de Castro,
Fernando Antonio Lucena Aiube,
Corresponding Author
Carlos Henrique Dias Cordeiro de Castro
UERJ – Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil
Correspondence
Carlos Henrique Dias Cordeiro de Castro, UERJ – Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil.
Email: [email protected]
Search for more papers by this authorFernando Antonio Lucena Aiube
UERJ – Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil
Search for more papers by this authorFirst published: 02 September 2022
Funding information:
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
Abstract
This paper evaluates the inflation forecasting models of a small open economy. We compare forecasts of Brazilian inflation (consumer price index—IPCA) based on the generalized autoregressive score (GAS) approach with the forecasts reported in the Focus Bulletin (the Brazilian Central Bank's weekly consensus survey report), as well as with other benchmark models. We selected a restricted number of the variables used most in the literature to compose the forecasting models. Furthermore, based on a confidence set, we applied a wide range of combination methods to the forecast components aiming at improving the forecasting power. Point and density forecast tests were performed to verify the performance of the models. The results showed that the GAS models had good predictive performance for the IPCA, and it was possible to improve their accuracy by combining them with other models.
CONFLICTS OF INTEREST
The authors hereby declare that they have no competing interests.
Open Research
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.
REFERENCES
- Aiolfi, M., & Timmermann, A. (2006). Persistence in forecasting performance and conditional combination strategies. Journal of Econometrics, 135(1–2), 31–53. https://doi.org/10.1016/j.jeconom.2005.07.015
- Amisano, G., Geweke, J. (2013). Prediction using several macroeconomic models. Working Paper Series 1537, European Central Bank. https://doi.org/10.2139/ssrn.2252579
10.2139/ssrn.2252579 Google Scholar
- Armstrong, J. S. (1989). Combining forecasts: The end of the beginning or the beginning of the end? International Journal of Forecasting, 5(4), 585–588. https://doi.org/10.1016/0169-2070(89)90013-7
- J. S. Armstrong (Ed.). (2001). Principles of forecasting: A handbook for researchers and practitioners (Vol. 30). Springer Science & Business Media. https://doi.org/10.1007/978-0-306-47630-3
10.1007/978-0-306-47630-3 Google Scholar
- Arruda, E., Ferreira, R., & Castelar, I. (2011). Modelos lineares e não lineares da curva de phillips para previsão da taxa de inflação no brasil. Revista Brasileira de Economia, 65, 237–252.
- Babaie-Kafaki, S., & Roozbeh, M. (2017). A revised Cholesky decomposition to combat multicollinearity in multiple regression models. Journal of Statistical Computation and Simulation, 87(12), 2298–2308. https://doi.org/10.1080/00949655.2017.1328599
- Barbosa-Filho, N. H. (2008). Inflation targeting in Brazil: 1999–2006. International Review of Applied Economics, 22(2), 187–200. https://doi.org/10.1080/02692170701880684
10.1080/02692170701880684 Google Scholar
- Bates, J. M., & Granger, C. W. (1969). The combination of forecasts. Journal of the Operational Research Society, 20(4), 451–468. https://doi.org/10.1057/jors.1969.103
- Bergmeir, C., Hyndman, R. J., & Koo, B. (2018). A note on the validity of cross-validation for evaluating autoregressive time series prediction. Computational Statistics & Data Analysis, 120, 70–83. https://doi.org/10.1016/j.csda.2017.11.003
- Bisaglia, L., & Grigoletto, M. (2021). A new time-varying model for forecasting long-memory series. Statistical Methods and Applications, 30(1), 139–155.
- Blasques, F., Gorgi, P., & Koopman, S. J. (2017). Accelerating GARCH and score-driven models: Optimality, estimation and forecasting. Estimation and Forecasting (July 18, 2017). https://doi.org/10.2139/ssrn.3004598
10.2139/ssrn.3004598 Google Scholar
- Blasques, F., Koopman, S. J., Lucas, A., & Schaumburg, J. (2016). Spillover dynamics for systemic risk measurement using spatial financial time series models. Journal of Econometrics, 195(2), 211–223. https://doi.org/10.1016/j.jeconom.2016.09.001
- Blazsek, S., Carrizo, D., Eskildsen, R., & Gonzalez, H. (2018). Forecasting rate of return after extreme values when using AR-t-GARCH and QAR-Beta-t-EGARCH. Finance Research Letters, 24, 193–198. https://doi.org/10.1016/j.frl.2017.09.006
- Blazsek, S., Chavez, H., & Mendez, C. (2016). Model stability and forecast performance of Beta-t-EGARCH. Applied Economics Letters, 23(17), 1219–1223. https://doi.org/10.1080/13504851.2016.1145343
- Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307–327. https://doi.org/10.1016/0304-4076(86)90063-1
- Box, G. E. P., & Jenkins, G. M. (1976). Time series analysis: Forecasting and control. Holden-Day.
- Capistrán, C., Timmermann, A., & Aiolfi, M. (2010). Forecast combinations, Working Papers.
- Cogley, T., & Sargent, T. J. (2005). Drifts and volatilities: Monetary policies and outcomes in the post WWII US. Review of Economic Dynamics, 8(2), 262–302. https://doi.org/10.1016/j.red.2004.10.009
- Cogley, T., & Sargent, T. J. (2015). Measuring price-level uncertainty and instability in the United States, 1850–2012. Review of Economics and Statistics, 97(4), 827–838. https://doi.org/10.1162/rest_a_00498
- Costa Filho, A. E. (2016). Inflação e incerteza inflacionária no Brasil. Economia Aplicada, 20(4), 355–381. https://doi.org/10.11606/1413-8050/ea125774
10.11606/1413-8050/ea125774 Google Scholar
- Crane, D. B., & Crotty, J. R. (1967). A two-stage forecasting model: Exponential smoothing and multiple regression. Management Science, 13(8), B–501. https://doi.org/10.1287/mnsc.13.8.B501
10.1287/mnsc.13.8.B501 Google Scholar
- Creal, D., Koopman, S. J., & Lucas, A. (2008). A general framework for observation driven time-varying parameter models (No. 08-108/4). Tinbergen Institute Discussion paper. https://doi.org/10.2139/ssrn.1297183
10.2139/ssrn.1297183 Google Scholar
- Creal, D., Koopman, S. J., & Lucas, A. (2013). Generalized autoregressive score models with applications. Journal of Applied Econometrics, 28(5), 777–795. https://doi.org/10.1002/jae.1279
- Dawid, A. P., & Sebastiani, P. (1999). Coherent dispersion criteria for optimal experimental design. Annals of Statistics, 27, 65–81. https://doi.org/10.1214/aos/1018031101
- Del Negro, M., & Primiceri, G. E. (2015). Time varying structural vector autoregressions and monetary policy: A corrigendum. The Review of Economic Studies, 82(4), 1342–1345. https://doi.org/10.1093/restud/rdv024
- Delle Monache, D., & Petrella, I. (2017). Adaptive models and heavy tails with an application to inflation forecasting. International Journal of Forecasting, 33(2), 482–501. https://doi.org/10.1016/j.ijforecast.2016.11.007
- Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association, 74(366a), 427–431.
10.1080/01621459.1979.10482531 Google Scholar
- Diebold, F. X., & Mariano, R. S. (1995). Comparing predictive accuracy. Journal of Business & Economic Statistics, 13, 253–265.
- Diebold, F. X., Gunther, T. A., & Tay, S. A. (1998). Evaluating density forecasts with applications to financial risk management. International Economic Review, 39(4), 863–883.
- Elliott, G., Gargano, A., & Timmermann, A. (2013). Complete subset regressions. Journal of Econometrics, 177(2), 357–373. https://doi.org/10.1016/j.jeconom.2013.04.017
- Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the Econometric Society, 50, 987–1007. https://doi.org/10.2307/1912773
- Ewey, W. K., & West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55(3), 703–708. https://doi.org/10.2307/1913610
- Faust, J., & Wright, J. H. (2013). Forecasting inflation. In Handbook of economic forecasting (Vol. 2) (pp. 2–56). Elsevier. https://doi.org/10.1016/B978-0-444-53683-9.00001-3
10.1016/B978-0-444-53683-9.00001-3 Google Scholar
- Fernandes, M., Medeiros, M. C., & Scharth, M. (2014). Modeling and predicting the CBOE market volatility index. Journal of Banking and Finance, 40(1), 1–10. https://doi.org/10.1016/j.jbankfin.2013.11.004
- Figueiredo, E., & Marques, A. (2009). Inflação inercial como um processo de longa memória: análise a partir de um modelo ARFIMA-FIGARCH. Estudos Economicos, 39, 437–458. https://doi.org/10.1590/S0101-41612009000200008
10.1590/S0101-41612009000200008 Google Scholar
- Gaglianone, W. P., Issler, J. V., & Matos, S. M. (2017). Applying a microfounded-forecasting approach to predict Brazilian inflation. Empirical Economics, 53(1), 137–163. https://doi.org/10.1007/s00181-016-1163-8
- Garcia, M. G., Medeiros, M. C., & Vasconcelos, G. F. (2017). Real-time inflation forecasting with high-dimensional models: The case of Brazil. International Journal of Forecasting, 33(3), 679–693. https://doi.org/10.1016/j.ijforecast.2017.02.002
- Genre, G., Kenny, A., Meyler, A., & Timmermann, A. (2013). Combining expert forecasts: Can anything beat the simple average? International Journal of Forecasting, 29(1), 108–121. https://doi.org/10.1016/j.ijforecast.2012.06.004
- Geweke, J., & Amisano, G. (2010). Comparing and evaluating Bayesian predictive distributions of asset returns. International Journal of Forecasting, 26(2), 216–230. https://doi.org/10.1016/j.ijforecast.2009.10.007
- Giacomini, R., & Rossi, B. (2010). Forecast comparisons in unstable environments. Journal of Applied Econometrics, 25(4), 595–620. https://doi.org/10.1002/jae.1177
- Giacomini, R., & White, H. (2006). Tests of conditional predictive ability. Econometrica, 74(6), 1545–1578. https://doi.org/10.1111/j.1468-0262.2006.00718.x
- Gneiting, T., & Raftery, A. E. (2005). Weather forecasting with ensemble methods. Science, 310(5746), 248–249. https://doi.org/10.1126/science.1115255
- Gneiting, T., Balabdaoui, F., & Raftery, A. E. (2007). Probabilistic forecasts, calibration and sharpness. Journal of the Royal Statistical Society B, 69(2), 243–268. https://doi.org/10.1111/j.1467-9868.2007.00587.x
- Gorgi, P., Koopman, S. J., & Li, M. (2019). Forecasting economic time series using score-driven dynamic models with mixed-data sampling. International Journal of Forecasting, 35(4), 1735–1747. https://doi.org/10.1016/j.ijforecast.2018.11.005
- Granger, C. W., & Ramanathan, R. (1984). Improved methods of combining forecasts. Journal of Forecasting, 3(2), 197–204. https://doi.org/10.1002/for.3980030207
- Hansen, P. R., & Lunde, A. (2005). A forecast comparison of volatility models: Does anything beat a GARCH (1, 1)? Journal of Applied Econometrics, 20(7), 873–889. https://doi.org/10.1002/jae.800
- Hansen, P. R., Lunde, A., & Nason, J. M. (2011). The model confidence set. Econometrica, 79(2), 453–497. https://doi.org/10.3982/ecta5771
- Harvey, A. (2013). Dynamic models for volatility and heavy tails: With applications to financial and economic time series. Cambridge University Press. https://doi.org/10.1017/CBO9781139540933
10.1017/CBO9781139540933 Google Scholar
- Harvey, A., & Tirthankar C. (2008). “Beta-t-(E)GARCH”, Discussion Paper University of Cambridge CWPE 08340.
- Harvey, A. C., & Chakravarty, T. (2008). Beta-t-(E)GARCH. Cambridge University, Faculty of Economics. WP-8340.
- Hendry, D. F., & Clements, M. P. (2004). Pooling of forecasts. The Econometrics Journal, 7(1), 1–31. https://doi.org/10.1111/j.1368-423X.2004.00119.x
10.1111/j.1368-423X.2004.00119.x Google Scholar
- Hong, Y., & Li, H. (2005). Nonparametric specification testing for continuous-time models with applications to term structure of interest rates. Review of Financial Studies, 18(1), 37–84. https://doi.org/10.1093/rfs/hhh006
- Hsiao, C., & Wan, S. K. (2014). Is there an optimal forecast combination? Journal of Econometrics, 178, 294–309. https://doi.org/10.1016/j.jeconom.2013.11.003
- Iversen, J., Laséen, S., Lundvall, H., & Söderström, U. (2016). Realtime forecasting for monetary policy analysis: The case of Sveriges Riksbank, Working Paper 318, Sveriges Riksbank Working Paper Series.
- Jose, V. R. R., & Winkler, R. L. (2008). Simple robust averages of forecasts: Some empirical results. International Journal of Forecasting, 24(1), 163–169. https://doi.org/10.1016/j.ijforecast.2007.06.001
- Kapetanios, G., Labhard, V., & Price, S. (2008). Forecasting using bayesian and information-theoretic model averaging: An application to UK inflation. Journal of Business & Economic Statistics, 26(1), 33–41. https://doi.org/10.1198/073500107000000232
- Kascha, C., & Ravazzolo, F. (2010). Combining inflation density forecasts. Journal of Forecasting, 29(1–2), 231–250. https://doi.org/10.1002/for.1147
- Knüppel, M. (2015). Evaluating the calibration of multi-step-ahead density forecasts using raw moments. Journal of Business & Economic Statistics, 33(2), 270–281. https://doi.org/10.1080/07350015.2014.948175
- Kohlscheen, E. (2012). Uma nota sobre erros de previsão da inflação de curto-prazo. Revista Brasileira de Economia, 66, 289–297. https://doi.org/10.1590/S0034-71402012000300002
10.1590/S0034-71402012000300002 Google Scholar
- Koopman, S. J., Lucas, A., & Scharth, M. (2016). Predicting time-varying parameters with parameter-driven and observation-driven models. The Review of Economics and Statistics, 98(1), 97–110. https://doi.org/10.1162/REST_a_00533
- Kwiatkowski, D., Phillips, P. C., Schmidt, P., & Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics, 54(1–3), 159–178. https://doi.org/10.1016/0304-4076(92)90104-Y
- Lucas, A., & Zhang, X. (2016). Score-driven exponentially weighted moving averages and value-at-risk forecasting. International Journal of Forecasting, 32(2), 293–302. https://doi.org/10.1016/j.ijforecast.2015.09.003
- McNees, S. K. (1992). The uses and abuses of ‘consensus’ forecasts. Journal of Forecasting, 11(8), 703–710. https://doi.org/10.1002/for.3980110807
- Medeiros, M., Vasconcelos, G., & Freitas, E. (2016). Forecasting Brazilian inflation with high-dimensional models. Brazilian Review of Econometrics, 36, 68–100.
- Mincer, J. A., & Zarnowitz, V. (1969). The evaluation of economic forecasts. In Economic forecasts and expectations: Analysis of forecasting behavior and performance (pp. 3–46). NBER.
- Nelson, B. D. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59, 347–370. https://doi.org/10.2307/2938260
- Newbold, P., & Granger, C. W. (1974). Experience with forecasting univariate time series and the combination of forecasts. Journal of the Royal Statistical Society: Series A (General), 137(2), 131–146. https://doi.org/10.2307/2344546
- Newey, W. K., & West, K. D. (1987). A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix. Econometrica, 55(3), 703–708.
- Nowotarski, J., Raviv, E., Trück, S., & Weron, R. (2014). An empirical comparison of alternative schemes for combining electricity spot price forecasts. Energy Economics, 46, 395–412. https://doi.org/10.1016/j.eneco.2014.07.014
- Nystrup, P., Madsen, H., & Lindström, E. (2017). Long memory of financial time series and hidden Markov models with time-varying parameters. Journal of Forecasting, 36(8), 989–1002. https://doi.org/10.1002/for.2447
- Pettenuzzo, D., & Timmermann, A. (2015). Internet appendix for “model instability and forecasting performance”.
- Phillips, P. C., & Perron, P. (1988). Testing for a unit root in time series regression. Biometrika, 75(2), 335–346. https://doi.org/10.1093/biomet/75.2.335
- Primiceri, G. E. (2005). Time varying structural vector autoregressions and monetary policy. The Review of Economic Studies, 72(3), 821–852. https://doi.org/10.1111/j.1467-937x.2005.00353.x
- Rossi, B., & Inoue, A. (2012). Out-of-sample forecast tests robust to the choice of window size. Journal of Business and Economic Statistics, 30(3), 432–453. https://doi.org/10.1080/07350015.2012.693850
- Rossi, B., & Sekhposyan, T. (2016). Forecast rationality tests in the presence of instabilities, with applications to Federal Reserve and survey forecasts. Journal of Applied Econometrics, 31(3), 507–532. https://doi.org/10.1002/jae.2440
- Rossi, B., & Soupre, M. (2017). Implementing tests for forecast evaluation in the presence of instabilities. The Stata Journal, 17(4), 850–865. https://doi.org/10.1177/1536867X1801700405
- Samuels, J., & Sekkel, R. (2017). Model confidence sets and forecast combination. International Journal of Forecasting, 33, 48–60. https://doi.org/10.1016/j.ijforecast.2016.07.004
- Stock, J., & Watson, M. (1996). Evidence on structural instability in macroeconomic time series relations. Journal of Business and Economic Statistics, 14(1), 14–30.
- Stock, J. H., & Watson, M. W. (2004). Combination forecasts of output growth in a seven-country data set. Journal of Forecasting, 23(6), 405–430. https://doi.org/10.1002/for.928
- Stock, J. H., & Watson, M. W. (2007). Why has U.S. inflation become harder to forecast? Journal of Money, Credit and Banking, 39, 3–33. https://doi.org/10.1111/j.1538-4616.2007.00014.x
- Thomson, M. E., Pollock, A. C., Önkal, D., & Gönül, M. S. (2019). Combining forecasts: Performance and coherence. International Journal of Forecasting, 35(2), 474–484. https://doi.org/10.1016/j.ijforecast.2018.10.006
- Timmermann, A. (2006). Forecast combinations. In G. Elliott, C. Granger, & A. Timmermann (Eds.), Handbook of economic forecasting (Vol. 1, pp. 135–196). Elsevier. https://doi.org/10.1016/S1574-0706(05)01004-9
- Triches, D., & Feijó, F. T. (2017). Uma estimação da Curva de Phillips híbrida para o Brasil no regime de metas de inflação. Economia Aplicada, 21(1), 29–43. https://doi.org/10.11606/1413-8050/ea150953
10.11606/1413-8050/ea150953 Google Scholar
- Weiss, C. E., Raviv, E., & Roetzer, G. (2018). Forecast combinations in R using the ForecastComb package. R Journal, 10(2), 262–281.
- West, K. D., & McCracken, M. W. (1998). Regression-based tests of predictive ability. International Economic Review, 39, 817–840.
- Wright, J. H. (2009). Forecasting US inflation by Bayesian model averaging. Journal of Forecasting, 28(2), 131–144. https://doi.org/10.1002/for.1088
- Zhang, B. (2019). Real-time inflation forecast combination for time-varying coefficient models. Journal of Forecasting, 38(3), 175–191. https://doi.org/10.1002/for.2563
