Forecasting inflation in Argentina

Garegnani, María Lorena; Gómez Aguirre, Maximiliano Working Paper Forecasting inflation in Argentina IDB Working Paper Series, No. IDB-WP-891 Provided in Cooperation with: Inter-American Development Bank (IDB), Washington, DC Suggested Citation: Garegnani, María Lorena; Gómez Aguirre, Maximiliano (2018) : Forecasting inflation in Argentina, IDB Working Paper Series, No. IDB-WP-891, Inter-American Development Bank (IDB), Washington, DC, https://doi.org/10.18235/0001160 This Version is available at: https://hdl.handle.net/10419/208111 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/3.0/igo/legalcode Forecasting Inflation in Argentina Lorena Garegnani Maximiliano Gómez Aguirre IDB WORKING PAPER SERIES Nº IDB-WP-891 June 2018 Department of Research and Chief Economist Inter-American Development Bank June 2018 Forecasting Inflation in Argentina Lorena Garegnani Maximiliano Gómez Aguirre Central Bank of Argentina Cataloging-in-Publication data provided by the Inter-American Development Bank Felipe Herrera Library Garegnani, María Lorena. Forecasting inflation in Argentina / Lorena Garegnani, Maximiliano Gómez Aguirre. p. cm. — (IDB Working Paper Series ; 891) Includes bibliographic references. 1. Inflation (Finance)-Argentina-Forecasting. 2. Inflation (Finance)-ArgentinaEconometric models. I. Gómez Aguirre, Maximiliano. II. 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The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the Inter-American Development Bank, its Board of Directors, or the countries they represent. http://www.iadb.org 2018 Abstract* In 2016 the Central Bank of Argentina began to announce inflation targets. In this context, providing authorities with good estimates of relevant macroeconomic variables is crucial for making pertinent corrections in order to reach the desired policy goals. This paper develops a group of models to forecast inflation for Argentina, which includes autoregressive models and different scale Bayesian VARs (BVAR), and compares their relative accuracy. The results show that the BVAR model can improve the forecast ability of the univariate autoregressive benchmark’s model of inflation. The Giacomini-White test indicates that a BVAR performs better than the benchmark in all forecast horizons. Statistical differences between the two BVAR model specifications (small and large-scale) are not found. However, looking at the RMSEs, one can see that the larger model seems to perform better for longer forecast horizons. JEL classifications: C11, C13, C32, C53 Keywords: Bayesian Vector Autoregressive, Forecasting, Prior specification, Marginal likelihood, Small-scale and large-scale models * This study was undertaken within the framework of CEMLA’s Joint Research Program 2017 coordinated by the Central Bank of Colombia. The authors thank counseling and technical advisory provided by the Financial Stability and Development (FSD) Group of the Inter-American Development Bank in the process of writing this document. The opinions expressed in this publication are those of the authors and do not reflect the views of CEMLA, the FSD group, the Inter-American Development Bank or the Central Bank of Argentina. 2 1. Introduction Long-term nominal commitments such as labor contracts, mortgages and other debt are widespread features of modern economies. Forecasting how the general price level will evolve over the life of a commitment is an essential part of private sector decision-making. The existence of long-term nominal obligations is also among the primary reasons economists generally believe that monetary policy is not neutral, at least over moderate horizons. Central banks aim is to keep inflation stable, and perhaps also to keep output near an efficient level. With these objectives, the New Keynesian model makes explicit that optimal policy will depend on optimal forecasts (e.g., Svensson, 2005), and further, that policy will be most effective when it is well understood by the public. Under “inflation targeting,” central banks have generally released forecasts in quarterly “Inflation Reports” in order to be more transparent in their actions. The costs and benefits of transparency are widely debated, but the need for a central bank to be concerned with inflation forecasting is broadly agreed. In short, inflation forecasting is of the foremost importance to households, businesses, and policymakers. During the year 2016, the Central Bank of Argentina has begun to announce inflation targets. In this context, providing the authorities with good estimates of relevant macroeconomic variables turns out to be crucial to making the pertinent corrections in order to reach the desired policy goals. A standard tool in macroeconomics that is widely employed in forecasting is Vector Autoregressive (VAR) analysis. VARs are flexible time series models that can capture complex dynamic relationships among macroeconomic aggregates. However, their dense parameterization often leads to unstable inference and inaccurate out-of-sample forecasts, particularly for models with many variables, due to the estimation uncertainty of the parameters. Litterman (1980) and Doan, Litterman, and Sims (1984) have proposed to combine the likelihood function (the data) with some informative prior distributions (the researcher’s belief about the values of coefficients) to improve the forecasting performance of VAR models, introducing a Bayesian approach into VAR modeling. In any Bayesian inference, a fundamental yet challenging step is prior specification, which influences posterior distributions of the unknown parameters and, consequently, the forecasts 3 (Geweke, 2005). Fortunately, the literature has proposed some methodologies to set how informative the prior distributions should be. Regarding prior selection, Litterman (1980) and Doan, Litterman, and Sims (1984) set the tightness of the prior by maximizing the out-of-sample forecasting performance of a small-scale model. Many authors follow this strategy, such as Robertson and Tallman (1999) and Wright (2009), and Giannone et. al (2014), who minimize the Root Mean Square Error (RMSE) of the forecasts. On the other hand, Banbura, Giannone and Reichlin (2008) propose controlling the overfitting caused by the considerable number of variables in the model by selecting the shrinkage of the coefficients in such a way as to provide appropriate fitting “in-sample.” Within this second selection strategy, we can find authors such as Giannone, Lenza and Primiceri (2012), Bloor and Mathenson (2009), Carriero, Clark and Marcellino (2015) and Koop (2011). Banbura, Giannone and Reichlin (2008) showed that by applying Bayesian VAR methodology, they were able to handle large unrestricted VARs models and therefore they demonstrated that VAR framework can be applied to empirical problems that require the analysis of more than a few sets of time series. The authors showed that a Bayesian VAR is a viable alternative to factor models or panel VARs for analysis of large dynamic systems. This paper develops a group of models to forecast inflation for Argentina, which includes autoregressive models, and different scale Bayesian VARs (BVAR), and compares their relative accuracy. The paper is organized as follows. Section 2 presents the methodological aspects related to the application of Bayesian analysis in a VAR framework, and Section 3 presents a brief description of the data. Section 4 goes through the empirical results, and finally, Section 5 concludes. 2. Bayesian VAR Methodology A VAR model has the following structure 𝐲𝐲t=𝐜𝐜+𝐁𝐁𝟏𝟏𝐲𝐲t−1+. . . +𝐁𝐁p𝐲𝐲t−p+𝛆𝛆t,(1) 4 where 𝐲𝐲t is a n × 1 vector of endogenous variables, 𝛆𝛆t ~ N(𝟎𝟎,𝚺𝚺) is a n × 1 vector of exogenous shocks, c is a n × 1 vector of constants, 𝐁𝐁𝟏𝟏 to 𝐁𝐁p are n × n matrices and 𝚺𝚺 is n × n covariance matrix. The BVAR coefficients are a weighted average of the prior mean (researcher’s belief) and the maximum likelihood (ML) estimators (inferred from the data), with the inverse covariance of the prior and the ML estimators as weights. Consider the following posterior distribution for the VAR coefficients 𝛃𝛃|𝛀𝛀 ~ N(𝛃𝛃0,𝛀𝛀−𝟏𝟏ξ) (2) where the vector 𝛃𝛃0 is the prior mean (whose elements will represent the coefficient in equation (1), the matrix Ω is the known variance of the prior and ξ is a scalar parameter controlling the tightness of the prior information. Even though Ω could have many shapes, gamma and Wishart distributions are frequently used in the literature, since they ensure a normally distributed posterior.2 The conditional posterior of 𝛃𝛃 can be obtained by multiplying the prior by the likelihood function. The posterior takes the form 𝛃𝛃|𝛀𝛀,𝐲𝐲 ~ N �𝛃𝛃 �(ξ),𝐕𝐕 �(ξ)�,(3) where 𝛃𝛃 �(ξ)≡vec �𝐁𝐁 �(ξ)�,(4) and 𝐁𝐁 �(ξ)≡(𝐱𝐱′𝐱𝐱 𝚺𝚺−1+(𝛀𝛀ξ)−1)−1(𝐱𝐱′𝐲𝐲𝚺𝚺−1+(𝛀𝛀ξ)−1𝛃𝛃0),(5) 𝐕𝐕 �(ξ)≡(𝐱𝐱′𝐱𝐱 𝚺𝚺−1+(𝛀𝛀ξ)−1)−1.(6) Vectors 𝐲𝐲 and 𝐱𝐱 represent observed data while 𝛃𝛃0 is a matrix where each column corresponds to the prior mean of each equation. It is important to note that if we choose a large value for ξ, the prior will have little weight into the posterior. This translates to a large volatility of the prior and not enough information coming from the prior. On the other hand, if the ξ is set to a small value (i.e., close to zero), the 2 If the posterior distributions are in the same family as the prior probability distribution, the prior and posterior are then called conjugate distributions. 5 prior becomes more informative and the posterior mean moves towards the prior mean. To see this point, we can express (5) as follows: 𝐁𝐁 �(ξ)≡𝛀𝛀 �[𝛀𝛀 0 −1𝛃𝛃0+(𝚺𝚺−1⊗𝐱𝐱′)𝐲𝐲 ] (7) and 𝛀𝛀 �=[𝛀𝛀0 −1+ 𝚺𝚺−1 ⊗𝐱𝐱′𝐱𝐱]−1 (8) If the second element between brackets in equation (7) is multiplied by (𝐱𝐱′𝐱𝐱)−1(𝐱𝐱′𝐱𝐱), we obtain the following equation 𝐁𝐁 �(ξ)≡𝛀𝛀 ��𝛀𝛀 0 −1𝛃𝛃0] + 𝛀𝛀 �[𝚺𝚺−1⊗𝐱𝐱′𝐱𝐱 (𝐱𝐱′𝐱𝐱)−1𝐱𝐱′𝐲𝐲 �(9) 𝐁𝐁 �(ξ)≡𝛀𝛀 ��𝛀𝛀 0 −1𝛃𝛃0] + 𝛀𝛀 �[𝚺𝚺−1⊗𝐱𝐱′𝐱𝐱 𝛃𝛃ols�(10) As can be seen, the posterior is a weighted average between the prior and the Ordinary Least Square (OLS) estimators,3 where the weights are the reciprocal of the prior covariance matrix and the reciprocal of the OLS covariance matrix respectively. As a result, if the information contained in the data is good enough to describe the process behind it, the posterior will move towards the OLS estimators. However, it is important to underscore that, even if the available series are adequate to describe the data generating process, the researcher could still formulate hypothesis about the distribution of the parameters based on his own beliefs. That would imply ignoring the information contained in the data, and usually that kind of decisions are based on strong beliefs. The issue mentioned in the last paragraph demonstrates the need to be cautious about choosing the prior mean and the hyperpriors. In the following subsections, these aspects are discussed in more detail. 2.1 Level or Growth Rate It is unclear a priori whether transforming variables into their growth rates can enhance the forecast performance of a BVAR model. On one hand, the level specification can better accommodate the existence of long-run (cointegrating) relationships across the variables, which would be omitted in a VAR in differences. On the other hand, Clements and Hendry (1996) have shown that in a 3 The OLS estimators of a VAR coincide exactly with the ML estimators conditional on the initial values. 12 Table 2. List of Hyperparameter Values Hyperparameters Values Large-Scale Model Small-Scale Model Autoregressive Coefficient: 0 0 Overall Tightness (λ1): 0.05 0.23 Lag Decay (λ3): 2 2 Exogenous Variable Tightness 1 1 Lag Length 1 1 The hyperparameter λ1 is equal to 0.05 for the large-scale model while the hyperparameter λ1 for the small-scale is 0.23. From a practical point of view, this means that the “true” value of the coefficients estimated (posterior) is probably farther from the prior mean in the small-scale model than in the large-scale one. Another aspect to consider about λ1, is the fact that this hyperparameter impacts on the distribution of the parameters of lagged endogenous and exogenous variables of each equation in the system. In this sense, with more shrinkage for example, it is less probable that the posterior coefficients of the lagged endogenous and exogenous variables depart from the prior. As can be seen in Table 2, the posterior coefficients of the variables in the large-scale model are less probable to depart from the prior than the small-scale ones. Models with many variables will tend to have a better in-sample fit even when λ1 is set to a “loose” value. The posteriors obtained for the smalland the large-scale model of the inflation equation in each type of model are shown in the Appendix. 4.1.2 Forecasting Exercise Our forecasting exercise is conducted in the following way. We estimate the hyperparameters considering the whole sample, through the maximization of the marginal likelihood; and then, we compute the forecasts. As mentioned above, the data set goes from January 2004 to July 2017. We compute one, three and six-step-ahead forecasts with rolling windows. The size of the estimation sample is the same for each forecast horizon. Out-of-sample forecast accuracy is measured in terms of RMSE of the forecasts. Therefore, we obtained three RMSEs for each model. 13 Relative forecast accuracy is analyzed in Table 3, by computed the different combinations of RMSE ratios. On average, the BVAR presents better accuracy than the benchmark independently of the forecast horizon. For immediate horizons, the small-scale model slightly outperforms the large-scale model, but the large-scale model outperforms the small-scale model for longer forecast horizons. Table 3. Relative Forecast Accuracy One Step Ahead Three Step Ahead Six Step Ahead Ratio Small ModelBenchmark Ratio Large ModelBenchmark Ratio Large Model-Small Model Ratio Small ModelBenchmark Ratio Large ModelBenchmark Ratio Large Model-Small Model Ratio Small ModelBenchmark Ratio Large ModelBenchmark Ratio Large ModelSmall Model 0.77 0.90 1.69 0.78 0.77 1.02 0.87 0.82 0.94 In the next subsection, we analyze these results with a Giacomini-White test. 4.2 Forecast Evaluation To evaluate the predictive performance of the different models, we used the tests described earlier. Each column of Table 4 contains the probability value of Giacomini-White test statistic for the different models. Table 4. Giacomini-White Test Forecast Horizon Large BVAR vs. Benchmark Small BVAR vs. Benchmark Difference Between BVAR Models One-Step-Ahead 0.03 0.01 0.29 Three-StepsAhead 0.00 0.00 0.49 Six-Steps-Ahead 0.09 0.05 0.41 14 The result of the Giacomini-White test show that at a 5% significance level, the large BVAR model outperforms the benchmark for one step and three-step-ahead forecast horizon, while the small BVAR outperforms the benchmark at a 5% significance level for all forecast horizons. The last column of the table shows the Giacomini-White test applied to the differences in predictive ability between the smalland large-scale BVAR models, but in this case the differences are not significant for all forecast horizons. 5. Conclusions This paper assesses the performance of Bayesian VAR to forecast inflation in Argentina. We considered a Normal Wishart BVAR specification for a smalland a large-scale model of differentiated variables setting the prior mean according to standard recommendations in previous studies. The overall tightness hyperprior and the lag length of the different models were set by optimization of the marginal likelihood. We found that large-scale models have narrower priors, giving more weight to the priors mean than small-scale models. Overall, the results show that the BVAR model can improve the forecast ability of the univariate autoregressive benchmark’s model of inflation. The Giacomini-White test indicates that a BVAR performes better than the benchmark in all forecast horizons. Statistical differences between the two BVAR model specifications (small and large-scale) are not found. However, looking at the RMSEs, one can see that the larger model seems to perform better for longer forecast horizons. 15 References Banbura M., D. Giannone and L. 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Small BVAR Characteristics Endogenous variables: Inflation, Interest Rate, Real Activity Exogenous variables: Constant, Dummy 2016-11 Estimation sample: July 2004 to July 2017 Sample size (omitting initial conditions): 156 Number of lags included in regression: 1 Prior: Normal-Wishart Autoregressive coefficient: 0 Overall tightness: 0.23 Lag decay: 2 Exogenous variable tightness: 1 Table 2A. Small BVAR Inflation Equation Coefficient Values Median SD LB UB INF(-1) I(-1) Y(-1) Constant d112016 0.468 0.901 2.631 0.280 -0.197 0.066 0.640 3.500 0.071 0.144 0.338 -0.356 -4.237 0.140 -0.479 0.598 2.157 9.499 0.420 0.086 Sum of squared residuals: 91.05 R-squared: 0.291 Adj. R-squared: 0.272 19 Table 3A. Large BVAR Characteristics Endogenous variables Inflation, Interest Rate, Real Activity, Multilateral Exchange Rate, Industrial Employment, Cement Sales, Asphalts Sales, Imports of Intermediate Goods, Total Exports, M2, Core Inflation, Construction Employment, Consumer Confidence Index, Supermarket Sales, Stock Market Index Exogenous variables: Constant, Dummy 2016-11 Estimation sample: July 2004 to July 2017 Sample size: 156 Number of lags: 1 Prior: Normal-Wishart Autoregressive coefficient: 0 Overall tightness: 0.05 Lag decay: 2 Exogenous variable tightness: 1 20 Table 4A. Large BVAR Inflation Equation Coefficient Values Sum of squared residuals: 89.33 R-squared: 0.304 Adj. R-squared: 0.224 Median SD LB UB INF(-1) I(-1) Y(-1) E(-1) EMPI(-1) CEM(-1) ASPH(-1) IMP(-1) EXP(-1) M2(-1) INFC(-1) EMPC(-1) ICC(-1) SUP(-1) STK(-1) Constant d112016 0.145 0.436 1.177 7.261 16.644 -0.680 0.083 0.125 0.091 4.093 0.183 -1.452 -0.011 2.243 0.133 0.056 -0.014 0.045 0.407 2.131 3.431 11.611 0.556 0.411 0.477 0.491 2.410 0.047 2.933 0.013 1.322 1.110 0.039 0.042 0.057 -0.362 -3.005 0.528 -6.143 -1.771 -0.723 -0.810 -0.873 -0.637 0.091 -7.207 -0.036 -0.351 -2.045 -0.021 -0.096 0.234 1.235 5.359 13.994 39.431 0.410 0.888 1.061 1.055 8.823 0.275 4.303 0.013 4.837 2.310 0.132 0.067