Inflation forecasting in turbulent times

Ertl, Martin et al. Working Paper Inflation forecasting in turbulent times IHS Working Paper, No. 56 Provided in Cooperation with: Institute for Advanced Studies (IHS), Vienna Suggested Citation: Ertl, Martin et al. (2024) : Inflation forecasting in turbulent times, IHS Working Paper, No. 56, Institut für Höhere Studien - Institute for Advanced Studies (IHS), Vienna This Version is available at: https://hdl.handle.net/10419/306330 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. 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/4.0/ IHS Working Paper 56 September 2024 Inflation Forecasting in Turbulent Times Martin Ertl Ines Fortin Jaroslava Hlouskova Sebastian P. Koch Robert M. Kunst Leopold Sögner Author(s) Martin Ertl, Ines Fortin, Jaroslava Hlouskova, Sebastian P. Koch, Robert M. Kunst, Leopold Sögner Editor(s) Robert M. Kunst Title Inflation Forecasting in Turbulent Times Institut für Höhere Studien - Institute for Advanced Studies (IHS) Josefstädter Straße 39, A-1080 Wien T +43 1 59991-0 F +43 1 59991-555 www.ihs.ac.at ZVR: 066207973 License „Inflation Forecasting in Turbulent Times“ by Martin Ertl, Ines Fortin, Jaroslava Hlouskova, Sebastian P. Koch, Robert M. Kunst, Leopold Sögner is licensed under the Creative Commons: Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0/) All contents are without guarantee. Any liability of the contributors of the IHS from the content of this work is excluded. All IHS Working Papers are available online: https://irihs.ihs.ac.at/view/ihs_series/ser=5Fihswps.html This paper is available for download without charge at: https://irihs.ihs.ac.at/id/eprint/7048/ Inflation Forecasting in Turbulent Times Martin Ertl Ines Fortin Jaroslava Hlouskova Sebastian P. Koch Robert M. Kunst Leopold S¨ogner * Abstract Recently, many countries were hit by a series of macroeconomic shocks, most notably as a consequence of the COVID-19 pandemic and Russia’s invasion in Ukraine, raising inflation rates to multi-decade highs and suspending well-documented macroeconomic relationships. To capture these tail events, we propose a mixed-frequency Bayesian vector autoregressive (BVAR) model with t-distributed innovations or with stochastic volatility. While inflation, industrial production, oil and gas prices are available at monthly frequencies, real gross domestic product (GDP) is observed at a quarterly frequency. Thus, we apply a mixed-frequency framework using the forward-filtering-backward-sampling algorithm to generate monthly real GDP growth rates. We forecast inflation in those euro area countries which extensively import energy from Russia and therefore have been heavily exposed to the recent oil and gas price shocks. To measure the forecast performance of our mixed-frequency BVAR model, we compare these inflation forecasts with those generated by a battery of competing inflation forecasting models. The proposed BVAR models dominate the competition for all countries in terms of the log predictive density score. Keywords: Bayesian VAR, mixed-frequency, forward-filtering-backward-sampling, inflation forecasting JEL classification: C5, E3 * All authors: Institute for Advanced Studies (IHS), Josefst¨adter Straße 39, 1080 Vienna, Austria. Leopold S¨ogner has a further affiliation with the Vienna Graduate School of Finance (VGSF). The authors would like to thank participants of the 2023 Meeting of the Austrian Economic Association (NOeG), Joshua C.C. Chan, Oscar Fernandez, Sylvia Fr¨uhwirth-Schnatter, Helmut Hofer, and two anonymous referees for helpful comments that have contributed to improving the paper. The authors gratefully acknowledge support from Peter Griessl, Christine Lietz, Johannes Nemeth, and Yannic Prohaska. Leopold S¨ogner acknowledges support by the Cost Action HiTEc – CA21163. 1 1 Introduction The COVID-19 pandemic, resulting supply side disruptions, the quick economic recovery, and the energy price shock following Russia’s invasion of Ukraine had unforeseen consequences on inflation dynamics and have posed major challenges for inflation forecasting. Evidence has emerged that parameter estimation in time series models widely used for macroeconomic forecasting has become more difficult due to the COVID-19 shock and its aftermath. For euro area inflation, Bobeica and Hartwig (2023) document that parameter estimates of Bayesian vector autoregressions (BVAR) were strongly affected. They propose to use a fat-tailed distribution for the error terms. To improve the accuracy of euro area inflation forecasts, they also recommend estimating larger models with a tighter prior (compared to standard BVAR specifications) and including off-model information for forecasts, such as information from the ECB Survey of Professional Forecasters (see also Kr¨uger et al., 2017; Banbura et al., 2021). Other work addressing the recent tail events focuses on the US economy, such as Lenza and Primiceri (2022), Carriero et al. (2022) and Schorfheide and Song (2021). More specifically, Lenza and Primiceri (2022) modify the innovation variance for the pandemic period. They exploit the fact that we know the exact timing of the increase in the innovations’ variance during the COVID19 period (March and subsequent months in 2020). Whereas this might be true for the pandemic, it is harder to disentangle the exact timing of the heterogeneous effects of rising energy prices on inflation rates in different countries. Countries have faced differing dependencies on energy supply from Russia, and governments have been implementing different policies to mitigate rapidly rising prices. For the period after May 2020, the authors simply assume that the residual variance will decay at a monthly rate of 20%. Carriero et al. (2022) suggest allowing for Student-t distributed innovations and outliers in a vector autorregressive (VAR) model with stochastic volatility. Extreme observations are viewed as outliers that are characterized by transitory increases in volatility, in which case it may be desirable to reduce their influence on model estimates. Their model augments the standard stochastic volatility specification with an outlier state. For the treatment of fat-tailed errors in stochastic volatility, they use t-distributed innovations. Antolin-Diaz et al. (2021) also allow for short-lived outliers that do not lead to a persistent rise in the stochastic volatility process in a dynamic factor model for nowcasting US GDP. Alternatively, Schorfheide and Song (2021) reconsider a mixed-frequency VAR to generate macroeconomic forecasts for the US during COVID-19. The recommendation is to exclude extreme observations during a few months of the pandemic to improve the forecasting performance. However, this assumes that the timing of outliers is known ex-ante and does not address the subsequent period of uncertainty properly. Furthermore, Clark et al. (2023) apply Bayesian machine learning techniques to account for possible non-linearity. The authors demonstrate that Bayesian regression trees have strong forecasting properties in both the overall level and in the 2 tails, respectively. In this paper, we consider a small-scale Bayesian VAR framework with different error variance specifications to forecast inflation in turbulent times for selected European countries. Working with the most recent inflation data, our experience is similar to the evidence described in the above-mentioned literature. Forecasting inflation using VAR models based on longer historical time series until the year 2019 (pre-COVID-19) results in a rapid decay of inflation rates down to rates observed before the recent sharp increase. By contrast, estimating VARs (with Gaussian errors) with the full 2004 to 2023 data set can result in non-stable (explosive) inflation forecasts. Both results are implausible and unsatisfactory. Therefore, we explore two different specifications of volatility. First, we choose an approach of forecasting post-pandemic inflation that is closely related to that of Bobeica and Hartwig (2023) using t-distributed disturbances. However, we consider a monthly instead of the quarterly frequency in order to use timelier and finer information on inflation dynamics. Thus, employing (quarterly) gross domestic product (GDP) results in a mixed-frequency problem. Further, we use the gas price as an additional energy variable. Second, we consider models with stochastic volatility to capture extreme events. It is a well-known fact that Bayesian VARs with timevarying volatility often provide better point and density forecasts of macroeconomic variables than models with homoscedastic errors terms; see, e.g., Clark (2011) and Clark and Ravazzolo (2015). Specifically, we consider the error terms to be generated by a factor stochastic volatility model as proposed in Kastner (2019). We include data on inflation, industrial production, and GDP from six euro area countries, namely Austria, Belgium, Finland, Germany, Italy, and Slovakia, which depend strongly on natural gas imports from Russia. To capture exogenous shocks affecting inflation we do not only include the oil price but also the gas price. We consider monthly observations from February 2004 to February 2024, with GDP only observed at a quarterly frequency, while the other data are available at monthly frequencies. Our underlying econometric model is a vector autoregressive model relying on monthly variables. That is, also for GDP the underlying model applies monthly growth rates (see, for instance Proietti and Giovannelli, 2021, for frequentist monthly GDP estimates). To perform parameter estimation this article follows a Bayesian approach. In working with Student-t distributed noise terms we mainly follow Bobeica and Hartwig (2023), but – in contrast to them – we apply a Minnesota type prior to the autoregressive parameter matrices. The same prior for the autoregressive matrices is also applied in the case of stochastic volatility. In addition, we have to account for mixed sampling frequencies. We adapt forwardfiltering-backward-sampling, as proposed in Fr¨uhwirth-Schnatter (1994), to obtain samples from the posterior distribution of the unobserved monthly GDP growth rates. We conduct a comprehensive empirical analysis including the period of sharp inflation increases and decreases between mid-2021 and early-2024 in the six euro area countries, which we consider the “turbulent times” in this paper. We find that the proposed variant of a mixed-frequency 3 Bayesian VAR with fat-tailed errors and the alternative variant with stochastic volatility provide better out-of-sample point and density forecast accuracies than a battery of popular competing inflation forecasting models. The competing models include a univariate version of the proposed model, a version of the Bayesian VAR model with only monthly variables, disregarding GDP, a univariate autoregressive model, and homoscedastic and heteroscedastic versions of an unobserved components model. To evaluate the inflation forecasts we employ traditional measures, such as the mean absolute error and the root mean squared error, as well as log predictive density scores. This article is organized as follows: Section 2 briefly describes the VAR model. Section 3 introduces the mixed-frequency problem. Then, Section 4 describes our Bayesian approach, in particular, the priors. Details are provided in a separate appendix. Section 5 discusses the performance of different models and then presents forecasts and an impulse response analysis for six European countries. The last section concludes. 2 The Model In this article we jointly model industrial production, IPt, inflation, Inflt, the real gross domestic product, GDPt, the gas price, pgas,t, and the oil price, poil,t, by using a vector autoregressive (VAR) model of order p. We consider data at a monthly frequency, and index tdenotes the time index. For each country, we stack the variables into the five-dimensional column vector yt= (∆ ln IPt, Inflt,∆ ln GDPt,∆ ln pgas,t,∆ ln poil,t)⊤∈R˜ kwhere ˜ k= 5. Then we get1 yt=a+ p X j=1 Ajyt−j+εt.(1) In the following we assume that the growth rates of the oil and the gas prices are not affected by ∆ ln IPt,Inflt, ∆ ln GDPtand therefore set the corresponding elements of Ajto zero (see also Equation (11) in Appendix A). In this article the noise term εteither follows a Student-t distribution (Case 1), or is generated by a stochastic volatility model (Case 2). Case 1: Following Bobeica and Hartwig (2023), the noise term εtfollows an iid multivariate Student-t distribution with mean zero, covariance matrix Σ, where 0 <Σ<∞, and νdegrees of freedom. From Bayesian literature (see, e.g., Geweke, 1993; Bobeica and Hartwig, 2023) a Student-t distributed noise term εtwith νdegrees of freedom can be obtained by drawing εtfrom 1In this article we apply the following notation: ∆xtdenotes xt−xt−1and ∆ ln xtabbreviates ln xt−ln xt−1 (that is, growth rates are calculated as logarithmic growth rates). For vectors and matrices we use boldface. If not otherwise stated, the vectors considered are column vectors. 0a×band 1a×bstands for a×bmatrix of zeros and ones and 0ais used to abbreviate 0a×1.⊗denotes the Kronecker product and Inthe identity matrix of dimension n×n. vec(M) vectorizes the matrix M, while vech(M) vectorizes the lower triangular part of a symmetric matrix M. N(·,·), IG (·,·), and W(·,·) denotes the multivariate normal, the inverse Gamma, and the Wishart distribution, respectively. U(ν, ¯ν) abbreviates a uniform distribution on the interval [ν, ¯ν]. ∝stands for proportional to. 4 a multivariate normal with mean zero and covariance matrix Σt:= λtΣ, where λtis sampled from an inverse Gamma distribution IG ν 2,ν 2. Case 2: Alternatively, we consider the noise terms εtto be generated by a factor stochastic volatility model as proposed in Kastner (2019). That is, for a ˜ kט k-dimensional matrix Σtwe assume Σt=ΛVtΛ⊤+ΣUt ,(2) where Λis a ˜ k×r-loading matrix, ris the number of volatility factors Vt=diag (exp(h1t),..., exp(hrt)) ∈Rr×r,ΣUt =diag (exp(hr+1t), . . . exp(hr+˜ kt)∈R˜ kט k, and each hjt,j= 1,...,˜ k+r, follows a stable first order autoregressive process with normally distributed noise terms. Then, εt=Σ1/2 tηt, where ηtfollows a ˜ k-dimensional standard normal distribution. The parameter vector θcollects all the parameters of the VAR considered in (1), that is a, and the vectorized parameter matrices Aj,j= 1, . . . , p. For t-distributed innovations it also contains vech (Σ), λ1, . . . , λT, as well as ν, while for the stochastic volatility model it contains all the parameters of the factor stochastic volatility models defined in Kastner (2019). We choose p such that the autocorrelations of the residuals are insignificant, i.e., p= 4. The VAR system defined in (1) results in the matrix polynomial a(z) = Ik−A1z−· · ·−Apzp, z∈C. We assume that the stability condition (the determinant of a(z)= 0, for all |z| ≤ 1) is met. Let Ldenote the lag operator. Then, yt=a(L)−1εt,t∈Z, provides us with the unique (weakly) stationary (and causal) solution of (1) (see, e.g., Deistler and Scherrer, 2018, Theorem 4.4). In addition, we conducted a panel VAR analysis. However, the forecasting performance turned out to be better in the country-by-country setup. That is why we focus in the main text on the country-specific VARs. 3 Data and mixed-frequency We use industrial production, inflation and real gross domestic product for the six countries Austria, Belgium, Finland, Germany, Italy, and Slovakia. The selected countries are all part of the European Economic and Monetary Union (EMU) and also depend strongly on gas imports (from Russia).2Thus, they are particularly vulnerable to gas price shocks and natural candidates for analyzing oil and gas price shocks as potential drivers of (energy) inflation. We do not consider, for instance, countries like Portugal or Spain that import little or zero natural gas from Russia. The inflation rates of the Baltic countries may have been affected much more by the Russian invasion in Ukraine due to a generally broader economic interaction with Russia, Belarus and Ukraine and are therefore also not considered. France and the Netherlands are not in the country 2For estimates of the number and diversity of gas supply sources, see, for instance, the European Union Agency for the Cooperation of Energy Regulators, https://aegis.acer.europa.eu/chest/dataitems/214/view, last accessed 29.11.2023. 5 list, as the former extensively implemented anti-inflationary measures while the latter changed its method of calculating inflation3during the course of the year 2023 (June 2023).4 While industrial production, inflation, and GDP are obviously country-specific, we use international price quotations for Brent oil as well as for TTF gas, thereby implicitly neglecting minor differences in country-specific wholesale prices. Industrial production and GDP are published seasonally adjusted while the harmonized index of consumer prices (HICP) is not. Instead of seasonally adjusting the HICP and using month-over-month percentage changes, we opt to work with price changes on a year-over-year basis (annual inflation), which effectively acts as some sort of seasonal adjustment. All variables measured in prices are denominated in Euro with the exception of the oil price, which is originally measured in US Dollar and then converted to Euro using the US Dollar/Euro exchange rate. The inflation rate is measured in percent. Further, we apply the following data transformations: we calculate month-over-month logarithmic growth rates for industrial production, oil and gas prices, and quarter-over-quarter growth rates for GDP. That is, we get the transformed variables ∆ ln IPt, ∆ ln pgas,t, ∆ ln poil,t observed on a monthly basis, and ln GDPq−ln GDPq−1, observed on a quarterly basis, where q, q + 1, . . . denotes a quarterly time scale. For final estimation we consider the period February 2004 to February 2024. The starting date of our sample is determined by the availability of gas prices.5 The data, its sources and transformations are summarized in Table 1. Variable Abbreviation Transformation Source Dataset or Code Industrial production IPt∆ ln IPtEurostat sts inpr m HICP inflation rate InfltEurostat prc hicp manr Real gross domestic product GDPq∆ ln GDPqEurostat namq 10 gdp TTF NL natural gas future pgas,t ∆ ln pgas,t Refinitiv Eikon TRNLTTD Brent oil price in Euro poil,t ∆ ln poil,t Refinitiv Eikon OILBREN/USEURSP US Dollar/Euro exchange rate Refinitiv Eikon USEURSP Table 1: Included variables. trepresents monthly frequency, qrepresents quarterly frequency. Note that inflation is calculated as the year-over-year growth rate of the Harmonized Index of Consumer Prices (HICP). For oil and gas prices as well as the exchange rate we use monthly averages of daily quotes. Observational Scheme: Equation (1) describes the data generating process for yton a monthly basis. The variables observed at a monthly frequency are called fast variables, yf t, while the variable GDP growth observed at a quarterly frequency is called a slow variable, ys t, with the sampling rate of the slow variable being three. In the data described above the growth rate of industrial production, ∆ ln IPt, inflation, Inflt, the change of the gas price, ∆ ln pgas,t, and the change of the oil price, ∆ ln poil,t, are observed at a monthly basis and are therefore fast variables. By contrast, GDP is observed at a quarterly rate and is a slow variable. Monthly real 3Switching from including only new energy contracts to a method that reflects all (new and existing) contracts. 4See, for instance, Armendariz et al. (2023). 5Our transformations ensure stationarity, which is confirmed by unit root tests. 6 −10 −5 0 5 10 M.5.sv M.5.t M.4.sv M.4.t M.1.sv M.1.t M.1.uscv M.1.ar M.1.uc Figure 2: Inflation forecasts errors for Austria. The violin plot (two-sided kernel density plots) summarizes the distribution of the forecast errors (i.e., forecasted inflation minus observed inflation) of all included models over all horizons (h= 1,...,6). The scale of the vertical axis was limited to ±10 to exclude extreme forecast errors (in the case of M.4.sv and M.4.t models). Densities are mostly highest for forecast errors just below zero. Negative forecast errors, that is, an underestimation of inflation, are observed more often than positive ones in our evaluation sample. The two unobserved components models show a tendency towards a bimodal forecast error distribution. 5.2 Forecasts This section presents the inflation forecasts for the six countries considered. Figure 3 presents inflation forecasts of the (country-by-country) BVAR models M.5.sv for Austria, Belgium, Germany, Finland, Italy, and Slovakia for the period June 2023 to January 2026, i.e., for 32 months ahead. The solid black lines are posterior median estimates based on 8,000 MCMC samples, and the four types of blue areas represent the 90%, 60%, 50%, and 30% forecasting intervals, respectively. As the realized values of inflation span until February 2024, we can thus observe 13 how well (or not) inflation was forecasted. Note that for Austria and Slovakia the nine-months ahead forecasts are inside the 90% forecasting intervals, while for Finland and Italy none of the inflation forecasts are within the 90% forecasting intervals. Austria, Germany, and Italy behave very similarly regarding past and forecasted inflation rates. With respect to the realized values we observe in the case of Italy a sharpe decline of inflation, namely from approximately 8% in June 2023 to approximately 0.5% in February 2024. The highest inflation rates are observed in the fourth quarter of 2022 reaching rates between 11.6% (in Germany and Austria) and 12.6% (in Italy). Also the forecasts with regard to the level of inflation at the end of the forecasting horizon as well as the forecasting intervals are very much alike. Belgium, Finland and Slovakia are different with respect to past as well as forecasted inflation rates. The strong increase as well as the sharp decrease of the Belgium inflation rate might be affected by Belgium HICP measurement.9The forecast for Belgium first declines below the 2% inflation target of the European Central Bank (ECB) and then approaches this target from below. Finland stands out with comparably low inflation rates. This does not come as a surprise taking into account that gas in Finland is used almost entirely by the industrial sector10 (e.g., pulp production) and only very marginally by households.11 The inflation development in Slovakia is different, because its peak is the largest and occurs later than in other countries. Also the forecast stands out, as it has much broader forecasting intervals with generally higher inflation rates. Figure 4 presents six snapshots of two-years ahead inflation forecasts for Austria with forecasts starting at six different time points, namely at July 2021, January and July 2022, January and July 2023, as well as January 2024. In all six cases inflation is forecasted to decline and the inflation forecasts decrease faster when the starting points of inflation forecasts are part of the more turbulent time period when inflation in Austria was highest (July 2022 and January 2023). We also observe that the forecasting intervals are largest during more turbulent times suggesting larger forecast uncertainty. 5.3 Impulse responses Figures 5 and 6 present impulse response functions of inflation with respect to a (positive) onestandard-deviation shock in the oil price change (i.e., about 10%) and with respect to a (positive) one-standard-deviation shock in the gas price change (i.e., about 16%) over 24 months for model M.5.sv. Estimates are obtained using the generalized impulse response analysis for vector autoregressive models as presented in Pesaran and Shin (1998).12 Note that the calculation of the 9Note, that in Belgium only new energy contracts are included in HICP measurement and not all contracts (existing and new) (see, e.g., Jonckheere, 2022). 10See, for instance, Vaden et al. (2022). 11According to Eurostat the HICP weight of gas consumption (i.e. by Finish households) is zero. 12This approach does not require orthogonalization of shocks and is invariant to the ordering of the variables in the VAR. 14 generalized impulse response function requires estimates of the covariance matrix Σt, which is time dependent for a model with stochastic volatility. When applying the stochastic volatility model we use the samples α(m)and the samples Σ(m) tto obtain the generalized impulse response function. The time point used is June 2023.13 The solid black lines are again median estimates and the four types of blue areas represent 90%, 60%, 50% and 30% credible intervals. We observe a positive though small impact of an increase in the oil price on inflation for all six countries, with inflation first increasing and then gradually decreasing. The inflation impulse responses peak in all countries in the first year (after the shock), and the earliest inflation peaks occur for Germany and Finland, while the latest one occurs for Slovakia. Finally, the largest uncertainty (in terms of the width of the credible intervals) can be observed for Slovakia. In principle, the effects implied by a shock in the gas price are similar to the ones implied by a shock in the oil price, only smaller. We observe a positive impact of an increase in the gas price on inflation. The effect is largest for Slovakia although surrounded also by the highest uncertainty. Note that the recently observed increases in inflation are much larger than the shocks of one standard deviation applied in the impulse response functions shown in Figures 5 and 6. The prices of oil and gas rose by over 40% during the most turbulent times, while the shocks assumed in the impulse response functions are around 10% and 16%, respectively. 13For the stochastic volatility model the impulse responses depend on time. We show them for June 2023. The ones for February 2024 (last month) are rather similar. 15 Austria Belgium 5%-95% 20%-80% 25%-75% 35-65% Median Observed 2021-01 2022-01 2023-01 2024-01 2025-01 2026-01 -2 0 2 4 6 8 10 12 14 16 5%-95% 20%-80% 25%-75% 35-65% Median Observed 2021-01 2022-01 2023-01 2024-01 2025-01 2026-01 -2 0 2 4 6 8 10 12 14 16 Germany Finland 5%-95% 20%-80% 25%-75% 35-65% Median Observed 2021-01 2022-01 2023-01 2024-01 2025-01 2026-01 -2 0 2 4 6 8 10 12 14 16 5%-95% 20%-80% 25%-75% 35-65% Median Observed 2021-01 2022-01 2023-01 2024-01 2025-01 2026-01 -2 0 2 4 6 8 10 12 14 16 Italy Slovakia 5%-95% 20%-80% 25%-75% 35-65% Median Observed 2021-01 2022-01 2023-01 2024-01 2025-01 2026-01 -2 0 2 4 6 8 10 12 14 16 5%-95% 20%-80% 25%-75% 35-65% Median Observed 2021-01 2022-01 2023-01 2024-01 2025-01 2026-01 -2 0 2 4 6 8 10 12 14 16 Figure 3: Inflation forecasts and forecasting intervals. The figure shows inflation forecasts and forecasting intervals for 32 months ahead for Austria, Belgium, Germany, Finland, Italy, and Slovakia from June 2023 to January 2026. The estimation sample is February 2004 to May 2023. The forecasts are based on 8,000 samples, 2,000 burn-in steps. 16 July 2021 January 2022 5%-95% 20%-80% 25%-75% 35-65% Median Observed 2021-01 2022-01 2023-01 2024-01 -2 0 2 4 6 8 10 12 14 16 5%-95% 20%-80% 25%-75% 35-65% Median Observed 2021-01 2022-01 2023-01 2024-01 -2 0 2 4 6 8 10 12 14 16 July 2022 January 2023 5%-95% 20%-80% 25%-75% 35-65% Median Observed 2021-01 2022-01 2023-01 2024-01 2025-01 -2 0 2 4 6 8 10 12 14 16 5%-95% 20%-80% 25%-75% 35-65% Median Observed 2021-01 2022-01 2023-01 2024-01 2025-01 -2 0 2 4 6 8 10 12 14 16 July 2023 January 2024 5%-95% 20%-80% 25%-75% 35-65% Median Observed 2021-01 2022-01 2023-01 2024-01 2025-01 2026-01 -2 0 2 4 6 8 10 12 14 16 5%-95% 20%-80% 25%-75% 35-65% Median Observed 2021-01 2022-01 2023-01 2024-01 2025-01 2026-01 -2 0 2 4 6 8 10 12 14 16 Figure 4: Inflation forecasts and forecasting intervals for Austria. The figure shows inflation forecasts and forecasting intervals for Austria for six different starting points (July 2021, January and July 2022, January and July 2023, as well as January 2024). The estimation sample ranges from February 2004 to the month previous to the indicated starting points. The forecasts are based on 8,000 samples, 2,000 burn-in steps. 17 Austria Belgium 5%-95% 20%-80% 25%-75% 35-65% Median 0 6 12 18 24 -0.02 0 0.02 0.04 0.06 5%-95% 20%-80% 25%-75% 35-65% Median 0 6 12 18 24 -0.02 0 0.02 0.04 0.06 Germany Finland 5%-95% 20%-80% 25%-75% 35-65% Median 0 6 12 18 24 -0.02 0 0.02 0.04 0.06 5%-95% 20%-80% 25%-75% 35-65% Median 0 6 12 18 24 -0.02 0 0.02 0.04 0.06 Italy Slovakia 5%-95% 20%-80% 25%-75% 35-65% Median 0 6 12 18 24 -0.02 0 0.02 0.04 0.06 5%-95% 20%-80% 25%-75% 35-65% Median 0 6 12 18 24 -0.02 0 0.02 0.04 0.06 Figure 5: Generalized impulse responses of inflation with respect to the oil price. The figure shows generalized impulse response functions of inflation (in %) with respect to a onestandard-deviation shock in the oil price (i.e., ≈10%), for Austria, Belgium, Germany, Finland, Italy, and Slovakia for 24 months. We apply samples of the covariance matrix Σ(m) tfor the time point June 2023. 18 Austria Belgium 5%-95% 20%-80% 25%-75% 35-65% Median 0 6 12 18 24 -0.02 0 0.02 0.04 0.06 5%-95% 20%-80% 25%-75% 35-65% Median 0 6 12 18 24 -0.02 0 0.02 0.04 0.06 Germany Finland 5%-95% 20%-80% 25%-75% 35-65% Median 0 6 12 18 24 -0.02 0 0.02 0.04 0.06 5%-95% 20%-80% 25%-75% 35-65% Median 0 6 12 18 24 -0.02 0 0.02 0.04 0.06 Italy Slovakia 5%-95% 20%-80% 25%-75% 35-65% Median 0 6 12 18 24 -0.02 0 0.02 0.04 0.06 5%-95% 20%-80% 25%-75% 35-65% Median 0 6 12 18 24 -0.02 0 0.02 0.04 0.06 Figure 6: Generalized impulse responses of inflation with respect to the gas price. The figure shows generalized impulse response functions of inflation (in %) with respect to a onestandard-deviation shock in the gas price (i.e., ≈16%), for Austria, Belgium, Germany, Finland, Italy, and Slovakia for 24 months. We apply samples of the covariance matrix Σ(m) tfor the time point June 2023. 19 6 Conclusions A series of macroeconomic shocks hit many countries between the years 2020 and 2022, primarily triggered by the COVID-19 pandemic and Russia’s invasion in Ukraine, which raised inflation rates across Europe to multi-decade highs and put well-documented relationships among macroeconomic variables under scrutiny. In particular, inflation forecasting became much more difficult. We propose a mixed-frequency Bayesian vector autoregressive model and, accounting for the recent tail events, we assume Student-t distributed innovations or, alternatively, stochastic volatility. We include the variables inflation, industrial production, gross domestic product, oil and gas prices. We forecast inflation in selected euro area countries, which have been heavily exposed to energy supply from Russia and, thus, to the recent oil and gas price shocks. We compare the forecast performance of our model with the forecast performance of several competing models of inflation in the out-of-sample period from July 2021 to February 2024. In the out-of-sample forecast evaluation it turns out that with respect to log predictive density scores the mixed-frequency BVAR models dominate the competing models. When the forecast performance is measured by MAE and RMSE, then the best forecast accuracy, except for Germany and Finland, occurs again for mixed-frequency BVAR models, though univariate models are strong competitors. Against pre-COVID-19 evidence (see, e.g., Koop and Korobilis, 2019), BVAR forecasts in a panel set-up are strongly dominated by our country-specific BVAR models, which might be due to the described country heterogeneity. Our results rather support the recent emphasis on fat-tailed noise terms for inflation modeling in the post-pandemic world as well as the vast evidence that stochastic volatility is pivotal for inflation forecasting. In our forecasting exercise, we present inflation forecasts starting in June 2023, a time of still high inflation in most countries, until January 2026. For Austria, Germany, Finland, and Italy inflation forecasts behave similarly, they slowly decrease to rates between approximately 2.5% and 3% in January 2026. The inflation trajectory for Belgium is different, since it falls below 2% and, afterwards, convergences smoothly towards levels close to the European Central Bank’s 2% inflation target. Finally, the inflation forecast for Slovakia exhibits the highest uncertainty. When forecasting inflation for Austria in different points in time, we demonstrate that the highest forecasting uncertainty occurs in the most turbulent time periods. The methodology developed in this article can be extended in several ways: First, one can split up HICP inflation into its components. Separate modeling of these components allows to infer shocks of oil and gas prices on these components of inflation. For example, the transmission of oil and gas prices on the energy component is of particular interest. Second, instead of a reduced form VAR one may consider a structural VAR, with the goal to identify structural shocks and to model the instantaneous effects, e.g., of energy prices on inflation in more detail. Third, the structural stability of the relationship between the variables considered can be further investigated, for example, whether there are significant changes in the relationship between energy prices and 20 inflation during turbulent times. 21 A The Panel Model This Appendix considers a panel VAR. Due to the forecasting performance of the panel model, country specific models are considered in the main text. To get the country specific analogs of the panel model (10) simply set n= 1. We consider a panel vector autoregressive (VAR) model of order pas a starting point (in this section we mainly follow L¨utkepohl, 2006; Kilian and L¨utkepohl, 2017): yit =ai+ p X j=1 Ay,jyit−j+εit .(10) The time series dimension is t= 1, . . . , T, while i= 1, . . . , n denotes the cross-sectional dimension. The variables yit ∈Rk, the intercept terms are allowed to be country dependent, ai,i= 1, . . . , n, while the autoregressive matrices Ay,j are the same for all countries i= 1, . . . , n. The noise terms are εit ∈Rk,i= 1, . . . , n. In addition, we include common variables yct ∈Rkc. In the empirical application discussed in Section 5, k= 3 and kc= 2. The vector yit contains growth rates in industrial production, inflation and the monthly GDP growth rate. The common variables are the growth rates in the oil and gas prices. Let yt:= y⊤ 1t,...,y⊤ nt,y⊤ ct⊤∈Rnk+kc,a:= a⊤ 1,...,a⊤ n,a⊤ c⊤∈Rnk+kc, and εt:=ε⊤ 1t,..., ε⊤ nt,ε⊤ ct⊤∈Rnk+kc.14 Then we describe the country models including the common variables by one joint VAR system, that is yt=a+ p X j=1 (In⊗Ay,j)Ayc,j 0 Ac,j  | {z } Aj∈R(nk+kc)×(nk+kc) yt−j+εt,(11) where Ayc,j is a kn ×kcmatrix. In (11) we imposed the simplifying assumption that yit,i= 1, . . . , n, do not Granger cause yct. The noise term εtfollows an iid multivariate Student-t distribution with mean zero and covariance matrix Σand νdegrees of freedom. By following, e.g., Geweke (1993), by means of sampling from a normal with mean zero and covariance matrix Σt=λtΣ, where 0 <Σ<∞and λt∼ IG ν 2,ν 2, we obtain samples from a multivariate t-distribution. In addition, we also consider stochastic volatility (Kastner, 2019). The VAR system defined in (11) results in the polynomial a(z) = Ik−A1z−· · ·−Apzp,z∈C. We assume that the stability condition (the determinant of a(z)= 0, for all |z| ≤ 1) is met. Let Ldenote the lag operator. Then, yt=a(L)−1εt,t∈Z, provides us with the unique stationary (and causal) solution of (1) (see, e.g., Deistler and Scherrer, 2018, Theorem 4.4). Let αbe obtained by stacking a1,...,an,acand Ay,1,Ayc,1,...,Ay,p,Ayc,p,Ac,1,...,Ac,p column wise. By means of Zit := (y⊤ it−1,y⊤ ct−1,...,y⊤ it−p,y⊤ ct−p)⊤∈R(k+kc)p,Zct := (y⊤ ct−1,...,y⊤ ct−p)⊤∈ 14In the main text, n= 1, k= 3, kc= 2, and ˜ k=nk +kc= 5. 22 results in the following state space form, where we consider one slow flow variable (GDP). For each country ithe monthly GDP growth rate is contained in the third coordinate of yit.16 Let us introduce the following notation: w+ t=Htxt+Qtηt,ηt∼ N (0nk+kc,Ink+kc) xt+1 =Fxt+1 +Gεt,εt∼ N (0nk+kc,Σt),where (19) Hit =Hobs := Ik−10k−1×k20k−1×k(p−k−1)+1 01×k−111×k⊗e⊤ 1k01×k(p−k−1)+1  | {z } [k×kp] for t∈NZand e1k= (1,0,...,0)⊤∈Rk×1 Hit =Hnotobs := Ik−10k−1×k(p−1)+1 01×k−101×k(p−1)+1  | {z } [k×kp] for t /∈NZ Hct of dimension kc×kcpis obtained conformingly Ht=In⊗Hit 0 0 Hct  Qit =0[k×k]for t∈NZ,Qit = (0k×k−1ekk) | {z } [k×k] for t /∈NZ where ekk = (0,...,0,1)⊤∈Rk×1. Note that Qct of dimension kc×kcis obtained conformingly; in our application Qct =02×2. Qt=In⊗Qit 0 0 Qct .(20) Recall that those coordinates of ytobserved every period are called the fast variables, while the coordinates only observed at NZ,N > 1, are called the slow variables. Define xt|T:= Exit|Y+ T, Πt|T:= Cov xtx⊤ t|Y+ Tand Πt,t−1|T:= Cov xtx⊤ t−1|Y+ T,θdenotes the model parameters (for the Kalman filter as well as the Kalman smoother, see, e.g., Shumway and Stoffer, 1982; Deistler and Scherrer, 2018). Let Kt:= FΠt|t−1H⊤ tΣ−1 t|t−1, Πt+1|t=Vxt+1 −xt+1|t=FΠt|t−1F⊤+GΣG⊤−KtΣt|t−1K⊤ t, xt+1|t=Fxt|t−1+Ktw+ t−w+ t|t−1, w+ t+1|t=Ht+1xt+1|t, Σt+1|t=Vw+ t+1 −w+ t+1|t=Ht+1Πt+1|tH⊤ t+1 +Qt+1IQ⊤ t+1 ,(21) where V(·) denotes a variance. The system is started at some x1|0= (1,x0)⊤and Π1|0= diag 1,1⊤ pnk+pkc, such that w+ 1|0=H1z1|0and Σ1|0=H1Π1|0H⊤ 1+˜ Q1˜ Q⊤ 1. Note that for adapted in a straightforward way. In the following we mainly focus on our application, where one slow flow variable is considered. 16Matrices Hit and Qit are constructed for the case when only one slow variable is considered, namely GDP growth. This (flow) variable is the third coordinate of wit. 29 variables which cannot be observed, denoted w+,notobs jt ,E(w+,notobs jt ) = 0, V(w+,notobs jt −w+,notobs jt|t−1) = V(w+,notobs jt ) = 1, and Cov(w+,notobs jt , yit) = 0, for all i=jand all t, where Cov(·,·) denotes the covariance matrix. For those time points where w+ t=wtwe get V(wt−wt|t−1) = HtΠt|t−1H⊤ t. In the following forecasts we consider wt+h(and not w+ t+h), where Ht+h=In⊗Hobs 0 0 Hct  for all h > 0. The h-step ahead forecasts, h≥1, follow from xt+h|t=Fxt+h−1|t, Πt+h|t=Vxt+h−xt+h|t=FΠt+h−1|tF⊤+GΣG⊤, wt+h|t=Ht+hxt+h|t, Σt+h|t=Vwt+h−wt+h|t=Ht+hΠt+h|tH⊤ t+h such that for h= 0 we get xt|t=xt|t−1+Πt|t−1H⊤ tΣ−1 t|t−1w+ t−w+ t|t−1, Πt|t=Vxt−xt|t=Πt|t−1−Πt|t−1H⊤ tΣ−1 t|t−1HtΠt|t−1.(22) Note that xt|t=Ext|Y+ tand Πt|t=Vxt−xt|t=Vxt−Ext|Y+ t. For the fast variables we get xjt|t=Exjt|Y+ t=xjt =wjt, that is the conditional expectation is the actual observation of the variable j, while for the slow variables we get xjt|t=Exjt|Y+ twhich is obtained by the above recursions. The lagged coordinates contained in xtfollow from these terms in a deterministic way. Only those elements of Πt|treferring to covariances of slow variables are non-zero. This directly follows from the properties of conditional expectation. In addition, xt|t=xt|t−1+Πt|t−1H⊤ tΣ−1 t|t−1w+ t−w+ t|t−1 =xt|t−1+Πt|t−1H⊤ tHtΠt|t−1H⊤ t−1w+ t−w+ t|t−1 =1, w+ 1t, w+ 2t, x3t|t−1, w+ 4t, w+ 5t, x6t|t−1, . . . , w+ (n−1)k+1,t, w+ (n−1)∗k+2,t, xnk,t|t−1, w+ c1t, w+ c2t⊤ =1, y1t, y2t, x3t|t−1, y4t, y5t, x6t|t−1, . . . , y(n−1)k+1,t, y(n−1)∗k+2,t, xnk,t|t−1, yc1t, yc2t⊤.(23) For those coordinates where wjt is an observed fast variable, the conditional expectation given the past and the current observations is simply yjt (this follows again from the properties of conditional expectation). The variables xjt|t−1(= Exjt|Y+ tas mentioned before) follow from the above recursion (23). The Kalman-smoothing equations for t=T−1,...,2,1 are Bt+1 =Πt|t−1F⊤−H⊤ tK⊤ tΠ−1 t+1|t xt|T=xt|t+Bt+1 xt+1|T−xt+1|t Πt|T=Πt|t+Bt+1 Πt+1|T−Πt+1|tB⊤ t+1 .(24) Since xjt|t=xjt for the fast variables, also xjT|t=xjt and the corresponding variance terms in 30 Πt|Tare zero. Since xjt|t=xjT |tthe rows of Bt+1 referring to fast variables have to be zero. Assuming that the noise terms conditional on x0are normally distributed, it follows from Fr¨uhwirth-Schnatter (1994) or Fr¨uhwirth-Schnatter (2006)[p. 419] that the missing values can be recursively drawn from a (degenerated) normal distribution with mean vector ¯ xt|Tand covariance matrix ¯ Πt|T,t=T, T −1,...,1,0. In the following we slightly adapt the proof of Fr¨uhwirthSchnatter (1994) to sample yt=Syxt;Syis a nk +kc×1 + nkp +kcpselector matrix (see also (17)) where SyS⊤ y=Ink+kc.17 By the Bayes theorem we get πxt|yt+1,...,yT,Y+ t,θ∝π(Syxt+1|xt,θ)πxt|Y+ t,θ(25) For t /∈NZ+ 1, the last density πxt|Y+ t,θis a normal density with mean vector xt|tand covariance matrix Πt|t. The density π(Syxt+1|xt,θ) = π(yt+1|xt,θ) is a normal density with mean vector SyFxtand covariance matrix SyGΣtG⊤S⊤ y. Since xt+1 contains yt+1,yt,...,yt−p+2, [xt+1](1+(nk+kc):1+p(nk+kc)) deterministically follows from [xt](2:1+(p−1)(nk+kc)). Hence, xt+1 follows a singular normal distribution with mean vector Fxtand covariance matrix GΣtG⊤. From the appendix in Fr¨uhwirth-Schnatter (1994) we know that by “completing the square” in the corresponding state space model we arrive at a normal distribution with a mean vector of the form ¯ xt|T= (I−Bt+1F)xt|t+Bt+1xt+1 and a covariance matrix of the form ¯ Pt|T= (I−Bt+1F)Πt|t. For our application this result and the relationship between xtand yt,...,yt−N+1 shows that yt follows a normal distribution with mean vector ¯ yt|Tand covariance matrix Sy¯ Pt|TS⊤ y. Hence, for t /∈NZ+ 1, samples of ytfollow from: yt|yt+1,...,yT,Y+ T∼ N ¯ yt|T,Sy¯ Pt|TS⊤ y,where ¯ yt|T=Sy(I−Bt+1F)xt|t+Bt+1ST yyt+1 =Syxt|t+SyBt+1S⊤ yyt+1 −SyFxt|t=Syxt|t+SyBt+1S⊤ yyt+1 −Syxt+1|t, ¯ Pt|T= (I−Bt+1F)Πt|t, Bt+1 =Πt|tF⊤FtΠt|tF⊤+GΣtG⊤−1,(26) where in our application xtdirectly follows from yt,...,yt−N+1. By plugging in terms obtained above and additional calculations we get ¯ Pt|T= (I−Bt+1F)Πt|t=Πt|t−Bt+1FΠt|t =Πt|t−Πt|tF⊤FtΠt|tF⊤+GΣtG⊤−1 | {z } =Π−1 t+1|tby (22) FΠt|t =Πt|t−Πt|tF⊤Π−1 t+1|tΠt+1|tΠ−1⊤ t+1|tFΠ⊤ t|t =Πt|t−Bt+1Πt+1|tB⊤ t+1 =Πt|t−Bt+1Vxt+1 −xt+1|tB⊤ t+1 .(27) Thus, for those periods twhere t /∈NZ+ 1, samples of ytfollow from (26). Since Πt|tis a sparse matrix, we sample from a singular normal distribution. Finally, for t∈NZ+ 1, yt+1,...,yt+N−1and w+ t+N−1=wt+N−1allow to calculate yjt by 17To simplify notation we often do not distinguish between samples obtained by means of (26) and the random variables yt. 31 means of yjt =wj,t+N−1−yjt+1 −· · ·−yjt+N−1for all slow coordinates j. Hence, in formal terms in this case the conditional distribution of yt|yt+1, . . . yT,Y+ tis a Dirac distribution with point mass on the observed fast variables and yjt =wj,t+N−1−yjt+1 − · · ·−yjt+N−1for the slow coordinates j. Hence, for the periods t∈NZ+ 1 and slow variables with index j(that is, for the first month of the corresponding quarter in our application), we get xjt|T=Exjt|Y+ T,xt+1,xt+2, . . . = y+ j,t+N−1−yj,t+1 − · · · − yj,t+N−1, for those periods t=s+ 1, t∈NZ, after swhere ws=w+ s. The variance of this term is zero. The lagged coordinates contained in xtfollow from these terms in a deterministic way. For our application this implies: We observe a quarterly growth rate of GDP, e.g., for the first quarter. The monthly GDP growth rates for March and February follow from samples obtained by means of (26). The monthly GDP for January directly follows from the quarterly GDP growth rate (from January to March) minus the monthly growth rates sampled first for March and then for February. B.2 Convergence and Mixing This section analyzes the convergence and mixing properties of our Bayesian sampler for the models with five variables. We consider the M1posterior draws θ(m) j,m=M0+ 1, . . . , M, M=M0+M1, for each country i= 1, . . . , n. In the following M0= 2,000 and M1= 8,000. Mixing of the Chain: To investigate the mixing behavior of the chain we derive the effective sample size c Meff jas, e.g., defined in Gelman et al. (2013)[Chapter 11.5]; here the coda package in Rwas applied. Table 3 presents the average effective sample sizes (i.e., we take the sample mean of the effective sample sizes obtained for the corresponding parameters contained in α,vech(Σ), etc.). In the Case 1, where the noise terms follow a t-distribution, we consider the parameter subvectors α,vech(Σ), λ, and ν. For εtgenerated by a stochastic volatility model (Case 2) we consider the parameter subvector αand samples of the volatility matrix Σtat t=T(too keep the amount of MCMC output to be stored low we only store the last value of the volatility process (vech (Σt))t=0,1,...,T ). For some parameters the effective sample size is larger than M1which can be explained by the estimation of the long run covariance matrix to obtain c Meff j. In both cases we observe for the parameters α, the volatility parameters, and ν, that the average effective sample size is larger than 700 based on 8,000 MCMC draws. Only for λthe effective sample size remained relatively low, which can be explained by the relatively high persistence of the samples of λt. When applying the stochastic volatility model we observe that the average effective sample size is at least 5,000. Convergence: Since the main focus of this paper is on forecasting, we run the Bayesian sampler with different seeds and compare all the fan charts (= distributions of Bayesian point forecasts) for the inflation forecasts by means of visual inspection. Here we observed that the fan charts strongly overlap also for different seeds. Therefore, we can conclude that the Bayesian sampler has sufficient convergence and mixing behaviour. C Unobserved components model with stochastic volatility We use the unobserved components model with stochastic volatility described in Kroese et al. (2014) as one of the benchmark models in the forecast evaluation. The observable variable, in our 32 Case 1: Case 2: t-distributed noise stochastic vol. αΣλναΣT Austria 1163 2012 101 1962 6261 7297 Belgium 719 1959 92 1962 5087 7023 Germany 864 2060 101 1820 5611 6824 Finland 1034 2030 120 1962 6134 6829 Italy 843 1998 67 2051 8048 7906 Slovakia 1096 2001 159 1962 5258 6918 Table 3: Average effective sample size c Meff j. case Inflt, depends on the unobserved component, τt, and a stochastic volatility term, Inflt=τt+e˜ ht/2ϵt,(28) where ϵt∼iid N(0,1). The unobserved component and log-volatility, ˜ ht, both follow a random walk, that is τt=τt−1+ut, ˜ ht=˜ ht−1+νt,(29) where ut∼iid N(0, ω2 τ) and νt∼iid N(0, ω2 ˜ h). 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