Real‐Time Forecasting Using Mixed‐Frequency VARs With Time‐Varying Parameters

Heinrich, Markus; Reif, Magnus Article — Published Version Real‐Time Forecasting Using Mixed‐Frequency VARs With Time‐Varying Parameters Journal of Forecasting Provided in Cooperation with: John Wiley & Sons Suggested Citation: Heinrich, Markus; Reif, Magnus (2025) : Real‐Time Forecasting Using Mixed‐ Frequency VARs With Time‐Varying Parameters, Journal of Forecasting, ISSN 1099-131X, Wiley, Hoboken, NJ, Vol. 44, Iss. 7, pp. 2055-2066, https://doi.org/10.1002/for.3276 This Version is available at: https://hdl.handle.net/10419/330162 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/4.0/ Journal of Forecasting, 2025; 44:2055–2066 https://doi.org/10.1002/for.3276 2055 Journal of Forecasting RESEARCH ARTICLE OPEN ACCESS RealTime Forecasting Using MixedFrequency VARs With TimeVarying Parameters MarkusHeinrich1 | MagnusReif2 1Stadtwerke Kiel, University of Kiel, Kiel, Germany | 2Deutsche Bundesbank, CESifo, Frankfurt,Germany Correspondence: Magnus Reif ([email protected]) Received: 9 March 2023 | Revised: 30 October 2024 | Accepted: 21 March 2025 Keywords: Bayesian methods| forecasting| mixedfrequency models| nowcasting| timevarying parameters ABSTRACT This paper provides a detailed assessment of the realtime forecast accuracy of a wide range of vector autoregressive models that allow for both structural change and indicators sampled at different frequencies. We extend the literature by evaluating a mixedfrequency timevarying parameter vector autoregressive model with stochastic volatility. Monte Carlo simulation shows that the novel model is wellsuited to estimate missing monthly observations in an environment that is subject to parameter instability. In a realtime forecast exercise, the model delivers accurate nowand forecasts and, on average, outperforms its competitors. Particularly, inflation and unemployment rate forecasts are more precise. JEL Classification: C11, C53, C55, E32 1 | Introduction Macroeconomists and, in particular, macroeconomic forecasters face two major challenges. First, there are structural changes within an economy. Second, in real time, forecasters need to process unbalanced datasets due to indicators sampled at different frequencies and indicatorspecific publication lags. Concerning structural change, it is commonly found that particularly modeling timevarying volatilities enhances VARbased inference and estimation, while fluctuations in the VAR coefficients are frequently considered to be less vital (for example, Chan and Eisenstat2017; D’Agostino etal.2013). Since the onset of the Great Recession, which probably caused important structural shifts, modeling timevarying links between variables has attracted anew interest. Concerning unbalanced datasets, literature stresses the merits of mixedfrequency approaches in computing precise nowand forecasts, and tracking the current state of the economy in real time (for instance, Kuzin etal.2011; Schorfheide and Song2015; Götz and Hauzenberger 2021). However, evidence regarding the forecast performance of models allowing for structural shifts in a mixedfrequency setting is rather sparse. This study aims at filling this gap by providing a detailed assessment of the realtime forecast accuracy of a bundle mixedfrequency models that allow for structural change. To this end, we estimate nonlinear and linear VARs with and without mixedfrequencies, including a fullyfledged model incorporating timevarying parameters, stochastic volatilities, and mixedfrequencies—a MFTVPSVVAR. This analysis enables us to trace out the relative impact of the models' mixedfrequency part and the timevariation in the models' coefficients on the forecast accuracy. Our comparison relies on realtime outofsample nowand forecast accuracy of both point and density forecasts concerning four key US macroeconomic variables: GDP, industrial production, CPI, and the unemployment rate. Overall, our forecast comparison provides two major findings. First, modeling structural change and intraquarterly dynamics is beneficial for point and density forecasts, notably for now and shortterm forecasts. In particular, the accuracy of inflation and unemployment rate forecasts can be substantially increased. Second, the MFTVPSVVAR delivers competitive point and density forecasts—on average over all variables it outperforms each competitor. Our results moreover suggest This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2025 The Author(s). Journal of Forecasting published by John Wiley & Sons Ltd. 2056 Journal of Forecasting, 2025 that the combination of mixedfrequencies, stochastic volatility, and timevarying parameters is particularly beneficial for inflation nowcasts computed with only little information about the respective quarters. In those cases, the MFTVPSV provides the largest gains in forecast accuracy. Inspecting the mixedfrequency models' forecasts during the Great Recession reveals that allowing for timevariation in the VAR coefficients and stochastic volatility is superior relative to only one of these specifications for inflation and the unemployment rate. Estimation of the models' TVPSV part follows Del Negro and Primiceri (2015). However, we estimate the hyperparameters that relate to the amount of timevariation in the parameters following AmirAhmadi etal.(2020).1 Estimation of the models' MF part is based on Mariano and Murasawa (2010) and Schorfheide and Song(2015). 2 | Data and Forecast Setup We use a dataset consisting of four macroeconomic indicators, three of which are sampled at monthly frequency and one is observed quarterly. The quarterly series is US real GDP; the monthly series are industrial production (IP), the consumer price index (CPI), and the unemployment rate. GDP, IP, and CPI enter the models in log first differences times 100. The unemployment rate remains untransformed. For the VARs estimated on quarterly frequency, the monthly indicators enter the models as quarterly averages. We obtain realtime data from the Archival FRED (ALFRED) database. The sample runs from January 1960 until September 2017. The first 8 years are used to specify priors such that the estimation starts in January 1968. We assess the predictions regarding the intraquarterly inflow of information using three information sets. We assume that the forecasts are generated around the middle of each month, when the current indicator releases are available. The first information set (I1) relates to the first month of each quarter such that the forecaster has information up to the middle of January, April, July, or October. In these months, the researcher has observations on IP growth, inflation, and unemployment until the end of the respective previous quarter and a first and preliminary estimate of GDP referring to the previous quarter. Concerning the second (third) information set I2 (I3), one (two) additional information for the monthly indicators and the first (second) GDP revision are available. As the quarterly VARs cannot cope with the monthly data flow, we estimate them in each recursion based on the balanced information set I1, which accounts for new information only in terms of data revisions. We evaluate our forecasts for data vintages using an expanding window from January 1990 until September 2017. The predictions are evaluated based on quarterly averages. In the case of IP and CPI, this implies that we use the predicted monthly growth rates to reconstruct the monthly levels of the series, which in turn are used to compute the quarterly growth rates. For GDP growth, we use the aggregation rule given by (5) to obtain quarterly GDP growth. To abstract from benchmark revisions, we evaluate GDP growth forecasts based on the second available estimate, that is the forecast for period t+h is evaluated with the realization taken from the vintage published in t+h+2 . As the remaining variables are revised only rarely and slightly, we evaluate the forecast based on the latest vintage. The maximum forecast horizon hmax is set to 4 quarters. Thus, the mixedfrequency models generate forecasts for hm=1, …, 12 months. Forecasts for horizons larger than one are obtained iteratively. 3 | Models We evaluate the forecast performance of mixedfrequency, stochastic volatility, and timevarying parameter VARs, as well as the forecast performance of combinations of these features. For the stochastic volatility models, we use random walk stochastic volatility, which is a parsimonious and competitive specification (Clark and Ravazzolo2015).2 Throughout the paper, n=nq+nm , where n,nq , and nm denote the number of total, quarterly, and monthly variables, respectively. Finally, p denotes the lag order. 3.1 | A VAR With TimeVarying Parameters and Stochastic Volatility For the VAR with timevarying parameters and stochastic volatility (TVPSVVAR), we assume that the vector of variables (yt) satisfies where B0,t is a n×1 vector of timevarying intercepts, Bi,t are matrices of timevarying coefficients and Ωt is the timevarying n×n variancecovariance matrix. Let Ωt = At Σ tA� t , where the diagonal elements of Σt are the stochastic volatilities and the lowertriangular elements of At are the contemporaneous relations between the variables. Moreover, let 𝜎t be the vector of the diagonal elements of Σt,at the vector of lowertriangular elements stacked by rows of At , and 𝛽t the vector of stacked VAR coefficients. We assume that where Q =diag ( q2 𝛽1,…,q2 𝛽k𝛽 ) .3 We obtain an SVVAR by excluding timevariation in the VAR coefficients, and a TVPVAR by excluding timevariation in the variancecovariance matrix. 3.2 | A MixedFrequency VAR Estimation of the mixedfrequency VAR (MFVAR) follows the Bayesian statespace approach of Schorfheide and Song(2015). Let y t =[y� q,t,y� m,t]� , where ym,t collects the monthly variables and yq,t denotes the quarterly variables at monthly frequency. As the quarterly variables are observed only in the last month of each quarter, yq,t contains missing observations for the first and second month of each quarter. We construct the measurement (1) yt=B0,t+ p ∑ i=1 Bi,tyt−i+𝜀t,𝜀t∼N(0, Ωt ) (2) log 𝜎 t =log𝜎 t−1 +e t ,e t =(e 1,t ,…,e n,t ) � ∼N(0, Ψ ) (3) a t =a t− 1+ 𝜐t , 𝜐t =( 𝜐 � 1,t,…, 𝜐 � n,t)�∼N(0, Φ) (4) 𝛽t=𝛽t−1+𝜒t,𝜒t∼N(0, Q) 2057 equation by assuming that quarterly GDP growth can be disaggregated into unobserved monthly GDP growth (Mariano and Murasawa2003): Combining the unobserved with the observed monthly variables in  y t =[ y� q,t,y� m,t]� , we define the state vector by z t =[ y� t,…, y� t−p+1]� and write the measurement equation as: Assuming that GDP growth is ordered first in the model, Ht is given by: The missing observations in zt are replaced by estimated states using simulation smoothing with a timevarying dimension of the statespace system (Durbin and Koopman2001). The transition equation of the MFVAR in statespace form is given by: where 𝜇 and F contain the intercepts and ARcoefficients, respectively. S is a pn ×pn variancecovariance matrix where the first n×n elements equal Ω and all remaining entries are zero. We obtain the MFSVVAR by setting the first n×n elements of S to Ωt using the laws of motion in (2) and (3). The MFTVPVAR is obtained by allowing F to vary over time according to (4). Including both specifications leads to the MFTVPSVVAR. 3.3 | Estimation Procedure and Prior Specification We estimate the models using Bayesian estimation techniques. The mixedfrequency models are estimated with 4 lags; the quarterly models are estimated with 2 lags.4 Prior specifications follow Clark (2011), Del Negro and Primiceri (2015), and Schorfheide and Song (2015). Additionally, for the hyperparameters that relate to the amount of timevariation in 𝛽t,ait , and log𝜎it , we follow AmirAhmadi etal.(2020) by implementing another layer of priors for those hyperparameters. A detailed description is provided in the supporting information. 3.4 | Monte Carlo Evidence In this section, we account for the novelty of the MFTVPSVVAR and conduct a Monte Carlo simulation to assess the finite sample properties of the model. We focus on a VAR(4) without intercept term and consider four data generating processes (DGPs)—constant coefficients (DGP I), a single break in both the variancecovariance matrix and the coefficients (DGP II), drifting coefficients and variances (DGP III), and constant coefficients with outlier observations (DGP IV).5 We use the respective VAR process along with the disaggregation constraint (5) to mimic a dataset consisting of two monthly and one quarterly series. We assess accuracy of the MFTVPSVVAR by computing the RMSEs of the estimated missing observations against the actual simulated values and compare the resulting figures with those derived from a (constantcoefficient) MFVAR. Table 1 shows that the MFTVPSVVAR provides more accurate estimates for DGPs II and III. In particular, for the driftingcoefficient case (DGP III), the model strongly outperforms the MFVAR. For DGP I, the MFVAR provides, as expected, the best fit. Interestingly, the MFTVPSVVAR seems to be inadequate for DGP IV. Inspecting the estimated coefficients and variances reveals that the MFTVPSVVAR interprets the outlierinduced hikes in volatility to be persistent, which leads to comparably poor estimates for the VAR's covariance matrix.6 In sum, these results suggest that the MFTVPSVVAR is wellequipped to estimate missing monthly observation in our dataset, which is likely subject to some kind of parameter instability (see, for example, Chan and Eisenstat2017, and the online Appendix). 4 | Results For the point forecasts, which we evaluate in terms of root mean squared errors (RMSE), we first assess the models' nowcast accuracy. Afterwards, we evaluate the accuracy of the point forecasts and predictive densities with respect to the subsequent quarters. We evaluate the predictive densities using the continuous ranked probability score (CRPS).7 We provide results for the entire sample (1995Q1–2017Q4) and for a shorter sample period (2008Q1–2017Q4) to assess whether a possible structural break around the Great Recession affects the forecast performance. (5) Δ 3logYq,t=yq,t= 1 3  yq,t+ 2 3  yq,t−1+ yq,t−2+ 2 3  yq,t−3+ 1 3  yq,t− 4 (6) yt=Htzt (7) H t= [ H� 1,tH� 2,t ]� (8) H 1,t= [ 1∕30 1×n−12∕30 1×n−110 1×n−12∕30 1×n−11∕30 1×n−101×(p−4)n ], (9) H 2,t= [ 0n−1×1In−10n−1×pn ] (10) zt=𝜇+Fzt−1+𝜐t,𝜐t∼N(0, S) TABLE 1 | RMSEs for estimated missing observations from Monte Carlo simulation. DGP I DGP II DGP III DGP IV MFVAR 0.96 0.82 1.09 1.02 MFTVPSVVAR 1.04 0.78 0.74 1.32 Note: DGP I: VAR with constant coefficients, DGP II: VAR with single break in coefficients and variances, DGP III: VAR with drifting coefficients and variances, DGP IV: VAR with outlier observations. RMSEs are based on 10 simulations per DGP. For each DGP, we set T=500 . 2058 Journal of Forecasting, 2025 TABLE 2 | Realtime nowcast RMSEs. 1990–2017 2008–2017 Model I1 I2 I3 I1 I2 I3 GDP growth MFTVPSVVAR 0.90 0.86 0.88 0.82 0.80 0.83 MFSVVAR 0.89 0.82 0.81 ∗ 0.81 0.76 0.75 MFTVPVAR 0.85 ∗∗ 0.78 0.79 ∗ 0.78 ∗ 0.69 ∗ 0.72 MFVAR 0.92 0.86 0.87 0.84 0.79 0.82 QTVPSVVAR 0.99 1.01 0.99 1.00 1.01 0.98 QTVPVAR 0.98 0.99 0.99 0.97 0.98 0.98 QVAR 1.03 1.03 ∗ 1.03 ∗ 1.01 1.02 1.02 QSVVAR 0.59 0.59 0.59 0.66 0.66 0.66 IP growth MFTVPSVVAR 0.90 0.80 0.59 ∗∗ 0.84 0.68 0.52 ∗ MFSVVAR 0.86 0.78 0.59 ∗∗ 0.82 0.70 0.52 ∗ MFTVPVAR 0.86 0.78 0.58 ∗∗ 0.83 0.70 0.51 ∗ MFVAR 0.87 0.79 0.59 ∗∗ 0.84 ∗∗ 0.71 0.53 ∗ QTVPSVVAR 0.99 0.99 0.99 0.93 0.94 0.94 QTVPVAR 1.03 1.04 1.04 1.03 1.03 1.03 QVAR 1.06 ∗∗∗ 1.06 ∗∗∗ 1.07 ∗∗∗ 1.05 ∗∗ 1.06 ∗∗ 1.06 ∗∗ QSVVAR 1.13 1.10 1.09 1.34 1.34 1.34 Inflation MFTVPSVVAR 0.77 ∗ 0.48 ∗∗ 0.28 ∗∗∗ 0.75 ∗ 0.46 ∗∗ 0.26 ∗∗ MFSVVAR 0.86 0.52 ∗∗ 0.29 ∗∗∗ 0.84 0.49 ∗∗ 0.27 ∗∗ MFTVPVAR 0.82 ∗ 0.50 ∗∗ 0.28 ∗∗∗ 0.81 0.48 ∗∗ 0.27 ∗∗ MFVAR 0.90 0.53 ∗∗ 0.29 ∗∗∗ 0.87 0.50 ∗∗ 0.27 ∗∗ QTVPSVVAR 0.91 ∗∗ 0.91 ∗∗ 0.91 ∗∗ 0.90 ∗∗ 0.90 ∗∗ 0.91 ∗∗ QTVPVAR 0.96 ∗∗∗ 0.96 ∗∗∗ 0.96 ∗∗∗ 0.96 ∗∗∗ 0.96 ∗∗∗ 0.96 ∗∗∗ QVAR 1.04 ∗ 1.03 ∗ 1.04 ∗ 1.03 1.03 1.03 QSVVAR 0.59 0.59 0.59 0.75 0.75 0.75 Unemployment rate MFTVPSVVAR 0.80 ∗∗ 0.57 ∗∗∗ 0.37 ∗∗∗ 0.74 ∗∗ 0.52 ∗∗ 0.33 ∗∗∗ MFSVVAR 0.80 ∗∗ 0.61 ∗∗∗ 0.38 ∗∗∗ 0.76 ∗∗ 0.55 ∗∗ 0.33 ∗∗∗ MFTVPVAR 1.00 0.69 ∗∗∗ 0.37 ∗∗∗ 1.02 0.67 ∗∗ 0.33 ∗∗∗ MFVAR 0.79 ∗∗∗ 0.61 ∗∗∗ 0.38 ∗∗∗ 0.74 ∗∗ 0.56 ∗∗ 0.33 ∗∗∗ QTVPSVVAR 0.94 0.93 0.95 0.92 0.92 0.93 QTVPVAR 1.01 1.02 1.02 1.01 1.03 1.03 QVAR 1.03 1.03 ∗ 1.04 ∗ 1.02 1.03 1.03 QSVVAR 0.27 0.26 0.25 0.30 0.30 0.30 Note: RMSEs are in absolute terms for the benchmark model (bottom row of each panel) and as ratios for the remaining models. Ratios below unity mark that the model outperforms the benchmark. Bold figures indicate the best result for the variable and information set. ∗ , ∗∗ , and ∗∗∗ denote significance at the 10%, 5%, and 1% level, respectively, according to the Diebold–Marianotest with Newey–West standard errors. 2059 4.1 | Nowcast Evaluation Table2 presents the results for the nowcast exercise taking into account the information sets I1 to I3. It provides three main takeaways. First, the mixedfrequency models outperform the quarterly models. On average, over all information sets and variables, the best nowcast performance is obtained by the MFTVPSVVAR and the MFSVVAR, which improve on the benchmark (QSVVAR) by roughly 30%. Second, most of the time, the nonlinear MFmodels outperform the linear MFVAR, indicating that—apart from using monthly information—parameter instability is beneficial also in a mixedfrequency setting. Third, the MFmodels' relative performance improves with more information available, showing that the models are able to efficiently process the sequential data releases. For GDP growth, only MFmodels significantly outperform the benchmark. The best performance for both samples is obtained by the MFTVPVAR. This result suggests that, from a nowcasting perspective, it is more important to account for changes in output growth dynamics than to account for the decline in output growth volatility. Moreover, the mixedfrequency VARs' relative performance improves in the 2008– 2017 sample, while the quarterly models performance remains rather unchanged. Thus, it appears that modeling the intraquarterly flow of information has gained importance. In the case of IP growth, accounting parameter instability does not yield more precise forecasts. In fact, the mixedfrequency models provide in both samples virtually identical RMSEs. Against the backdrop of the strong drop of IP growth volatility (see Appendix D of the supporting information), this results is rather surprising. For inflation, the MFTVPSVVAR delivers the best performance; it improves on the benchmark by, on average, 50%. Regarding both samples, the results indicate that notably with little information about the current quarter the MFTVPSVVAR provides large gains in forecast accuracy relative to the competing models. The latter is particularly relevant because expert forecast are usually published in the second month of a quarter. With more information available, however, the differences towards the remaining MFmodels vanish. For the unemployment rate, the MFTVPSVVAR delivers the best nowcasts on average across information sets. However, it appears that nonlinearity is not as important as for the remaining variables as the differences to the MFVAR are negligible. The latter is maybe not surprising given that the fluctuations in volatility of the unemployment rate are less pronounced compared with the remaining variables (see Appendix D of the supporting information). In sum, the nowcast exercise provides strong evidence in favor of nonlinear models. For GDP and IP growth, allowing for timevarying parameters without stochastic volatility strongly improves forecast accuracy. For inflation and the unemployment rate the best forecast performance is obtained by models that account for both timevarying parameters and stochastic volatility. 4.2 | Forecast Evaluation The results in Table3 show that mixedfrequency VARs provide competitive forecasts even for higher horizons and for both samples.8 In the case of the unemployment rate, modeling withinquarter dynamics is particularly beneficial—at each horizon even the worst performing mixedfrequency VAR outperforms the best performing quarterly VAR. Moreover, the results reveal that the models' forecast performance substantially differs across variables. The best relative performance, over all variables and horizons, is delivered by the MFSVVAR and the MFTVPSVVAR; the corresponding RMSEs are roughly 10% lower than those of the benchmark. For GDP growth, the results provide three key findings. First, only the QTVPSVVAR significantly improves on the benchmark regarding both samples; for the shorter sample (right panel), the gains are even more pronounced. Second, the TVP models relative performance strongly improves in the shorter sample, suggesting that timevarying in the autoregressive coefficients has gained importance since the Great Recession. Third, also the mixedfrequency models performance strongly improves in the shorter sample (albeit in most cases insignificant) with the MFTVPSVVAR providing the best performance. For IP growth, the benchmark is hard to beat (for both samples); no model significantly improves on the benchmark. For inflation and the unemployment rate, the MFTVPSVVAR delivers the best performance on average over all horizon, suggesting that timevariation in each coefficient is crucial for these variables. Moreover, for the unemployment rate, the MFmodels consistently outperform the benchmark, while the quarterly models fail to do. Hence, the results provide evidence that both intraquarterly dynamics and timevariation in the VARcoefficients are particularly important for these variables. Thus, our results confirm the findings from previous studies based on quarterly models (see, among others, D’Agostino etal.2013; Barnett etal.2014) by use of mixedfrequency models. In sum, the results indicate that the gains in accuracy due to variations in the VARcoefficients are smaller than the gains induced by stochastic volatility. However, using models with both features provides the most accurate forecasts on average over all variables. Finally, the results provide evidence that modeling withinquarter dynamics is beneficial also regarding shortterm forecasts. 4.3 | Predictive Density Evaluation The results for the CRPS are displayed in Table4. Overall, the results point to the usefulness of withinquarter information in delivering well calibrated predictive densities; the mixedfrequency models provide better results on average over all variables and horizons than their quarterly counterparts. The MFTVPSVVAR provides the best performance with a CRPS reduction of about 10% (on average over all variables and horizons). This emphasizes the importance of parameter instability for generating accurate predictive densities. 2060 Journal of Forecasting, 2025 TABLE 3 | Realtime forecast RMSEs. 1990–2017 2008–2017 Model h=1 h=2 h=3 h=1 h=2 h=3 GDP growth MFTVPSVVAR 0.98 0.96 0.97 0.94 0.91 ∗∗ 0.89 ∗∗∗ MFSVVAR 0.99 1.00 0.98 0.99 0.99 0.97 ∗ MFTVPVAR 0.99 1.00 0.98 0.96 0.95 0.90 ∗∗ MFVAR 1.05 1.05 0.99 1.02 1.00 0.95 ∗∗ QTVPSVVAR 0.98 0.93 ∗∗∗ 0.94 ∗∗ 0.94 ∗∗∗ 0.88 0.86 ∗∗∗ QTVPVAR 0.99 0.93 ∗∗∗ 0.94 ∗∗ 0.98 ∗∗ 0.87 ∗∗∗ 0.86 ∗∗∗ QVAR 1.04 ∗∗∗ 1.00 1.00 1.01 0.98 1.00 QSVVAR 0.61 0.61 0.63 0.69 0.68 0.69 IP growth MFTVPSVVAR 1.04 1.05 ∗ 1.03 1.00 0.99 0.99 MFSVVAR 1.00 1.02 0.99 1.01 1.02 0.99 MFTVPVAR 1.05 1.04 ∗∗ 1.03 1.03 1.02 1.01 MFVAR 1.06 1.09 ∗∗∗ 1.04 ∗∗ 1.05 1.06 ∗ 1.03 QTVPSVVAR 1.03 1.01 1.01 0.99 0.98 1.00 QTVPVAR 1.02 1.00 1.00 1.02 0.98 0.98 QVAR 1.07 ∗∗∗ 1.05 ∗∗∗ 1.03 ∗∗∗ 1.05 ∗∗∗ 1.04 ∗∗ 1.03 ∗∗ QSVVAR 1.23 1.26 1.30 1.46 1.48 1.52 Inflation MFTVPSVVAR 0.81 ∗∗∗ 0.83 ∗∗∗ 0.85 ∗∗∗ 0.81 ∗∗∗ 0.84 ∗∗∗ 0.89 ∗∗∗ MFSVVAR 0.96 1.00 1.01 0.96 0.98 1.00 MFTVPVAR 0.91 ∗∗ 0.91 ∗∗∗ 0.89 ∗∗∗ 0.92 ∗ 0.94 ∗ 0.94 ∗∗ MFVAR 1.06 1.16 ∗∗∗ 1.31 ∗∗∗ 1.03 1.05 1.11 ∗∗∗ QTVPSVVAR 0.89 ∗∗∗ 0.87 ∗∗∗ 0.87 ∗∗∗ 0.89 ∗∗∗ 0.88 ∗∗∗ 0.90 ∗∗∗ QTVPVAR 0.93 ∗∗∗ 0.93 ∗∗∗ 0.91 ∗∗∗ 0.94 ∗∗∗ 0.95 ∗∗∗ 0.95 ∗∗∗ QVAR 1.07 ∗∗∗ 1.12 ∗∗∗ 1.18 ∗∗∗ 1.04 ∗∗∗ 1.07 ∗∗∗ 1.10 ∗∗∗ QSVVAR 0.65 0.65 0.62 0.81 0.79 0.73 Unemployment rate MFTVPSVVAR 0.81 ∗∗∗ 0.88 ∗∗ 0.92 0.78 ∗∗∗ 0.84 ∗∗ 0.87 ∗∗ MFSVVAR 0.84 ∗∗∗ 0.91 ∗ 0.94 0.81 ∗∗ 0.88 ∗ 0.91 MFTVPVAR 0.94 1.00 1.02 0.92 0.97 0.98 MFVAR 0.84 ∗∗∗ 0.91 0.95 0.80 ∗∗ 0.87 ∗∗ 0.91 QTVPSVVAR 0.97 1.00 1.01 0.94 ∗ 0.97 0.98 QTVPVAR 1.03 ∗∗ 1.04 ∗∗∗ 1.04 ∗∗∗ 1.02 ∗∗ 1.03 ∗∗ 1.04 ∗∗∗ QVAR 1.03 ∗∗∗ 1.02 ∗∗∗ 1.02 ∗∗∗ 1.03 ∗∗ 1.02 ∗∗ 1.01 QSVVAR 0.45 0.65 0.85 0.54 0.80 1.06 Note: See Table2 for a description. 2061 TABLE 4 | Realtime forecast CRPS. 1990–2017 2008–2017 Model h=1 h=2 h=3 h=1 h=2 h=3 GDP growth MFTVPSVVAR 0.99 0.97 0.99 0.93 0.91 0.90 MFSVVAR 0.99 1.01 0.99 0.99 1.01 0.98 MFTVPVAR 1.06 ∗ 1.12 ∗∗∗ 1.12 ∗∗∗ 1.00 1.06 1.04 MFVAR 1.07 ∗∗ 1.08 ∗∗ 1.03 1.04 1.03 0.97 QTVPSVVAR 0.99 0.97 0.99 0.95 0.92 0.93 QTVPVAR 1.04 1.02 1.02 1.01 0.93 0.93 QVAR 1.07 ∗∗∗ 1.05 ∗∗∗ 1.05 ∗∗∗ 1.03 1.02 1.03 QSVVAR 0.33 0.33 0.34 0.36 0.36 0.37 IP growth MFTVPSVVAR 1.03 1.06 ∗ 1.07 ∗ 0.95 0.98 1.03 MFSVVAR 1.00 1.03 1.02 1.01 1.03 1.02 MFTVPVAR 1.02 1.03 1.02 0.98 0.98 0.99 MFVAR 1.09 ∗∗ 1.11 ∗∗∗ 1.07 ∗∗ 1.05 1.05 1.02 QTVPSVVAR 1.06 ∗∗∗ 1.03 1.03 1.00 1.00 1.01 QTVPVAR 1.06 ∗∗∗ 1.04 1.04 1.04 1.00 0.99 QVAR 1.10 ∗∗∗ 1.06 ∗∗∗ 1.05 ∗∗∗ 1.05 1.02 1.02 QSVVAR 0.64 0.66 0.69 0.73 0.76 0.78 Inflation MFTVPSVVAR 0.81 ∗∗∗ 0.83 ∗∗∗ 0.81 ∗∗∗ 0.81 ∗∗∗ 0.85 ∗∗∗ 0.86 ∗∗∗ MFSVVAR 0.97 1.06 1.08 ∗∗∗ 0.95 1.03 1.06 MFTVPVAR 0.88 ∗∗ 0.88 ∗∗∗ 0.81 ∗∗∗ 0.90 ∗ 0.91 ∗ 0.86 ∗∗∗ MFVAR 1.06 1.20 ∗∗∗ 1.32 ∗∗∗ 0.99 1.04 1.07 QTVPSVVAR 0.88 ∗∗∗ 0.85 ∗∗∗ 0.83 ∗∗∗ 0.89 ∗∗∗ 0.87 ∗∗∗ 0.86 ∗∗∗ QTVPVAR 0.93 ∗∗∗ 0.91 ∗∗∗ 0.87 ∗∗∗ 0.92 ∗∗∗ 0.91 ∗∗∗ 0.88 ∗∗∗ QVAR 1.09 ∗∗∗ 1.12 ∗∗∗ 1.15 ∗∗∗ 1.04 1.05 1.04 QSVVAR 0.32 0.34 0.34 0.42 0.41 0.40 Unemployment rate MFTVPSVVAR 0.79 ∗∗∗ 0.85 ∗∗ 0.89 ∗ 0.75 ∗∗∗ 0.79 ∗∗∗ 0.83 ∗∗ MFSVVAR 0.84 ∗∗∗ 0.89 ∗∗ 0.93 ∗ 0.80 ∗∗∗ 0.85 ∗∗ 0.89 ∗∗ MFTVPVAR 1.02 1.11 ∗∗ 1.16 ∗∗ 1.02 1.09 1.14 ∗ MFVAR 0.84 ∗∗∗ 0.91 ∗∗ 0.94 0.80 ∗∗∗ 0.85 ∗∗ 0.88 ∗ QTVPSVVAR 0.96 0.99 1.01 0.92 ∗∗ 0.95 0.97 QTVPVAR 1.03 ∗ 1.04 ∗∗ 1.05 ∗∗ 1.02 1.03 1.04 QVAR 1.03 ∗∗ 1.01 1.00 1.01 1.00 0.98 QSVVAR 0.23 0.33 0.43 0.27 0.40 0.54 Note: Scores are in absolute terms for the benchmark (bottom row of each panel) and as ratios for the remaining models. Ratios below unity show that the model outperforms the benchmark. Bold figures mark the best result for the variable and horizon. ∗ , ∗∗ , and ∗∗∗ denote significance at the 10%, 5%, and 1% level, respectively, according to a ttest on the average difference in scores relative to the benchmark model with Newey–West standard errors. 2062 Journal of Forecasting, 2025 For GDP growth density forecasts, no model significantly improves on the benchmark. Only the MFTVPSVVAR and QTVPSVVAR provide an (insignificant) improvement. Thus, not only the mean of the predictive distribution is slightly more precise than those of the benchmark, but the entire distribution. As for point forecasts, IP growth density forecast from the benchmark are superior compared with the remaining models. Regarding inflation, the results indicate two outcomes. First, the QTVPSVVAR and the MFTVPSVVAR deliver the largest improvements on the benchmark. Hence, as for point forecasts, it is important to model timevariation in both the parameters and the residual variances to obtain precise predictive densities. Second, including timevariation in the parameters plays a vital role; both the MFTVPVAR and the QTVPVAR strongly improve on the benchmark. For the unemployment rate, the results support those from the point forecasts evaluation; the MFTVPSVVAR delivers the best performance, improving on the benchmark by up to 20%. Imposing parameter instability only in the VAR coefficients seems to deteriorate the accuracy of the predictive densities. In fact, the MFVAR and the QVAR outperform the TVP VARs. One reason for this results might be that the TVP VARs—particularly the MFTVPVAR—exaggerate the actual timevariation in the VAR coefficients. Moreover and in contrast to inflation, each mixedfrequency model improves both on the benchmark and on its quarterly counterpart. Thus, for generating precise predictive densities for the unemployment rate, it is crucial to include intraquarterly information and stochastic volatility. In summary, the results of the predictive density evaluation support the findings from the point forecast evaluation. Using mixedfrequency models is beneficial over all variables and horizons. It significantly improves results for inflation and the unemployment rate. In addition, we confirm the importance of stochastic volatility in density forecasting by use of mixedfrequency VARs. We provide evidence that combining FIGURE 1 | Inflation forecasts during the Great Recession. Rows refer to mixedfrequency models; columns refer to the forecast origins. Actual values (red line) and forecasts: mean (black line) and 60% and 90% bands from the predictive distributions.