Are low frequency macroeconomic variables important for high frequency electricity prices?
aa r X i v : . [ q -f i n . S T ] J u l Are low frequency macroeconomic variables important for high frequencyelectricity prices? ∗ Claudia Foroni a Francesco Ravazzolo b,c
Luca Rossini d,e a European Central Bank, Germany b Free University of Bozen-Bolzano, Italy c CAMP, BI Norwegian Business School, Norway d Vrije Universiteit Amsterdam, The Netherlands e Ca’ Foscari University of Venice, Italy
July 28, 2020
Abstract
We analyse the importance of low frequency hard and soft macroeconomic information, respectively theindustrial production index and the manufacturing Purchasing Managers’ Index surveys, for forecasting high-frequency daily electricity prices in two of the main European markets, Germany and Italy. We do that bymeans of mixed-frequency models, introducing a Bayesian approach to reverse unrestricted MIDAS models(RU-MIDAS). Despite the general parsimonious structure of standard MIDAS models, the RU-MIDAS hasa large set of parameters when several predictors are considered simultaneously and Bayesian inference isuseful for imposing parameter restrictions. We study the forecasting accuracy for different horizons (from 1day ahead to 28 days ahead) and by considering different specifications of the models. Results indicate thatthe macroeconomic low frequency variables are more important for short horizons than for longer horizons.Moreover, accuracy increases by combining hard and soft information, and using only surveys gives lessaccurate forecasts than using only industrial production data.
JEL codes:
C11, C53, Q43, Q47.
Keywords:
Density Forecasting, Electricity Prices, Forecasting, Mixed-Frequency VAR models, MIDASmodels. ∗ This paper has previously circulated with the title “Forecasting daily electricity prices with monthly macroeconomic variables”.We thank seminar and conference participants at University of Geneva, at University of Glasgow and at Maastricht University,the 23rd Applied Time Series Econometrics Workshop at Federal Reserve Bank of St. Louis, the 13th Netherlands EconometricStudy Group, the 2nd Workshop on Energy Economics at SKKU, the NBP Workshop on Forecasting at Central Bank of Poland,the 2nd IWEEE at Venice for helpful comments and suggestions to improve this work. This paper should not be reported asrepresenting the views of the European Central Bank (ECB). The views expressed are those of the authors and do not necessarilyreflect those of the ECB. This research used the SCSCF multiprocessor cluster system at Ca’ Foscari University of Venice. LucaRossini acknowledges financial support from the EU Horizon 2020 programme under the Marie Sk lodowska-Curie scheme (grantagreement no.796902). Introduction
Electricity markets have received increased attention in the literature since their deregulation in the late 90s.There are several reasons motivating such interest. First, electricity is not storable and therefore demandand supply must always match. To achieve this, sophisticated markets have been created, where the oneday-ahead hourly spot market is the main market in terms of volume. In the day-ahead spot market hourlyprices are set by matching demand and supply. This market offers large amount of data and requires forecastsof both demand and prices. Second, power grids are one of the most critical infrastructures and have a majorrole in sustainable development and economic growth. The recent innovation in energy production and, inparticular, the large increase in renewable energy resources (RES) have added complexity to the managementof the electricity system, see Gianfreda et al. (2020) for an application of RES to predict day-ahead prices.Moreover, smart grids are the future technologies in power grid development, management, and control,see Yu et al. (2011) and Yu and Xue (2016). They have revolutionized the regime of existing power grids,by employing advanced monitoring, communication and control technologies to provide secure and reliableenergy supply. And new technologies have changed energy consumption, making it necessary to use effectiveenergy management strategies based on electricity prices and electricity load forecasts. As a consequence,a growing literature has investigated these dynamics and built forecasting models for several markets allaround the world (Europe, United States, Canada, and Australia).The literature on load forecasting has focused on three horizons: short-term load forecasts (from onehour to one week); mid-term load forecasts (from one week to one month) and long-term load forecasts (fromone month to years); see, for example, Alfares and Nazeeruddin (2002) and Suganthi and Samuel (2012)for definitions and models. By contrast, the literature on price forecasting has mainly focused only onthe day-ahead spot market, see Weron (2014) for a recent and detailed review. Two possible reasons arethat the predictive power of predictors for day-ahead spot prices is usually short lived, and longer futuremarkets are subject to low liquidity and highly correlated to spot prices. This paper tries to fill this gap andintroduces a new methodology to produce mid term spot price forecasts, that is forecasts of day-ahead spotprices up to one month ahead. In order to accomplish this, it suggests applying lower frequency predictorsbased on macroeconomic variables containing more valuable information for mid-term horizons as opposedto the regressors usually applied in short-term price forecasting. Further, it develops a model to match themismatch in frequency between the daily prices and the monthly macro variables.In the last years, there is a growing interest in models that account for data of different frequencies forforecasting purposes. The focus in the literature has mostly been on improving the forecast of low-frequencyvariables by means of high-frequency information. In particular, different models have been introduced fordealing with the different sampling frequencies at which macroeconomic and financial indicators are available.The most common choice is to reduce the model to state space form and use the Kalman filter for forecasting(e.g. see Aruoba et al. (2009); Giannone et al. (2008a); Mariano and Murasawa (2002) and in a Bayesiancontext Eraker et al. (2015); Schorfheide and Song (2015)). As an alternative choice, Ghysels (2016) developsa class of mixed-frequency VAR model, where both low- and high-frequency variables are included in thevector of dependent variables (see Blasques et al., 2016, for an application in small-scale factor model). This lass of model is estimated by OLS, but the number of regressors tends to increases due to the stackingstructure of the model.In an univariate context, Ghysels et al. (2006) introduce MIDAS, which links directly low- to high-frequency data (see Clements and Galv˜ao, 2008, 2009, for macroeconomic forecasting), but it requires aform of NLS estimation, which improves the computational costs substantially in model with more thanone high-frequency explanatory variables. Foroni et al. (2015a) develop unrestricted MIDAS (U-MIDAS)model, which can be estimated by OLS and thus handle high-frequency explanatory variables. However, theU-MIDAS models have problems when the frequency mismatch is high and several regressors are included,thus leading to a Bayesian extension of the literature on MIDAS and U-MIDAS, see Foroni et al. (2015b) andPettenuzzo et al. (2016), and a stochastic volatility estimation method for U-MIDAS in density nowcasting(Carriero et al., 2015).Recently, new models have been proposed for forecasting high-frequency variables by means of low-frequency variables. An example is the paper of Dal Bianco et al. (2012), who analyse the forecasts of theeuro-dollar exchange rate at weekly frequency by means of macroeconomic fundamentals in a state-spaceform `a la Mariano and Murasawa (2009). Ghysels (2016) contributes by introducing a mixed-frequency VARmodel, which address both the prediction of high-frequency variables using low-frequency variables and viceversa. Furthermore, Foroni et al. (2018) introduce Reverse Unrestricted MIDAS (RU-MIDAS) and ReverseMIDAS (R-MIDAS) model for linking high-frequency dependent variable with low-frequency explanatoryvariables in univariate context.From a methodological innovation point of view, this paper proposes a Bayesian approach to RU-MIDASof Foroni et al. (2018) in order to incorporate low frequency information into models for the prediction ofhigh frequency variables. Our goal is to derive a model that allows to combine efficiently several predictors,possibly with different frequencies. The use of Bayesian inference allows to mitigate parameter uncertaintyand to compute probabilistic statements without any further assumption. Despite other mixed-frequencyspecifications could be incorporated, we decide to work with the RU-MIDAS because our focus is on longerhorizons, where relative predictability is lower and linear models usually perform accurately.Several papers have documented that surveys are useful for predicting macro variables, see e.g.Hansson et al. (2005), Abberger (2007), Claveria et al. (2007), Aastveit et al. (2016). As highlighted by,e.g., Evans (2005), Giannone et al. (2008b), Aastveit et al. (2014), an advantage of surveys is that theyare timely available and possibly contain forward looking information. We label them “soft” data, giventhey usually just represent the opinions or impressions of consumers or purchasing managers, who are askedto compare economic and financial conditions today with the recent past, and/or to forecast the economicenvironment in the near future. However, this sort of economic indicator surveys has not yet been comparedto hard data in a context similar to ours, so the reliability in our context is not proven yet.We assess the performance of the proposed approach by evaluating the relevance of hard and softmacroeconomic variables that are available at monthly frequency for forecasting the daily electricity pricesin two of the most important European countries, Germany and Italy. We predict the daily electricity priceat different horizons and we introduce different low frequency explanatory variables, such as the industrial See blog . roduction index evaluated at different levels, the Manufacturing Purchasing Managers’ Index surveys andthe oil prices. In the last years, a large and growing body of literature deals with the forecasting of dailyelectricity prices (see Weron, 2014, for a review). However, the main focus of the literature is on theforecasting of electricity prices influenced by variables with the same frequency, such as renewable energysources (Gianfreda et al., 2020) or weather forecasts (Huurman et al., 2012). This empirical applicationdraws on the literature using macroeconomic variables to improve the forecasting performance of singlefrequency models, due to the fact that macroeconomic variables are of interested in the diagnostic of electricityprices.The results show that there is a strong improvement in the forecasting if we add all monthlymacroeconomic variables (such as PMI surveys and IPIs) and different oil prices specification (daily ormonthly), at almost all horizons for Italy and at the short horizons for Germany. We find gains around20% at short horizons and around 8% at long horizons. The benchmark is almost never included in themodel confidence set. Interesting, accuracy increases by combining hard and soft information, and usingonly surveys gives marginally less accurate forecasts than using only industrial production data.The paper is organized as follows. Section 2 summarizes the RU-MIDAS models and the Bayesianapproach. Section 3 presents the data used in the paper. In Section 4, we present the forecasting of dailyelectricity prices by using daily and monthly macroeconomic variables. Section 5 concludes. Foroni et al. (2018) show the derivation of the reverse unrestricted MIDAS (RU-MIDAS) regression approachfrom a general dynamic linear model and its estimation procedure. Here we sketch the derivation, adaptingit to our case of monthly/daily observations. For the sake of simplicity, we assume the following two variablesof interest. Let us observe at high-frequency (HF) the variable x for t = 0 , k , . . . , k − k ,
1, while the variable y can be observed at low frequency (LF) every k periods for t = 0 , , , . . . .In our case, the variable x follows an AR( p ) process c ( L ) x t = d ( L ) y ∗ t + e xt , (1)where y ∗ is the exogenous regressor; d ( L ) = d L + . . . + d p L p , c ( L ) = I − c L − . . . c p L p and the errors arewhite noise. Furthermore, we assume that the starting values y ∗− p/k , . . . , y ∗− /k and x − p/k , . . . , x − /k are allfixed and equal to zero.It is possible to introduce the lag operator for the low and high-frequency variables. In particular, let usdefine Z , the LF lag operator such that Z = L k and Z j y t = y t − j ; and the polynomial in the HF lag operator, γ ( L ) with γ ( L ) d ( L ) containing only L k = Z . If we multiple Eq. (1) by γ ( L ) and ω ( L ), we have γ ( L ) c ( L ) ω ( L ) x t = γ ( L ) d ( L ) ω ( L ) y ∗ t + γ ( L ) ω ( L ) e xt , t = 0 , , , . . . (2)where ω ( L ) = ω + ω L + . . . + ω k − L k − represents the temporal aggregation scheme by means of a olynomial. Moreover, if Eq. (2) is represented as˜ c ( L ) x t = g ( Z ) y t + ˜ γ ( L ) e xt , t = 0 , , , . . . , (3)where y t = w ( L ) y ∗ t and g ( Z ) is the product of γ ( L ) and d ( L ) and function only of Z , Eq. (3) is called anexact reverse unrestricted MIDAS model. In particular, in Eq. (3), the high-frequency variable is a functionof its own lags, of the LF lags of the observable variable y and of the error terms. Thus, the HF periodinfluences the model specification. For each i = 0 , . . . , k −
1, a lag polynomial in the HF lag operator, γ i ( L ),can be defined and the product g i ( L ) = γ i ( L ) d ( L ) is a function only of power of Z . As seen above, if wemultiple Eq. (1) by γ i ( L ) and d ( L ), we have˜ c i ( L ) x t = g i ( L k + i ) y t + ˜ γ i ( L ) e xt , t = 0 + ik , ik . . . , i = 0 , . . . , k − γ i ( L ) cannot be determined exactly, it is possibleto use an approximate reverse unrestricted MIDAS (RU-MIDAS) models based on linear lag polynomial˜ a i ( L ) x t = b i ( L k + i ) y t + ξ it , t = 0 + ik , ik . . . , i = 0 , . . . , k − a i ( L ) and b i ( L k + i ) are larger enough such that ξ it is a white noise. Since the error terms ξ it are correlated across i , one could estimate the RU-MIDAS equations for different values of i by using asystem estimation method. In particular, Eq. (5) can be grouped in a single equation by adding a properset of dummy variables. In our empirical application, we consider a daily dependent variable and monthlyexplanatory variables such that the single-equation version of Eq. (5) is x t = α − X i =2 D i ! y t − + α D y t − + · · · + α D y t − ++ β , − X i =2 D i ! x t − + β , D x t − + · · · + β , D x t − ++ β , − X i =2 D i ! x t − + β , D x t − + · · · + β , D x t − + (6)+ β , − X i =2 D i ! x t − + β , D x t − + · · · + β , D x t − + v t t = 0 , , , . . . , where D , . . . , D are dummy variables taking value one in each last 28-th day, last 27-th day and firstday of the month respectively and v t is independent and identically distributed as a Normal distribution,where N (0 , σ ). It is possible to estimate the model in Eq. (6) by GLS to allow the possible correlation andheteroskedasticity. However, it may be difficult to estimate the model by using a frequentist approach, inparticular if there are several regressors. Thus we use a Bayesian approach to solve this issue. .1 Bayesian approach Contrary to most of the MIDAS literature, which follows a classic approach, in this paper we estimateour models with Bayesian techniques. Few papers so far have focused on the Bayesian estimation of regularMIDAS models (see, for example, Pettenuzzo et al. (2016) and Foroni et al. (2015b)). However, the Bayesianmethod has not yet been applied to the RU-MIDAS approach, as described in the previous section. Differentlythan the classical estimation, our Bayesian approach allows for estimation of complex nonlinear models withmany parameters, is useful for imposing parameter restrictions and, above all, allows to compute probabilisticstatements without any further assumption.In this paper, therefore, we focus on introducing the Bayesian estimation in the RU-MIDAS model. Wedefine prior information on the vector of coefficients and on the variance, using the independent Normal-Wishart prior as in Koop and Korobilis (2010) adapted to univariate time series, thus a Normal-Gammaprior.This section is devoted to the study of prior and posterior inference on the vector of coefficients of theautoregressive model and on the variance coefficient. In particular, we work with a prior which has ARcoefficients and variance coefficients being independent each other, thus it is called independent Normal-Gamma prior.The general prior for this kind of model, which does not involve the restrictions of the natural conjugateprior, is the independent Normal Gamma prior. Let us assume γ be the vector of the AR coefficients definedin equation (6) and made by α , . . . , α , β , , . . . , β , , β , , . . . , β , and σ be the variance coefficients,thus the independent prior can be represented as p ( γ, σ − ) = p ( γ ) p ( σ − ). In this case, the prior for γ is anormal distribution: γ ∼ N (cid:0) γ, V γ (cid:1) , (7)while the prior for the variance coefficients is a Gamma distribution σ − ∼ G a ( a, b ) (8)By using these priors, the joint posterior p ( γ, σ − | x ) has not a convenient form, but the conditional posteriordistribution have a closed form. In particular, the posterior distribution for the vector of AR coefficients is: γ | x, σ − ∼ N (cid:0) γ, V γ (cid:1) (9)where the posterior mean and posterior variance are: V γ = V − γ + 1 σ T X t =1 z t z t ! − γ = V γ V γ γ + 1 σ T X t =1 z t x t ! , where z t is the vector containing the explanatory variables y t − , . . . , y t − and the lagged dependentvariables x t − , . . . , x t − . oreover, the posterior distribution for the variance coefficients is: σ − | γ, x ∼ G a ( a, b ) (10)where the posterior hyperparameters are a = T + a b = b + T X t =1 ( x t − z t γ ) We estimate the Bayesian model described above using the Bayesian Markov chain Monte Carlo (MCMC)methods. We have used the Gibbs sampling algorithm for both prior distributions and all our results arebased on samples of 6.000 posterior draws, with a burn-in period of 1.000 iterations. Moreover, we choosethe prior hyperparameters such that the prior are not informative.Regarding the forecasting techniques adopted in the paper, we use the direct forecasting method (see,e.g. Marcellino et al., 2006) since the forecasting of the future values of the explanatory variable y is notrequired, although the model specification should change for each forecasting horizon considered. The longforecast horizons we target, up to 28 days ahead, might imply that short-term dynamics are less relevantand we investigate in section 4 different lag specifications. In this section we describe the two datasets analysed in the application. In particular, we consider two of themost important European countries from a macroeconomic and energy point of view, Germany and Italy,both parts of the G8 economies.We use daily day-ahead prices (in levels) to estimate models for electricity traded/sold in Germany andItaly. Moreover, we employ different monthly macroeconomic variables, which either differ by country, suchas industrial production index or Purchasing Managers’ Index (PMI); or are equal for all the countries, suchas the oil brent prices. The national electricity prices are obtained directly from the corresponding powerexchanges. In particular, the German daily auction prices of the power spot market is collected from the
European Energy Exchange
EEX, whereas the daily single national prices PUN are collected from the ItalianISO.In terms of macroeconomic variables, we consider the total industrial production index for Germany andItaly, and its main components: consumer goods (IPI-Cons, i.e. the consumer durable goods); electricity(IPI-Elec, i.e. the activity of providing electric power, natural gas, steam, hot water and the like througha permanent infrastructure (network) of lines, mains and pipes) and manufacturing (IPI-Manuf, i.e. theactivities in the manufacturing section involve the transformation of materials into new products) . Thedata are taken from Eurostat and are seasonally and calendar adjusted.The other macroeconomic variable consider is the Manufacturing PMI surveys, which is a measure of theperformance of the manufacturing sectors and it is derived from a survey of 500 industrial companies. On he other hand, the Italian PMI is based on surveys of about 400 industrial companies. The ManufacturingPMI is based on five industrial indexes with the following weights: New Orders (30%), Output (25%),Employment (20%), Suppliers’ Delivery Times (15%) and Stock of Items Purchased (10%) with the DeliveryTimes Index inverted so that it moves in a comparable direction. This index is of capital importance sincethe manufacturing sector dominates a large part of the total GDP and thus it is an important indicator ofthe business conditions and of the overall economic condition in the country. Moreover, a reading above 50indicates an expansion of the manufacturing sector compared to the previous month; on the other hand, avalue below 50 represents a contraction. We consider it an important soft macroeconomic indicator of futureeconomic conditions.We should emphasize that the monthly variables are released in different days of the month and thus weneed to keep attention when we will analyse them in the forecasting exercise. As an example, the Industrialproduction index is released on the first working day of the following month, thus the IPI for December2018 is released on the 2nd of January 2019; the IPI for January 2019 is released on the 1st of February,etc. Obviously if the first day of the month is a specific holiday or saturday or sunday, then the IPI willbe released in the first coming working day. For what concerns the surveys, the final release of the PMI istypically available the first day of the month after the one they refer to (i.e. the PMI of Dempber is releasedat the very beginning of January, etc.) [Insert Figure 1 here]The sample spans from 1 January 2006 to 31 December 2019 for both countries. We use the first sevenyears as estimation sample and the last seven years as forecast evaluation period. The historical dynamics ofthese series are reported in Fig. 1 for Germany and in Fig. 2 for Italy. Prices clearly show the new stylizedfact of “downside” spikes together with mean-reversion. On the other hand, in panel (b) of Fig. 1 and 2,we show the daily and monthly dynamics of the oil prices. The black (daily) and the red (monthly) linesshow the same course during the entire sample size. In particular, the oil price shows two strong falls, thefirst around the end of 2008 and the beginning of 2009; the second around the end of 2014. Regarding thefirst fall, the drop in oil prices that started in 2008 takes place against the backdrop of the global financialcrisis. In fact, the oil prices drop from historic highs of 141 .
06$ in July 2008 to 40 .
07$ in March 2009. Afteran increase of the oil prices in the following years, the second fall appears in the fourth quarter of 2014 asrobust global production exceeded demand, thus leading to a sharp decline.[Insert Figure 2 here]Regarding the other macroeconomic variable of interest, the industrial production index (IPI), it shows adifferent behaviour between the two countries. In fact, in Germany, the industrial production index followsthe first drop of the oil prices in 2008/2009, while it leads to a constant slow increase in the following yearsuntil 2018. In recent years (2018 and 2019), the IPI related to the different specifications has a slow decrease,with a particular strong fall for the IPI of electricity supply in 2019. On the other hand, the situation inItaly is completely different since after the fall in 2008, the situation remains the same or slightly decreases n the subsequent years, with a tiny increase at the end of 2017. Regarding the PMI Survey, the behaviorof the series follows the same movement across the two countries, with a huge fall in 2008, which representsan important contraction in both countries economies and a consequently increase in 2009 and a constantbehavior in the next years over 50, which indicates a small expansion of the economies. As stated for theIPI, the last years of the sample show evidence of contraction in both countries and in both economies ascan be seen for the GDP. In this section we present the results for the forecasting of daily electricity prices by means of differentmacroeconomic variables. In particular, the first estimation sample in the forecasting exercise extends fromJanuary 2006 to December 2012, and it is then extended recursively by keeping the size of the estimationwindow fixed to 7 years in such a way we perform a rolling window estimation. For each day of the evaluationsample, we compute forecasts from 1 to 28 days ahead, and we assess the goodness of our forecasts usingdifferent point and density metrics.
Regarding the accuracy of point forecasts, we use the root mean square errors (RMSEs) for each of the dailyprices and for each horizons. Whereas, to evaluate density forecasts, we use both the average log predictivescore, viewed as the broadest measure of density accuracy (see Geweke and Amisano, 2010) and the averagecontinuous ranked probability score (CRPS). The latter measure does a better job of rewarding values fromthe predictive density that are close and not equal to the outcome, thus it is less sensitive to outlier outcome(see, e.g. Gneiting and Raftery, 2007; Gneiting and Ranjan, 2011).As seen in Eq. (6), one can evaluate different RU-MIDAS model based on different lags order ofthe high-frequency variables and on the inclusion of different low-frequency variables. As suggested inWeron and Misiorek (2008), Raviv et al. (2015) and Gianfreda et al. (2020), we consider a RU-MIDAS modelwith lag order of the electricity prices equal to 7. In particular, this model includes only the first, second andthe seventh lag of the daily electricity prices; with an abuse of notation we will set p = 3 and consequentlyAR(3). Moreover, due to the seasonal components of the daily electricity prices, we include seasonal dummiesrepresenting each season of the year: spring, summer, autumn and winter, respectively. In the benchmarkmodels, called BAR(3), the estimation is provided by using a Normal-Gamma prior and the same prior asbeen used also for the Bayesian RU-MIDAS model, called B-RU-MIDAS.In our analysis, we focus also on another benchmark specification, the autoregressive model of order 1(BAR(1)), where only one lag of the daily electricity prices is included. Also for this benchmark model, weinclude seasonal dummies in the analysis.The main interest of the paper is forecasting daily electricity prices by using macroeconomic variables.Thus, we consider different macroeconomic explanatory variables in the construction of the models. In eachmodel and for each country, as explanatory variables, we include separately the monthly specification of theManufacturing PMI surveys or the three main industrial production indices (All-IPI). As further check, we dd a specification of the model that has both all the three main industrial production indices and the PMIsurveys. Moreover, the daily oil prices and the monthly oil prices has been included in the model specificationfor some specific cases. When we discuss the three main IPI, we consider IPI based on the manufacturingsector (IPI-Manuf), on the activity of providing electric power (IPI-Elec) and on Main Industrial Groupings(MIG) for consumer goods (IPI-Cons). As a robustness check, we analyse models where we include PMIsurveys and either ALL-IPI, only one of the index, or combinations of two indices (IPI-Cons-Elec, IPI-Cons-Manuf, IPI-Elect-Manuf) and we include or not the oil price specification as monthly or daily.In detail, in our tables we report the RMSE, average log predictive score and average CRPS for thebenchmark BAR(3) and BAR(1) with seasonal dummies and with a Normal-Gamma prior. For the otherBayesian RUMIDAS models with Normal-Gamma prior (B-RU-MIDAS), we report: the ratios of each model’sRMSE to the baseline BAR model, such that entries smaller than 1 indicate that the given model yieldsforecasts more accurate than those from the baseline; differences in score relative to BAR baseline, such thata positive number indicates a model beats the baseline; and ratios of each model’s average CRPS relative tothe baseline BAR model, such that entries smaller than 1 indicate that the given model performs better.To test the predictive accuracy, we apply Diebold and Mariano (1995) t-tests for equality of the averageloss (with loss defined as squared error, log score or CRPS). The asterisks denote if the differences inaccuracy are statistically different from zero, with one, two or three asterisks corresponding to significancelevel 10%, 5% and 1% respectively. We use p-values based on one-sided test, where the benchmark modelsare the null hypothesis and the other models are the alternatives. We also employ the Model ConfidenceSet procedure of Hansen et al. (2011) to jointly compare the predictive power of all models. We use the R package MCS detailed in Bernardi and Catania (2016) and differences are tested separately for each class ofmodels (meaning for each panel in the tables and for each horizon).
Point forecasts
We start by evaluating the point forecast of the different models and in the panel (A) of Table 1 and 2,we present the RMSEs for different mixed frequency models relative to the benchmark model, the so calledBayesian AR(3) with seasonal dummies and Normal-Gamma prior.[Insert Table 1 here]Focusing first on Germany, in Table 1 we observe that the RMSE remains broadly constant over thehorizons. Since we are predicting daily electricity prices, there is a strong improvement in the forecastingif we add monthly macroeconomic variables. In particular, the improvement is large in the first horizons,and in general for short-term forecasts, while at longer horizons, such that 21 and 28, the content of macroinformation is less relevant and we even see a decrease in the forecasting performance, even if gains arestill 10% relative to the benchmark. It is in general hard to rank the models with different macroeconomic The daily oil price has been interpolated over the weekends in order to have a full sample size. Regarding density forecasts, we use equal weights and not adopt weighting scheme as in Amisano and Giacomini (2007) ndicators, where the performance of the different model specifications in terms of point forecasting is rathersimilar. In particular, adding oil price with daily or monthly specification does not improve the forecastingresults with respect to the model that does not include oil price. As further results, the inclusion in theanalysis of only one of the two most important macroeconomic variables, PMI surveys or All-IPI, leads toa worst performance with respect to model that consider both variables, and using only surveys gives lessaccurate forecasts than using only industrial production data.However, what we find, is a strong evidence of statistically superior predictability by the alternativemodels to the benchmark at several horizons. The B-RU-MIDAS model with All-IPI, PMI surveys and oilprice gives the best statistic at one day ahead with a 20% reduction in RMSE, but also other version of B-RU-MIDAS without including the oil price provides economically sizeable gains at those horizons. Moreover,B-RU-MIDAS with all the IPI variables and PMI surveys provide also statistically gains at longer horizons,such that h = 21 ,
28 with no differences between the inclusion or not of the oil price (both daily or monthly).Looking at Germany, the model that includes all the macroeconomic variables and the monthly oil prices isconsidered the best models at the first two horizons, while increase the horizons lead to different results. Forexample at 21 step ahead, the best model becomes the one that include only the PMI surveys, while at 28step ahead, the model that includes all the macroeconomic variables and the oil prices return to be the bestmodel. [Insert Table 2 here]For the case of Italy, results are shown in Table 2. Contrary to the case of Germany, for the case ofItaly there is a strong movement of the RMSEs from the first horizon to the 28 horizon, moving from 8 . .
64. Moreover, the model that consider All-IPI, PMI Surveys and the daily oil prices analysed in thepaper tend to dominate in terms of forecasting performance. In particular, the B-RU-MIDAS with all the IPImacroeconomic variables and the daily oil prices leads to a reduction around 19% of the RMSE with respectto the benchmark model at the first horizons. While at the second and third horizon, the gains are in termsof 21%. On the other hand, when the horizon size increases, the B-RU-MIDAS models gain somewhat less,but still the reduction is around 9% from the benchmark. However, if we do not consider in the analysis themonthly PMI surveys or All-IPI we have a small reduction of just 4% from the benchmark. Differently fromGermany, in Italy the model that includes all the macroeconomic variables and the monthly oil price is notconsidered the best model across the horizon from evidence of statistically superior predictability in termsof Model Confidence Set. Whilst the best model across the horizons seems to be the model that includesonly all the IPIs (All-IPI). Moreover, in Italy from a statistically superior predictability point of view, theinclusion of oil price, in terms of both monthly or daily specification, does not need to better models at allthe horizons.Looking at Panel B of Table 1 and 2, we present the results for the RMSE for different models relativeto a benchmark that includes only the first lag of the electricity prices. For Germany, we can see thatthe RMSE moves from 13 to 14 .
51 across the horizons, which leads to worst results with respect to theautoregressive with 3 lags. This result is also confirmed from the ratios of the RMSE, where the best models re the ones that include the different specification of the oil price. These models gain around 28% from thebenchmark in the first 3 horizons and this improvement is also important for long term horizons. Hence, at21 or 28 horizons ahead, the inclusion of oil prices leads to improvements of around 20%. These gains arevisible also for the other models that includes only the monthly macroeconomic variables jointly or separable.Regarding the inclusion in the Superior set, the models with oil prices specification seems to be present inall the horizons, while the models with only monthly macroeconomic variables are less important at the firstand at the last horizon.Regarding Italy, Panel B of Table 2 shows the results for the second benchmark. Differently from Germany,the RMSE moves from 8 . . Density forecasts
We now focus on two different metrics for the density forecasts: the CRPS and the log predictive score, thesecond and third sub-panel of Panel (A) in Table 1 and 2, for Germany and Italy respectively, with baselinemodel AR(3). In general, the accuracy of density forecasts improves in the models with macroeconomicvariables, where we observe substantial low CRPS across the models and horizons. As before, we observegenerally higher CRPS values when the horizon increases from 1 to 28. In particular as in the point forecastanalysis, the B-RU-MIDAS with all the monthly IPI variables (All-IPI); PMI Surveys and both oil pricespecification (daily or monthly) gives the best statistics at one day ahead with a 23% reduction in averageCRPS in Germany. At longer horizons, as in the point forecast analysis, the inclusion of macro informationand oil price leads to lower gains, but still significant and around 8% better relative to the benchmarkmodels. Moreover, the inclusion of only one monthly macroeconomic variable (such as All-IPI or PMISurveys) separately leads to small improvements in the order of 19% at short horizons and 3% at longhorizons. As for point forecasting, using only surveys gives less accurate forecasts than using only industrialproduction data. For the inclusion in the Superior set of Models at 10%, at the first horizon the model withall monthly macroeconomic variables and monthly oil price is the best, while at the last horizon it becomesthe one with daily oil price. In the same direction for the other horizons, the models with both monthly ordaily oil price are the best models with respect to the benchmark.Looking at second sub-panel of Panel (A) of Table 2, we can see that the model with daily oil pricesand monthly macroeconomic variables outperforms the benchmark by 20% at short horizons, while at longerhorizons the improvements reduces to 9%. As stated for Germany and in the point forecasting exercise, the nclusion of only one macroeconomic variables leads to improvements with respect to the baseline AR(3)model of 19% at the short term horizons, while at long horizons these improvements drastically decrease to3%.Regarding the average log predictive score (see the third sub-panel of Panel (A) in Table 1 and 2), theresults change with respect to the average CRPS. In particular, for Germany, the average log predictivelikelihood shows smaller increases at all horizons except for the first and last horizons. The gains in termof log predictive score is higher in the models that include only the monthly macroeconomics variablesjointly moving from a 11% at the first horizons to a 9% at the last horizons. In this case, there are noevidences of superior predictability of a models over the others, except that the models that include monthlymacroeconomic variables and monthly oil prices leads to gains with respect to the benchmark models.For Italy, the gains of using macroeconomic variables is more stable over the horizons. Hence, theinclusion of monthly macroeconomic variables leads to an increase of the 18% of forecasting accuracy at thefirst horizon and of 7% at longer horizons. Moreover, the inclusion of one monthly macroeconomic variablesseparately from the other leads to better forecasting scenario over all the horizons and it is also confirmedby the Diebold-Mariano test and the statistically superior predictability set. Also if we include the oil pricespecification in term of monthly or daily price leads to improvements of the forecasting accuracy over thehorizons but at lower quantity.Regarding the second benchmark model, the AR(1), Panel (B) of Table 1 and 2 shows the results forthe two density forecasting measures. For Germany, as in the point forecasting measures, the CRPS movesfrom 6 .
15 for the AR(3) to 7 .
04 for the AR(1) at the first horizon and from 6 . . pecification of the industrial production index. Thus, we have studied models that include the PMI surveysand either All-IPI, only one of the index, or combinations of two indices (IPI-Cons-Elec, IPI-Cons-Manuf,IPI-Elect-Manuf). These model specifications have been studied also when the oil prices is included withmonthly or daily index. The results across the models do not change drastically if we include just one IPIor different combinations, thus we have decided to include them in the Supplementary Material. This paper analyses for the first time to the best of our knowledge the forecasting performances of mixedfrequency models for electricity prices. In particular, we use monthly macroeconomic variables for predictingdaily electricity prices in two of the most important European countries, Germany and Italy. The paperstudies how to incorporate low-frequency information from manufacturing Purchasing Managers’ Indexsurveys and industrial production index into models that forecasts high frequency variables, the dailyelectricity prices. Moreover, in the analysis, we have included different specification of the oil prices, measuredas monthly or daily variable.Our analysis of point and density forecasting performances covers different horizons (from one day to onemonth ahead) on the sample spanning from 2013 to 2019. Our results clearly indicate that the RU-MIDASspecifications with all monthly macroeconomic variables and the inclusion of oil prices dominate AR models,both in terms of point and density forecasting over all the horizons. Moreover, we find gains around 20%at short horizons and around 8% at long horizons, thus it turns out that the macroeconomic low frequencyvariables are more important for short horizons than for longer horizons. Moreover, the benchmark modelis almost never included in the model confidence set.We conclude that from an energy forecasting perspective these mixed frequency models seem to haveinteresting and important advantages over simpler models. Going forward, it would be interesting to studythe possible extension of these models to hourly data in order to include other variables of interest, such asrenewable energy sources, which are currently taking lead in the electricity generation.
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Germany Data Representation
Daily Series for Electricity Day-ahead Prices (top left), Monthly PMI Surveys (top right), Monthly Industrial Production index(IPI) for Consumer Goods (middle left), Monthly IPI for Electricity Prices (middle right), Monthly IPI for Manufacturing (bottomleft) and Daily (black) and Monthly (red) Oil Brent Prices (bottom right) observed in Germany from 01/01/2016 to 31/12/2019. / J an / / J an / / J an / / J an / / J an / / J an / / J an / / J an / -1000100200300 E l e c t r i c i t y P r i c e D e c - D e c - D e c - D e c - D e c - D e c - D e c - N o v - P M I S u r v e ys D e c - D e c - D e c - D e c - D e c - D e c - D e c - N o v - I P I C on s u m e r G ood s D e c - D e c - D e c - D e c - D e c - D e c - D e c - N o v - I P I E l e c t r i c i t y S upp l y D e c - D e c - D e c - D e c - D e c - D e c - D e c - N o v - I P I M anu f a c t u r i ng D e c - D e c - D e c - D e c - D e c - D e c - D e c - N o v - O il B r en t Italy Data Representation
Daily Series for Electricity Day-ahead Prices (top left), Monthly PMI Surveys (top right), Monthly Industrial Production index(IPI) for Consumer Goods (middle left), Monthly IPI for Electricity Prices (middle right), Monthly IPI for Manufacturing (bottomleft) and Daily (black) and Monthly (red) Oil Brent Prices (bottom right) observed in Italy from 01/01/2016 to 31/12/2019. / J an / / J an / / J an / / J an / / J an / / J an / / J an / / J an / E l e c t r i c i t y P r i c e D e c - D e c - D e c - D e c - D e c - D e c - D e c - N o v - P M I S u r v e ys D e c - D e c - D e c - D e c - D e c - D e c - D e c - N o v - I P I C on s u m e r G ood s D e c - D e c - D e c - D e c - D e c - D e c - D e c - N o v - I P I E l e c t r i c i t y S upp l y D e c - D e c - D e c - D e c - D e c - D e c - D e c - N o v - I P I M anu f a c t u r i ng D e c - D e c - D e c - D e c - D e c - D e c - D e c - N o v - O il B r en t Point (RMSE) and density (average CRPS and Predictive Likelihood) forecastingmeasure for Germany.
RMSE; average CRPS and average Predictive Likelihood (PL) for BAR(3) (Panel A) and BAR(1) (Panel B) baseline model andratios/difference for other models. The estimation and forecasting sample last 7 years each. The forecasting is provided for horizons h = 1 , , , , , , Panel A Horizon 1 2 3 7 14 21 28
RMSE
BAR(3) 11.359 12.546 12.903 11.154 11.588 11.977 12.167B-RU-MIDAS (All-IPI + Surveys) 0.804 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.819 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.815 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.799 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.798 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
Average
BAR(3) 6.155 6.829 6.977 5.934 6.144 6.368 6.472
CRPS
B-RU-MIDAS (All-IPI + Surveys) 0.777 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.810 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.805 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.771 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.773 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
Average
BAR(3) -3.954 -4.056 -4.082 -3.973 -4.097 -4.105 -4.152
Predictive
B-RU-MIDAS (All-IPI + Surveys) 0.113 0.078 ∗∗∗ ∗∗∗ -0.009 ∗∗∗ ∗∗∗ ∗∗∗
Likelihood
B-RU-MIDAS (Surveys) 0.061 0.082 ∗∗∗ ∗∗∗ -0.040 ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.070 0.079 ∗∗∗ ∗∗∗ -0.051 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.049 ∗∗∗ ∗∗∗ ∗∗∗ -0.005 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.035 0.063 ∗∗∗ ∗∗∗ -0.028 ∗∗∗ ∗∗∗ ∗∗∗
Panel B Horizon 1 2 3 7 14 21 28
RMSE
BAR(1) 13.050 13.983 14.339 13.140 13.723 13.905 14.515B-RU-MIDAS (All-IPI + Surveys) 0.735 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.737 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.735 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.725 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.723 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
Average
BAR(1) 7.039 7.536 7.746 6.995 7.374 7.442 7.736
CRPS
B-RU-MIDAS (All-IPI + Surveys) 0.718 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.722 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.719 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.705 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.706 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
Average
BAR(1) -4.144 -4.187 -4.220 -4.133 -4.178 -4.203 -4.281
Predictive
B-RU-MIDAS (All-IPI + Surveys) 0.156 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
Likelihood
B-RU-MIDAS (Surveys) 0.157 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.166 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.153 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.192 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
Notes: Refer to Section 2 for details on model formulations. B-RU-MIDAS indicates Bayesian RU-MIDAS with Normal-Gamma prior including lags, seasonaldummies and different exogenous variables. All forecasts are produced with recursive estimation of the models. For BAR(3) (Panel A) and BAR(1) (Panel B) baseline, the table reports RMSE; average CRPS and average PL; for all other models, table reportsratios/differences between score of current model and of benchmark. For RMSE and CRPS (PL); entries less than 1 (entries greater than 0) indicate thatforecasts from current model are more accurate than forecasts from baseline model. ∗∗∗ , ∗∗ and ∗ indicate score ratios/difference are significantly different from 1 at 1%, 5% and 10%, according to Diebold-Mariano test. Gray cells indicate models that belong to the Superior Set of Models delivered by the
MCS procedure at confidence level 10%.
Point (RMSE) and density (average CRPS and Predictive Likelihood) forecastingmeasure for Italy.
RMSE; average CRPS and average Predictive Likelihood (PL) for BAR(3) (Panel A) and BAR(1) (Panel B) baseline model andratios/difference for other models. The estimation and forecasting sample last 7 years each. The forecasting is provided for horizons h = 1 , , , , , , Panel A Horizon 1 2 3 7 14 21 28
RMSE
BAR(3) 8.106 8.867 9.061 8.412 9.378 10.093 10.635B-RU-MIDAS (All-IPI + Surveys) 0.816 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.819 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.815 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.822 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.809 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
Average
BAR(3) 4.412 4.833 4.925 4.536 5.033 5.370 5.713
CRPS
B-RU-MIDAS (All-IPI + Surveys) 0.806 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.810 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.805 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.806 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.796 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
Average
BAR(3) -3.625 -3.733 -3.738 -3.696 -3.804 -3.903 -3.913
Predictive
B-RU-MIDAS (All-IPI + Surveys) 0.161 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
Likelihood
B-RU-MIDAS (Surveys) 0.160 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.178 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.155 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
RMSE
BAR(1) 9.191 9.798 10.058 9.641 10.633 11.256 11.891B-RU-MIDAS (All-IPI + Surveys) 0.762 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.781 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.771 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.760 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.746 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
Average
BAR(1) 4.993 5.329 5.465 5.199 5.768 6.043 6.387
CRPS
B-RU-MIDAS (All-IPI + Surveys) 0.758 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.775 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.767 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.748 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.736 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
Average
BAR(1) -3.765 -3.821 -3.871 -3.817 -3.900 -3.958 -4.028
Predictive
B-RU-MIDAS (All-IPI + Surveys) 0.245 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
Likelihood
B-RU-MIDAS (Surveys) 0.199 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.249 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.223 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.244 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
See Notes in Table 1 nline Appendix for: “Are low frequency macroeconomic variablesimportant for high frequency electricity prices?” This Supplementary material provides further robustness checks for the forecasting of daily electricity priceswith different benchmark models and multiple exogenous variables.Table S.1 and S.3 show the RMSE with respect to the benchmark model AR(3) for Germany and Italyrespectively, thus all the models include the three lags of the electricity prices and the seasonal dummies.Moreover, each model includes monthly PMI Surveys, where specified, and different combinations of the IPIs;whereas indicated we include daily or monthly oil price in the analysis. In Table S.2 and S.4, we include theRMSE models with one lag of the electricity prices for Germany and Italy respectively.Table S.5 and S.7 show the average CRPS with respect to the benchmark model AR(3) for Germany andItaly respectively and the same arises for the AR(1) model in Table S.6 and S.8.In conclusion, Table S.9 and S.11 shows the average predictive likelihood for all the models with theinclusion of three lags of the electricity prices for Germany and Italy respectively. The same analysis isdescribed in Table S.10 and S.12, where only one lag of the electricity prices is included in the modelformulation.
Table S.1:
Point Forecasting measure (RMSE) for Germany.
RMSE for the BAR(3) baseline model and ratios for other models. The estimation and forecasting sample last 7 years each. Theforecasting is provided for horizons h = 1 , , , , , , Horizon 1 2 3 7 14 21 28BAR(3) 11.359 12.546 12.903 11.154 11.588 11.977 12.167B-RU-MIDAS (All-IPI + Surveys) 0.804 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.819 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.815 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys) 0.800 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys) 0.800 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys) 0.808 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys) 0.800 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys) 0.805 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys) 0.804 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.799 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Monthly Oil) 0.799 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Monthly Oil) 0.800 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Monthly Oil) 0.804 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Monthly Oil) 0.799 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Monthly Oil) 0.800 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Monthly Oil) 0.801 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.798 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Daily Oil) 0.797 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Daily Oil) 0.797 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Daily Oil) 0.802 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Daily Oil) 0.796 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Daily Oil) 0.800 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Daily Oil) 0.799 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
Notes: Refer to Section 2 for details on model formulations. B-RU-MIDAS indicates Bayesian RU-MIDAS with Normal-Gamma prior including lags,seasonal dummies and different exogenous variables. All forecasts are produced with recursive estimation of the models. For BAR(3) baseline, the table reports RMSE; for all other models, table reports ratios between score of current model and of benchmark. ForRMSE; entries less than 1 indicate that forecasts from current model are more accurate than forecasts from baseline model. ∗∗∗ , ∗∗ and ∗ indicate score ratios are significantly different from 1 at 1%, 5% and 10%, according to Diebold-Mariano test. Gray cells indicate models that belong to the Superior Set of Models delivered by the
MCS procedure at confidence level 10%.
Point Forecasting measure (RMSE) for Germany.
RMSE for the BAR(1) baseline model and ratios for other models. The estimation and forecasting sample last 7 years each. Theforecasting is provided for horizons h = 1 , , , , , , Horizon 1 2 3 7 14 21 28BAR(1) 13.050 13.983 14.339 13.140 13.723 13.905 14.515B-RU-MIDAS (All-IPI + Surveys) 0.735 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.737 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.735 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys) 0.731 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys) 0.730 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys) 0.743 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys) 0.730 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys) 0.737 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys) 0.737 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.725 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Monthly Oil) 0.727 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Monthly Oil) 0.727 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Monthly Oil) 0.735 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Monthly Oil) 0.725 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Monthly Oil) 0.728 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Monthly Oil) 0.727 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.723 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Daily Oil) 0.722 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Daily Oil) 0.722 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Daily Oil) 0.732 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Daily Oil) 0.720 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Daily Oil) 0.726 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Daily Oil) 0.724 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
See Notes in Table S.1
Point Forecasting measure (RMSE) for Italy.
RMSE for the BAR(3) baseline model and ratios for other models. The estimation and forecasting sample last 7 years each. Theforecasting is provided for horizons h = 1 , , , , , , Horizon 1 2 3 7 14 21 28BAR(3) 8.106 8.867 9.061 8.412 9.378 10.093 10.635B-RU-MIDAS (All-IPI + Surveys) 0.816 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.819 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.815 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys) 0.815 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys) 0.811 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys) 0.819 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys) 0.814 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys) 0.819 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys) 0.813 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.822 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Monthly Oil) 0.821 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Monthly Oil) 0.820 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Monthly Oil) 0.822 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Monthly Oil) 0.821 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Monthly Oil) 0.823 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Monthly Oil) 0.819 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.809 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Daily Oil) 0.811 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Daily Oil) 0.808 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Daily Oil) 0.810 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Daily Oil) 0.811 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Daily Oil) 0.811 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Daily Oil) 0.806 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
See Notes in Table S.1
Table S.4:
Point Forecasting measure (RMSE) for Italy.
RMSE for the BAR(1) baseline model and ratios for other models. The estimation and forecasting sample last 7 years each. Theforecasting is provided for horizons h = 1 , , , , , , Horizon 1 2 3 7 14 21 28BAR(1) 9.191 9.798 10.058 9.641 10.633 11.256 11.891B-RU-MIDAS (All-IPI + Surveys) 0.762 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.781 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.771 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys) 0.763 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys) 0.752 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys) 0.771 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys) 0.755 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys) 0.771 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys) 0.759 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.760 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Monthly Oil) 0.767 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Monthly Oil) 0.758 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Monthly Oil) 0.765 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Monthly Oil) 0.760 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Monthly Oil) 0.766 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Monthly Oil) 0.758 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.746 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Daily Oil) 0.753 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Daily Oil) 0.745 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Daily Oil) 0.750 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Daily Oil) 0.747 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Daily Oil) 0.751 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Daily Oil) 0.743 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
See Notes in Table S.1
Density Forecasting measure (average CRPS) for Germany.
Average CRPS for the BAR(3) baseline model and ratios for other models. The estimation and forecasting sample last 7 yearseach. The forecasting is provided for horizons h = 1 , , , , , , Horizon 1 2 3 7 14 21 28BAR(3) 6.155 6.829 6.977 5.934 6.144 6.368 6.472B-RU-MIDAS (All-IPI + Surveys) 0.777 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.810 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.805 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys) 0.773 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys) 0.773 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys) 0.782 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys) 0.773 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys) 0.778 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys) 0.779 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.771 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Monthly Oil) 0.770 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Monthly Oil) 0.771 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Monthly Oil) 0.777 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Monthly Oil) 0.769 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Monthly Oil) 0.772 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Monthly Oil) 0.773 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.773 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Daily Oil) 0.771 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Daily Oil) 0.771 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Daily Oil) 0.778 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Daily Oil) 0.769 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Daily Oil) 0.774 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Daily Oil) 0.775 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
Notes: Refer to Section 2 for details on model formulations. B-RU-MIDAS indicates Bayesian RU-MIDAS with Normal-Gamma prior including lags,seasonal dummies and different exogenous variables. All forecasts are produced with recursive estimation of the models. For BAR(3) baseline, the table reports average CRPS; for all other models, table reports ratios between score of current model and of benchmark.For CRPS; entries less than 1 indicate that forecasts from current model are more accurate than forecasts from baseline model. ∗∗∗ , ∗∗ and ∗ indicate score ratios are significantly different from 1 at 1%, 5% and 10%, according to Diebold-Mariano test. Gray cells indicate models that belong to the Superior Set of Models delivered by the
MCS procedure at confidence level 10%.
Density Forecasting measure (average CRPS) for Germany.
Average CRPS for the BAR(1) baseline model and ratios for other models. The estimation and forecasting sample last 7 yearseach. The forecasting is provided for horizons h = 1 , , , , , , Horizon 1 2 3 7 14 21 28BAR(1) 7.039 7.536 7.746 6.995 7.374 7.442 7.736B-RU-MIDAS (All-IPI + Surveys) 0.718 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.722 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.719 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys) 0.714 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys) 0.713 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys) 0.727 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys) 0.712 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys) 0.720 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys) 0.721 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.705 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Monthly Oil) 0.707 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Monthly Oil) 0.707 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Monthly Oil) 0.717 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Monthly Oil) 0.704 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Monthly Oil) 0.708 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Monthly Oil) 0.707 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.706 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Daily Oil) 0.705 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Daily Oil) 0.705 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Daily Oil) 0.716 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Daily Oil) 0.702 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Daily Oil) 0.709 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Daily Oil) 0.707 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
See Notes in Table S.5
Density Forecasting measure (average CRPS) for Italy.
Average CRPS for the BAR(3) baseline model and ratios for other models. The estimation and forecasting sample last 7 yearseach. The forecasting is provided for horizons h = 1 , , , , , , Horizon 1 2 3 7 14 21 28BAR(3) 4.412 4.833 4.925 4.536 5.033 5.370 5.713B-RU-MIDAS (All-IPI + Surveys) 0.806 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.810 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.805 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys) 0.805 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys) 0.802 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys) 0.809 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys) 0.805 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys) 0.809 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys) 0.804 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.806 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Monthly Oil) 0.806 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Monthly Oil) 0.804 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Monthly Oil) 0.807 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Monthly Oil) 0.806 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Monthly Oil) 0.807 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Monthly Oil) 0.804 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.796 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Daily Oil) 0.799 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Daily Oil) 0.796 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Daily Oil) 0.797 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Daily Oil) 0.798 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Daily Oil) 0.797 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Daily Oil) 0.793 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
See Notes in Table S.5
Density Forecasting measure (average CRPS) for Italy.
Average CRPS for the BAR(1) baseline model and ratios for other models. The estimation and forecasting sample last 7 yearseach. The forecasting is provided for horizons h = 1 , , , , , , Horizon 1 2 3 7 14 21 28BAR(1) 4.993 5.329 5.465 5.199 5.768 6.043 6.387B-RU-MIDAS (All-IPI+ Surveys) 0.758 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.775 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.767 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys) 0.759 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys) 0.748 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys) 0.769 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys) 0.751 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys) 0.767 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys) 0.756 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.748 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Monthly Oil) 0.755 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Monthly Oil) 0.746 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Monthly Oil) 0.754 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Monthly Oil) 0.749 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Monthly Oil) 0.754 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Monthly Oil) 0.747 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.736 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Daily Oil) 0.744 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Daily Oil) 0.736 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Daily Oil) 0.741 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Daily Oil) 0.738 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Daily Oil) 0.741 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Daily Oil) 0.734 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
See Notes in Table S.5
Density Forecasting measure (average Predictive Likelihood) for Germany.
Average Predictive Likelihood for the BAR(3) baseline model and differences for other models. The estimation and forecastingsample last 7 years each. The forecasting is provided for horizons h = 1 , , , , , , Horizon 1 2 3 7 14 21 28BAR(3) -3.954 -4.056 -4.082 -3.973 -4.097 -4.105 -4.152B-RU-MIDAS (All-IPI + Surveys) 0.113 0.078 ∗∗∗ ∗∗∗ -0.009 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.061 0.082 ∗∗∗ ∗∗∗ -0.040 ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.070 0.079 ∗∗∗ ∗∗∗ -0.051 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys) 0.038 0.094 ∗∗∗ ∗∗∗ -0.027 ∗∗∗ ∗∗∗ -0.003 0.083 ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys) 0.035 0.078 ∗∗∗ ∗∗∗ -0.049 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys) 0.046 0.047 ∗∗∗ ∗∗∗ -0.068 ∗∗∗ ∗∗∗ -0.010 0.068 ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys) 0.080 0.078 ∗∗∗ ∗∗∗ -0.053 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys) 0.059 0.059 ∗∗∗ ∗∗∗ -0.069 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys) 0.087 0.078 ∗∗∗ ∗∗∗ -0.060 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.049 ∗∗∗ ∗∗∗ ∗∗∗ -0.005 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Monthly Oil) 0.033 0.103 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ -0.010 0.051 ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Monthly Oil) 0.009 0.078 ∗∗∗ ∗∗∗ -0.047 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Monthly Oil) 0.046 0.048 ∗∗∗ ∗∗∗ -0.040 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Monthly Oil) 0.008 ∗∗∗ ∗∗∗ ∗∗∗ -0.006 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Monthly Oil) 0.028 0.040 ∗∗∗ ∗∗∗ -0.024 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Monthly Oil) 0.024 0.082 ∗∗∗ ∗∗∗ -0.007 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.035 0.063 ∗∗∗ ∗∗∗ -0.028 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Daily Oil) 0.048 0.040 ∗∗∗ ∗∗∗ -0.052 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Daily Oil) 0.059 0.068 ∗∗∗ ∗∗∗ -0.071 ∗∗∗ ∗∗∗ -0.003 0.057 ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Daily Oil) 0.015 0.059 ∗∗∗ ∗∗∗ -0.040 ∗∗∗ ∗∗∗ -0.026 0.028 ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Daily Oil) 0.034 0.044 ∗∗∗ ∗∗∗ -0.041 ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Daily Oil) 0.069 0.085 ∗∗∗ ∗∗∗ -0.036 ∗∗∗ -0.010 ∗∗∗ -0.004 0.048 ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Daily Oil) 0.031 0.057 ∗∗∗ ∗∗∗ -0.021 ∗∗∗ ∗∗∗ ∗∗∗
Notes: Refer to Section 2 for details on model formulations. B-RU-MIDAS indicates Bayesian RU-MIDAS with Normal-Gamma prior including lags,seasonal dummies and different exogenous variables. All forecasts are produced with recursive estimation of the models. For BAR(3) baseline, the table reports average Predictive Likelihood; for all other models, table reports differences between score of current modeland of benchmark. For PL; entries greater than 0 indicate that forecasts from current model are more accurate than forecasts from baseline model. ∗∗∗ , ∗∗ and ∗ indicate score differences are significantly different from 1 at 1%, 5% and 10%, according to Diebold-Mariano test. Gray cells indicate models that belong to the Superior Set of Models delivered by the
MCS procedure at confidence level 10%.
Density Forecasting measure (average Predictive Likelihood) for Germany.
Average Predictive Likelihood for the BAR(1) baseline model and differences for other models. The estimation and forecastingsample last 7 years each. The forecasting is provided for horizons h = 1 , , , , , , Horizon 1 2 3 7 14 21 28BAR(1) -4.144 -4.187 -4.220 -4.133 -4.178 -4.203 -4.281B-RU-MIDAS (All-IPI + Surveys) 0.156 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.157 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.166 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys) 0.153 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys) 0.205 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys) 0.166 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys) 0.189 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys) 0.191 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys) 0.191 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.153 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Monthly Oil) 0.211 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Monthly Oil) 0.186 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Monthly Oil) 0.195 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Monthly Oil) 0.222 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Monthly Oil) 0.183 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Monthly Oil) 0.183 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.192 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Daily Oil) 0.158 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys + Daily Oil) 0.184 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys + Daily Oil) 0.176 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Daily Oil) 0.207 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Daily Oil) 0.198 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Daily Oil) 0.202 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
See Notes in Table S.9
Density Forecasting measure (average Predictive Likelihood) for Italy.
Average Predictive Likelihood for the BAR(3) baseline model and differences for other models. The estimation and forecastingsample last 7 years each. The forecasting is provided for horizons h = 1 , , , , , , Horizon 1 2 3 7 14 21 28BAR(3) -3.625 -3.733 -3.738 -3.696 -3.804 -3.903 -3.913B-RU-MIDAS (All-IPI + Surveys) 0.161 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.160 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.178 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Con + Surveys) 0.155 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys) 0.149 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys) 0.173 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys) 0.162 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys) 0.164 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys) 0.163 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.155 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ -0.004 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
See Notes in Table S.9
Table S.12:
Density Forecasting measure (average Predictive Likelihood) for Italy.
Average Predictive Likelihood for the BAR(1) baseline model and differences for other models. The estimation and forecastingsample last 7 years each. The forecasting is provided for horizons h = 1 , , , , , , Horizon 1 2 3 7 14 21 28BAR(1) -3.765 -3.821 -3.871 -3.817 -3.900 -3.958 -4.028B-RU-MIDAS (All-IPI + Surveys) 0.245 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (Surveys) 0.199 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI) 0.249 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys) 0.216 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec + Surveys) 0.215 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Manuf + Surveys) 0.205 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys) 0.225 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys) 0.215 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys) 0.233 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Monthly Oil) 0.223 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Monthly Oil) 0.209 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Monthly Oil) 0.233 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Monthly Oil) 0.243 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (All-IPI + Surveys + Daily Oil) 0.244 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons + Surveys + Daily Oil) 0.258 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Elec + Surveys + Daily Oil) 0.265 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Cons-Manuf + Surveys + Daily Oil) 0.263 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗
B-RU-MIDAS (IPI-Elec-Manuf + Surveys + Daily Oil) 0.272 ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗