Spanish GDP short-term point and density forecasting using a mixed-frequency dynamic factor model

Fresoli, Diego Article Spanish GDP short-term point and density forecasting using a mixed-frequency dynamic factor model SERIEs - Journal of the Spanish Economic Association Provided in Cooperation with: Spanish Economic Association Suggested Citation: Fresoli, Diego (2024) : Spanish GDP short-term point and density forecasting using a mixed-frequency dynamic factor model, SERIEs - Journal of the Spanish Economic Association, ISSN 1869-4195, Springer, Heidelberg, Vol. 15, Iss. 2, pp. 145-177, https://doi.org/10.1007/s13209-024-00297-3 This Version is available at: https://hdl.handle.net/10419/326953 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/ SERIEs (2024) 15:145–177 https://doi.org/10.1007/s13209-024-00297-3 ORIGINAL ARTICLE Spanish GDP short-term point and density forecasting using a mixed-frequency dynamic factor model Diego Fresoli1 Received: 23 September 2022 / Accepted: 26 February 2024 / Published online: 10 April 2024 © The Author(s) 2024 Abstract We have assessed the effect of data releases when constructing short-term point and density forecasts of the Spanish gross domestic product growth. For this purpose, we considered a real-forecasting exercise in which we defined several pseudo-data vintages that had a mixture of monthly and quarterly frequencies and were unbalanced towards the end of the sample. We implemented a mixed-frequency dynamic factor model to deal with data features and to produce gross domestic product forecasts. We evaluated the predictive content of data releases from point and density forecast perspectives, the latter aspect of the analysis being previously unexplored in the literature producing Spanish gross domestic product short-term forecasts. We observed significant improvements in point forecasts as information is released throughout the quarter, confirming existing results. Additionally, our findings indicated substantial enhancements in the accuracy of density forecasts as new data releases materialized. Keywords Short-term gross domestic product (GDP) point forecast ·Density forecast ·Mixed-frequency dynamic factor model JEL Classification C32 ·C530 ·E39 ·E370 1 Introduction One of the well-known facts about gross domestic product (GDP) is that its values are released with a delay that exceeds 3 weeks for the first advance estimate of the previous quarter. For this reason, many economic agents, central banks, government statistical offices, and other financial institutions spend much of their time and resources conThe replication material for the study is available at https://zenodo.org/records/10691459. BDiego Fresoli [email protected] 1Facultad de Ciencias Económicas y Empresariales, Dpto. Anáisis Económico: Economía Cuantitativa, Universidad Autonóma de Madrid, Madrid, Spain 123 146 SERIEs (2024) 15:145–177 structing GDP short-term forecasts. For this purpose, they use a variety of monthly indicators, more frequently and more timely available than the target quarterly GDP. The combination of the quarterly GDP and monthly indicators used as predictors generates a mixed-frequency nature of data. Furthermore, as variables are asynchronous, withtheirlatestdatamadeavailable at differentpointsintimeandwithdifferentdelays, a mix of available and missing observations comes up towards the end of the sample, a pattern known as unbalancedness or ragged edge data structure. Hence, constructing GDP short-term forecasts involves coping with missing observations as an essential part of data processing. In Spain’s case, monitoring and tracking the short-term evolution of GDP is often supported by the dynamic factor model (DFM) and its state-space representation. These are, for instance, the Spain-STING (short-term indicator of growth) and MIDPred (integrated model of prediction), which are used by institutions like the Bank of Spain and the Independent Authority for Fiscal Responsibility, respectively, or the FASE model (factor analysis of the Spanish economy) which is originally designed to oversee and forecast GDP within the Ministry of Economy; see Camacho and Quiros (2011), Cuevas and Quilis (2012), Cuevas et al. (2017), and Pareja et al. (2020). Moreover, the MICA-BBVA (Model of economic and financial Indicators used to monitor the Current Activity by Banco Bilbao Vizcaya Argentaria) model has also been proposed to obtain GDP projections through real and several financial indicators; see Camacho and Doménech (2012). An essential part of the task of all these real-time forecasting applications is the construction of Spanish GDP short-term point estimates and evaluating their accuracy. A common finding is the existence of accuracy gains as new data become available during the quarter. Nevertheless, in an environment of increasing economic uncertainty, users of GDP forecasts demand point projections for its short-term evolution and, more assiduously, the complete future characterization provided by its short-term density forecasts; see, for instance, Mazzi et al. (2014), Aastveit et al. (2014,2018), Bäurle et al. (2021), Mitchell et al. (2022), and Çakmakl i and Demircan (2022). Consequently, it is crucial to assess the information content of data at each point in time from the perspective of the GDP short-term density forecasts. In the present paper, we have aimed to address the information when constructing Spanish GDP short-term forecasts, using the standard evaluation approach through point forecasts and, more relevantly and innovatively, density forecasts. For this purpose, we have combined thirty-two monthly indicators from the Spanish economy, including real-activity, financial and survey variables, and the quarterly employment and GDP growth rates within a mixed-frequency model-based forecasting methodology. The data-rich framework has enabled us to cover most of the indicators previously considered in the referenced models for Spain. As we did not have access to published historical records of the variables, we have assumed that the publication calendar on a recent date, i.e. beginning of October 2023, has been stable over the entire period and used this to define patterns of the missing observations towards the end of the sample.1 Based on the data release timings, we first considered three temporal blocks, the early, the middle, and the late, which we further disaggregated into sub-blocks according to 1We cannot address the effect of the data revisions on GDP forecast performance through pseudo-data vintages generated in this way, leaving it open for future research. 123 SERIEs (2024) 15:145–177 147 their economic content. Finally, the unbalancedness patterns of the temporal blocks and economic sub-blocks have been replicated in all quarters of the1995–2023 period, producing pseudo-real-time data vintages that we have used to address the information inflow; see, for instance, Giannone et al. (2008) for USA, Kuzin et al. (2011)forthe Euro area, and Cuevas et al. (2017) for Spain. Apart from some specific nuances, the essence of the model we have used is the same as those referenced for Spain. We have constructed GDP forecasts using factors that summarize the overall state of the business cycle. The factor extraction is based on principal components (PC) and DFM; see Ba´nbura and Modugno (2014). The advantage of implementing a DFM approach is that it enables a large panel of variables to be modelled in a parsimonious way. Even more importantly, the DFM and its statespace representation deal with the mixed-frequency and unbalanced nature of the data. For this purpose, we ran the Kalman filter to fill any missing observations throughout, and at the end of the sample, and to produce a point forecast and conditional variance. We have used the latter with a normal assumption to provide a conditional distribution of the future GDP given the data availability, or density forecast. We have performed a recursive estimation scheme incorporating the information flow and obtained point forecasts, as most papers on short-term Spanish GDP evolution have. Furthermore, we have investigated the performance of short-term density forecasts obtained by mixed-frequency DFM, an aspect almost always overlooked by the literature. Our findings confirmed the existence of valuable information released during the quarter from point forecasts perspective. Moreover, we observed substantial enhancements in the accuracy of Spanish GDP short-term density forecasts obtained through mixed-frequency DFM as the quarter progresses, and new information is incorporated into the data. It is worth stressing that the literature concerning Spanish GDP short-term forecasting has focused solely on constructing and evaluating short-term point forecasts. The presentpaperadds the perspectiveofanevaluation throughtheGDPshort-termdensity forecasts that can be generated through DFM and its state-space representation. By doing this, we have not sought an analysis of the relative merits of the different models to the phenomenon of Spanish GDP short-term forecasting. The real-time forecasting exercise and the evaluation of point and density forecasts for different data vintages may as well be employed in the methodological frameworks of the referenced papers. We have organized the remainder of the paper as follows. The second section describes the data. The third explains how we artificially generated the pseudo-data vintages and describes the notation related to them. The fourth section presents the mixed-frequency DFM, its state-space representation, and how the Kalman filter produces point and density forecasts under data irregularities. Then, it discusses the forecast design and how we evaluated point and density forecasts. The fifth section presents and discusses the main findings. We have concluded in the sixth section. 2 Data description Table 1shows the basic set of indicators, consisting of thirty-two monthly variables and the quarterly employment and GDP growth rates. The target variable is the quar123 148 SERIEs (2024) 15:145–177 terly GDP for which we have considered its real (chained volume index) version. The indicators used as predictors of GDP represent a broad spectrum of the business cycle activities. Accordingly, we have included hard indicators related to real economic activity. These represent the labour market (such as registered unemployment and social security contributors), domestic economic activity (the manufacture industrial production index, electric power consumption, apparent consumption of cement, etc.), and the foreign sector (real export and import). In addition, we have considered the financial sector through the interest rate spread and the credit to companies and households; see Camacho and Doménech (2012) who found that financial indicators are specially relevant in producing a GDP forecast during periods of recession. Finally, we have also considered some of the soft indicators that potentially capture agents’ sentiment and confidence about the short-term future of the economy. These indicators cover opinion polls, i.e. survey-based indicators such as the purchasing managers’ index, economic sentiment and industry production perspective indicators. In selecting indicators, we have been guided by the following criteria. Firstly, although it was not a requirement, we attempted, as much as possible, to have a balanced pattern of observations at the beginning of the sample period.2Thus, all monthly indicators have observations spanning from 1995, except for the registered unemployment, large companies’ sales variables, whose figures began in January 1996, the purchasing managers’ index, overnights, turnover index service, and turnover index industry, whose first observations were made in August 1998, January 1999, 2000 and 2002, respectively. Secondly, as far as possible, we wanted to have a rich data framework covering most of the indicators previously considered in the literature that has dealt with Spanish GDP short-term forecasting. Consequently, we have considered indicators previously used in the large-scale approach by Cuevas and Quilis (2012), or the small-scale DFMs of the Spanish economy by Camacho and Quiros (2011), Cuevas et al. (2017) and Pareja et al. (2020).3 We considered the seasonally adjusted variables to identify a systematic measure underlying the pure economic fluctuations of the GDP growth rate.4Moreover, the type of DFM implemented in this paper requires stationarity in order to identify and estimate factors, which materialize through appropriate transformations that remove secular trends. We have obtained stationary fluctuations for most indicators by taking the first differences of their level or logarithm. The exceptions are the spread rate, and two soft indicators obtained from surveys, the purchasing managers’ index and the industrial production perspective indicator, for which we have considered their levels, 2In the forecasting exercise we have carried out, the model parameters are estimated recursively, starting with a given sample period. The purpose of having a balanced panel is to avoid variables contributing significantly different observations to the estimation of the model parameters. 3Nevertheless, there are some exceptions with respect to Cuevas and Quilis (2012), wehave notconsidered: (i) the availability of consumer and capital goods, whose first observation starts in January 2005, and therefore, we have left them aside to avoid having too many missing at the beginning of the sample; (ii) industrial order book index and real gross wage data, for which data is no longer freely accessible. On the other hand, we could not reproduce MICA data because some of the financial variables it included are not freely available. 4We collected seasonally adjusted variables if they were already available in this format. When they were not, we obtained seasonally adjusted series using TRAMO-SEATS. In Table 1, the variables with codes ending in a capital letter D are already adjusted, while we have corrected those without it. 123 SERIEs (2024) 15:145–177 149 as well as the registered unemployment and the credit to companies and households, for which we have induced stationarity by taking a second difference of its logarithm. Finally, the quarterly employment and GDP exhibit trend behaviours that led us to consider their first difference of their log transformations as an approximation of their quarterly growth rate.5 In the present paper, we have analysed two data frameworks in a real-time GDP forecasting exercise in the following sections. We have considered a data-rich environment, including all variables in Table 1. This comprehensive dataset has enabled us to analyse the information content over the quarter of a broad set of indicators encompassing those previously used in the literature constructing Spanish GDP nowcasts. It is the primary data framework for which we have conducted most of the short-term forecast evaluation analysis. On the other hand, for comparison, we have also considered a small-scale data framework, reproducing the indicators and data transformation of the MIDPred; see Cuevas et al. (2017).6The variables with an ‘X’ in Table 1correspond to those in the MIDPred data framework. 3 Pseudo-real-time data vintages Ideally, we would like to work with data vintages constructed with records of variables at the points in time when they were released, also including any revisions to previously published values that may have been made. Nevertheless, we cannot work with pure real-time vintages since we cannot access historical records of monthly indicators, employment and GDP. Instead, we have relied on pseudo-real-time vintages that were generated artificially to resemble the publication pattern of variables. The data were collected at the beginning of October 2023, just after the first advance of the third quarter of 2023, together with the publication dates of all indicators. The sixth column in Table 1provides the publication calendar obtained in October 2023, i.e. the approximate number of days after the beginning of a month. The critical assumption behind the pseudo-real-time vintages is that the publication calendar has always been the same for all months of the sample period 1995–2023, or, if it has changed, the timing basically remained unaltered. This assumption has served as the purpose of investigating the consequences that a rather new publication schedule has for GDP short-term forecasting, assuming that such a calendar will not undergo further alterations. We have not sought a historical assessment of the information content of past data vintages, but rather an analysis of what is implied by a recent publication schedule, keeping in mind forecasts that may be made with a similar schedule in the time to come. Of course, using pseudo-real-time data vintages did not enable us to tackle the effect of data revisions on forecast performance. We will leave this aspect for future research. 5The choice of data transformation applied to each indicator is based on an analysis of its trend behaviour and augmented Dickey–Fuller (ADF) test for the level (or logarithms of its level) and its first differences. For more detailed information on ADF test results and data transformations, see Table 2in Appendix B. 6Although we have focused on variables included in the MIDPred for comparison, those in the SpainSTING data framework or the approach by Pareja et al. (2020) may also be considered. 123 150 SERIEs (2024) 15:145–177 Table 1 List and information of monthly indicators and GDP: n, variable name, label, source code, unit of measurement, publication calendar (approximate number of days after the beginning of a month), composition of temporal blocks (i.e. early, middle, and late), composition of economic sub-blocks, (i.e. PMI, TRAIN, FUEL, etc.), and publication delay, in months, to which the last data release refers n Variable Label Code Unit Pub. calendar Broad block Economic sub-block Pub. delay 1 Purchasing Managers Index, Service xPMI S 688124 Points +1 days EARLY PMI 1 month 2 Entry of Tourists (Frontur) TOUR 247100 Thousands of people +2 days EARLY TOUR 2 months 3 Train Traffic. Goods. Total. TRAIN 242200 Million Passengers/Km +2 days EARLY TRAIN 2 months 4 Gasoline Consumption Auto GAS 256100 Thousands Tm +2 days EARLY FUEL 2 months 5 Diesel Consumption DIESEL 256200 Thousands Tm +2 days EARLY FUEL 2 months 6 Electric Power Consumption (mainland) xENERGY 255100 Millions KWh +2 days EARLY ENERGY 1 month 7 Registered unemployment. Total. UNEM 170000N.D Thousands +2 days EARLY LABOUR 1 month 8 Registered Contracts. CONTRACTS 173100 Number +2 days EARLY LABOUR 1 month 9 Social Security Affiliates xSSAFI 190000 Thousands +2 days EARLY LABOUR 1 month 10 Industrial Production Index. xIPI 229000CC.D Index (2015=100) +5 days EARLY IPI 2 months 11 Industrial Production Index. Manufacture IPIM 229200CC.D Index (2015=100) +5 days EARLY IPI 2 months 12 Credit to Companies and Households xCREDIT 807104 Millions e+ 5 days EARLY FIN 2 months 13 Spread: Diff. between interest rate 1 year and 3 months (*) SPREAD D_DTES00B7 -D_DTES00U7 % + 5 days EARLY FIN 1 month 14 Sea Traffic. Goods. Total SEA 244200 Thousands Tm +6 days EARLY SEA 2 months 15 Apparent Cement Consumption xCEMENT 236000 Thousands Tm + 14 days MIDDLE CEMENT 1 month 16 Air Traffic. Passengers. Total.. AEREO 245100.D Thousands + 14 days MIDDLE AEREO 1 month 17 Car Registration CAR 271200 Number + 15 days MIDDLE VEHICLE REG 1 month 123 SERIEs (2024) 15:145–177 151 Table 1 continued n Variable Label Code Unit Pub. calendar Broad block Economic sub-block Pub. delay 18 Truck Registrations TRUCK 282100 Number + 15 days MIDDLE VEHICLE REG 1 month 19 Large Companies Sales xSALES 25D100 Index (2013=100) + 15 days MIDDLE SALES 2 months 20 Large Companies Sales. Compensation to Employees xSALES C 25M500 Index (2013=100) + 15 days MIDDLE SALES 2 months 21 Construction Index. Total IND CONST 235500 Index (2015=100) +20 days MIDDLE CONSTRUCT 2 month 22 Turnover Index Industry TURNOVER I 223100.D Index (2015=100) +21 days MIDDLE TURNOVER 2 months 23 Turnover Index Service TURNOVER S 249000.D Index (2015=100) +21 days MIDDLE TURNOVER 2 months 24 Exports (constant prices) EXPORTS 502R00 Thousands e+ 22 days MIDDLE FOREIGN 2 months 25 Imports (constant prices) x IMPORTS 506R00 Thousands e+ 22 days MIDDLE FOREIGN 2 months 26 Number of Overnight Stays. Residents Abroad. OVERNIGHTS 241421 Thousands People + 23 days MIDDLE OVERNIGHTS 1 month 27 Employed Labour Force Survey xEMPLOYMENT 120000 /1h20000 Thousands People + 26 days LATE EMPLOYMENT 3 months 28 Gross Domestic Product. Index of chained volumes. xGDP 940000D Index (2015=100) + 29 days LATE GDP 3 months 29 Economic Sentiment Indicator S ESI EI_BSSI_M_R2 Net % + 29 days LATE SURVEY 0 months 30 Industry Production Perspective S INDUPP EI_BSIN_M_R Net % + 29 days LATE SURVEY 0 months 31 Construction of Residential Building Permits (Num.) BUILDING 230111 Thousands e+ 30 days LATE BUILDING 2 months 32 Real estate mortgages. Urban real estate (Value.) MORTGAGE 258020 Thousands e+ 30 days LATE BUILDING 2 months 33 Retail trade index large business RETAIL-LS 272801.D Index (2015=100) + 30 days LATE RETAIL 1 month 34 Retail Trade Index (without gas station) REATIL 272701G.D Index (2015=100) + 30 days LATE RETAIL 1 month Xmarks the indicators considered by the MIDPred model. (*) Monthly interest rate (Spanish Treasury bonds, sec. market) is constructed as averages of daily observations. We provide the data sources in Appendix A. 123 152 SERIEs (2024) 15:145–177 Using the assumed publication calendar, we grouped the variables according to the timeofthemonthinwhichtheirnewfigureswerepublished,whichledustodistinguish three temporal blocks of publication that included indicators whose values are publishedearly(E) inthemonth,i.e. approximatelyinthefirst thirdofit,in the middle(M), i.e. in the second third, and late (L) in the month, i.e. in the last third. The seventh column in Table 1shows the variables that have made up each temporal block. Finally, the last column of Table 1shows the delay in months, to which the last data release refers. We observed that the indicators’ published values refer to different periods. In blocks ‘E’ and ‘M’, data releases refer either to 1-lag or 2-lag months for registered unemployment and industrial production index, for example. On the other hand, in block ‘L’, the values released also refer to the current month, as is the case of soft survey-based economic sentiment and industrial production perspective indicators. As a result, each block is characterized by a certain pattern of available and missing observations. The publication delay of the temporal blocks are replicated in the first month (0q/1m), second (0q/2m)and third (0q/3m)of the current quarter, generating a certain pattern of available and missing observations. In addition, we considered the last month of the preceding quarter (−1q/3m), which illustrates the evolution of the information before the current quarter began, i.e. it gives a time perspective, and helps to seize the value of the information released in subsequent months, and the blocks ‘E’ and ‘M’ of the following quarter’s first month (+1q/1m)that illustrate the data availability just before the release of the GDP first advance. Note that each month has three temporal blocks of data releases, with the exception of the last month, which has only two. As a result, we had υ=1,2, ..., 14 vintages, i.e. three for the first four months, and two for the last month. The main issue of the temporal blocks is that they end up including variables with very diverse definitions. To overcome this issue, we have further divided them according to the economic content and temporarily ordered them, respecting their publication dates as much as possible.7Therefore, we split the vintage ‘E’ into six sub-blocks: purchasing managers’ index (PMI), consumption of electric power (ENERGY), labour market (LABOUR), entry of tourist (TOUR), train traffic (TRAIN), consumption of gasoline and diesel (FUEL), spread and credit (FIN), industrial production (IPI), and sea traffic (SEA); ‘M’ in cement consumption (CEMENT), air traffic (AEREO), car and truck registration (VEHICLE R), large companies sales and compensation (SALES), construction production (CONSTRUCT), turnover service and industry indexes (TURNOVER),and theforeignsector(FOREIGN); finally, ‘L’includes nights stays (OVERNIGHTS), employment, GDP, the survey-based indicator as economic sentiment and industrial perspective (SURVEY), activity related to the retail trade (RETAIL), and building permits and mortgage (BUILDING). In any case, the results were robust to changes in the order chosen for the sub-blocks. The eight column in Table 1shows the resulting economic sub-blocks within each temporal block. Grouping the indicators according to their economic content enabled us to investigate within the temporal blocks. We used economic sub-blocks in months within quarters, following the same reasoning explained above for the temporal blocks. There 7In the case of any ambiguity due to similar publication dates, we ordered the economic sub-blocks taking into account the period of the released value, putting first those data releases that showed the largest time delay. 123 SERIEs (2024) 15:145–177 159 ALS =1 W W  w=1 log ˆ φυ˜ Y(w) 2,T∗+h∗,(12) where ˆ φυ(·)denotes the normal density forecast of Y(w) 2,T∗+h∗. Indeed, it is large when the density forecasts assign a high probability to the observed GDP growth rates and, consequently, can be used to rank them; see Bao et al. (2007). Note that in Eqs. (11) and 12, the point and density forecast depend on the estimated model and the information set. As a result, MSFE and ALS reflect the uncertainty associated with parameter estimation and model instability, and the information flow. Furthermore, we have provided statistical tests to assess the predictive content of new data releases. The crucial point is whether the information inflow improves the forecast performance. Relevant new information, when included in an expanding information set, should lead to a forecast with lower MSPE. On the contrary, information lacking a predictive value should alter neither the point forecast nor the resulting MSPE. We implemented a Mincer–Zarnowitz type of regression to test the predictive content of new information, as follows: e(w) T∗+h∗|υ=β0+β1ˆ ˜ Y(w) 2,T∗+h∗|υ+1+uT∗+h∗,w=1, ..., W,(13) where ˆ ˜ Y(w) 2T∗+h∗|υ+1=ˆ ˜ Y(w) 2T∗+h∗|υ+1−ˆ ˜ Y(w) 2,T∗+h∗|υ, is the forecast revision; see Mincer and Zarnowitz (1969), and Elliott and Timmermann (2016), pp. 355–358. The no relevant new information hypothesis can be expressed in terms of the slope coefficient β1, which must be zero. Rejecting the null hypothesis means that the forecast error obtained using the information in υis correlated with the new information in υ+1that is comprised in the forecast revision. Therefore, we would conclude that the inflow of information is relevant to the GDP forecast, and including it in the information set can lead to a MSPE reduction. We implemented a similar argument to test whether new information significantly changes ALS. The regression used is of the same type as (13) but it has as dependent variable the scaled forecast error e(w) T∗+h∗|υ/V(˜ Y2,T∗+h∗|υ). Once more, rejecting the null of a slope coefficient equal to zero implies that the new information improves the expected log-scored. 13 Finally, we assessed the density forecast in terms ofitscalibration.Awell-calibrated density forecast reasonably estimates the unknown conditional distribution of the future variable (Gneiting et al. (2007) and Mitchell and Wallis (2011)). We used the probability integral transform numbers (PITs), the inverse forecast cumulative distribution evaluated at the realized observations, and the inverse PITs (INTs). If the density forecast shows conformity with the true and unknown one, i.e. it is wellcalibrated, the PITs are i.i.d. uniform over the interval (0,1) and, thus, the INTs are i.i.d. standard normal; see Diebold et al. (1998) and Berkowitz (2001). We considered the likelihood ratio test devised by Berkowitz (2001), which is based on the INTs and exploits their three characteristics under the null hypothesis of correct calibration: zero 13 See Appendix Dfor a formal argument of why rejecting the null hypothesis of β1=0 implies a MSFE (ALS) improvement. 123 160 SERIEs (2024) 15:145–177 mean, unit variance, and absence of serial correlation. Berkowitz’s test statistics is B=−2L(0,1,0)−L(ˆc,ˆσ2,ˆρ)∼χ2 3, where L(c,σ2,ρ) is the exact log-likelihood when we assume that INT t=c+ ρINT t−1+et,et i.i.d. ∼N(0,σ2), and hats denote estimated parameters. Accordingly, L(0,1,0)is the log-likelihood function obtained under the assumption that INTs are i.i.d. normal zero mean and unit variance; see Berkowitz (2001). An inspection of the constant, variance and autoregressive coefficient estimates can reveal the reasons behind the quality of calibration, i.e. causes that lead away from the null hypothesis of conformity between the estimated density forecast and the true and unknown one. The asymptotic chi-squared distribution of Berkowitz’s statistics is obtained assuming that the sample size is large enough so that the estimation errors of the constant, variance, and autoregressive coefficient vanish. Nevertheless, in our case, the sample size (i.e. number of the INTs) was small, so this assumption was difficult to maintain. Also, estimation errors underlies the INTs as they are based on a MFDFM with parameters fixed at some estimates. Thus, parameter uncertainty makes the asymptotic distribution of the Berkowitz’s statistic very conservative. For this reason, we also constructed its bootstrap distribution to tackle parameter and estimation errors and obtained its critical values of interest; see Hall and Wilson (1991) and Kreiss and Franke (1992). 5 Empirical results To implement the MFDFM, we first need to set the number of common factors (r) and the lag order of their dynamics (p). Here, we have followed the referenced literature producing Spanish GDP short-term forecasts and chosen r=1 and p=2 to present the results of the real-time forecasting exercise; see Camacho and Quiros (2011), Cuevas and Quilis (2012), Cuevas et al. (2017), and Pareja et al. (2020).14 To characterize the factor, i.e. to give it an economic interpretation, we considered as an illustration the estimation results when we used the complete data set covering the period 1995Q2–2023Q2. The first factor explained 57% of the total variability and described the business cycle, as suggested by Fig. 1, with large positive weights for social security affiliates (SSAFI), industrial production perspective (S INDUPP), and purchasingmanagers’index(PMI),andlargenegativeweightforthespreadrate(INT), and medium-size positive weights for large companies’ sales (SALES), turnover index 14 An anonymous referee has commented on that the chance of some indicators have undergone big structural changes is high, even more so considering the period starting in 2020. Under big structural breaks, increasing the number of factors may be necessary to construct more accurate forecasts; see Breitung and Eickmeier (2011) and Chen et al. (2014), who point out that big structural breaks in the loading of the indicators may increase the dimension of the factor space. Although more factors result in slightly different quantitative results, they yield the same conclusions as when a DFM with one factor and a lag order of two is used. Therefore, for the sake of comparison and simplicity, we have set r=1andp=2. The results for alternative values of rand pare available upon request. 123 SERIEs (2024) 15:145–177 161 Fig. 1 Estimated loadings obtained using a MFDFM with r=1andp=2 and no modelling of the idiosyncratic errors applied to the large dataset. The estimation period is 1995Q2–2023Q2 Fig. 2 Quarterly Spanish GDP growth rate and estimated common component obtained using a obtained using a MFDFM with r=1andp=2 and no modelling of the idiosyncratic errors applied to the large dataset. The estimation period 1995Q2–2023Q2 service (TURNOVER S), industrial production indexes (IPI and IPIM), retail trade index (RETAIL), and consumption of cement (CEMENT). Furthermore, Fig. 2shows the estimated re-centred and scaled common component ¯ ˜ Y2+σ˜ Y2×ˆ Y2×(ˆ Ft+2ˆ Ft−1+3ˆ Ft−2+2ˆ Ft−3+ˆ Ft−4) tracks quite well the GDP growth rate and it is effective in capturing in advance the movement of the latter. Both are highly correlated contemporaneously, with an estimated correlation coefficient of 0.95. They are also dynamically correlated, with estimated cross-correlation coefficients between GDP and the common component’s oneand two-quarter lags of 0.92 and 0.86, respectively. 123 162 SERIEs (2024) 15:145–177 5.1 Point and density forecast performance Figure 3plots the root MSPE (RMSPE) over the temporal blocks and their economic disaggregation obtained for the large data framework with the MFDFM without modelling the dynamics of the idiosyncratic errors. For comparison, we have also included the point accuracy when we estimated and forecasted the GDP growth rate using MIDPred approach. Furthermore, we also reported the AR(1) forecast as a benchmark. The first panel of Fig. 3shows that publication releases throughout the quarter are essential as they reduce the RMSPE. However, the rates at which RMSPE decreases are not constant for all releases. We observe that the steepest declines in point forecast accuracy attained by the MFDFM occur at vintages −1q/3m, and 0q/1m, then dampen before disappearing towards mid +0q/3m. The first panel of Fig. 3also suggests that when the purpose is to produce Spanish GDP short-term estimates, the rich data environment and the MIDPred approaches provide somewhat similar performance for all data vintages. Nevertheless, these approaches systematically outperform the AR(1), attaining better accuracy than the benchmark for all publication releases. Interesting to note is that MDDFM are comparatively more attractive than the AR(1) for data vintages before the release of the first advance of GDP than for the subsequent releases. ThefirstpanelofFig.3alsoreportstheobservedteststatisticsforthenullhypothesis of no informational gain between consecutive vintages. We have reported these test statistics for the large data frame approach. In the first panel in Fig. 3, we observed that those vintages with steep drops in MSPE are associated with positive and significant observedstatistics,meaningarejection ofthenullof noinformationalgain.Incontrast, containeddropsinMSPEbetweenconsecutivevintagesshowstatisticsthataresmall in magnitude and non-significant, i.e. with no strong evidence against the null. Therefore, we corroborated that referring to the pronounced reductions in MSPE and the presence of significant informational gains are equivalent. We can analyse which type of series contributes the most to the improvement of forecast performance using the second panel in Fig. 3, which reports the RMSPE, and the observed test statistic for the economic sub-blocks. The first aspect to highlight is that the overall behaviour of the point forecast accuracy. Moreover, we observed that MFDFM approaches are better than AR(1) for all vintages, although much more attractive before the release of the latest GDP data in the current quarter’s first month. Remarkably, the closest approach between the performances of the MFDFM methods and the AR(1) models is produced when the advanced of the previous quarter’s GDP is released. The labour market (LABOUR) data has a pronounced impact on MSPE. The LABOUR sub-block includes the registered unemployment, social security affiliates, and registered contracts, indicators that are published as soon as the month begins, with the latest values referring mainly to the previous month. Also important in terms of RMSPE reduction is SURVEY, composed of survey-based indicators, which are released at the end of the month and are the most timely since they are the first to have information about it. The purchasing managers’ index service (PMI S) also provides pronounced drops in RMSPE. Beyond LABOUR, SURVEY, and PMI S, we observed a substantial reduction in the RMSPE when the latest GDP data is released towards the 123 SERIEs (2024) 15:145–177 163 end of the quarter’s first month. Another sub-group that provides additional point forecast accuracy, although less pronounced, is IPI, which gives positive and statistically significant MSPE reductions in the first two months. It is worth noting that the existing literature on Spanish GDP short-term forecasts has consistently observed improvements in the accuracy of point forecasts as new data related to GDP and its predictors becomes available throughout the quarter. Furthermore, our findings have emphasized the significance of LABOUR, SURVEY, IPI, and PMI indicators, which have already been recognized as relevant for producing Spanish GDP short-term forecasts, while also prompting questions about the contributions of retail sales, overnight stays, energy consumption, and large companies’ sales. We have further evaluated the predictive content of the information flow for GDP short-term destiny forecasts. The panels in Fig. 4report the ALS for the Spanish GDP growth rate density forecasts for the MFDFM approaches and the AR(1) model over thetemporal blocksandtheir economicdisaggregation.Theyalsoinclude the observed teststatisticsforthenullhypothesisofnorelevantinformationforthedensityforecasts, obtained for large data frame. Once more, a rejection of the null hypothesis points to a potential improvement in the density forecast that we could achieve by recomputing the point forecast and, consequently, the density, leading to an improvement in log score. The first panel in Fig. 4shows that the ALS increases as the new data becomes available, i.e. Spanish GDP short-term density forecasts improve with the information inflow. More specifically, the ALS varies only slightly in −1q/3m, then changes most sharply during the 0q/1mand 0q/2m, stabilizing its growth from 0q/3monward. Remarkably, most of the significant temporal blocks in the first panel in Fig. 4are also relevant in the first one in Fig. 3when we focus on point forecast accuracy. The second panel of Fig. 4shows positive changes in density forecast accuracy during 0q/1mthat moderate after it, confirming the results of the first panel. The most relevant economic vintages in terms of log score improvement are the LABOUR, SURVEY, GDP, and IPI, which sometimes produce positive moderate, and significant changes in the log score, in that order. Furthermore, Fig. 4suggests that the large dataset and the MIDPred approaches provide similar ALS, perhaps slightly better for the former than the latter from mid 0q/1monwards. The comparison of the ALS of MFDFM methods and the AR(1) model also revealed, on one side, that the former provides, on average, much better density forecasts and, on the other, that the difference between the two is more pronounced until the release of the first GDP advance of the previous quarter. Moreover, this point in the current quarter produces the closest approximation between the ALS of both models. An explanation is that before the release of the latest GDP data, the more timely monthly indicators signal the evolution of the business cycle, replacing the most recent and unavailable GDP information. Accordingly, monthly predictors produce substantial relative gains with respect to an AR model in which only GDP information is exploited. The latest GDP figure brings information on business cycle conditions that were summarized, until that time in the monthly indicators and produces an approximation of the performances of the MFDFM and the AR model. 123 164 SERIEs (2024) 15:145–177 Fig. 3 RMSPE over temporal blocks (first) and economic sub-groups (second) between −1q/3mand +1q/1mfor quarterly GDP growth rate obtained using a: (i) MFDFM with r=1andp=2 and no modellingoftheidiosyncratic errors applied to thelargedataset(continuous-dottedblue line), (ii) a MFDFM with r=1andp=2 and AR(1) idiosyncratic errors applied to the small-data set, i.e. MIDPred model (continuous blue line), and, (iii) univariate AR(1) model (dashed blue line). The vertical bars show the observedtest statistics obtained usinga MFDFM withr=1andp=2 andno modelling ofthe idiosyncratic errors applied to the large dataset for the null hypothesis of no information gain between the data vintage on the x-axis and the one immediately preceding it, and the vertical area outside the dash horizontal lines represent a 5% Rejection Region (color figure online) 123 SERIEs (2024) 15:145–177 165 Fig. 4 ALS over temporal blocks (first) and economic sub-groups (second) between −1q/3mand +1q/1m for quarterly GDP growth rate obtained using a: (i) MFDFM with r=1andp=2 and no modelling of the idiosyncratic errors applied to the large dataset (continuous-dotted blue line), (ii) a MFDFM with r=1 and p=2 and AR(1) idiosyncratic errors applied to the small-data set, i.e. MIDPred model (continuous blue line), and (iii) univariate AR(1) model (dashed blue line). The vertical bars show the observed test statistics obtained using a MFDFM with r=1andp=2 and no modelling of the idiosyncratic errors applied to the large dataset for the null hypothesis of no information gain between the data vintage on the x-axis and the one immediately preceding it, and the vertical area outside the dash horizontal lines represent a 5% Rejection Region (color figure online) 123 166 SERIEs (2024) 15:145–177 In summary, there are point forecast performance measures revealed the importance of the information flow Spanish GDP growth rate nowcasts. Likewise, with some nuances, they point out that the most pronounced improvement are attained towards the end of the current quarter’s first month. This moment in the quarter emerges as the period of separation between data vintages that produces, firstly, the greatest improvement in forecast performances and, secondly, the largest difference between the MFDFM approaches and the AR(1) model. An explanation is that before the release of the latest GDP data, the timely monthly indicators signal the evolution of the business cycle, providing a summary of the unavailable previous quarter’s GDP information. Accordingly, relevant monthly predictors produce a substantial improvement in forecastaccuracymeasuresandgeneratethe mostevident relativegain with respect to an AR model in which only the GDP information has been exploited. Finally, those indicators with a high information content are those in the LABOUR, SURVEY, PMI S, and IPI. 5.2 Density forecast calibration We now turn to the calibration of density forecasts. Figure 5shows Berkowitz’s observed test statistics obtained for density forecasts over economic data vintages and its 5% asymptotic and bootstrap critical values. An observed test statistic larger than the critical value at a given significance level implies that we cannot reject the null hypothesis of good calibration of the density forecast. But beyond this rigid interpretation, we implement the critical values as a reference to globally differentiate data vintages that generate an inadequate density forecast calibration from those in which there is no evidence against a good one. In Fig. 5, we observe less and less evidence against good calibration as more and more data becomes available in the first three months. In other words, the GDP short-term density forecasts steadily improve and approach the region of no evidence against good calibration with the flow of information. It is also remarkable that the steepest changes in the observed Boccur throughout 0q/1m, generating an apparent clustering between the vintages for which we clearly reject the null of good calibration hypothesis and those for which there is no strong evidence against it. Berkowitz’s test signals a violation of the hypothesis of good calibration, but it does not provide reasons for it. The latter need to be found in the characteristics that INTs should have and may lack. Thereby, we included the second panel in Fig. 5which plots the estimated values ˆc,ˆσ2, and ˆρunderlying the test. Firstly, the estimated constant is always negative, showing little approximation towards its null value of zero with data releases. A negative constant indicates a density forecast giving high probabilities to large values of the future outcome, i.e. it has a large mean, signalling a positive bias of point forecasts. Secondly, the estimated variance decreases with data releases, especially since the beginning of the current quarter, indicating that the reason for the closer proximity between the density forecasts and the unknown ones archived with data releases is a better estimation of the conditional variance. Furthermore, the fact that the estimated auto-correlation of INTs diminishes over data vintages indicates 123 SERIEs (2024) 15:145–177 167 Fig. 5 First, Berkowitz’s observed test statistics for the null of good density forecast calibration over data vintages between −1q/3mand +1q/1mobtained using a MFDFM withr=1andp=2 and no modelling of the idiosyncratic errors applied to the large dataset applied. The vertical areas above the horizontal continuous lines and the dash line represent the 5% asymptotic and bootstrap rejection regions. Second, estimates of constant (ˆc), variance ˆσ2and autoregressive coefficient (ˆρ) for each economic sub-block that the mixed-frequency DFM used to construct GDP forecasts better captures the GDP dynamics, as more information becomes available. 15 15 It is remarkable that observed Berkowitz test statistics are rather large for data vintages between −1q/3m and late 0q/1m, just before the release of last quarter’s GDP value. A reason is that before that point in time, density forecasts are two-period ahead concerning the last observation available of GDP. For more than one-period ahead forecasts, the PITs and INTs may exhibit serial correlation, and the large observed 123 168 SERIEs (2024) 15:145–177 Overall, the estimated values of ˆc,ˆσ2and ˆρindicate that the less evidence against the null of conformity between the density forecast and the unknown achieved with data releases is mainly due to a reduction in the dispersion and the auto-correlation of the INTs. In other words, calibration improves with data releases because the density forecasts better capture the conditional variance of the unknown ones. As an illustration, Figs. 6and 7show the GDP growth rates density forecasts and its realized value for several quarters. The first notable impression from these figures is that GDP short-term density forecasts evolve with the data releases throughout the quarter, as the information flow affect the conditional mean and variance estimates underlying the conditional normal densities. The centre of the latter, i.e. the point forecast, moves as if trying to catch the realized GDP growth rate, and the dispersion narrows with new data releases. For example, Fig. 6displays the situation during the great economic recession of 2008/2009. The fourth quarter of 2008 and the first quarter of 2009 show a continuous adjustment towards the left in short-term density forecasts as the information flow proceeds, but not enough because the realized GDP growth rates are well located on the left tail of the densities. In the second and third quarters of 2009, the situation reverses, and the flow of information pushes densities from the left to GDP. Moreover, the reduction in dispersion with data releases is apparent. Changes in GDP density forecasts with the information flow are also evident in panels of Fig. 7, which describe the quarters in 2019, although comparatively less pronounced than those seen in Fig. 6. A sharp contrast in these figures shows us that the responses of the conditional means and variances are much more pronounced in periods of economic turbulence than in periods of relative calm when the releases of the latest data reveal more abrupt changes in the evolution of the economic indicators. Finally, it is noteworthy that realized GDP growth rates are generally below the centre of its short-term density forecasts in all but the second and third quarters of 2009. Hence, the probability of observing a value less than the observed GDP for most densities is smaller than 0.5, i.e. PITs are smaller than 0.5, implying negative INTs. This exemplifies what happens on average for data vintages of all quarters and is the reason behind the negative estimates of constants in models for INTs. 6 Conclusion We have investigated the predictive content of data releases for short-term Spanish GDP forecasting. We used information from a wide group of monthly variables that were then implemented as predictors of the Spanish GDP growth rate within a MFDFM. We based our analysis on a recursive estimation scheme, using pseudo realtime data vintages that replicated the publication calendar of the monthly indicator and the GDP. Then, we evaluated from the more common perspective of point forecasts and the more novel one of density forecasts. As time passes and monthly indicators and GDP are released there are notable improvements in short-term GDP forecasts. However, a significant portion of the Bstatistics before the release of the previous quarter’s GDP value can be originated in the fact that we have requested absence of auto-correlation. 123 SERIEs (2024) 15:145–177 175 :EF(YT∗+h∗(ωυ+1), YT∗+h∗)−EF(YT∗+h∗(ωυ), YT∗+h∗)=0.(15) Taking expectations on both sides of equation (14), we obtain that (15) implies that EF(YT∗+h∗(ωυ), YT∗+h∗)( YT∗+h∗(ωυ+1)−YT∗+h∗(ωυ))=0.(16) The last can be viewed as an orthogonality condition. Under the hypothesis of no change in the expected loss, F(·)(i.e. in words of Granger, “the generalized forecast error”) is uncorrelated to the change in point forecasts. The orthogonality condition (16) put forward Mincer-Zarnowitz type of regression of the following form: FYT∗+h∗|υ,YT∗+h∗=β0+β1YT∗+h∗|v+1−YT∗+h∗|υ+uT∗+h∗, and testing whether the slope coefficient β1is equal to zero. The rejection of null hypothesis means that the additional relevant information in υ+1improves the point forecast and, consequently, helps achieve a lower expected loss. Anintuitiveexplanationisasfollows.TheforecasterroreT∗+h∗|υistheunexpected component YT∗+h∗with respect to the information set υ,comprised in the point forecast. On the other hand, YT∗+h∗|υ+1is the part of the new forecast orthogonal to the previous one, thus corresponding to the change in point forecasts that can be attributed to the new information in v+1. If the forecast error eT∗+h∗|υrelates to YT∗+h∗|υ+1, then it means that υ+1contains new relevant information that can lead to a MSPE improvement. The final form of the Mincer-Zarnowitz regression, i.e. its dependent variable, depends on the loss function. Consider first the quadratic loss function Q(YT∗+h∗|υ+1,YT∗+h∗|j+1)=(YT∗+h∗−YT∗+h∗|j)2, in which case we have that Q(YT∗+h∗|υ+1,YT∗+h∗|υ+1)=−2(YT∗+h∗−YT∗+h∗|υ). 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