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Evaluation of Mixed Frequency Approaches for Tracking Near-Term Economic Developments in North Macedonia

Evaluation of Mixed Frequency Approaches for Tracking Near-Term Economic Developments in North... Aggregate demand forecasting, also known as nowcasting when it applies to current quarter assessment, is of notable interest to policy makers. This paper concentrates on the empirical methods dealing with mixed- frequency data. In particular, it focuses on the MIDAS approach and its later extension, the Bayesian MF- VAR. The two strategies are evaluated in terms of their accuracy to nowcast Macedonian GDP growth, using same monthly frequency data set. The results of this study indicate that the MIDAS regressions demonstrate comparable forecasting performance to that of MF-VAR model. Moreover, it is interesting to note that the two approaches are reciprocal, since in general, their combined forecast demonstrates clear superiority in predicting business cycle turning points. Additionally, the MF-VAR model showed higher precision in times of increased uncertainty. Keywords: MF-VAR, Bayesian estimation, MIDAS, forecast pooling, forecast evaluation JEL Classification: E37, C53 Gani Ramadani (corresponding author) Senior Advisor Monetary Policy and Research Department, 1. Introduction National Bank of the Republic of North Macedonia From their sampling frequency perspective, eco- E-mail: ramadanig@nbrm.mk nomic data vary considerably. In decision-making Address: Blvd Kuzman Josifovski Pitu 1, process, we are faced with both, the problem of data 1000 Skopje, Republic of North Macedonia frequency and the publication delay. In this regard, ORCID: https://orcid.org/0000-0001-5861-6981 the State Statistical Office of the Republic of North Macedonia, following the international practices, re- Magdalena Petrovska leases quarterly figure for Gross Domestic Product Senior Advisor (GDP) with a delay of slightly more than two months. Monetary Policy and Research Department, Having in mind that these data are a crucial asset for es- National Bank of the Republic of North Macedonia tablishing and implementing policies, their rough esti- E-mail: petrovskam@nbrm.mk mation is therefore necessary. Estimation can be based on monthly readings, such as the industrial production Vesna Bucevska, PhD volume index, external trade, value indices of turnover Full professor of Econometrics and Financial in the retail trade, etc. This whole process is known as Econometrics nowcasting. The fundamental postulate of nowcasting Faculty of Economics-Skopje, lies in exploiting the information released early and Ss. Cyril and Methodius University eventually at higher frequency than the variable of in- E-mail: vesna.bucevska@eccf.ukim.edu.mk terest so that one may get an “early estimate” before the Copyright © 2021 by the School of Economics and Business Sarajevo 43 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA official data becomes accessible (Bańbura et al., 2013). the specifications. This gives rise to concern of how to quantify em- Initially, MIDAS models have been applied in the pirically the relationships between variables sampled financial domain (see for instance Ghysels, Santa Clara, at various frequencies. The most naive approach is to and Valkanov (2006). Recently, numerous applications stick with the lowest frequency in the data, but in this involve the MIDAS approach as a forecasting plat- case a loss of possibly valuable high-frequency infor- form for quarterly GDP, like for example Clements and mation is unavoidable. However, the recent literature Galvão (2008) and Clements and Galvão (2009). Later displays a growing interest in the value added that di- additions are Foroni, Marcellino, and Schumacher rect modelling of mixed-frequency could provide. (2011); Kuzin, Marcellino and Schumacher (2011); This paper gives an overview of some of the key Drechsel and Scheufele (2012a); Andreou, Ghysels, approaches employed in the field literature to cope and Kourtellos (2013); Ferrara, Marsilli, and Ortega with mixed-frequency data: mixed-frequency Vector (2014); Duarte (2014); and Aastveit, Foroni and Autoregression (MF-VAR) in a Bayesian framework, Ravazzolo (2016), amongst others. Furthermore, launched by Schorfheide and Song (2015), as well as Foroni, Marcellino and Stevanovic (2018) show ana- the less computationally intensive counterpart, i.e. un- lytically, in Monte Carlo simulations, the relevance of restricted mixed-data sampling (U-MIDAS) approach considering the moving average (MA) component in introduced by Foroni, Marcellino, and Schumacher MIDAS and U-MIDAS models thus closing the gap in (2011). To this end, our study is exploratory and inter- the respective literature. Andreou et al. (2019), on the pretative in nature. Analogously, the primary research other hand show how the group factor context ap- challenge was to verify which of the considered ap- plies to mixed‐frequency data panels. proaches generalises better and is more capable of This study also falls into a relatively new and thus producing reliable GDP nowcasts. Therefore, two increasing body of literature on mixed frequency VAR empirically testable statements were defined: by em- models that accommodate a state space approach. ploying MF-VAR approach, accurate and efficient now - The main idea assumes reformulation of each lower casts of North Macedonia’s GDP can be acquired; by frequency series into a partially latent high frequen- employing U-MIDAS approach, accurate and efficient cy series. The Kalman filter or, the Gibbs sampler in a nowcasts of North Macedonia’s GDP can be acquired. Bayesian framework, then allow a partially latent VAR To this end, after discussing in a nutshell the two process to be estimated. See Mittnik and Zadrozny employed modelling methodologies, we proceed (2005); Kuzin, Marcellino and Schumacher (2011); Bai, comparing them in an exhaustive empirical exercise. Ghysels and Wright (2013); and Foroni and Marcellino Specifically, we revolve around comparison of the re - (2014) as a leading research on state space type MF- sulting models in terms of the proposed predictions, VAR models adopting a non-Bayesian version of using same high frequency (HF) data set. To this end, the Kalman filter. On the other hand, Mariano and the variable of interest is the Macedonian quarterly Murasawa (2010) have a pioneering contribution for a GDP growth rate. state space type MF-VAR using the expectation–maxi- A theoretical comparison of these two classes of mization (EM) algorithm, and Chiu et al. (2011) and models points out that, U-MIDAS is more parsimoni- Schorfheide and Song (2015) for MF-VARs cast in state ous than MF-VAR, and as a direct forecasting tool dis- space form using the Gibbs sampler (for greater cov- plays greater robustness to misspecification (Kuzin, erage and extension of these literature please refer to Marcellino, and Schumacher 2011). This study aims at Mikosch and Neuwirth 2015). documenting this status. Moreover, we assess wheth- er the forecast accuracy improves when combining these two models. 3. Data With respect to the potential high-frequency indi- cators to draw from, we consider a broad framework 2. Literature review of time series routinely employed in the process of Ghysels, Santa-Clara and Valkanov (2004) pio- GDP nowcasting, starting from economic sentiment neered one of the most competitive univariate tools indicators to hard data . The selection process was in- suited to handle the mixed-frequency data, i.e. the so tended to bring about the “best” subset of predictors. called mixed-data sampling method (MIDAS). Mixed- To this end, all of the variables were subject to pre- data sampling (MIDAS) models operate with time se- filtering based on vigorous one-by-one testing within ries at various frequencies. In this structure distributed the bridge equations set-up as a naïve approach to lag polynomials are employed to ensure parsimony in handle the mixed-frequency data . 44 SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA The shortlisted variables chosen from a broader set In order to specify the measurement equation, the of similar alternatives are actually those who pass the authors have to define the aggregation equation. in-sample selection based on their recent forecasting Following, Kapetanios, Marcellino and Petrova (2018), performance. In other words, the informed variable taking GDP growth as an example, the disaggregation selection procedure that we follow reflects our metrics of the quarterly GDP growth, 𝑦𝑦 , observed every 𝑡𝑡 = � � based on minimum relative forecasting errors. 3; 6; 9; …; 𝑇𝑇 , into the month-on-month GDP growth, In addition, all of the considered variables are 𝑦𝑦 , never observed, is based on the following seasonally adjusted, as well as transformed ensuring aggregation equation: their stationarity (e.g. trending variables are expressed ∗ ∗ ∗ as growth rates). The monthly data releases follow 𝑦𝑦 �𝑦𝑦 �𝑦𝑦 �𝑦𝑦 ��Λ 𝑧𝑧 � � � � � �� � � � � ��� ��� similar timing. This allows us to reproduce the same To this end, the quarterly variable is treated as the pattern of missing reading at the end of each recursive sample, so to imitate the data availability in real-time. three-month average of the monthly process. The ragged-edge overview of the dataset is presented Following Foroni and Marcellino (2013), since 𝑦𝑦 in Table A.1 of the Appendix. In addition, depending is observed only every third month, there is a need of on the publication timetable, we assume that data are a selection matrix that equals the identity matrix if 𝑡𝑡 accessible at the earliest at the month-end. corresponds to the last month of the quarter and is Furthermore, the quarterly variable that we empty otherwise. Therefore, the measurement consider, i.e., the real GDP is obtainable in the third equation can be specified as: month after the end of the referent quarter. For 𝑦𝑦 � ��� Λ 𝑧𝑧 instance, the GDP reading for 2017Q4 becomes � � � 𝑥𝑥 � � acquirable in March 2018. where M is the selection matrix. The problem of dimensionality is surpassed by introducing of a Minnesota prior that shrinks the VAR 4. Methodology coefficients toward univariate random walk 4.1. MF-VAR methodology representations. (adaptation from Kapetanios, Marcellino and Petrova 2018). The methodological explanation covering the MF-VAR model is an adaptation from Claudia Foroni’s doctoral thesis which provides a compendium of individual 4.2. The U-MIDAS approach mixed-frequency approaches along with a very intuitive understanding of the differences between MIDAS regressions are perceived as a widespread them. alternate to the multivariate state-space framework In the subsequent paragraphs, we describe the elaborated in the previous sub-section. This main characteristics of the Bayesian MF-VAR approach, econometric technique, as a very general type of ARDL following Schorfheide and Song (2015) as a most model, is based on both a regression structure and a quintessential study in the field literature. weight function which tracks the high frequency lags Namely, these authors cast a MF-VAR in state space of the regressors (Marsilli 2014). The majority of the form. Furthermore, in order to conduct Bayesian formulas and back up explanations employed in this inference for model parameters and unobserved section were adapted from Barsoum and Stankiewicz monthly variables, they make use of Markov chain (2013). Monte Carlo (MCMC) methods. Following Foroni and The elementary form of the MIDAS model Marcellino (2013), the state equation of the model has employed to get an h−step ahead forecast might be a VAR(p) representation, treating quarterly series as expressed following Clements and Galvão (2008): monthly series with missing observations, written as it ��� � follows: 𝑦𝑦 �� �� ��𝐿𝐿 �𝜃𝜃�𝑥𝑥 �� � � � � ��� Let for all 𝑡𝑡 the latent month-on-month GDP ��� ��� ����� Where ��𝐿𝐿 �𝜃𝜃� � ����𝜃𝜃�𝐿𝐿 is the growth 𝑦𝑦 and the corresponding monthly indicator ��� sum of weights assigned to K lags of the independent 𝑥𝑥 follow a VAR(p) process th variable (the lag polynomial). ����𝜃𝜃� is the k weight of the K-lag polynomial, shaped by a certain function � � � � 𝑧𝑧 �� Φ 𝑧𝑧 �� Φ �𝜐𝜐 � � � � � � ��� � of 𝜃𝜃 parameters (as for instance an exponential 𝜐𝜐 �����������Σ�� function). 𝐿𝐿 denotes the lag operator so that SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 45 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA � � ��� As we are set to do GDP forecasting, we want to 𝐿𝐿 𝑥𝑥 �𝑥𝑥 . 𝑡𝑡 is the time index for y, as a ��� ������� include autoregressive elements in the U-MIDAS lower frequency variable, while m is the time index for model. Simply introducing an AR part into the the variable with higher frequency, i.e. x. Q describes previous regression we obtain variables observed on a quarterly and M on a monthly ��� basis. 𝑦𝑦 �𝛽𝛽 ��𝑦𝑦 � �𝛽𝛽 𝑥𝑥 �� Having in mind the non-linearity of the lag � � ��� ��� ������� � ��� polynomials, the non-linear least square (NLS) is a typical estimation method for MIDAS models. Each of the U-MIDAS regressions employed to However, in some instances the form of the lag predict Macedonian real GDP growth, for up to one polynomial may be overly restrictive in comparison quarter ahead (i.e., a “nowcast” of the current quarter) with the underlying data generating process. Thus a use single indicator. With only one indicator in each model with no restrictions on the weights of the lag representation and a restricted number of lags, the polynomial was launched by Foroni, Marcellino and coefficients in equation above can be estimated Schumaher (2011) (adaptation from Bersoum and without internalising the degrees of freedom problem Stankiewicz 2013). These authors advanced a new (adaptation from Leboeuf and Morel 2014). parametrization scheme for the MIDAS based on a Our U-MIDAS specification includes 3 lags of the linearization of the distributed lag function called monthly variables in total, extending over the quarter unrestricted MIDAS (U-MIDAS), where all the for which we have the last reading of real GDP growth parameters are estimated using OLS. The U-MIDAS as well as data along the first quarter to forecast, model exploit a linear lag polynomial that can be provided they are obtainable. As the separate monthly expressed as (adaptation from Marsilli 2014): readings of the regressors are published over a quarter, the model representation changes slightly ��� � (adaptation from Leboeuf and Morel 2014). 𝑦𝑦 �𝛽𝛽 � �𝛽𝛽 𝑥𝑥 �� � ��� ������� � In what follows, we provide a real-time ��� demonstration of the U-MIDAS model for the The dependent variable y is represented by an Macedonian GDP as an analogy to the example of the equation that contains an intercept 𝛽𝛽 and a lag U-MIDAS for euro-area GDP employed by Leboeuf and polynomial weighted by parameters 𝛽𝛽 . To this end, ��� Morel (2014). all the parameters 𝛽𝛽 of this polynomial need to be ( 1) ( 2) ( 3) ��� 𝑀𝑀 𝑀𝑀 𝑀𝑀 Let 𝑋𝑋t , 𝑋𝑋t and 𝑋𝑋t be monthly variables in estimated as no structure is set on the shape of the the first, second and third month of quarter 𝑡𝑡 , for weights of the lag polynomial (adaptation from ( ) 𝑄𝑄 which we are generating a nowcast of 𝑡𝑡𝑌𝑌 (real GDP Bersoum and Stankiewicz 2013). ( 1) 𝑀𝑀 growth in quarter 𝑡𝑡 ). In other words, 𝑋𝑋t is a quarterly The lags of the explanatory variable are time series containing all first monthly values of the represented by the measure 𝑚𝑚 , where m = 3 specifies ( 2) 𝑀𝑀 variable X for each quarter over past horizon. 𝑋𝑋t is a the number of observations of the higher-frequency quarterly time series encompassing all second indicator (e.g. monthly variable 𝑥𝑥 ) for each observation monthly values of a variable X for each quarter over of the lower-frequency variable (e.g. quarterly variable ( 3) 𝑀𝑀 past horizon. 𝑋𝑋t is a quarterly time series including 𝑦𝑦 ). That is, if e.g. 𝑦𝑦 is the reading of the dependent all third monthly values of the variable X for each variable for the first quarter of 2017 (March 2017), then quarter over past horizon (adaptation from Leboeuf 𝑥𝑥 represents the observation of the explanatory ��� and Morel 2014). variable for December 2016 (1 quarter before), 𝑥𝑥 ����� For the U-MIDAS approach now we have monthly for November 2016 (4 months before), whereas variables to be transformed into quarterly variables. As 𝑥𝑥 for January 2016 (14 months before) and so ������ in one quarter there are 3 months, each monthly forth (Bersoum and Stankiewicz 2013). indicator will be transformed into 3 variables with One of the obvious setbacks of the U-MIDAS lies in quarterly frequency. ( 1) the fact that, when the discrepancy in frequencies 𝑀𝑀 𝑋𝑋t - only data for months 1,4,7,10 are taken ( 2) between the variables in the model is large, its 𝑀𝑀 𝑋𝑋t - only data for months 2,5,8,11 are taken ( 3) performance plunges significantly due to the rapid 𝑀𝑀 𝑋𝑋t - only data for months 3,6,9,12 are taken increase of the number of parameters. To this end, this Then, just for an illustration, if X has a 1 month method do not fit all kinds of empirical applications publication delay: (adaptation from Bersoum and Stankiewicz 2013). • In the first month, the model for Y nowcasting However, for many macroeconomic analyses the use contains a constant, one lag of Y and 3 months of data of the U-MIDAS model may be advantageous. on variable X 46 SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA ( ) ( 1) ( ) 5 𝑄𝑄 𝑀𝑀 𝑄𝑄 𝑌𝑌 =𝛽𝛽 +𝜑𝜑 𝑌𝑌 + a crisis year like 2017 . Therefore, the model is 𝑡𝑡 1 1 𝑡𝑡 −1 ( 1) ( 2) ( 3) ( 1) 𝑀𝑀 𝑀𝑀 𝑀𝑀 𝑀𝑀 𝛾𝛾 𝑋𝑋 +𝛾𝛾 𝑋𝑋 +𝛾𝛾 𝑋𝑋 +𝜔𝜔 evaluated based on both, its ability to approximate 2,1 𝑡𝑡 −1 2,2 𝑡𝑡 −1 2,3 𝑡𝑡 −1 𝑡𝑡 history, and its usefulness to capture the turning i.e. months 10,11 & 12 points. • In the second month, the representation is the same as in the first one, but the first month of the ( 1) 𝑀𝑀 current quarter (𝑋𝑋t ) is added to the model 5.1. Main findings from the empirical exercise specification: ( ) ( 2) ( ) ( 1) ( 2) ( This section outlines the average performance of the 𝑄𝑄 𝑀𝑀 𝑄𝑄 𝑀𝑀 𝑀𝑀 𝑀𝑀 𝑌𝑌 =𝛽𝛽 +𝜑𝜑 𝑌𝑌 +𝛾𝛾 𝑋𝑋 +𝛾𝛾 𝑋𝑋 +𝛾𝛾 𝑋𝑋 𝑡𝑡 1 1 𝑡𝑡 −1 1,1 𝑡𝑡 2,2 𝑡𝑡 −1 2,3 𝑡𝑡 −1 3) ( 2) employed modelling frameworks. To this end, we 𝑀𝑀 +𝜔𝜔 𝑡𝑡 report the RMSE performance of the MF-VAR for i.e. months, 1, 11 & 12 nowcasting quarterly GDP growth at one period ahead horizon, against the U-MIDAS pooled forecasts. In • In the third, two months of the current quarter (𝑡𝑡 ) addition, we report the combination forecast of these and one month of the previous quarter (𝑡𝑡 -1) are two models using inverse mean square error (IMSE) included (3 months of the variable X in total): weighting scheme . The RMSE result for the ( ) ( 3) ( ) ( 1) ( 2) 𝑄𝑄 𝑀𝑀 𝑄𝑄 𝑀𝑀 𝑀𝑀 𝑌𝑌 =𝛽𝛽 +𝜑𝜑 𝑌𝑌 +𝛾𝛾 𝑋𝑋 +𝛾𝛾 𝑋𝑋 + 𝑡𝑡 1 1 𝑡𝑡 −1 1,1 𝑡𝑡 1,2 𝑡𝑡 benchmark bridge equations model is presented as ( 3) ( 3) 𝑀𝑀 𝑀𝑀 𝛾𝛾 𝑋𝑋 +𝜔𝜔 2,3 𝑡𝑡 −1 𝑡𝑡 well. The results aligned with the bridge equations i.e. months 1, 2 & 12 model, U-MIDAS pooled forecasts and the MF-VAR ( ) 𝑀𝑀𝑀𝑀 where 𝜔𝜔 𝑡𝑡 for 𝑀𝑀 =1,2 or 3, is the error term of the individual forecasts, as well as their combination regressions. forecasts are summarized in the Table 1 below. The Also, in U-MIDAS, the weights assigned to each results, are obtained recursively, based on log month are completely data-driven, reflecting the difference approximation as well as on seasonally concept that each month of data has different adjusted figures, for the evaluation sample 2015 Q4 - importance in forecasting GDP. U-MIDAS has another 2017 Q4 (third months of the quarter). appealing feature fo forecasting in short-term horizon. Considering the fact that we evaluate the models Namely, unlike bridge equations approach, for not only statistically but also regarding their ability for instance, it does not necessitate a forecast of missing pseudo out-of-sample projecting of turning points, we months and consequently does not require any might say that the benchmark model, (i.e. bridge assumptions about the development of the indicators equations framework) is outperformed by the mixed- in the following months (adaptation from Leboeuf and frequency approaches in the respective evaluation Morel 2014). period because it showed very clear difficulty not only in predicting the cyclical declines, but also in recognizing the presence of negative growth rates. Analogously, in an in-sample nowcasting exercise, the 5. Evaluation of the models by means U-MIDAS pooling performs pretty well: it shows a of empirical comparison comparable performance with respect to the MF-VAR, although not superior one. Namely, in evaluating Empirical comparison is supposed to be a standard forecast performances, we conduct the Diebold- when evaluating different models regarding their Mariano test (Diebold and Mariano, 1995) and usefulness in regular projection rounds. Consequently, compare predictive accuracy between the model in the pages that follow we are assessing how the nowcasts (for more insight please refer to the Table A.2 results of the two modelling frameworks will in the Appendix). The results obtained indicate that generalize to the independent data set. Put differently, the U-MIDAS regressions show statistically we attempt to discover whether the relatively simpler comparable forecasting performance to that of the U-MIDAS could allow for some predictive gains over MF-VAR model. However, MF-VAR delivered better the MF-VAR model at very near term. forecasts over the first half of 2017 marked as a period Moreover, in practice, when deciding to establish of increased domestic political uncertainty. On the some model as a tool in the regular economic analysis, other hand, U-MIDAS produced better predictions besides low RMSE, model’s forecasts smooth response than the alternative during the stable times (please to news is also a very relevant issue. To this end, we are refer to Figure 1). looking for a strong evidence in improving forecasting performance in both, a “normal” period like 2016, and SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 47 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA Table 1. Overview on nowcast pooling based on twelve single indicator bridge equations,U-MIDAS regressions , MF-VAR individual nowcasts and combined forecast Combined actual GDP Bridge equations U-MIDAS MF-VAR (U-MIDAS & MF-VAR) growth hm=1 hm=1 hm=1 hm=1 2015Q4 0.82 0.14 0.62 1.46 1.20 2016Q1 0.37 0.47 0.75 -0.25 0.06 2016Q2 0.18 0.77 0.77 -0.83 -0.33 2016Q3 0.89 0.68 0.92 1.14 1.07 2016Q4 1.23 0.42 0.67 1.15 1.00 2017Q1 -2.09 0.05 -0.06 -0.36 -0.27 2017Q2 -0.88 1.70 1.88 -1.03 -0.12 2017Q3 1.72 1.17 1.41 2.45 2.12 2017Q4 2.10 -0.10 -0.10 0.51 0.32 RMSE 1.41 1.39 0.94 0.93 Source: Authors’ calculations Figure 1. Actual versus predicted quarterly GDP growth (obtained recursively, based on log difference approximation, seasonally adjusted figures) Source: Authors’ calculations Consequently, it is interesting to point out that forecast exhibit a tendency to be superior in forecast- the two approaches (i.e. U-MIDAS and MF-VAR) are in ing the turning points of the business cycle (please fact complementing, since in general, their combined see Figure 2). 48 SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA Figure 2. Actual versus combined forecast of quarterly GDP growth (obtained recursively, based on log difference approximation, seasonally adjusted figures) Source: Authors’ calculations 6. Conclusion Evaluation of the models regarding their utility in MF-VAR model delivered more accurate predictions, regular forecasting rounds should be backed-up by in times of increased uncertainty, when reliable as- an empirical comparison. Analysed from the perspec- sessments of the current situation are most needed. tive of our central research question, the study results However, this particular annotation should be taken show that there is no statistically significant difference as indicative rather than definitive, given the relatively in the forecasts produced by the two mixed-frequency short test period covering only one such episode. This approaches for the pseudo-out-of-sample period. issue is broadly aligned with one of the main limita- More precisely, the results obtained point out that tions of the study, i.e. lack of longer time series for the U-MIDAS regressions show statistically compara- some of the variables. ble forecasting performance to that of MF-VAR model The results of this comparative study may be prac- and that the two approaches are actually reciprocal, tical to institutional forecasters and economic agents, given that their combined forecast in general shows as information on where the economy is heading is a superiority in projecting the turning points of the particularly valuable. The MF-VAR and the U-MIDAS business cycle. However, in our empirical exercise, the are obvious choices for nowcasting in practice. SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 49 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA Endnotes References 1 Most of them are part of the regular NBRNM’s current Aastveit, K.A., Foroni, C. and Ravazzolo, F. 2016. Density economic analysis framework. In this regard, speak- forecasts with MIDAS models.Journal of Applied ing generally, the selection of series is based on the Econometrics, 32(4): 783–801. existingcontributions in the field literature, as it is also Abdić, A., Resić, E., Abdić, A., and Rovčanin, A. 2020. a notion in Abdic et al. (2020). In parallel, the data Nowcasting GDP of Bosnia and Herzegovina: a compari- segment containing quarterly and monthly series is drawn from four main sources: State Statistical Office, son of forecast accuracy models. South East European National Bank of the Republic of North Macedonia, Journal of Economics and Business,15 (2): 1-14. Eurostat and the European Commission. Andreou, E., Ghysels, E. and Kourtellos, A. 2013. Should Macroeconomic Forecasters Use Daily Financial Data 2 Considering their estimation simplicity and their and How?.Journal of Business & Economic Statistics, transparency, bridge equations are extensively used in policy institutions, in particular, central banks. 31(2):240-251. Andreou,  E., Gagliardini,  P., Ghysels, E. and  Rubin,  M. 2019. 3 Forecasts of the high frequency indicators within Inference in Group Factor Models With an Application the bridge equations framework are obtained by to Mixed‐FrequencyData.Econometrica,  10.3982/ using some particular time series models (best- fit - ECTA14690, 87(4):1267-1305. ting ARIMA models based on minimum AIC criterion, limiting the number of AR and MA terms to 3). The Bai, J., Ghysels, E. and Wright, J.H. 2013. State Space subsequent forecast values are aggregated and Models and MIDAS Regressions.Econometric Reviews, added into the bridge equations in order to get the 32(7):779-813. low-frequency variable forecast. Bańbura, M., Giannone, D., Modugno, M. and Reichlin, L. 2013. Now-casting and the real-time data flow, Working 4 That has been pioneered by Ghysels, Santa-Clara, and Valkanov (2004). Paper SeriesNo. 1564, European Central Bank. Barsoum, F., and Stankiewicz, S. 2013. Forecasting GDP 5 Domestic political uncertainty-related risk stands out Growth Using Mixed-Frequency Models With Switching in this year’s risk landscape. Regimes. Working Paper Series of the Department of 6 We employ the inverse value of the MSE in order to Economics 2013-10, University of Konstanz. penalise the model that is associated with greater Chiu, C.W., Eraker, B., Foerster, A.T., Kim, T.B. and Seoane, H.D. uncertainty. 2011. Estimating VAR’s Sampled at Mixed or Irregular 7 We apply quadratic loss function for the Diebold- Spaced Frequencies: A bayesian approach, Research Mariano test. Working Paper RW 11-11, Federal Reserve Bank of 8 We consider simple mean as a combination scheme - Kansas City. the most exploited method in the literature. Clements, M. and Galvão, A. 2008. Macroeconomic fore- casting with mixed frequency data: forecasting output growth in the United States. Journal of Business and Economic Statistics, 26:546–554. Clements, M. and Galvão, A. 2009. Forecasting US Output Growth Using Leading Indicators: An Appraisal Using MIDAS Models. Journal of Applied Econometrics,24(7): 1187–1206. Diebold, F. X. and Mariano,R. S. 1995. Comparing predic- tive accuracy.Journal of Business & Economic Statistics, 13:253–263. Drechsel, K. and Scheufele, R. 2012. Bottom-up or Direct? Forecasting German GDP in a Data-rich Environment, Working Papers, No 2012-16, Swiss National Bank. Duarte, C. 2014. Autoregressive augmentation of MIDAS re- gressions. Working Papers w201401, Banco de Portugal, Economics and Research Department. Ferrara, L., Marsilli, C. and Ortega, J-P. 2014. Forecasting Growth During the Great Recession: Is Financial Volatility the Missing Ingredient?.Economic Modelling, 36: 44-50,Working Paper No. 454, Banque de France. 50 SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA Foroni, C. and Marcellino, M. 2013. A survey of economet- Kuzin, V., Marcellino, M. and Schumacher, C. 2011. MIDAS vs. ric methods for mixed-frequency data. Department of Mixed frequency VAR: Nowcasting GDP in the euro-area. Economics 2013/02. European University Institute. International Journal of Forecasting, 27(2):529–542. Foroni, C. and Marcellino, M. 2014. A comparison of mixed Leboeuf, M. and Morel, L. 2014. Forecasting Short-Term frequency approaches for nowcasting Euro area mac- Real GDP Growth in the Euro Area and Japan Using roeconomic aggregates.International Journal of Unrestricted MIDAS Regressions. Discussion Papers, Forecasting, 30(3):554-568. Bank of Canada. Foroni, C., Marcellino, M. and Schumacher, C. 2011. Marsilli, C. 2014. Mixed-Frequency Modeling and Economic U-MIDAS: MIDAS regressions with unrestricted lag poly- Forecasting. Economics and Finance. Université de nomials. Discussion Paper Series 1: Economic Studies 35, Franche-Comté, 2014. English. NNT: 2014BESA2023. Deutsche Bundesbank, Research Centre. tel-01645421. Foroni, C., Marcellino, M. and Stevanovic, D. 2018. Mixed Mikosch, H. and Neuwirth, S. 2015. Real-time forecasting Frequency Models with MA Components. Deutsche with a MIDAS VAR. BOFIT Discussion Papers 13/2015, Bundesbank Discussion Paper No. 02/2018. Bank of Finland, Institute for Economies in Transition. Ghysels, E., Santa-Clara, P. and Valkanov, R. 2004. The MIDAS Mittnik, S. and Zadrozny, P.A. 2004. Forecasting Quarterly Touch: Mixed Data Sampling Regression Models. CIRANO German GDP at Monthly Intervals Using Monthly IFO Working Papers 2004s-20, CIRANO. Business Conditions Data. CESifo Working Paper Series Ghysels, E., Santa-Clara, P . and Valkanov, R. 2005. There is a risk- No. 1203, CESifo Group Munich. return trade-off after all. Journal of Financial Economics, Mariano, R. and Murasawa, Y. 2010. A Coincident Index, 76:509–548. doi:10.1016/j.jfineco.2004.03.008 Common Factors, and Monthly Real GDP, Oxford Bulletin Ghysels, E., Santa-Clara, P. and Valkanov, R. 2006. Predicting of Economics and Statistics, 72(1):27-46. volatility: getting the most out of return data sam- Schorfheide, F. and Song, D. 2015. Real-Time Forecasting pled at different frequencies. Journal of Econometrics, With a Mixed-Frequency VAR.Journal of Business & 131:59–95. Economic Statistics, 33(3):366–380. Kapetanios, G., Marcellino, M. and Petrova, K. 2018. Analysis of the most recent modelling techniques for big data with particular attention to Bayesian ones. Eurostat Statistical Working Papers, 2018 edition. SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 51 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA APPENDIX Table A.1. Employed variables and the corresponding publication lags Main releases Publishing lag Frequency Number of employees – Total –Industry 1 month monthly Turnover recorded in capital goods industries 2 months monthly Industrial production index – Total – Germany 1 month monthly Manufacture of motor vehicles, trailers and semi-trailers in EU-28 1 month monthly PPI – Exporting industries (PPI=Producer Price Index) 1 month monthly Hours worked - Construction 2 months monthly Industrial production index - Manufacture of other non-metallic mineral products 1 month monthly M2-Denar part 1 month monthly Real average monthly net-wage 2 months monthly Tourism-overnight stays 2 months monthly EC ESI–Macedonia (EC ESI=European Commission Economic Sentiment Indicator) 1 month Monthly EC ESI–Germany (EC ESI=European Commission Economic Sentiment Indicator) 1 month Monthly Gross Domestic Product at constant prices (millions of Denar) 1 quarter quarterly Notes: The publication lags reflect the number of missing values at the end of each quarter Table A.2. Diebold-Mariano test (HLN adjusted) U-MIDAS vs. MF-VAR Null hypothesis: Both forecasts have the same accuracy Accuracy Statistic <> prob > prob < prob Abs Error 0.762745 0.4675 0.7662 0.2338 Sq Error 1.208683 0.2613 0.8694 0.1306 Bridge equations vs. U-MIDAS Null hypothesis: Both forecasts have the same accuracy Accuracy Statistic <> prob > prob < prob Abs Error 1.14736 0.2844 0.8578 0.1422 Sq Error 0.321752 0.7559 0.6221 0.3779 Bridge equations vs. MF-VAR Null hypothesis: Both forecasts have the same accuracy Accuracy Statistic <> prob > prob < prob Abs Error 1.124426 0.2934 0.8533 0.1467 Sq Error 1.439139 0.1881 0.906 0.094 Source: Authors’ calculations 52 SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png South East European Journal of Economics and Business de Gruyter

Evaluation of Mixed Frequency Approaches for Tracking Near-Term Economic Developments in North Macedonia

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de Gruyter
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© 2021 Gani Ramadani et al., published by Sciendo
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2233-1999
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2233-1999
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10.2478/jeb-2021-0013
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Abstract

Aggregate demand forecasting, also known as nowcasting when it applies to current quarter assessment, is of notable interest to policy makers. This paper concentrates on the empirical methods dealing with mixed- frequency data. In particular, it focuses on the MIDAS approach and its later extension, the Bayesian MF- VAR. The two strategies are evaluated in terms of their accuracy to nowcast Macedonian GDP growth, using same monthly frequency data set. The results of this study indicate that the MIDAS regressions demonstrate comparable forecasting performance to that of MF-VAR model. Moreover, it is interesting to note that the two approaches are reciprocal, since in general, their combined forecast demonstrates clear superiority in predicting business cycle turning points. Additionally, the MF-VAR model showed higher precision in times of increased uncertainty. Keywords: MF-VAR, Bayesian estimation, MIDAS, forecast pooling, forecast evaluation JEL Classification: E37, C53 Gani Ramadani (corresponding author) Senior Advisor Monetary Policy and Research Department, 1. Introduction National Bank of the Republic of North Macedonia From their sampling frequency perspective, eco- E-mail: ramadanig@nbrm.mk nomic data vary considerably. In decision-making Address: Blvd Kuzman Josifovski Pitu 1, process, we are faced with both, the problem of data 1000 Skopje, Republic of North Macedonia frequency and the publication delay. In this regard, ORCID: https://orcid.org/0000-0001-5861-6981 the State Statistical Office of the Republic of North Macedonia, following the international practices, re- Magdalena Petrovska leases quarterly figure for Gross Domestic Product Senior Advisor (GDP) with a delay of slightly more than two months. Monetary Policy and Research Department, Having in mind that these data are a crucial asset for es- National Bank of the Republic of North Macedonia tablishing and implementing policies, their rough esti- E-mail: petrovskam@nbrm.mk mation is therefore necessary. Estimation can be based on monthly readings, such as the industrial production Vesna Bucevska, PhD volume index, external trade, value indices of turnover Full professor of Econometrics and Financial in the retail trade, etc. This whole process is known as Econometrics nowcasting. The fundamental postulate of nowcasting Faculty of Economics-Skopje, lies in exploiting the information released early and Ss. Cyril and Methodius University eventually at higher frequency than the variable of in- E-mail: vesna.bucevska@eccf.ukim.edu.mk terest so that one may get an “early estimate” before the Copyright © 2021 by the School of Economics and Business Sarajevo 43 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA official data becomes accessible (Bańbura et al., 2013). the specifications. This gives rise to concern of how to quantify em- Initially, MIDAS models have been applied in the pirically the relationships between variables sampled financial domain (see for instance Ghysels, Santa Clara, at various frequencies. The most naive approach is to and Valkanov (2006). Recently, numerous applications stick with the lowest frequency in the data, but in this involve the MIDAS approach as a forecasting plat- case a loss of possibly valuable high-frequency infor- form for quarterly GDP, like for example Clements and mation is unavoidable. However, the recent literature Galvão (2008) and Clements and Galvão (2009). Later displays a growing interest in the value added that di- additions are Foroni, Marcellino, and Schumacher rect modelling of mixed-frequency could provide. (2011); Kuzin, Marcellino and Schumacher (2011); This paper gives an overview of some of the key Drechsel and Scheufele (2012a); Andreou, Ghysels, approaches employed in the field literature to cope and Kourtellos (2013); Ferrara, Marsilli, and Ortega with mixed-frequency data: mixed-frequency Vector (2014); Duarte (2014); and Aastveit, Foroni and Autoregression (MF-VAR) in a Bayesian framework, Ravazzolo (2016), amongst others. Furthermore, launched by Schorfheide and Song (2015), as well as Foroni, Marcellino and Stevanovic (2018) show ana- the less computationally intensive counterpart, i.e. un- lytically, in Monte Carlo simulations, the relevance of restricted mixed-data sampling (U-MIDAS) approach considering the moving average (MA) component in introduced by Foroni, Marcellino, and Schumacher MIDAS and U-MIDAS models thus closing the gap in (2011). To this end, our study is exploratory and inter- the respective literature. Andreou et al. (2019), on the pretative in nature. Analogously, the primary research other hand show how the group factor context ap- challenge was to verify which of the considered ap- plies to mixed‐frequency data panels. proaches generalises better and is more capable of This study also falls into a relatively new and thus producing reliable GDP nowcasts. Therefore, two increasing body of literature on mixed frequency VAR empirically testable statements were defined: by em- models that accommodate a state space approach. ploying MF-VAR approach, accurate and efficient now - The main idea assumes reformulation of each lower casts of North Macedonia’s GDP can be acquired; by frequency series into a partially latent high frequen- employing U-MIDAS approach, accurate and efficient cy series. The Kalman filter or, the Gibbs sampler in a nowcasts of North Macedonia’s GDP can be acquired. Bayesian framework, then allow a partially latent VAR To this end, after discussing in a nutshell the two process to be estimated. See Mittnik and Zadrozny employed modelling methodologies, we proceed (2005); Kuzin, Marcellino and Schumacher (2011); Bai, comparing them in an exhaustive empirical exercise. Ghysels and Wright (2013); and Foroni and Marcellino Specifically, we revolve around comparison of the re - (2014) as a leading research on state space type MF- sulting models in terms of the proposed predictions, VAR models adopting a non-Bayesian version of using same high frequency (HF) data set. To this end, the Kalman filter. On the other hand, Mariano and the variable of interest is the Macedonian quarterly Murasawa (2010) have a pioneering contribution for a GDP growth rate. state space type MF-VAR using the expectation–maxi- A theoretical comparison of these two classes of mization (EM) algorithm, and Chiu et al. (2011) and models points out that, U-MIDAS is more parsimoni- Schorfheide and Song (2015) for MF-VARs cast in state ous than MF-VAR, and as a direct forecasting tool dis- space form using the Gibbs sampler (for greater cov- plays greater robustness to misspecification (Kuzin, erage and extension of these literature please refer to Marcellino, and Schumacher 2011). This study aims at Mikosch and Neuwirth 2015). documenting this status. Moreover, we assess wheth- er the forecast accuracy improves when combining these two models. 3. Data With respect to the potential high-frequency indi- cators to draw from, we consider a broad framework 2. Literature review of time series routinely employed in the process of Ghysels, Santa-Clara and Valkanov (2004) pio- GDP nowcasting, starting from economic sentiment neered one of the most competitive univariate tools indicators to hard data . The selection process was in- suited to handle the mixed-frequency data, i.e. the so tended to bring about the “best” subset of predictors. called mixed-data sampling method (MIDAS). Mixed- To this end, all of the variables were subject to pre- data sampling (MIDAS) models operate with time se- filtering based on vigorous one-by-one testing within ries at various frequencies. In this structure distributed the bridge equations set-up as a naïve approach to lag polynomials are employed to ensure parsimony in handle the mixed-frequency data . 44 SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA The shortlisted variables chosen from a broader set In order to specify the measurement equation, the of similar alternatives are actually those who pass the authors have to define the aggregation equation. in-sample selection based on their recent forecasting Following, Kapetanios, Marcellino and Petrova (2018), performance. In other words, the informed variable taking GDP growth as an example, the disaggregation selection procedure that we follow reflects our metrics of the quarterly GDP growth, 𝑦𝑦 , observed every 𝑡𝑡 = � � based on minimum relative forecasting errors. 3; 6; 9; …; 𝑇𝑇 , into the month-on-month GDP growth, In addition, all of the considered variables are 𝑦𝑦 , never observed, is based on the following seasonally adjusted, as well as transformed ensuring aggregation equation: their stationarity (e.g. trending variables are expressed ∗ ∗ ∗ as growth rates). The monthly data releases follow 𝑦𝑦 �𝑦𝑦 �𝑦𝑦 �𝑦𝑦 ��Λ 𝑧𝑧 � � � � � �� � � � � ��� ��� similar timing. This allows us to reproduce the same To this end, the quarterly variable is treated as the pattern of missing reading at the end of each recursive sample, so to imitate the data availability in real-time. three-month average of the monthly process. The ragged-edge overview of the dataset is presented Following Foroni and Marcellino (2013), since 𝑦𝑦 in Table A.1 of the Appendix. In addition, depending is observed only every third month, there is a need of on the publication timetable, we assume that data are a selection matrix that equals the identity matrix if 𝑡𝑡 accessible at the earliest at the month-end. corresponds to the last month of the quarter and is Furthermore, the quarterly variable that we empty otherwise. Therefore, the measurement consider, i.e., the real GDP is obtainable in the third equation can be specified as: month after the end of the referent quarter. For 𝑦𝑦 � ��� Λ 𝑧𝑧 instance, the GDP reading for 2017Q4 becomes � � � 𝑥𝑥 � � acquirable in March 2018. where M is the selection matrix. The problem of dimensionality is surpassed by introducing of a Minnesota prior that shrinks the VAR 4. Methodology coefficients toward univariate random walk 4.1. MF-VAR methodology representations. (adaptation from Kapetanios, Marcellino and Petrova 2018). The methodological explanation covering the MF-VAR model is an adaptation from Claudia Foroni’s doctoral thesis which provides a compendium of individual 4.2. The U-MIDAS approach mixed-frequency approaches along with a very intuitive understanding of the differences between MIDAS regressions are perceived as a widespread them. alternate to the multivariate state-space framework In the subsequent paragraphs, we describe the elaborated in the previous sub-section. This main characteristics of the Bayesian MF-VAR approach, econometric technique, as a very general type of ARDL following Schorfheide and Song (2015) as a most model, is based on both a regression structure and a quintessential study in the field literature. weight function which tracks the high frequency lags Namely, these authors cast a MF-VAR in state space of the regressors (Marsilli 2014). The majority of the form. Furthermore, in order to conduct Bayesian formulas and back up explanations employed in this inference for model parameters and unobserved section were adapted from Barsoum and Stankiewicz monthly variables, they make use of Markov chain (2013). Monte Carlo (MCMC) methods. Following Foroni and The elementary form of the MIDAS model Marcellino (2013), the state equation of the model has employed to get an h−step ahead forecast might be a VAR(p) representation, treating quarterly series as expressed following Clements and Galvão (2008): monthly series with missing observations, written as it ��� � follows: 𝑦𝑦 �� �� ��𝐿𝐿 �𝜃𝜃�𝑥𝑥 �� � � � � ��� Let for all 𝑡𝑡 the latent month-on-month GDP ��� ��� ����� Where ��𝐿𝐿 �𝜃𝜃� � ����𝜃𝜃�𝐿𝐿 is the growth 𝑦𝑦 and the corresponding monthly indicator ��� sum of weights assigned to K lags of the independent 𝑥𝑥 follow a VAR(p) process th variable (the lag polynomial). ����𝜃𝜃� is the k weight of the K-lag polynomial, shaped by a certain function � � � � 𝑧𝑧 �� Φ 𝑧𝑧 �� Φ �𝜐𝜐 � � � � � � ��� � of 𝜃𝜃 parameters (as for instance an exponential 𝜐𝜐 �����������Σ�� function). 𝐿𝐿 denotes the lag operator so that SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 45 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA � � ��� As we are set to do GDP forecasting, we want to 𝐿𝐿 𝑥𝑥 �𝑥𝑥 . 𝑡𝑡 is the time index for y, as a ��� ������� include autoregressive elements in the U-MIDAS lower frequency variable, while m is the time index for model. Simply introducing an AR part into the the variable with higher frequency, i.e. x. Q describes previous regression we obtain variables observed on a quarterly and M on a monthly ��� basis. 𝑦𝑦 �𝛽𝛽 ��𝑦𝑦 � �𝛽𝛽 𝑥𝑥 �� Having in mind the non-linearity of the lag � � ��� ��� ������� � ��� polynomials, the non-linear least square (NLS) is a typical estimation method for MIDAS models. Each of the U-MIDAS regressions employed to However, in some instances the form of the lag predict Macedonian real GDP growth, for up to one polynomial may be overly restrictive in comparison quarter ahead (i.e., a “nowcast” of the current quarter) with the underlying data generating process. Thus a use single indicator. With only one indicator in each model with no restrictions on the weights of the lag representation and a restricted number of lags, the polynomial was launched by Foroni, Marcellino and coefficients in equation above can be estimated Schumaher (2011) (adaptation from Bersoum and without internalising the degrees of freedom problem Stankiewicz 2013). These authors advanced a new (adaptation from Leboeuf and Morel 2014). parametrization scheme for the MIDAS based on a Our U-MIDAS specification includes 3 lags of the linearization of the distributed lag function called monthly variables in total, extending over the quarter unrestricted MIDAS (U-MIDAS), where all the for which we have the last reading of real GDP growth parameters are estimated using OLS. The U-MIDAS as well as data along the first quarter to forecast, model exploit a linear lag polynomial that can be provided they are obtainable. As the separate monthly expressed as (adaptation from Marsilli 2014): readings of the regressors are published over a quarter, the model representation changes slightly ��� � (adaptation from Leboeuf and Morel 2014). 𝑦𝑦 �𝛽𝛽 � �𝛽𝛽 𝑥𝑥 �� � ��� ������� � In what follows, we provide a real-time ��� demonstration of the U-MIDAS model for the The dependent variable y is represented by an Macedonian GDP as an analogy to the example of the equation that contains an intercept 𝛽𝛽 and a lag U-MIDAS for euro-area GDP employed by Leboeuf and polynomial weighted by parameters 𝛽𝛽 . To this end, ��� Morel (2014). all the parameters 𝛽𝛽 of this polynomial need to be ( 1) ( 2) ( 3) ��� 𝑀𝑀 𝑀𝑀 𝑀𝑀 Let 𝑋𝑋t , 𝑋𝑋t and 𝑋𝑋t be monthly variables in estimated as no structure is set on the shape of the the first, second and third month of quarter 𝑡𝑡 , for weights of the lag polynomial (adaptation from ( ) 𝑄𝑄 which we are generating a nowcast of 𝑡𝑡𝑌𝑌 (real GDP Bersoum and Stankiewicz 2013). ( 1) 𝑀𝑀 growth in quarter 𝑡𝑡 ). In other words, 𝑋𝑋t is a quarterly The lags of the explanatory variable are time series containing all first monthly values of the represented by the measure 𝑚𝑚 , where m = 3 specifies ( 2) 𝑀𝑀 variable X for each quarter over past horizon. 𝑋𝑋t is a the number of observations of the higher-frequency quarterly time series encompassing all second indicator (e.g. monthly variable 𝑥𝑥 ) for each observation monthly values of a variable X for each quarter over of the lower-frequency variable (e.g. quarterly variable ( 3) 𝑀𝑀 past horizon. 𝑋𝑋t is a quarterly time series including 𝑦𝑦 ). That is, if e.g. 𝑦𝑦 is the reading of the dependent all third monthly values of the variable X for each variable for the first quarter of 2017 (March 2017), then quarter over past horizon (adaptation from Leboeuf 𝑥𝑥 represents the observation of the explanatory ��� and Morel 2014). variable for December 2016 (1 quarter before), 𝑥𝑥 ����� For the U-MIDAS approach now we have monthly for November 2016 (4 months before), whereas variables to be transformed into quarterly variables. As 𝑥𝑥 for January 2016 (14 months before) and so ������ in one quarter there are 3 months, each monthly forth (Bersoum and Stankiewicz 2013). indicator will be transformed into 3 variables with One of the obvious setbacks of the U-MIDAS lies in quarterly frequency. ( 1) the fact that, when the discrepancy in frequencies 𝑀𝑀 𝑋𝑋t - only data for months 1,4,7,10 are taken ( 2) between the variables in the model is large, its 𝑀𝑀 𝑋𝑋t - only data for months 2,5,8,11 are taken ( 3) performance plunges significantly due to the rapid 𝑀𝑀 𝑋𝑋t - only data for months 3,6,9,12 are taken increase of the number of parameters. To this end, this Then, just for an illustration, if X has a 1 month method do not fit all kinds of empirical applications publication delay: (adaptation from Bersoum and Stankiewicz 2013). • In the first month, the model for Y nowcasting However, for many macroeconomic analyses the use contains a constant, one lag of Y and 3 months of data of the U-MIDAS model may be advantageous. on variable X 46 SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA ( ) ( 1) ( ) 5 𝑄𝑄 𝑀𝑀 𝑄𝑄 𝑌𝑌 =𝛽𝛽 +𝜑𝜑 𝑌𝑌 + a crisis year like 2017 . Therefore, the model is 𝑡𝑡 1 1 𝑡𝑡 −1 ( 1) ( 2) ( 3) ( 1) 𝑀𝑀 𝑀𝑀 𝑀𝑀 𝑀𝑀 𝛾𝛾 𝑋𝑋 +𝛾𝛾 𝑋𝑋 +𝛾𝛾 𝑋𝑋 +𝜔𝜔 evaluated based on both, its ability to approximate 2,1 𝑡𝑡 −1 2,2 𝑡𝑡 −1 2,3 𝑡𝑡 −1 𝑡𝑡 history, and its usefulness to capture the turning i.e. months 10,11 & 12 points. • In the second month, the representation is the same as in the first one, but the first month of the ( 1) 𝑀𝑀 current quarter (𝑋𝑋t ) is added to the model 5.1. Main findings from the empirical exercise specification: ( ) ( 2) ( ) ( 1) ( 2) ( This section outlines the average performance of the 𝑄𝑄 𝑀𝑀 𝑄𝑄 𝑀𝑀 𝑀𝑀 𝑀𝑀 𝑌𝑌 =𝛽𝛽 +𝜑𝜑 𝑌𝑌 +𝛾𝛾 𝑋𝑋 +𝛾𝛾 𝑋𝑋 +𝛾𝛾 𝑋𝑋 𝑡𝑡 1 1 𝑡𝑡 −1 1,1 𝑡𝑡 2,2 𝑡𝑡 −1 2,3 𝑡𝑡 −1 3) ( 2) employed modelling frameworks. To this end, we 𝑀𝑀 +𝜔𝜔 𝑡𝑡 report the RMSE performance of the MF-VAR for i.e. months, 1, 11 & 12 nowcasting quarterly GDP growth at one period ahead horizon, against the U-MIDAS pooled forecasts. In • In the third, two months of the current quarter (𝑡𝑡 ) addition, we report the combination forecast of these and one month of the previous quarter (𝑡𝑡 -1) are two models using inverse mean square error (IMSE) included (3 months of the variable X in total): weighting scheme . The RMSE result for the ( ) ( 3) ( ) ( 1) ( 2) 𝑄𝑄 𝑀𝑀 𝑄𝑄 𝑀𝑀 𝑀𝑀 𝑌𝑌 =𝛽𝛽 +𝜑𝜑 𝑌𝑌 +𝛾𝛾 𝑋𝑋 +𝛾𝛾 𝑋𝑋 + 𝑡𝑡 1 1 𝑡𝑡 −1 1,1 𝑡𝑡 1,2 𝑡𝑡 benchmark bridge equations model is presented as ( 3) ( 3) 𝑀𝑀 𝑀𝑀 𝛾𝛾 𝑋𝑋 +𝜔𝜔 2,3 𝑡𝑡 −1 𝑡𝑡 well. The results aligned with the bridge equations i.e. months 1, 2 & 12 model, U-MIDAS pooled forecasts and the MF-VAR ( ) 𝑀𝑀𝑀𝑀 where 𝜔𝜔 𝑡𝑡 for 𝑀𝑀 =1,2 or 3, is the error term of the individual forecasts, as well as their combination regressions. forecasts are summarized in the Table 1 below. The Also, in U-MIDAS, the weights assigned to each results, are obtained recursively, based on log month are completely data-driven, reflecting the difference approximation as well as on seasonally concept that each month of data has different adjusted figures, for the evaluation sample 2015 Q4 - importance in forecasting GDP. U-MIDAS has another 2017 Q4 (third months of the quarter). appealing feature fo forecasting in short-term horizon. Considering the fact that we evaluate the models Namely, unlike bridge equations approach, for not only statistically but also regarding their ability for instance, it does not necessitate a forecast of missing pseudo out-of-sample projecting of turning points, we months and consequently does not require any might say that the benchmark model, (i.e. bridge assumptions about the development of the indicators equations framework) is outperformed by the mixed- in the following months (adaptation from Leboeuf and frequency approaches in the respective evaluation Morel 2014). period because it showed very clear difficulty not only in predicting the cyclical declines, but also in recognizing the presence of negative growth rates. Analogously, in an in-sample nowcasting exercise, the 5. Evaluation of the models by means U-MIDAS pooling performs pretty well: it shows a of empirical comparison comparable performance with respect to the MF-VAR, although not superior one. Namely, in evaluating Empirical comparison is supposed to be a standard forecast performances, we conduct the Diebold- when evaluating different models regarding their Mariano test (Diebold and Mariano, 1995) and usefulness in regular projection rounds. Consequently, compare predictive accuracy between the model in the pages that follow we are assessing how the nowcasts (for more insight please refer to the Table A.2 results of the two modelling frameworks will in the Appendix). The results obtained indicate that generalize to the independent data set. Put differently, the U-MIDAS regressions show statistically we attempt to discover whether the relatively simpler comparable forecasting performance to that of the U-MIDAS could allow for some predictive gains over MF-VAR model. However, MF-VAR delivered better the MF-VAR model at very near term. forecasts over the first half of 2017 marked as a period Moreover, in practice, when deciding to establish of increased domestic political uncertainty. On the some model as a tool in the regular economic analysis, other hand, U-MIDAS produced better predictions besides low RMSE, model’s forecasts smooth response than the alternative during the stable times (please to news is also a very relevant issue. To this end, we are refer to Figure 1). looking for a strong evidence in improving forecasting performance in both, a “normal” period like 2016, and SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 47 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA Table 1. Overview on nowcast pooling based on twelve single indicator bridge equations,U-MIDAS regressions , MF-VAR individual nowcasts and combined forecast Combined actual GDP Bridge equations U-MIDAS MF-VAR (U-MIDAS & MF-VAR) growth hm=1 hm=1 hm=1 hm=1 2015Q4 0.82 0.14 0.62 1.46 1.20 2016Q1 0.37 0.47 0.75 -0.25 0.06 2016Q2 0.18 0.77 0.77 -0.83 -0.33 2016Q3 0.89 0.68 0.92 1.14 1.07 2016Q4 1.23 0.42 0.67 1.15 1.00 2017Q1 -2.09 0.05 -0.06 -0.36 -0.27 2017Q2 -0.88 1.70 1.88 -1.03 -0.12 2017Q3 1.72 1.17 1.41 2.45 2.12 2017Q4 2.10 -0.10 -0.10 0.51 0.32 RMSE 1.41 1.39 0.94 0.93 Source: Authors’ calculations Figure 1. Actual versus predicted quarterly GDP growth (obtained recursively, based on log difference approximation, seasonally adjusted figures) Source: Authors’ calculations Consequently, it is interesting to point out that forecast exhibit a tendency to be superior in forecast- the two approaches (i.e. U-MIDAS and MF-VAR) are in ing the turning points of the business cycle (please fact complementing, since in general, their combined see Figure 2). 48 SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA Figure 2. Actual versus combined forecast of quarterly GDP growth (obtained recursively, based on log difference approximation, seasonally adjusted figures) Source: Authors’ calculations 6. Conclusion Evaluation of the models regarding their utility in MF-VAR model delivered more accurate predictions, regular forecasting rounds should be backed-up by in times of increased uncertainty, when reliable as- an empirical comparison. Analysed from the perspec- sessments of the current situation are most needed. tive of our central research question, the study results However, this particular annotation should be taken show that there is no statistically significant difference as indicative rather than definitive, given the relatively in the forecasts produced by the two mixed-frequency short test period covering only one such episode. This approaches for the pseudo-out-of-sample period. issue is broadly aligned with one of the main limita- More precisely, the results obtained point out that tions of the study, i.e. lack of longer time series for the U-MIDAS regressions show statistically compara- some of the variables. ble forecasting performance to that of MF-VAR model The results of this comparative study may be prac- and that the two approaches are actually reciprocal, tical to institutional forecasters and economic agents, given that their combined forecast in general shows as information on where the economy is heading is a superiority in projecting the turning points of the particularly valuable. The MF-VAR and the U-MIDAS business cycle. However, in our empirical exercise, the are obvious choices for nowcasting in practice. SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 49 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA Endnotes References 1 Most of them are part of the regular NBRNM’s current Aastveit, K.A., Foroni, C. and Ravazzolo, F. 2016. Density economic analysis framework. In this regard, speak- forecasts with MIDAS models.Journal of Applied ing generally, the selection of series is based on the Econometrics, 32(4): 783–801. existingcontributions in the field literature, as it is also Abdić, A., Resić, E., Abdić, A., and Rovčanin, A. 2020. a notion in Abdic et al. (2020). In parallel, the data Nowcasting GDP of Bosnia and Herzegovina: a compari- segment containing quarterly and monthly series is drawn from four main sources: State Statistical Office, son of forecast accuracy models. South East European National Bank of the Republic of North Macedonia, Journal of Economics and Business,15 (2): 1-14. Eurostat and the European Commission. Andreou, E., Ghysels, E. and Kourtellos, A. 2013. Should Macroeconomic Forecasters Use Daily Financial Data 2 Considering their estimation simplicity and their and How?.Journal of Business & Economic Statistics, transparency, bridge equations are extensively used in policy institutions, in particular, central banks. 31(2):240-251. Andreou,  E., Gagliardini,  P., Ghysels, E. and  Rubin,  M. 2019. 3 Forecasts of the high frequency indicators within Inference in Group Factor Models With an Application the bridge equations framework are obtained by to Mixed‐FrequencyData.Econometrica,  10.3982/ using some particular time series models (best- fit - ECTA14690, 87(4):1267-1305. ting ARIMA models based on minimum AIC criterion, limiting the number of AR and MA terms to 3). The Bai, J., Ghysels, E. and Wright, J.H. 2013. State Space subsequent forecast values are aggregated and Models and MIDAS Regressions.Econometric Reviews, added into the bridge equations in order to get the 32(7):779-813. low-frequency variable forecast. Bańbura, M., Giannone, D., Modugno, M. and Reichlin, L. 2013. Now-casting and the real-time data flow, Working 4 That has been pioneered by Ghysels, Santa-Clara, and Valkanov (2004). Paper SeriesNo. 1564, European Central Bank. Barsoum, F., and Stankiewicz, S. 2013. Forecasting GDP 5 Domestic political uncertainty-related risk stands out Growth Using Mixed-Frequency Models With Switching in this year’s risk landscape. Regimes. Working Paper Series of the Department of 6 We employ the inverse value of the MSE in order to Economics 2013-10, University of Konstanz. penalise the model that is associated with greater Chiu, C.W., Eraker, B., Foerster, A.T., Kim, T.B. and Seoane, H.D. uncertainty. 2011. 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Autoregressive augmentation of MIDAS re- gressions. Working Papers w201401, Banco de Portugal, Economics and Research Department. Ferrara, L., Marsilli, C. and Ortega, J-P. 2014. Forecasting Growth During the Great Recession: Is Financial Volatility the Missing Ingredient?.Economic Modelling, 36: 44-50,Working Paper No. 454, Banque de France. 50 SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA Foroni, C. and Marcellino, M. 2013. A survey of economet- Kuzin, V., Marcellino, M. and Schumacher, C. 2011. MIDAS vs. ric methods for mixed-frequency data. Department of Mixed frequency VAR: Nowcasting GDP in the euro-area. Economics 2013/02. European University Institute. International Journal of Forecasting, 27(2):529–542. Foroni, C. and Marcellino, M. 2014. A comparison of mixed Leboeuf, M. and Morel, L. 2014. 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Analysis of the most recent modelling techniques for big data with particular attention to Bayesian ones. Eurostat Statistical Working Papers, 2018 edition. SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021 51 EVALUATION OF MIXED FREQUENCY APPROACHES FOR TRACKING NEAR-TERM ECONOMIC DEVELOPMENTS IN NORTH MACEDONIA APPENDIX Table A.1. Employed variables and the corresponding publication lags Main releases Publishing lag Frequency Number of employees – Total –Industry 1 month monthly Turnover recorded in capital goods industries 2 months monthly Industrial production index – Total – Germany 1 month monthly Manufacture of motor vehicles, trailers and semi-trailers in EU-28 1 month monthly PPI – Exporting industries (PPI=Producer Price Index) 1 month monthly Hours worked - Construction 2 months monthly Industrial production index - Manufacture of other non-metallic mineral products 1 month monthly M2-Denar part 1 month monthly Real average monthly net-wage 2 months monthly Tourism-overnight stays 2 months monthly EC ESI–Macedonia (EC ESI=European Commission Economic Sentiment Indicator) 1 month Monthly EC ESI–Germany (EC ESI=European Commission Economic Sentiment Indicator) 1 month Monthly Gross Domestic Product at constant prices (millions of Denar) 1 quarter quarterly Notes: The publication lags reflect the number of missing values at the end of each quarter Table A.2. Diebold-Mariano test (HLN adjusted) U-MIDAS vs. MF-VAR Null hypothesis: Both forecasts have the same accuracy Accuracy Statistic <> prob > prob < prob Abs Error 0.762745 0.4675 0.7662 0.2338 Sq Error 1.208683 0.2613 0.8694 0.1306 Bridge equations vs. U-MIDAS Null hypothesis: Both forecasts have the same accuracy Accuracy Statistic <> prob > prob < prob Abs Error 1.14736 0.2844 0.8578 0.1422 Sq Error 0.321752 0.7559 0.6221 0.3779 Bridge equations vs. MF-VAR Null hypothesis: Both forecasts have the same accuracy Accuracy Statistic <> prob > prob < prob Abs Error 1.124426 0.2934 0.8533 0.1467 Sq Error 1.439139 0.1881 0.906 0.094 Source: Authors’ calculations 52 SOUTH EAST EUROPEAN JOURNAL OF ECONOMICS AND BUSINESS, VOLUME 16 (2) 2021

Journal

South East European Journal of Economics and Businessde Gruyter

Published: Dec 1, 2021

Keywords: MF-VAR; Bayesian estimation; MIDAS; forecast pooling; forecast evaluation; E37; C53

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