Predictor Preselection for Mixed‐Frequency Dynamic Factor Models: A Simulation Study With an Empirical Application to GDP Nowcasting

Franjic, Domenic; Schweikert, Karsten Article — Published Version Predictor Preselection for Mixed‐Frequency Dynamic Factor Models: A Simulation Study With an Empirical Application to GDP Nowcasting Journal of Forecasting Provided in Cooperation with: John Wiley & Sons Suggested Citation: Franjic, Domenic; Schweikert, Karsten (2024) : Predictor Preselection for Mixed‐ Frequency Dynamic Factor Models: A Simulation Study With an Empirical Application to GDP Nowcasting, Journal of Forecasting, ISSN 1099-131X, Wiley, Hoboken, NJ, Vol. 44, Iss. 2, pp. 255-269, https://doi.org/10.1002/for.3193 This Version is available at: https://hdl.handle.net/10419/319320 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. http://creativecommons.org/licenses/by/4.0/ Journal of Forecasting, 2025; 44:255–269 https://doi.org/10.1002/for.3193 255 Journal of Forecasting RESEARCH ARTICLE OPEN ACCESS Predictor Preselection for MixedFrequency Dynamic Factor Models: A Simulation Study With an Empirical Application to GDP Nowcasting DomenicFranjic | KarstenSchweikert Core Facility Hohenheim and Institute of Economics, University of Hohenheim, Stuttgart, Germany Correspondence: Domenic Franjic ([email protected]) Received: 17 October 2023 | Revised: 20 June 2024 | Accepted: 20 August 2024 Funding: The authors received no specific funding for this work. Keywords: elastic net | highdimensional | softthresholding | targeted predictors | variable selection ABSTRACT We investigate the performance of dynamic factor model nowcasting with preselected predictors in a mixedfrequency setting. The predictors are selected via the elastic net as it is common in the targeted predictor literature. A simulation study and an application to empirical data are used to evaluate different strategies for variable selection, the influence of tuning parameters, and to determine the optimal way to handle mixedfrequency data. We propose a novel crossvalidation approach that connects the preselection and nowcasting step. In general, we find that preselecting provides more accurate nowcasts compared with the benchmark dynamic factor model using all variables. Our newly proposed crossvalidation method outperforms the other specifications in most cases. Jel classification: C32, C53, E37. 1 | Introduction The mixedfrequency dynamic factor model (DFM) has become a workhorse model for macroeconomic forecasting and nowcasting (Bok etal. 2018; Giannone, Reichlin, and Small2008). Through a combination of factor analysis and Kalman smoothing (Doz, Giannone, and Reichlin2011; Stock and Watson2002a,2002b), the model can handle big data sets,1 constructed from mixedfrequency predictors, while also exploiting the often shorter publication lags of the predictor variables. This is especially useful when forecasting a not yet released lower frequency variable, often current quarter GDP, with the help of partially available quarterly and monthly economic indicators (see, for example, Bańbura etal. 2013; Giannone, Reichlin, and Small2008). By construction, the framework invites for the use of increasingly larger data sets to include predictors from all sectors of the economy. However, concerns have been raised that the resulting factors are less useful for forecasting particularly when the idiosyncratic errors are crosscorrelated (Boivin and Ng 2006). Therefore, Bai and Ng(2008) propose to employ a set of targeted predictors (TPs) for the factor analysis. More specifically, predictors are preselected using the elastic net (EN) before the estimation of a factor model and the construction of a forecast.2 The concept is extended to a mixedfrequency nowcasting framework by Bessec(2013) and Siliverstovs(2017). The literature provides ambiguous results on whether the TP approach can substantially improve the forecasting accuracy (Bulligan, Marcellino, and Venditti 2015; Castle, Clements, and Hendry2013; Eickmeier and Ng2011; Kim and Swanson 2014, 2018). Particularly, it is unknown how well the TP approach performs in mixedfrequency data structures that are common in empirical applications (Bańbura et al. 2013). In previous studies, the preselection step of the TP approach is mostly implemented as originally proposed by Bai This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2024 The Author(s). Journal of Forecasting published by John Wiley & Sons Ltd. Journal of Forecasting, 2025 and Ng(2008), where the number of variables aimed to be extracted is set to 30 (see, e.g., Bai and Ng2008; Eickmeier and Ng2011; Kopoin, Moran, and Pare2013; Schumacher2010). This is in contrast to the more common approach in the literature on penalized regressions, where the tuning parameters are crossvalidated using a part of the sample (Hastie, Tibshirani, and Wainwright2015; Zou and Hastie2005; Zou, Hastie, and Tibshirani2007). Consequently, the following issues in terms of model specification arise: It is unclear which algorithm should be used to solve the EN problem. Available choices are, for example, coordinate descent or LARSEN. Although the minimizer of the EN objective function is theoretically unique, the corresponding parameter vector, and hence the placement and number of zeros, is generally not unique. Depending on this choice, it must be decided whether the tuning parameters are determined based on (i) crossvalidation (CV)3 or (ii) the maximum number of predictor variables to be selected from the full panel of predictor variables. Then, there is the additional question of how to handle mixedfrequency data before feeding it into the EN. The current literature does not provide a conclusive statement on the performance of TP nowcasting. Only few studies report evidence for TP models to substantially improve over benchmark models when applied to different real world data sets (see, e.g., Girardi, Golinelli, and Pappalardo 2017; Kopoin, Moran, and Pare2013; Schumacher2010). Instead, most studies report mixed results (Bulligan, Marcellino, and Venditti2015; Castle, Clements, and Hendry2013; Eickmeier and Ng2011; Kim and Swanson2014,2018). A reason for this could lie in suboptimal specification of the preselection step. The current literature lacks studies that evaluate the influence of important tuning parameters and guidelines on how these parameters should be validated. Most of the research is conducted by comparing the TP performance against other models in selected empirical data sets without analyzing in detail how the tuning of the preselection step influences the nowcasting performance. When turning to the issue of handling mixedfrequency data in the TP framework, the literature provides even less guidance for applied researchers. To the knowledge of the authors, there is no largescale simulation study that investigates the abovementioned issues exhaustively. The paper at hand thus aims to contribute to the literature in the following way: (i) It is investigated whether any of the TP specifications can improve the nowcast accuracy of different benchmark models in a controlled mixedfrequency setting, and (ii) we study the effects that the choice of the tuning parameters in the preselection step has on the overall model performance. Particularly, we compare the performance of data driven specifications against selecting a fixed number of predictors. (iii) We propose a new TP CV strategy based on prior nowcast errors that seems to perform well compared with the existing approaches. Additionally, we assess the predictive accuracy of those methods for empirical data sets that differ in the number of predictors and spatial resolution.4 Our findings from the simulation study and the empirical application allow for multiple conclusions. Most notably, we find that for each DGP and real data set investigated, at least one TP specification can improve the nowcast accuracy compared with the benchmark models. Selecting a fixed number of predictors performs reasonably well, particularly if the data set contains many irrelevant variables. However, we can show that the TP models with a data driven preselection step frequently outperform the models with a fixed number of selected predictors. The data driven (or crossvalidated) models work well in misspecified or noisy systems and achieve the best results in the empirical application. The proposed TP specification that links the preselecting and nowcasting steps by crossvalidating based on the nowcast error yields the most consistent results, outperforming the benchmark DFM in controlled settings and in our empirical application. The remainder of this paper is organized as follows. Section 2 discusses the necessary basics for DFM nowcasting, followed by an outline of the model specifications and their implementation. Section3 describes the setup and reports the results from our simulation study. In Section4, the empirical data sets are introduced and the relative performance of the estimators are evaluated in a real world context. Section5 summarizes the findings of this study. 2 | Methodology 2.1 | DFM In the following, we briefly describe the mixedfrequency DFMs estimated with the twostep procedure proposed by Giannone, Reichlin, and Small(2008) and Doz, Giannone, and Reichlin (2011). Initially, we consider a vector of stationary monthly variables zt ={ zh,t } H h=1 and a DFM characterized by the following equations: Further, if allowing for missing data at the right end of the time index due to different publication lags, socalled ragged edges, it is assumed that Here, f� t ={f r,t } R r=1 is the vector of static factors at time t , and 𝚲 ={𝜆h,r} H,R h=1,r=1 is the factor loadings matrix. 𝝃t ={ 𝜉h,t } H h=1 is the idiosyncratic error vector at time t with diagonal covariance matrix 𝚺𝝃 . The matrix 𝚼 ={𝜐r , q} R,Q r=1,q=1 has full rank Q , and the vector of primitive shocks 𝝐 t={𝜖q,t} Q q=1 is assumed to be a white noise process with zero mean and covariance matrix IQ . Furthermore, the roots of the lag polynomial 𝚽P( 𝕃 ) , where 𝚽p ={𝜙 r,s,p } R,R r=1,s=1 , for all p=1…,P , are assumed to lie outside the unit circle. The mixedfrequency structure of the DFM is implemented according to Mariano and Murasawa(2003). We assume that the target variable, for example, log GDP, is integrated of order one. Further, we assume that quarterly observations are recorded (1) zt=𝚲ft+𝝃t,𝚽 P (𝕃)ft=𝚼𝝐t 𝔼(𝝃t𝝃� t)=𝚺𝝃=diag(𝜎2 1,𝝃,…,𝜎2 H,𝝃) 𝔼 (𝝃 t 𝝃� t−s )=0, ∀t,s>0, 𝔼(𝝃 t( 𝚼𝝐 t−w) �)=0, ∀w,t . (2) 𝜎 2 h,𝝃= { 𝜎2 h>0, if zh,tis available ∞, otherwise . 256 in the last month of the quarter. xt,q denotes a quarterly variable, and xt−tm,m , for tm=0,1∕3,2∕3 , is the corresponding latent monthly variable at the last, second, and first month of quarter t . Analogously to our empirical application, we use log GDP for our illustration. Since log GDP is a flow variable, the quarterly observations lngdpt,q are related to unobserved monthly observations lngdpt,m via and it holds for the quarterly growth rates, Δ3ln gdpt,q = ln gdpt,q − ln gdpt − 1,q , that Equation(4) expresses the quarterly growth rates as a function of the monthly growth rates and thereby provides us with the aggregation weights for the mixedfrequency DFM.5 As part of the model selection process, we determine the number of common factors by optimizing the information criterion provided in Bai and Ng(2002). To ensure some form of dimensionality reduction, the maximum value for  R to be considered is min(⌊0.5N∗ m⌋, 15) for the simulated data and min(⌊0.5N∗ m⌋, 5) for the real world data, where N∗ m is the number of monthly TPs and ⌊ ⋅ ⌋ is the floor function. The number of primitive shocks Q is chosen to be estimated using the procedure in Bai and Ng(2007). For the order of the VAR processes of the common factor, we follow the convention used in Marcolino de Mattos etal.(2019), where a value for  P is retrieved by computing the AIC, BIC, HQC, and final prediction error and choosing the most parsimonious model indicated by these measures. 2.2 | Preparing MixedFrequency Data for the Preselection Step Since the EN is not able to handle mixedfrequency data sets automatically, it requires a preliminary step creating a balanced set of variables. The TP literature provides two ideas in this regard. The first approach proposed by Siliverstovs(2017) is a skipsampling or blocking approach often used in MIDAS models. In this case, the monthly observations of a predictor variable are transformed into three quarterly variables, thereby aligning the frequencies of the quarterly variable of interest and the monthly predictors. For example, unemployment rates as a monthly predictor variable is transformed in such a way, tracking the unemployment rate at the first, second, and third month of a quarter. Skipsampling the monthly variables has some obvious drawbacks. The complete data matrix Xs is of dimension ((Nq+3Nm)×𝜏) , where 𝜏<T is the point in time for which all predictors are observed. Given the number of monthly observations, Nm , skipsampling thus increases the dimensionality of the problem considerably, which can lead to high computational costs. Moreover, since the variables are highly correlated, increasing the overall number of variables also increases the likelihood of including irrelevant variables. Alternatively, Bessec (2013) aggregates the monthly variables into pseudo quarterly indicators. The aggregation approach has the advantage that the dimensionality of the problem is not further increased at the cost of losing some information from the higher frequency predictors. This study differs from Bessec (2013) in that the monthly variables are not averaged over a quarter. Instead, we follow the ideas of Mariano and Murasawa(2003) and aggregate the monthly variables using the geometric mean of the three monthly variables of each quarter.6 Hence, the aggregation scheme in this step is conceptually similar to the one used in the mixedfrequency DFM. 2.3 | Preselection with the EN The EN was first introduced in Zou and Hastie(2005) and represents a combination of the Ridge and LASSO regularization. To solve the EN problem, we consider two popular algorithms, namely, LARSEN (Zou and Hastie2005) and coordinate descent (Friedman, Hastie, and Tibshirani2010), and put special emphasis on the tuning parameters for these algorithms.7 Consider the balanced vintage 𝜏 . For the variable selection step, let the target variable be the quarterly time series with index one and define y:=x1,q={x1,t,q}𝜏 t=1∈X𝜏,q . Let X ={xn,t} M,𝜏 n=1,t=1 , a samefrequency predictor matrix, be retrieved from the predictors in 𝜏 by one of the methods described above, where M is either equal to N−1 or Nq+3Nm−1 depending on whether aggregation or skipsampling is used. The EN problem is defined as where 𝜷 ={𝛽 n } M n=1 is the coefficient vector. The parameters l1 and l2 are, respectively, the LASSO (Tibshirani1996) and Ridge (Hoerl and Kennard1988) tuning parameter. It is also common to define 𝛼 = l 1 (l1+l2) and recast(5) into Here, 𝛼∈[0,1] is a mixing, or weight, parameter between the two size constraints. The first algorithm is referred to as the LARSEN algorithm (Efron et al. 2004; Zou and Hastie 2005) and is most commonly used in the TP literature (see, e.g., Bai and Ng2008; Bessec2013; Siliverstovs2017). Despite its popularity in the TP literature, the procedure has a potential drawback when used in a purely data driven framework. Eventually, at some point along the solution path, each variable moves into the active set. For larger data sets, this requires more steps to calculate the full solution path, and solving the problem becomes computationally costly. This is of special concern with respect to skipsampling, where the number of variables in the model matrix is drastically increased. (3) ln gdp t,q= 1 3 (ln gdpt,m+ln gdpt−1∕3,m+ln gdpt−2∕3,m) , (4) Δ 3ln gdpt,q= 1 3Δln gdpt,m+ 2 3Δln gdpt−1∕3,m+Δln gdpt−2∕3, m +2 3 Δln gdpt−1,m+1 3 Δln gdpt−4∕3,m. (5) min�‖ y−X � 𝜷 ‖2 +l 1‖ 𝜷 ‖1 +l 2‖ 𝜷 ‖2�, (6) min�‖ y−X�𝜷 ‖ 2+l � 𝛼 ‖ 𝜷 ‖ 1+ (1 − 𝛼) 2‖ 𝜷 ‖ 2 ��. 257 An alternative algorithm, the socalled coordinate descent, has been proposed by Friedman, Hastie, and Tibshirani(2010). The procedure has the advantage that it is generally faster than the LARSEN approach when applied to large cross sections. It might thus be possible to use coordinate descent in the TP framework to achieve similar forecasting error improvements with less computational costs compared to the LARSEN algorithm. The EN tuning parameters need to be determined by the researcher and several strategies are offered in the literature. For the coordinate descent algorithm, we follow the default settings of the glmnet package by Friedman, Hastie, and Tibshirani (2010). For the LARSEN algorithm, the l2 grid is chosen to be a sequence of 99 logarithmically spaced values between ln(1.001) and ln(10) , where 0 is added to incorporate the pure LASSO fit. The maximum number of steps k is set to the number of variables of the given data set.8 For CV, an expanding window time series CV procedure according to Hyndman and Athanasopoulos ((2018), Ch. 5.10) is implemented. Different to the classical CV approach applied to i.i.d. data, this specific approach preserves the timeseries structure of the underlying observations. For the TP specification with fixed parameters l2 and k , we use the values reported by Bai and Ng(2008), that is, l2=0.25 and k=30 . This specification is often found in empirical studies.9 2.4 | Model Specifications We begin with a description of the first TP specification family that uses information contained in the data to tune the EN parameters and therefore guide the variable selection process. The main intuition behind these data driven parameterizations is that the TP models might be more effective if they were specifically trained to the underlying data instead of relying on a fixed number of TPs. Furthermore, it must be assumed that in a real world scenario, the set of variables leading to the smallest nowcasting error might be timevarying. The data driven specifications thus provide a more flexible framework in this regard when compared to the fixed value approach. To construct the nowcast at time T , we first build the balanced panel with variables in X 𝜏 ,q and X𝜏,m by either aggregating or skipsampling. As above, 𝜏<T is the most recent date for which the panel 𝝉 is balanced. Second, we use the variables contained in X𝜏 and the target variable y𝜏 to solve the EN problem either via LARSEN or coordinate descent. In both cases, we validate the parameters of the EN via timeseries CV using a onestepahead forecast. Then,  ∗ T= { X∗ q,T,X∗ m,T } is the vintage of TPs constructed from variables for which the solution of the EN problem in the previous step has indicated nonzero coefficients. Now, we can use ∗ T to construct a nowcast of yT using the mixedfrequency DFM. This is done using a repeated variable selection step with an expanding window prior to each nowcast.10 Note that in total, there are four specifications under investigation for this TP specification family. For simplicity, we employ the following terminology. For each specification, the first part of its name is either an “A- ” if the mixedfrequency data is aggregated or an “S- ” if it is skipsampled. The following part indicates whether the EN is solved via coordinate descend (“CD- ”) or the LARSEN algorithm (“LE- ”). This is followed by “TPN” for a target predictor nowcast.11 In addition to the family of data driven models, the fixed value procedures are investigated. Restricting the model to always selecting a fixed number of predictors might lead to improvements in terms of nowcasting performance by way of a more stable variable selection step. The terminology above is also applied to the two fixed value specifications. The skipsampling approach is denoted by “SFTPN” and aggregation is denoted by “AFTPN.” We also consider two new types of specifications that we denote by nowcast validated TP specifications. In these data driven specifications, the final validation of the set of TPs is not achieved by the EN based on an insample fit but rather it is based on the nowcasting error of the previous quarter. The main intuition is that the EN eventually leads to a set of TPs retrieved via a linear projection of the target variable onto the predictor space. However, factor models are used in a second step to link the predictors to the variable of interest, and a set of predictors leading to the lowest nowcasting error may not generally coincide with the set of predictors identified by the EN. Thus, a natural extension to the models above is to use the EN as a first pass filter, given a grid of values for the mixing parameter, generating 100 candidate sets, and then evaluate these sets using the previous period's nowcasting error. Moreover, if the validation is conducted as a pseudonowcasting exercise, that is the publication lags are accounted for in the nowcast step, this approach allows for a variable selection step that implicitly accounts for the release structure of all variables. The nowcast validation procedure slightly differs from the previous two model families in the second step. Here, we use X𝜏−1 and y𝜏−1 to solve the EN problem via coordinate descent for a grid of 𝛼 values 𝜶 , where l is cross validated. Let :={𝛼∈𝜶,𝜏} be the set of vintages, where each vintage corresponds to a value in 𝜶 , and consists of the variables for which the solution of the EN problem reports nonzero coefficients given the corresponding 𝛼 value. We use  to construct a nowcast of y𝜏 for each unique vintage in the set.12 When doing so, we impose the release structure of the underlying data observed at T onto each 𝛼∈𝜶,𝜏 to explicitly account for the importance of the timeliness of the release of each predictor. The TPs set is then found as the set resulting in the smallest RMSFE. The two versions of these models are distinguished by their handling of mixedfrequency data and are denoted by “ANTPN” and “SNTPN,” respectively. The results of five benchmark models are also reported. Specifically, we analyze the forecasting errors of an AR(1), ARMA, and naive constant growth model. Furthermore, the nowcast results of a DFM using all variables, referred to as crowded DFM or CDFM, is the most important benchmark for the TP specifications. In the simulation study, we are able to compute nowcasts from a DFM that only uses the relevant variables, the socalled oracle DFM (ODFM). This is done to compare the results of using all the variables with those of using only the ones stemming from the same process as the target variable (CDFM vs. ODFM). 258 Journal of Forecasting, 2025 3 | Simulation 3.1 | Setup In our simulation study, four different DGPs are investigated. Inspired by the ideas of Boivin and Ng(2006), each DGP is composed of two different models, which are structurally identical, for example, both series admit a factor model of similar dimensions with the same dynamic structure but are parameterized differently. The models are only connected by their idiosyncratic errors, which are all drawn from the same multivariate distribution with a nondiagonal covariance matrix, that is, the idiosyncratic error are correlated across models. Additionally, each DGP is considered using either a relatively high or low signaltonoise ratio (SNR) in the measurement equation. For simplicity, we denote the data stemming from the same process as the target variable by relevant (subscript “re”) while the other variables are referred to as irrelevant (subscripts “ir”). The main reasons for using this procedure are the following: First, crosscorrelation of the error terms is the main driver of why a preselection step was initially considered. Therefore, we also draw from crosscorrelated errors in the same process. Second, when using large data sets, it is difficult to assume that all available variables are relevant, that is, stem from a single model. Thus, we sample from two different models to provide a more realistic scenario. Lastly, considering different SNRs is important, since the effect of an increase in the SNR on the forecasting performance is ambiguous. For example, it could be that a lower SNR increases the need for variable preselection, since the effects of including irrelevant predictors are stronger than in the case of a high SNR. On the other hand, when the SNR is low, the EN might not be able to correctly identify the important predictors.13, 14 The first and second DGPs are a combination of factor models that closely resemble the theoretical underlying DFM in the nowcast step. The first DGP (DGPI) is a combination of two fivefactor models, where their underlying structure is taken from Bai and Ng(2007). The second DGP (DGPII) is chosen to be a more parsimonious single factor model with each factor following a white noise process. Let xt,re ={x n,t,re } N re n=1 and xt,ir ={x n,t,ir } N ir n=1 be the sets of relevant and irrelevant variables at time t . The DGPs are formally defined as and The motivation for using the above DGPs is twofold. First, since the DGPs closely resemble the processes in the Giannone, Reichlin, and Small(2008) framework, it can be assumed that the nowcasting step estimates the true parameterization of the model more precisely if the variables stemming from the relevant processes have been identified correctly. However, it can be expected that the variable selection step is unstable, because the factor driven dependence structure has to be approximated linearly in the preselection step. Comparing DGPI and DGPII, it might be possible to infer on the role of the complexity of the underlying DGP on the variable selection and nowcasting accuracy. The third DGP considered (DGPIII) is conceptually different from the ones above. Here, the data set contains variables that are generated from two VARMA (1,1) processes, where the errors are crosscorrelated. Contrary to the DGPs described above, the TP approach is expected to perform comparatively well when applied to data generated by DGPIII since the model is linear in the predictors by construction. Particularly, the variable selection is expected to be more stable. The nowcasting step, however, might create more problems for the DFM since the model might not be approximated well by a small number of common factors. Therefore, including irrelevant variables and crosscorrelated errors might heavily influence the nowcasting step. We expect that for the VARMA parameterization, the TP approach should result in considerably more precise nowcasts compared to the CDFM.15 The last DGP(DGPIV) is a combination of two symmetric DFMs where the measurement equation includes lags of the latent factors which follow a VMA process. Again, the structure of the DFMs was taken from Bai and Ng(2007). Formally, the data are generated via Both processes in DGPIV are highly parameterized and highly dynamic DFMs that differ from the underlying model of the Giannone, Reichlin, and Small(2008) framework, where the factors have an autoregressive lag structure and the observed variables are not dependent on the lags of the factors explicitly. Especially since the nowcasting framework treats the lags of the factors as additional static factors, it might be expected that, in general, the nowcasting procedure is less precise. Furthermore, since the factors follow a VMA process, it is likely that P is estimated to be large thus further increasing the dimensionality of the estimated model. The variable selection step is assumed to be very unstable since a linear fit of the highly complex DGP is difficult to achieve. Summarizing, (DGPI) x t,re =𝚲 re f t,re +𝝃 t,re f t,re =𝚽 re f t−1,re +𝝂 t,re xt,ir =𝚲irft,ir +𝝃t,ir ft,ir =𝚽irft−1,ir +𝝂t,ir 𝝂 t,re =𝚼re𝝐t,re, and 𝝂t,ir =𝚼ir𝝐t,ir 𝝃t:=(𝝃� t,re,𝝃� t,ir)�∼(0N,𝚺𝝃), where N=Nre +N ir R=5, Q=3, and P=1. (DGPII) x t,re =𝝀 re f t,re +𝝃 t,re f t,re =𝜖 t,re xt,ir =𝝀irft,ir +𝝃t,ir ft,ir =𝜖t,ir 𝝃t:= ( 𝝃� t,re,𝝃� t,ir ) � ∼(0N,𝚺𝝃), where N=Nre +N ir R=1, Q=0, and P=0. (DGPIII) x t,re =𝚷 re x t−1,re +Θ re 𝝃 t−1,re +𝝃 t,re xt,ir =𝚷irxt−1,ir +Θir𝝃t−1,ir +𝝃t,ir 𝝃t:= ( 𝝃� t,re,𝝃� t,ir ) � ∼(0N,𝚺𝝃), where N=Nre +N ir (DGPIV) x t,re = ( 𝚲0,re +𝚲1,re𝕃+𝚲2,re𝕃 2) ft,re +𝝃t,re ft,re =Ψre𝝐t−1,re +𝝐t,re xt,ir =(𝚲0,ir +𝚲1,ir𝕃+𝚲2,ir𝕃2)ft,ir +𝝃t,ir ft,ir =Ψir𝝐t−1,ir +𝝐t,ir 𝝃t:= ( 𝝃� t,re,𝝃� t,ir ) � ∼(0N,𝚺𝝃), where N=Nre +N ir R=6, Q=2, and P=∞. 259 it is expected that the TP approach, if at all, only leads to small improvements of the nowcasting accuracy in this case. For the parameterization of the DGPs, we follow Bai and Ng (2007) and draw the factor loading matrices and vectors 𝚲 re ={𝜆n,r,re} N re ,R n=1,r=1 , 𝚲 ir ={𝜆n,r,ir} N ir ,R n=1,r=1 and 𝝀re ={𝜆 n,re } N re n=1 , 𝝀ir ={𝜆 n,ir } N ir n=1 from a standard normal multivariate distribution. This is also the case for the factor loadings corresponding to the lagged factors in DGPIV. Furthermore, for all error vectors of the factor processes it holds that 𝝐 t,re ∼(0 N re ,I N re ) , 𝝐 t,ir ∼(0 N ir ,I N ir ) , 𝜖t,re ∼ (0,1) , and 𝜖t,ir ∼ (0,1) . For DGPI,DGPII, and DGPIV, the covariance matrix of the idiosyncratic error 𝚺𝝃 , having nonzero offdiagonal elements, is constructed by 𝚺𝝃=INRIN , where R is a randomly drawn correlation matrix using the rcorrmatrix function of the clusterGeneration package by Qiu and Joe (2020). For DGPIII, IN is replaced by 0.005 ⋅ IN for the construction of 𝚺𝝃 , such that the simulated series have a similar magnitude compared to those drawn from the other DGPs before standardization. For the low SNR case, the variances are multiplied by a factor of 5. The starting vector for each factor, or in the case of DGPIII for the variables, is chosen to be zero, or the zero vector, and a burnin period of 200 observations is used. For DGPI and DGPIV, it is chosen that 𝚽re = 𝚽ir = diag(0.2,0.375,0.55,0.725,0.9) and Ψre =Ψ ir =diag(0.2,0.9) , respectively (Bai and Ng2007). Further, the matrices 𝚼re and 𝚼ir are calculated via 𝚼re =AreSreAre and 𝚼ir =AirSirAir , where Are and Air are random orthonormal matrices computed via the randortho function provided by the pracma package by Borchers(2021), and Sre and Sir are diagonal matrices of rank Q with elements drawn from (0.8,0.12) . To ensure stationarity in DGPIII, 𝚷re =PreDrePre and 𝚷ir =PirDirPir , where Pre and Pir are matrices with random entries drawn from (−0.5,0.5) , and Dre and Dir are diagonal with random entries drawn from (0.1,0.8) . To guarantee invertability, Θre and Θir are constructed analogously. Since the study aims to simulate a real world nowcasting exercise, a quarter of both the relevant and irrelevant variables is aggregated via(4) to resemble quarterly observations. While this does not influence the extraction of the factors in the Giannone, Reichlin, and Small(2008) procedure, it is expected to have an impact on the variable selection step. Considering the lag structure, only the target variable is lagged by 90 days. We refrain from imposing a random lag structure on the predictors since this would only influence the relative TP performance if the publication lags were longer than the one for the target variable, which is rarely the case in practice.16 The data is drawn for (T,Nre,Nir)={(200,150, 50), (200,100, 100), (200,50, 150)} to evaluate the TP procedure for different combinations of relevant and irrelevant variables. The expanding nowcasting window spans the last 100 observations while the first 100 are used as the initial training set.17 3.2 | Results To evaluate the performance of the different TP specifications, the RMSFE taken over the complete nowcasting period is reported. The results are found in Tables1 and 2 where the relative performance with respect to the CDFM is highlighted. We arrive at four general conclusions. First, in most cases, it is found that the ODFM model outperforms the CDFM model which is itself outperformed by some of the TP specifications. This gives clear evidence that (i) TP seems to be a viable option to improve nowcast accuracy, and (ii) the optimal set of TP is not always equal to the set of relevant variables. In other words, even if the true model specification was known, targeting predictors might still improve the nowcasting performance exploiting the dependence structure in a given data set. Second, the TP procedure achieves the worst nowcasting results for Nre =50 and Nir =150 . Both in terms of absolute and relative performance, a clear increase in the RMSE is found for all TP specifications. While this is to be expected, because having more irrelevant variables in the data set on average leads to a higher likelihood of selecting more irrelevant variables, it clearly shows that the TP procedure should be used with care. Simply adding (possibly irrelevant) variables to the data set does not boost the TP performance. Third, using CV methods seems to outperform choosing a fixed number of predictors. While the AFTPN provides the lowest RMSFE for 5 out of 12 DGP configurations in the high SNR setting, it is also highly volatile resulting in RMSFE increases by up to 81%. The nowcast validated specifications only achieve the lowest RMSFE in 4 out of 12 cases. However, the SNTPN specification is very reliable beating the benchmark CDFM in 9 out of 12 DGP configurations. In the low SNR setting, the nowcast validated specifications provide the lowest RMSFE in 9 out of 12 configurations clearly outperforming the fixed specifications. This hints towards the importance of using the information contained in the data when applying the TP framework, instead of relying on ad hoc solutions, that is, selecting the number of predictors beforehand. Finally, decreasing the signaltonoise ratio considerably worsens the overall performance of the TP framework. The TP procedure often results in an RMSFE improvement of only about 1%. Note, however, that in this setting the CDFM provides results that are often only marginally better than the more parsimonious AR(1), ARMA, and naive mean benchmarks, if at all. Therefore, using TP models can still be a viable alternative to improve the nowcasting performance of mixedfrequency DFMs in noisier settings. Turning to the relative performance of each specification in more detail, we find that most TP specifications are able to provide lower RMSFEs than the most important benchmark model (CDFM). The SNTPN specification is found to be the one that outperforms the CDFM benchmark most often, beating the benchmark in 9 out of 12 cases. Taking the average over the DGPs, it is found that the ALETPN, SLETPN, and SNTPN provide a lower RMSFE than the CDFM. Accounting for the expectedly bad performance in the case of DGPIV (see Table1) and only taking averages over DGPI – DGPIII, it is found that each specification results in a lower RMSFE by at least 4%. The single best performing specification is found to be the ANTPN 260 Journal of Forecasting, 2025 TABLE 1 | Relative root mean squared error of 100 targeted predictors nowcasts for N=200 variables generated with a high signaltonoise ratio. Nrel = 150, Nir = 50 Nrel = 100, Nir = 100 Nrel = 50, Nir = 150 Model DGPI DGPII DGPIII DGPIV DGPI DGPII DGPIII DGPIV DGPI DGPII DGPIII DGPIV Naive 1.22 1.06 1.37 4.25 1.19 1.06 1.10 4.18 1.22 1.06 0.99 3.86 AR 1.17 1.06 1.37 4.24 1.14 1.06 1.10 4.17 1.16 1.06 1.02 3.85 ARMA 1.17 1.06 1.37 4.24 1.14 1.06 1.10 4.17 1.16 1.06 1.02 3.85 ODFM 1.02 1.00 1.00 0.95 0.98 1.00 0.77 0.97 0.99 1.00 1.00 0.95 CDFM 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 ACDTPN 0.90 0.95 0.97 1.77 0.94 1.02 0.94 1.47 0.99 0.99 0.95 1.63 ALETPN 1.00 0.99 1.00 0.99 0.93 0.99 0.77 1.01 0.98 0.98 1.00 1.23 SCDTPN 0.97 1.02 0.88 1.37 1.02 1.02 0.89 1.23 0.93 1.00 0.99 1.20 SLETPN 1.02 1.00 0.96 0.98 0.98 0.99 0.78 0.97 1.00 1.00 1.00 0.96 AFTPN 0.92 0.81 1.14 1.81 0.86 0.79 0.73 1.23 0.86 0.95 0.98 1.17 SFTPN 0.93 0.91 1.09 1.40 0.90 0.87 0.77 1.56 0.94 0.96 1.01 1.03 ANTPN 0.94 1.03 0.79 1.21 1.05 1.01 0.65 1.20 0.88 1.00 0.91 1.27 SNTPN 0.98 0.99 0.84 0.97 0.94 1.01 0.67 1.10 0.94 0.96 0.95 1.17 Note: The error measure is reported relative to the benchmark CDFM. Bold printed values indicate the lowest achieved RMSFE. The model acronyms correspond to the following specifications: AR: An AR(1) model; ARMA: An ARMA model with lag order determined recursively via the AIC; Naive: A naive unconditional mean model; ODFM: A DFM using only the relevant variables; CDFM: A DFM using all variables; ACDTPN (SCDTPN): The set of targeted predictors is retrieved via the EN, which is solved using coordinate descent, with EN parameters retrieved via CV. The data is aggregated (skipsampled); ALETPN (SLETPN): The EN is solved via LARSEN; ANTPN (SNTPN): A TP model where the set of targeted predictors is retrieved using the nowcasting error from the previous period; AFTPN (SFTPN): A TP model where the set of targeted predictors is retrieved via the EN, where the EN parameters are set so that 30 predictors are selected. The variables are newly selected for each nowcast. 261 TABLE 2 | Relative root mean squared error of 100 targeted predictors nowcasts for N=200 variables generated with a low signaltonoise ratio. Nrel = 150, Nir = 50 Nrel = 100, Nir = 100 Nrel = 50, Nir = 150 Model DGPI DGPII DGPIII DGPIV DGPI DGPII DGPIII DGPIV DGPI DGPII DGPIII DGPIV Naive 1.01 0.99 1.33 1.22 1.00 0.99 1.15 1.10 0.99 1.00 1.00 1.00 AR 0.98 0.99 1.32 1.21 0.97 0.99 1.16 1.08 0.96 1.00 1.05 0.99 ARMA 0.98 0.99 1.32 1.21 0.97 0.99 1.16 1.08 0.96 1.00 1.05 0.99 ODFM 1.04 1.00 1.00 0.97 0.99 1.00 0.81 0.91 0.98 1.01 0.99 0.92 CDFM 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 ACDTPN 1.02 1.03 0.96 1.00 1.01 0.98 0.91 0.98 0.99 1.00 0.99 0.94 ALETPN 0.99 1.00 0.99 0.98 0.99 1.00 0.81 1.04 1.00 1.00 1.00 1.00 SCDTPN 1.01 1.01 0.92 1.04 1.01 1.00 0.88 1.00 1.00 1.00 0.99 1.00 SLETPN 1.00 1.01 0.99 1.05 1.00 1.00 0.81 1.03 0.99 1.00 1.00 1.00 AFTPN 1.02 1.03 1.09 1.08 0.98 0.98 0.77 0.96 0.97 1.00 0.97 1.00 SFTPN 1.01 1.00 1.03 1.11 1.01 0.97 0.80 0.98 0.99 1.01 1.00 0.93 ANTPN 1.06 0.99 0.79 1.03 1.05 0.97 0.68 0.96 0.97 1.00 0.93 1.00 SNTPN 0.99 0.99 0.77 0.99 1.00 0.98 0.71 0.99 1.01 0.98 0.94 0.96 Note: The error measure is reported relative to the benchmark CDFM. Bold printed values indicate the lowest achieved RMSFE. The model acronyms correspond to the following specifications: AR: An AR(1) model; ARMA: An ARMA model with lag order determined recursively via the AIC; Naive: A naive unconditional mean model; ODFM: A DFM using only the relevant variables; CDFM: A DFM using all variables; ACDTPN (SCDTPN): The set of targeted predictors is retrieved via the EN, which is solved using coordinate descent, with EN parameters retrieved via CV. The data is aggregated (skipsampled); ALETPN (SLETPN): The EN is solved via LARSEN; ANTPN (SNTPN): A TP model where the set of targeted predictors is retrieved using the nowcasting error from the previous period; AFTPN (SFTPN): A TP model where the set of targeted predictors is retrieved via the EN, where the EN parameters are set so that 30 predictors are selected. 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