ESSAYS ON FORECASTING MACROECONOMIC VARIABLES USING MIXED FREQUENCY DATA

Transcription

1 ESSAYS ON FORECASTING MACROECONOMIC VARIABLES USING MIXED FREQUENCY DATA By KIHWAN KIM A dissertation submitted to the Graduate School New Brunswick Rutgers, The State University of New Jersey in partial ful llment of the requirements for the degree of Doctor of Philosophy Graduate Program in Economics written under the direction of Norman R. Swanson and approved by New Brunswick, New Jersey May, 2016

2 ABSTRACT OF THE DISSERTATION Essays on Forecasting Macroeconomic Variables using Mixed Frequency Data By KIHWAN KIM Dissertation Director: Norman R. Swanson This dissertation investigate the forecasting performance of mixed frequency factor models with mixed frequency dataset. First chapter consider the mixed frequency factor approach used in ADS (2009) to construct their co-incident activity index, and ask the question of whether a class of mixed frequency indexes is useful for predicting the future values of quarterly U.S. real GDP growth and monthly industrial production, unemployment and In my rst chapter, I consider the mixed frequency factor approach used in ADS (2009) to construct their co-incident activity index, and ask the question of whether a class of mixed frequency indexes is useful for predicting the future values of quarterly U.S. real GDP growth and monthly industrial production, unemployment and in ation. My forecasting assessment of the mixed frequency factor model is performed in conjunction with standard prediction models such as autoregressions, multivariate distributed lag models, and di usion index models of the variety examined by Stock and Watson (2002a). The main ndings of the study are as follows. First, prediction models using only mixed frequency indexes show their best performance at short-term GDP forecasting horizons, and are particularly good during recessions. Second, prediction models using both mixed frequency indexes and di usion indexes forecast monthly variables more accurately than models using single frequency type indexes. Third, model combinations perform relatively poorly in real GDP forecasting contexts, although they perform better when applied to the prediction of monthly variables. Fourth, survey information can be conveniently exploited with higher frequency variables such as daily and weekly variables, and mixed frequency ii

3 indexes using such survey information are sometimes useful in forecasting lower frequency variables. In the second chapter, I evaluate the predictive performance of hybrid models for forecasting four economic variables. The hybrid approach takes into account the notion that simple autoregressions and sophisticated factor models predictive abilities may change according to the state of the economy. I nd that hybrid prediction models produce better forecasts than standard models and than combination models, in most cases, using the same menu of models discussed above. For example, in one-quarter ahead GDP forecasts, the best hybrid model reduces the mean squared forecast error of the best model combinations and the linear models by 14 and 11 percent, on average, respectively. More speci cally, the mean squared forecast error of autoregression is reduced by approximately 35 percent. In 12-month ahead predictions of in ation, the best hybrid model improves the best model combinations and the linear models by 11 percent and 16 percent, on average, respectively. This number again increases, in this case to 36 percent, when comparing only with autoregression. One reason for these ndings is that hybrid prediction models more e ectively utilize survey information. iii

4 Acknowledgments I would like to thank Norman R. Swanson, my dissertation adviser. His intuitive advice has inspired my research and his warm support has been helping me go through my life in Rutgers. I am especially grateful that he has been patient to me. I would also like to thank my dissertation committee, John Landon-Lane and Xiye Yang. John has been giving me his enthusiastic lectures about bayesian estimation. Xiye has supported me in every detailed ways during my job market period. Their valuable comments have founded my research greatly. I also have been thanking to my advisor, since my undergraduate, Chang-Jin Kim. His being has been pushing me forward, even when I was too passive. Lastly, I would like to thank my family. I cannot thank enough to my parents and parents-in-law for the unconditional love and sacri ce. To my wife, Kotbee, and my daughter, Elyse, thank you for giving me the most beautiful life in New Jersey. iv

5 Dedication To Kotbee and Elyse v

6 Table of Contents Abstract : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Acknowledgments : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Dedication : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : List of Tables : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : List of Figures : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ii iv v vii x 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. An Assessment of the Forecasting Performance of Mixed Frequency Factor Models : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Introduction Factor Modelling Frameworks Data and Forecasting Methodology Empirical Results Conclusions Appendix An Evaluation of the Forecasting Performance of Mixed Frequency Factor Hybrid Models : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Introduction Hybrid Methodology Ex-post Experimental Hybrid Models Concluding Remarks Appendix References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 225 vi

7 List of Tables 2.1. Top Proxies for the ADS Index based on R Target Variables Prediction Models Panel A. Model Speci cations Paenl B. Mnemonic De nitions Indicator Sets for Construction of Mixed Frequency Indexes Panel A. Variable De nition Panel B. Indicator Sets for Construction of Mixed Frequency Indexes Summary of MSFE- ve "Best" Models Across Model Speci cations.. 39 Panel A. GDP s Top ve MSFE "Best" Prediction Models Panel B. IP s Top ve MSFE "Best" Prediction Models Panel C. UR s Top ve MSFE "Best" Prediction Models Panel D. CPI s Top ve MSFE "Best" Prediction Models Average of RMSFE of Groups of Prediction Models One-Quarter Ahead GDP Forecasts during Business Cycles A1: Relative Mean Square Forecast Errors for GDP A2: Relative Mean Square Forecast Errors for IP A3: Relative Mean Square Forecast Errors for UR A4: Relative Mean Square Forecast Errors for CPI vii

8 3.1. Summary of MSFE- ve "Best" Models Across Model Speci cations Panel A. GDP s Top ve MSFE "Best" Prediction Models Panel B. IP s Top ve MSFE "Best" Prediction Models Panel C. UR s Top ve MSFE "Best" Prediction Models Panel D. CPI s Top ve MSFE "Best" Prediction Models Cuto s of Real GDP Growth for Each Target Variable A1 : GDP t+1 Hybrid Forecasts for the Whole Sample A2 : GDPt+2 A Hybrid Forecasts for the Whole Sample A3 : GDPt+4 A Hybrid Forecasts for the Whole Sample A4 : GDPt+8 A Hybrid Forecasts for the Whole Sample A5 : GDPt+2 I Hybrid Forecasts for the Whole Sample A6 : GDPt+4 I Hybrid Forecasts for the Whole Sample A7 : GDPt+8 I Hybrid Forecasts for the Whole Sample A8 : IP t+1 Hybrid Forecasts for the Whole Sample A9 : IPt+3 A Hybrid Forecasts for the Whole Sample A10: IPt+6 A Hybrid Forecasts for the Whole Sample A11: IPt+12 A Hybrid Forecasts for the Whole Sample A12: IPt+3 I Hybrid Forecasts for the Whole Sample A13: IPt+6 I Hybrid Forecasts for the Whole Sample A14: IPt+12 I Hybrid Forecasts for the Whole Sample A15: UR t+1 Hybrid Forecasts for the Whole Sample A16: URt+3 A Hybrid Forecasts for the Whole Sample viii

9 3.1. A17: URt+6 A Hybrid Forecasts for the Whole Sample A18: URt+12 A Hybrid Forecasts for the Whole Sample A19: URt+3 I Hybrid Forecasts for the Whole Sample A20: URt+6 I Hybrid Forecasts for the Whole Sample A21: URt+12 I Hybrid Forecasts for the Whole Sample A22: CP I t+1 Hybrid Forecasts for the Whole Sample A23: CP It+3 A Hybrid Forecasts for the Whole Sample A24: CP It+6 A Hybrid Forecasts for the Whole Sample A25: CP It+12 A Hybrid Forecasts for the Whole Sample A26: CP It+3 I Hybrid Forecasts for the Whole Sample A27: CP It+6 I Hybrid Forecasts for the Whole Sample A28: CP It+12 I Hybrid Forecasts for the Whole Sample ix

10 List of Figures 2.1. ADS Index and MF Index Historical Breakdowns of ADS Index One-Quarter Ahead GDP Predictions Panel A. Plots of Forecasts Panel B. Plots of RMSFE Forecast Gains of Factors over Autoregression Forecast Gains During Business Cycles Forecast Breakdown by Indicator s Contribution Two-Quarter Ahead GDP Prediction Panel A. Plots of Average of GDP Prediction Panel B. Plots of Increments of GDP Prediction Panel C. Plots of RMSFEs of Average Prediction Panel D. Plots of RMSFEs of Increments Prediction Eight-Quarter Ahead GDP Prediction Panel A. Plots of Average of GDP Prediction Panel B. Plots of Increments of GDP Prediction Panel C. Plots of RMSFEs of Average Prediction Panel D. Plots of RMSFEs of Increments Prediction One-Month Ahead IP Prediction Panel A. Plots of Prediction in 1989: : Panel B. Plots of Prediction in 2000: : Panel C. Plots of Prediction in 2007: : Panel D. Plots of RMSFEs x

11 2.10. Three- and Twelve-Month Ahead IP Prediction Panel A. 3-Month Ahead Average Predictions Panel B. 3-Month Ahead Increments Predictions Panel C. 12-Month Ahead Average Predictions Panel D. 12-Month Ahead Increments Predictions Three- and Twelve-Month Ahead UR Prediction Panel A. 3-Month Ahead Average Predictions Panel B. 3-Month Ahead Increments Predictions Panel C. 12-Month Ahead Average Predictions Panel D. 12-Month Ahead Increments Predictions Three- and Twelve-Month Ahead CPI Prediction Panel A. 3-Month Ahead Average Predictions Panel B. 3-Month Ahead Increments Predictions Panel C. 12-Month Ahead Average Predictions Panel D. 12-Month Ahead Increments Predictions Plots of MSFEs of AR(SIC) Predictions Panel A. h- Step Ahead GDP Predictions Panel B. h- Step Ahead IP Predictions Panel C. h- Step Ahead UR Predictions Panel D. h- Step Ahead CPI Predictions E ects of Adding Di rent Frequencies or Survey Information to the Indicator Sets xi

12 3.1. NBER Recession periods and GDP Cuto of Panel A. NBER Recession Periods Panel B. GDP Cuto of Panel C. Overlaps of Two Above Periods One-Quarter Ahead GDP Forecasts Panel A. Plots of Forecasts Panel B. Plots of RMSFEs Time Varying RMSFEs of GDP Forecasts Panel A. 2-Quarter, Average Panel B. 2- Quarter, Increments Panel C. 4-Quarter, Average Panel D. 4-Quarter, Increments Time Varying RMSFEs of IP Predictions Panel A. 3-Month, Average Panel B. 3-Month, Increments Panel C. 12-Month, Average Panel D. 12-Month, Increments Time Varying RMSFEs of UR Prediction Panel A. 3-Month, Average Panel B. 3-Month, Increments Panel C. 12-Month, Average Panel D. 12-Month, Increments Time Varying RMSFEs of CPI Prediction Panel A. 3-Month, Average Panel B. 3-Month, Increments Panel C. 12-Month, Average Panel D. 12-Month, Increments NBER Recession periods and GDP Cuto of Panel A. NBER Recession Periods Panel B. GDP Cuto of Panel C. Overlaps of Two Above Periods xii

13 1 Chapter 1 Introduction This dissertation investigates the forecasting performance for four macroeconomic variables in two essays. Predictions using various indexes from factor models are assessed. The general idea of factor models is that a small number of latent processes underlie the information contained in largescale datasets. Also, mixed frequency factor models are a subgroup of factor models, in which factors from indicators of di ering frequencies are used in model speci cation. More speci cally, in mixed frequency factor models, latent factor processes of daily frequencies are assumed to characterize data generating processes of variables of interest. This is potentially important, because higher frequency information has been shown in the empirical literature to improve prediction or interpolation of lower frequency variables. Essays focus on the performance of the various factors from these models in the out-of-forecasting context. In the rst chapter, I consider the mixed frequency factor approach used in ADS (2009) to construct their co-incident activity index, and ask the question of whether a class of mixed frequency indexes is useful for predicting the future values of quarterly U.S. real GDP growth and monthly industrial production, unemployment and in ation. My forecasting assessment of the mixed frequency factor model is performed in conjunction with standard prediction models such as autoregression, multivariate distributed lag models, and di usion index models of the variety examined by Stock and Watson (2002). The ADS index of the federal reserve bank of Philadelphia is one of the MF indexes I constructed. Main ndings are as follows. First, prediction models using MF indexes shows good performance in the one-quarter ahead GDP forecasts, especially during recession episodes. In addition, predictive improvement primarily comes from monthly payroll and industrial production. Second, in the forecasts of monthly variables, prediction models using both mixed frequency indexes and di usion indexes predict more accurately than models using one sort of indexes. Third, model combination perform poorly in real GDP while those are all listed in the top model in each in ation forecasts. Fourth, within the ADS model, survey information about the economy in the near future can be conveniently exploited with higher frequency data such as daily spread and weekly initial claims and sometimes provide predictive information for the target variables. One of the ndings in the rst chapter is that predictions in one-quarter ahead GDP from factor

14 2 forecasting models and multivariate distributed lag models are signi cantly more accurate during recession periods than those from autoregression. Reversely, during expansion periods, predictions from autoregression are more accurate than those from factor forecasting models or multivariate distributed lag models. These ndings are in line with or extension of the ndings in the literature (D Agostino, Giannone and Surico 2006, D Agostino and Giannone 2012, Kim and Swanson 2014). Taking advantage of these features, I construct a class of hybrid prediction models. In the second chapter, I examined the properties of the hybrid models with all prediction models considered in the rst chapter. I nd that hybrid models improve forecasts from the best single model and the best model combination in most of the cases. Especially, in one-quarter ahead GDP forecasts, the best model in hybrid models improves the best model in model combinations and in single models by 14 percent and by 11 percent, respectively. It also reduces mean squared forecast errors of autoregression by around 35 percent. In 12-month ahead average of in ation forecasts, the best hybrid model improves the best model combination and the best single model by 11 percent and 16 percent, respectively. Also, it improves autoregression by around 36 percent. Furthermore, hybrid prediction models in in ation forecasts utilize the predictive contents about the economic downturns in survey information more usefully than single prediction models. These empirical ndings provide the usefulness of the hybrid models I constructed, in the context of forecasting.

15 3 Chapter 2 An Assessment of the Forecasting Performance of Mixed Frequency Factor Models 2.1 Introduction Economic agents, such as consumers, investors, rms, central banks or governments, make decisions often based on the predictions of economic variables. The accuracy of the forecasts for decision making can have important consequences. In that regard, econometricians have been developing forecast models and making use of various datasets in order to improve the accuracy of predictions. Among the advances of prediction models, it is factor augmented prediction models that have been extensively studied in the recent years. The idea behind the factor model is that among many variables, some common factors are assumed to exist and align latently. Since Geweke (1977) and Sargent and Sims (1977) introduction of factor models to economics, the studies about composite coincident indexes of Stock and Watson (1989, 1991) opened wide discussions in the literature. In particular, Mariano and Murasawa (2003) and Nunes (2005) developed mixed frequency factor models by including quarterly real gross domestic product to monthly variables. Aruoba, Diebold and Scotti (2009, henceforth ADS), progressed mixed frequency factor models to the higher frequency setup by assuming that underlying factor evolves on a daily basis. The authors include weekly and daily information to monthly and quarterly variables and construct a timely business condition index. This literature is often categorized by small factor modeling approaches. Key papers have been contributed by the following: Engle and Watson (1981), Watson and Engle (1983), Stock and Watson (1989, 1991), Quah and Sargent (1993), Diebold and Rudebusch (1996), Gregory, Head and Raynauld (1997), Chauvet (1998), Kim and Nelson (1998), Mariano and Murasawa (2003, 2010), Kose, Otrok and Whiteman (2003), Giannone, Reichlin and Sala (2004), Nunes (2005), Giannone, Reichlin and Small (2008), Aruoba, Diebold and Scotti (2009), Aruoba and Diebold (2010), Camacho and Pérez-Quirós (2010), Camacho, Pérez-Quirós and Poncela (2012). Furthermore, factor models have been developed in a di erent direction, which is using large cross-sectional datasets, wherein variables are measured at the same lower -monthly or quarterlyfrequency. In theoretical perspective, large factor models have been studied in Forni and Reichlin

16 4 (1998), Ding and Hwang (1999), Stock and Watson (1999, 2002a,b, 2005, 2006, 2009, 2013), and Forni, Hallin, Lippi and Reichlin (2000, 2005), Bai (2003), Bai and Ng (2002, 2006, 2007, 2013), Hallin and Liska (2007), Onatski (2009, 2010), Doz, Giannone and Reichlin (2011, 2012), Forni and Lippi (2011), Corradi and Swanson (2014). Large factor models have more distinctive points than small factor models. First is that the factors can be consistently estimated via principal component estimation as well as maximum likelihood (e.g. Stock and Watson 2002b, Forni, Hallin, Lippi and Reichlin 2005, Doz, Giannone and Reichlin 2011). Second is that the number of factors from large factor models to be implemented in the application is much smaller than the number of indicators. For example, among the factors extracted from 143 numbers of macro- nancial datasets, only a small number of factors are sometimes to be used in the speci cation and estimation of the prediction models. Indeed, contemporary literature nds that factor augmented models or only factor prediction models yield predictions that often outperform those based on the speci cation of standard econometric models (see e.g. Boivin and Ng 2005, 2006, Bai and Ng 2008, Armah and Swanson 2010, 2011, D Agostino and Giannone 2012, Kim and Swanson 2014, Bräuning and Koopman 2014, Stock and Watson 1999, 2002a, 2005, 2006, 2012 and the references cited therein). That is, a few number of factors ("di usion indexes") extract useful information of the large dataset for forecasting. A literature related to mixed frequency factor models has been growing with a point of nowcasting and of short-term forecasting for key macro variables. The general merit of mixed frequency factor models, compared to large factor models, is that the higher frequency information can be added timely to improve prediction or interpolation of lower frequency variables. For example, monthly industrial production or all employees on non-agricultural payroll information in August can be used to predict the third quarter of real GDP. With the addition of higher frequency information, mixed frequency factor models have been shown to be successful in nowcasting, backcasting and short-term forecasting. Key papers are Mariano and Murasawa (2010), Modugno (2013), Banbura and Modugno (2014), Camacho, Pérez-Quirós and Poncela (2014), Camacho and Martinez-Martin (2014), Marcellino, Porqueddu and Venditti (2015). Two related literatures handling mixed frequency dataset have been developing, though my focus is di erent from the two. First is the MIDAS approach, and second is the mixed frequency VAR approach. Proposed by Ghysels, Santa-Clara and Valkanov (2006) and Ghysels, Sinko and Valkanov (2006) in the context of predicting volatility of nancial time series, MIDAS (MIxed DAta Sampling) models handle mixed frequency data in a regression setup with a tightly parameterized way and is easily applicable for direct forecasting. Clements and Galvao (2008) make use of MIDAS models to macroeconomic forecasting. Mixed frequency VAR models on nowcasting or short-term forecasting

17 5 have been studied, since Giannone, Reichlin and Small (2008) developed a mixed frequency VAR model in the now-casting context. Two literatures also make use of factors into their models. Key papers in two literatures include: Ghysels, Santa-Clara and Valkanov (2006) and Ghysels, Sinko and Valkanov (2006), Clements and Galvao (2008), Giannone, Reichlin and Small (2008), Armesto, Hernández-Murillo, Owyang and Piger (2009), Camacho and Perez-Quiros (2010), Mariano and Murasawa (2010), Marcellino and Schumacher (2010), Angelini et al (2011), Bai, Ghysels and Wright (2011), Banbura and Rünstler (2011), Camacho (2013), Foroni and Marcellino (2014), Schofheide and Song (2014), Marcellino, Porqueddu and Venditti (2015). Among the papers discussed above, some are noteworthy for my paper. Boivin and Ng (2006) nds that the factors extracted from smaller dataset by principal component method result in better forecasting performance than the factors from the larger dataset. This suggests that important information used for forecasting is contained in the smaller dataset. Although I are not using principal component method in my mixed frequency index construction, I focus on the small number of key indicators. In order to test daily nancial index s forecastibility to the daily real activity measure, Andreou, Ghysels and Koutellos (2013) benchmark ADS index as a daily real activity measure. Using MIDAS regression, the ADS index shows better performance in the short-term forecast of real GDP growth against Random Walk as well as autoregression for the forecasting periods from second quarter in 2001 to the third quarter in For relating to the ADS index with survey information, Balke, Fulmer and Zhang (2015) constructs the daily real activity index with Beige Book information. The Beige Book information is written in as texts and the authors developed a procedure to transform this text into quantitative numbers. It is shown that Beige Book provides useful information about the current status of the economy during recessions for a short period. Camacho and Martinez-Martin (2014) studies small factor models based on Aruoba and Diebold (2010), which handle monthly and quarterly variables, as well as survey and nancial indicators, and nds that the Index considered in the study has comparable predictive ability to the index from Aruoba and Diebold (2010). My study advanced that of Camacho and Martinez-Martin (2014). I utilized weekly or daily data series into monthly-quarterly indicators setup to quantify the forecasts contribution of higher frequency variables and survey information. Most closely related study to mine is Modugno (2013). While the author makes use of a daily mixed frequency factor model to the context of nowcasting in ation with a focus on the role of speci c daily nancial series, I are applying various daily real activity measures to the prediction of four macro variables: real gross domestic product, industrial production, unemployment rate, and consumer price index. ADS (2009) re nes the mixed frequency dynamic factor models having monthly and quarterly indicators into daily setup, which can conveniently deal with the higher frequency information.

18 6 Including weekly initial claims to the monthly employees on non-agricultural payroll and quarterly real GDP growth, authors show that this timely business condition index mimics market uctuations particularly well and captures higher frequency movements which cannot be acquired by monthly NBER Business Cycle dates. Furthermore, the ADS index in federal reserve bank of Philadelphia based on ADS (2009) has been constructed and updated on a weekly basis when new data arrives, which provide economic agents with the timely measure of the business conditions. In this paper, I address the forecasting performances of ADS (2009) model as well as ADS index. Through the mixed frequency dynamic factor model of ADS (2009), can I extract useful information for forecasting from higher frequency indicators, such as weekly initial claims or daily spread, or from survey information. In this paper, to answer these questions, I compare the forecastibility of various mixed frequency (MF) indexes including ADS index. For comparisons of models, several benchmark models including autoregression, multivariate distributed lag models and standard di usion index model are given. Also, I make prediction models using both MF indexes and di usion indexes. This study is posing the questions of how the small-mixed-frequency dynamic factor model using higher frequency data or survey information performs, compared to or along with the large factor model using monthly frequency dataset in Stock and Watson (2002a,b) in the forecasting perspective. As ADS (2009) suggests, the comparison of di usion indexes from "large same frequency data set" and my mixed frequency indexes from "small-mixed-frequency data set" is meaningful in the context of forecasting, since this comparison may signal the importance of higher frequency information and of mixed frequency small factor model of ADS (2009). I have constructed various MF indexes within the indicator sets, which range from daily government yield spread to semi-annual survey information, as well as several modi cations of models. To a lesser extent, I target two versions of a variable: average of a target for forecasting periods and increments of a target from the previous period. Prediction models using nonsmoothed MF indexes or smoothed MF indexes are also given. Lastly, I classify the prediction models based on whether they have autoregressive term or not. My results can be summarized as follows. First, prediction models using only MF indexes show best performance at the short-term GDP forecasts, especially during recession episodes. In addition, the primary contribution of short-term GDP forecasts comes from the monthly payroll and industrial production. However, the prediction model using MF index extracted from weekly initial claims, two more monthly variables and quarterly real GDP, attains the best model. Also, including daily information to this dataset makes the short-term GDP forecast less accurate. Second, some prediction models using only MF indexes show moderate performance in the long term forecasts. Especially, in 8-quarter ahead increments of GDP forecasts, all best ve models improve autoregression around 3.5% with rejection of 10% Diebold-Mariano (DM) one-sided test. One feature that the best ve

19 7 models share is that in the factor construction set, they have a daily spread information in common. Third, for average of industrial production and unemployment rate, prediction models using both MF indexes and di usion indexes forecasts better than prediction models using single indexes. For increments of the targets, prediction models using both MF indexes and di usion indexes are less frequently listed in the top ve models than prediction models using MF indexes. In in ation forecasting, prediction models using di usion indexes forecast more accurately than prediction models not having those. Fourth, model combinations perform poorly in real activity variables, while those are all listed in the single best model in each in ation forecasts. Fifth, survey information about the economy in the near future can be conveniently exploited with higher frequency data in ADS (2009) model. Also, prediction models using MF indexes, which have survey information in those, provide predictive information for short-term GDP, industrial production by itself and in ation with di usion indexes. Sixth, when I target the real activity variables and forecasting horizons are over six month, higher ranking prediction models using smoothed MF indexes predict the targets more accurately than higher ranking prediction models using nonsmoothed MF indexes. Seventh, the predictive contributions of autoregressive term di er by target variables and forecast horizons. Especially, in the case of CPI forecasts, with di usion indexes, autoregressive components always yield negative forecast gains compared to models not having autoregressive component except one-month ahead forecasts. The structure of the chapter is organized as follows. In Section 2.2, I present two factor modelling frameworks and estimation methods, based on the approximate factor model (Stock and Watson 2002b, 2006) and a mixed frequency dynamic factor model (ADS 2009). In Section 2.3, I outline data used in the empirical illustration and the manner constructing various MF indexes and corresponding prediction models. In Section 2.4, I present the ndings based on empirical analysis. I conclude in Section 2.5 with some remarks. 2.2 Factor Modelling Frameworks In this section, I revisit two factor models discussed in a number of key papers sited above. Since the primary focus of my research agenda is to compare the factors from the large factor model with the factors from the small mixed frequency factor model, I will present two factor models of Stock and Watson (2006) and ADS (2009) in sequence. The former is the static factor modeling approach wherein principal components is used to estimate latent factors. The static principal component are consistently estimated with large N and T (Stock and Watson 2002b). The static principal components are called Di usion Indexes (DI) followed by Stock and Watson (2002a). For

20 8 convenience, I will refer to this model as the same frequency large factor model. Thereafter, I discuss the mixed frequency dynamic factor model presented in ADS (2009), which is estimated by maximum likelihood estimation. I will name the factors estimated from this model as Mixed Frequency (MF) Indexes. The main di erence between the two approaches, other than using same frequency dataset and mixed frequency dataset, is that the former nonparametrically estimates the factors while the latter speci es the process of factor such as autoregressive process or ARMA process Same Frequency Large Factor Model Following Stock and Watson (2006), suppose X t has a dynamic factor model (henceforth DFM) representation with q common dynamic factors, f t. X it = i (L) 0 f t + e it ; (2.1) for i = 1,2,...,N, and t = 1,2,...,T, where X it is a single datum, f t is the q1 vector of unobserved factors, i (L) are q1 vector lag polynomials in nonnegative powers of L, and e it is an idiosyncratic shock. That is, N series of data are assumed to be composed of two parts, common components, i (L)f t ;and idiosyncratic errors e it, for each i: Furthermore, and E(f t e is ) = 0 for all i; t; s; (2.2) E(e it e js ) = 0 for all i; j; t; s; i 6= j: (2.3) That is, the factors and idiosyncratic errors are assumed to be uncorrelated at all leads and lags and the idiosyncratic error terms are taken to be mutually uncorrelated at all leads and lags. Under this assumption, I call the DFM the exact DFM, which can be weakened by allowing some degree of serial correlation (the approximate DFM). Note that I do not impose parametric assumptions on idiosyncratic disturbances. In this nonparametric case, I can use the principal components method to estimate the factors and factor loadings after assuming identifying assumptions, as discussed in detail in the above papers. From equation (2.1), under the assumption that the lag polynomials has nite dimension, p, I can transform the exact DFM into the static DFM as follows. X t = F t + e t ; (2.4) where F t = (f 0 tf 0 t 1:::f 0 t p+1) 0 is r 1;where rpq. Here r is the number of static factors. is a factor loading matrix on the r static factors consisting of zeros and the coe cients of i (L). Since

21 9 F t consists of r static factors, I call equation (2.4) static DFM representation (Stock and Watson 2006). The static factors can be estimated as the principal components of the normalized data X t : Let us outline the estimation procedure. Following Stock and Watson (2006), let k (k< minfn; T g) be an arbitrary number of factors, N<T, be the Nk matrix of factor loadings, ( 1 ; 2 ; :::; N ) 0 ; and F be a Tk matrix of factors (F 1,F 2,...,F T ): From equation (2.4), estimates of and F t are obtained by solving the following optimization problem : V = min F; s:t 0 = I k 1 T TX (X t F t ) 0 (X t F t ); (2.5) t=1 I treat F 1 ; :::;F T as xed parameters to be estimated after normalizing. Given b ; the solution to equation (2.5) satisfy that b F t = ( b 0 b ) 1b 0 X t : Substituting this into equation (2.5) yields V = min 1 T TX Xt(I 0 ( 0 ) 1 0 )X t s:t: 0 = I k t=1 = max tr(( 0 ) P XX (0 ) 1 2 s:t: 0 = I k = max 0 P XX s:t: 0 = I k ; where P XX = T 1 P T t=1 X tx 0 t. This optimization is solved by setting b to the eigenvectors of matrix X 0 X corresponding to its k largest eigenvalues. The estimator of factors is b F t = b 0 X t. For choosing the number of factors, Bai and Ng (2002) is suggested as a method. After estimating b and b F t, let ^V (k) = T 1 P T t=1 (X t b b F t ) 0 (X t b b Ft ) be the sum of squared residuals from regressions of X t on the k factors and IC(k) = log( ^V (k)) + k( N+T NT ) log(c2 NT ) be the information criterion where C NT = minf p N; p T g: The consistent estimates of the true number of factors is then ^k = arg min 0kk IC(k) where k is the maximum number of factors Mixed Frequency Small Factor Model Stock and Watson (1989) constructed a latent factor from four monthly variables ( Industrial Production, real manufacturing, trade and sales, the number of employees on nonagricultural payroll and personal income less transfer payment) and named it a composite coincident index of business cycle. With the motivation that monthly composite coincident index has short of real GDP related interpretation, by adding quarterly real GDP to monthly variables, Mariano and Murasawa (2003) and Nunes (2005) extend the single index model to the mixed frequency setup. Evans (2005) and ADS (2009) construct the timely measure of the economy in the daily basis. Camacho, Perez-Quiros and Poncela (2014), develop the mixed frequency factor model by including one-time Markov switching mean of the monthly factor process. Furthermore, Marcellino, Porqueddu and Venditti (2015)

22 10 incorporates stochastic volatility in the factor process in the model of Mariano and Murasawa (2003). Among the models cited above, I make use of the mixed frequency dynamic factor model presented in ADS (2009), which assumes that a daily MF index comove among variables in di erent frequencies. Following ADS (2009), I assume that the latent dynamics of a factor, mf t, follows a zero-mean AR(p) process at a daily basis. That is, the subscript t is the daily time. I call mf t the mixed frequency (MF) index. Let mf t = 1 mf t p mf t p + e t ; (2.6) where e t is white noise with unit variance. Thus, the single daily factor is assumed to be comove in the set of macroeconomic indicators. Suppose I have the J number of indicators. Let yt i denote the ith datum at t date, which depends linearly on mf t and possibly also on various exogenous variables wt 1 ; : : : ; wt k and/or lags of yt, i thus general measurement equation between the indicator yt i and the factor mf t is y i t = c i + i mf t + i1 w 1 t + + ik w k t for i = 1; :::; J; + i1 y i t in y i t n + u i t; (2.7) where the u i t are contemporaneously and serially uncorrelated innovations. At time t; the ith indicator y i can be possibly missing. For example, if y i t is quarterly real GDP, it is assumed that y i t is observed in the last day of the quarter. To handle this problem systematically, ADS (2009) distinguishes between stock and ow variables, and between observed data and missing data. Suppose that ey i t denotes a stock variable observed at a lower frequency. At any time t, if y i t is observed, then ey i t = y i t. And if it is not observed, then ey i t = NA: Thus, the stock variable at time t is 8 < y ey t i t i = : NA ; if y i t is observed ; otherwise : (2.8) Combining equation (2.7) and equation (2.8), the measurement equation for a stock variable is 8 < c ey t i i + i mf t i + i1 ey t i i = + + ipey t i n + u i t ; if yt i is observed : (2.9) : NA ; otherwise Unlike a stock variable, a ow variable is assumed to exist at higher frequency but observed at lower frequencies, thus can be interpreted as an intraperiod sum of the corresponding daily values, so that a ow variable is de ned as 8 XD i >< y ey t i t i j+1 ; if yt i is observed = j=1 >: N A ; otherwise : (2.10)

23 11 where D i denotes the number of days in a period. Combining equation (2.7) and equation (2.10), the measurement equation for a ow variable observed in a certain period is 8 < P Di ey t i i=1 = c P Di i + i i=1 mf t i+1 + i1 ey t i D i + + in ey t i nd i + u i t ; if yt i is observed : : N A ; otherwise (2.11) I use the sum of state variables for the period ( P D i i=1 mf t i+1) in the measurement equation in case of ow variable, as in ADS (2009). That is, in the quarterly real GDP growth measurement equation, all daily factors from the rst day to the last day of the quarter are summed and plugged into the measurement equation. Note that di erent temporal aggregation schemes between a lower frequency ow variable and a daily state variable may be considered. For di erent temporal aggregation schemes, refer to Mariano and Murasawa (2003), Mitchell et al (2005) and Proietti and Moauro (2006). Here, equation (2.6) is the state equation and equations (2.9) and (2.11) are the measurement equations. Together, these equations constitute a state-space model. Under the assumption that errors in state and measurement equations are distributed by normal distribution, I can estimate the model via maximum likelihood using Kalman ltering and prediction error decomposition. More speci cally, as in ADS (2009), the factor dynamics and the relation between the factor and the data can be represented in the following way. Four indicators are chosen, real GDP as a quarterly ow indicator, the number of employees on nonagricultural payrolls as a monthly stock indicator, initial claims for unemployment insurance as a weekly ow variable, yield spread between 10-year and 3-month U.S. Treasury yields as a daily stock variable. The measurement equations for daily spread, which is missing for weekend and holidays, with autoregressive term and no exogenous variables, 8 < ey Spread t = : c mf t + 1 ey i t 1 + u 1 t ; if y Spread t NA ; otherwise is observed The measurement equations for a weekly initial claims for unemployment insurance is, which is : ow variable and missing for six days, with autoregressive term, 8 < ey t IC = : P 7 i=1 c P 7 i=1 mf t NA i ey t IC W + u2 t ; if y IC t ; otherwise is observed : The measurement equations for monthly employees of nonagricultural payroll, which is stock variable and observed at one day per month, with autoregressive term being included, is 8 < ey P ay t = : c mf t + 3 ey P ay t M + u3 t ; if y P ay t is observed NA ; otherwise :

24 12 The measurement equations for quarterly real GDP, which is ow variable and observed at one day during a quarter is 8 < P Q ey t GDP i=1 = c P Q i=1 mf t : NA i ey t GDP Q + u4 t ; if y GDP t ; otherwise is observed : The state equation is assumed to follow zero mean AR(1) process. That is, mf t = mf t 1 + e t : The above measurement equations and state equation can be summarized in a vector form with Normal distribution on errors ey t 1 ey t 2 ey t 3 ey t 4 3 = Measurement Equation mf t mf t q or u 1 t ey 2 t ey 3 t ey 4 t W M Q u 2t u 3 t u 4t y t = Z t t + w t + " t mf t+1 mf t mf t 1. mf t q+1 u 1 t+1 3 = State Equation mf t mf 5 4 t mf t 1 mf t 2 u 1 t q e t t ; t+1 = T t t + R t where where " t ; 4 H t 0 5 A ; 0 Q t H t = 2 2t ; Q = ; 3t t

25 13 and u jt and 2 jt signify, respectively, the measurement error and the variance thereof, in case of ow variable j. The sum of daily MF indexes for the week or the quarter is plugged into the measurement equation. In the case of stock variable such as monthly payroll and daily spread, one MF index at the day is plugged in the measurement equation. The coe cient matrix, T t ; in the state equation is time-varying because the MF index is evolving on a daily basis and the number of days in the quarter is time-varying. The big dimension of state vector make the estimation of ADS model not readily attainable, especially in the case of forecasting application. The authors adopts Harvey cumulator, which the mixed frequency factor literature sometimes uses. De ne the C Di t a period D i as cumulator variable for C Di t = I t C Di t 1 + mf t where I t is an indicator variable de ned as = I t C Di t 1 + 1mf t mf t " t ; 8 < It Di = : 0; if t is the rst day of the period D i 1; otherwise : The measurement equation in real GDP, for example, is equivalent to 8 < ey t GDP = : P Q i=1 c C Q t N A + 4 ey t GDP Q + u4 t ; if y GDP t ; otherwise: is observed : Now, the state space system is represented by four-by-one state vector, which is much smaller dimension than ninety three by one of original model mf t+1 Ct+1 W Ct+1 M C Q t+1 3 = It W It M I Q t mf t C W t C M t C Q t [e t ] ; 7 5 t+1 = T t + R t ;

26 ey t 1 ey t 2 ey t 3 3 = c 1 P W i=1 c 2 c mf t C W t C M t ey 4 t P Q i=1 c C Q t ey t 1 ey t 2 ey t 3 D W M u 1t u 2t u 3 t 3 ; ey 4 t Q u 4t Y t = C + Z t + w t +" t : Kalman Filter and Signal Extraction Now turn to the estimation of this state space model. For the illustrative use of the Kalman lter, the above two equations are rewritten as below : Y t = C+ Z t + w t + " t t = T t t 1 + R t ; where " t N(0; Q) and t N(0; H): Y t is a vector of indicators, possibly having missing observations. t is the latent state vector. Also, it contains the mixed frequency factor mf t, the weekly, the monthly, and the quarterly cumulator. In the vector of exogenous variables w t, one autoregressive term of each indicator is included as in ADS (2009). The matrix T t is time varying in every day as the implementation of the Harvey cumulator explained above. Under error normality, the Kalman lter can be used to estimate this system (see e.g., Anderson and Moore 1979, Harvey 1989, Hamilton 1994, Kim and Nelson 1999, Durbin and Koopman 2001). Following Kim and Nelson (1999), Y t [yt;y 1 t;:::; 2 yt N ]; Y tjt 1 = E[Y t jy t 1 ]; tjt 1 = Y t Y tjt 1 ; F tjt 1 = cov[ tjt 1 ]; tjt = E( t jy t ); P tjt = cov( t jy t ); tjt 1 E( t jy t 1 ); and P tjt 1 = cov( t jy t 1 ): The Kalman lter consists of following six equations: four prediction equations and two updating equations. For any t; with no missing observation, tjt 1 = T t t 1jt 1 ; (2.12) P tjt 1 = ZP t 1jt 1 Z 0 + RHR 0 ; (2.13) tjt 1 = Y t Y tjt 1 = Y t C Z tjt 1 w t ; (2.14) F tjt 1 = ZP tjt 1 Z 0 + Q; (2.15) tjt = tjt 1 + P tjt 1 Z 0 F 1 tjt 1 tjt 1; (2.16) P tjt = P tjt 1 P tjt 1 Z 0 F 1 tjt 1 ZP tjt 1: (2.17)

27 15 Two prediction steps are associated with the state equation and the two more prediction steps are performed using the measurement equations. Given initial choices of state vector 0j0 and its covariance matrix, it is projected that its future value of state vector and its covariance matrix in (2.12) and (2.13). In (2.14) and (2.15), the vector of prediction errors and its covariance matrix are obtained after comparing the realized observations with predictions for them. Lastly, it updates the information to the state and its covariance matrix through (2.16) and (2.17). If some observations are missing, the measurement equation in vector form is modi ed as the observed number of data changes. That is, measurement equations associated with missing data are slipped away from the measurement equation in vector form. Y t = C + Z t + t w t + u t ; (2.18) u t N (0; Q ): Thus, in the Kalman lter, prediction steps are performed using the modi ed measurement equations (2.18). For any t; with some missing observations, tjt 1 = T t t 1jt 1 ; P tjt 1 = ZP t 1jt 1 Z 0 + RHR 0 ; tjt 1 = Y t Y tjt 1 = Y t C Z tjt 1 w t ; (2.19) F tjt 1 = Z P tjt 1 Z 0 + Q ; (2.20) tjt = tjt 1 + P tjt 1 Z 0 F 1 tjt 1 tjt 1 ; P tjt = P tjt 1 P tjt 1 Z 0 F 1 tjt 1 Z P tjt 1 : If all data are missing at period t, only prediction steps using state equation are required, tjt 1 = T t t 1jt 1 ; P tjt 1 = ZP t 1jt 1 Z 0 + RHR 0 : Estimation With i.i.d. normal distributions of errors in measurement equation and state equation, Maximum likelihood estimation is applied using the prediction error decomposition to the linear state space model. Speci cally, when the number of all data N are observed at any time t, the log-likelihood is incrementally increased by the following amounts, LogL = 1 2 [N log 2 + (log jf tjt 1j + 0 tjt 1 F 1 tjt 1 tjt 1)]:

28 16 For some data missing, the log-likelihood at time t is LogL = 1 2 [N log 2 + (log jf tjt 1 j + 0 tjt 1 F 1 tjt 1 tjt 1 )]:, where N < N is the number of observations, tjt 1 of equation (2.19), F tjt 1 of equation (2.20). If all indicators are missing, the incremental change of likelihood is zero. Thus, for t = 1; :::; T, the sum of all incremental (log) likelihood result in the log likelihood of this model. In extracting an MF index, two steps are necessary. First is to estimate the parameters in the models. Using ML estimation, with initial choices of parameters, state vector and its covariance matrix, I nd the estimates of parameters, which is maximizing the log-likelihood. With the estimates of parameters, one can extract an MF index in the state vector by running the Kalman lter. For the initial choice of state vector 0j0 and of its covariance matrix P 0j0, under the conditions of all state variables being stationary, the unconditional mean of state vector, E( t ); and its covariance matrix, E(P t ) are chosen. Following Durbin and Koopman (2001), the xed interval smoothing algorithm is given as below. If all data are observed at period t and with r T = 0, where tjt = tjt + P t r t 1 ; P tjt = P tjt P tjt N t 1 P tjt ; r t 1 = Z 0 tf 1 t v t + L 0 tr t ; N t 1 = Z 0 tf 1 t Z t + L 0 tn t /L t ; L t = T t K t Z t ; v t = Y t E(Y t jy t 1 ): If some observations are missing at period t, tjt = tjt + P t r t 1; P tjt = P tjt P tjt N t 1P tjt ; r t 1 = Z 0 t F 1 t v t + L 0 t r t ; N t 1 = Z 0 t F 1 t Z t + L 0 t N t /L t ;

29 17 where L t = T t K t Z t ; v t = Y t E(Y t jy t 1 ): If all data are missing at period t, tjt = tjt + P t r t 1 ; P tjt = P tjt P tjt N t 1 P tjt ; r t 1 = Ttr 0 t ; N t 1 = TtN 0 t T t : 2.3 Data and Forecasting Methodology Data I use two classes of U.S. dataset. First is the indicator set for constructing MF indexes. The frequencies of dataset range from semi-annual to daily. Also, since the number of indicators is smaller than that of the dataset of large factor models, I name it the "small-mixed-frequency dataset". Second is the dataset for construction of di usion indexes, having 143 number of monthly variables, which I refer to as "large-same-frequency dataset" for convenience. The "small-mixed-frequency dataset" is chosen mostly based on ADS (2009), with an addition of a small subset of survey data for future real gross domestic product. Most data series are obtained from FRED (Federal Reserve Economic Data) maintained by federal reserve bank of St.Louis Fed. Survey data series are obtained from Real-Time Data Research Center in federal reserve bank of Philadelphia. The description of the "small-mixed-frequency dataset" follow the frequency of the dataset. The lowest frequency data that I have in my MF index construction is Livingston Survey, which is available every six month and extensively studied in the literature. In my implementation, I include the mean and medium of two-step ahead (two-quarter ahead) forecast of the real GDP growth from 1971:1 to 2012:2. Since the survey is performed by individual forecasters, original survey data set is maintained by individuals. The real-time data center transforms this panel dataset into the aggregate number and I take the transformed aggregates in my forecasting experiment. I write 1 in 1971:1 to signify the rst half period of the year 1971, since this Livingston Survey is performing twice a year. Next is two quarterly frequency data series, which are Survey of Professional Forecasters survey

30 18 dataset and the real gross domestic product. The Survey of Professional Forecasters dataset is the oldest quarterly survey. This survey information is obtained more timely than that of Livingston survey. From the fourth quarter in 1968 to the fourth quarter in 2012, the mean and median of one-quarter ahead forecast of real GDP growth are contained in the indicator set. The real gross domestic product (GDP) in my MF index construction is seasonally adjusted from the rst quarter in 1960 to the second quarter in Monthly indicators are index of industrial production ( ), total non-agricultural employees on payroll ( ), real manufacturing, trade and sales ( ), real personal income less transfer payment ( ), consumer price index for all items ( ). All series are seasonally adjusted. Weekly series is the initial claims for unemployment insurance from January 7, 1967 to June 29, Daily series is the government bond spread, the di erence between 10-year Treasury-bond and 3-month Treasury-bill, from January 2, 1962 to June 28, Second class of dataset for constructing di usion indexes is the "large-same-frequency dataset," which Kim and Swanson (2014) maintained from January in 1960 to December in This dataset contains monthly 143 number of macro- and nancial variables. The series are chosen to be the following categories of macroeconomic time series: industrial productions, employments, manufacturing and trade sales, housing starts and sales, inventories, orders or un lled orders, stock price indexes, exchange rates, interest rate and its spreads, money or credit related quantity, and price related indexes such as consumer price index or personal consumption expenditure. This set also has a group of survey information. One index is Michigan consumer expectation and six indexes are regarding the Purchasing Manager s index (or National Association of Purchasing Manager s indexes). The asymptotic theory about di usion indexes assumes that the dataset for estimating factors is transformed to be stationary. After making the dataset stationary, all series are standardized to have sample mean zero and unit variance. The Appendix in Stock and Watson (2006) give more detailed descriptions of the series and speci c transformations used Forecasting Methodology My primary goal is to compare the prediction performance of my MF indexes relative to the standard benchmark models in the forecasting literature. In order to assess the forecasting performance of the MF indexes, I concentrate on the direct multistep-ahead predictions with recursive estimation windows. In Table 2.3, I present the speci cations and short summary of all prediction models. The benchmark models considered in my paper are the AR(SIC) model, Multivariate Distributed Lag (DL) models, Multivariate Autoregressive Distributed Lag (DLAR) models, standard Di usion Index (DI) models, and autoregressive Di usion Index (DIAR) models.

31 19 Univariate Autoregression AR(SIC) model is the formal benchmark model in the forecasting literature. Its parsimony and forecasting performance is well-known and often found that it is hard to beat. In my case, the autoregressive lags are selected by Schwarz Information Criteria, and I abbreviate it as SIC, hereafter. by t+h = bc h + bp h X i=1 b h i y t i ; where bc h and b h i are estimated by OLS and bp h is chosen by SIC: Multivariate Autoregression The Multivariate Distributed Lag (DL) model and Multivariate Autoregressive Distributed Lag (DLAR) models are prediction models using exogenous variables (and their lags). The regressors in DL or DLAR model are the same variables used in extracting MF Indexes. That is, I use the "small-mixed-frequency dataset" excluding survey data in the exogenous variables set. That is, dataset has SPR1, IC1, Pay, IP, RM, PI, CPI, described in table 4. The frequency is adjusted according to the target variable s frequency. For example, for predicting quarterly growth of real GDP, every last observation of a quarter is selected to make quarterly series from daily spread, weekly initial claims and monthly series. In estimating regression, I rst choose variables by SIC. Then, I select the lag of autoregressive component by SIC and estimate the lags of rst regressor, and add another exogenous variable and choose lags of it. by t+h = bc h + JX Xbp h;j j=1 i=1 h b h;j bq i x j t i + X i=1 b h i y t i ; where bc h, bh;j i s and b h i s are estimated by OLS and bp h;j and bq h are chosen by SIC: Di usion Index model Final benchmark models are two models associated with the standard di usion indexes. The di usion indexes (or static principal components), estimated from the large cross-sectional dataset, have been successful and studied extensively in forecasting literature as well as in macro-or macro- nance related literature. One model is a prediction model using only di usion indexes and those lags (DI model) and the other is the model with the di usion indexes with autoregressive term of target variable and those lags (DIAR model). Unlike DL or DLAR model, which uses data series directly to prediction, the prediction from the DI or DIAR model is made from two steps. First, I extract di usion indexes from the large-same-frequency dataset using principal component method. Second, with the estimated di usion indexes, I make a prediction with or without autoregressive terms. For predicting quarterly real GDP, I choose the last observation of each quarter and applied it to the estimation of prediction model. by t+h = bc h + KX bp h kx k=1 j=1 b h;k j DI k t j + bqx i=1 b h i y t i ;

32 20 where bc h ; bh;k j and b h i are estimated by OLS and bp h k and bq are chosen by SIC:For predicting growth of the real GDP, I use rst principal component as a forecaster (K = 1) while I use rst two principal components when I form a prediction of monthly variables (K = 2). The choice of number of lags is followed by the literature s suggestion, especially using DIs in the forecasting contexts, e.g., in Stock and Watson 2002a and D Agostino and Giannone As well as the several benchmark models above, I make predictions using the Mixed Frequency (MF) indexes. As in the case of DI model, I make a prediction with or without autoregressive term and abbreviate the prediction models as MFAR or MF model, respectively. Also, the prediction of the MF or MFAR models consists of two steps. First, I extract a MF Index in the mixed frequency dynamic factor model. Second, I make an OLS regression of a target variable onto the MF index with or without autoregressive term. by t+h = bc h + bp h X b h j MF t j + bq h X j=1 i=1 b h i y t i ; where bc h ; bh j and b h i are estimated by OLS and bp h and bq h are chosen by SIC: The reason why I distinguish autoregressive term of a target variable in addition to factor only models such as DI or MF model is that, in the literature, factor only model performs sometimes well enough ( Stock and Watson 1999, 2002a) or adding autoregressive terms in the model leads to improved predictions. (e.g. Clements and Galvao, 2008). I want to quantify the e ect of adding autoregressive terms in every cases of my forecasting experiments. I construct MF indexes from the variations according to the two data related criteria and two model related criteria. First is the modi cations of indicator sets. In Panel B of Table 4, I present the indicator sets I construct MF indexes. In the panel, I have di erent frequencies ranging from biannual to daily frequency in my variables set for the construction of MF indexes. I make MF indexes with or without survey data, quarterly GDP, weekly IC, and daily spread. I do this to assess the utility of much lower or higher frequency or survey information, such as quarterly Survey of professional Forecasters, quarterly GDP, weekly initial claims, or daily government spread data can be. In addition, I prepare indicator sets by the frequency. I have MF Index from only daily (D1), only weekly (W1), only monthly (four versions), only quarterly indicator (Q1). Also, I have monthly and quarterly: Pay-GDP (MQ1), Pay-IP-GDP (MQ2), Pay-IP-RM-GDP (MQ3), Pay-IP-RM-PI-GDP (MQ5) as well as weekly and monthly: IC1-Pay (WM1), IC1-Pay IP (WM2), IC1-Pay IP-RM (WM3), IC1-Pay IP-RM-PI (WM4). All sets from A to T have Weekly-Monthly-Quarterly (WMQ) indicators or Daily-Weekly-Monthly-Quarterly (DWMQ) indicators. Second is the transformations (log-di erence) of the weekly indicators. All variables in the

33 21 indicator sets, except weekly initial claims, are transformed to be stationary by taking log-di erence. Thus, I wonder if the persistency of a higher frequency variable among stationary low frequency variables in the state space model can a ect the estimation of a latent factor when applying Kalman lter or smoother. Thus, I explore these possibilities in the out-of-sample forecasting perspective. Third is whether an MF index is smoothed or not in the Kalman lter. Kalman smoother makes an MF index using full information from all data used in the estimation, say T 1. That is, via Kalman smoother, the estimate of the MF index at t < T 1 (a state variable in the state space model) can be provided with the information from t + 1 to T 1. Thus, if the states are well adjusted to the future value of the indicators, it will be sometimes helpful in the forecasting target variables. Furthermore, in the nowcasting or interpolation literature, Kalman smoother has an important role in estimating the missing values of the variables in the system. This process is directly associated with the smoothed index (state variable). Thus, the prediction performance of a smoothed MF index is related to the reliability of the interpolated variables in the out-of-sample as well as its own forecasting performance. Fourth is, when extracting an MF index, whether an observation of a low frequency variable is set to the rst or last day of a month and/or of a quarter. For example, for a monthly industrial production in December, should a datum be set at December 1 or December 31 to extract a factor? This seemingly odd exercise has an empirically appealing reason. This stems from a fact, which will be presented below, that the correlation between the actual ADS index in Philadelphia Fed and my MF index from the same indicators matched in the rst day of each month and quarter is.97. This is much higher than the correlation of 0.78 between the ADS index and MF index, when indicators are set to the last day of each period. ADS (2009) assumes that all quarterly and monthly indicators are set to the last day of each quarter and month. This empirical similarity with ADS index explores this possibility. Setting it at the rst day in a quarter or a month of indicators before estimating an MF index implies that I use unavailable data in advance. Excluding some nancial daily series and survey information, weekly, monthly and quarterly data that I use are all released even after the last day of each period and revised after the rst release. These experiments with fully revised data may shed some light on the importance of timely estimate nowcasting, short-term forecasting or survey information. As soon as I have estimate of the current status of a target variable and use it in the index construction, it may produce better forecast in some contexts. Also, it raises di erent model speci cation. For example, when I put observation at the last day of a month, the sum of daily indexes for the month is plugged in the measurement equation. However, when I put monthly data series at the rst day of a month, the sum of daily indexes in the former case is replaced with the

34 22 rst daily index of the month, which results in slacker model speci cation than the former. Next, I present how I make a prediction using di erent frequencies of dataset. First, I consider the case that I have same-frequency data in the target variable and the forecasters. In my case, when I predict monthly target variable, I use monthly exogenous variables or monthly di usion indexes. In this case, I divide the data set into two subsamples. The rst subsample has R observations for the estimation of the prediction model. The second subsample has P observations for prediction sample, hence, a total of T = R + P: Speci cally, for making a rst h-step ahead prediction, after estimating di usion Indexes using rst R samples, I estimate the prediction model with the di usion indexes using up to R h sample. I plug estimated indexes to the prediction equation above to form by 1 T +hjt : At R + 1, I repeat the same procedure to form a second forecast, by T 2 +hjt : By iterating estimation of factors and of prediction model with these factors from R + h to the R + P observations, I construct a set of forecasts, b YT +hjt = fby 1 T +hjt ; by2 T +hjt estimated by OLS and the number of lags is estimated using SIC. h+1 ; :::byp T +hjt g. In all cases, the prediction equation is Second, for mixed-frequency dataset, I divide the dataset based on the speci c date. For example, in the case of predicting quarterly real GDP growth from the rst quarter in 1987 to the fourth quarter in 2012, I divide my sample based on the rst quarter in Before estimating the prediction equation, using dataset up to the fourth quarter of 1986, I estimate the MF index. Since MF index is daily frequency, I choose last observation of each quarter from the estimated MF index series to make quarterly index. Then, using this quarterly series with real GDP growth, I estimate the prediction model. The quarterly index is plugged into the prediction model to make the rst forecast estimates by T 1 +hjt : A set of P same way as in the rst case. h + 1 predictions in h-step ahead prediction is formed in the Let X be the set of exogenous variables, DI be the set of di usion indexes and MF be the set of various MF indexes. The following linear prediction models include each prediction model: by T +hjt = b h + bp h X j=1 b h0 j b Z T j+1 + bq h X j=1 b h j y T j+1 ; (2.21) where b Z 2 fx,di,mfg; bp and bq are selected by SIC and maximum length of lags is con ned to 11. All models are presented in Table 3 with short descriptions of models. The target variables are quarterly real GDP and three monthly macro variables: IP, UR and CPI. Real GDP, IP and CPI are log-di erenced and UR is di erenced to be predicted. In addition, I make predictions for two versions of a target variable that both methods are studied in the literature. First is the average of a target variable for forecast periods, yt A +hjt, and second is the incremental change of a target variable from previous period, y I T +hjt +h 1 : The reason I experiment two cases is

35 23 I wonder if the same model is good at forecasting two versions of targets. y A T +hjt = log(y T +h ) log(y T ); y I T +hjt +h 1 = log(y T +h ) log(y T +h 1 ); where Y T +h are real GDP, IP and CPI. For unemployment rate, y A T +hjt = Y T +h Y T ;and y I T +hjt +h 1 = Y T +h Y T +h 1 are used. This is given in Table 2.2. In the forecasting literature, it is known that the simple forecast combinations sometimes predict better than the individual forecasts (Bates and Granger 1969, Clemen 1989, Timmermann 2006). Thus, after estimating all models, I perform ten cases of model combinations. At each case, I impose the equal weight on each prediction model. The descriptions of models is presented in Panel A in Table 2.3. First is the average of the benchmark models predictions, that is, AR, DL, DLAR, DI, and DIAR model. Second and third is the average of all predictions from nonsmoothed and smoothed MF models. Fourth and fth is the average of all predictions from nonsmoothed and smoothed MFAR models. Sixth and seventh model combination are regarding the nonsmoothed and smoothed MFDI models. Eighth and ninth is the average of predictions from nonsmoothed and smoothed MFDIAR models. Tenth model is the average of all models predictions. By comparisons from these experiments, I can see, on average, the e ect of MF indexes having a certain feature such as smoothed or nonsmoothed MF indexes or having autoregressive terms, for example. For evaluations of predictions for each model, I use mean square forecast errors (MSFEs), i.e. MSFE = 1 P h + 1 TX t=r+h (y t+h by t+h ) 2 ;, where T = R + P. In particular, predictions of each model are compared with predictions of AR(SIC) model by the ratio of two model s MSFEs (RMSFE). RMSFE = MSFE i MSFE 0 ;, where MSFE i is mean squared forecast errors from model i and MSFE 0 is mean squared forecast errors from AR(SIC) model. Then, I perform the Diebold-Mariano (DM: 1995) predictive accuracy tests. Suppose I have two predictions from two models, fby 1;tg h P t=r+h and fbyh i;t gp t=r+h from model 0 and model i; where h is the forecast horizon. Here, AR(SIC) model make a set of predictions, fby 1;tg h P t=r+h and other prediction model i generate fbyh i;t gp t=r+h ; and corresponding prediction errors fb" h 1;tg P t=r+h and fb"h i;tg P t=r+h : The null hypothesis of DM test statistic is that, given loss measure, two models have equal predictive accuracy in a pointwise sense. That is, H 0 : E f(" h 0;t) f(" h i;t) = 0;

36 24 where " h 0;t and " h i;t are true prediction error of model 0 and i; respectively, and f() is the loss function. In my case, the loss function is quadratic. Thus, the DM test statistic is DM = p P p d ; b 2 d where d = 1 P P T t=r+h b d t ; b d t = (b" h 1;t) 2 (b" h i;t) 2 ; b" h 1;t and b" h i;t are estimates of " h 1;t and " h i;t and b2 d is the HAC standard error for b d t : The DM test statistic has a asymptotic standard normal distribution under the assumption that the parameter estimation error vanishes and two prediction models are non-nested. If DM statistic shows negative sign, the benchmark AR(SIC) model yields a lower point MSFE than model i s prediction. In my case, I implement the one-sided DM test statistic. 2.4 Empirical Results Preliminary Analysis on ADS Index I present the graphs of my MF index I constructed and of ADS index maintained by Philadelphia fed in Figures 2.1. The six indicators, which are used to construct my MF index, are real gross domestic product, industrial production, employees on non-farm payroll, real manufacturing trade and sales, real personal income less transfer payment, and initial claims for unemployment insurance. I take the same transformations as in Aruoba (2013). That is, level data is used for initial claims and log-di erencing are taken for the rest of variables before constructing ADS index. The model speci cation is the same as ADS (2009), which is presented in Sec 2.2. More concretely, level of the weekly initial claims is the rst indicator, which is a ow variable, and the weekly sum of states and one autoregressive lag are included in the measurement equation. The log-di erenced monthly and quarterly indicators are all ow variables and have their own single lag term with the sum of states for a month or a quarter. The state equation is assumed to be daily AR(1) process with zero mean. For identi cation of the MF index, the variance of error terms in the factor process is xed to be one. I subtract the sample mean of indicators from dataset before estimation to reduce the burden of estimation. I have the same procedure for constructing my MF index as in ADS (2009) and Aruoba (2013). In Figure 2.1, the ADS index as of is presented with my MF Index. ADS index is updated in real-time and the index I constructed is estimated with fully revised data. Also, ADS index released in is the index constructed from indicators available up to , while MF index make use of indicators available up to There is a time lags in the usage of the indicators. Although I take these facts into account, the ADS index seems to be di erent in correlation from ADS (2009) model. Especially, when monthly and quarterly indicators are set to the rst day of month and of quarter, respectively, the correlation between ADS index

37 25 and my nonsmoothed MF index is After smoothing the MF index with the Kalman smoother, the correlation declines to If I set indicators to the last day of each period in ADS (2009), the correlation between ADS index and non-smoothed MF index is reduced to.78. By empirical similarity with ADS index, I refer to the rst day non-smoothed MF index as ADS index and apply the same manner to the other indicator sets. In Figure 2.2, breakdowns of the ADS index are plotted with its six indicators. The historical breakdowns are followed by Stock and Watson (1989). The decomposition of ADS index is to estimate the index with all other indicators except one being set to zero, in order to dismantle the e ect of one indicator to the index. It is industrial production that shows the largest correlation, which is 0.93, between ADS index and its breakdowns. Also, the breakdown experiment reveals that the level of weekly initial claims in constructing MF index may attribute to noisy index, compared to the smooth indexes from the lower frequency variables. To disentangle the components that make up our MF Index in a di erent angle, I use R 2 to see how much an indicator accounts for the ADS Index s variation. Table 2.1 present this ranking of R 2 to the ADS Index. Payroll and industrial production is almost equally contributed to the ADS Index. This statistic also indicates that the weekly information can explain MF Index more than some monthly information does, since the ADS Index is the least explained by the personal income less transfer payment. To investigate the ranking of indicators from another point, I use t-statistic in Bai and Ng (2008), which is applied and extended in Armah and Swanson (2010) and Kim and Swanson (2014). The statistic is designed to rank variables in terms of their contribution to overall factor variation in the di usion index environment. Using t-statistic, Bai and Ng (2008) select an observable proxy variable, which is most similar to the di usion indexes. By using observable variables, instead of estimated di usion indexes, the problems arising from the estimation errors associated with the construction of di usion indexes can be reduced. Here I rank observable proxies for ADS index among six indicators by t-statistic in panel B of Table 2.1. Industrial production is shown to contribute the largest portion to the ADS Index. The t-statistic suggests that quarterly real GDP and weekly initial claims contribute the least to the ADS Index Prediction Results This section presents the out-of-sample prediction results of the MF Indexes and the benchmark models. For the indicator sets from A to F, the MF indexes have two modi cations: 1. An MF index is smoothed or not. 2. Quarterly and/or monthly indicators in the index construction sets are matched in the rst day or in the last day of the quarter and/or the month. In addition, since

38 26 the prediction models are two folds, with or without autoregressive component, one indicator set has eight forecasts. With di usion indexes, it becomes 16 forecasts for each dataset. Since the number of indicator sets are six from A to F, the total 96 forecasts are made from these sets. From G to N and MQ4, these sets contain survey information and I do not perform First and Last experiment, since the survey datasets are released in the middle of the periods. From O to T, these indicator sets are prepared to see the e ects of transformation (log-di erence) of weekly IC. Thus, fourteen indicator sets from G to T have two variations, four forecasting models (with or without autoregressive component and/or di usion Indexes), make total of 112 forecasts. Also, I prepare a variation of indicator set by the frequency, that is described in previous section. Including all these prediction models, models having MF indexes are 344 number of prediction models. By comparing these models, I can examine how an indicator of having possibly di erent frequencies can contribute to the forecast accuracy of the target variable, on average, for example. Table 2.3 and 2.4 show the summaries of the forecasting performances of all models. Table 2.5 illustrates the best top- ve models with their names and RMSFE at each horizon and target variable. Table 2.6 shows the distributions of forecast RMSFEs according to the groups of models categorized by my experiments. In the rst column, MF, MFAR, MFDI and MFDIAR presents the set of all RMSFEs from MF, MFAR, MFDI and MFDIAR models. This set is partly shown at the 2.5, 50, 97.5 percentiles of all RMSFEs, shown as numerical entries on the right side of each model s name. Survey is named after the group of all prediction models having survey information. Thus, it contains the set of all RMSFEs from survey related prediction models. Speci cally, all prediction results relating to the datasets, MQ4, and from G to J, are gathered and RMSFEs corresponded to the three percentiles (2.5, 50, 97.5) are shown. NonSurvey stand for all prediction results from the dataset M2, D and F, in which just survey information is excluded from the dataset MQ4, from G to J. All summary results are based on results provided in the Appendix Table 2.5 A1 to Table 2.5 A4. At a rst look of Panel A in Table 2.5, prediction models using only MF indexes performs best in one-step ahead and moderate in 8-step ahead (GDP A t+8) and are improved with di usion index in the 2- or 4-step ahead average forecasts. However, most of the 4- and 8-step ahead increments of GDP forecasts from factor prediction models fail to improve simple AR(SIC) model with some exceptions (Table 2.5 A1). These exceptions in 8-step ahead increments of GDP forecasts show moderate improvements over AR(SIC) model. In addition, model combinations surprisingly perform poorly, since only one case appear at 8-step ahead increments of GDP forecast among total 35 cases. To a lesser extent, some nonsmoothed MF Indexes show better performance at 1-step ahead prediction while smoothed MF Indexes seem to predict more accurately in the horizons longer than one.

39 27 The ADS index (MF_C_NSF) performs slightly better than AR(SIC) model up to two-step ahead forecast. Speci cally, in 1-step ahead, ADS index improves 12 percent over AR(SIC) model. In 2-step ahead average of GDP forecasting, the ADS index without autoregressive term outperforms AR(SIC) model by 14 percent. In the increments of the target, the index slightly underperforms or on a par with autoregression except two-step ahead forecast. Compared with DI or DIAR, the ADS index has similar predictive accuracy to the DI predictions in the GDP forecasts. In one-quarter ahead forecasts, the best model is MF_B_NSL (0:778), the model using nonsmoothed MF index, which is extracted from set B of IC1, Pay, IP, RM, and GDP. This model improves 22.2 percent over AR(SIC) model. That is, MF indexes without an autoregressive terms of target variable can be good at short term GDP forecast, suggesting that the information in the MF Indexes is capturing not only contemporaneous but also dynamic information useful for forecasting the short-term GDP. The common features that the best ve models share in the short-term GDP forecasts are that all models make use of MF indexes which are: 1. nonsmoothed 2. constructed from weekly-monthly-quarterly indicators and 3. constructed from indicators without personal income less transfer payment. The gure 2.3 shows one-step ahead GDP predictions from AR(SIC), DI, MF_B_NSL (MF in gure), MFDI_B_NSL (MF+DI in gure) and MQ4_NSL (SPF added MF in gure). The gure in the top panel presents the prediction plots of each model. Of all models, MF_B_NSL model forecasts the Great Recession period most sharply although it is a bit lagging. The DI model and other SPF containded MF model are well forecasting all recession periods. However, AR(SIC) model least accurately forecasts all recession periods. In the bottom panel, the gure shows the time varying RMSFEs of each model compared to AR(SIC) model. As seen in the gure in panel A, during recession periods, all models RMSFEs are declining. It suggests that the relative forecastibility of each model di er by economic episodes, and especially the factor prediction models predict more accurately than simple autoregression. However, during expansion periods, each model cannot improve AR(SIC) model except from 2000 to Especially, the DI model s prediction performance is worsened during 90s. From the second quarter in 1990 until prior to the Great Recession, survey contained MF_MQ4_NSL model is better than other models. In Table 2.7, I present one-quarter ahead GDP predictions during expansion and recession episodes. The expansion and recession episodes are following the NBER Business Cycle dating committee. Interestingly, during expansion, even the best performing MF_B_NSL model, are worse than autoregression by 5.6 percent. In addition, most models such as DI and DL models show less accurate forecasts than autoregression during NBER expansion periods. However, during recession episodes, MF_B_NSL improves AR(SIC) model over 55.5 percent. This result seems to stem from

40 28 the poor performance of AR(SIC) model during recession based on two facts. First, many MF models as well as other benchmark models improves similar extent over AR(SIC) model during recession periods. Second, the MSFE of AR(SIC) model during recession is more than ve times of MSFE during expansion. In gure 4, I stack the forecast gains of MF_B_NSL, and the two benchmark models by the business cycles. The gure indicates that the relatively better performance of the MF_B_NSL model during expansion than other benchmark models, results in the best model in the whole sample. In other words, all forecast gains of this MF model is concentrated on the recession periods. The gure 2.4 shows the time varying mean squared forecast errors of AR(SIC) model in the forecasting periods from the rst quarter in 1987 to fourth quarter in As expected, the Great Recession makes AR(SIC) model s prediction much worse than the prediction of the period prior to it. The ndings are in line with and extend the observations in D Agostino and Giannone (2012) to the recent economic episodes. D Agostino and Giannone (2012) observe that after the mid of 1980s,i.e., during the Great Moderation periods, the performance of the factor forecasting models have become worse relative to the simple autoregression. Also, DI model s forecastibilties are concentrated on the periods prior to the mid 1980s. In my case, the forecasting performance of my factor forecasting models concentrate on two periods; before 1990s and the Great Recession. From 1990 to 2007, the simple autoregression is performing better than factor forecasting model. For other horizons and other target variables, the tendency is getting weaker. If asked, the tables will be provided. I can dismantle one-step ahead forecasts by its indicator s contribution. Speci cally, using the prediction results from MF models among M1, M2, M3 and MQ3, the sources of the short-term forecast can be traced back to each indicator. Figure 2.6 stack the forecast gains from each MF index constructed on these indicators. The MF_MQ3_NSL, which have monthly Pay, IP, RM and quarterly GDP in the indicator set and PAY+IP+RM+GDP in the gure 2.6, produce 19.8 percent of forecast gains without weekly IC. Thus, weekly IC variable may result in the forecast gain of around 2.5 percent. By comparing MF_MQ3_NSL(0:802), MF_M3_NSL(0:826) and MF_M2_NSL(0:789); quarterly GDP contribute around 2.4 percent gain and RM contribute negative 3.7 percent. The MF index from monthly Pay only, MF_M1_NSL and shown as PAY in Figure 2.6, improves AR(SIC) model up to 18 percent. The MF index from Pay and IP improves 21.1 percent. Most of the forecast gain in the short-term GDP forecast seems to come from the two monthly variables. Lastly, instead of using real GDP, when I have the one-step ahead survey information of GDP growth of SPF, and extracting MF Index with two monthly variables of Pay and IP, MF_MQ4_NSL predicts the short-term GDP (0:786); which is comparable to the best top- ve models. Without quarterly GDP, adding weekly IC to two monthly indicators, the short-term forecast is

41 29 worsened by 7.6 percent, compared to the forecast from MF index using GDP, Pay, IP and IC. By comparing another three cases of weekly-monthly indicators with three cases of monthly indicators (only for nonsmoothed MF Indexes), adding weekly IC to monthly variables make forecasts less accurate in the short-term. Including quarterly GDP to monthly indicators improves short-term forecasts in three out of four cases. Also, adding quarterly GDP to weekly and monthly indicators all improves the short-term forecasts of MF Indexes using only monthly indicators. Also, with weekly to monthly and quarterly indicators improves four out of six cases. However, MF index using only weekly IC shows much less accurate short-term GDP forecast. That is, for the proper adaptation of weekly IC to the prediction of quarterly GDP, it seems necessary to include quarterly GDP in the factor construction set. In two-quarter ahead forecasts, prediction models using both di usion index and MF Indexes are listed 9 out of 10 cases. Also, models in the best ve models use MF indexes having four indicators at most in the factor construction set. In 4- and 8-step ahead average of GDP, it is notable that using log-di erenced weekly initial claims seems to be in e ect. The best ve models are all models using MF indexes having log-di erenced weekly initial claims in the indicator sets. Also, in nine among these ten cases, prediction models are using smoothed MF indexes. The forecast gains of using the transformed one, compared with the case of using the level of initial claims, range from 8.8 to 26.6 percent gains by comparing MF_Q_SL, MFAR_Q_SL, MF_T_SL, MFDI_T_SL, MFDIAR_T_SL with MF_C_SL, MFAR_C_SL, MF_F_SL, MFDI_F_SL, MFDIAR_F_SL, respectively. 4-step ahead increments of GDP predictions from MF models are poorly performed among all GDP forecasts, since most of the models rarely improve AR(SIC) model. Compared to 4-step ahead increments of GDP forecasts, 8-step ahead forecasts are slightly better. The best ve models improve AR(SIC) model by around 3.5 % with rejection of the Diebold-Mariano one-sided test at ten percent level. One feature that best ve models share is that they contain the daily spread. I turn to the cases of three monthly variables. In these cases, prediction models using di usion indexes with MF indexes are often listed in the best models at each horizon and target variable. Especially, for IP and UR, MFDI or MFDIAR models perform better in average of a target forecasts than in the increments. In the increments forecasts, MF or MFAR models are more frequently listed in the best ve models than models containing di usion indexes. For CPI forecasts, MFDI models are mostly listed in the top ve models with all model s combination. First monthly variable I forecast is the industrial production. Unlike the case of GDP forecasts, in the best ve models having no survey related model and only one model combination, eight prediction models having survey information and ve model combinations are ranked. Among eight

42 30 survey related prediction models, seven cases are MF or MFAR models. That suggests that the survey information is useful in the MF index construction and sometimes provide better predictive information than di usion indexes does. In one-month ahead predictions, the four best models are MFDIAR models and one MF model. Among them, two models have survey information. Especially, without di usion indexes and autoregressive term, MF_G_SL can predict comparably to the top models. From the panel D of gure 2.7, MF_G_SL predict one-step ahead IP better than top four models prior to the the Great Recession. In comparison with the case of short-term GDP forecasts ( ve MF or MFAR models being listed in the top- ve models), only one MF model is ranked in one-month ahead IP forecasts. This implies that combining the information contained in both MF and di usion indexes are more useful in one-month ahead IP forecasting than using that of only one index. This tendency lasts up to 3-month ahead average of IP forecasts. In these cases, combining MF indexes and DI indexes improve DI or DIAR model up to 9 percents. However, from six-month to 12-month ahead forecasts, MF or MFAR models are listed 16 among 20 cases (including CMA5). From gure 8, the di usion index model predicts less accurately in the case of 3-month ahead increments of IP forecasts than the case of average, compared to AR(SIC) model. Also, its forecasting performance is less pronounced in the long-term forecasts than in the short-term forecasts. In 6- to 12-month ahead increments of IP forecasts, my MF models are hard to improve AR(SIC) model (Appendix Table 2.5 A2), since some best performing models show weak improvements over AR(SIC) model. The prediction models using MF indexes with di usion indexes are the most pronounced in the case of average of unemployment among my target variables. Some features are notable compared to the previous GDP and IP cases. Similar to the case of GDP, having only one model combination ranked in the best ve models, three cases of model combinations are listed in 35 cases of the best ve models. The log-di erence transformation of weekly IC also is in e ect from 6- to 12-month ahead forecasts, as prediction models using those MF indexes are listed 19 among 20 cases. Similar to IP forecasts, MF or MFAR models are performing better in predicting increments of unemployment rate than models having di usion indexes. Also, its forecast improvements is weak as in IP or GDP case. However, unlike the cases of GDP and IP, prediction models using MF indexes from only weekly and/or monthly indicators with di usion indexes and autoregressive term are listed four among ten cases in the 1- and 3-month ahead average of unemployment. This suggests that the short-term predictive content in weekly information is more useful in the case of unemployment than the cases of GDP or IP. Next, I present the results in the case of forecasting level of in ation based on the consumer price index. One speci cation of the Phillips curve forecast is when the target is an average of CPI for a

43 31 forecasting horizon h, h t+h = c + (L)u t + (L) t + " t+h; (2.22) where h t+h = (100) ln(p t+h=p t ) is the h-month CPI in ation, u t is the unemployment rate. If I forecast the increments of in ation from the previous month, h t+h is replaced with (100) ln(p t+h=p t+h 1 ). Two speci cations are considered in this study. First is the forecasting regression having only unemployment and its lag as forecasters. Second is the standard Phillips curve speci cation that consider autoregressive lags together with unemployment as regressors. The information criterion I use to select the number of lags is Schwarz Information Criterion. Both speci cations of Phillips curve forecasts produce less accurate forecast than AR(SIC) model for all cases. For DL or DLAR model, they show less accurate forecasts than AR(SIC) model. Also, every predictions from MF models are less accurate than AR(SIC) or DI model. With autoregressive component, MF indexes, Phillips curve and DL model show signi cant improvements, resulting in closing the gap between AR(SIC) model and those models for any horizons. Figure 2.12 shows this tendency in 3- to 12-month ahead CPI forecasts. MF models, like other models suggested in the gures, make much less accurate forecasts during 1990s period, while those improve during 2000s. Considering observation from AR(SIC) model during these periods, which produces smaller MSFE before 2000 and become less accurate after 2001 s recession, this inverted V-shape caused by AR(SIC) model is applied to models presented in the gures. However, prediction models having di usion indexes or AR component seems to predict CPI in ation more accurately than other prediction models having no di usion indexes or AR component at longer than 3-month horizon. In Table 2.7, forecast combination using groups of models are suggested. In CPI forecasts, forecast combination using all MFDI models-having MF index, di usion indexes and those lags-show more accurate prediction on average than di erent model combinations. The forecast performance of DI model is especially notable for 12-month ahead average of CPI, i.e., the level of in ation (Stock and Watson 1999). Also, this performance is achieved without autoregressive term. Rather, with autoregressive component, DIAR or MFDIAR models often show less accurate forecasts than DI or MFDI models. In gure 2.12, among all models presented, MFDIAR model in four cases shows comparable performance to AR(SIC) model from 1990 to the Great Recession period. However, in all four cases, MFDIAR model becomes less accurate than AR(SIC) model and MFDI model after the Great Recession period. Considering that gures present accumulative relative MSFE from estimation starting point, this change is interesting and abrupt. For MFDI model, which shows less accurate forecasts than AR(SIC) model during the same periods- from 1990 to 2007, it becomes more accurate

44 32 than AR(SIC) model and MFDIAR model beyond the Great Recession period. It may imply that autoregressive term of CPI in ation during With di usion indexes, MF indexes sometimes improve the autoregression in the forecast horizons longer than three monthst. Especially, MFDI prediction models using di usion indexes with MF indexes containing only weekly or daily information such as W1 and D1, are ranked 9 among 35 cases in best ve prediction models. In addition, prediction models using MF indexes containing survey information as well as di usion indexes are listed three cases. However, overall improvements of adding MF indexes are limited in the case of in ation forecasting. In gure 2.10, it is con rmed that the performance of prediction models using only MF index (MF model) or DI model predict less accurately than AR(SIC) model from 1990 to Model combinations in the case of in ation forecasting are listed as the single best model at every cases, and the forecast gains are up to 5 percent compared to the second best model. Note that the all models combinations (CMA10) are better than the model combinations of the factor forecasting models (CMA2-9). The model combinations of factor forecasting models is predicting less accurately than the model combination of benchmark models or of all models. The contributions, on average, of autoregressive term di er by target variables and horizons. MFAR models on average give positive forecast gains over MF models, except 1.2 negative gain in the case of 8-quarter ahead increments of GDP forecast. Also, adding autoregressive term to MFDI models on average give positive forecast gains in every GDP forecasts. Adding di usion index make MF model predict more accurately, on average, in the one-quarter ahead forecast. From 2-quarter ahead, the contribution of di usion index to MF indexes become negative, on average. In CPI forecasts, with the presence of di usion indexes, autoregressive component makes forecasts less accurate, except one-month ahead forecasts. Apart from the analysis based on the model combination, which is based on equal weight average of all models, since model combination can reduce the individual model s estimation bias or forecast swings, the distributional plots of each group of models in Table 6 may show di erent picture. For example, the 2.5 percentile of nonsmoothed MF indexes predict one-quarter ahead GDP more smoothly than a smoothed MF index can, which is captured in the short-term best ve models unlike the case of model combinations. However, in 2-, 4- and 8-step ahead forecasts for both targets, 2.5 percentile of smoothed MF indexes predict accurately than that of nonsmoothed indexes, suggesting that higher ranking models with smoothed MF indexes predict better than the ones from nonsmoothed MF indexes in the medium to longer term GDP forecast. Together with IP, UR and CPI cases, from six-month ahead (two-quarter ahead in GDP) forecasts, smoothed MF Indexes in the 2.5 percentile predict more accurately than nonsmoothed MF indexes in the 2.5 percentile. Note

45 33 that in CPI forecasts, almost every MFDI models are more accurate than MFDIAR models at every horizons except one-month ahead forecasts (Appendix Table A6 Panel D. MFDI-MFDIAR). The following provides auxiliary observations. First is that weekly initial claims seem to have predictive contents on the two groups of four variables. Speci cally, MF indexes having weekly or weekly-monthly information are listed nine cases in the UR and CPI forecasting while there are zero cases for the GDP and IP forecasting. Next is, that prediction models using the smoothed MF index from only quarterly GDP is listed three times out of ten cases of 12-month ahead IP forecasts. This observation may suggest that the information in the quarterly GDP using Kalman smoother may provide useful information regarding the longer term forecasts of IP. As of the two versions of target variables, it is harder to improve AR(SIC) model in the case of increments of targets than that of average, and the speci cations of best models are not always alike. In addition, in the increments forecasts for real GDP, IP and unemployment, models having di usion indexes such as DI, MFDI or MFDIAR models are much less often listed in the top- ve models than MF or MFAR models are. 2.5 Conclusions In summary, I discussed the forecasting performances of the MF indexes extracted from the mixed frequency dynamic factor model using "small-mixed-frequency dataset" in Aruoba, Diebold, and Scotti (2009). Also, the static principal components using "large-same-frequency dataset" in Stock and Watson (2002a,b, 2006) are compared and implemented with my MF indexes. In extracting MF indexes from ADS (2009), I experiment with di erent variations of data series and of models. With two classes of factors, I assess these factor prediction models performances in forecasting quarterly U.S. real GDP and three monthly variables: industrial production, unemployment and CPI in ation rate. The model using only MF indexes show the best performance in the one-quarter ahead forecasts of real GDP growth, especially during recession periods. In the 8-quarter ahead forecasts, the model using only MF indexes show moderate performance by itself. When forecasting three monthly variables, MF indexes complements di usion indexes in a synergetic way. That is, models using MF indexes and di usion indexes predict the monthly average of target variables more accurately than models using one sort of indexes. Also, survey information about the shortterm future economic conditions is usefully implemented in the mixed frequency factor model with monthly or higher frequency data. Prediction models using MF indexes from survey information sometimes provides useful predictive information.

46 34 Table 2.1: Top Proxies for the ADS Index based on R 2 Ranking Indicator for MF index construction R 2 1: All Employees on non-agricultural Payroll 0:70 2: Industrial Production 0:69 3: Real Manufacturing, Trade and Sales 0:37 4: Real Gross Domestic Product 0:36 5: Initial Claims for Unemployment Insurance 0:24 6: Real Personal Income less Transfer Payments 0:21 () Notes: This table shows ranking of how much each indicator can explain the variability of MF index. The indicators are weekly intial claims, four monthly industrial production, payroll, real manufacturing trade sales, and real personal income less transfer payment, and one quarterly real GDP. The R-squares are presented in the right side of indicators.

47 35 Table 2.2: Target Variables Frequency Series Abbreviation Y A t+h Y I t+h Quarterly Real Gross Domestic Product GDP 100 ln(z t+h =Z t) 100 ln(z t+h =Z t+h 1 ) Monthly Industrial Production IP 100 ln(z t+h =Z t) 100 ln(z t+h =Z t+h 1 ) Unemployment Rate UR Z t+h Z t Z t+h Z t+h 1 Consumer Price Index CPI 100 ln(z t+h =Z t) 100 ln(z t+h =Z t+h 1 ) () Notes: All data are U.S.

48 36 Table 2.3: Prediction Models Panel A : Model Speci cations* Model Description Autoregressive model with lags selected by SIC AR(SIC) y T +hjt = b h p 0 + P b h i y T i=1 i+1 Distribute Lag model selected by SIC DL y T +hjt = b h 0 + P K pp b j;h i X j T i+1 j=1 i=1 Autoregressive Distributed Lag model selected by SIC DLAR y T +hjt = b h p 0 + P b k;h P i y T i+1 + k pp b j;h i X j T i+1 i=1 j=1 i=1 Di usion Index model DI y T +hjt = b h 0 + P K pp b j;h i DI j T i+1 j=1 i=1 Di usion Index model with AR component DIAR y T +hjt = b h p 0 + P b h i y P T i+1 + K pp b j;h i DI j T i+1 i=1 k=1 i=1 MF : Mixed Frequency Models MF_X_NSF y T +hjt = b h p 0 + P b h i X_NSF T i+1 i=1 MF_X_SF y T +hjt = b h p 0 + P qp b h i y T i+1 + b h i X_SF T i+1 i=1 i=1 MF_X_NSL y T +hjt = b h p 0 + P b h i X_NSL T i+1 i=1 MF_X_SL y T +hjt = b h p 0 + P qp b h i y T i+1 + b h i X_SL T i+1 i=1 i=1 MFAR: Mixed Frequency Models with AR component MFAR_X_NSF y T +hjt = b h p 0 + P qp b i y T i+1 + b h i X_NSF T i=1 i=1 i+1 MFAR_X_SF y T +hjt = b h p 0 + P b i X_SF T i+1 i=1 MFAR_X_NSL y T +hjt = b h p 0 + P qp b h i y T i+1 + b h i X_NSL T i+1 i=1 i=1 MFAR_X_SL y T +hjt = b h p 0 + P qp b h i y T i+1 + b h i X_SL T i+1 i=1 i=1 MFDI: Mixed Frequency Models with DI component MFDI_X_NSF y T +hjt = b h p 0 + P b h i X_NSF P T i+1 + k qp b j;h i DI j T i+1 i=1 j=1 i=1 MFDI_X_SF y T +hjt = b h p 0 + P b h i X_NSF P T i+1 + k qp b j;h i DI j T i+1 i=1 j=1 i=1 MFDI_X_NSL y T +hjt = b h p 0 + P b h i X_NSF P T i+1 + k qp b j;h i DI j T i+1 i=1 j=1 i=1 MFDI_X_SL y T +hjt = b h p 0 + P b h i X_NSF P T i+1 + k qp b j;h i DI j T i+1 i=1 j=1 i=1 MFDIAR: Mixed Frequency Models with DI and AR components MFDIAR_X_NSF y T +hjt = b h p 0 + P pp b h i y T i+1 + b h i X_NSF P T i+1 + k qp b j i DIj T i+1 i=1 i=1 j=1 i=1 MFDIAR_X_SF y T +hjt = b h p 0 + P pp b h i y T i+1 + b h i X_SF P T i+1 + k qp b j i DIj T i+1 i=1 i=1 j=1 i=1 MFDIAR_X_NSL y T +hjt = b h p 0 + P pp b h i y T i+1 + b h i X_NSL P T i+1 + k qp b j i DIj T i+1 i=1 i=1 j=1 i=1 MFDIAR_X_SL y T +hjt = b h p 0 + P pp b h i y T i+1 + b h i X_SL P T i+1 + k qp b j i DIj T i+1 i=1 i=1 j=1 i=1

49 37 Table 2.3: Panel A. (Cont.) Model Description Forecast Combinations NP y T +hjt = 1 by N ;model averaging with equal weights,using selected models from Panel A: T +hjt i=1 CMA1 All benchmark models i.e, AR, DL, DLAR, DI and DIAR CMA2 All nonsmoothed MF models CMA3 All smoothed MF models CMA4 All nonsmoothed MFAR models CMA5 All smoothed MFAR models CMA6 All nonsmoothed MFDI models CMA7 All smoothed MFDI models CMA8 All nonsmoothed MFDIAR models CMA9 All smoothed MFDIAR models CMA10 All models Panel B. Mnemonic De nitions First Day Last Day Smoothed X_SF X_SL Nonsmoothed X_NSF X_NSL () Notes: Prediction models are presented with respect to the average of a target variable for the forecasting horizon h. Prediction models are grouped into six categories. First group is benchmark prediction models, i.e, AR, DL, DLAR, DI and DIAR. Second group is denoted as MF, which we refer to as MF models, using only Mixed Frequency indexes in the prediction model. Third group is denoted MFAR, MFAR models which make use of autoregressive (AR) component with MF index. Fourth group is denoted MFDI, which make use of mixed frequency index with di usion indexes in the prediction model. Fifth group is MFDIAR, which add AR component to MFDI models. Sixth group is in regard to forecast combinations. In our applications, model averaging with equal weights for a group of prediction models. For mnemonics of X_NSF, X_SF, X_NSL, or X_SL, X stands for an indicator set to construct a mixed frequency index and X=A,B,:::,T,D1,:::,INF, described in Panel B in Table 4. _NSF,_SF,_NSL or _SL represent two modi cations, First is whether the index is smoothed or not. Second is that whether monthly or quarterly frequency data are set to " rst" day or "last" day of a month or a quarter. _NSF is mnemonics, which divide into NS and F. NS is an abbreviation of "Nonsmoothed", that is, nonsmoothed mixed frequency index. F means an indicator set, where monthly and/or quarterly data are set at the "First" day of the month or the quarter. _SF is mnemonics, which divide into S and F. S is an abbreviation of "Smoothed", that is, smoothed mixed frequency index. F means an indicator set, where monthly and/or quarterly data are set at the "First" day of the month or the quarter. _NSL is mnemonics, which divide into NS and L. NS is an abbreviation of "Nonsmoothed", that is, nonsmoothed mixed frequency index. L means an indicator set, where monthly and/or quarterly data are set at the "Last" day of the month or the quarter. Again, _SL is mnemonics, which divide into S and L. S is an abbreviation of "smoothed", that is, smoothed mixed frequency index. L means an indicator set, where monthly and/or quarterly data are set at the "Last" day of the month or the quarter. That is, L means that all monthly indicators data are set to the "Last" day of the month and quarterly data are set to the "Last" day of the quarter. That is, F means that all monthly indicators data are set to the "First" day of the month and quarterly data are set to the "First" day of the quarter. In the previous sentence, we use terms such as "and/or" in labelling "monthly and/or quarterly" to construct the index for forecasting purposes. For forecasting a quarterly variable, both monthly and quarterly data can set at the rst day. However, for monthly variables, only monthly indicators can be set to the rst day of the month. If we set quarterly variable at the rst day of the quarter and extracting an index, this index cannot be used for forecasting. Thus, for forecasting quarterly variables, the First dataset sets both quarterly and monthly data to the rst day of each period. For forecasting monthly targets, only monthly indicators are matched at the rst day of the month. This "First" and "Last" experiment is performed in the indicator sets from A to F. All other cases of indicator sets are having _NSL or _SL. Related Kalman lter and smoothing algorithm of index is given in Sec 2.2.

50 38 Table 2.4: Indicator Sets for Construction of Mixed Frequency Indexes Panel A. Variable De nition Frequency Variables Abbreviation Transformation Daily Government Bond Spread SPR X t X t 1 Weekly Initial Claims for Unemployment Insurance Level IC1 X t Growth Rate IC2 ln(x t) ln(x t 1 ) Monthly Payroll Pay ln(x t) ln(x t 1 ) Industrial Production IP ln(x t) ln(x t 1 ) Real Manufacturing Trade & Sales RM ln(x t) ln(x t 1 ) Real Personal Income less Transfer Payment PI ln(x t) ln(x t 1 ) Consumer Price Index CPI ln(x t) ln(x t 1 ) Quarterly Real GDP GDP ln(x t) ln(x t 1 ) Real GDP, mean, SPF SPF1 X t Real GDP, median, SPF SPF2 X t Real GDP, mean, Livingston LIV1 X t Real GDP, median, Livingston LIV2 X t () Notes: Spread is di erence between 10 year Treasury Constant Maturity rate and 3 month Treasury Bill. Both rates are all denoted in percentage and not seasonally adjusted. The range is from January 02 in 1962 to June 28 in Initial Claims of the dataset is modi ed in another way. First, we use the level of Initial Claims as in ADS (2009). Second, the weekly growth rate of initial claims. from 1967 January 7 to 2013 June 29 is used. Growth rate of real GDP in one-quarter ahead forecast is taken from Survey of Professional Forecasters. Growth rate of real GDP in two-quarter ahead forecast is taken from Livingston Survey. All other variables span from 1960 January to 2013 June. Panel B. Indicator Sets for Construction of Mixed Frequency Indexes Set Variables used in Index Construction Set Variables used in Index Construction D1 SPR W1 IC1 WM4 IC1,Pay, IP, RM, PI M1 Pay Q1 GDP M2 Pay, IP MQ1 Pay, GDP M3 Pay, IP, RM MQ2 Pay, IP, GDP M4 Pay, IP, RM, PI MQ3 Pay, IP, RM, GDP WM1 IC1,Pay MQ4 Pay, IP, SPF1 WM2 IC1,Pay,IP MQ5 Pay, IP, RM, PI, GDP WM3 IC1,Pay, IP, RM INF IC1,Pay, IP, RM, PI, CPI, GDP A IC1, Pay, IP, GDP K SPR, IC1, Pay, IP, RM, PI, GDP, SPF1 B IC1, Pay, IP, RM, GDP L SPR, IC1, Pay, IP, RM, PI, GDP, SPF2 C IC1, Pay, IP, RM, PI, GDP M SPR, IC1, Pay, IP, RM, PI, GDP, LIV1 D SPR, IC1, Pay, IP, GDP N SPR, IC1, Pay, IP, RM, PI, GDP, LIV2 E SPR, IC1, Pay, IP, RM, GDP O IC2, Pay, IP, GDP F SPR, IC1, Pay, IP, RM, PI, GDP P IC2, Pay, IP, RM, GDP G IC1, Pay, IP, RM, PI, GDP, SPF1 Q IC2, Pay, IP, RM, PI, GDP H IC1, Pay, IP, RM, PI, GDP, SPF2 R SPR, IC2, Pay, IP, GDP G IC1, Pay, IP, RM, PI, GDP, LIV1 S SPR, IC2, Pay, IP, RM, GDP J IC1, Pay, IP, RM, PI, GDP, LIV2 T SPR, IC2, Pay, IP, RM, PI, GDP () Notes: Each mixed frequency (MF) index is constructed in the mixed frequency factor model in Section 2.2. This table shows the various indicator sets for constuction of MF indexes. For example, Variable abbreviation is described in Panel A. Note that the Mixed Frequency Indexes are all daily basis. For example, Q1 means that the daily mixed frequency index extracted from one quarterly GDP.

51 39 Table 2.5: Summary of MSFE- ve "Best" models Across Model Speci cations* Panel A : GDP s Top ve MSFE "Best" Prediction Models Ranking Model RMSFE GDP t+1 1 MF_B_NSL 0:778 2 MFAR_O_NSL 0:779 3 MFAR_B_NSL 0:780 4 MF_O_NSL 0:782 5 MFAR_MQ2_NSL 0:783 GDP A t+2 1 MFDI_MQ1_SL 0:775 2 MFDI_MQ2_SL 0:780 3 MF_A_NSL 0:791 4 MFDIAR_MQ2_SL 0:791 5 MFDIAR_MQ1_SL 0:793 GDP A t+4 1 MFDIAR_Q_SL 0:839 2 MFAR_Q_SL 0:841 3 MFDI_Q_SL 0:848 4 MF_Q_SL 0:851 5 MFDI_O_SL 0:858 GDP A t+8 1 MF_Q_SL 0:866 2 MFAR_Q_SL 0:883 3 MF_T_SL 0:884 4 MFDI_T_SL 0:890 5 MFDIAR_T_SL 0:890 GDP I t+2 1 MFDIAR_MQ1_SL 0:877 2 MFDI_MQ1_SL 0:879 3 MFDI_O_SL 0:908 4 MFDIAR_O_SL 0:910 5 MFDI_MQ2_SL 0:911 GDP I t+4 1 MFAR_INF_SL 0:989 2 MFAR_Q1_SL 0:993 3 MFDIAR_T_SL 0:993 4 MFAR_T_SL 0:994 5 MFAR_C_SF 0:994 GDP I t+8 1 MF_F_SF 0:964 2 MF_D_SF 0:965 3 MF_D1_NSL 0:967 4 MF_D1_SL 0:968 5 MF_E_SF 0:968

52 40 Table 2.5: (Cont.) Panel B : IP s Top ve MSFE "Best" Prediction Models Ranking Model RMSFE IP t+1 1 MFDIAR_B_NSF 0:845 2 MFDIAR_H_SL 0:847 3 MFDIAR_C_NSF 0:850 4 MFDIAR_C_SL 0:852 5 MF_G_SL 0:852 IP A t+3 1 MFDIAR_P_SL 0:794 2 CMA9 0:805 3 MFDIAR_S_SL 0:831 4 MFDIAR_C_NSF 0:834 5 MFDIAR_B_NSF 0:837 IP A t+6 1 MFAR_B_SF 0:877 2 MFDIAR_Q_SL 0:880 3 CMA9 0:883 4 MF_A_SF 0:884 5 MFAR_M4_SL 0:887 IP A t+12 1 MF_M_SL 0:900 2 MFAR_M_SL 0:907 3 MFDIAR_Q1_SL 0:929 4 MFDI_Q1_SL 0:930 5 CMA5 0:944 IP I t+3 1 MFDI_Q_SL 0:981 2 CMA10 0:984 3 MF_H_NSL 0:986 4 CMA7 0:987 5 MF_G_NSL 0:987 IP I t+6 1 MFAR_S_SL 0:971 2 MF_S_SL 0:973 3 MF_B_SF 0:974 4 MF_C_SF 0:976 5 MFAR_MQ3_SL 0:978 IP I t+12 1 MF_M_SL 0:983 2 MFAR_M_SL 0:983 3 MF_Q1_SL 0:988 4 MF_K_SL 0:992 5 MFAR_K_SL 0:993

53 41 Table 2.5: (Cont.) Panel C : UR s Top ve MSFE "Best" Prediction Models Ranking Model RMSFE UR t+1 1 MFDIAR_W1_NSL 0:809 2 MFDIAR_R_NSL 0:815 3 MFDIAR_M_NSL 0:820 4 MFDIAR_B_SL 0:822 5 MFDIAR_W1_SL 0:823 UR A t+3 1 MFDIAR_R_SL 0:665 2 MFDIAR_WM1_SL 0:670 3 MFDI_WM1_SL 0:670 4 CMA9 0:671 5 MFDI_R_SL 0:674 UR A t+6 1 MFDIAR_T_SL 0:663 2 MFDI_T_SL 0:671 3 MFDIAR_Q_SL 0:674 4 MFDI_Q_SL 0:691 5 MFAR_S_SL 0:701 UR A t+12 1 MF_T_SL 0:769 2 MFAR_T_SL 0:772 3 MF_Q_SL 0:775 4 MFDIAR_T_SL 0:775 5 MFDI_T_SL 0:779 UR I t+3 1 MFDI_M_SL 0:890 2 CMA7 0:894 3 MF_E_SF 0:897 4 CMA10 0:898 5 MF_MQ3_NSL 0:899 UR I t+6 1 MFAR_T_SL 0:947 2 MF_T_SL 0:947 3 MFDI_T_SL 0:948 4 MF_Q_SL 0:949 5 MFAR_Q_SL 0:949 UR I t+12 1 CMA10 0:961 2 MF_T_SL 0:962 3 MF_S_SL 0:963 4 MF_Q_SL 0:963 5 MF_K_SL 0:964

54 42 Table 2.5: (Cont.) Panel D : CPI s Top ve MSFE "Best" Prediction Models Ranking Model RMSFE CPI t+1 1 CMA MFDIAR_S_NSL MFAR_MQ5_SL MFAR_I_SL MFAR_D_SL CPI A t+3 1 CMA MFDI_WM3_SL MFDI_WM4_SL MFDI_MQ1_SL MFDI_MQ1_NSL CPI A t+6 1 CMA10 0:793 2 MFDI_W1_NSL 0:854 3 MFDI_WM3_SL 0:856 4 MFDI_MQ5_SL 0:856 5 MFDI_MQ1_SL 0:859 CPI A t+12 1 CMA10 0:693 2 MFDI_WM3_SL 0:801 3 MFDI_MQ5_SL 0:802 4 MFDI_WM4_SL 0:810 5 MFDI_D1_SL 0:811 CPI I t+3 1 CMA10 0:885 2 MFDI_WM3_SL 0:915 3 MFDI_WM4_SL 0:915 4 MFDI_F_NSF 0:916 5 MFDI_D1_NSL 0:917 CPI I t+6 1 CMA10 0:925 2 CMA1 0:968 3 MFDI_W1_SL 0:970 4 MFDI_MQ5_SL 0:970 5 MFDI_D1_SL 0:973 CPI I t+12 1 CMA10 0:905 2 MFDI_K_SL 0:944 3 MFDI_F_SF 0:945 4 MFDI_S_SL 0:947 5 MFDI_E_SF 0:948 () Notes: This table shows the best ve models for each target variable and each horizon. Prediction results are presented in two versions of target variables: average of the target for a forecast horizon h and increments of the target from the previous period. In Table 2.2, speci c tranformation is given. Numerical entries are relative Mean Square Forecast Error (RMSFE) compared to the MSFE of AR (SIC) model. Single, double and triple stars signify, respectively, the rejection at 10, 5 and 1 percent level of the one-sided Diebold-Mariano (1995) predictive accuracy test. More prediction results are provided in Appendix Table 5 A1 A4.

55 43 Table 2.6: Average of RMSFEs of Groups of Prediction Models h GDP A t+h GDP I t+h M 0:871 0:879 0:990 1:056 0:984 1:057 1:054 WM 0:974 0:955 0:964 1:136 1:025 1:032 1:047 MQ 0:907 0:898 0:973 1:052 0:982 1:055 1:053 WMQ 0:893 0:894 0:962 1:058 0:997 1:042 1:043 DWMQ 0:958 0:936 0:958 1:154 1:004 1:031 1:057 MF 0:966 0:964 0:992 1:058 1:041 1:045 1:025 MFAR 0:948 0:924 0:964 1:056 1:001 1:030 1:037 MFDI 0:923 0:938 0:983 1:103 1:041 1:061 1:050 MFDIAR 0:930 0:935 0:976 1:103 1:021 1:043 1:059 h IP A t+h IP I t+h M 0:958 0:955 0:971 1:038 1:023 1:005 1:032 WM 1:071 0:952 0:983 1:048 1:017 1:018 1:027 MQ 0:954 0:966 0:996 1:047 1:029 1:020 1:029 WMQ 0:952 0:941 0:977 1:035 1:018 1:008 1:029 DWMQ 0:943 0:959 0:993 1:043 1:021 1:010 1:029 MF 0:985 0:997 0:991 1:004 1:038 1:006 1:006 MFAR 0:990 0:984 0:990 0:999 1:017 1:006 1:011 MFDI 0:939 0:938 0:995 1:089 1:025 1:017 1:046 MFDIAR 0:917 0:910 0:980 1:074 1:028 1:016 1:047 UR A t+h UR I t+h M 0:890 0:772 0:825 0:920 0:934 0:979 1:001 WM 0:889 0:752 0:801 0:906 0:934 0:991 0:995 MQ 0:889 0:775 0:829 0:928 0:935 0:991 1:002 WMQ 0:868 0:755 0:801 0:911 0:929 0:977 0:999 DWMQ 0:871 0:775 0:827 0:910 0:938 0:982 0:996 MF 0:908 0:833 0:880 0:910 0:945 0:986 0:985 MFAR 0:895 0:819 0:852 0:894 0:945 0:984 1:000 MFDI 0:876 0:738 0:801 0:941 0:925 0:982 1:000 MFDIAR 0:853 0:732 0:797 0:940 0:934 0:985 1:007 CPI A t+h CPI I t+h M 1:103 1:110 1:106 1:076 1:010 1:058 1:009 WM 1:096 1:080 1:073 1:060 0:997 1:039 1:009 MQ 1:097 1:106 1:098 1:072 1:007 1:051 1:005 WMQ 1:106 1:110 1:110 1:088 1:015 1:057 1:010 DWMQ 1:099 1:094 1:099 1:081 1:008 1:050 1:015 MF 1:320 1:273 1:282 1:386 1:046 1:107 1:057 MFAR 0:998 1:044 1:092 1:032 1:015 1:046 1:019 MFDI 1:077 0:998 0:906 0:879 0:939 0:994 0:961 MFDIAR 1:007 1:086 1:117 1:031 1:035 1:056 1:009 () Notes: This table shows the average of RMSFEs in a group of models. The mnemonics of the group of models are presented in the rst column. M is the average of all RMSFEs from prediction models using MF indexes constructed in monthly indicator sets. WM is the average of all RMSFEs from prediction models using MF indexes constructed in Weekly-monthly indicator sets. MQ is the average of all RMSFEs from prediction models using MF indexes constructed in monthly-quarterly indicator sets. WMQ is the average of all RMSFEs from prediction models using MF indexes constructed in weekly-monthly-quarterly indicator sets. DWMQ is the average of all RMSFEs from prediction models using MF indexes constructed in Daily-weekly-monthly-quarterly indicator sets. In all these sets, predictions using MF indexes, di usion indexes, or autoregressive components are included. MF is the average of all RMSFEs from prediction models using only MF indexes. MFAR is the average of all RMSFEs from MFAR models. MFDI is the average of all RMSFEs from MFDI models. MFDIAR is the average of all RMSFEs from MFDIAR models. In each set, predictions using smoothed or nonsmoothed MF indexes are contained.

56 44 Table 2.7: One-Quarter ahead GDP Forecasts during Business cycles Target GDP t+1 Target GDP t+1 Business Cycles Expansion Recession Business Cycles Expansion Recession DL 1:854 0:463 MF_MQ3_NSL 1:118 0:403 DLAR 1:823 0:432 MFAR_MQ3_NSL 1:132 0:405 DI 1:263 0:482 MFDI_MQ3_NSL 1:137 0:411 DIAR 1:270 0:492 MFDIAR_MQ3_NSL 1:146 0:410 MF_Q1_NSL 1:038 1:029 MF_MQ3_SL 1:247 0:345 MFAR_Q1_NSL 1:004 1:044 MFAR_MQ3_SL 1:243 0:358 MFDI_Q1_NSL 1:329 0:500 MFDI_MQ3_SL 1:269 0:366 MFDIAR_Q1_NSL 1:344 0:499 MFDIAR_MQ3_SL 1:259 0:377 MF_Q1_SL 1:082 0:926 MF_MQ4_NSL 0:984 0:536 MFAR_Q1_SL 1:088 1:022 MFAR_MQ4_NSL 0:983 0:542 MFDI_Q1_SL 1:190 0:511 MFDI_MQ4_NSL 1:304 0:573 MFDIAR_Q1_SL 1:176 0:518 MFDIAR_MQ4_NSL 1:301 0:574 MF_M1_NSL 0:977 0:622 MF_MQ4_SL 1:132 0:422 MFAR_M1_NSL 0:975 0:641 MFAR_MQ4_SL 1:143 0:434 MFDI_M1_NSL 1:335 0:575 MFDI_MQ4_SL 1:146 0:403 MFDIAR_M1_NSL 1:331 0:579 MFDIAR_MQ4_SL 1:155 0:413 MF_M1_SL 1:093 0:700 MF_MQ5_NSL 1:324 0:419 MFAR_M1_SL 1:149 0:751 MFAR_MQ5_NSL 1:312 0:421 MFDI_M1_SL 1:192 0:444 MFDI_MQ5_NSL 1:287 0:432 MFDIAR_M1_SL 1:208 0:463 MFDIAR_MQ5_NSL 1:288 0:442 MF_M2_NSL 0:982 0:546 MF_MQ5_SL 1:946 0:429 MFAR_M2_NSL 0:981 0:552 MFAR_MQ5_SL 1:870 0:420 MFDI_M2_NSL 1:310 0:575 MFDI_MQ5_SL 1:569 0:421 MFDIAR_M2_NSL 1:308 0:576 MFDIAR_MQ5_SL 1:572 0:424 MF_M2_SL 1:137 0:426 MF_W1_NSL 1:419 1:044 MFAR_M2_SL 1:149 0:438 MFAR_W1_NSL 1:359 0:884 MFDI_M2_SL 1:153 0:403 MFDI_W1_NSL 1:349 0:509 MFDIAR_M2_SL 1:163 0:411 MFDIAR_W1_NSL 1:365 0:518 MF_M3_NSL 1:162 0:403 MF_W1_SL 1:610 1:045 MFAR_M3_NSL 1:168 0:408 MFAR_W1_SL 1:567 0:975 MFDI_M3_NSL 1:172 0:400 MFDI_W1_SL 1:403 0:539 MFDIAR_M3_NSL 1:181 0:409 MFDIAR_W1_SL 1:417 0:547 MF_M3_SL 1:213 0:372 MF_WM1_NSL 0:928 1:122 MFAR_M3_SL 1:223 0:380 MFAR_WM1_NSL 0:988 0:850 MFDI_M3_SL 1:254 0:378 MFDI_WM1_NSL 1:267 0:485 MFDIAR_M3_SL 1:258 0:380 MFDIAR_WM1_NSL 1:274 0:495 MF_M4_NSL 1:221 0:394 MF_WM1_SL 1:326 0:628 MFAR_M4_NSL 1:234 0:399 MFAR_WM1_SL 1:359 0:582 MFDI_M4_NSL 1:223 0:398 MFDI_WM1_SL 1:396 0:510 MFDIAR_M4_NSL 1:234 0:404 MFDIAR_WM1_SL 1:405 0:503 MF_M4_SL 1:267 0:398 MF_WM2_NSL 1:011 0:680 MFAR_M4_SL 1:286 0:398 MFAR_WM2_NSL 1:038 0:608 MFDI_M4_SL 1:253 0:396 MFDI_WM2_NSL 1:265 0:487 MFDIAR_M4_SL 1:290 0:399 MFDIAR_WM2_NSL 1:272 0:498 MF_MQ1_NSL 0:970 0:646 MF_WM2_SL 1:505 0:451 MFAR_MQ1_NSL 0:946 0:661 MFAR_WM2_SL 1:473 0:453 MFDI_MQ1_NSL 1:338 0:523 MFDI_WM2_SL 1:573 0:375 MFDIAR_MQ1_NSL 1:356 0:525 MFDIAR_WM2_SL 1:554 0:372 MF_MQ1_SL 1:075 0:671 MF_WM3_NSL 1:095 0:590 MFAR_MQ1_SL 1:205 0:744 MFAR_WM3_NSL 1:111 0:516 MFDI_MQ1_SL 1:142 0:460 MFDI_WM3_NSL 1:268 0:472 MFDIAR_MQ1_SL 1:134 0:504 MFDIAR_WM3_NSL 1:273 0:483 MF_MQ2_NSL 0:995 0:517 MF_WM3_SL 1:514 0:383 MFAR_MQ2_NSL 0:988 0:525 MFAR_WM3_SL 1:478 0:386 MFDI_MQ2_NSL 1:288 0:525 MFDI_WM3_SL 1:603 0:376 MFDIAR_MQ2_NSL 1:280 0:526 MFDIAR_WM3_SL 1:601 0:382 MF_MQ2_SL 1:185 0:401 MF_WM4_NSL 1:240 0:587 MFAR_MQ2_SL 1:192 0:414 MFAR_WM4_NSL 1:238 0:526 MFDI_MQ2_SL 1:171 0:402 MFDI_WM4_NSL 1:286 0:470 MFDIAR_MQ2_SL 1:181 0:426 MFDIAR_WM4_NSL 1:290 0:480 MSFE 0:205 1:137 MSFE 0:205 1:137

57 45 Table 2.7: (Cont.) Target GDP t+1 Target GDP t+1 Business Cycles Expansion Recession Business Cycles Expansion Recession MF_WM4_SL 1:545 0:420 MF_C_NSF 1:232 0:420 MFAR_WM4_SL 1:514 0:398 MFAR_C_NSF 1:250 0:426 MFDI_WM4_SL 1:628 0:381 MFDI_C_NSF 1:265 0:420 MFDIAR_WM4_SL 1:628 0:386 MFDIAR_C_NSF 1:279 0:426 MF_D1_NSL 1:061 1:503 MF_C_SF 1:355 0:385 MFAR_D1_NSL 1:031 0:955 MFAR_C_SF 1:427 0:435 MFDI_D1_NSL 1:259 0:472 MFDI_C_SF 1:343 0:367 MFDIAR_D1_NSL 1:269 0:481 MFDIAR_C_SF 1:286 0:402 MF_D1_SL 1:117 1:513 MF_D_NSL 1:222 0:473 MFAR_D1_SL 1:076 0:968 MFAR_D_NSL 1:205 0:474 MFDI_D1_SL 1:271 0:467 MFDI_D_NSL 1:271 0:475 MFDIAR_D1_SL 1:284 0:477 MFDIAR_D_NSL 1:278 0:486 MF_INF_NSL 1:226 1:009 MF_D_SL 1:313 1:096 MFAR_INF_NSL 1:198 0:927 MFAR_D_SL 1:267 0:821 MFDI_INF_NSL 1:244 0:503 MFDI_D_SL 1:277 0:484 MFDIAR_INF_NSL 1:257 0:517 MFDIAR_D_SL 1:285 0:492 MF_INF_SL 1:488 0:862 MF_D_NSF 1:229 0:895 MFAR_INF_SL 1:522 0:961 MFAR_D_NSF 1:394 0:971 MFDI_INF_SL 1:479 0:780 MFDI_D_NSF 1:256 0:488 MFDIAR_INF_SL 1:483 0:829 MFDIAR_D_NSF 1:276 0:492 MF_A_NSL 1:077 0:462 MF_D_SF 1:306 1:017 MFAR_A_NSL 1:068 0:459 MFAR_D_SF 1:227 1:207 MFDI_A_NSL 1:235 0:507 MFDI_D_SF 1:300 0:990 MFDIAR_A_NSL 1:210 0:508 MFDIAR_D_SF 1:203 1:153 MF_A_SL 1:079 0:749 MF_E_NSL 1:285 0:452 MFAR_A_SL 1:091 0:765 MFAR_E_NSL 1:270 0:437 MFDI_A_SL 1:275 0:480 MFDI_E_NSL 1:291 0:442 MFDIAR_A_SL 1:269 0:485 MFDIAR_E_NSL 1:295 0:452 MF_A_NSF 1:057 0:469 MF_E_SL 1:425 1:197 MFAR_A_NSF 1:050 0:485 MFAR_E_SL 1:343 0:892 MFDI_A_NSF 1:130 0:475 MFDI_E_SL 1:303 0:501 MFDIAR_A_NSF 1:129 0:479 MFDIAR_E_SL 1:315 0:510 MF_A_SF 1:355 0:481 MF_E_NSF 1:229 0:518 MFAR_A_SF 1:455 0:582 MFAR_E_NSF 1:232 0:517 MFDI_A_SF 1:362 0:475 MFDI_E_NSF 1:250 0:411 MFDIAR_A_SF 1:317 0:516 MFDIAR_E_NSF 1:271 0:410 MF_B_NSL 1:046 0:440 MF_E_SF 1:285 0:461 MFAR_B_NSL 1:053 0:436 MFAR_E_SF 1:315 0:494 MFDI_B_NSL 1:083 0:457 MFDI_E_SF 1:285 0:476 MFDIAR_B_NSL 1:088 0:451 MFDIAR_E_SF 1:299 0:519 MF_B_SL 1:001 0:689 MF_F_NSL 1:430 0:435 MFAR_B_SL 1:004 0:696 MFAR_F_NSL 1:407 0:424 MFDI_B_SL 1:177 0:490 MFDI_F_NSL 1:326 0:435 MFDIAR_B_SL 1:174 0:495 MFDIAR_F_NSL 1:330 0:444 MF_B_NSF 1:140 0:420 MF_F_SL 1:417 0:991 MFAR_B_NSF 1:149 0:426 MFAR_F_SL 1:347 0:820 MFDI_B_NSF 1:137 0:421 MFDI_F_SL 1:289 0:487 MFDIAR_B_NSF 1:145 0:429 MFDIAR_F_SL 1:302 0:494 MF_B_SF 1:290 0:391 MF_F_NSF 1:224 0:518 MFAR_B_SF 1:354 0:452 MFAR_F_NSF 1:227 0:520 MFDI_B_SF 1:266 0:395 MFDI_F_NSF 1:248 0:411 MFDIAR_B_SF 1:229 0:434 MFDIAR_F_NSF 1:268 0:411 MF_C_NSL 1:292 0:372 MF_F_SF 1:267 0:461 MFAR_C_NSL 1:320 0:374 MFAR_F_SF 1:297 0:500 MFDI_C_NSL 1:289 0:377 MFDI_F_SF 1:263 0:477 MFDIAR_C_NSL 1:312 0:381 MFDIAR_F_SF 1:279 0:530 MF_C_SL 0:981 0:696 MF_G_NSL 1:426 0:387 MFAR_C_SL 0:986 0:712 MFAR_G_NSL 1:424 0:391 MFDI_C_SL 1:117 0:487 MFDI_G_NSL 1:366 0:394 MFDIAR_C_SL 1:116 0:494 MFDIAR_G_NSL 1:367 0:403 MSFE 0:205 1:137 MSFE 0:205 1:137

58 46 Table 2.7: (Cont.) Target GDP t+1 Target GDP t+1 Business Cycles Expansion Recession Business Cycles Expansion Recession MF_G_SL 1:118 0:700 MFDI_N_NSL 1:344 0:429 MFAR_G_SL 1:115 0:685 MFDIAR_N_NSL 1:346 0:438 MFDI_G_SL 1:122 0:474 MF_N_SL 1:501 0:963 MFDIAR_G_SL 1:138 0:481 MFAR_N_SL 1:391 0:770 MF_H_NSL 1:390 0:385 MFDI_N_SL 1:310 0:473 MFAR_H_NSL 1:391 0:389 MFDIAR_N_SL 1:322 0:479 MFDI_H_NSL 1:353 0:388 MF_O_NSL 1:025 0:476 MFDIAR_H_NSL 1:355 0:396 MFAR_O_NSL 1:016 0:480 MF_H_SL 1:061 0:712 MFDI_O_NSL 1:234 0:509 MFAR_H_SL 1:060 0:703 MFDIAR_O_NSL 1:213 0:511 MFDI_H_SL 1:105 0:479 MF_O_SL 1:150 0:492 MFDIAR_H_SL 1:117 0:488 MFAR_O_SL 1:148 0:510 MF_I_NSL 1:407 0:403 MFDI_O_SL 1:153 0:487 MFAR_I_NSL 1:392 0:400 MFDIAR_O_SL 1:150 0:507 MFDI_I_NSL 1:317 0:426 MF_P_NSL 1:244 0:396 MFDIAR_I_NSL 1:321 0:435 MFAR_P_NSL 1:256 0:402 MF_I_SL 1:169 0:804 MFDI_P_NSL 1:239 0:394 MFAR_I_SL 1:153 0:726 MFDIAR_P_NSL 1:252 0:401 MFDI_I_SL 1:183 0:462 MF_P_SL 1:234 0:388 MFDIAR_I_SL 1:196 0:469 MFAR_P_SL 1:231 0:378 MF_J_NSL 1:205 1:760 MFDI_P_SL 1:282 0:381 MFAR_J_NSL 1:166 1:128 MFDIAR_P_SL 1:302 0:379 MFDI_J_NSL 1:337 0:500 MF_Q_NSL 1:195 0:424 MFDIAR_J_NSL 1:344 0:514 MFAR_Q_NSL 1:232 0:419 MF_J_SL 1:011 1:641 MFDI_Q_NSL 1:203 0:430 MFAR_J_SL 1:000 0:999 MFDIAR_Q_NSL 1:235 0:424 MFDI_J_SL 1:274 0:493 MF_Q_SL 1:453 0:522 MFDIAR_J_SL 1:282 0:503 MFAR_Q_SL 1:483 0:530 MF_K_NSL 1:283 0:513 MFDI_Q_SL 1:467 0:528 MFAR_K_NSL 1:280 0:482 MFDIAR_Q_SL 1:509 0:544 MFDI_K_NSL 1:272 0:460 MF_R_NSL 1:033 0:513 MFDIAR_K_NSL 1:276 0:469 MFAR_R_NSL 1:033 0:548 MF_K_SL 1:363 0:414 MFDI_R_NSL 1:278 0:505 MFAR_K_SL 1:343 0:412 MFDIAR_R_NSL 1:277 0:510 MFDI_K_SL 1:507 0:341 MF_R_SL 1:142 0:528 MFDIAR_K_SL 1:469 0:383 MFAR_R_SL 1:068 0:605 MF_L_NSL 1:506 0:393 MFDI_R_SL 1:232 0:510 MFAR_L_NSL 1:495 0:397 MFDIAR_R_SL 1:209 0:509 MFDI_L_NSL 1:407 0:400 MF_S_NSL 1:190 0:413 MFDIAR_L_NSL 1:409 0:409 MFAR_S_NSL 1:177 0:395 MF_L_SL 1:220 0:745 MFDI_S_NSL 1:228 0:403 MFAR_L_SL 1:208 0:714 MFDIAR_S_NSL 1:234 0:415 MFDI_L_SL 1:185 0:482 MF_S_SL 1:144 0:570 MFDIAR_L_SL 1:201 0:490 MFAR_S_SL 1:124 0:473 MF_M_NSL 1:383 0:609 MFDI_S_SL 1:316 0:415 MFAR_M_NSL 1:385 0:600 MFDIAR_S_SL 1:318 0:423 MFDI_M_NSL 1:264 0:479 MF_T_NSL 1:208 0:435 MFDIAR_M_NSL 1:270 0:485 MFAR_T_NSL 1:216 0:433 MF_M_SL 1:371 0:556 MFDI_T_NSL 1:237 0:447 MFAR_M_SL 1:437 0:613 MFDIAR_T_NSL 1:248 0:449 MFDI_M_SL 1:446 0:455 MF_T_SL 1:473 0:546 MFDIAR_M_SL 1:466 0:470 MFAR_T_SL 1:467 0:591 MF_N_NSL 1:493 0:422 MFDI_T_SL 1:565 0:582 MFAR_N_NSL 1:462 0:411 MFDIAR_T_SL 1:568 0:586 MSFE 0:205 1:137 MSFE 0:205 1:137 () Notes: This table shows the forecast accuracy of all models during economic episodes, relative to AR(SIC) model in one-quarter ahead U.S. real GDP growth. Bold entries represents point MSFE ve "best" models in a given episode. Single, double and triple stars denote the rejection at the 10, 5 and 1 percent level of the one-sided Diebold and Mariano (1995) predictive accuracy test, respectively.

59 47 Figure 2.1: ADS Index and MF Index 6 4 Correlation : ADS Index MF Index () Notes: The ADS index and the mixed frequency (MF) index we construct are plotted with shaded recession periods declared by NBER. One weekly variable is the unemployment initial claims. Four monthly variables are all employees on non-agricultural payroll, industrial production, real manufacturing, trade and sales, and real personal income less transfer payment. One quarterly real GDP is used. No tranformation is taken to the weekly initial claims, while log-di erencing is taken to the rest of variables. This transformation of each variable is followed by Aruoba (2013). The period for constructing MF index is from 1960:3 to 2012:6. The MF Index is not smoothed. Further modelling details are given in Section 2.2. For more descriptions of ADS index are explained in Section 2.4.

60 48 Figure 2.2: Historical Breakdowns of ADS Index Correlation : Correlation : ADS Index Historical Breakdown to Initial Claims for Jobless ADS Index Histprical Breakdown to Employment Payroll Correlation : Correlation : ADS Index Historical Breakdown to IP ADS Index Historical Breakdown to Real Manufacturing Trade & Sales Correlation : Correlation : ADS Index Historical Breakdown to Personal Income less Transfer ADS Index Historical Breakdown to GDP () Notes: The ADS index and its breakdown are plotted. In Stock and Watson (1989), the authors let zero to all other variables except a indicator and extract the factor from one indicator. We also decompose the ADS index into its component s contribution by extracting the factor from each single indicator.

61 49 Figure 2.3: One-Quarter Ahead GDP Predictions Panel A. Plots of Forecasts Panel B. Plots of RMSFEs () Notes: The gure in the top panel shows one-quarter ahead GDP forecasts by prediction models. The shaded areas are recession periods followed by NBER. The gure in the bottom panel presents the time varying relative MSFEs (Mean Squared Forecast Errors) of one-step ahead GDP forecasts by prediction models. The benchmark model is AR(SIC) model and its performance is set to one. The time varying MSFEs of AR(SIC) models are presented in gure MF is a model using only MF index. This index is constructed from indicator set B, and the mnemonic of model is MF_B_NSL. MFDI is denoted by a MFDI model, plotting the predictions from MFDI_B_NSL model. SPF added MF means MQ4_NSL. Model Combination is the average of all predictions using smoothed MF with autoregressive components (CMA5). For more detailed mnemonics and descriptions of prediction models are given in Table 2.3 and 2.4.

62 50 Figure 2.4: Forecast Gains of Factors over Autoregression () Notes: The histogram present the forecast gains of forecast models in gure 2.3, over AR(SIC) model in onequarter ahead GDP forecast during all period. Prediction models are explained in Figure 2.3. Forecast gains is used to measure the relative forecast predictiveness, and the forecast comparisons are made based on AR(SIC) model s forecast accuracy. For example, DI model shows 8.3 percent more accurate forecasts than AR(SIC) model, measured in MSFE. DI and MF index, respectively, are mnemonics for di usion index model and mixed frequency index model. MF index + DI is the MFDI model using MF index and di usion index (DI) as regressors. Survey added MF index stands for an mixed frequency index model constructed using survey information, Survey of Professional Forecasters. Model Combination is the average of forecast models. Descriptions.of these mnemonics and model speci cations are given in Table 2.3 and 2.4.

63 51 Figure 2.5: Forecast Gains During Business Cycles Panel A. Forecast Gains During Expansions Panel B. Forecast Gains During Recessions () Notes: Left and right histogram present the forecast gains over AR(SIC) model in one-quarter ahead GDP predictions during expansion and recession periods, respectively, by prediction models. All models are explained in gure 2.3 and 2.4. The business cycles are followed by the NBER s business cycle dating committee. More descriptions of the mnemonics and model speci cations are given in Table 2.3 and 2.4. More details of forecasting results during economic episodes are given in Table 2.8.

64 52 Figure 2.6: Forecast Breakdown by Indicator s Contribution () Notes: The histogram present the forecast gains of forecast models in gure 2.3, over AR(SIC) model in onequarter ahead GDP forecast during all period. Prediction models are explained in Figure 2.3. Forecast gains is used to measure the relative forecast predictiveness, and the forecast comparisons are made based on AR(SIC) model s forecast accuracy. For example, DI model shows 8.3 percent more accurate forecasts than AR(SIC) model, measured in MSFE. DI and MF index, respectively, are mnemonics for di usion index model and mixed frequency index model. MF index + DI is the MFDI model using MF index and di usion index (DI) as forecaster together. Survey added MF index stands for an mixed frequency index model constructed using survey information, Survey of Professional Forecasters. Model Combination is the average of forecast models. Descriptions.of these mnemonics and model speci cations are given in Table 2.3 and 2.4.