Words Are the New Numbers: A Newsy Coincident Index of Business Cycles

Thorsrud, Leif Anders Working Paper Words Are the New Numbers: A Newsy Coincident Index of Business Cycles Working Paper, No. 21/2016 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Thorsrud, Leif Anders (2016) : Words Are the New Numbers: A Newsy Coincident Index of Business Cycles, Working Paper, No. 21/2016, ISBN 978-82-7553-953-1, Norges Bank, Oslo, https://hdl.handle.net/11250/2495606 This Version is available at: https://hdl.handle.net/10419/210110 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/deed.no Words are the new numbers: A newsy coincident index of business cycles Norges BaNk research 21 | 2016 Leif Anders Thorsrud WorkiNg PaPer Norges BaNk Working PaPer xx | 2014 rapportNavN 2 Working papers fra Norges Bank, fra 1992/1 til 2009/2 kan bestilles over e-post: [email protected] fra 1999 og senere er publikasjonene tilgjengelige på www.norges-bank.no Working papers inneholder forskningsarbeider og utredninger som vanligvis ikke har fått sin endelige form. hensikten er blant annet at forfatteren kan motta kommentarer fra kolleger og andre interesserte. synspunkter og konklusjoner i arbeidene står for forfatternes regning. Working papers from Norges Bank, from 1992/1 to 2009/2 can be ordered by e-mail: [email protected] Working papers from 1999 onwards are available on www.norges-bank.no norges Bank’s working papers present research projects and reports (not usually in their final form) and are intended inter alia to enable the author to benefit from the comments of colleagues and other interested parties. Views and conclusions expressed in working papers are the responsibility of the authors alone. ISSN 1502-819-0 (online) ISBN 978-82-7553-953-1 (online) Words are the new numbers: A newsy coincident index of business cycles∗ Leif Anders Thorsrud† December 21, 2016 Abstract I construct a daily business cycle index based on quarterly GDP and textual information contained in a daily business newspaper. The newspaper data are decomposed into time series representing newspaper topics using a Latent Dirichlet Allocation model. The business cycle index is estimated using the newspaper topics and a time-varying Dynamic Factor Model where dynamic sparsity is enforced upon the factor loadings using a latent threshold mechanism. The resulting index is shown to be not only more timely but also more accurate than commonly used alternative business cycle indicators. Moreover, the derived index provides the index user with broad based high frequent information about the type of news that drive or reflect economic fluctuations. JEL-codes: C11, C32, E32 Keywords: Business cycles, Dynamic Factor Model, Latent Dirichlet Allocation (LDA) ∗This Working Paper should not be reported as representing the views of Norges Bank. The views expressed are those of the authors and do not necessarily reflect those of Norges Bank. I thank Hilde C. Bjørnland, Fabio Canova, Pia Glæserud, Juan F. Rubio-Ram´ırez, Maximilian Rohrer, and Christian Schumacher for valuable comments. Vegard Larsen provided helpful technical assistance for which I am grateful. Comments from participants at the Joint Research Workshop of Norges Bank and Deutsche Bundesbank and the CAMP Workshop on Commodities, business cycles and monetary policy also helped improve the paper. This work is part of the research activities at the Centre for Applied Macro and Petroleum economics (CAMP) at the BI Norwegian Business School. †Norges Bank and Centre for Applied Macro and Petroleum economics, BI Norwegian Business School. Email: [email protected] 1 1 Introduction Policy makers and forecasters need to assess the state of the economy in real time to devise appropriate policy responses and condition on an updated information set. However, in real time, our main measure of economic activity, GDP growth, is not observed as it is compiled on a quarterly frequency and published with a considerable lag, usually up to at least one month. To mediate these caveats, various more timely indicators (like financial and labor market data) are monitored closely, and coincident indexes constructed.1 However, these common approaches face at least two drawbacks. First, the relationships between the timely indicators typically monitored, e.g., financial market data, and GDP growth are inherently unstable (see, e.g., Stock and Watson (2003)). Second, due to limited availability of high frequency data, the type of data from which coincident indexes often are constructed is constrained. As a result, changes in any coincident index constructed from such series do generally not give the index user broad information about what’s leading to the changes in the index. For example, changes in financial returns can be observed daily and are commonly believed to be due to new information about future fundamentals, but the changes themselves do not reveal what this new information is. For policy makers in particular, as reflected in the broad coverage of various financial and macroeconomic data in monetary policy reports and national budgets, understanding why an index changes might be as important as the movement itself. Related to this, the indicators often used are typically obtained from structured databases and professional data providers. In contrast, the agents in the economy likely use a plethora of high-frequency information to guide their actions and thereby shape aggregate economic fluctuations. It is not a brave claim to assert that this information is highly unstructured and does not come (directly) from professional data providers, but more likely reflect information shared, generated, or filtered through a large range of channels, including media. In this paper, I propose a new coincident index of business cycles aimed at addressing the drawbacks discussed above. In the tradition of Mariano and Murasawa (2003) and Aruoba et al. (2009), I estimate a latent daily coincident index using a Bayesian timevarying Dynamic Factor Model (DFM) mixing observed daily and quarterly data. To this, I make two contributions. First, the daily data set comes from a novel usage of textual information contained in a daily business newspaper, represented as topic frequencies across time. Thus, words are the new numbers, and the name: A newsy coincident index of 1Stock and Watson (1988) and Stock and Watson (1989) provide early examples of studies constructing coincident indexes using single frequency variables and latent factors, while Mariano and Murasawa (2003) extent this line of research to a mixed frequency environment using monthly and quarterly data. Later contributions mixing even higher frequency data, e.g., daily, with quarterly observations are given by, e.g., Evans (2005) and Aruoba et al. (2009). 2 business cycles (NCI ). In turn, this innovation allows for decomposing the changes in the latent daily business cycle index into the (time-varying) news components it constitutes, and therefore also say something more broadly about why (in terms of news topics) the index changes at particular points in time. My hypothesis is simple: To the extent that the newspaper provides a relevant description of the economy, the more intensive a given topic is represented in the newspaper at a given point in time, the more likely it is that this topic represents something of importance for the economy’s current and future needs and developments. Instead of relying on a limited set of conventional high frequency indicators to measure changes in business cycle conditions, I use a primary source for new broad based information directly - the newspaper.2 Second, building on the Latent Threshold Model (LTM) idea introduced by Nakajima and West (2013), and applied in a factor model setting in Zhou et al. (2014), the DFM is specified using an explicit threshold mechanism for the time-varying factor loadings. This enforces sparsity on the system, but also explicitly takes into account that the relationship between the latent daily business cycle index and the indicators used to derive it might be unstable (irrespective of whether newspaper data or more standard high frequent data is used to derive the index). My main results show that both innovations listed above are important. I demonstrate, using Receiver Operating Characteristic (ROC) curves, that compared to more traditional business cycle indicators and coincident indexes, the NCI provides a more timely and trustworthy signal about the state of the economy. This gain is achieved through the combined usage of newspaper data and allowing for time-variation in the factor loadings. Moreover, the NCI contains important leading information, suggesting that the NCI would be a highly useful indicator for turning point predictions and nowcasting. Decomposing the NCI into the individual news topic contributions it constitutes reveals that on average, across different business cycle phases, news topics related to monetary and fiscal policy, the stock market and credit, and industry specific sectors seem to provide the most important information about business cycle conditions. Finally, the sign and timing of their individual contributions map well with the historical narrative we have about recent business cycle phases. In using newspaper data the approach taken here shares many features with a growing number of studies using textual information to predict and explain economic outcomes, but extends this line of research it into the realm of coincident index construction. For 2Economic theory suggests that news might be important for explaining economic fluctuations because it contains new fundamental information about the future (see, e.g., Beaudry and Portier (2014)). Alternatively, as in, e.g., Angeletos and La’O (2013), news is interpreted as some sort of propagation channel for sentiment. Results reported in Larsen and Thorsrud (2015) indicate that information in the newspaper, represented as topic frequencies, contain new fundamental information about the future. 3 example, Tetlock (2007) classifies textual information using negative and positive word counts, and links the derived time series to developments in the financial market; Baker et al. (2013) construct an uncertainty index based on the occurrence of words in newspapers associated with uncertainty and link it to policy-related economic uncertainty; Choi and Varian (2012) use Google Trends and search for specific categories to construct predictors for present developments in a wide range of economic variables.3 In this paper, textual information is utilized using a Latent Dirichlet Allocation (LDA) model. The LDA model statistically categorizes the corpus, i.e., the whole collection of words and articles, into topics that best reflect the corpus’s word dependencies. A vast information set consisting of words and articles can thereby be summarized in a much smaller set of topics facilitating interpretation and usage in a time-series context.4 Compared with existing textual approaches, the LDA approach offers several advantages. In terms of word counting, which words are positive and which negative obviously relates to an outcome. A topic does not. A topic has content in its own right. Moreover, the LDA is an automated machine learning algorithm, so (subjectively) choosing the words or specific categories to search for is not needed. Instead, the LDA automatically delivers topics that best describe the whole corpus. This permits us to examine if textual information in the newspaper is representative for economic fluctuations, and if so, identify the type of new information (in terms of topics) that might drive or reflect economic fluctuations. In Larsen and Thorsrud (2015), it is shown that individual news topics extracted using a LDA model adds marginal predictive power for a large range of economic aggregates at a quarterly frequency. Here I build on this knowledge and use similar topics to construct the daily NCI. The perhaps most closely related paper to this is Balke et al. (2015). They use customized text analytics to decompose the Beige Book, a written description of economic conditions in each of the twelve districts banks of the Federal Reserve System in the U.S., into time series and construct a coincident index for the U.S. business cycle. They find that this textual data source contains information about current economic activity not contained in quantitative data. Their results are encouraging and complement 3Bloom (2014) provides a summary of the literature which constructs aggregate uncertainty indexes based on (among other things) counting pre-specified words in newspapers. See Tetlock (2014) for a short overview of the usage of textual data in the finance literature. In macroeconomics, there is a growing literature utilizing textual data to examine the effects of central bank’s communication (see, e.g., Apel and Blix Grimaldi (2012) and the references therein). 4Blei et al. (2003) introduced the LDA as a natural language processing tool. Since then the methodology has been heavily applied in the machine learning literature and for textual analysis. Surprisingly, in economics, it has hardly been applied. See, e.g., Hansen et al. (2014), Hansen and McMahon (2015), and Larsen and Thorsrud (2015) for exceptions. 4 my findings. However, the Beige Book is published at an irregular frequency, and not all countries have Beige Book-type information. In contrast, most countries have publicly available newspapers published (potentially) daily.5Finally, as alluded to above, in contrast to existing studies using textual data, with the news topic approach one can decompose the daily changes in the coincident index into news topic contributions. The rest of this paper is organized as follows. Section 2describes the newspaper data, the topic model, and the estimated news topics. The mixed frequency and time-varying DFM is described in Section 3. Results are presented in Section 4. Section 5concludes. 2 Data The raw data used in this analysis consists of a long sample of the entire newspaper corpus for a daily business newspaper and quarterly GDP growth for Norway. I focus on Norway because it is a small and open economy and thereby representative of many western countries, and because small economies, like Norway, typically have only one or two business newspapers, making the choice of corpus less complicated. Here, I simply choose the corpus associated with the largest and most read business newspaper, Dagens Næringsliv (DN), noting that DN is also the fourth largest newspaper in Norway irrespective of subject matter. DN was founded in 1889, and has a right-wing and neo-liberal political stance. Importantly, however, the methodology for extracting news from newspaper data, and analyze whether or not it is informative about business cycle developments, is general and dependent neither on the country nor newspaper used for the empirical application. To make the textual data applicable for time series analysis, the data is first decomposed into time series of news topics using a Latent Dirichlet Allocation (LDA) model. In general, topic modeling algorithms are statistical methods that analyze the words of the original texts to discover the themes that run through them and the themes’ connection to one another. Although topic models are well known, and have been massively applied, in the machine learning literature, their usage in the field of economics has been rare. Blei (2012) provides a nice layman’s introduction to topic modeling. The newspaper corpus and the LDA specification in this paper is similar to that described in Larsen and Thorsrud (2015). Still, as the usage of textual data and the application of a LDA model are relatively new in economics, I provide a summary of the computations below. I then 5In relation to this, the U.S. is in many aspects a special case when it comes to quantitatively available economic data, simply because there is so much available at a wide variety of frequencies. For most other countries, this is not the case. The usage of daily newspaper data can potentially mitigate such missing information. 5 examine the mapping between the estimated news topics and GDP growth using simple principal components analysis, before presenting the proposed time-varying and mixed frequency Dynamic Factor Model (DFM) in the subsequent section. 2.1 The news corpus, the LDA and topics The DN news corpus is extracted from Retriever’s “Atekst” database, and covers all articles published in DN from May 2, 1988, to December 29, 2014. In total this amounts to Na= 459745 articles, well above one billion words, more than a million unique tokens, and a sample of Td= 9741 days. This massive amount of data makes statistical computations challenging, but as is customary in this branch of the literature, some steps are taken to clean and reduce the raw dataset before estimation. A description of how this is done is given in Appendix C. I note here that around 250 000 unique tokens are kept after the filtering procedure. The “cleaned”, but still unstructured, DN corpus is decomposed into news topics using a Latent Dirichlet Allocation (LDA) model. The LDA model is an unsupervised topic model introduced by Blei et al. (2003) that clusters words into topics, which are distributions over words, while at the same time classifying articles as mixtures of topics.6 By unsupervised learning algorithm we mean an algorithm that can learn/discover an underlying structure in the data without the algorithm being given any labeled samples to learn from. The term “latent” is used, because the words, which are the observed data, are intended to communicate a latent structure, namely the meaning of the article. The term “Dirichlet” is used because the topic mixture is drawn from a conjugate Dirichlet prior. Figure 1illustrates the LDA model graphically. The outer box, or plate, represents the whole corpus as Mdistinct documents (articles). N=PM m=1 Nmis the total number of words in all documents, and Kis the total number of latent topics. Letting boldfont variables denote the vector version of the variables, the distribution of topics for a document is given by θm, while the distribution of words for each topic is determined by ϕk. Both θmand ϕkare assumed to have conjugate Dirichlet distributions with (hyper) parameter (vectors) αand β, respectively. Each document consists of a repeated choice of topics Zm,n and words Wm,n, drawn from the Multinomial distribution using θmand ϕk. The circle associated with Wm,n is gray colored, indicating that these are the only observable variables in the model. At an intuitive level, the best way to understand the LDA model is likely to make a thought experiment of how the articles in the newspaper (the corpus) were generated. 6This latter point is important, because it distinguishes the LDA model from other often used text classifying algorithms where each article is assumed to be described by only one single topic. 6 and unobservable variables assumed to be stationary with zero mean, decomposed as follows: yt= y∗ 1,t y2,t!(5) where y∗ 1,t is a Nq×1 vector of unobserved daily output growth rates, mapping into quarterly output growth rates as explained below, and y2,t is a Nd×1 vector of daily newspaper topic variables, described in Section 2.2.N=Nq+Nd, and zj,t is a N×q matrix with dynamic factor loadings for j= 0,1,· · · , s, and sdenotes the number of lags used for the dynamic factors at. The dynamic factors, containing the daily business cycle index, follow a VAR(h) process given by the transition equation in (4b), where ωt∼i.i.d.N(0,Ω). Finally, equation (4c) describes the time series process for the N×1 vector of idiosyncratic errors et. It is assumed that these evolve as independent AR(p) processes with ut∼i.i.d.N(0,U), and that utand ωtare independent. The model’s only time-varying parameters are the factor loadings (zj,t), which are restricted to follow independent random walk processes. Apart from the usage of newspaper data, the DFM described above is fairly standard. Similar specifications have been applied in recent work by Lopes and Carvalho (2007), Del Negro and Otrok (2008), Ellis et al. (2014), and Bjørnland and Thorsrud (2015). Some of these studies also include stochastic volatility in the DFM. In a mixed frequency setting for example, Marcellino et al. (2013) estimate a DFM (using monthly and quarterly data) without time-varying parameters, but with stochastic volatility. I abstract from this property here to focus on the innovations introduced in this paper. Two extensions are applied here: First, sparsity is enforced on the system through the time-varying factor loadings using a latent threshold mechanism. Second, since the variables in the ytvector are observed at different frequency intervals, cumulator variables are used to ensure consistency in the aggregation from higher to lower frequencies and make estimation feasible. Below I elaborate on these two extensions. A full description of the model, and its extensions, is given in Appendix E.11 11It follows from the above discussion that there is a conceptually close resemblance between the LDA model described in Section 2.1, and factor models commonly used in economics. In both instances, some set of observed variables are assumed to be determined by a (predefined) number of common latent variables. As such, one could envision a model where the observables, words and output growth in terms of this analysis, and their relationship to latent factors where estimated jointly within one model. I am, however, not aware of existing models in the literature that combine time series with textual data in this manner. Incorporating the mixed frequency and latent threshold dynamics into such model would complicate the problem further. Thus, as the first investigation of this sort, I opt for the simpler two-step approach in this analysis. 13 3.1 Enforcing sparsity and identification Following the Latent Threshold Model (LTM) idea introduced by Nakajima and West (2013), and applied in a DFM setting in Zhou et al. (2014), sparsity is enforced on the system through the time-varying factor loadings using a latent threshold. For example, for one particular element in z0,t,zi,0,t, the LTM structure can be written as: zi,0,t =z∗ i,0,tςi,0,t ςi,0,t =I(|z∗ i,0,t| ≥ di,0) (6) where z∗ i,0,t =z∗ i,0,t−1+wi,0,t (7) with wi,0,t ∼i.i.d.N(0, σ2 i,0,w). In (6)ςi,0,t is a zero one variable, whose value depends on the indicator function I(|z∗ i,0,t| ≥ di,0). If |z∗ i,0,t|is above the the threshold value di,0, then ςi,0,t = 1, otherwise ςi,0,t = 0. In general, the LTM framework is a useful strategy for models where the researcher wants to introduce dynamic sparsity. For example, as shown in Zhou et al. (2014), allowing for such mechanism uniformly improves out-of-sample predictions in a portfolio analysis due to the parsimony it induces. Here, the LTM concept serves two purposes. First, if estimating the factor loadings without allowing for time variation, the researcher might conclude that a given topic has no relationship with at, i.e., that zi,0:s= 0, simply because, on average, periods with a positive zi,0:s,t cancels with periods with a negative zi,0:s,t. By using the time-varying parameter formulation above, this pitfall is avoided. Second, it is not very likely that one particular topic is equally important throughout the estimation sample. A topic might be very informative in some periods, but not in others. The threshold mechanism potentially captures such cases in a consistent and transparent way, safeguards against over-fitting, and controls for the fact that the relationship between the indicators and output growth might be unstable, confer the discussion in Section 1.12 As is common for all factor models, the factors and factor loadings in (4) are not identified without restrictions. To separately identify the factors and the loadings, the following identification restrictions on z0,t in (4a) are enforced: z0,t ="˜ z0,t ˆ z0,t#,for t= 0,1, . . . , T (8) Here, ˜ z0,t is a q×qidentity matrix for all t, and ˆ z0,t is left unrestricted. Bai and Ng (2013) and Bai and Wang (2012) show that these restrictions uniquely identify the dynamic factors and the loadings, but leave the VAR(h) dynamics for the factors completely unrestricted. 12The same arguments naturally applies when constructing coincident indexes using more conventional indicators (like financial and labor market data). 14 3.2 Introducing mixed frequency variables Due to the mixed frequency property of the data, the ytvector in equation (4a) contains both observable and unobservable variables. Thus, the model as formulated in (4) can not be estimated. However, following Harvey (1990), and since y∗ 1,t is a flow measure, the model can be reformulated such that observed quarterly series are treated as daily observations with missing observations. To this end, the ytvector is decomposed as in equation (5). Assuming further that the quarterly variables, e.g., output growth defined in Section 2.3, are observed at the last day of each quarter, we can define: ˜y1,t =  Pm j=0 y∗ 1,t−jif ˜y1,t is observed NA otherwise (9) where ˜y1,t is treated as the intra-period sum of the corresponding daily values, and m denotes the number of days since the last observation period. Because quarters have uneven number of days, ˜y1,t is observed on an irregular basis. Accordingly, mwill vary depending on which quarter and year we are in. This variation is however known, and easily incorporated into the model structure. Given (9), temporal aggregation can be handled by introducing a cumulator variable of the form: C1,t =β1,tC1,t−1+y∗ 1,t (10) where β1,t is an indicator variable defined as: β1,t =   0 if tis the first day of the period 1 otherwise (11) and y∗ 1,t maps to the latent factor, at, from equation (4b). Thus, ˜y1,t =C1,t whenever ˜yt,1is observed, and treated as a missing observation in all other periods. Because of the usage of the cumulator variable in (10), one additional state variable is introduced to the system. Importantly, however, the system will now be possible to estimate using standard filtering techniques handling missing observations. Details are given in Appendix E. Some remarks are in order. First, although mappings between mixed frequency variables have been applied extensively in both mixed frequency VARs and factor models, see Foroni and Marcellino (2013) for an overview, the cumulator approach has been exploited less regularly. For the purpose of this analysis it offers a clear advantage because it expands the number of state variables in the system only marginally. In contrast, using the mixed frequency approaches in, e.g., Mariano and Murasawa (2003) and Aruoba et al. (2009), would have expanded the number of state variables in the model by over 180 and 90, respectively. Such large number of states pose significant challenges for estimation, 15 making it almost infeasible in a Bayesian context.13 Second, introducing (flow) variables of other frequencies than daily and quarterly into the system is not difficult. For each new frequency one simple constructs one new cumulator variable, specific for that frequency, and augment the system accordingly. 3.3 Model specification and estimation In the model specification used to produce the main results one latent daily coincident index is identified. This latent daily coincident index is assumed to follow an AR(10) process, thus, q= 1 and h= 10. I do not allow for lags of the dynamic factors in the observation equation (4a) of the system, i.e., s= 0. Conceptually it would have been straightforward to use higher values for sfor the Ndrows in (4a) associated with the observable daily observations. However, for the Nqrows associated with the quarterly variables, setting s > 0 would conflict with the temporal aggregation described in Section 3.2. For all the Nelements in et(see equation 4c), the AR(p) dynamics are restricted to one lag, i.e., p= 1. To avoid end point issues due to data revisions with the latest vintage of output, I restrict the estimation sample to the period 1989-01-01 to 2013-3112. Finally, based on simple correlation statistics between the news topic time series and output growth I truncate the Df 1data set to include only the 20 most correlated (in absolute value) topics, see Table 3in Appendix A. This latter adjustment is done to ease the computational burden, but, as seen from Figure 4, unsupervised PCA estimates of the topic time series result in almost identical factor estimates irrespective of whether 20 or 80 topics are used, suggesting that 20 topics are enough.14 The time-varying DFM is estimated by decomposing the problem of drawing from the joint posterior of the parameters of interest into a set of much simpler ones using Gibbs simulations. The Gibbs simulation employed here, together with the prior specifications, are described in greater detail in Appendix E. The results reported in this paper are all based on 9000 iterations of the Gibbs sampler. The first 6000 are discarded and only every sixth of the remaining are used for inference.15 13For example, Aruoba et al. (2009) employ Maximum Likelihood estimation, and note that one evaluation of the likelihood takes roughly 20 seconds. As Bayesian estimation using MCMC (see Section 3.3) requires a large number of iterations, the problem quickly becomes infeasible in terms of computation time. 14Still, the truncation is admittedly somewhat arbitrary. Noting that comparable coincident index models already proposed in the literature also resort to some type of variable selection prior to estimation, I leave it for future research to devise potentially more optimal methods to truncate the topics data set. 15As shown in Appendix E.7, and in Appendix E.8 for a simulation experiment, the convergence statistics seem satisfactory. 16 (a) NCI and GDP a (b) 2001:Q1 - 2001:Q3 (c) 2002:Q3 - 2003:Q1 (d) 2008:Q2 - 2009:Q3 Figure 5. GDP ais recorded at the end of each quarter, but reported on a daily basis in the graphs using previous end-of-period values throughout the subsequent quarter. NCI is the standardized measure of the daily business cycle index. Recession periods, defined by a MS-FMQ model (see Section 4.1), are illustrated using gray color shading. Figures 5b to 5d focus on three specific periods where output is illustrated using GDP . The indicators are normalized to zero on the first day of the first quarter displayed. OSEBX is the cumulative return over the period, and Spread is the difference between the 10 year and 3 month money market interest rate. 4 A newsy coincident index of the business cycle Figure 5reports the estimated NCI. As clearly seen in the upper part of the figure, the index tracks the general economic fluctuations closely. Compared to the simple PCA estimates reported in Figure 4, the NCI seems to provide a better fit: It captures the low growth period in the early 1990s, the boom and subsequent bust around the turn of the century, and finally the high growth period leading up to the Great Recession. Note, however, that in Norway, the downturn in the economy following the Norwegian banking crisis in the late 1980s was just as severe as the downturn following the global financial crisis in 2008. An (informal) example of the importance of having timely information about the state of the economy is given in Figures 5b to 5d. They show the benefits of the NCI relative to using two timely and often-used indicators: the stock index (OSEBX ) and 17 yield spreads (Spreads) (see, e.g., Estrella and Mishkin (1998)) around three important turning points in the Norwegian economy the last decades. For example, as seen in Figure 5d, between the second and third quarter of 2008 output growth declined considerably. During the month of August 2008, and in particular following Lehman Brothers collapse on September 15, 2008, the stock index, the yield spread, and the NCI plummet. Since the actual number for GDP growth in the third quarter of 2008 was not known before late 2008, both leading indicators and the NCI would have been useful for picking up the change in economic conditions prior to what we now know turned out to be a recession in this example. However, Figure 5d, and in particular Figures 5b and 5c, also show the problem with relying on the indicators alone: Their relationship with output growth is unstable. During the recession period in the early 2000s for example, see Figure 5b, the spread did not signal any downturn at all. Likewise, for this period the changes in the stock index did not turn significantly negative before almost one quarter after the recession started. In contrast, for all three recession periods reported in Figure 5, the NCI provides a more or less timely signal of the downturn. 4.1 Business cycles and index evaluation Making a formal evaluation of the NCI is challenging. By construction, the quarterly sum of the daily NCI will equal the observed quarterly growth rates in GDPa(plus a measurement error, c.f. Section 3.2), while daily business cycle conditions, on the other hand, are not observed. Alternatively, in the tradition of Burns and Mitchell (1946), and later work by, e.g., Bry and Boschan (1971) and Hamilton (1989), to mention just two of many, aggregate economic activity can be categorized as phases of expansions and contractions, and one can assess the index’s ability to classify such phases. This is the route I take here. Following Travis and Jord`a (2011), I use Receiver Operating Characteristic (ROC) curves and the area under the curve (AUROC) statistic to score the NCI ’s ability to classify the state of the economy.16 Here, I do so along four dimensions: How well it 16In economics, Travis and Jord`a (2011) introduced the ROC methodology to classify economic activity into recessions and expansions. An ideal binary classifier would always indicate a recession when a recession actually occurs (true positive), while never indicate a recession when it does not occur (false positive). In Figure 6a, for example, such a classifier would be depicted by a point in the upper left corner. A model not performing any better than random guessing would end up at the 45 degree line. Thus, using the ROC one can easily compare the trade-offs (cost/benefit) one faces when using different models or indicators for classification. The AUROC is an often used summary statistic within the ROC framework. By definition the AUROC can not exceed 1, perfect classification, or be lower than 0.5. I compute the AUROC score non-parametrically using the algorithm described in Travis and Jord`a (2011), and refer to their work for an overview of the ROC technicalities and advantages in terms of scoring business cycle 18 (a) NCI and different reference cycles (b) NCI and quarterly classification (c) NCI at various lead lengths (d) NCI and alternatives Figure 6. Receiver Operating Characteristics curves (ROC). Figure 6a reports the NCI ’s ability of classifying business cycle phases across four different business cycle chronologies. In Figures 6b to 6d the MS-FMQ chronology is used as the reference cycle. Figure 6b reports the results when classification is scored at a quarterly frequency. Figure 6c reports the results when the NCI is lagged p={0,40,...,200} days. Figure 6d compares the performance of the daily NCI against a set of daily and monthly alternatives. For the monthly indicators, LFS and BCI, daily numbers are obtained using previous end-of-period values throughout the subsequent month. categorizes business cycles using different reference cycles; how well it categorizes business cycles at a different level of time aggregation; how well it categorizes business cycles at different lags; and finally, how well it categorizes business cycles compared to other often used and observable alternatives. See Section 4.4 for evaluations of the NCI relative to other estimated coincident indexes. Figure 6a assesses the NCI ’s classification ability against four different business cycle chronologies, developed by Aastveit et al. (2016) for the Norwegian economy.17 Each chronologies. Still, as the true underlying state of the economy is never observed, even retrospectively, and since the categorization of economic activity does not follow any universally agreed upon law, there will be an inherent uncertainty also with this type of evaluation. An evaluation of a different sort, but perhaps more hefty, can be obtained by running a real-time out-of-sample forecasting experiment. 17In contrast to in, e.g., the U.S., which has an official business cycle dating committee (NBER), no such 19 chronology is constructed using different methodologies to extract the unobserved phases: uniand multivariate Bry-Boschan approaches (BB-GDP and BB-ISD), a univariate Markow-switching model (MS-GDP), and a Markov-Switching factor model (MS-FMQ). Aastveit et al. (2016) provide a description of these approaches and the data used. The resulting quarterly classifications, and additional model details, are summarized in Table 2in Appendix A.18 As seen from Figure 6a, irrespective of which reference cycle that is used to define the Norwegian business cycle, the NCI yields a true positive rate of roughly 80 percent, at the cost of only 25 percent false positives. The AUROC measures are also between 0.85 and 0.87 in all four cases, signaling very good classification. While these results are strong, but not perfect, it should be remembered that the NCI might provide an estimate of the economy’s phases that is closer to the unknown truth than any of the other reference cycles I use to evaluate it. Moreover, the classification models typically used are at the quarterly (or monthly) frequency, while the NCI allows for daily classification. Aggregating the NCI to a quarterly time series, by simply computing the mean growth rate for each quarter, we observe that the index’s classification ability becomes even better, see Figure 6b. When using the MS-FMQ as the reference cycle, for example, an AUROC of 0.92 is achieved at the quarterly frequency against 0.87 at the daily frequency. Compared with the results reported for quarterly Norwegian data in Aastveit et al. (2016), and U.S. data in Travis and Jord`a (2011), this score is very competitive.19 The results reported in Figure 5indicated that the NCI had leading properties. This is confirmed more formally in Figure 6c. Lagging the NCI 40 days yields a higher AUROC score than actually using the NCI as a contemporaneous classifier for the business cycle. The performance of the NCI does not really start to drop before it is lagged almost one quarter (80 days), suggesting that the NCI would be a highly useful indicator for turning point predictions and nowcasting. Traditionally, coincident indexes are constructed using a number of observable daily and monthly variables. In Figure 6d, the classification properties of some of these variables (see Appendix Afor data descriptions) are compared to the NCI. The best performing observable indicator in terms of ROC curve scoring is the daily Spread followed by the monthly labor force survey (LFS). Using stock returns or the business confidence indicator institution or formal dating exists for Norway. 18Daily classifications are obtained by assuming that the economy remains in the same phase on each day within the quarterly classification periods. 19Using the reference cycle generated by the MS-FMQ model for Norwegian data, Aastveit et al. (2016) show that the BB-GDP model gets an AUROC of 0.93. Using U.S. data, and comparing various leading indicators and coincident indexes, Travis and Jord`a (2011) show that the best performing coincident index is the one developed by Aruoba et al. (2009). This index receives an AUROC of 0.96 when the NBER business cycle chronology is used as a reference cycle. 20 (OSEBX and BCI ) are almost no better than random guessing in terms of classifying the business cycle, confirming the impression from Figure 5. It is noteworthy that the PCA estimated news index (see Section 2.3) performs better than any of the other alternatives. At the cost of 40 percent false positive rates, it can give almost 100 percent true positive rates. Still, the AUROC score for the PCA estimated news index is well below the NCI ’s. In sum, the results presented above suggest that the NCI adds value. Although other alternatives also provide information that is competitive relative to the NCI, these alternatives are not necessarily available on a daily frequency and they do not provide the users of such information any broader rational in terms of why the indicators fall or rise. As shown in the next section, the NCI does. 4.2 News and index decompositions Figure 7illustrates how changes in the NCI can be decomposed into the contributions from the individual news topics, and thereby address what type of new information underlies changes in business cycle conditions.20 To economize on space, I only report nine of the topics contributing to the NCI estimate. The 11 remaining topics are reported in Figure 9in Appendix B. Three distinct results stand out. First, the topics listed in Figure 7do, for the most part, reflect topics one would expect to be important for business cycles in general, and for business cycles in Norway in particular. Examples of the former are the Monetary policy,Fiscal policy,Wage payments/Bonuses,Stock market,Funding, and Retail topics, while the Oil production and Oil service topics are examples of the latter.21 The remaining topics (see Figure 9in Appendix B) are typically related to general business cycle sensitive sectors (reflected by, e.g., Airline industry and Automobiles topics) and technological developments (reflected by, e.g., IT-technology and Startup topics). Still, although most topics are easily interpretable and provide information about what is important for the current state of the economy, some topics either have labels that are less informative, or reflect surprising categories. An example is the Life topic, reported in Figure 9. That said, such exotic or less informative named topics, are the exception rather than the rule. It is also the case that a given newspaper article contains many topics at the same time. To the extent that different topics, meaningful or not from an economic point of view, stand close to each 20Technically, these results are constructed using the Kalman Filter iterations and decomposing the state evolution at each updating step into news contributions (see Appendix E.5). The decompositions reported in Figure 7are based on running the Kalman Filter using the posterior median estimates of the hyperparameters and the time-varying factor loadings (at each time t). 21Norway is a major petroleum exporter, and close to 50 percent of its export revenues are linked to oil and gas. See Bjørnland and Thorsrud (2015), and the references therein, for a more detailed analysis of the strong linkages between the oil sector and the rest of the mainland economy. 21 other in the decomposition of the corpus (see Figure 2) they might covary and therefore both add value in terms of explaining the current state of the economy. Second, while some topics seem to be important almost every period throughout the sample, other topics only contribute significantly at certain time periods. The Oil service topic provides an example: Almost throughout the whole sample, until the mid 2000s, its contribution is close to zero. After 2004, however, its contribution becomes highly positive. Similar observations can also be confirmed for the Stock market topic, in particular. The extended periods of zero contribution are partly due to the threshold mechanism used when estimating the time-varying factor loadings. I return to this discussion in Section 4.3. Third, the timing of when specific topics become important, either positively or negatively, resonates well with what we now know about the economic developments the last two decades. Without dredging too deep into the historical narrative of the Norwegian business cycle, I give three examples: It is by now well recognized that the extraordinary boom in the Norwegian economy during the 2000s was highly oil-driven. The large positive contributions from the two oil topics, Oil service and Oil production, reflect this.22 It is also well known that Norwegian (cost) competitiveness has declined considerably during the two last decades. According to the National Accounts statistics, annual wages and salaries increased considerably during especially two periods: the mid-1990s and the mid-late 2000s. Both patterns are clearly visible in the graph showing how media coverage of the Wage payments/Bonuses topic contributes to the index fluctuations. Finally, we see from the bottom graph in Figure 7that the Funding topic, a newspaper topic focused on words associated with credit and loans, contributed especially negatively during the Great Recession period. Again, this resonates well with the historical narrative, given what we today know about the Great Recession episode. Some might find it tempting to interpret the news topics, and their contribution to the NCI, as some type of causal relationship between news and economic fluctuations. Technically, within the current framework, this is not a valid interpretation because the decompositions reported in Figure 7are based on predictive properties. Instead, the newspaper topics should simply be interpreted as a broad panel of different high frequent economic indicators, informative about the current state of the economy. 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(2014). Information Transmission in Finance. Annual Review of Financial Economics 6(1), 365–384. Tetlock, P. C., M. Saar-Tsechansky, and S. Macskassy (2008). More Than Words: Quantifying Language to Measure Firms’ Fundamentals. Journal of Finance 63(3), 1437–1467. Travis, J. B. and s. Jord`a (2011). Evaluating the Classification of Economic Activity into Recessions and Expansions. American Economic Journal: Macroeconomics 3(2), 246–77. Zhou, X., J. Nakajima, and M. West (2014). Bayesian forecasting and portfolio decisions using dynamic dependent sparse factor models. International Journal of Forecasting 30, 963–980. 31 Appendices Appendix A Data, reference cycles and topics The newspaper corpus and data used for gross domestic product (GDP) are described in Sections 2.1 and 2.3, respectively. Different classifications of the business cycle into phases of expansions and recessions are listed in Table 2. A summary of all the estimated news topics, and the most important words associated with each topic, are reported in Table 3. The remaining data used in this analysis are obtained from Reuters Datastream, and are as follows: Spread is constructed as the difference between the 10-year benchmark rate and the interbank three month offered rate. OSEBX is (log) daily returns computed using the Oslo Stock Exchange Benchmark index. Both the Spread and OSEBX variables are recorded on a daily frequency. Missing observations, during weekends, are filled using simple linear interpolation. Like for the daily newspaper topic time series, prior to estimation I smooth the series using a 60-day (backward-looking) moving average filter, and standardize the resulting variables. BCI is the seasonally adjusted industrial confidence indicator for the manufacturing sector in Norway, and LFS is the seasonally adjusted labor force. Both variables are recorded on a monthly frequency, and transformed to (log) monthly growth rates. The series are smoothed using a two-month (backward-looking) moving average filter and standardized prior to estimation. Table 2. Reference cycles 1986 to 2014 (as estimated in Aastveit et al. (2016)). The different chronologies build on: a Bry-Boschan approach using GDP growth (BB-GDP); a univariate Markow-switching model using GDP growth (MS-GDP); the Bry-Boschan approach applied to a coincident index based on inverse standard deviation weighting (BB-ISD); and a multivariate Markow-swithing model (MS-FMQ). For both the BB-ISD and MS-FMQ models six quarterly variables are included: the Brent Blend oil price, employment in mainland Norway, household consumption, private real investment in mainland Norway, exports of traditional goods and GDP for mainland Norway. See Aastveit et al. (2016), and the references therein, for more formal model, data, and estimation descriptions. BB-GDP MS-GDP BB-ISD MS-FMQ 1986 - 1989 Peak 1987:Q2 1986:Q2 1987:Q4 1987:Q2 Trough 1989:Q3 1989:Q1 1990 - 1994 Peak 1991:Q1 Trough 1991:Q4 1991:Q4 1991:Q4 1995 - 2001 Peak 2001:Q1 2001:Q1 2001:Q1 2001:Q1 Trough 2001:Q3 2001:Q3 2001:Q3 2001:Q3 2002 - 2003 Peak 2002:Q2 2002:Q3 Trough 2002:Q4 2003:Q1 2004 - 2010 Peak 2008:Q2 2007:Q4 2007:Q4 2008:Q2 Trough 2008:Q3 2010:Q1 2009:Q1 2009:Q3 32 Table 3. Estimated topics and labeling. The topics are labeled based on the meaning of the most important words (see the text for details). The “Corr” column reports the topics’ correlation (using the Df 1data set, see Section 2.2) with linearly interpolated daily GDP a(the correlation rank is reported in parenthesis). The words are translated from Norwegian to English using Google Translate. Topic Label Corr First words Topic 0 Calender 0.03(69) january, march, october, september, november, february Topic 1 Family business 0.18(19) family, foundation, name, dad, son, fortune, brothers Topic 2 Institutional investing 0.10(39) fund, investments, investor, return, risk, capital Topic 3 Justice 0.04(65) lawyer, judge, appeal, damages, claim, supreme court Topic 4 Surroundings 0.18(18) city, water, meter, man, mountain, old, outside, nature Topic 5 Housing 0.14(29) housing, property, properties, apartment, square meter Topic 6 Movies/Theater 0.08(50) movie, cinema, series, game, producer, prize, audience Topic 7 Argumentation 0.11(34) word, besides, interesting, i.e., in fact, sure, otherwise Topic 8 Unknown 0.09(42) road, top, easy, hard, lift, faith, outside, struggle,fast Topic 9 Agriculture 0.03(68) industry, support, farmers, export, production, agriculture Topic 10 Automobiles 0.18(17) car, model, engine, drive, volvo, ford, møller, toyota Topic 11 USA 0.09(47) new york, dollar, wall street, president, usa, obama, bush Topic 12 Banking 0.00(80) dnb nor, savings bank, loss, brokerage firm, kreditkassen Topic 13 Corporate leadership 0.05(59) position, chairman, ceo, president, elected, board member Topic 14 Negotiation 0.04(61) solution, negotiation, agreement, alternative, part, process Topic 15 Newspapers 0.22( 9) newspaper, media, schibsted, dagbladet, journalist, vg Topic 16 Health care 0.00(77) hospital, doctor, health, patient, treatment, medication Topic 17 IT systems 0.17(24) it, system, data, defense, siem, contract, tanberg, deliver Topic 18 Stock market 0.23( 8) stock exchange, fell, increased, quote, stock market Continued on next page 33 Table 3 – continued from previous page Topic Label Corr First words Topic 19 Macroeconomics 0.07(53) economy, budget, low, unemployment, high, increase Topic 20 Oil production 0.18(20) statoil, oil, field, gas, oil company, hydro, shelf, stavanger Topic 21 Wage payments 0.26( 7) income, circa, cost, earn, yearly, cover, payed, salary Topic 22 Norwegian regions 0.17(23) trondheim, llc, north, stavanger, tromsø, local, municipality Topic 23 Family 0.04(64) woman, child, people, young, man, parents, home, family Topic 24 Taxation 0.03(71) tax, charge, revenue, proposal, remove, wealth tax, scheme Topic 25 EU 0.04(62) eu, eea, commission, european, brussel, membership, no Topic 26 Norwegian industry 0.20(13) hydro, forest, factory, production, elkem, industry, produce Topic 27 Unknown 0.07(54) man, he, friend, smile, clock, evening, head, never, office Topic 28 Norwegian groups 0.09(45) orkla, storebrand, merger, bid, shareholder, acquisitions Topic 29 UK 0.06(57) british, london, great britain, the, of, pound, england Topic 30 Narrative 0.03(72) took, did, later, never, gave, stand, happened, him, began Topic 31 Shipping 0.10(36) ship, shipping, dollar, shipowner, wilhelmsen, fleet, proud Topic 32 Projects 0.10(38) project, nsb, development, fornebu, entrepreneurship Topic 33 Oil price 0.11(32) dollar, oil price, barrel, oil, demand, level, opec, high Topic 34 Sports 0.00(78) olympics, club, football, match, play, lillehammer, sponsor Topic 35 Organizations 0.10(41) leader, create, organization, challenge, contribute, expertise Topic 36 Drinks 0.13(30) wine, italy, taste, drinks, italian, fresh, fruit, beer, bottle Topic 37 Nordic countries 0.04(63) swedish, sweden, danish, denmark, nordic, stockholm Topic 38 Airline industry 0.21(12) sas, fly, airline,norwegian, braathens, airport, travel Topic 39 Entitlements 0.02(73) municipality, public, private, sector, pension, scheme Continued on next page 34 Table 3 – continued from previous page Topic Label Corr First words Topic 40 Employment conditions 0.08(51) cut, workplace, measures, salary, labor, working, employ Topic 41 Norwegian politics 0.05(60) høyere, party, ap, labor party, stoltenberg, parlament, frp Topic 42 Funding 0.31( 3) loan, competition, creditor, loss, bankruptcy, leverage Topic 43 Literature 0.01(76) book, books, read, publisher, read, author, novel, wrote Topic 44 Statistics 0.27( 6) count, increase, investigate, share, average, decrease Topic 45 Watercraft 0.01(75) ship, boat, harbor, strait, shipowner, on board, color Topic 46 Results 0.31( 4) quarter, surplus, deficit, tax, group, operating profit, third Topic 47 TV 0.12(31) tv, nrk, channel, radio, digital, program, media Topic 48 International conflicts 0.10(40) war, africa, irak, south, un, army, conflict, troops, attack Topic 49 Political elections 0.02(74) election, party, power, politics, vote, politician, support Topic 50 Music 0.09(46) the, music, record, of, in, artist, and, play, cd, band, song Topic 51 Oil service 0.19(14) rig, dollar, contract, option, offshore, drilling, seadrill Topic 52 Tourism 0.21(11) hotel, rom, travel, visit, stordalen, tourist, guest Topic 53 Unknown 0.16(26) no, ting, think, good, always, pretty, actually, never Topic 54 Aker 0.11(35) aker, kværner, røkke, contract, shipyard, maritime Topic 55 Fishery 0.16(27) fish, salmon, seafood, norway, tons, nourishment, marine Topic 56 Europe 0.08(49) german, russia, germany, russian, west, east, french, france Topic 57 Law and order 0.06(56) police, finance guards, aiming, illegal, investigation Topic 58 Business events 0.00(79) week, financial, previous, friday, wednesday, tdn, monday Topic 59 Supervision 0.10(37) report, information, financial supervision, enlightenment Topic 60 Retail 0.31( 2) shop, brand, steen, rema, reitan, as, group, ica, coop Continued on next page 35 Table 3 – continued from previous page Topic Label Corr First words Topic 61 Startup 0.28( 5) bet, cooperation, establish, product, party, group Topic 62 Food 0.19(15) food, restaurant, salt, nok, pepper, eat, table, waiter Topic 63 Listed stocks 0.11(33) shareholder, issue, investor, holding, stock exchange listing Topic 64 Asia 0.09(43) china, asia, chinese, india, hong kong, south, authorities Topic 65 Art 0.09(44) picture, art, exhibition, gallery, artist, museum, munch Topic 66 Disagreement 0.08(52) criticism, express, asserting, fault, react, should, alleging Topic 67 Debate 0.15(28) degree, debate, context, unequal, actually, analysis Topic 68 Life 0.18(21) man, history, dead, him, one, live, church, words, strokes Topic 69 Distribution 0.18(22) customer, post, product, offers, service, industry, firm Topic 70 Telecommunication 0.08(48) telenor, mobile, netcom, hermansen, telia, nokia, ericsson Topic 71 IT technology 0.21(10) internet, net, pc, microsoft, technology, services, apple Topic 72 Monetary policy 0.33( 1) interest rate, central bank, euro, german, inflation, point Topic 73 Education 0.04(66) school, university, student, research, professor, education Topic 74 Government regulations 0.03(70) rules, authorities, competition, regulations, bans Topic 75 Trade organizations 0.16(25) lo, nho, members, forbund, strike, organization, payroll Topic 76 Fear 0.04(67) fear, emergency, hit, severe, financial crisis, scared Topic 77 Fiscal policy 0.19(16) suggestions, parliamentary, ministry, selection, minister Topic 78 Energy 0.05(58) energy, emissions, statkraft, industry, environment Topic 79 Foreign 0.07(55) foreign, abroad, japan, japanese, immigration, games 36 Appendix B Additional results Figure 9. News topics and their (median) contribution to NCI estimates across time. The news topic contributions are standardized and illustrated using different colors. GDP a, graphed using a black dotted line, is recorded at the end of each quarter, but reported on a daily basis using previous end-of-period values throughout the subsequent quarter. Recession periods, defined by a MS-FMQ model (see Section 4.1) are illustrated using grey shading. 37 Appendix C Filtering the news corpus To clean the raw textual data set, a stop-word list is first employed. This is a list of common words one does not expect to have any information relating to the subject of an article. Examples of such words are the,is,are, and this. The most common Norwegian surnames and given names are also removed. In total, the stop-word list together with the list of common surnames and given names removed roughly 1800 unique tokens from the corpus. Next, an algorithm known as stemming is run. The objective of this algorithm is to reduce all words to their respective word stems. A word stem is the part of a word that is common to all of its inflections. An example is the word effective whose stem is effect. Finally, a measure called tf-idf, which stands for term frequency - inverse document frequency, is calculated. This measures how important all the words in the complete corpus are in explaining single articles. The more often a word occurs in an article, the higher the tf-idf score of that word. On the other hand, if the word is common to all articles, meaning the word has a high frequency in the whole corpus, the lower that word’s tf-idf score will be. Around 250 000 of the stems with the highest tf-idf score are kept, and used as the final corpus. Appendix D LDA estimation and specification The LDA model was developed in Blei et al. (2003). Here the estimation algorithm described in Griffiths and Steyvers (2004) is implemented. First, recall that the corpus consists of Mdistinct documents. N=PM m=1 Nmis the total number of words in all documents, Kis the total number of latent topics, and Vis the size of the vocabulary. Each document consists of a repeated choice of topics Zm,n and words Wm,n. Let tbe a term in V, and denote P(t|z=k), the mixture component, one for each topic, by Φ={ϕk}K k=1. Finally, let P(z|d=m) define the topic mixture proportion for document m, with one proportion for each document Θ = {θm}M m=1. The goal of the algorithm is then to approximate the distribution: P(Z|W;α, β) = P(W,Z;α, β) P(W;α, β)(12) using Gibbs simulations, where αand βare the (hyper) parameters controlling the prior conjugate Dirichlet distributions for θmand ϕk, respectively. A very good explanation for how this method works is found in Heinrich (2009). The description below provides a brief summary only. With the above definitions, the total probability of the model can be written as: P(W,Z,Θ,Φ;α, β) = K Y k=1 P(ϕi;β) M Y m=1 P(θm;α) N Y t=1 P(zm,t|θm)P(wm,t|ϕzm,t ) (13) 38 letters denoted with an underscore reflect the prior, then: σ2 i,u| · · · ∼ IG(¯vu,¯σ2 i,u) (32) where ¯vu=T+¯ Tuand ¯σ2 i,u = [¯ σ2 i,u¯ Tu+PT t=1(˜yd i,t −˜at,dzi,t)0(˜yd i,t −˜at,dzi,t)]/¯vu, and: σ2 i,w| · · · ∼ IG(¯vw,¯σ2 i,w) (33) where ¯vw=T+¯ Twand ¯σ2 i,w = [¯ σ2 i,w¯ Tw+PT t=1(z∗ i,t −z∗ i,t−1)0(z∗ i,t −z∗ i,t−1)]/¯vw. Note here that for σ2 i,u, the simulations are done for i= 1, . . . , Nd, while for σ2 i,w they are only done for i= 2, . . . , Ndbecause z1,t = 1 for all tby restriction. E.3 Block 3: F,Ω|A Conditional on A, the transition equation in (23b) is independent of the rest of the system. While the first and second equations of (23b) do depend on the estimates of Φ, the part of the transition equation associated with at,d is independent of the rest of the components in (23b). Accordingly, Φand σ2 ωd, the element in the lower right corner of Ω, can first be simulated independently from the rest of the parameters in (23b), and then σ2 ωm,σ2 ωq,zm, and zqcan be simulated conditionally on Φand σ2 ωd. To simulate Φand σ2 ωd, we employ the independent Normal-Gamma prior. Accordingly, continuing with letting letters denoted with an underscore reflect the prior, the conditional posterior of Φis: Φ| · · · ∼ N(¯ Φ, ¯ VΦ)I[s(Φ)] (34) with ¯ VΦ= (¯ VΦ−1+ T X t=1 a0 t−1,dσ−2 ωdat−1,d)−1(35) ¯ Φ=¯ VΦ(¯ VΦ−1 ¯ Φ+ T X t=1 a0 t−1,dσ−2 ωdat,d) (36) and I[s(Φ)] is an indicator function used to denote that the roots of Φlie outside the unit circle. Further, the conditional posterior of σ2 ωdis: σ2 ωd| · · · ∼ IG(vωd, σ2 ωd) (37) with ¯vωd=T+¯ Tωd, and ¯σ2 ωd= [¯ σ2 ωd¯ Tωd+PT t=1(at−at−1Φ)0(at−at−1Φ)]/¯vωd. Once Φand σ2 ωdare drawn, we can construct, for j={q, m}: Ct,j −βt,jCt−1,j ≡C∗ t,j =zjat,d +ωt,j (38) and draw from the conditional posterior of zjand σ2 ωj. Using again the independent Normal-Gamma prior: zj| · · · ∼ N(¯zj,¯ Vzj) (39) 45 with ¯ Vzj= (¯ Vz−1 j+ T X t=1 a0 t,dσ−2 ωjat,d)−1(40) ¯zj=¯ Vzj(¯ Vz−1 j ¯ zj+ T X t=1 a0 t,dσ−2 ωjC∗ t,j) (41) Finally, the conditional posterior of σ2 ωjis: σ2 ωj| · · · ∼ IG(¯vωj,¯σ2 ωj) (42) with ¯vωj=T+¯ Tωj, and ¯σ2 ωj= [¯ σ2 ωj¯ Tωj+PT t=1(C∗ t,j −at,dzj)0(C∗ t,j −at,dzj)]/¯vωj. E.4 Block 4: E|Y,A,Zand P|E,U For each observation of the Nddaily variables, we have that: ei,t =yi,t −zi,tat,d (43) Thus, conditional on Y,Aand Z,Eis observable. As above, since Eis independent across the Ndequations, we can sample the elements of Pin (23c) one equation at the time. As this is done in the same manner as in equations (34) to (36) of Block 3 (with the obvious change of notation), I do not repeat the computations here. E.5 The Carter and Kohn algorithm and observation weights Consider a generic state space system, written in companion form, and described by: yt=Ztat+ut∼N(0,U) (44a) at=F at−1+Rωt∼N(0,Q) (44b) where we assume that the hyper-parameters θ={U,F,R,Q}, and Ztare known, and we wish to estimate the latent state atfor all t= 1, . . . , T. To do so, we can apply Carter and Kohn’s multimove Gibbs sampling approach (see Carter and Kohn (1994)). First, because the state space model given in equation (44) is linear and (conditionally) Gaussian, the distribution of atgiven Yand that of atgiven at+1 and Yfor t=T− 1,...,1 are also Gaussian: aT|Y∼N(aT|T,PT|T), t =T(45a) at|Y,at+1 ∼N(at|t,at+1 ,Pt|t,at+1 ), t =T−1, T −2,· · · ,1 (45b) 46 where aT|T=E(aT|Y) (46a) PT|T=Cov(aT|Y) (46b) at|t,at+1 =E(at|Y,at+1) = E(at|at|t,at|t+1) (46c) Pt|t,at+1 =Cov(at|Y,at+1) = Cov(at|at|t,at|t+1) (46d) Given a0|0and P0|0, the unknown states aT|Tand PT|Tneeded to draw from (45a) can be estimated from the (conditionally) Gaussian Kalman Filter as: at|t−1=F at−1|t−1(47a) Pt|t−1=F Pt−1|t−1F0+Q(47b) Kt=Pt|t−1Z0 t(ZtPt|t−1Z0 t+U)−1(47c) at|t=at|t−1+Kt(yt−Ztat|t−1) (47d) Pt|t=Pt|t−1−KtZtPt|t−1(47e) That is, at t=T, equation 47d and 47e above, together with equation 45a, can be used to draw aT|T. Moreover, at|t,at+1 for t=T−1, T −2,· · · ,1 can also be simulated based on 45b, where at|t,at+1 and Pt|t,at+1 are generated from the following updating equations: at|t,at+1 =at|t+Pt|tF0(F Pt|tF0+Q)−1(at+1 −F at|t) (48a) Pt|t,at+1 =Pt|t+Pt|tF0(F Pt|tF0+Q)−1F Pt|t(48b) When computing the news topic contributions in Figures 7and 9, I decompose the state vector into a history of forecast error contributions. For simplicity, I use the notation introduced in Appendix E.5 to describe how this is done. At each time interval t, the forecast error in predicting ytis given by vt=yt−Ztat|t−1. In computing at|t, equation (47d) above, the Kalman gain Ktis used to weight each forecast error when computing the updated state estimate. If the predictions of the ith observable at time tare perfect, vi,t = 0 and this observation does not contribute to potential updates from at|t−1to at|t. If the predictions of the ith observable at time tare not perfect, vi,t 6= 0, the observation will influence the updated state estimate as long as it is given weight through the Kt matrix. As the updating equation in 47d has a recursive structure, the time evolution of at|tcan easily be decomposed into a set of weighted forecast error contributions, resulting in the decompositions shown in Figures 7and 9. E.6 Prior specification To implement the MCMC algorithm, and estimate the model, prior specifications for the initial state variables a0,Z0, and for the hyper-parameters Ω,U,W,Ft,P, and d 47 are needed. The prior specifications used for the initial states take the following form: a0∼N(0, I ·100), and Z0∼N(0, I ·100). The priors for the hyper-parameters Φand Φ, which are part of the Ftand Pmatrices, respectively, are set to: ¯ Φ∼N(ˆ ΦOLS, V (ˆ ΦOLS)) ¯ Φi∼N(0,0.5) for i= 1, . . . , Nd where ˆ ΦOLS are the OLS estimates of an AR(h) using the first principal component of the daily news dataset as dependent variable. V(ˆ ΦOLS) is a diagonal matrix where the non-zero entries are the variance terms associated with the ˆ ΦOLS elements. To draw zq and zm, which are part of the Ftmatrix, I use: ¯ zj∼N(1,1) for j={q, m}. The priors for the hyper-parameters Ω,U, and W, are all from the Inverse-Gamma distribution, where the first element in each prior distribution is the shape parameter, and the second the scale parameter: ¯ σ2 i,w ∼IG(¯ Tw, κ2 w) where ¯ Tw= 8000 and κw= 0.003 for i= 2, . . . , Nd;¯ σ2 i,u ∼IG(¯ Tu, κ2 u) where ¯ Tu= 100 and κu= 0.1 for all i= 1, . . . , Nd;¯ σ2 ωj∼ IG(¯ Tωj, κ2 ωj) where ¯ Tωj= 1000 for j={q, m, d}, and κωq= 0.003, and κωm=κωd= 0.1. In sum, as the full sample size T= 8769 observations, these priors are very informative for the variance terms associated with the time-varying factor loadings, but less so for the other parameters. Note, however, that the prior variance associated with the quarterly cumulator variable error term, σ2 ωq, is assumed to be considerably lower than the other variance terms. Finally, to draw the latent threshold, d, using the algorithm described in Appendix E.2, the Kparameter needs to be defined. Kcontrols our prior belief concerning the marginal sparsity probability. For example, assuming that a time-varying parameter follows Bt∼ N(0, v2), and marginalizing over Bt, it can be shown that Pr(|Bt|= 0) = 2Φ(d v)−1, where Φ is the standard normal CDF. Defining K=d vas the standardized scaling parameter with respect to the threshold, it can be seen that K= 3 implies a marginal sparsity probability exceeding 0.99. As described in Nakajima and West (2013), a neutral prior will support a range of sparsity values in order to allow the data to inform on relevant values, and they suggest that setting K= 3 is a reasonable choice.26 However, in contrast to Nakajima and West (2013), where the time-varying parameters follows AR(1) dynamics, the timevarying factor loadings in (28) follows independent random walk processes. The random walk is non-stationary, and does not have a marginal distribution. For this reason I have experimented with estimating the model using different values for K, finding that higher values for K, coupled with the rather tight priors for the variance of the factor loadings, results in worse model performance (in terms of ROC and AUROC scoring). Accordingly, 26Note that when combined with the priors over the other hyper-parameters in the model, the implied marginal prior for each threshold will not be uniform (see Nakajima and West (2013) for details). 48 Table 4. Convergence statistics. The AutoCorr row reports the 10th-order sample autocorrelation of the draws, the RNE row reports the relative numerical efficiency measure, proposed by Geweke (1992), while the IRL row reports the i-statistic, proposed by Raftery and Lewis (1992). For each entry we report the mean value together with the minimum and maximum value obtained across all parameters in parentheses. Parameters Statistic UΩF P W d AutoCorr −0.0 (−0.1,0.1) 0.1 (0.1,0.1) 0.1 (−0.1,0.1) −0.0 (−0.1,0.1) 0.0 (−0.0,0.1) 0.0 (−0.2,0.1) RNE 1.1 (0.3,1.9) 0.3 (0.2,0.5) 0.3 (0.7,1.7) 1.0 (0.6,1.6) 0.7 (0.4,1.0) 0.9 (0.4,1.5) IRL 1.9 (1.9,1.9) 1.4 (1.4,1.4) 1.4 (1.2,1.2) 1.2 (1.4,1.4) 1.0 (1.0,1.0) 1.0 (1.0,1.0) K= 0.05, in the estimations conducted in this analysis. E.7 Convergence of the Markov Chain Monte Carlo Algorithm Table 4summarizes the main convergence statistics used to check that the Gibbs sampler mixes well. In the first row of the table the mean, as well as the minimum and maximum, of the 10th-order sample autocorrelation of the posterior draws is reported. A low value indicates that the draws are close to independent. The second row of the table reports the relative numerical efficiency measure (RNE), proposed by Geweke (1992). The RNE measure provides an indication of the number of draws that would be required to produce the same numerical accuracy if the draws represented had been made from an i.i.d. sample drawn directly from the posterior distribution. An RNE value close to or below unity is regarded as satisfactory. Autocorrelation in the draws is controlled for by employing a 4 percent tapering of the spectral window used in the computation of the RNE. The last row, labelled IRL, reports the mean of the i-statistic. This statistic was proposed by Raftery and Lewis (1992). In essence, it measures the ratio of two other statistics: the total number of draws needed to achieve the desired accuracy for each parameter, and the number of draws that would be needed if the draws represented an i.i.d. chain, see Raftery and Lewis (1992) for details.27 Values of IRL exceeding 5 indicate convergence problems with the sampler. As can be seen from the results reported in Table 4, the sampler seems to have converged. That is, the mean autocorrelations are all very close to zero, and the minimum or maximum values obtained seldom exceed 0.1 in absolute value. Moreover, the mean RNE statistic does not exceed unity by a large margin for any of the parameters. Finally, the 27The parameters used for computing these diagnostics are as follows: quantile = 0.025; desired accuracy = 0.025; required probability of attaining the required accuracy = 0.95. 49 Table 5. Estimates and true DGP parameters. The numbers reported in parenthesis are standard deviations. See also Figure 10. F1Ω3,3U3,3U4,4U5,5U6,6 Estimated 0.99 0.48 1.29 1.05 0.99 1.00 (0.01) (0.01) (0.02) (0.02) (0.02) (0.02) True 0.99 0.5 1 1 1 1 IRL statistics are always well below 5. Additional convergence results can be obtained on request. E.8 A simulation experiment To control the estimation procedure, and verify the code, I run a simple simulation experiment. Artificial data is generated from a data generating process like the one described in Appendix E, with T= 8769 daily observations. Nq= 1, Nm= 1, and Nd= 8, such that N= 10. Quarterly and monthly observations are attributed across some generic year, quarters and months, such that the artificial sample contains roughly 100 and 300 observable quarterly and monthly observations, respectively. Hyper-parameters used to simulate the data are set as follows: All diagonal elements in W,U, and Ωare set to 0.001, 1, and 0.5, respectively. The threshold parameter dis set equal to 0.3 for all of the time-varying factor loadings. The autoregressive process for the law of motion for the latent daily factor, at, is specified with one lag and Φ= 0.99. The autoregressive processes for the idiosyncratic errors are specified with Φi= 0 for i= 1, . . . , Nd. Finally, zj= 1 for j={q, m}, and the latent state variables in A0and Z0are initialized at zero. The prior specifications used for estimation are in expectation all set equal to the true values, but for neither specification is the degrees of freedom parameters set higher than 100. Figure 10 reports the estimated latent daily factor alongside the simulated factor. As is clearly seen in the figure, they are very close to being perfectly correlated. In the figure four of the estimated time-varying factor loadings, together with their simulated counterparts, are also illustrated. Again, the estimated and simulated processes are very similar. As seen from the figures, the estimation procedure is also capable of identifying the true threshold value with a large degree of precision. Table 5reports the posterior median and standard deviation of the parameter estimates for Φ1,Ω3,3, and Ui,i for i={3,4,5,6}. All estimates are precisely estimated and very close to their true values. 50 Index z3,3z4,3 z5,3z6,3 Figure 10. The upper graph reports the simulated daily index together with its estimate. The subsequent graphs report the simulated factor loadings for daily observations 3 to 6, together with the true thresholds (d) and the estimated loadings and thresholds. 51