No news in business cycles

No News in Business Cycles Mario Forni∗ Universit`a di Modena e Reggio Emilia CEPR and RECent Luca Gambetti† Universitat Autonoma de Barcelona Luca Sala‡ Universit`a Bocconi and IGIER February 21, 2011 Abstract This paper uses a structural, large dimensional factor model to evaluate the role of ‘news’ shocks (shocks with a delayed effect on productivity) in generating the business cycle. We find that (i) existing small-scale VECM models are affected by ‘non-fundamentalness’ and therefore fail to recover the correct shock and impulse response functions; (ii) news shocks have a limited role in explaining the business cycle; (iii) their effects are in line with what predicted by standard neoclassical theory; (iv) the bulk of business cycle fluctuations are explained by shocks unrelated to technology. JEL classification: C32, E32, E62. Keywords: structural factor model, news shocks, invertibility, fundamentalness. ∗Contact: Dipartimento di Economia Politica, via Berengario 51, 41100, Modena, Italy. Tel. +39 0592056851; e-mail: [email protected] †The financial support from the Spanish Ministry of Science and Innovation through grant ECO200909847 and the Barcelona Graduate School Research Network is gratefully acknowledged. Contact: Office B3.1130 Departament d’Economia i Historia Economica, Edifici B, Universitat Autonoma de Barcelona, Bellaterra 08193, Barcelona, Spain. Tel. +34 935814569; e-mail: [email protected] ‡Contact: Universit´a Bocconi, Via Roentgen 1, 20136, Milan, Italy. Tel. +39 0258363062; e-mail: [email protected] 1 1 Introduction In recent years there has been a renewed interest in the idea that business cycles could be generated by changes in expectations (this idea dates back to Pigou, 1927). The literature has focused on shocks having delayed effects on technology, the so-called ‘news shocks’. The seminal paper by Beaudry and Portier (2006) finds that positive news shocks have a positive impact on stock prices, consumption, investment and hours worked and account for more than half of output fluctuations (see Figure 10 in Beaudry and Portier, 2006).1 These results do not square with standard neoclassical one-sector models, in which good news about future technology trigger a wealth effect that affects positively consumption but negatively hours, output and investment on impact. Beaudry and Portier (2007), Jaimovich and Rebelo (2009), Schmitt-Grohe and Uribe (2008) propose models that can reconcile the theory with the above results. Existing evidence has been obtained by using small-scale VAR or VECM models. This is problematic, because when structural shocks have delayed effects on macroeconomic variables, VAR models using such variables may be affected by non-fundamentalness (Lippi and Reichlin, 1994, Leeper, Walker and Yang, 2008, Forni and Gambetti, 2010b, Feve, Matheron and Sahuc, 2009). Non-fundamentalness means that the variables used by the econometrician do not contain enough information to recover the structural shocks and the related impulse response functions. The question is essentially whether the structural MA representation of such variables can be inverted or not. If not, the variables do not have a VAR representation in the structural shocks, implying that such shocks cannot be obtained by estimating a VAR with these variables.2 To get an intuition of the problem, assume that the news shock affects total factor productivity (TFP) with a one-period delay. Clearly, by observing TFP at time twe get information about news arrived in t−1, but do not learn anything about the current shock. Coupling TFP with a series affected by the shock on impact (like stock prices) does not necessarily solve the problem, as shown in Section 2. In this paper we present new evidence on the effects of news shocks by estimating a large-dimensional factor model with US quarterly data. Large factor models, including Factor Augmented VARs (FAVARs), can be used for structural economic analysis just like VAR models, as in Giannone, Reichlin and Sala (2004), Bernanke, Boivin and Eliasz (2005), Stock and Watson (2005), Forni, Giannone, Lippi and Reichlin (2009), Forni 1Beaudry and Lucke (2009) and Beaudry, Portier and Dupaigne (2008) confirm the same empirical findings. 2A partial list of references on non-fundamentalness includes Lippi and Reichlin (1993), Hansen and Sargent (1991), Chari, Kehoe and McGrattan (2005), Fernandez-Villaverde, Rubio-Ramirez, Sargent and Watson (2005), Giannone, Reichlin and Sala (2006). 2 and Gambetti (2010a).3Their advantage in the present context is that they are not affected by the non-fundamentalness problem, as shown in Forni, Giannone, Lippi and Reichlin (2009).4The intuition is that large factor models, unlike VARs, include a large amount of information (virtually all available macroeconomic series), so that insufficient information is unlikely. As a matter of fact, factor models have been successful in explaining well known VAR puzzles like the ‘price’ puzzle and the ‘exchange rate’ puzzle (Bernanke, Boivin and Eliasz, 2005, Forni and Gambetti, 2010a). In addition, the factor model enables us to verify whether a given VAR information set is affected by nonfundamentalness or not. Our testing procedure is explained in Section 3.5. Our results are the following. First, we estimate a two-shock factor model and apply the above test to the two variables in the benchmark model of Beaudry and Portier (TFP and stock prices). We find that the structural MA representation of TFP and stock prices is non-fundamental. Then we identify the news shock as in Beaudry and Portier (2006), by assuming a zero impact effect on TFP and find that the impulse responses and variance decompositions obtained with the factor model are completely different from those obtained by imposing the same identification scheme to a bivariate VECM. In particular, the effects on stock prices are much smaller. Then we focus on our preferred factor model specification (a six-shock specification). We identify the news shock by imposing both a zero impact effect and a maximal longrun effect on TFP. The latter condition corresponds to the idea that news shocks should explain the main bulk of technology in the long-run. We find that: (i) hours worked, investment and output have negative impact responses, whereas consumption and stock prices are essentially unaffected on impact; (ii) investment, consumption, output and stock prices increase gradually as TFP increases; (iii) news shocks account for about 20-25% of business-cycle fluctuations in investment, consumption and GDP. Such effects are essentially in line with what predicted by a standard neoclassical model. Finally, we identify a standard technology shock, having non-zero impact effect on productivity, by imposing that no other shock affects TFP contemporaneously. We find that the news and the technology shocks explain together almost all of TFP volatility at all frequencies, but only 25-35% of business-cycle fluctuations in investment, consumption and GDP, leaving substantial room for sources of volatility unrelated to technology. 3Large ‘generalized’ or ‘approximate’ dynamic factor models are specifically designed to handle a large amount of information. Early references are Forni and Reichlin (1998), Forni, Hallin, Lippi and Reichlin (2000), Forni and Lippi (2001), Stock and Watson (2002a, 2002b), Bai and Ng (2002). 4This result holds true provided that economic agents can see the structural shocks, as assumed in most of the current theoretical literature. A recent noticeable exception is Lorenzoni (2009), where agents can only observe technology ‘news’ disturbed by an aggregate ‘noise’. We are not concerned with this interesting case in the present paper. 3 Overall, our results are fairly similar to those obtained by Barsky and Sims (2009) with a six-variable VAR including inflation, a short term interest rate, consumption and a consumer sentiment index, in addition to TFP and stock prices. Consistently with this, our test is not able to reject fundamentalness for such variables. The paper is structured as follows. In Section 2 we provide a simple analytical example that shows how non-fundamentalness can arise in the presence of news shocks. In Section 3 we present the factor model, argue why it is not subject to the non-fundamentalness problem, and describe our fundamentalness test. Section 4 presents empirical results. Section 5 concludes. 2 Non-fundamentalness and News Shocks In this Section we present a simple example, in which non-fundamentalness appears as a consequence of the presence of news shocks. Measured TFP, θt, is assumed to follow the non-stationary process: θt=θt−1+εt−2+ut(1) where εtis the news shock and utis the ‘standard’ technology shock, affecting TFP on impact. Agents observe the shock εtat time tand react to it immediately, while the shock will affect TFP only at time t+ 2. Therefore the econometrician will not be able to identify εtby observing θt. The representative consumer maximizes Et ∞ X t=0 βtCt, where Ctis consumption and βis a discount factor, subject to the constraint Ct+PtSt+1 = (Pt+θt)St, where Ptis the price of a share, Stis the number of shares and (Pt+θt)Stis the total amount of resources available at time t. The equilibrium value for asset prices is given by: Pt=Et ∞ X j=1 βjθt+j Considering (1), the above equation can be solved to get the following structural MA representation ∆θt ∆Pt!= L21 β2 1−β+βL β 1−β! εt ut!.(2) 4 The determinant is −β2 1−β−βz +β 1−βz2 which vanishes for z= 1 and z=−β. As β < 1, the moving average is non invertible and the two shocks utand εtare non-fundamental for the variables ∆Ptand ∆θt. Not even a very forward-looking variable like stock prices conveys enough information to recover the shock. 3 The structural factor model In this paper we use the factor model presented in Forni, Giannone, Lippi and Reichlin (2009, FGLR henceforth).5Here we provide a short presentation of the model, discuss the relation with non-fundamentalness and explain our fundamentalness test. 3.1 Representation We assume that each macroeconomic variable xit is the sum of two mutually orthogonal unobservable components, the common component χit and the idiosyncratic component ξit: xit =χit +ξit.(3) The idiosyncratic components are poorly correlated in the cross-sectional dimension.6 They arise from shocks or sources of variation which considerably affect only a single variable or a small group of variables. For variables related to particular sectors, like industrial production indexes or production prices, the idiosyncratic component may reflect sector specific variations; for strictly macroeconomic variables, like GDP, investment or consumption, the idiosyncratic component can be interpreted as a measurement error.7 The common components account for the bulk of the co-movements between macroeconomic variables, being linear combinations of a relatively small number rof factors 5FGLR is a special case of the generalized dynamic factor model proposed by Forni, et al. (2000, 2004, 2005) and Forni and Lippi (2001, 2010). This model differs from the traditional dynamic factor model of Sargent and Sims (1977) and Geweke (1977) in that the number of cross-sectional variables is infinite and the idiosyncratic components are allowed to be mutually correlated to some extent, along the lines of Chamberlain (1983), Chamberlain and Rothschild (1983) and Connor and Korajczyk (1988). Closely related models have been studied by Forni and Reichlin (1998), Stock and Watson (2002a, 2002b, 2005), Bai and Ng (2002, 2007), Bai (2003) and Bernanke et al. (2005). 6See FGLR, Assumption 5 for a precise statement. 7Altug, (1989), Sargent, (1989), and Ireland (2004) show that the model can be interpreted as the linear solution of a DSGE model with measurement error. 5 f1t, f2t,· · · , frt, not depending on i: χit =a1if1t+a2if2t+· · · +arifrt =aift.(4) The dynamic relations between the macroeconomic variables arise from the fact that the vector ftfollows the relation ft=N(L)ut,(5) where N(L) is a r×qmatrix of rational functions in the lag operator Land ut= (u1tu2t· · · uqt)0is a q-dimensional vector of orthonormal white noises, with q≤r. Such white noises are the structural macroeconomic shocks.8 The discussion in Section 3.4 motivates the assumption that N(z) is zeroless, i.e. rank(N(z)) = qfor any z, which implies fundamentalness. This ensures that fthas the finite order VAR representation (Anderson and Deistler, 2008) D(L)ft=t=Rut,(6) where D(L) is a r×rmatrix of polynomials such that D(L)−1R=N(L) and R=N(0). Combining equations (3) to (6), the model can be written in dynamic form xit =bi(L)ut+ξit,(7) where bi(L) = aiD(L)−1R. (8) The entries of the q-dimensional vector bi(L) are the impulse response functions. 3.2 Identification Representation (7) is not unique, since the impulse response functions and the related primitive shocks are not identified. In particular, if His any orthogonal q×qmatrix, then χit =ci(L)vt where ci(L) = bi(L)H0and vt=Hut. However, assuming mutually orthogonal structural shocks, post-multiplication by H0is the only admissible transformation, i.e. the impulse response functions are unique up to orthogonal transformations, just like in structural VAR models (FGLR, Proposition 2). 8In the large dynamic factor model literature they are sometimes called the “common” or “primitive” shocks or “dynamic factors” (whereas the entries of ftare the “static factors”). Equations (3) to (5) need further qualification to ensure that all of the factors are loaded, so to speak, by enough variables with large enough loadings (see FGLR, Assumption 4); this “pervasiveness” condition is necessary to have uniqueness of the common and the idiosyncratic components, as well as the number of static factors rand dynamic factors q. 6 As a consequence, structural analysis in factor models can be carried on along lines very similar to those of standard structural VAR analysis. Specifically q(q−1)/2 restrictions have to be imposed on the matrix of impulse response functions Bn(L) = (b1(L)0b2(L)0· · · bn(L)0)0, with nthe number of variables, to pin down all the elements of H. If the researcher is interested in identifying just a single shock, the target is to determine the entries of a single column of the matrix H, say H1, which is enough to obtain the first column of Bn(L), say Bn1(L). 3.3 Estimation Estimation proceeds through the following steps. 1. Starting with an estimate ˆr, the static factors are estimated by means of the first ˆrprincipal components of the variables in the dataset, and the factor loadings by means of the associated eigenvectors. Precisely, let ˆ Γxbe the sample variancecovariance matrix of the data: the estimated loading matrix ˆ An= (ˆa0 1ˆa0 2· · · ˆa0 n)0 is the n×rmatrix having on the columns the normalized eigenvectors corresponding to the first largest ˆreigenvalues of ˆ Γx, and the estimated factors are ˆ ft=ˆ A0 n(x1tx2t· · · xnt)0.9 2. ˆ D(L) and ˆtare obtained by running a VAR(ˆp) with ˆ ftwhere the number of lags ˆpis chosen according to some criterion. 3. Let ˆ Γbe the sample variance-covariance matrix of ˆt. Having an estimate ˆqof the number of dynamic factors, an estimate of a non-structural representation of the common components is obtained by using the spectral decomposition of ˆ Γ. Precisely, let ˆµ j,j= 1,...,ˆq, be the j-th eigenvalue of ˆ Γ, in decreasing order, ˆ M the q×qdiagonal matrix with qˆµ jas its (j, j) entry, and ˆ Kthe r×qmatrix with the corresponding normalized eigenvectors on the columns. The estimated matrix of non-structural impulse response functions is ˆ Cn(L) = ˆ Anˆ D(L)−1ˆ Kˆ M.(9) To account for estimation uncertainty, the following non-overlapping block bootstrap technique is adopted. Let X= [xit] be the T×nmatrix of data. Such matrix is partitioned into Ssub-matrices Xs(blocks), s= 1, . . . , S, of dimension τ×n,τbeing the 9The factors are identified only up to linear transformations. What is estimated is a basis of the factor space. 7 integer part of T/S.10 An integer hsbetween 1 and Sis drawn randomly with reintroduction Stimes to obtain the sequence h1, . . . , hS. A new artificial sample of dimension τS ×nis then generated as X∗=hX0 h1X0 h2· · · X0 hSi0and the corresponding impulse response functions, ˆ Cn(L), are estimated and the identifying assumptions are imposed to get H1and the corresponding impulse response functions ˆ Bn1(L) = ˆ Cn(L)H1. A set of structural impulse response functions is obtained by repeating drawing, estimation and identification. Confidence bands are obtained by taking the relevant percentiles of the point-wise distributions. 3.4 Tall systems and fundamentalness Here we discuss why the assumption of fundamentalness is justified in the factor model. Let us go back to equation (5) ft=N(L)ut, where N(L) is a (r×q) matrix of rational functions in the lag operator L, with r≥q. Under what conditions are the shocks utfundamental for ft? A necessary and sufficient condition is that the rank of N(z) be qfor all zsuch that |z|<1 (see e.g. Rozanov, 1967, Ch. 1, Section 10, and Ch. 2, p. 76). Let us first focus on the particular case r=q, and interpret ftas a vector of observable variables to be used in a VAR. The above fundamentalness condition reduces to the requirement that the determinant of N(z) does not vanish within the unit circle in the complex plane. If this condition holds, then the shock utcan be found using a VAR for ft. In general, however, there is no guarantee that the condition holds, as shown in Section 2. Now let us turn to the case r > q, which is the normal case in the factor model. In such case N(z) is a “tall”, rectangular matrix. Its rank is less than qfor some z, i.e. the shock is non-fundamental, only if all of the (q×q) sub-matrices of N(z) are singular. Clearly this is a very special case, since it requires r q!−1 equalities to be satisfied. Therefore, in general, when r > q,N(z) has rank qfor all zand the representation can be assumed fundamental. As a very elementary example, consider the case q= 1, r= 2, f1t=ut+ 2ut−1, f2t= 2ut−1. Here utis non-fundamental for both f1tand f2t, and cannot be found as a linear combination of present and past values of a single factor. However, utis fundamental for the vector ft, since ut=f1t−f2t. Observe that fundamentalness of representation (5) implies fundamentalness of the 10Note that τhas to be large enough to retain relevant lagged autoand cross-covariances. 8 system χt=Bn(L)ut, where χt= (χ1t· · · χnt)0and Bn(L) = AnD(L)−1R,An= (a0 1a0 2· · · a0 n)0(provided that Anhas full column rank). 3.5 Testing for fundamentalness While the whole system Bn(L) is fundamental, the q-dimensional square submatrices of Bn(L) corresponding to selected subsets of variables can be singular for values of z within the unit circle (without hurting consistency of estimation). Precisely, considering aq-dimensional vector of integers I, with elements Ii,i= 1, . . . , q,utis fundamental for the subvector χIt = (χI1t· · · χIqt)0=BI(L)utif det BI(z) does not vanish within the unit circle. A test for fundamentalness of a particular square subsystem can then be performed by looking at the estimated distribution of the modulus ρof the smallest root. We reject the null of fundamentalness (ρ≥1) against the alternative of non-fundamentalness (ρ < 1) at the significance level αas long as the frequency of values larger than 1 is smaller than α. Rejection of fundamentalness implies that an hypothetical VAR model using χIt would be misspecified. In principle, such an implication cannot be directly extended to the true VAR setting, where xIt is used in place of χIt. In practice however the idiosyncratic components are usually very small, so that rejection (acceptance) of fundamentalness provides a useful indication against (in favor of) a particular VAR specification. 4 Empirics 4.1 Data and model specification Our data set is composed of 116 US quarterly series, covering the period 1959-I to 2007IV. Most series are taken from the FRED database. A few stock market and leading indicators are taken from Datastream. Some series have been constructed by ourselves as transformations of the original FRED series. The series include both national accounting data like GDP, investment, consumption and the GDP deflator, TFP and consumers sentiment which are available only at quarterly frequency, and series like industrial production indices, CPI, PPI and employment, which are produced monthly. Monthly data have been temporally aggregated to get quarterly figures. As required by the model, the data are transformed to obtain stationarity. Following Stock and Watson (2005), prices and nominal variables are taken in second differences of logs, rather than in first differences of logs, and interest rates in first differences, rather 9 no.series Transf. Mnemonic Long Label 46 2 AWOTMAN Average Weekly Hours: Overtime: Manufacturing 47 2 CIVPART Civilian Participation Rate 48 5 CLF16OV Civilian Labor Force 49 5 CE16OV Civilian Employment 50 5 USPRIV All Employees: Total Private Industries 51 5 USGOOD All Employees: Goods-Producing Industries 52 5 SRVPRD All Employees: Service-Providing Industries 53 5 UNEMPLOY Unemployed 54 5 UEMPMEAN Average (Mean) Duration of Unemployment 55 2 UNRATE Civilian Unemployment Rate 56 5 HOUST Housing Starts: Total: New Privately Owned Housing Units Started 57 2 FEDFUNDS Effective Federal Funds Rate 58 2 TB3MS 3-Month Treasury Bill: Secondary Market Rate 59 2 GS1 1-Year Treasury Constant Maturity Rate 60 2 GS10 10-Year Treasury Constant Maturity Rate 61 2 AAA Moody’s Seasoned Aaa Corporate Bond Yield 62 2 BAA Moody’s Seasoned Baa Corporate Bond Yield 63 2 MPRIME Bank Prime Loan Rate 64 6 BOGNONBR Non-Borrowed Reserves of Depository Institutions 65 6 TRARR Board of Governors Total Reserves, Adjusted for Changes in Reserve 66 6 BOGAMBSL Board of Governors Monetary Base, Adjusted for Changes in Reserve 67 6 M1SL M1 Money Stock 68 6 M2MSL M2 Minus 69 6 M2SL M2 Money Stock 70 6 BUSLOANS Commercial and Industrial Loans at All Commercial Banks 71 6 CONSUMER Consumer (Individual) Loans at All Commercial Banks 72 6 LOANINV Total Loans and Investments at All Commercial Banks 73 6 REALLN Real Estate Loans at All Commercial Banks 74 6 TOTALSL Total Consumer Credit Outstanding 75 6 CPIAUCSL Consumer Price Index For All Urban Consumers: All Items 76 6 CPIULFSL Consumer Price Index for All Urban Consumers: All Items Less Food 77 6 CPILEGSL Consumer Price Index for All Urban Consumers: All Items Less Energy 78 6 CPILFESL Consumer Price Index for All Urban Consumers: All Items Less Food & Energy 79 6 CPIENGSL Consumer Price Index for All Urban Consumers: Energy 80 6 CPIUFDSL Consumer Price Index for All Urban Consumers: Food 81 6 PPICPE Producer Price Index Finished Goods: Capital Equipment 82 6 PPICRM Producer Price Index: Crude Materials for Further Processing 83 6 PPIFCG Producer Price Index: Finished Consumer Goods 84 6 PPIFGS Producer Price Index: Finished Goods 85 6 OILPRICE Spot Oil Price: West Texas Intermediate 86 5 USSHRPRCF US Dow Jones Industrials Share Price Index (EP) NADJ 87 5 US500STK US Standard & Poor’s Index if 500 Common Stocks 88 5 USI62...F US Share Price Index NADJ 89 5 USNOIDN.D US Manufacturers New Orders for Non Defense Capital Goods (BCI 27) 90 5 USCNORCGD US New Orders of Consumer Goods & Materials (BCI 8) CONA 16 no.series Transf. Mnemonic Long Label 91 1 USNAPMNO US ISM Manufacturers Survey: New Orders Index SADJ 92 5 USVACTOTO US Index of Help Wanted Advertising VOLA 93 5 USCYLEAD US The Conference Board Leading Economic Indicators Index SADJ 94 5 USECRIWLH US Economic Cycle Research Institute Weekly Leading Index 95 2 GS10-FEDFUNDS 96 2 GS1-FEDFUNDS 97 2 BAA-FEDFUNDS 98 5 GEXPND/GDPDEF Government Current Expenditures/ GDP deflator 99 5 GRECPT/GDPDEF Government Current Receipts/ GDP deflator 100 2 GDEF Governnent Real Expend-Real Receipts 101 5 GCEC1 Real Government Consumption Expenditures & Gross Investment, 1 Decimal 102 1 Fernald’s TFP growth CU adjusted 103 1 Fernald’s TFP growth 104 5 DOW JOONES/GDP DEFL 105 5 S&P500/GDP DEFL 106 1 Fernald’s TFP growth - Investment 107 1 Fernald’s TFP growth - Consumption 108 1 Fernald’s TFP growth CU - Investment 109 1 Fernald’s TFP growth CU - Consumption 110 1 Personal Finance Current 111 1 Personal Finance Expected 112 1 Business Condition 12 Months 113 1 Business Condition 5 Years 114 1 Buying Conditions 115 1 Consumer’s sentiment: Current Index 116 1 Consumer’s sentiment: Expected Index 17 References [1] Altug, S., 1989, Time-to-Build and Aggregate Fluctuations: Some New Evidence, International Economic Review 30, 889-920. 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[47] Stock, J.H. and M.W. Watson, 2005, Implications of Dynamic Factor Models for VAR Analysis, NBER Working Papers no. 11467. [48] Uhlig, H., 2005, What are the effects of monetary policy on output? Results from an agnostic identification procedure, Journal of Monetary Economics 52, 381-419. 21 Tables jVariables (Ij) Two Shocks 1 TFP (102) Stock P (105) 2 TFP (103) Stock P (105) Six Shocks 1 TFP (102) Stock P (105) Non Dur. C (12) Inv. (7) Hours (27) GDP (1) 2 TFP (103) Stock P (105) Non Dur. C (12) Inv. (7) Hours (27) GDP (1) 3 TFP (102) Stock P (105) Non Dur. C (12) GDP (1) CPI (75) 3M T-Bill (58) 4 TFP (102) Stock P (105) Non Dur. C (12) Hours (27) CPI (75) 3M T-Bill (58) 5 TFP (102) Stock P (105) Non Dur. C (12) Sentiment (116) CPI (75) 3M T-Bill (58) Table 1: Subsets of variables (I) used in the test described in Section 3.5. The numbers in brackets correspond to those in the Appendix. jMean Median 68% 90% 95% Point est. Two Shocks 1 0.531 0.515 0.768 1.060 1.086 0.481 2 0.711 0.812 0.940 1.102 1.128 0.861 Six Shocks 1 0.692 0.763 0.934 1.084 1.125 0.459 2 0.636 0.665 0.878 1.023 1.066 0.279 3 0.645 0.666 0.835 1.051 1.083 0.755 4 0.557 0.546 0.712 0.966 1.041 0.294 5 0.856 0.952 1.072 1.161 1.192 1.099 Table 2: Moduli of the smallest root of the submatrices BI(L) defined in Table 1. 22 Variables Horizons 0 4 8 40 Factor model TFP (102) 0.0 6.5 7.2 7.4 Stock Prices (105) 16.1 55.2 61.2 63.4 VECM TFP (102) 0 0.7 0.6 33.9 Stock Prices (105) 99.7 97.6 96.5 93.4 Table 3: Explained forecast error variance (percentages) at various horizons in the two-shock factor model and the bivariate VAR for the common components using the Cholesky identification (levels). The numbers in brackets correspond to those in the Appendix. 23 Variables Horizons (a) % Total Variance % Variance 2-8 Years 0 4 8 40 (b) (c) News shock TFP (102) 0.0 11.1 17.6 29.9 7.8 14.6 GDP (1) 6.2 11.0 11.3 15.3 15.2 19.9 Consumption (11) 2.9 17.0 27.0 40.2 25.0 25.0 Investment (7) 8.0 14.1 12.3 12.5 20.0 20.3 Hours (27) 26.1 14.4 17.5 15.7 21.9 19.9 Stock Prices (105) 6.9 7.0 8.1 9.8 10.0 10.1 Sentiment current (115) 7.0 14.7 21.5 24.4 24.4 22.1 Sentiment expected (116) 26.4 31.1 36.0 37.9 37.9 32.3 Prices (75) 19.5 23.9 20.6 15.5 23.7 27.1 3M T-Bill (58) 28.5 25.5 20.5 18.4 25.4 25.6 Technology shock TFP (102) 100.0 85.7 79.2 69.2 80.4 76.3 GDP (1) 64.2 22.4 20.8 23.6 38.1 12.0 Consumption (11) 30.7 16.2 16.0 16.5 17.8 9.2 Investment (7) 8.7 2.6 2.4 3.8 6.1 2.9 Hours (27) 0.7 0.9 1.0 0.9 2.2 1.6 Stock Prices (105) 0.5 0.8 1.0 1.3 2.7 1.4 Sentiment current (115) 3.3 3.0 3.6 4.0 4.0 3.0 Sentiment expected (116) 13.8 8.0 7.8 7.9 7.9 6.1 Prices (75) 1.0 1.1 1.4 1.6 2.9 1.4 3M T-Bill (58) 2.6 1.5 1.4 1.3 4.0 1.7 Table 4: Variance decomposition. (a) Fraction of the variance of the forecast error for the levels of the variables at different horizon (b) Percentage of variance of the variables transformed to get stationarity explained by the shock (c) Percentage of cyclical variance (of periodicity between 2 to 8 years) explained by the shock. The numbers in brackets correspond to those in the Appendix. 24 Figures Figure 1: Impulse response functions in the two-shocks model. Left column: technology shock, right column: news shock. Upper row: response of TFP; Lower row: rsponses of stock prices. Solid: factor model (median). Dotted: factor model 68% confidence bands. Dashed: VECM for the common components. 25