Testing for sufficient information in structural VARs

Testing for Sufficient Information in Structural VARs∗ Mario Forni† Universit`a di Modena e Reggio Emilia CEPR and RECent Luca Gambetti‡ Universitat Autonoma de Barcelona February 22, 2011 Abstract We derive necessary and sufficient conditions under which a set of variables is informationally sufficient, i.e. it contains enough information to estimate the structural shocks with a VAR model. Based on such conditions, we suggest a procedure to test for informational sufficiency. Moreover, we show how to amend the VAR if informational sufficiency is rejected. We apply our procedure to a VAR including TFP, unemployment and per-capita hours worked. We find that the three variables are not informationally sufficient. When adding missing information, the effects of technology shocks change dramatically. JEL classification: C32, E32, E62. Keywords: Structural VAR, non-fundamentalness, information, FAVAR models, technology shocks. ∗We thank Fabio Canova, Marco Lippi and Luca Sala for helpful discussion. †Financial support from Fondazione Cassa di Risparmio di Modena is gratefully acknowledged. 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: luca.gamb[email protected] 1 1 Introduction Since Sims (1980)’s seminal paper, Structural Vector Autoregression (SVAR) models have become extremely popular for structural and policy analysis. The idea behind these models is that structural economic shocks can be found as linear combinations of the residuals of the linear projection of a vector of variables onto their past values, i.e. are innovations with respect to the econometrician’s information set. Therefore, an obvious requirement for the analysis to be meaningful is that such an information set conveys all of the relevant information. This is implicitly assumed in any VAR application. But is this assumption always sensible? Unfortunately the answer is no. The basic problem is that, while agents typically have access to rich information, VAR techniques allow to handle a limited number of variables. If the econometrician’s information set does not span that of the agents the structural shocks are non-fundamental and cannot be obtained from a VAR (Hansen and Sargent, 1991, Lippi and Reichlin, 1993, 1994, Chari, Kehoe and Mcgrattan, 2008). Fernandez-Villaverde et al. (2007) derives a simple condition to check whether the shocks of a DSGE model are recoverable from a VAR and shows theoretical cases in which VAR techniques fail. Fiscal foresight and news shocks are two examples, see Leeper, Walker and Yang, (2008) and Yang (2008). Forni and Gambetti (2010), Forni and Gambetti and Sala (2010) and Gambetti (2010). At now there are no testing procedures to verify whether a specific VAR suffers from this informational problem. The contribution of this paper is twofold. First we theoretically characterize necessary and sufficient conditions under which a set of variables is informationally sufficient in a VAR, i.e. it contains enough information to estimate the structural shocks. Second, we propose a testing procedure based on such conditions. When informational sufficiency is rejected we propose a strategy to amend the VAR to fill the informational gap. We derive two main results under the general assumption the economy admits a state space representation. First, we provide a necessary and sufficient condition for informational sufficiency. The condition requires that there are no state variables that Granger cause the variables included in the VAR.1The intuition is that the state 1The precise relation between our sufficient information condition and Condition 1 of Villaverde et al. (2007) is explained in Section 2.3. An essential difference is that our condition can be tested without resorting to any particular economic model. 2 variables contain all of the relevant information; therefore, if they do not help to predict a vector, such vector must contain the same information. Second, we show that, even if the VAR is not informational sufficient, still a single shock of interest can be correctly estimated. In order for this to be the case, the shock must be orthogonal to the past of the state variables. Such conditions can be tested empirically. Based on the former result, we suggest the following testing procedure. First, we estimate the space spanned by the state variables of the economy by using the principal components of a large dataset, containing all available macroeconomic information. Second, we test whether the estimated principal components Granger cause the variables included in the VAR. The variables are informationally sufficient if and only if the null hypothesis of no Granger causality is not rejected. The latter result can be used to verify whether, even if the VAR is not informationally sufficient, a particular shock of interest can still be estimated. The test works as follows. First, we identify and estimate the structural shock. Second, we perform a test of orthogonality between the estimated shock and the lags of the principal components. If the null of orthogonality is rejected, then the shock obtained from the VAR cannot be structural. If a set of variables is not sufficient, we suggest to estimate either a structural factor model like Forni et al. (2009) or a VAR augmented by by the principal components, i.e. the FAVAR model proposed by Bernanke Boivin and Eliasz (2005), where number of principal components is determined by applying a sequence of sufficient information tests. As an application we study technology shocks in the US. We test whether a smallscale VAR model, such as those typically used to study the effects of technology shocks, is informationally sufficient. Specifically, we use a VAR with total factor productivity, the unemployment rate and per-capita hours worked. We find that these three variables are Granger caused by the first two principal components of a large dataset of US macrcoeconomic variables. Therefore we add such principal components to the VAR and show that the remaining principal components do not Granger cause the augmented VAR, meaning that the information conveyed in the augmented VAR is sufficient. Finally, we identify the technology shock as the only one driving total factor productivity in the long run, in both the original and the augmented VAR. Differences in the results 3 in the two models are dramatic. While in the original VAR technology shocks increase hours and reduce unemployment, in the augmented VAR results are reversed: hours reduce and unemployment increases. In the augmented model, investment and GDP react very sluggishly to the shock, prices fall and the real wage increases. Overall the result are hard to reconcile with the view that technology shocks are an important source of business cycle fluctuations. The remainder of the paper is organized as follows. Section 2 presents theoretical results, as well as our proposed testing procedures. Section 3 discusses the application. Section 4 concludes. 2 Theory 2.1 The macroeconomy Let us start from the following MA representation of the macroeconomy. Assumption 1 (MA representation). The n-dimensional vector xtof stationary macroeconomic time series satisfies xt=F(L)ut,(1) where utis a q-dimensional, orthonormal white noise vector of structural macroeconomic shocks and F(L)is an n×qmatrix of impulse response functions, i.e. squaresummable linear filters in the non-negative powers of the lag operator L, such that rank (F(z)) = qfor some complex number z. Representation (1) can be thought of as the representation of a macroeconomic equilibrium. Consider for instance the state-space representation studied in Villaverde, Rubio-Ramirez, Sargent and Watson (2007), i.e. st=Ast−1+But(2) xt=Cst−1+Dut(3) where stis an r-dimensional vector of stationary “state” variables, q≤r≤n,A,B, Cand Dare conformable matrices of parameters, Bhas a left inverse B−1such that B−1B=Iq. Pre-multiplying (2) by B−1we get ut=B−1(I−AL)st. Substituting this into (3) and rearranging gives xt=DB−1+ (C−DB−1A)Lst.(4) 4 Stationarity of stensures invertibility of (2), so that st= (I−AL)−1But. Combining this with (4) we get the MA representation xt=DB−1+ (C−DB−1A)L(I−AL)−1But,(5) which is a special case of (1). The assumption on the rank (F(L)) ensures that the representation is not redundant in the sense that there is another representation with a smaller number of shocks. 2.2 Sufficient information The SVAR econometrician observes xt, possibly with error. Precisely, Assumption 2. (Econometrician’s information set) The econometrician information set X∗ tis given by the closed linear space spanned by present and past values of the variables in x∗ t(in symbols X∗ t= span(x∗ 1t, . . . , x∗ nt)), where x∗ t=xt+ξt=F(L)ut+ξt,(6) ξtbeing a (possibly zero) vector of measurement errors, orthogonal to ujt−k,j= 1, . . . , q, any k, and ξt−k,k > 0. In practice the number of observable variables nis very large, so that the econometrician needs to reduce it in order to estimate a VAR. The VAR information set is then spanned by an s-dimensional sub-vector of x∗ t, or more, generally, an s-dimensional linear combination of x∗ t, say z∗ t=W x∗ t(with snot necessarily equal to q). Assumption 3 (VAR information set). The information set of the VAR is Z∗ t= span(z∗ 1t−k, . . . , z∗ st−k, k ≥0),z∗ t=W x∗ t,Wbeing s×n. Now, consider the theoretical projection equation of z∗ ton its past history, i.e. z∗ t=P(z∗ t|Z∗ t−1) + t.(7) The SVAR methodology consists in (a) estimating a VAR to get t; (b) attempting to get the structural shocks as linear combinations of the estimated entries of t. Hence a key property of z∗ tand the related information set, is that the entries of tspan the structural shocks, i.e. the information in the history of z∗ tis sufficient to estimate the shocks. We call such property “sufficient information”. 5 Definition 1 (Sufficient information). We say that z∗ tand the related information set Z∗ tcontain “sufficient information” if and only if there exist a matrix Msuch that ut=Mt. Let us stress that sufficiency, defined in this way, is related only to the variables in z∗ tand has nothing to do with the choice of a proper identification scheme. The correct identification of Mis a further problem, which does make sense only if sufficiency holds true. 2.3 Sufficient information and fundamentalness From (6) and the definition of z∗ twe get z∗ t=W F (L)ut+W ξt=zt+W ξt.(8) Structuralness is related to “fundamentalness” of the MA representation in (8).2 Let us first recall the concept of fundamentalness. Definition 2 (Fundamentalness). We say that utis fundamental for wt=Hxt, and the MA representation wt=HF (L)utis fundamental, if and only if ut∈ Wt= span(w1t−k, . . . , wmt−k, k ≥0) (i.e. Ut= span(u1t−k, . . . , uqt−k, k ≥0) = Wt). The following proposition holds: Proposition 1.The information in z∗ tis sufficient if and only if (a) zjt ∈ Z∗ tfor any jand (b) utis fundamental with respect to zt. Proof. If (a) and (b) hold true, then ut∈ Z∗ t=Et= span(1t−k, . . . , st−k, k ≥0). Being orthogonal to Et−1,utbelongs to span(1t, . . . , st). On the other hand, let us assume that z∗ tis sufficient, i.e. ut=Mt. Then (a) holds, because zjt ∈ Utand Ut⊆ Z∗ t. As for (b), let St= span(z1t−k, . . . , zst−k, W ξt−k, k ≥0). Now, ujt ∈ St−1), j= 1, . . . , q, since it belongs to Z∗ tand Z∗ t⊆ St. But ujt is orthogonal to ξt−k,k≥0 by Assumption 2. Hence ujt ∈ Zt,j= 1, . . . , q. QED Proposition 1 says that, for z∗ tbeing sufficient, there must be a linear transformation of z∗ twhich is free of measurement errors and have a fundamental representation in the structural shocks. 2Some important references about fundamentalness are Hansen and Sargent (1991), Lippi and Reichlin (1993, 1994), Chari, Kehoe and McGrattan (2008), Fernandez-Villaverde et al. (2007). 6 To conclude this section, let us observe that, in the particular case of F(L) being a matrix of rational functions, fundamentalness of utfor wt, along with fundamentalness of the associated MA representation wt=HF (L)utis equivalent to the following condition (see e.g. Rozanov, 1967, Ch. 2). Condition R.The rank of HF (z)is qfor all zsuch that |z|<1. Considering equation (5) and the case wt=xt, condition R is satisfied if and only if Dis invertible and the eigenvalues of A−BD−1Care strictly less than one in modulus, which is Condition 1 of Villaverde et al. (2007). 2.4 Testable implications of sufficient information Proposition 2.If x∗ tGranger causes z∗ t, then z∗ tis not informationally sufficient. Proof. Assume that z∗ tis sufficient, so that ut=Mt. Then ujt−k∈ Z∗ t−1for k > 0. It follows that P(z∗ t|Z∗ t−1) = W(F1ut−1+F2ut−2+. . .) and t=W F0ut+W ξt. Hence tis orthogonal to both ut−k,k > 0, and, by serial uncorrelation of ξt(Assumption 2), ξt−k,k > 0. Therefore t⊥x∗ t−k,k > 0 and x∗ tdoes not Granger cause z∗ t. QED The intuition is that, if a set of variables is sufficient, than it contains all of the existing information, so that no other variable or set of variables can Granger cause it. Proposition 2 can be of some usefulness in practice.3In particular, if the econometrician believes that a given variable in x∗ t, say vt, conveys relevant information, he can check whether vtGranger causes z∗ tas a vector. If vtGranger causes z∗ t, the VAR with z∗ tis misspecified. Observe that, according to Proposition 2, identification is not required to perform the test, consistently with the fact that sufficient information, as observed above, is independent of the identification scheme. On the other hand, Proposition 2 has an important limitation in that, being only a necessary condition, it can be used to reject sufficiency but not to validate it. Clearly, testing all of the variables in x∗ twould be close to a validation, but unfortunately this is not feasible, since in practice x∗ tis of high dimension. On the one hand, we cannot use all of the variables simultaneously; on the other hand, testing each one of them separately would yield, with very high probability, to reject sufficiency even if z∗ tis informationally sufficient, owing to Type I error. 3Proposition 2 is derived (within somewhat different settings) in Forni and Reichlin (1996) and Giannone and Reichlin (2006). 7 We can provide a sufficient condition by assuming the state space representation above, i.e. by replacing Assumption 1 with the more restrictive Assumption 10: Assumption 10(ABCD representation). The vector xtof macroeconomic time series satisfies equations (2) and (3). It is easily seen from equations (6) and (4) that x∗ tfollows the static factor model x∗ t=Gft+ξt,(9) where G=DB−1C−DB−1Aand ft=s0 ts0 t−10. In addition, we need to assume that the history of the structural shocks helps predicting z∗ t, or, equivalently, that z∗ tis autocorrelated to some extent (since otherwise nothing can Granger cause it). Assumption 4 (Autocorrelation of z∗ t). There exists a summable sequence {ck}∞ k=1 such that R=WP∞ k=1 ckFkhas rank q. The following proposition establishes a necessary and sufficient condition for informational sufficiency. Proposition 3.Let Kbe any non-singular p×pmatrix, pbeing the dimension of ft. z∗ tis informationally sufficient if and only if gt=Kftdoes not Granger cause z∗ t. Proof. Let us assume that z∗ tis sufficient, i.e. ut=Mt. Then tis orthogonal to ut−k,k > 0 and therefore to gt−k,k > 0. Hence P(z∗ t|Z∗ t−1) = P(z∗ t|z∗ jt−k, git−k, j = 1, . . . , s, i = 1, . . . , p, k > 0), so that gtdoes not Granger cause z∗ t. Regarding the opposite implication, let us assume that gtdoes not Granger cause z∗ t. We have P(z∗ t|Z∗ t−1) = P(z∗ t|z∗ jt−k, git−k, j = 1, . . . , s, i = 1, . . . , p, k > 0). But the latter projection is equal to P(z∗ t|ujt−k, j = 1, . . . , q, k > 0) = WP∞ k=1 Fkut−k=ζt, since ζt belongs to span(z∗ jt−k, git−k, j = 1, . . . , s, i = 1, . . . , p, k > 0) and z∗ t−ζtis orthogonal to such space because of Assumption 2. On the other hand, ζt=P(z∗ t|Z∗ t−1) = P∞ k=1 Akt−k. Projecting both sums on span(it−k, uit−k, i = 1, . . . , s, j = 1, . . . , r) we get W Fkut−k=Akt−kfor all k, so that W Fkut=Aktfor all kand R= (WP∞ k=1 ckFk)ut= (P∞ k=1 Ak)t. Assumption 4 ensures that Rhas a left inverse, so that ut=R−1(P∞ k=1 Ak)t. QED The intuition for sufficiency is that, under Assumption 1’, the factors contain all of the information available in the system; therefore they Granger cause every predictable vector, unless such vector contain the same information. 8 Proposition 3 is useful in that, besides providing a sufficient condition, allows us to summarize the signals in the large dimensional vector xtinto a relatively small number of factors (the entries of gt). Such factors are unobservable, but, under suitable assumptions, can be consistently estimated by the principal components ˆgt, as both the number of variables and the number of time observations go to infinity (Stock and Watson, 2002; Forni, Giannone, Lippi and Reichlin, 2009). 2.5 Testing for sufficient information Proposition 3 provides the theoretical basis for the following testing procedure. 1. Take a large data set x∗ tcapturing all of the relevant macroeconomic information. 2. Set a maximum number of factors Pand compute the first Pprincipal components of x∗ t. 3. Perform Granger causation tests to see whether the first hprincipal components, h= 1, . . . , P, Granger cause z∗ t. If the null of no Granger causality is never rejected, z∗ tis informationally sufficient. Otherwise, sufficiency is rejected. If informational sufficiency is rejected, we cannot use the VAR for global identification. However, partial identification could still provide correct results, as shown in the following subsection. 2.6 Structuralness of a single shock Even if informational sufficiency is rejected, z∗ tcould be sufficient to get a single shock of interest, say u1t, or a subset of shocks u1t, . . . , ujt,j < q. This is important in that for many applications the econometrician is interested in identifying just a single shock. To see this, consider the following example z∗ 1t=u1t+u2t−1 z∗ 2t=u1t−u2t−1 In this case z∗ tis not sufficient for utby Proposition 1. In fact, since the determinant of the MA filter has a zero in zero, the MA representations non fundamental by 9 Appendix: Data Transformations: 1=levels, 2= first differences of the original series, 4 = logs of the original series, 5= first differences of the logs of the original series . no.series Transf. Mnemonic Long Label 1 5 GDPC1 Real Gross Domestic Product, 1 Decimal 2 5 GNPC96 Real Gross National Product 3 5 NICUR/GDPDEF National Income/GDPDEF 4 5 DPIC96 Real Disposable Personal Income 5 5 OUTNFB Nonfarm Business Sector: Output 6 5 FINSLC1 Real Final Sales of Domestic Product, 1 Decimal 7 5 FPIC1 Real Private Fixed Investment, 1 Decimal 8 5 PRFIC1 Real Private Residential Fixed Investment, 1 Decimal 9 5 PNFIC1 Real Private Nonresidential Fixed Investment, 1 Decimal 10 5 GPDIC1 Real Gross Private Domestic Investment, 1 Decimal 11 5 PCECC96 Real Personal Consumption Expenditures 12 5 PCNDGC96 Real Personal Consumption Expenditures: Nondurable Goods 13 5 PCDGCC96 Real Personal Consumption Expenditures: Durable Goods 14 5 PCESVC96 Real Personal Consumption Expenditures: Services 15 5 GPSAVE/GDPDEF Gross Private Saving/GDP Deflator 16 5 FGCEC1 Real Federal Consumption Expenditures & Gross Investment, 1 Decimal 17 5 FGEXPND/GDPDEF Federal Government: Current Expenditures/ GDP deflator 18 5 FGRECPT/GDPDEF Federal Government Current Receipts/ GDP deflator 19 2 FGDEF Federal Real Expend-Real Receipts 20 1 CBIC1 Real Change in Private Inventories, 1 Decimal 21 5 EXPGSC1 Real Exports of Goods & Services, 1 Decimal 22 5 IMPGSC1 Real Imports of Goods & Services, 1 Decimal 23 5 CP/GDPDEF Corporate Profits After Tax/GDP deflator 24 5 NFCPATAX/GDPDEF Nonfinancial Corporate Business: Profits After Tax/GDP deflator 25 5 CNCF/GDPDEF Corporate Net Cash Flow/GDP deflator 26 5 DIVIDEND/GDPDEF Net Corporate Dividends/GDP deflator 27 5 HOANBS Nonfarm Business Sector: Hours of All Persons 28 5 OPHNFB Nonfarm Business Sector: Output Per Hour of All Persons 29 5 UNLPNBS Nonfarm Business Sector: Unit Nonlabor Payments 30 5 ULCNFB Nonfarm Business Sector: Unit Labor Cost 31 5 WASCUR/CPI Compensation of Employees: Wages & Salary Accruals/CPI 32 1 COMPNFB Nonfarm Business Sector: Compensation Per Hour 33 5 COMPRNFB Nonfarm Business Sector: Real Compensation Per Hour 34 1 GDPCTPI Gross Domestic Product: Chain-type Price Index 35 1 GNPCTPI Gross National Product: Chain-type Price Index 36 1 GDPDEF Gross Domestic Product: Implicit Price Deflator 37 1 GNPDEF Gross National Product: Implicit Price Deflator 16 no.series Transf. Mnemonic Long Label 38 5 INDPRO Industrial Production Index 39 5 IPBUSEQ Industrial Production: Business Equipment 40 5 IPCONGD Industrial Production: Consumer Goods 41 5 IPDCONGD Industrial Production: Durable Consumer Goods 42 5 IPFINAL Industrial Production: Final Products (Market Group) 43 5 IPMAT Industrial Production: Materials 44 5 IPNCONGD Industrial Production: Nondurable Consumer Goods 45 1 AWHMAN Average Weekly Hours: Manufacturing 46 1 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 1 UEMPMEAN Average (Mean) Duration of Unemployment 55 1 UNRATE Civilian Unemployment Rate 56 5 HOUST Housing Starts: Total: New Privately Owned Housing Units Started 57 1 FEDFUNDS Effective Federal Funds Rate 58 1 TB3MS 3-Month Treasury Bill: Secondary Market Rate 59 1 GS1 1-Year Treasury Constant Maturity Rate 60 1 GS10 10-Year Treasury Constant Maturity Rate 61 1 AAA Moody’s Seasoned Aaa Corporate Bond Yield 62 1 BAA Moody’s Seasoned Baa Corporate Bond Yield 63 1 MPRIME Bank Prime Loan Rate 64 5 BOGNONBR Non-Borrowed Reserves of Depository Institutions 65 5 TRARR Board of Governors Total Reserves, Adjusted for Changes in Reserve 66 5 BOGAMBSL Board of Governors Monetary Base, Adjusted for Changes in Reserve 67 5 M1SL M1 Money Stock 68 5 M2MSL M2 Minus 69 5 M2SL M2 Money Stock 70 5 BUSLOANS Commercial and Industrial Loans at All Commercial Banks 71 5 CONSUMER Consumer (Individual) Loans at All Commercial Banks 72 5 LOANINV Total Loans and Investments at All Commercial Banks 73 5 REALLN Real Estate Loans at All Commercial Banks 74 5 TOTALSL Total Consumer Credit Outstanding 75 5 CPIAUCSL Consumer Price Index For All Urban Consumers: All Items 76 5 CPIULFSL Consumer Price Index for All Urban Consumers: All Items Less Food 77 5 CPILEGSL Consumer Price Index for All Urban Consumers: All Items Less Energy 17 no.series Transf. Mnemonic Long Label 78 5 CPILFESL Consumer Price Index for All Urban Consumers: All Items Less Food & Energy 79 5 CPIENGSL Consumer Price Index for All Urban Consumers: Energy 80 5 CPIUFDSL Consumer Price Index for All Urban Consumers: Food 81 5 PPICPE Producer Price Index Finished Goods: Capital Equipment 82 5 PPICRM Producer Price Index: Crude Materials for Further Processing 83 5 PPIFCG Producer Price Index: Finished Consumer Goods 84 5 PPIFGS Producer Price Index: Finished Goods 85 5 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 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 1 GS10-FEDFUNDS 96 1 GS1-FEDFUNDS 97 1 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&P/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 117 4 Per-capita hours worked (HOANBS/Civilian Polulation 16 and over) 18 References [1] Bai, J., and S. 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[28] Yang, S.S. (2008). Quantifying tax effects under policy foresight. Journal of Monetary Economics, 52(8):1557-1568. 21 Tables A B j z∗ tw1 tw2 tz∗ tw1 tw2 t 1 0.000 − − 0.005 − − 2−0.480 − − 0.055 − 3 0.405 0.475 −0.113 0.977 4−0.620 0.375 −0.091 0.452 5−0.125 0.250 −0.115 0.581 6−0.105 0.500 −0.142 0.641 7−0.125 0.545 −0.126 0.186 8−0.285 0.785 −0.027 0.197 9−0.125 0.705 − − 0.216 10 −0.085 0.450 − − 0.207 11 −0.050 0.660 − − 0.148 12 − − 0.355 − − 0.186 13 − − 0.395 − − 0.239 14 − − 0.560 − − 0.279 15 − − 0.720 − − 0.337 Table 1: p-values A: Test for informational sufficiency B: Test for structuralness of the technology shock. 22 Figures Figure 1: Impulse response functions 23 Figure 2: Impulse response functions. 24