Can GDP measurement be further improved? Data revision and reconciliation

Transcription

1 Can GDP measuremen be furher improved? Daa revision and reconciliaion Jan P.A.M. Jacobs Universiy of Groningen, Universiy of Tasmania, CAMA and CIRANO Samad Sarferaz KOF Swiss Economic Insiue, ETH Zurich, Swizerland Jan-Egber Surm KOF Swiss Economic Insiue, ETH Zurich, Swizerland and CESifo, Germany Simon van Norden HEC Monréal, CIRANO and CIREQ Augus 2018 Absrac Recen years have seen many aemps o combine expendiure-side esimaes of U.S. real oupu (GDE) growh wih income-side esimaes (GDI) o improve esimaes of real GDP growh. We show how o incorporae informaion from muliple releases of noisy daa o provide more precise esimaes while avoiding some of he idenifying assumpions required in earlier work. This relies on a new insigh: using muliple daa releases allows us o disinguish news and noise measuremen errors in siuaions where a single vinage does no. Our new measure, GDP ++, fis he daa beer han GDP +, he GDP growh measure of Aruoba e al. (2016) published by he Federal Reserve Bank of Philadephia. Hisorical decomposiions show ha GDE releases are more informaive han GDI, while he use of muliple daa releases is paricularly imporan in he quarers leading up o he Grea Recession. JEL classificaion: E01, E32 Keywords: naional accouns, oupu, income, expendiure, news, noise Preliminary versions of his paper were presened a he 10h Inernaional Conference on Compuaional and Financial Economerics (CFE 2016), he XIII Conference on Real-Time Daa Analysis, Mehods and Applicaions, Banco de España, and he ESCoE Conference on Economic Measuremen 2018, London. We hank Dean Croushore, Gabriele Fiorenini, Adrian Pagan, Alexander Rahke and Enrique Senana for helpful commens. 1

2 1 Inroducion Unlike many oher naions, U.S. naional accouns feaure disinc esimaes of real oupu based on he expendiure approach (GDE) and he income approach (GDI), see Figure 1. As poined ou by Sone, Champernowne and Meade (1942), while in heory hese wo approaches should give idenical esimaes, measuremen errors cause discrepancies o arise. 1 These discrepancies are someimes imporan. Chang and Li (2015) examine he impac of using GDI raher han GDE in nearly wo dozen recen empirical papers published in major economic journals; hey find subsanive differences in roughly 15% of hem. Nalewaik (2012) finds ha GDI leads o quicker deecion of U.S. recessions han GDE. Figure 1: U.S. GDP growh: Expendiure side vs. income side 10 5 GDE 0 GDI The same applies o he producion-based esimae of oupu. See e.g. he sudy of Rees, Lancaser and Finlay (2015) on Ausralian GDP. 1

3 While several sudies have ried o deermine which measure should be preferred in various conexs, Weale (1992) and Diebold (2010) argue ha reconciling hem is a more useful response as i should incorporae more informaion. Fixler and Nalewaik (2009) poin ou, however, ha reconciliaion radiionally relies on he assumpion ha measuremen errors are noise, which in urn forces he reconciled esimae of he laen variable ( rue GDP in his case) o be less variable han any of he individual series being reconciled. They insead propose ha measuremen errors may also include a news componen. While his causes a loss of idenificaion, hey glean informaion from he revision of GDE and GDI o place bounds on relaive conribuions of news and noise in a leas-squares framework. Aruoba e al. (2012) consider he problem from a forecas combinaion perspecive, assuming news errors and imposing priors in lieu of idenificaion wihou revisions, while Aruoba e al. (2016) consider alernaive idenifying assumpions and propose he addiion of an insrumenal variable. Almuzara e al. (2018) invesigae a dynamic facor model wih coinegraion resricions. Aruoba e al. (2016) is he basis for he GDP + measure published by he Federal Reserve Bank of Philadelphia. 2 However, while heir approach ignores he possibiliy of daa revision, Figure 2 shows ha he published series is subjec o imporan revisions, which complicaes is inerpreaion and use in policy decisions. Separaely, Jacobs and van Norden (2011) and Kishor and Koenig (2012) propose sae-space frameworks ha allow esimaion of boh news- and noise-ype measuremen errors in daa revision, bu do no consider problems of daa reconciliaion. In his paper we exend Jacobs and van Norden (2011, henceforh JvN) o consider he problem of reconciliaion and idenificaion in which here are muliple esimaes of he common underlying variable, all of which are subjec o revision. Allowing for boh news and noise measuremen errors, he resul is a modeling framework subsanially more general han hose previously proposed. We show ha idenificaion of hese wo ypes of measuremen errors is made possible by modeling daa revisions as well as he dynamics of he series. We provide a hisorical decomposiion of GDE and GDI ino news and noise shocks, and we compare hose series o our improved GDP esimae, GDP ++. We find ha GDP ++ is more persisen han eiher GDE or 2 See hp:// 2

4 Figure 2: GDP + in real-ime 6 5 GDP + Oc GDP + May GDP + Nov GDP + Jan Various vinages of GDP + according o he esimaes of he Federal Reserve Bank of Philadelphia. GDI. While boh series appear o conain boh news and noise shocks, news shocks have a larger share in GDE han in GDI. The paper is srucured as follows. In Secion 2 we presen our economeric framework. We show ha our sysem is idenified using real-ime daa and news-noise assumpions. In Secion 3 we describe our daa and esimaion mehod. Resuls are shown in Secion 4 and Secion 5 concludes. Formal proofs of some resuls relaed o idenificaion are presened in an Appendix. 3

5 2 Economeric Framework In his secion, afer esablishing some noaion, we describe our economeric framework. We begin by briefly reviewing he univariae news and noise model of JvN before generalizing i o he problem of daa reconciliaion. We hen compare he resuls o he GDP + model of Aruoba e al. (2016) and discuss heir differences for he idenificaion of news and noise measuremen errors. We follow he sandard noaion in his lieraure by leing y +j be an esimae published a ime + j of some real-valued scalar variable y a ime. We define y as a l 1 vecor of l differen vinage esimaes of y +i, i = 1,..., l so y [ ]. y +1, y +2,..., y +l For sae-space models, we follow he noaion of Durbin and Koopman (2001) y = Z α + ε (1) α +1 = T α + R η (2) where y is l 1, α is m 1, ε is l 1 and η is r 1; ε N(0, H) and η N(0, I r ). Boh error erms are i.i.d. and orhogonal o one anoher A Sae-Space model of Measuremen Error wih News and Noise JvN denoe he unobserved rue value of a variable as ỹ, so ha is measuremen error u y ι l ỹ, where ι l is an l 1 vecor of ones. They model hese measuremen errors as he sum of news and noise measuremen errors. Measuremen errors are said o be noise ( ) ζ +i when hey are orhogonal o he rue values ỹ, so ha y +i = ỹ + ζ +i, cov(ỹ, ζ +i ) = 0. (3) 3 For more deailed assumpions, see Durbin and Koopman (2001, Secion 3.1 and 4.1. For convenience we omi consans from he model in his exposiion. 4

6 Noise implies ha revisions (y +i+1 are described as news (ν +i ) if and only if y +i ) are generally forecasable. Measuremen errors ỹ = y +i + ν +i, cov(y +j, ν +i ) = 0 j i (4) If daa revisions are pure news errors, curren and pas vinages of he series will be of no use in forecasing fuure daa revision. In heir sae-space model JvN impose ε 0 l 1 and pariion he sae vecor α ino four componens α = [ỹ, φ, ν, ζ ], (5) of lengh 1, b, l and l respecively, where φ is used o capure he dynamics of he rue values while ν and ζ are he news and noise measuremen errors, respecively. They similarly pariion Z = [Z 1, Z 2, Z 3, Z 4 ] (6) where Z 1 = ι l (a l 1 vecor of 1 s), Z 2 = 0 l b (an l b marix of zeros), Z 3 = I l, and Z 4 = I l (boh l l ideniy marices). Their measuremen equaion (1) hen simplifies o y = Z α = ỹ + ν + ζ = Truh + News + Noise. (7) They conformably pariion he marix T as T 11 T T 21 T T =, (8) 0 0 T T 4 where T 11 is a scalar, and {T 12, T 21, T 22, T 3, T 4 } are 1 b, b 1, b b, l l and l l; 0 is a conformably defined marix of zeros. The (b + 1) (b + 1) block in he upper lef simply capures he dynamics of ỹ while T 3 and T 4 capure he dynamics of he news and noise shocks. If measuremen errors are independen across ime periods (bu no vinages), hen T 3 T 4 0 l l. As we will see below, in he special case where ỹ is assumed o follow 5

7 ] an AR (p) process, his will impose p = b + 1, he row vecor [T 11 T 12 will conain he auoregressive coefficiens and he remainder of he upper lef (b + 1) (b + 1) par will be composed of zeros and ones. 4 The essenial difference beween news and noise errors is capured in he (1 + b + 2l) (1 + 2l) marix R, which is pariioned as follows R 1 R 3 0 R R = 0 U l diag(r 3 ) 0, (9) 0 0 R 4 where U l is a l l marix wih zeros below he main diagonal and ones everywhere else, R 3 = [σ ν1, σ ν2,..., σ νl ], where σ νi is he sandard error of he measuremen error associaed wih i-h esimae y +i, diag(r 3 ) is a l l marix wih elemens of R 3 on is main diagonal, and R 4 is an l l marix. Finally, he error erm is pariioned as η = [, η e, η ν, η ζ] where η e refers o errors associaed wih he rue values, and η ν and η ζ are he errors for news and noise, respecively. JvN noe ha (if he model is idenified, a quesion we deal wih below) his framework permis convenional echniques o be used o esimae he model parameers, allow for missing observaions, esimae and forecas he unobserved rue values ỹ ogeher wih heir confidence inervals, and es hypoheses. 2.2 Daa Reconciliaion We now show how he above framework may be adaped o he case where we have wo alernaive esimaes of he same underlying rue value ỹ, boh of which are subjec o revision. We define Y as a 2l 1 vecor of l differen vinage esimaes for he 2 variables y1 +i [ y1 +1, y , y1 +l, y2 +1 and y2 +i, i = 1,..., l, for a paricular observaion, so Y ], y2 +2,..., y2 +l, a vecor of lengh 2l. Our sae-space model 4 For deails, see Jacobs and van Norden (2011). 6

8 now becomes Y = Z α (10) α +1 = T α + R η (11) We again pariion he sae vecor α ino four componens α = [ỹ, φ, ν, ζ ], (12) which are now of lengh 1, b, 2l and 2l respecively, and we similarly pariion Z = [Z 1, Z 2, Z 3, Z 4 ] (13) where Z 1 = ι 2l (a 2l vecor of ones), Z 2 = 0 2l b (a 2l b marix of zeros), and Z 3 = Z 4 = I 2l (boh are 2l 2l ideniy marices). The measuremen equaion (10) herefore again simplifies o Y = Z α = ỹ + ν + ζ = Truh + News + Noise. The marix T is pariioned much as before T 11 T T 21 T T =, (14) 0 0 T T 4 The upper lef block (consising of T 11, T 12, T 21 and T 22 ) is precisely he same as in (8) above; his is because i solely deermines he dynamics of ỹ, which are unchanged. However, he addiion of a new series increases he dimension of T 3 and T 4 from l l o 2l 2l. R is now a (1 + b + 4l) (1 + 4l) marix where we separae he news and noise 7

9 measuremen errors for he wo variables R 1 R 3 R R U l diag(r 3 ) R = 0 0 U l diag(r 4 ) R 5 0, (15) R 6 ] where he row vecor R 3 = [σ ν, σ 11 ν,..., σ 12 ν corresponds o he news in y1 while R 11 4 = ] [σ ν, σ 21 ν,..., σ 22 ν corresponds o he news in y2. diag(r 2l 3 ) and diag(r 4 ) are l l diagonal marices wih he elemens of R 3 and R 4 on heir main diagonals, while R 5 and R 6 are l l diagonal marices. Finally, we pariion η = [, η e, η ν 1, η ν 2, η ζ 1, η ζ 2 ] where ηe refers o errors associaed wih he rue values, and η νi and η ζi are he errors for news and noise measuremen errors in variable i. To illusrae, consider he following very simple case. Le y1 GDE (he growh rae of real gross domesic expendiure), y2 GDI (he growh rae of real gross domesic income), l = 2 (we only consider wo vinages, he 1s and 2nd releases) and we ll assume ha he growh rae of rue real oupu ỹ follows an AR (1). Then (10) becomes GDE 1s GDE 2nd GDI 1s GDI 2nd ỹ ỹ 1 = ν ζ ỹ ν GDE, ζ GDE, ỹ 0 ν GDE, ζ GDE,2 0 0 = + + ỹ 0 0 ν GDI, ζ GDI,1 0 ỹ ν GDI, ζ GDI,2 = Truh + News + Noise. 8

10 and (11) becomes ỹ +1 ỹ ν +1 ρ = 0 0 T 3 0 ỹ ỹ 1 ν + R η, ζ T 4 ζ where R = σ e σν GDE1 σν GDE2 σν GDI1 σν GDI σν GDE1 σν GDE σν GDE σν GDI1 σν GDI σν GDI σζ GDE σζ GDE σζ GDI σ GDI2 η = [ e, ν GDE1, ν GDE2, ν GDI1, ν GDI2, ζ GDE1, ζ GDE2, ζ GDI1, ζ GDI2 ] ζ 2.3 Idenificaion and GDP + Aruoba e al. (2016) consider he problem of idenificaion in a special case of he GDE/GDI example considered above where only a single vinage is available (l = 1). Their unresriced 9

11 model may be wrien as 5 GDE = GDI ỹ ỹ 1 η E (16) ỹ +1 ỹ η E +1 η I +1 = ρ η I ỹ ỹ 1 η E η I σ yy σ ye σ yi σ Ey σ EE σ EI σ Iy σ IE σ II e y e E e I (17) and hey show ha i is no idenified. They propose adding a hird (insrumenal) variable which is correlaed wih ỹ bu no wih η E or η I, suggesing ha household survey daa may be suiable for his purpose. We argue ha he model may be idenified insead by increasing he number of vinages analysed and assuming ha measuremen errors are he sum of news and noise measuremen errors as characerized above. We explore his poin in he remainder of his secion by comparing he available number of sample momens o he number of free parameers in he model. In he Appendix we provide a more rigorous proof of idenificaion in a slighly simpler model using he mehods of Komunjer and Ng (2011). The essenial insigh comes from he form of he R marix in (15). News and noise measuremen errors have ighly consrained behaviour across successive daa vinages; Noise errors are assumed o be uncorrelaed across vinages and wih innovaions in rue values, while news errors mus be correlaed wih one anoher, wih innovaions in rue values, and heir variances mus be decreasing as series are revised. If we have wo series o reconcile (here GDE and GDI) and l vinages of each, we have 2 l (2 l + 1)/2 observable cross momens as well as 2 l firs-order auocorrelaion coefficiens, for a oal of l (2 l + 3) momens. The only free parameers in he above model, however, are he auocorrelaion coefficien ρ and he (1 + 4 l) non-zero elemens of 5 See Aruoba e al. (2016), equaions (A.1) and (A.2). Their model furher differs from he model above in ha (a) hey model only he sum of news and noise shocks, and (b) hey assume ha T 3 = T 4 = 0, a condiion ha we will also impose, below. 10

12 R, for a oal of 2 (1 + 2 l). This implies ha he number of available momens increases wih l 2 while he number of free parameers increases only wih l. 6 In he special case where we use only a single daa release, l = 1, we have 2 (1+2 1) = 6 free parameers o esimae, bu only 1 ( ) = 5 available momens wih which o do so. This is consisen wih he lack of idenificaion noed by Aruoba e al. (2016). However, if we use l = 2 daa vinages, we have 2 ( ) = 10 free parameers and 2 ( ) = 14 momens wih which o idenify hem. For l = 3 we have 27 momens wih which o esimae 14 parameers and for l = 4 (he case we consider below) we have 44 momens wih which o esimae 18 parameers. This suggess ha as we add more daa releases, we poenially have he abiliy o generalize he model furher sill. The univariae daa revision model of JvN envisages wo such ypes of generalizaion. 1. We may wish o relax some of he zero resricions on R. In paricular, i may be desirable o allow for news shocks o be correlaed across he wo variables, or o allow for noise shocks o be correlaed across daa releases. 2. We may wish o relax some of he zero resricions on he ransiion marix in (11) o allow for measuremen errors o be correlaed across calendar periods. (JvN refer o hese as spillover effecs.) In he Appendix, we briefly explore he possibiliies for idenificaion wih some of hese generalizaions. We now urn o consider he revisions in he available daa. 3 Daa and Esimaion 3.1 Daa We use monhly vinages of quarerly expendiure-based and income-based esimaes of GDP from he Bureau of Economic Analysis (BEA) covering he period 2003Q1 2014Q3. 6 Noe ha we have ignored any free parameers in T 3 and T 4 in hese calculaions. We reurn o his, below. One mus also keep in mind ha idenificaion by daa revision requires ha he daa are in fac revised. If no, we effecively reurn o he underidenified case of l = 1. 11

13 For GDE we employ he Advance, he Third, he 12h and he 24h releases and Second/Third, 12h and he 24h releases for GDI. Due o a lag in source daa availabiliy he BEA does no prepare Advance esimaes for GDI. The iniial esimaes for GDI are presened wih he Second GDI esimae. Esimaes for fourh quarer GDI are presened in he Third esimae only Esimaion We employ Gibbs Sampling mehods o obain poserior simulaions for our model s parameers (see, e.g., Kim and Nelson 1999). We use conjugae and diffuse priors for he coefficiens and he variance covariance marix, resuling in a mulivariae normal poserior for he coefficiens and an invered Wishar poserior for he variance covariance marix. For he prior for he coefficiens resriced o zero we assume he mean o be zero and variance o be close o zero. Our Gibbs sampler has he following srucure. We firs iniialize he sampler wih values for he coefficiens and he variance covariance marix. Condiional on daa, he mos recen draw for he coefficiens and for he variance covariance marix, we draw he laen sae variables α for = 1,..., T using he procedure described in Carer and Kohn (1994). In he nex sep, we condiion on daa, he mos recen draw for he laen variable α and for he variance covariance marix, drawing he coefficiens from a mulivariae normal disribuion. Finally, condiional on daa, he mos recen draw for he laen variables and he coefficiens, we draw he variance covariance marix from an invered Wishar disribuion. We cycle hrough 100K Gibbs ieraions, discarding he firs 90K as burn-in. Of hose 10K draws we save only every 10h draw, which gives us in oal 1000 draws on which we base our inference. Convergence of he sampler was checked by sudying recursive mean plos and by varying he saring values of he sampler and comparing resuls. 7 See Fixler e al. (2014) for a more deailed discussion of he GDE-GDI vinage hisory. 12

14 4 Resuls Here we compare our measure of GDP o releases of GDE and GDI in four differen ways: (i) in graphs, (ii) looking a hisorical decomposiions, (iii) by invesigaing dynamics, and (iv) by calculaing relaive conribuions. To disinguish beween he rue unknown values of GDP and our model s esimaes of hese values, we refer o our model s esimaes as GDP ++. Figure 3: GDP ++ vs. GDE True value Advance Second 12h release 24h release The blue line represens he poserior mean of GDP (he rue value) and he shaded area around he blue line indicaes 90% of poserior probabiliy mass. The green line represens he advance esimae, he purple line is he second esimae, he red line he 12h release and he orange line he 24h release of expendiure side GDP growh. 13

15 4.1 Comparison of GDP ++ and releases of GDE and GDI In Figure 3 we compare GDP ++ and is shaded poserior ranges (90% of probabiliy mass) o he four releases of GDE we employed in he esimaion, he Advance, hird, he 12h and he 24h release. There is some evidence ha he releases are more volaile han he rue values of GDP. We observe ha he releases are ouside he poserior bounds for some periods. This observaion holds especially for he Advance release and he 24h release; in some periods, like e.g. 2010Q1, he Advance release and he 24h release are on differen sides of he poserior range. Figure 4: GDP ++ vs. GDI True value Second/Third 12h 24h release The blue line represens he poserior mean of GDP, he rue value, and he shaded area around he blue line indicaes 90% of poserior probabiliy mass. The purple line is he second/hird esimae, he red line he 12h release and he orange line he 24h release of income side GDP growh. 14

16 Figure 4 shows GDP ++ ogeher wih shaded poserior ranges (90% of probabiliy mass) and he hree releases of GDI we employed in he esimaion, he Second/Third, he 12h and he 24h release. The releases flucuae around he poserior bounds of he rue values. The GDI releases are more volaile han our esimaes GDP ++. The releases of GDI are also much more volaile han he releases of GDE. Noe ha he sample pahs of GDP M and GDE and GDI in Aruoba e al. (2016, Figure 3) show a differen picure han our Figures 3 and 4. GDE differs more from heir GDP measure han GDI. 4.2 Hisorical decomposiion Our economeric framework (10-11) allows he hisorical decomposiion of GDE and GDI in erms of news and noise measuremen errors. We illusrae he decomposiion for GDE. Suppose, we have l releases of GDE GDE 1 = ρgdp η G + η 1 Eζ GDE 2 = ρgdp η G + η 1 Eν + η 2 Eζ. =. GDE l = ρgdp η G + η 1 Eν η l 1 Eν + ηl Eζ. Then he oal revision of GDE can be wrien as GDE l GDE 1 = ηeν η l 1 Eν + ηeζ l ηeζ 1 }{{}}{{} News Noise (18) where every elemen on he righ-hand side of he equaion is par of he sae vecor whose esimaes may be recovered using sandard echniques. 15

17 Figure 5: Hisorical Decomposiion GDE 3 2 Toal revision News Noise GDI 3 2 Toal revision News Noise Hisorical decomposiion of he oal revision (24h release minus second esimae) ino news and noise. The red bars depic he share of news and he green bars he share of noise in oal revision (grey line). The hisorical decomposiion is based on he decomposiion described in (18). 16

18 The oucomes of he hisorical decomposiions are shown in Figure 5. The op panel shows oal revisions in GDE wih news and noise shares, he boom panel oal GDE revisions wih news and noise shares. We observe ha oal revisions in GDI, he boom panel, are larger han oal revisions in GDE, a sylized fac which can also be disilled from he previous wo figures. The wo panels sugges ha he news share in oal GDE revisions is larger han he noise share while he opposie seems o hold for oal revisions in GDI. This observaion is consisen wih Fixler and Nailewaik (2009), who also rejec he pure noise assumpion in GDI. I also appears ha GDI was paricularly noisy around he sar of 2008 and afer Dynamics of GDP ++ and oher GDP measures In Figure 6 we depic he (ρ, σ 2 ) pairs summarizing he dynamics of our rue GDP esimae across all draws. We conras he (ρ, σ 2 ) pairs corresponding o our GDP ++ esimae o he (ρ, σ 2 ) pairs obained when using a news measuremen error only or a noise measuremen error only version of our model, he benchmark model esimaed in Aruoba e al. (2016) and when fiing an AR(1) model o GDE and GDI. Figure 6 reveals ha our real-ime daa based esimae of GDP is somewha less persisen han he GDP + measure of Aruoba e al. (2016), bu exhibis a higher persisence han he esimaes for GDE and GDI. 8 We also find ha he poserior mean of he innovaion variance of our GDP ++ is much smaller han he innovaion variances of GDE, GDI and he benchmark model of Aruoba e al. (2016). The innovaion variance of GDP ++ is also smaller han he innovaion variance of he models esimaed wih news and noise measuremen errors only, which in urn are higher han he innovaion variance of GDP +. The combinaion of a ρ ha is close o hose implied by he various models esimaed in Aruoba e al. (2016) and a σ 2 ha is much smaller han he ones implied by Aruoba e al. (2016) leads o a higher forecasabiliy of he GDP ++ measure. 8 We hank Dongho Song for making his Malab code available online. 17

19 2 Figure 6: GDP Dynamics GDPE GDPI 5 GDP News 4 3 GDP Noise GDP ++ GDP The grey shaded area consiss of (ρ, σ 2 ) pairs across draws from our sampler and he blue do is he poserior mean of he (ρ, σ 2 ) pairs across draws. The black dos represen he poserior mean of he (ρ, σ 2 ) pairs of he news only and noise model, respecively. The red do is he poserior mean of he (ρ, σ 2 ) pairs of GDP + using he benchmark specificaion (ζ = 0.8) described in Aruoba e al. (2016). The green dos are (ρ, σ 2 ) pairs, resuling from AR(1) models fied o expendiure side and income side GDP growh, respecively. The sampling period for re-esimaing he Aruoba e al. (2016) model and for fiing he AR(1) models o he wo GDP measures is 2003Q3 2014Q3 (released on Ocober 28, 2016). 4.4 Relaive conribuions of GDE and GDI o GDP ++ To assess he relaive imporance of GDI and GDE a differen releases, we use he Kalman gains. They represen he weigh ha he esimaed value places on esimaes of various releases. The oucomes are lised in Table 1. The resuls show ha he weighs assigned o differen releases vary grealy as we change he assumed srucure of he measuremen error. When hey are assumed o be pure News, he second panel of he able shows ha 98% of he weigh is pu on he las release of GDE. Once we allow for he possibiliy of noise errors, however, more weigh is assigned o GDI and weighs are spread over more releases. The earlies releases of GDE receive 18

20 Table 1: Kalman Gains Weigh on GDE GDI News and Noise Advance Second/Third h h Release News Only Advance Second/Third h h Release Noise Only Advance Second/Third h h Release less weigh han he laer releases, while he opposie is rue for GDI. In all cases, we also find ha GDE releases are more imporan for explaining GDP han GDI releases, in conras o Aruoba e al. (2016). 5 Conclusion We have described a new approach o daa reconciliaion ha explois muliple daa releases on each series. This helps boh wih he idenificaion of measuremen errors and wih opimally exracing informaion from muliple noisy series. We used his o propose a new measure of U.S. GDP growh using real-ime daa on GDE and GDI. Our measure GDP ++ is shown o be more persisen han GDE and GDI and has smaller residual variance. In addiion i has a similar auoregressive coefficien bu smaller residual variance han he GDP measure GDP + of Aruoba e al. (2016). Hisorical decomposiions of GDE and GDI measuremen errors reveal a larger news share in GDE han in GDI. 19

21 Appendix This appendix firs analyzes he idenificaion of he univariae sae space sysem in Jacobs and van Norden (2011), using he procedure described in Komunjer and Ng (2011) and used by Aruoba e al. (2016). Thereafer we discuss he possibiliies for idenificaion in more general reconciliaion models by comparing he number of free parameers o he number of available momen condiions. Idenificaion in he univariae, wo vinage JvN framework The sae space form of he Jacobs and van Norden (JvN) model wih wo vinages and no spillovers can be expressed as y1 = y ỹ ν 1 ν 2 ζ 1 ζ 2 ρ = ỹ ν 1 ν 2 ζ 1 ζ 2, (A.1) ỹ 1 ν 1 1 ν 2 1 ζ 1 1 ζ η,ỹ η 1,ν η 2,ν η 1,ζ η 2,ζ, (A.2) where y i for i = 1, 2 denoes he differen releases, ỹ is he rue value of he variable of ineres, ν i and ζ i for i = 1, 2 are he news and he noise componens and η i,ν and η i,ζ for i = 1, 2 are he news and he noise shocks, [η,ỹ η 1,ν η 2,ν η 1,ζ η2,ζ ] N(0, H) wih H = diag(σỹ, 2 σν1, 2 σν2, 2 σζ1 2, σ2 ζ2 ), where diag denoes a diagonal marix. 20

22 The sysem in (A.1) and (A.2) can also be wrien as y1 y 2 = 1 ỹ [ ] ỹ = ρỹ ω,ỹ ω 1,ν ω 2,ν ω 1,ζ ω 2,ζ ω,ỹ ω 1,ν ω 2,ν ω 1,ζ ω 2,ζ, (A.3), (A.4) where ω,ỹ = η,ỹ + η 1,ν + η 2,ν, ω 1,ν = η 1,ν η 2,ν, ω 2,ν = η 2,ν, ω 1,ζ = η1,ζ, ω2,ζ = η2,ζ and [ω,ỹ ω 1,ν ω 2,ν ω 1,ζ ω2,ζ ] N(0, Σ) wih variance-covariance marix Σ defined as Σỹỹ Σỹν1 Σỹν2 0 0 Σ ν1ỹ Σ ν1ν1 Σ ν1ν2 0 0 Σ = Σ ν2ỹ Σ ν2ν1 Σ ν2ν Σ ζ1ζ1 0, (A.5) Σ ζ2ζ2 where Σỹỹ = σ 2 ỹ + σ 2 ν1 + σ 2 ν2, Σỹν1 = σ 2 ν1 σ 2 ν2, Σỹν2 = σ 2 ν2, Σ ν1ν1 = σ 2 ν1 + σ 2 ν2, Σ ν1ν2 = σ 2 ν2, (A.6) Σ ν2ν2 = σ 2 ν2, Σ ζ1ζ1 = σ 2 ζ1, Σ ζ2ζ2 = σ 2 ζ2, 21

23 which implies Σỹν1 = Σ ν1ν1, Σỹν2 = Σ ν2ν2, (A.7) Σ ν2ν2 = Σ ν1ν2. Moreover, consider he following resricion σ 2 ν2 = 0, (A.8) which is jusified due o he fac ha he second release news shock corresponds o informaion ouside he sample and is hus no needed. Aruoba e al. (2016) show ha a sae space sysem described in Equaions (A.3) and (A.4) is no idenified wih Σ unresriced and idenified under cerain resricions on elemens of Σ. We now invesigae wheher he resricions implied by JvN lead o an idenified sysem following he procedure described in Aruoba e al. (2016). Theorem 1. Suppose ha Assumpions 1, 2, 4-NS and 5-NS of Komunjer and Ng (2011) hold. Then according o Proposiion 1-NS of Komunjer and Ng (2011), he sae space model described in (A.1) and (A.2) is idenified given he resricions implied by (A.1), (A.2) and (A.8). Proof of Theorem 1. We begin by rewriing he sae space model in (A.3) and (A.4) o mach he noaion used in Komunjer and Ng (2011) x +1 = A(θ)x + B(θ)ɛ +1 z +1 = C(θ)x + D(θ)ɛ +1, (A.9) (A.10) where x = ỹ, z = [y 1 y 2 ], ɛ = [ω,ỹ ω,ν 1 ω,ν 2 ω,ζ 1 ω2,ζ ], A(θ) = ρ, B(θ) = [ ], C(θ) = [ρ ρ], D(θ) = and θ = [ρ σỹ 2 σν1 2 σν2 2 σζ1 2 σ2 ζ2 ]. 22

24 Given ha Σ is posiive definie and 0 ρ < 1, Assumpion 1 and 2 of Komunjer and Ng (2011) are saisfied. Given ha D(θ)ΣD(θ) is nonsingular also Assumpion 4-NS of Komunjer and Ng (2011) is saisfied. Rewriing he sae space model in (A.9) and (A.10) ino is innovaion represenaion gives ˆx = A(θ)ˆx + K(θ)a +1 z +1 = C(θ)ˆx + a +1, (A.11) (A.12) where K(θ) is he Kalman gain and a +1 is he one-sep ahead forecas error of z +1 wih variance Σ a (θ). The Kalman gain and he variance of he one-sep ahead forecas error for his sysem can be expressed as K(θ) = (pρc + Σ BD )(pcc + Σ DD ) 1 Σ a (θ) = pcc + Σ DD, (A.13) (A.14) where p is he variance of he sae vecor, solving he following Riccai equaion p = pρ 2 + Σ BB (pρc + Σ BD )(pcc + Σ DD ) 1 (pρc + Σ DB ). (A.15) and Σ BB = BΣB, Σ BD = BΣD, Σ DD = DΣD wih Σ BB = Σỹỹ, ] Σ BD = [Σỹỹ + Σỹν1 Σỹỹ + Σỹν2, (A.16) Σ DD = Σ ỹỹ + 2Σỹν1 + Σ ν1ν1 + Σ ζ1ζ1.. Σỹỹ + Σỹν1 + Σ ν2ỹ + Σ ν2ν1 Σỹỹ + 2Σỹν2 + Σ ν2ν2 + Σ ζ2ζ2 23

25 By using he definiions in (A.6), he expressions in (A.16) can also be wrien as [ Σ BB = σỹ 2 + σν1 2 + σν2, 2 Σ BD = Σ DD = σ2 ỹ + σζ1 2.. σỹ 2 σỹ 2 + σν1 2 + σζ2 2 σ 2 ỹ ] σỹ 2 + σν1 2, (A.17) Assumpion 5-NS of Komunjer and Ng (2011) relaes o he conrollabiliy and observabiliy of sae space sysems. The sae space sysem in (A.3) and (A.4) is conrollable if marix [K(θ) A(θ)K(θ)] has full row rank and i is observable if he marix [C(θ) A(θ) C(θ) ] has full column rank and is hus said o be minimal. To show ha Assumpion 5-NS is saisfied, firs noe ha Σ BB Σ BD Σ 1 DD Σ DB is he Schur complemen of Ω, he variance covariance marix of he join disribuion of x +1 and z +1, wih respec o Σ DD where Ω = Σ BB Σ DB Σ BD Σ DD. Because Ω is a posiive definie marix, is Schur complemen is also posiive definie hus leading o Σ BB Σ BD Σ 1 DD Σ DB > 0. Now o show ha his inequaliy leads o p > 0, we use he following lemma Lemma 1. Assume A and (A + B) are inverible and ha rank(b) = 1, hen (A + B) 1 = A r(ba 1 ) A 1 BA 1. We can now use Lemma 1 o rewrie Equaion (A.15) as p = pρ 2 + Σ BB (pρc + Σ BD )Σ 1 DD (pρc + Σ DB) + p g (pρc + Σ BD )Σ 1 DD CC Σ 1 DD (pρc + Σ DB), (A.18) where g = 1 + pr(cc Σ 1 DD ). Afer some manipulaions we find he following quadraic equaion ap 2 + bp + c = 0, 24 (A.19)

26 wih a = r(cc Σ 1 DD ), b = (ρ Σ BD Σ 1 DD Σ DB) 2 + r(cc Σ 1 DD )(Σ BB Σ BD Σ 1 DD Σ DB) 1, c = Σ BB Σ BD Σ 1 DD Σ DB. The necessary and sufficien condiions for p > 0 are b 2 4ac > 0 and b b 2 4ac 2a > 0. The firs condiion leads o b 2 + 4r(C Σ 1 DD C)(Σ BB Σ BD Σ 1 DD Σ DB) > 0 and he second o r(c Σ 1 DD C)(Σ BB Σ BD Σ 1 DD Σ DB) > 0 Since Σ DD is posiive definie (hus r(c Σ 1 DD C) > 0) boh condiions are saisfied if Σ BB Σ BD Σ 1 DD Σ DB > 0. Given also ha A(θ) = ρ 0 and C(θ) 0, we obain K(θ) 0 and hus Assumpion 5-NS is saisfied. Now Proposiion 1-NS of Komunjer and Ng (2011) can be applied, which implies ha wo vecors θ 0 = [ρ σ 2 ỹ,0 σ 2 ν1,0 σ 2 ν2,0 σ 2 ζ1,0 σ 2 ζ2,0] and θ 1 = [ρ σ 2 ỹ,1 σ 2 ν1,1 σ 2 ν2,1 σ 2 ζ1,1 σ 2 ζ2,1] are observaionally equivalen iff here exiss a scalar τ 0 such ha A(θ 1 ) = τa(θ 0 )τ 1 K(θ 1 ) = τk(θ 0 ) C(θ 1 ) = C(θ 0 )τ 1 Σ a (θ 1 ) = Σ a (θ 0 ). (A.20) (A.21) (A.22) (A.23) Given ha A(θ) = ρ, i follows from Equaion (A.20) ha ρ 0 = ρ 1 and hus we can deduce from (A.22) ha γ = 1. Hence, by using Equaions (A.13) and (A.14), he condiions 25

27 (A.21) and (A.23) can be expressed as K 1 = K 0 = (p 0 ρc + Σ BD0 )(pcc + Σ DD0 ) 1 Σ a1 = Σ a0 = p 0 CC + Σ DD0, (A.24) (A.25) where p 0 solves he following Riccai equaion p 0 = p 0 ρ 2 + Σ BB0 K 0 (p 0 ρc + Σ DB0 ). (A.26) Equaions (A.24) o (A.26) are saisfied if and only if p 1 (1 ρ 2 ) Σ BB1 = p 0 (1 ρ 2 ) Σ BB0 p 1 ρc + Σ BD1 = p 0 ρc + Σ BD0 p 1 CC + Σ DD1 = p 0 CC + Σ DD0. (A.27) (A.28) (A.29) Wihou loss of generaliy le Σỹỹ,1 = Σỹỹ,0 + δ(1 ρ 2 ) (A.30) leading o Σ BB,1 = Σ BB,0 + δ(1 ρ 2 ). (A.31) We now proceed by spliing he analysis ino wo cases. Case 1: δ = 0. From (A.27) we obain p 1 = p 0. (A.28) hence implies σ 2 y,1 = σ2 y,0 and σ 2 ν1,1 = σ 2 ν1,0 and given ha Σỹỹ,1 = Σỹỹ,0 i follows σ 2 ν2,1 = σ 2 ν2,0. (A.29) implies ha Σ DD1 = Σ DD0 and hus σ 2 ζ1,1 = σ2 ζ1,0 and σ2 ζ2,1 = σ2 ζ2,0, leading o he fac ha θ 1 = θ 0. Case 2: δ 0. From (A.27) we obain p 1 = p 0 + δ. From (A.28) i follows σ 2 ỹ,1 = σ 2 ỹ,0 δρ 2 and σ 2 ν1,1 = σ 2 ν1,0. (A.32) Moreover, (A.27) gives σ 2 ν2,1 = σ 2 ν2,0 + δ. (A.33) 26

28 Finally, he equaions in (A.29) lead o σ 2 ζ1,1 = σ 2 ζ1,0 and σ 2 ζ2,1 = σ 2 ζ2,0. (A.34) Noe ha (A.6) and (A.32) o (A.34) resul ino Σỹỹ,0 + δ(1 ρ 2 ) Σỹν1,0 δ Σỹν2,0 δ 0 0 Σ ν1ỹ,0 δ Σ ν1ν1,0 + δ Σ ν1ν2,0 + δ 0 0 Σ 1 = Σ ν2ỹ,0 δ Σ ν2ν1,0 + δ Σ ν2ν2,0 + δ Σ ζ1ζ1 0. (A.35) Σ ζ2ζ2 Finally, from (A.33) and (A.34) i follows ha δ = 0. Idenificaion in generalized reconciliaion models We now reurn o he use of momen condiions o discuss possible approaches o idenificaion in daa reconciliaion models wih muliple daa releases. We only consider he reconciliaion of exacly wo daa series, bu oherwise consider linear dynamic daa generaing processes more general han any ha we have seen used in he lieraure. Specifically, we consider models of he form Y 0 = Z α Y 1 (A.36) and where α = T α 1 + R η (A.37) Y i is a 1 l vecor conaining l releases of series i = {0, 1} esimaes of he unobserved rue value ỹ. α is a (p + 4 l) 1 laen sae vecor ha we may pariion as α = ] p [Ỹ 1 ν 0 ν 1 ζ 0 ζ 1 27

29 Ỹ p 1 is a p 1 vecor [ỹ,..., ỹ p+1 ] ν i for i = {0, 1} is a l 1 vecor of news shocks conained in Y i ζ i for i = {0, 1} is a l 1 vecor of noise shocks conained in Y i η N (0, Σ) is a (1 + 4 l) 1 vecor of i.i.d. mean zero normally disribued shocks.wih diagonal covariance marix Σ. Z is a (2 l) (p + 4 l) marix of he form ] Z [1 (2 l) 1 0 (2 l) (p 1) I 2 l I 2 l 1 a b is a marix of dimension a b composed enirely of 1 s 0 a b is a marix of dimension a b composed enirely of 0 s I a is a a a ideniy marix T is a (p + 4 l) (p + 4 l) block diagonal marix of he form T T p 0 p (4 l) 0 (4 l) p T S (A.38) T p is a p p marix T p = ρ 1, ρ 2,... I p 1 ρ p 0 (p 1) 1 (A.39) T S is a (4 l) (4 l) arbirary diagonal marix 28

30 R is a (p + 4 l) (1 + 4 l) marix of he form l 1 1 l 0 1 l 0 1 l 0 (p 1) 1 0 (p 1) l 0 (p 1) l 0 (p 1) l 0 (p 1) l 0 l 1 U l Ψ 0 l l 0 l l R = 0 l 1 0 l l U l 0 l l 0 l l 0 l 1 0 l l 0 l l I l l Φ (A.40) 0 l 1 0 l l 0 l l 0 l l I l l U l is a l l marix wih 0 s below he main diagonal and 1 s everywhere else Ψ, Φ are unresriced l l marices. The model esimaed in he paper is he special case of he above where 1. p = 1 2. T S = 0 (4 l) (4 l) 3. Φ = Ψ = 0 l l Relaxing he firs condiion allows us o consider model where he dynamics of he unobserved rue values follow an AR(p) process raher han simply an AR(1). Allowing for higher-order auocorrelaions adds an addiional p 1 free parameers o he model, bu also adds an addiional 2 (p 1) sample auocorrelaions ha may be used for idenificaion. Relaxing he second condiion allows wha JvN refer o as spillover effecs. This permis revisions o he values for calendar period o be correlaed wih revisions o calendar period 1. This may occur, for example, when revisions end o shif measured growh from one quarer o an adjascen quarer, or when he incorporaion of lower frequency daa sources (e.g. annual ax reurns) shif muliple periods in he same direcion. This adds an addiional 4 l free parameers o he model. However, i also brings ino play an addiional 2 (l 1) momens capuring he 1s-order auocorrelaions of he revisions of our wo series. 29

31 Relaxing he hird condiion allows for he possibiliy ha measuremen errors of eiher ype may be correlaed across he wo series. Conemporaneous correlaions (i.e. measuremen errors ha affec he same release of each series) are capured by he diagonals of hese wo marices. Evidence ha informaion ends o be incorporaed ino releases of y 0 before (afer) hose of y 1 implies ha here should be non-zero enries of Ψ above (below) he main diagonal. Conemporaneous correlaions would add an addiional 2 l free parameers o he model, while unresriced correlaions would add an addiional 2 l 2 free parameers. However, we have already assumed he use of all l (2 l + 1) conemporaneous cross-momens of he various vinages of boh series, so here is no offseing gain in he number of momens available for idenificaion. 30

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