The innovation long-run risk component

F anceschini, Fabio
Wo king Pape
The inno a ion long- un isk componen
Quade ni - Wo king Pape DSE, No. 1215
P o ided in Coope a ion wi h:
Uni e si y o Bologna, Depa men o Economics
Sugges ed Ci a ion: F anceschini, Fabio (2025) : The inno a ion long- un isk componen , Quade ni
- Wo king Pape DSE, No. 1215, Alma Ma e S udio um - Uni e si à di Bologna, Dipa imen o di
Scienze Economiche (DSE), Bologna,
h ps://doi.o g/10.6092/unibo/amsac a/8613
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ISSN 2282-6483
The Inno a ion
Long-Run Risk Componen
Fabio F anceschini
Quade ni - Wo king Pape DSE N° 1215
The Inno a ion Long-Run Risk Componen
Fabio F anceschini ∗
5 h No embe 2025
[click he e o he la es e sion]
Abs ac
This pape p o ides obus empi ical e idence ha shocks o agg ega e Resea ch and
De elopmen (R&D) ha e pe sis en e ec s on mac oeconomic dynamics and ep esen a
signi ican isk o in es o s, as p edic ed by he ‘long- un isk’ li e a u e. The analysis
ocuses on a single a iable, ‘e ec i e R&D’, which cap u es he en i e con ibu ion o
R&D o p oduc i i y g ow h, lexibly accoun ing o knowledge spillo e s and p oduc
p oli e a ion e ec s. De ia ions o e ec i e R&D om i s equilib ium le el can be
empi ically iden i ied le e aging he e o co ec ion e m in he coin eg a ion ela ionship
among R&D, o al ac o p oduc i i y, and he labo o ce. In US da a, s uc u al e ec i e
R&D shocks a ec p oduc i i y and consump ion g ow h a es beyond business cycle
ho izons and a e associa ed wi h a signi ican isk p emium in a c oss sec ion o s ock
and bond po olios (a ound 2% annually), wi h cash- low sensi i i ies p o ing a key
de e minan .
Keywo ds: R&D, Long- un isk, Asse P icing, Coin eg a ion
JEL Codes: E32, E44, G12, O30
∗Depa men o Economics, Uni e si y o Bologna (I aly); ancesc[email p o ec ed].
This pape was a chap e o my Ph.D. hesis and was unded by he Eu opean Union - Nex Gene a ionEU, in
he amewo k o he “GRINS -G owing Resilien , INclusi e and Sus ainable p ojec ” (PNRR - M4C2 - I1.3 -
PE00000018 - CUP J33C22002910001). Special hanks o my ad iso s Max C oce and Ma ín Gonzalez-Ei as
o many help ul discussions. I also hank Emanuele Bacchiocchi, F ede ico Belo, Je emy Boccan uso, Lau a
Bo azzi, Ma co B ian i, S e lana B yzgalo a, Giuseppe Ca alie e, Enzo Dia, Luca Fanelli, Luca Gemmi,
Howa d Kung, Filippo Massa i, An onio Minni i, Elisa Ossola, Oli ie o Pallanch, And ea Renze i and Alwyn
Young as well as semina pa icipan s a he Uni e si y o Bologna, he Uni e si y o Milano-Bicocca, he
INSPIRE semina se ies, and he 27
h
INFER Annual Con e ence o commen s and discussions om which I
g ea ly bene i ed. The iews and opinions exp essed a e solely hose o he au ho and do no necessa ily
e lec hose o he Eu opean Union, no can he Eu opean Union be held esponsible o hem.
1
Non- echnical summa y
This pape demons a es ha luc ua ions in agg ega e Resea ch and De elopmen (R&D)
ha e pe sis en e ec s on he b oade economy wi h signi ican implica ions o asse p ices.
Speci ically, as R&D shocks inc ease expec ed g ow h in p oduc i i y and consump ion a
ho izons beyond he business cycle, in es o s demand highe compensa ion om asse s whose
alues a e mo e sensi i e o R&D dynamics, as hese asse s ampli y unce ain y abou u u e
consump ion s eams.
The analysis cen e s on a ans o ma ion o R&D called ‘e ec i e R&D’, which exp esses
R&D in uni s o p oduc i i y gains while accoun ing o spillo e s om pas inno a ions and
p oduc p oli e a ion. Cons uc ed using qua e ly US da a om 1947 o 2024, shocks o
his measu e – isola ed om p oduc i i y g ow h independen shocks – a e shown o a ec
p oduc i i y and consump ion g ow h o up o 20 yea s.
These shocks a e hen es ed as a isk ac o . Fi s , using 183 s ock and bond po olios
wi h a me hodology ha ex ac s all la en isk ac o s o add ess omi ed a iable bias,
he pape p o ides obus e idence ha e ec i e R&D shocks ep esen a p iced isk ac o .
Second, examining how asse s’ cash lows espond o hese shocks e eals ha his sensi i i y
is a c ucial channel h ough which he isk p emium eme ges. This analysis also yields
desc ip i e e idence suppo ing se e al insigh s: e u ns o i m-le el R&D a e highe when
economy-wide R&D in es men is ele a ed, and i m cha ac e is ics ela ed o unding
capaci y and g ow h oppo uni ies sys ema ically ela e o sensi i i y o agg ega e R&D.
O e all, his pape makes ou main con ibu ions. I es ablishes he empi ical signi icance
o an R&D-based ac o iden i ied by mac oeconomic heo y (a wo old con ibu ion), lexibly
connec s asse p icing o endogenous g ow h models h ough he ‘e ec i e R&D’ measu e
and p o ides an econome ic amewo k o eco e i , o e s desc ip i e e idence on how i m
cha ac e is ics a ec sensi i i y o agg ega e R&D, and suppo s he pe sis ence ampli ica ion
mechanism o R&D widely adop ed in ecen mac oeconomic li e a u e.
2
1 In oduc ion
In es men s in Resea ch and De elopmen (R&D) ha e a p o ound and well-documen ed
impac on he economy. A he agg ega e le el, R&D has been linked o slow-mo ing
luc ua ions in mac oeconomic quan i ies such as p oduc i i y and consump ion (Comin and
Ge le 2006; E ans e al. 1998), and heo y has shown ha his nexus ca ies subs an ial
implica ions o asse p ices (Kung and Schmid 2015). The unde lying mechanism is ha o
he ‘Long-Run Risk’ (LRR) amewo k om Bansal and Ya on (2004), who i s demons a ed
how pe sis ence in consump ion g ow h enables mac oeconomic models o accoun o he la ge
equi y isk p emium obse ed in inancial ma ke s, a p emium ha a exceeds p edic ions
based on obse ed consump ion g ow h ola ili y alone (Meh a and P esco 1985). This
amewo k has since become a cen al e e ence in mac o- inance esea ch,
1
bu has also aced
sus ained c i icism,2making igo ous empi ical alida ion o models buil on i essen ial.
This pape p o ides obus empi ical e idence ha shocks o US agg ega e R&D ha e
pe sis en e ec s on p oduc i i y and consump ion g ow h, and ca y subs an ial implica ions
o asse p ices. Speci ically, I ocus on a heo e ical measu e o agg ega e R&D – ‘e ec i e
R&D’ – ha uniquely summa izes he o al impac o R&D on p oduc i i y g ow h, accoun ing
o i ually any deg ee o spillo e and p oduc p oli e a ion e ec s. I hen p opose an
empi ical amewo k o eco e i s ime se ies and he associa ed s uc u al shocks, illus a ing
hei pe sis en e ec s on he economy. Finally, I show ha s uc u al shocks o e ec i e
R&D se e as a isk ac o wi h signi ican powe in p icing he c oss-sec ion o inancial
asse s. This cons i u es a wo old no el y, as i in oduces he i s empi ically signi ican
isk ac o ha is (1) based on agg ega e R&D, and (2) cons uc ed om s uc u al shocks
in a heo e ically iden i ied sys em.
In he long- un isk amewo k, agen s ha e ecu si e p e e ences as in Eps ein and
Zin (1989), which a e cha ac e ized by an a e sion o swings in o ecas s o dis an u u e
consump ion g ow h. Pe sis en shocks can induce such luc ua ions wi h e y small a iance,
making hem a powe ul heo e ical ea u e by allowing models o accoun o sizable isk
p emia wi h li le added complexi y, bu also an elusi e one o alida e empi ically, since
low- equency p ocesses a e challenging o iden i y in ini e samples. A g owing li e a u e
has ackled he empi ical challenge di ec ly, ei he by de ising e ined empi ical s a egies
(B yzgalo a e al. 2025; Dew-Becke and Giglio 2016; Gou ie oux and Jasiak 2024; O u
e al. 2013; Scho heide e al. 2018) o by cons uc ing new da a (Liu and Ma hies 2022).
Ano he s and has de i ed he LRR pi o al condi ions as educed o ms o iche s uc u al
models, yielding addi ional es able implica ions (e.g. Kal enb unne and Lochs oe 2010).
Wi hin his amewo k, C oce (2014) showed ha he p edic able componen o p oduc i i y
g ow h is pe sis en and does ansmi o consump ion g ow h. He e med his componen
he ‘p oduc i i y long- un isk componen ’, which Kung and Schmid (2015) la e a ionalized
1Applica ions include, among o he s, exchange a e dynamics (Colaci o and C oce 2011), clima e change
asse p icing (Bansal e al. 2021), e m s uc u e models (Ai e al. 2018), and oil p ice dynamics (Ready
2018).
2
Mos no ably Cons an inides and Ghosh (2011), Beele and Campbell (2012), and Eps ein e al. (2014).
3

Figu e 1: Consump ion g ow h is US eal o al consump ion pe capi a om BEA; ‘TFP’ is he
u iliza ion-adjus ed, excluding R&D capi al, TFP om Fe nald (2012). O bo h is plo ed he 6
h
componen o O u e al. (2013) decomposi ion, which il e he luc ua ions wi h hal -li e be ween 8
and 16 yea s. Excess e ec i e R&D is in le els, no g ow h a es.
as o igina ing om pe sis en luc ua ions in R&D in es men . This pape p o ides di ec
empi ical e idence on he mac oeconomic mechanism and inancial p edic ions de eloped in
his heo e ical amewo k, while con ibu ing a no el econome ic p ocedu e, g ounded in
mac oeconomic heo y, o iden i y a p ocess ha cap u es e isions o long- un expec a ions
o inno a ion – an ‘inno a ion long- un isk componen ’. The esul s in his s udy depa
om he p edic ions in Kung and Schmid (2015) only ega ding he speci ic mechanics behind
R&D shock p opaga ion, which empi ically does no appea o a ise om he pe sis ence o
e ec i e R&D i sel , bu a he om i s in e ac ion wi h ex e nal mac oeconomic ac o s.
Figu e 1displays he es ima ed e ec i e R&D, along wi h he p ocesses i is expec ed o d i e,
i.e., he componen s wi h hal -li es be ween 8 and 16 yea s o consump ion and p oduc i i y
g ow h a es, whose como emen is appa en .
E ec i e R&D exp esses agg ega e R&D in uni s o expec ed p oduc i i y g ow h, as
p esc ibed by endogenous g ow h heo y. I is a log-linea combina ion o agg ega e R&D,
he s ock o ideas and a p oduc a ie y measu e, cap u ing he nonlinea impac o R&D
on p oduc i i y g ow h. I s de ini ion is o malized wi hin a heo e ical amewo k ha
illus a es he minimal condi ions linking agg ega e R&D o asse p ices, while in en ionally
abs ac ing om he op imal choices d i ing R&D dynamics, lea ing he la e o empi ical
analysis. This heo e ical amewo k elies p ima ily on a p oduc ion unc ion o ideas ha
nes s Kung and Schmid (2015) as a special case, while mo e lexibly accommoda ing bo h
ully and semi-endogenous g ow h mechanisms. This g ea e lexibili y, oge he wi h a
close alignmen o he econome ic app oach and he unde lying heo y, allows he esul ing
4
e ec i e R&D measu e o add ess some undesi able s a is ical p ope ies o he empi ical
measu e in Kung and Schmid (2015).
3
In pa icula , I show ha hei e ec i e R&D measu e
exhibi s pe sis ence ha , while consis en wi h adi ional calib a ions o consump ion LRR,
is subs an ially highe han ha o he p oduc i i y componen i is in ended o d i e.
Mo eo e , i s p oximi y o a uni oo aises conce ns o spu ious in e ence in s anda d
econome ic applica ions. By con as , he e ec i e R&D measu e de eloped he e gene a es
highly pe sis en e ec s while emaining highly s a iona y, enabling mo e eliable applica ion
o es ablished asse p icing c oss-sec ional es s. Ano he ela ed measu e is ha o Kogan
e al. (2017), which o e s mo e g anula in o ma ion and complemen s e ec i e R&D by
ocusing on he ou come o he inno a ion p ocess – success ul inno a ions – a he han i s
inpu .
The empi ical iden i ica ion o e ec i e R&D elies on he widely suppo ed assump ion
o p oduc i i y g ow h s a iona i y, which implies ha he linea combina ion o ending
a iables cons i u ing e ec i e R&D o ms a coin eg a ion ela ionship ha can be es ima ed.
Fo echnical easons ela ed o he iming o he a iables, his ela ionship is es ima ed using
a single-equa ion app oach (Phillips and Hansen 1990). Fo obus ness, he coin eg a ion
model is es ima ed using p oduc i i y le els ins ead o he s ock o ideas, he eby in oducing
p oduc i i y d i e s ex e nal o R&D in o he e o -co ec ion e m, which o ms a ‘g oss’
e ec i e R&D measu e. Iden i ica ion o ne e ec i e R&D dynamics is pu sued in se e al
ways: (1) linea i y o bo h he mapping om g oss o ne e ec i e R&D and he VAR model
ep esen ing sys em dynamics ensu es s uc u al shocks a e no dis o ed; (2) all es ima es a e
pe o med using bo h he g oss measu e and a ne measu e, eco e ed elying on a ecu si e
o mula based on a one-s ep p oduc i i y o ecas eg ession wi h obus con ols ha ha e
been shown in p e ious wo k o cap u e p oduc i i y g ow h dynamics; (3) he dynamic
implica ions om he VAR a e es ed and ex ended o consump ion g ow h by employing
he e ec i e R&D s uc u al shocks om he VAR in a local p ojec ion exe cise (Jo dà
2005; Mon iel Olea and Plagbo g-Mølle 2021). A c ucial esul is ha , al hough e ec i e
R&D is highly s a iona y, i s shocks consis en ly a ec p oduc i i y and consump ion g ow h
well beyond he business cycle, likely o e ho izons o en yea s. These a e signi ican ly
longe ho izons compa ed wi h he e idence in Kung and Schmid (2015), while con olling
o ex e nal ac o s, using me hods obus o small-sample bias, and allowing o iche
mul i a ia e dynamics. The pe sis ence o e ec s om R&D luc ua ions has also been
s udied by Anzoa egui e al. (2019), who emphasized he ole o g adual echnology adop ion
in gene a ing pe sis ence, and many o he s udies which ha e le e aged i as a channel o
p opaga ing business cycle shocks.
4
Rela i e o hese s udies, his pape ex ends he analysis
by ocusing on he inancial implica ions o R&D pe sis ency and by pi o ing on he unique
R&D measu e ha e lec s he ime- a ying e iciency o R&D.
The c oss-sec ional asse p icing es s ely on he e ec i e R&D s uc u al shocks om
3They e e o ‘e ec i e R&D’ as ‘R&D in ensi y’.
4
Fo example, Benigno and Fo na o (2018) illus a e i s ole in p opaga ing nega i e shocks, An olin-Diaz
and Su ico (2025) o mili a y spending shocks, Beqi aj e al. (2025) o mone a y shocks, Aksoy e al. (2019)
o demog aphic changes.
5
he mac oeconomic analysis, which a e o hogonalized o p oduc i i y g ow h shocks. This
ensu es ha R&D shocks a e isola ed om he main con ounding ac o in hese educed- o m
es s. Combined wi h he p ocedu e om Giglio and Xiu (2021), which exploi s a wide
c oss-sec ion o asse s o mi iga e conce ns abou omi ed isk ac o bias, his yields highly
eliable es ima es o he isk p emia associa ed wi h e ec i e R&D shocks. The p emia a e
insigni ican o con empo aneous shocks bu highly signi ican and consis en o olling
sums o e mul i-yea ho izons, yielding oughly 2% annually – consis en wi h in es o s’
unde eac ion o R&D news documen ed in Ebe ha e al. (2004). Fu he mo e, a key
ea u e o long- un isk models is ha isk is ansmi ed h ough asse cash lows. This
is es ed ollowing Bansal e al. (2005) using s ock po olios so ed by i m cha ac e is ics
p e iously linked o R&D in es men , as well as by indus y po olios. These esul s also
p o ide desc ip i e e idence b idging he co po a e inance li e a u e on R&D in es men
and he asse p icing li e a u e on i m-speci ic R&D isk p emia. Beyond Kung and Schmid
(2015), only Hsu (2009) link agg ega e R&D and inancial ma ke s. This pape ex ends bo h
s udies by p o iding no el explici c oss-sec ional isk p emium es ima es and by employing
an R&D measu e wi h a igh e mapping o heo y.
F om a echnical pe spec i e, he heo e ical de ini ion o e ec i e R&D is g ounded only in
a de ini ion o To al Fac o P oduc i i y (TFP) con ibu o s and a ‘lab-equipmen ’ p oduc ion
unc ion o ideas inspi ed by Jones (1999), making he esul s b oadly applicable.
5
While he
amewo k allows R&D e iciency in gene a ing new ideas o be dilu ed by bo h dec easing
e u ns o pas ideas and an expanding a ie y o p oduc s, i does no explici ly assess
hei ela i e con ibu ion o i ing he da a, wi h p oduc a ie y signi icance depending
on which measu e o R&D expendi u e is employed. On he empi ical side, coin eg a ion
models ha e been widely applied o s udy he ela ionship be ween R&D and echnological
p og ess in mac oeconomic s udies (Bo azzi and Pe i 2007; Ha and Howi 2007; He ze
2022b; K use-Ande sen 2023; Madsen 2008). Howe e , hese wo ks mainly ocus on o eign
spillo e s and he dis inc ion be ween ully- and semi-endogenous economies, whe eas his
pape emphasizes he dynamic p ope ies o R&D and i s inancial implica ions. Mo eo e ,
none o hese s udies employ a single-equa ion app oach, which, on he o he hand, is
equen ly used in he empi ical mac o- inance li e a u e (e.g., Le au and Lud igson 2001;
Melone 2021) and is adop ed he e.
The pape p oceeds as ollows. Sec ion 2ou lines he heo e ical amewo k, de ining
e ec i e R&D and i s key mac oeconomic and inancial p edic ions. Sec ion 3desc ibes
he econome ic amewo k p oposed o iden i y e ec i e R&D and es he associa ed
p edic ions. Sec ion 4p esen s he coin eg a ion esul s and o ecas ing eg essions ha
yield he e ec i e R&D measu es. Sec ion 5documen s he impac o e ec i e R&D on
mac oeconomic quan i ies, and Sec ion 6 epo s he c oss-sec ional asse p icing esul s.
Sec ion 7concludes.
5
The ‘lab-equipmen ’ class o models, in oduced in Rome (1987), uses inal ou pu goods o p oduce
ideas, in con as o labo -based models à la Rome (1990).
6
2 Theo e ical amewo k
2.1 Pe sis en mac oeconomic shocks and inancial ma ke s
In a disc e e- ime, a bi age- ee economy, he expec ed excess e u n o any asse
𝑖
o e he
isk- ee a e
𝑅𝑓
𝑡
is p opo ional o he co a iance be ween i s e u n
𝑅𝑖𝑡
and he S ochas ic
Discoun Fac o (SDF) 𝑀𝑡:6
E𝑡[𝑅𝑖𝑡+1]−𝑅𝑓
𝑡=−𝑅𝑓
𝑡⋅Co 𝑡[𝑀𝑡+1,𝑅𝑖𝑡+1]. (1)
To discipline and be e unde s and he dynamics o he SDF, asse p icing heo y o en
ela es he SDF o he in e empo al ma ginal a e o subs i u ion (IMRS) o a ep esen a i e
agen wi h p e e ences o e consump ion s eams. Shocks o s a e a iables ha ma e o
in es o consump ion become he key d i e s o asse alua ions.
As illus a ed by Bansal and Ya on (2004), pe sis en mac oeconomic quan i ies s ongly
a ec he IMRS and, consequen ly, asse alua ions, when he ep esen a i e agen has ecu -
si e p e e ences as in Eps ein and Zin (1989). These p e e ences sepa a e he in e empo al
elas ici y o subs i u ion (IES) om isk a e sion, and imply ha he agen is sensi i e bo h
o con empo aneous consump ion shocks
𝜀𝑐,𝑡+1 =ln 𝐶𝑡+1−E𝑡[ln𝐶𝑡+1], (2)
and o shocks o long- un consump ion p ospec s
𝜀𝑥,𝑡+1 ={E𝑡+1−E𝑡}(∞
∑
𝑗=1(𝜅𝑥)𝑗⋅Δln 𝐶𝑡+1+𝑗), (3)
as e lec ed in he log SDF,
ln𝑀𝑡+1 =E𝑡[ln 𝑀𝑡+1]−𝑏𝑐𝜀𝑐,𝑡+1−𝑏𝑥𝜀𝑥,𝑡+1,(4)
whe e
𝜅𝑥∈(0,1)
is a unc ion o he equilib ium consump ion–weal h a io, and
𝑏𝑐>0
and
𝑏𝑥>0
a e he loadings on con empo aneous and long- un consump ion shocks, espec i ely.
The pe sis ence o mac oeconomic shocks plays a cen al ole in his amewo k because he
mo e pe sis en ly a shock a ec s consump ion g ow h, he la ge he luc ua ions i gene a es
in 𝜀𝑥,𝑡+1, and hence he s onge i s impac on he IMRS.7
Assuming ha
𝜀𝑐,𝑡+1
and
𝜀𝑥,𝑡+1
a e also he key s ochas ic de e minan s o asse e u n
6A s anda d e e ence is Coch ane (2005).
7
Fo ins ance, i consump ion g ow h simply ollowed
Δln 𝐶𝑡+1 =𝜌𝑐Δln 𝐶𝑡+
𝜀𝑡+1
wi h
|𝜌𝑐|<1
and

𝜀𝑡+1 i.i.d., hen he long- un shock would amoun o 𝜀𝑥,𝑡+1 =1
1−𝜅𝑥𝜌𝑐
𝜀𝑡+1.
7
Ne e heless, he in luence o he ini ial condi ion decays exponen ially h ough he weigh s
𝜅𝑡𝑠
,
which decline apidly o e ime unde he baseline speci ica ion. Fu he mo e, as illus a ed
below, he main analysis can be conduc ed equi alen ly using ei he
𝑠𝑡
o
𝑠𝑡
, and bo h
measu es a e employed o ensu e obus ness o he indings. Sec ion A.3 p o ides a de ailed
discussion o he eco e y’s accu acy and p ecision, de i ing analy ical exp essions o he
s anda d e o s o he eco e ed se ies ha accoun o bo h he unobse abili y o he ini ial
condi ion and sampling a iabili y.
To ob ain a eliable es ima e o
𝛾1
, wo assump ions a e made. Fi s , i is assumed ha a
se o pe asi e mac oeconomic ac o s
𝐟𝑡
spans he non-inno a ion componen o TFP g ow h,
so ha
𝑎𝑡=𝐜′𝐟𝑡
o some ec o
𝐜
. This assump ion is suppo ed by he ex ensi e li e a u e
on he p edic able pa o TFP g ow h (Ai e al. 2018; C oce 2014), o which
𝑎𝑡
ep esen s
he non-idea componen ; acco dingly, i should al eady be cap u ed by he in o ma ion se
o he p edic o s p e iously used. Second, he e ec s o he i s lag o e ec i e R&D on
he ex e nal componen
𝑎𝑡
a e negligible in magni ude. This assump ion is concep ually
mo i a ed by he ex e nal componen ’s likely dependence on nume ous agg ega e economic
ac o s, and is s ongly co obo a ed by he empi ical indings p esen ed in subsequen
analysis. The e o e, while eedback mechanisms may e en ually ampli y he con ibu ion o
e ec i e R&D o he luc ua ions o
𝑎𝑡
, he immedia e impac is expec ed o be negligible.
Unde hese assump ions, 𝛾1is iden i ied by 𝑏𝑠in he es ima ion o (8) as
Δln𝑍𝑡+1 =𝑏0+𝑏𝑠𝑠𝑡+𝐛′𝑓𝐟𝑡+𝑢𝑡+1.(20)
Since his es ima ion ocuses solely on he pa ame e
𝑏𝑠
a he han he ull dynamics o he
R&D impac , his o ecas ing eg ession is also es ima ed as a single equa ion a he han
wi hin a sys em. Rela i e o a mul i a ia e app oach, his me hod ades a modes inc ease
in es ima ion a iance o a subs an ial educ ion in bias, inco po a ing nume ous con ol
a iables in
𝐟𝑡
, including lagged alues o he dependen a iables, wi h lag leng hs selec ed
using s anda d in o ma ion c i e ia.
An al e na i e app oach o eco e
𝑠
is o es ima e he coin eg a ion be ween
ln𝑆𝑡
,
ln𝑍𝑡
,
ln𝐿𝑡
, and he ac o s
𝐟𝑡
di ec ly. Howe e , his app oach equi es es ima ing a long- un
co a iance ma ix wi h 144 elemen s using ewe han 280 obse a ions, which is challenging
and may lead o imp ecise and uns able coin eg a ing pa ame e es ima es.
3.2 Inno a ion shocks and long- un dynamics
Following he es ima ion o e ec i e R&D, he dynamics o
𝑎𝑡
,
Δln𝑍𝑡
, and
𝑠𝑡
(o
𝑠𝑡
) a e
s udied join ly, as a sys em, o achie e wo objec i es. Fi s , his app oach enables an in e-
g a ed assessmen o he long- un impac o e ec i e R&D, accoun ing o dynamic eedbacks
and con empo aneous co ela ions ha uni a ia e models would o e look. Second, i allows
iden i ica ion o s uc u al shocks o he inno a ion componen . The iden i ica ion s a egy,
de ailed below, isola es shocks o inno a ion e o s ha a e o hogonal o luc ua ions in
o he mac oeconomic a iables o he sys em, mos impo an ly p oduc i i y g ow h. This
14

sepa a ion is c ucial o minimize con amina ion om o he sou ces o isk in es ima es o he
isk p emium om educed- o m asse p icing es s using e ec i e R&D as a isk ac o .
The s ochas ic p ocesses om Sec ion 2a e hen ex ended o inco po a e a bi a y
pe sis ence in
𝑎𝑡
(con olled by
𝜌𝑎
) and eedback e ec s be ween
𝑎𝑡
and
𝑠𝑡
(go e ned by
𝜃𝑠
and
𝜃𝑎
). Impo an ly, he shocks o e ec i e R&D ha a e no de e mined by ex e nal
ac o s,
𝜀𝑠,𝑡
, a e assumed o ha e no e ec on he con empo aneous le el o he non-idea
componen . This is a common assump ion in he li e a u e (see, o ins ance, Mo an and
Que al o (2018)), since he inno a ion p ocess a ely has ou comes wi hin such a sho ime
ame o be conside ed con empo aneous.14
The esul ing s uc u al sys em is
𝑎𝑡+1 =𝜃𝑠𝑠𝑡+𝜌𝑎𝑎𝑡+𝑏𝑎𝑎𝜀𝑎,𝑡+1 (21a)
Δln𝑍𝑡+1 =(𝛾1+𝜃𝑠) 𝑠𝑡+(𝜌𝑎−1)𝑎𝑡+𝑏𝑎𝑎𝜀𝑎,𝑡+1 (21b)
𝑠𝑡+1 =𝜌𝑠𝑠𝑡+𝜃𝑎𝑎𝑡+𝑏𝑎𝑠𝜀𝑎,𝑡+1+𝑏𝑠𝑠𝜀𝑠,𝑡+1,(21c)
whe e
𝜀𝑎,𝑡+1
and
𝜀𝑠,𝑡+1
a e i.i.d. shocks om a s anda dized no mal dis ibu ion, and
𝑏𝑎𝑎
,
𝑏𝑎𝑠
, and
𝑏𝑠𝑠
a e ee pa ame e s con olling he ola ili y o he shocks and hei c oss-
e ec s. A simila sys em can be w i en o
𝑠𝑡
by subs i u ing
𝑠𝑡= 𝑠𝑡+𝛼𝑍𝑎𝑡
in o he abo e
equa ions. No ably, each a iable con inues o be d i en by he same unde lying s uc u al
shocks whe he
𝑠
o
𝑠
is used; only he educed- o m coe icien s di e , allowing he same
iden i ica ion scheme o eco e he same s uc u al shocks. Mo e de ails o he sys em a e
p o ided in Appendix A.2.
In his wo k, he es ima ion o
(21)
is app oached by ocusing on pa simonious 2-
a iable Vec o Au o eg ession (VAR) models ha app oxima e he VAR–mo ing-a e age
ep esen a ion ob ained om subs i u ing he ex e nal ac o ou o he 3- a iable sys ems:
Δln𝑍𝑡+1 =(𝛾1+𝜃𝑠) 𝑠𝑡+𝜌𝑎Δln𝑍𝑡−(𝜃𝑠+𝜌𝑎𝛾1) 𝑠𝑡−1+𝑏𝑎𝑎𝜀𝑎,𝑡+1−𝑏𝑎𝑎𝜀𝑎,𝑡 (22a)
𝑠𝑡+1 =𝜌𝑠𝑠𝑡−𝜃𝑎𝜌𝑎
1−𝜌𝑎Δln𝑍𝑡+(𝜃𝑎𝜃𝑠+𝜃𝑎𝜌𝑎(𝛾1+𝜃𝑠)
1−𝜌𝑎) 𝑠𝑡−1+
𝑏𝑎𝑠𝜀𝑎,𝑡+1+𝑏𝑠𝑠𝜀𝑠,𝑡+1+𝜃𝑎𝑏𝑎𝑎
1−𝜌𝑎𝜀𝑎,𝑡,(22b)
whe e he numbe o lags is chosen based on s anda d in o ma ion c i e ia. A Cholesky
decomposi ion o he educed- o m esiduals’ co a iance ma ix, wi h
𝑠𝑡
o de ed las , iden i ies
he s uc u al shocks o e ec i e R&D,
𝜀𝑠,𝑡
, om he es ima es o
(22)
. As an icipa ed, he
same holds i he sys em is exp essed in e ms o
𝑠𝑡
ins ead o
𝑠𝑡
, since only he coe icien s
change. C ucially, his app oach lea es he ex e nal componen unobse ed, which may
in oduce se ial co ela ion in he educed- o m esiduals i i s dynamics a e poo ly e lec ed
in he wo- a iable sys em, po en ially in alida ing s uc u al iden i ica ion. Howe e , he
adequacy o his speci ica ion can be assessed ex pos h ough s anda d diagnos ic es s.
14
The implausibili y o con empo aneous ou comes is u he suppo ed by he signi ican lag in he di usion
o new echnologies, as a gued in Ro embe g (2003) and Anzoa egui e al. (2019).
15
A na u al way o inco po a e a wide in o ma ion se in he sys em could be app oaching
he sys em
(21)
wi h a Fac o -Augmen ed VAR in which he pe asi e mac oeconomic ac o s
𝐟𝑡
ha ha e been p e iosuly a gued o span he p oduc i i y ex e nal componen we e o be
included among he endogenous a iables in place o
𝑎𝑡
. Howe e , as wi h he coin eg a ion
p oblem, he apidly inc easing numbe o pa ame e s in his app oach leads o subs an ial
cos s in e ms o es ima ion noise (11 endogenous a iables imply 121 coe icien s pe lag in
he VAR) and a high isk o o e i ing.
Robus ness o he impulse esponse unc ions (IRFs) is ensu ed by he use o ecu si e
esidual boo s ap s anda d e o s o VAR es ima es (Lü kepohl 2005), complemen ed wi h
local p ojec ion (LP) es ima es ollowing Jo dà (2005) and Mon iel Olea and Plagbo g-Mølle
(2021). LPs exploi lags o bo h he mac oeconomic ac o s and he dependen a iable,
p o iding esul s ha a e gene ally mo e obus o misspeci ica ion hough mo e ola ile han
hose om he VAR. Following he s anda d app oach, o analyze he dynamic esponses o
p oduc i i y and consump ion g ow h a es o e ec i e R&D shocks wi h local p ojec ions,
cumula i e esponses a e es ima ed using he ollowing speci ica ion:
ℎ
∑
𝑗=1Δ𝑦𝑡+𝑗 =𝑏𝑦,ℎ,0+𝑏𝑦,ℎ,𝑠⋅𝜀𝑠,𝑡+𝑘
∑
𝑙=0(′𝑦,ℎ,𝑓,𝑙𝑡−𝑙+𝑏𝑦𝑦,ℎ,𝑙⋅𝑦𝑡−𝑙), (23)
whe e
𝑘
is se o i e, exceeding he annual da a equency o bols e obus ness; he esul s
a e obus o al e na i e nea by choices o
𝑘
. Con enien ly, he coe icien s
𝑏𝑦,ℎ,𝑠
p ecisely
e lec he impac o R&D shocks on he dynamics o long- un expec a ions, unca ed a
ho izon
ℎ
; ha is, hey app oxima ely indica e how s ongly hese shocks a ec he long- un
isk ac o 𝜀𝑥,𝑡, as de ined in (3).
3.3 Asse p icing es s
The key p edic ion om he heo e ical amewo k in Sec ion 2is ha i shocks o e ec i e
R&D in luence he long- un dynamics o consump ion g ow h (and agen s ha e ecu si e
p e e ences), hen
𝜆𝑥
in
(5)
should be posi i e and signi ican . Al hough he co a iance-based
na u e o asse p icing would, in p inciple, allow e ec i e R&D o se e as he long- un isk
ac o in le els – as is common in he mac o- inance li e a u e, since es ima ion p ocedu es
ne ou p edic able componen s – his analysis ocuses on s uc u al shocks o e ec i e R&D
as he isk ac o , while also epo ing esul s based on he le els. This choice p o ides a
mo e s ingen es o he heo y and ensu es consis ency wi h he es s o he mac oeconomic
p edic ions.
The key conce n in es ima ing ac o models such as
(5)
is omi ed a iable bias, which
a ises when he es ima ion model ails o cap u e all p iced sou ces o isk in he economy.
This conce n is pa icula ly ele an he e because he heo e ical amewo k unde lying
his wo k, like mos asse -p icing models, is delibe a ely s ylized and does no explici ly
accoun o all sys ema ic isks. Recen wo k by Giglio and Xiu (2021) add esses his issue
by p oposing a me hodology ha imp o es he obus ness o isk p emia es ima es h ough
16
explici con ol o omi ed a iables.
To u he in es iga e he sou ces o inno a ion long- un isk p emia, his s udy eplica es
he analysis o Bansal e al. (2005). Thei me hodology, speci ically designed o he Long-
Run Risks amewo k, ocuses on s ocks’ cash- low isks a he han o al e u n a ia ion,
isola ing he channel h ough which long- un isk ac o s heo e ically gene a e equi y isk
p emia. While his app oach is less s ingen han he Giglio and Xiu (2021) me hodology,
i p o ides complemen a y e idence ha acili a es compa ison wi h exis ing s udies and
con ibu es o b oade discussions on he economics o R&D. The emainde o his sec ion
p esen s he wo me hodological app oaches in de ail, while he nex sub-sec ion desc ibes
he es asse s.
A obus app oach o isk p emia es ima ion
In he con ex o a s anda d ac o s uc u e o e u ns such as
𝐑𝑡−𝑅𝑓
𝑡−1 =𝜷𝝀+𝜷𝐯𝑡+𝐮𝑡,(24)
when he numbe o obse a ions and es asse s go o in ini y, he ‘ ue’ isk ac o s in
he economy
𝐯𝑡
can be eco e ed by he P incipal Componen Analysis up o an a bi a y
o a ion
𝐯𝑡=𝐻𝐯𝑡
, whe e
𝐻
is a ull- ank ma ix. Then, Giglio and Xiu (2021) ocus on an
obse able ac o
𝑥𝑡
o in e es ha is a ine in he ‘ ue’ ac o s, wi h measu emen e o
𝑤𝑡
,
𝑥𝑡=𝜁0+𝜻𝑣𝐯𝑡+𝑤𝑡,(25)
and show ha he isk p emia associa ed wi h i can be e ec i ely es ima ed wi hou bias.
In his amewo k, he isk p emium associa ed wi h
𝑥𝑡
amoun s o
𝜻𝑣𝝀
, which co esponds
o he expec ed excess e u n o an asse wi h a be a o one wi h espec o
𝑥𝑡
and ze o wi h
espec o any o he independen ac o .
The key o i s es ima ion is ha
𝜻𝑣𝐻−1
can be ob ained by eg essing
𝑥𝑡
on
𝐯𝑡
, while
𝐻𝝀
can be ob ained by eg essing
𝐑𝑡
on
𝜷𝐻−1
, he la e being es ima ed by eg essing
e u ns on he ‘ ue’ ac o s. This deli e s all he necessa y elemen s o eco e 𝜻𝑣𝝀, since
𝜻𝑣𝐻−1𝐻𝝀=𝜻𝑣𝝀. (26)
In his se ing, a c ucial modeling choice conce ns he numbe o p incipal componen s
ea ed as he ‘ ue’ isk ac o s o he economy. This analysis conside s mul iple speci ica ions
o he numbe o ac o s, guided by he s anda d app oach o Bai and Ng (2002) and he
c i e ia p oposed in Alessi e al. (2010).
15
Equally impo an is he b ead h o he es
asse s: he wide he span o economic s a es hey ep esen , he mo e obus he con ol o
omi ed isk ac o s will be. Mo eo e , o allow he in o ma ion con ained in he shocks o
be inco po a ed in o p ices g adually (Ebe ha e al. 2004), he inno a ion long- un isk
15The la e consis s o wo c i e ia and is implemen ed by aking he median ac oss 300 epe i ions.
17
ac o is examined bo h as con empo aneous shocks o e ec i e R&D
𝜀𝑠,𝑡
and as olling sums
o hese shocks. Speci ically, esul s a e epo ed o olling sums wi h ho izons o 1 qua e
(i.e., con empo aneous shocks), wo yea s (ma ching he analysis on cash lows), and ou
yea s (co esponding o he ypical leng h o a business cycle).
A adi ional es ima ion app oach
Bansal e al. (2005) applies he s anda d p ocedu e o Fama and Macbe h (1973) wi h
wo modi ica ion. Fi s , hey igno e he isk associa ed wi h sho - e m luc ua ions in
consump ion g ow h, based on he empi ical inding ha such luc ua ions accoun o a
negligible po ion o he equi y isk p emium (Meh a and P esco 1985). This obse a ion
is p ecisely wha mo i a es he ocus on long- un isks, whose p emia a e p edic ed o be
la ge by o de s o magni ude. Second, hey measu e asse isk by he sensi i i y o cash- low
g ow h o he isk ac o s a he han by e u n sensi i i ies. This choice aligns mo e closely
wi h he heo e ical o mula ion o Long-Run Risk and o he consump ion-based models, in
which e u n be as a e endogenously de e mined by he sensi i i y o cash lows o he isk
ac o s.
The model hey es is:
E𝑡[𝑅𝑖𝑡+1]−𝑅𝑓
𝑡=𝜆𝑥𝛽𝑖
𝑥,𝐷,(27)
whe e he di idend-be a, 𝛽𝑖
𝑥,𝐷, is es ima ed om he uni a ia e ime-se ies eg ession
Δln𝐷𝑖𝑡=𝛽𝑖
0,𝐷+𝛽𝑖
𝑥,𝐷⋅1
𝐻𝐻
∑
𝑙=1𝜀𝑠,𝑡−𝑙+𝑢𝑖𝑡.(28)
Bansal e al. (2005) uses he aw se ies o consump ion g ow h as eg esso , so he mo ing
a e age p ima ily se es o il e ou high- equency luc ua ions and iden i y shi s in long-
un consump ion p ospec s. In con as , since shocks o e ec i e R&D a e shown in he
mac oeconome ic analysis o ha e pe sis en e ec s on consump ion g ow h, hey cap u e
changes in long- un consump ion p ospec s di ec ly, wi hou he need o il e ing. To assess
his, he sensi i i y o he esul s o he agg ega ion ho izon is examined, epo ing ou comes
bo h o he adi ional ho izon in Bansal e al. (2005) (
𝐻=8
qua e s) and o
𝐻=1
qua e . Fu he de ails on his amewo k a e p o ided in Appendix A.4.
3.4 Da a
Mac oeconomic da a
The baseline measu e o R&D in his s udy is he eal US qua e ly p i a e R&D expendi u es
(chained 2017 dolla s), consis en wi h closely ela ed s udies (Beqi aj e al. 2025; Kung
and Schmid 2015; Mo an and Que al o 2018). P i a e R&D e lec s he p o i -maximizing
inno a ion decisions a he co e o mos endogenous g ow h models mo e di ec ly han o al
R&D, since go e nmen expendi u es ope a e h ough dis inc ins i u ional mechanisms,
18
likely ma e ializing in a undamen ally di e en knowledge p oduc ion unc ion. The o al
R&D se ies is used as a obus ness check. Bo h se ies a e p o ided by he Bu eau o Economic
Analysis (BEA) ia he Fede al Rese e Economic Da a (FRED) online da abase and span
1947 Q1 o 2025 Q2.16
To al Fac o P oduc i i y is ob ained ollowing Fe nald (2012). The baseline se ies
is he qua e ly TFP g ow h adjus ed o u iliza ion and excluding R&D om capi al,
while obus ness checks use he aw TFP se ies. The u iliza ion adjus men is p e e ed
because emo ing cyclical u iliza ion dynamics – like any non-idea- ela ed ac o – enhances
he signal- o-noise a io o he unde lying echnological componen o p oduc i i y, he eby
imp o ing he p ecision o he es ima es; he R&D capi al is excluded because i s cons uc ion
h ough simple cumula ion and dep ecia ion o R&D lows is inconsis en wi h he knowledge
p oduc ion unc ion s udied in his wo k. Di e ences be ween he se ies mainly a ise om
he u iliza ion adjus men , wi h minimal impac om he exclusion o R&D om capi al.
The se ies span 1947 Q2 o 2025 Q1, and le els a e ob ained by cumula ing he g ow h
a es.17
The labo o ce is measu ed by he o al employmen le el, wi h non- a m employmen
used as a obus ness check. Bo h se ies a e p o ided mon hly by he Bu eau o Labo
S a is ics ia FRED. The qua e ly se ies is cons uc ed by aking he las alue o each
qua e , spanning 1948 Q1 o 2025 Q2.18
The p edic ing ac o s consis o wo dis inc se s, p e iously employed o o ecas TFP
g ow h in Ai e al. (2018): (1) nine iden i ied ac o s, ex ending hose used in Bansal and
Shalias o ich (2013); and (2) nine non-iden i ied ac o s di ec ly om Lud igson and Ng
(2009). These a e e e ed o wi h he sho hands ‘BS’ and ‘LN’, espec i ely. The BS
ac o s se o iginally comp ised he US Cycle Adjus ed P ice Ea nings (CAPE) a io, he
3-mon h T easu y-bill yield, and he 3- and 5-yea T easu y bond yields. Ai e al. (2018)
la e expanded his se o include he US s ock ma ke in eg a ed daily ola ili y. This wo k
addi ionally inco po a es he 10-yea T easu y bond yield, eal US co po a e p o i s, eal
US non inancial co po a e liquid asse s, and labo inpu g ow h, in o de o be e cap u e
mac oeconomic dynamics a longe ho izons, as well as aspec s ela ed o inancing condi ions,
p o i abili y, and p oduc p oli e a ion — all known o in luence p oduc i i y and R&D
decisions. These con ols span 1951 Q4 o 2025 Q1.
19
The LN ac o s se ins ead is o med
by he p incipal componen s o a wide a ay o mac oeconomic and inancial a iables. These
se ies ha e mon hly equency and a e agg ega ed as qua e ly a e ages, yielding a ime span
om 1960 Q1 o 2025 Q2.
20
As illus a ed in Appendix C, all BS a iables appea o exhibi
16
The baseline eal se ies is ob ained by de la ing he nominal R&D se ies
Y006RC1Q027SBEA
wi h he
de la o Y006RG3Q086SBEA. The o al se ies has ID Y694RX1Q020SBEA.
17
The da a a e p o ided online by he au ho and also include he u iliza ion adjus men and changes in
capi al excluding R&D.
18IDs: CE16OV and LNS12035019.
19
CAPE is om R. Shille ’s websi e; bill and bond yields a e om Finaeon un il he i s a ailabili y o
DGS3MO
,
DGS3
,
DGS5
, and
DGS10
on FRED; daily s ock ma ke da a is om CRSP. US co po a e p o i s and
liquid asse s a e he se ies CPROFIT and BOGZ1FL104001005Q on FRED, de la ed by GDPDEF om FRED.
20Publicly a ailable on S. Lud igson’s websi e.
19

uni - oo beha iou , whe eas he e is only weak e idence o non-s a iona i y among he LN
ac o s. The e o e, while he LN ac o s a e used in le els, he BS ac o s a e included in
i s di e ences o mi iga e he isk o spu ious eg ession.
Finally, consump ion is measu ed as eal o al pe sonal consump ion expendi u es pe
capi a, in chained 2017 US dolla s. The se ies is p o ided by he BEA ia FRED and span
1947 Q1 o 2025 Q2.21
Mo e de ails on he da a can be ound in Appendix B.1.
Tes asse s da a
The selec ion o es asse s o he obus isk p emia es ima ion is guided by he need
o co e as b oad a po ion o he economic s a e space as possible, ensu ing ha he
esul ing es ima es a e gene alizable and obus . Expanding on he app oach o B yzgalo a
e al. (2025), his wo k employs 153 anomaly s ock po olios om Jensen e al. (2021), 17
indus y-so ed s ock po olios om K. F ench’s da abase, and 13 bond po olios. The bond
po olios a e cons uc ed om he ze o-coupon yield da a o Gü kaynak e al. (2007) by
i ing Nelson-Siegel-S ensson cu es and sub ac ing he e u n on a h ee-mon h T easu y
bill. These po olios span ma u i ies o 6 mon hs and 1, 2, 3, 4, 5, 6, 7, 10, 15, 20, 25, and
30 yea s. The inal da ase co e s he pe iod om 1971 Q4 o 2024 Q4.
The es asse s employed o he Bansal e al. (2005) exe cise include only s ock po olios,
as in he o iginal s udy: 10 so ed by size, 10 by book- o-ma ke equi y, and 10 by pas -yea
e u ns. This se is e e ed o as he ‘legacy pool’. I is expanded wi h 2-by-3 po olios
join ly so ed by size (la ge s. smalle ma ke capi aliza ion ela i e o he median) and
a ious i m cha ac e is ics ela ed o R&D in es men o o m he ‘ex ended pool’, and
hen u he complemen ed by 17 indus y po olios o o m he ‘wide pool’. The i m
cha ac e is ics conside ed in he ex ended pool cap u e (i) he in ensi y o inno a i e e o s
( i m-speci ic R&D a io o ma ke capi aliza ion), (ii) inancing capaci y (le e age, u no e ,
and p o i abili y), and (iii) g ow h oppo uni ies (asse s g ow h and Tobin’s
𝑞
). These
dimensions ha e been associa ed wi h dispe sion in c oss-sec ional isk p emium and di e en
o ms o sensi i i y o inno a ion dynamics, he e o e a e likely o gene a e he e ogenei y
in exposu e o he long- un isk ca ied by agg ega e R&D. Speci ically, a ia ion in i ms’
R&D in ensi y a ec s he spillo e s and ‘ ishing ou ’ e ec s expe ienced by a i m, which
Jiang e al. (2016) showed o be p iced in inancial ma ke s, while inancing capaci y and
g ow h oppo uni ies in e ac wi h agg ega e R&D in es men by a ec ing i ms’ abili y o
eac o inno a ion shocks (Aghion e al. 2012; B own e al. 2009; Hall 2002; Hall e al. 2010;
Li 2011; Male ic 2018; Zhang 2014). Thei inclusion u he p o ides desc ip i e s a is ics
ha con ibu e o he discussion on he c oss-sec ional a ia ion in e u ns wi h i m-speci ic
R&D in ensi y, which, since i s i s documen a ion in Chan e al. (2001), emains deba ed
(Ahmed e al. 2025; Leung e al. 2020). The payou se ies is cons uc ed ollowing Bansal
e al. (2005) and Hansen e al. (2005), wi h de ails p o ided in Appendix B.2. Mon hly
21ID: A794RX0Q048SBEA.
20
s ock da a come om CRSP, and annual accoun ing da a om he Compus a Fundamen als
da ase . Mon hly e u ns a e compounded o ob ain qua e ly igu es and hen de la ed
using he consump ion de la o . The po olio e u n and cash- low g ow h a e se ies begin
a di e en da es: he long-pool and indus y-so ed po olios s a in 1967 Q1, while he
o he s begin in 1975 Q1; all se ies end in 2022 Q4. Key s a is ics o he o med se ies a e
epo ed in Appendix B.2.
4 The empi ical inno a ion componen
4.1 The g oss e ec i e R&D
The es ima ion esul s o he long- un ela ion in
(17)
a e shown in Table 1. The i s column
o he able shows he baseline speci ica ion, wi h he columns o he igh subs i u ing one
a iable a a ime wi h he al e na i e obus ness measu e. The las column uses he baseline
da a bu employs he IM-OLS me hod ins ead.
Fi s , he
𝛼𝑍
es ima es ha e he expec ed sign and a e always signi ican ly di e en om
ze o. Simply pu , his means ha R&D expendi u e and TFP le els inc ease oge he ; aw
R&D ises wi h he scale o he economy, as expec ed. A simila ly expec ed coe icien is
ha o labo ,
𝛼𝐿
, which sugges s some dilu ion in R&D’s powe o ad ance he echnological
on ie and implies a media ing e ec on how R&D expendi u e ela es o he echnological
on ie o ob ain a meaning ul measu e o e ec i e R&D. The only excep ion occu s when
o al R&D is used, highligh ing he possibili y ha public R&D may ope a e h ough a
di e en p oduc ion unc ion han p i a e R&D.
22
Fu he , he use o non a m employmen
does no esul in disce nible changes, while he use o aw TFP, al hough i does no ma e ially
a ec he es ima es, al e s he sho - e m luc ua ions in he esul ing e o co ec ion e m,
as can be seen om he plo s o all ECT ime se ies in Appendix C(Figu e 10). Visual
inspec ion o he ime se ies also e eals he conside able ins abili y o he IM-OLS es ima es,
as e lec ed by he la ge condi ion numbe o he coe icien s’ co a iance ma ix
𝜅
, e en
hough he es ima es essen ialy con i m he baseline esul s.
The lowe pa o Table 1 epo s desc ip i e s a is ics o he e o co ec ion e m
esul ing om he es ima ion, i.e., he ime se ies o
𝑠𝑡
. All es ima ed g oss e ec i e R&D
se ies a e ound o be s a iona y acco ding o bo h he ADF and KPSS es s, wi h none
exhibi ing a ime end o a squa ed ime end. The pe sis ency o he se ies, as measu ed
by an AR(1) i , is in line wi h he pe sis en componen o p oduc i i y g ow h ha is
ansmi ed o consump ion g ow h, which O u e al. (2013) and C oce (2014) has shown
ha ing a hal -li e be ween 2 and 16 yea s. As shown in Appendix C(Table 10), co ela ions
among ECTs om di e en speci ica ions ne e d op below 86%, consis en wi h he close
es ima ion esul s, excep o he IM es ima ion.
22Un epo ed es ima es a e in ac signi ican ly nega i e when only public R&D is used.
21
Table 1: Coin eg a ion esul s. S anda d e o s in pa en heses.
𝑇
is he numbe o obse a ions;
𝜅
is
he condi ion numbe o he coe icien s’ co a iance ma ix. S a is ics in he bo om pa o he able
e e o he e o co ec ion e m
𝑠
:
𝜎𝑠
is i s s anda d de ia ion;
𝑡𝑡
and
𝑡𝑡2
a e he ime end and
squa ed ime end coe icien s (wi h HAC s anda d e o s); ADF and KPSS a e s a iona i y es s
in le els; AR(1) is he coe icien om an AR(1) i ; HL a e he lowe and uppe bounds, a 95%
con idence, o he hal -li e implied by he AR(1) es ima e, in yea s.
Baseline 𝑆: To . R&D 𝑍: Raw TFP 𝑄: N.F. Empl. Es . Me h.: IM
𝛼𝑍3.526*** 4.197*** 3.655*** 3.349*** 2.821***
(0.439) (0.516) (0.490) (0.464) (0.552)
𝛼𝐿0.909*** −0.354 0.956*** 0.953*** 1.387***
(0.336) (0.395) (0.356) (0.325) (0.398)
T309 309 309 309 309
𝜅 3.4×1063.4×1063.3×1063.5×1061.1×108
𝑠𝑡
𝜎𝑠 0.130 0.144 0.128 0.129 0.253
0.00 0.00 0.00 0.00 0.00
20.00 0.00 0.00 0.00 0.00
ADF −2.57** −2.45** −2.92*** −2.45** −9.18***
KPSS 0.09 0.09 0.09 0.10 0.29
AR(1) 0.96 0.96 0.95 0.97 0.15
HL low 2.6 2.7 2.1 2.7 0.1
HL high 21.0 23.6 12.0 25.1 0.1
* p <0.1, ** p <0.05, *** p <0.01
4.2 The e ec i e R&D
The es ima ed
𝑠𝑡
se ies allows o he es ima ion o equa ion
(20)
, which yields
𝛾1
, he missing
pa ame e equi ed o eco e he e ec i e R&D se ies. In o ma ion c i e ia (Appendix C,
Figu e 8) selec one lag o speci ica ions based on he adjus ed TFP measu e and wo lags
o hose based on he aw TFP measu e. The co esponding esul s a e epo ed in Table 2.
The es ima ed R&D coe icien is consis en ly posi i e and s a is ically signi ican , con-
i ming a s ong empi ical ela ionship be ween inno a ion in ensi y and p oduc i i y g ow h.
I s magni ude closely ma ches he empi ical es ima es in Kung and Schmid (2015) and implies
an annualized inc ease o abou 0.8% in TFP g ow h ollowing a one-s anda d-de ia ion
inc ease in
𝑠𝑡
. Con ol a iables a e always join ly signi ican , and speci ica ions using
LN-based con ols consis en ly achie e highe explana o y powe , as e lec ed in highe R
2
alues and lowe in o ma ion c i e ia.
Table 2also epo s he es ima ed implied
𝜅𝑠
and he eco e ed
𝑠𝑡
se ies. Speci ically,
𝑠𝑡
is ob ained by applying equa ion
(19)
o each
𝑡
in he sample, using pa ame e es ima es
o
𝛼𝑍
om Table 1and
𝑏𝑠
om Table 2. The se ies a e immed a he beginning o he
sample, up o he i s
𝑡
such ha
𝜅𝑡𝑠<0.01
, ensu ing ha he in luence o he unobse ed
ini ial condi ion is negligible, as he omi ed po ion co esponds o only one-hund ed h o
22
Table 2: One-s ep p oduc i i y g ow h o ecas esul s. HAC s anda d e o s in pa en heses.
𝑇
is
he numbe o obse a ions;
𝑅2
is he goodness-o - i ;
𝑘
is he numbe o con ol a iables among
he eg esso s;
𝑊(𝑘)
is he Wald s a is ic o hei join signi icance. De ails on
𝜅𝑠
and i s s anda d
e o s a e in Sec ion A.3. The lowe panel epo s s a is ics o he eco e ed ime se ies
𝑠
(see
Sec ion 3.1):
𝜎𝑠
is i s s anda d de ia ion;
𝑡𝑡
and
𝑡𝑡2
a e he coe icien s o he linea and quad a ic
ime ends (wi h HAC signi icance le els); ADF and KPSS a e s a iona i y es s in le els; AR(1) is
he au o eg essi e coe icien om a i s -o de i ; HL a e he 95% con idence bounds o he implied
AR(1) hal -li e, in yea s.
Baseline 𝑆: To . R&D 𝑍: Raw TFP 𝑄: N.F. Empl.
BS LN BS LN BS LN BS LN
𝑏𝑠(%) 1.558*** 1.549*** 1.066*** 1.223*** 0.997*** 0.794** 1.507*** 1.520***
(0.429) (0.285) (0.318) (0.254) (0.355) (0.337) (0.418) (0.289)
𝑇 292 261 292 261 291 260 292 261
R2(%) 9.5 12.4 7.4 11.8 21.8 41.7 9.1 12.2
𝑘 10 10 10 10 20 20 10 10
𝑊(𝑘) 79.97*** 61.00*** 64.56*** 50.27*** 287.53*** 2775.10*** 73.09*** 61.99***
𝜅𝑎0.964 0.971 0.950 0.949 0.945 0.945 0.955 0.949
(0.014) (0.013) (0.016) (0.012) (0.017) (0.012) (0.014) (0.012)
𝑠𝑡(𝜅𝑡𝑎<0.01)
𝑇𝑠 226 225 207 220 183 151 219 219
𝜎𝑠 0.058 0.057 0.068 0.065 0.060 0.062 0.055 0.055
𝑡𝑡 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
𝑡𝑡20.00 0.00 0.00 0.00 0.00*** 0.00*** 0.00 0.00
ADF −3.91*** −3.95*** −3.17*** −3.49*** −3.95*** −3.50*** −3.56*** −3.56***
KPSS 0.09 0.09 0.18 0.13 0.22 0.19 0.14 0.14
AR(1) 0.71 0.70 0.72 0.68 0.68 0.70 0.70 0.70
HL low 0.4 0.3 0.4 0.3 0.3 0.3 0.3 0.3
HL high 0.8 0.8 0.8 0.7 0.7 0.8 0.7 0.7
* p <0.1, ** p <0.05, *** p <0.01
𝑠0
. Appendix A.3 p o ides a de ailed discussion o he unce ain y in his eco e y, a ising
bo h om he ini ial condi ion app oxima ion and he es ima ion noise, and i also includes
de ails on he compu a ion o he epo ed s anda d e o s. Figu e 11 in Appendix Cshows
all eco e ed se ies wi h con idence in e als, illus a ing ha p ecision g ea ly a ies ac oss
speci ica ions. The se ies om he baseline speci ica ion, in pa icula , exhibi he smalles
unce ain y, p ima ily e lec ing he lowe unce ain y in he es ima es o
𝑏𝑠
. Desc ip i e
s a is ics epo ed in he lowe panel o Table 2indica e b oadly consis en ime se ies
ola ili y ac oss speci ica ions. The eco e ed se ies display no end, a e s a iona y as
con i med by ADF and KPSS es s, and exhibi subs an ially lowe pe sis ence han he
𝑠𝑡
se ies. No ably, bo h he pe sis ence o
𝑠
and he magni ude o
𝑏𝑠
a e signi ican ly lowe
han hose assumed in he heo e ical model o Kung and Schmid (2015). This disc epancy is
no p oblema ic, as he econome ic amewo k de eloped in his wo k accoun s o eedback
e ec s be ween inno a ion and ex e nal componen s – a ea u e absen in ha model bu
23
Table 4: Risk p emia es ima ion ollowing Giglio and Xiu (2021). Each column epo s he isk
p emia associa ed wi h he espec i e speci ica ion o e ec i e R&D s uc u al shocks, based on he
co esponding es ima ions in Table 3. -s a is ics a e epo ed in squa e b acke s. The es asse s a e
183 po olios spanning 1971 Q4 o 2023 Q4. The numbe o ac o s (6, 14, and 22) co esponds o
op imal selec ions pe Alessi e al. (2010) and Bai and Ng (2002), wi h 22 being he smalles op imal
numbe om he la e me hod.
Baseline 𝑆: To . R&D 𝑍: Raw TFP 𝑄: N.F. Empl. 𝑠
Ho izon: 1 qua e
6 Fac o s 0.01 0.02 0.02 0.02 0.02
[0.93] [0.90] [1.22] [1.03] [1.28]
14 Fac o s 0.04 0.03 0.05 0.04 0.04
[1.32] [0.74] [1.35] [1.12] [1.08]
22 Fac o s −0.01 −0.09 0.01 −0.02 −0.04
[−0.21] [−1.30] [0.09] [−0.38] [−0.55]
Ho izon: 2 yea s
6 Fac o s 0.06 0.04 0.08*0.06 0.10
[1.63] [0.91] [1.87] [1.63] [1.44]
14 Fac o s 0.22** 0.12 0.27*** 0.19** 0.29*
[2.35] [1.24] [2.92] [2.05] [1.86]
22 Fac o s 0.15 0.11 0.23*0.10 0.18
[1.12] [0.73] [1.81] [0.74] [0.84]
Ho izon: 4 yea s
6 Fac o s 0.08 0.07 0.09 0.08 0.11
[1.33] [1.11] [1.33] [1.36] [0.97]
14 Fac o s 0.48*** 0.34** 0.52*** 0.45*** 0.69***
[3.28] [2.50] [3.52] [3.11] [2.75]
22 Fac o s 0.54*** 0.43** 0.61*** 0.48** 0.80**
[2.80] [2.23] [3.17] [2.52] [2.28]
Num.Obs. 213 213 213 213 213
* p <0.1, ** p <0.05, *** p <0.01
long- un isk speci ically, hese sensi i i ies a e all nega i e. Second, di idend-be as a e
nega i e ac oss mos i m-speci ic R&D-in ensi y le els. The excep ion is small, highly R&D-
in ensi e i ms, which exhibi a dis inc i e esponse depending on he na u e o agg ega e
R&D changes: hei payou s aise when inc eases a e unexpec ed, bu all when p edic able
componen s a e included (i.e. wi h espec o o e all e ec i e R&D changes). Thi d, among
small i ms, a iables ela ed o inancing capaci y – u no e , p o i abili y, and le e age –
co ela e posi i ely wi h sensi i i ies, whe eas hose cap u ing in es men oppo uni ies show
he opposi e pa e n, wi h Tobin’s q exhibi ing a pa icula ly s ong nega i e ela ion wi h
payou be as.
A ull s uc u al in e p e a ion o he obse ed pa e ns in di idend-be as is beyond he
scope o his wo k, bu he e idence appea s b oadly consis en wi h: (i) i ms bene i ing om
30

Table 5: Di idend be as o addi ional es asse s om he ex ended pool a a 1-qua e and 2-yea
ho izons (1975 Q1–2022 Q4). S ocks a e double-so ed in o
2×3
po olios by size (NYSE median)
and accoun ing cha ac e is ics: RD (R&D/ma ke cap), To (sales/asse s), P o (g oss p o i s/asse s),
L g (deb /asse s), AG (asse g ow h), and TQ (Tobin’s
𝑄
). Risk ac o s: Cons. (consump ion
g ow h), Raw/Adj. TFP ( aw/adjus ed o al ac o p oduc i i y),
𝑠
: shock/le el (e ec i e R&D
shock/le el).
Po olio Cons. Raw TFP Adj. TFP 𝑠: shock 𝑠: le el
Ho izon 1 8 1 8 1 8 1 8 1 8
RD(1-small) 0.09 0.18 0.05 0.39 −0.07 0.03 0.01 0.00 −0.59 −0.58
RD(2-small) 0.06 0.04 0.02 0.46 −0.11 0.19 0.04 −0.02 −1.16 −0.73
RD(3-small) 0.75 −0.68 0.59 1.17 −0.35 0.45 0.54 1.09 −1.55 −3.37
RD(1-big) 0.01 0.03 0.00 0.08 −0.02 −0.01 0.00 −0.03 −0.27 −0.29
RD(2-big) 0.05 0.12 0.01 0.23 −0.03 0.00 0.03 −0.07 −0.36 −0.28
RD(3-big) 0.03 0.09 −0.01 0.24 −0.07 −0.09 −0.02 −0.17 −1.20 −1.20
To(1-small) 0.05 0.13 0.05 0.27 −0.01 0.03 0.02 0.06 −0.25 −0.76
To(2-small) 0.10 0.29 −0.01 0.81 −0.13 0.22 0.01 0.28 0.20 0.99
To(3-small) 0.38 1.06 0.23 1.85 −0.15 0.08 0.12 0.75 2.64 3.53
To(1-big) 0.01 0.06 −0.01 0.08 −0.02 −0.02 −0.01 −0.05 −0.31 −0.40
To(2-big) 0.03 0.09 0.00 0.17 −0.03 −0.03 0.00 −0.07 −0.45 −0.32
To(3-big) 0.04 0.11 0.03 0.29 −0.05 0.02 0.03 0.02 −0.21 −0.24
P o (1-small) 0.00 −0.01 −0.02 0.20 −0.07 0.11 0.01 −0.18 −1.15 −0.82
P o (2-small) 0.23 0.58 0.11 0.97 −0.13 −0.12 0.06 0.42 1.00 0.56
P o (3-small) 0.38 1.34 0.19 2.29 −0.15 0.35 0.09 0.71 3.98 6.16
P o (1-big) 0.01 0.05 0.00 0.09 −0.02 −0.02 −0.01 −0.02 −0.25 −0.22
P o (2-big) 0.03 0.08 0.00 0.16 −0.03 −0.09 0.00 −0.12 −0.64 −0.67
P o (3-big) 0.05 0.13 0.02 0.34 −0.06 0.14 0.03 −0.05 −0.41 −0.07
L g(1-small) 0.01 0.15 −0.01 0.20 −0.05 −0.05 0.02 −0.09 −0.73 −0.68
L g(2-small) 0.18 0.47 0.08 1.03 −0.19 0.22 0.06 0.17 −0.86 0.00
L g(3-small) 0.10 0.30 0.04 0.75 −0.06 0.07 0.01 0.24 1.21 1.54
L g(1-big) 0.03 0.09 0.01 0.19 −0.04 −0.01 0.00 −0.07 −0.61 −0.52
L g(2-big) 0.02 0.08 0.00 0.14 −0.03 −0.06 0.00 −0.08 −0.36 −0.41
L g(3-big) 0.02 0.04 0.00 0.10 −0.02 0.02 0.00 −0.03 −0.19 −0.08
AG(1-small) 0.19 0.82 0.01 1.66 −0.13 0.18 0.04 0.67 2.84 3.12
AG(2-small) 0.18 0.43 0.08 1.02 −0.20 0.16 0.05 −0.07 −1.57 −0.27
AG(3-small) 0.09 0.19 0.05 0.38 −0.05 0.04 0.03 0.09 0.29 0.47
AG(1-big) 0.06 0.14 0.00 0.29 −0.09 −0.05 0.01 −0.14 −1.02 −0.84
AG(2-big) 0.02 0.05 0.01 0.21 −0.01 0.01 0.00 −0.06 −0.30 −0.49
AG(3-big) 0.01 0.07 0.00 0.08 −0.03 −0.02 0.00 −0.04 −0.29 −0.01
TQ(1-small) 0.35 0.87 0.05 2.43 −0.50 0.78 0.18 0.27 −1.51 1.92
TQ(2-small) 0.14 0.65 0.01 1.10 −0.12 −0.15 0.06 0.20 0.36 0.63
TQ(3-small) 0.06 0.14 0.05 0.24 −0.01 0.00 −0.01 −0.06 −0.13 −0.37
TQ(1-big) 0.03 0.15 −0.01 0.25 −0.06 −0.09 −0.02 −0.06 −0.51 −0.47
TQ(2-big) 0.01 0.09 −0.01 0.15 −0.03 −0.07 0.01 −0.05 −0.43 −0.52
TQ(3-big) 0.03 0.03 0.03 0.14 −0.03 0.10 0.01 −0.07 −0.35 0.03
31
pee s’ highe R&D in es men , unless hey simul aneously aise hei own R&D in es men
in esponse o p edic able agg ega e R&D inc eases; (ii) a g ea e abili y o gene a e in e nal
cash lows – highe u no e and p o i abili y – imp o ing he abili y o eac o agg ega e
inno a ion news, pa icula ly in smalle , ypically mo e cons ained i ms, al hough e idence
om le e age-so ed po olios is ambiguous;
24
(iii) agg ega e inno a ion ends o educe
payou g ow h o ‘leading’ i ms, i.e. he la ges and he as es -g owing ones.
Resul ing es ima es o isk p emia om he c oss-sec ional s ep a e epo ed in Table 6.
No ably, he p emia associa ed wi h consump ion cash- low isk di e ma kedly ac oss
ho izons: hey a e highly signi ican a he 1-qua e ho izon bu only ba ely signi ican in
he wide pool a he 2-yea ho izon. This decline is e en mo e appa en in he c oss-sec ional
R
2
, which alls om oughly 20–30% o a ange o 2–50% ac oss pools. The magni ude o
he p emia, a he ho izon common o Bansal e al. (2005), exceeds hei s bu emains well
wi hin he con idence in e als.
Risk p emia associa ed wi h p oduc i i y long- un isk, as measu ed based on mo ing
a e ages o aw TFP, a e much mo e consis en : p emia a e signi ican o wo o he h ee
es -asse pools a bo h ho izons and explana o y powe inc eases a he 2-yea ho izon.
By con as , es ima es based on adjus ed TFP a e less s able ac oss ho izons, exhibi ing
nega i e p emia a he 1-qua e ho izon and posi i e p emia a he 2-yea ho izon, wi h R
2
compa able o hose o aw TFP, excep in he legacy pool, whe e R2d ops by o e 50%.
Like p oduc i i y, he p emia associa ed wi h inno a ion long- un isk, based on he
s uc u al shocks, a e signi ican o he wo wides o he h ee es -asse pools and emain
so ac oss ho izons. By con as , o ming he isk ac o om e ec i e R&D le els yields
p emia ha a e smalle in magni ude and ne e signi ican a he 5% le el, highligh ing he
subs an ial ole o he p edic able componen o e ec i e R&D in p icing. None heless, h ee
o he six p emia es ima ed o e ec i e R&D in le els ha e -s a is ics abo e 1.2, indica ing
ha some isk is s ill cap u ed. O e all, hese esul s suppo he no ion ha a subs an ial
po ion o he p emia o holding long- un isk de i es om cash- low isk, consis en wi h
he s anda d long- un isk amewo k.
24
In gene al, highe le e age implies educed inancial slack o e en dis ess. This is di icul o econcile
wi h he inding ha high-le e age po olios do no exhibi he s onges consump ion sensi i i ies. A mo e
consis en in e p e a ion is ha le e age pa ly p oxies o be e access o ex e nal inance a he han
dis ess, possibly e lec ing omi ed i m cha ac e is ics co ela ed wi h le e age—an issue likely exace ba ed
by he coa se wo-by- h ee so ing used he e.
32
Table 6: Cash- low isk p emia.
𝑡
-s a is ics in squa e b acke s. Risk ac o s: Cons. (consump ion
g ow h), Raw/Adj. TFP ( aw/adjus ed o al ac o p oduc i i y),
𝑠
: shock/le el (e ec i e R&D
shock/le el). The legacy pool comp ises 16 es asse s (224 obse a ions, 1967 Q1–2022 Q4); he
ex ended pool, 51 es asse s (192 obse a ions, 1975 Q1–2022 Q4); and he wide pool, 68 es asse s
(192 obse a ions, 1975 Q1–2022 Q4).
Cons. Raw TFP Adj. TFP 𝑠: shock 𝑠: le el
Ho izon: 1 qua e
Legacy pool 0.68 0.18 −1.47 1.16 0.02
[1.60] [0.84] [−1.54] [1.34] [0.66]
R2(%) 30.99 0.87 58.96 38.69 19.84
MAPE (%) 0.19 0.25 0.12 0.18 0.23
Ex . pool 1.72** 1.97*** −2.47** 2.28** 0.06*
[2.58] [3.14] [−2.12] [2.31] [1.68]
R2(%) 24.85 20.44 17.95 16.77 4.89
MAPE (%) 0.41 0.44 0.43 0.46 0.48
Wide pool 1.45*** 1.38** −1.09*** 2.00** 0.03
[3.11] [2.39] [−3.48] [2.51] [0.97]
R2(%) 18.05 14.16 4.38 14.97 4.06
MAPE (%) 0.45 0.47 0.49 0.47 0.48
Ho izon: 2 yea s
Legacy pool 0.26 0.20 0.27 0.35 0.01
[1.42] [1.48] [0.67] [1.40] [0.70]
R2(%) 54.21 59.62 6.35 55.46 21.37
MAPE (%) 0.15 0.12 0.25 0.15 0.23
Ex . pool 0.32 0.40** 1.17** 0.71** 0.04
[1.65] [2.15] [2.15] [2.14] [1.63]
R2(%) 3.06 23.66 29.24 20.32 4.58
MAPE (%) 0.48 0.41 0.41 0.44 0.48
Wide pool 0.28*0.36** 0.42*0.60*0.03
[1.93] [2.53] [1.71] [1.92] [1.26]
R2(%) 2.17 18.61 12.23 17.33 4.96
MAPE (%) 0.49 0.45 0.47 0.46 0.48
* p <0.1, ** p <0.05, *** p <0.01
33
7 Conclusion
This pape ocuses on a heo e ical measu e o agg ega e R&D ha is designed o e lec he
con ibu ion o R&D in es men s o p oduc i i y g ow h dynamics. This measu e – e med
‘e ec i e R&D’ o he ‘inno a ion componen ’ – accoun s he media ing ole o idea spillo e s
and p oduc p oli e a ion in he e ec o R&D on p oduc i i y g ow h, consis en wi h bo h
ully- and semi-endogenous g ow h mechanisms.
A uni a ia e empi ical amewo k is in oduced o eco e luc ua ions in wo e sions
o his measu e: a g oss e ec i e R&D se ies, de i ed om he coin eg a ion ela ionship
among R&D, TFP, and labo o ce; and a ne measu e, cons uc ed, ecu si ely, elying a
one-pe iod TFP g ow h o ecas . Bo h se ies a e s a iona y in qua e ly U.S. da a, hough
hey di e ma kedly in pe sis ence: he g oss measu e exhibi s hal -li es o 3 o 21 yea s,
while he ne measu e displays a hal -li e o less han one yea .
Embedding ei he se ies in a VAR wi h p oduc i i y g ow h shows ha inno a ion
shocks gene a e pe sis en mo emen s in p oduc i i y g ow h, p opaga ing o e ho izons
o a decade. S uc u al iden i ica ion ensu es ha shocks o e ec i e R&D a e o hogonal
o con empo aneous p oduc i i y g ow h, while local p ojec ion exe cises con i m hese
dynamics and ex end hem o consump ion g ow h, which esponds o inno a ion shocks a
ho izons well beyond he business cycle, possibly up o 15 yea s.
The pape ’s p ima y con ibu ion is demons a ing ha shocks o he inno a ion compo-
nen cons i u e a signi ican c oss-sec ional isk ac o , consis en wi h long- un isk asse
p icing heo ies, associa ed o a subs an ial p emium, in he o de o 2% annually. This
inding is pa icula ly obus , le e aging s uc u al VAR iden i ica ion o elimina e spu ious
co ela ions wi h o he p oduc i i y- ela ed ac o s, employing ecen es ima ion echniques
o con ol o omi ed isk ac o s, and d awing on a b oad pool o 183 s ock- and bond-based
es asse s.
The analysis u he highligh s he impo ance o he cash- low channel in inno a ion
long- un isk and e eals he e ogeneous exposu es ac oss i m cha ac e is ics linked o R&D.
In pa icula , payou s o small, R&D-in ensi e i ms inc ease wi h agg ega e R&D shocks,
while hose o o he R&D-so ed po olios decline. Measu es o in e nal inancing capaci y
( u no e , p o i abili y) co ela e posi i ely wi h payou sensi i i ies, whe eas p oxies o
in es men oppo uni ies (asse g ow h, Tobin’s q) co ela e nega i ely.
Taken oge he , hese indings es ablish agg ega e inno a ion as a undamen al mac oe-
conomic isk ac o wi h dis inc asse p icing implica ions. Beyond his con ibu ion, he
empi ical me hodology de eloped he e p o ides a pla o m o u u e wo k explo ing in e na-
ional e idence, i m-le el esponses, and he di e en ial oles o al e na i e ypes o R&D,
among o he s.
34
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39
Figu e 6: Values o

𝑏𝑠
,
𝛼𝑍
(le panel), and
𝛼𝐿
( igh panel) ac oss he 16 speci ica ions conside ed in
his wo k. Hollow do s indica e speci ica ions whe e
𝛼
is no signi ican a he 10% le el. Fo
𝛼𝑍
,
he linea eg ession o
𝑏𝑠
yields a slope o
−0.55
wi h a he e oskedas ici y- obus
𝑝
- alue o
1.5%
.
Fo
𝛼𝐿
, he slope is no signi ican ly di e en om ze o a he
10%
le el when all speci ica ions a e
included (
𝑡=1.71
), bu equals
−8.63
wi h a
𝑝
- alue below
1%
when es ic ed o speci ica ions whe e
𝛼𝐿is signi ican ly di e en om ze o.
he pa ame e es ima es in he obse able componen , i.e. he easible eco e y es ima o
𝑠𝑡
:
Va (E[𝑠𝑡∣
𝜽,𝑋]∣𝑋)=Va ( 𝛼𝑍(𝑡−1
∑
𝑗=0 𝜅𝑠𝑗⋅Δln 𝑍𝑡−𝑗)+𝑡−1
∑
𝑗=0 𝜅𝑠𝑗Δ𝑠𝑡−𝑗∣𝑋) (65)
=Va (𝑡−1
∑
𝑗=0 𝜅𝑠𝑗⋅Δln 𝑆𝑡−𝑗− 𝛼𝐿(𝑡−1
∑
𝑗=0 𝜅𝑠𝑗Δln𝐿𝑡−𝑗)∣𝑋), (66)
whe e he i s equali y ollows om
𝑠0
being independen o bo h he pa ame e es ima es
and he da a sample, wi h ze o mean by de ini ion, while he second equali y esul s om
explici ly exp essing he i s di e ences o he g oss e ec i e R&D. Following he Del a
me hod, his is es ima ed as

Va (E[𝑠𝑡∣
𝜽,𝑋]∣𝑋)=∇ 𝑠𝑡( 𝛼𝑍, 𝛼𝐿,
𝑏𝑠)⊤⎡
⎢
⎢
⎣

𝜎2𝛼𝑍𝜎 𝛼𝑍, 𝛼𝐿0
𝜎 𝛼𝑍, 𝛼𝐿
𝜎2𝛼𝐿0
0 0 
𝜎2
𝑏𝑠
⎤
⎥
⎥
⎦∇𝑠𝑡( 𝛼𝑍, 𝛼𝐿,
𝑏𝑠), (67)
whe e he co a iances be ween
𝑏𝑠
and
𝛼•
a e conse a i ely assumed o be ze o, ollowing
ea lie a gumen s and he e idence in Figu e 6,
26
and he g adien e alua ed a he poin
es ima es is
∇𝑠𝑡( 𝛼𝑍, 𝛼𝐿,
𝑏𝑠)=⎡
⎢
⎢
⎣
∑𝑡−1
𝑗=0−
𝑏𝑠⋅𝑗⋅ 𝜅𝑠𝑗−1(Δln𝑆𝑡−𝑗− 𝛼𝐿Δln𝐿𝑡−𝑗)
−∑𝑡−1
𝑗=0 𝜅𝑠𝑗Δln𝐿𝑡−𝑗
∑𝑡−1
𝑗=0− 𝛼𝑍⋅𝑗⋅ 𝜅𝑠𝑗−1(Δln 𝑆𝑡−𝑗− 𝛼𝐿Δln𝐿𝑡−𝑗)⎤
⎥
⎥
⎦.(68)
26
This assump ion is mo e conse a i e wi h espec o
𝛼𝑍
han o
𝛼𝐿
, al hough he e idence does no
p o ide a s ong case o a posi i e co a iance in ei he case.
46

The second e m, on he o he hand, cap u es he unce ain y a ising om he unobse abili y
o he ini ial e ec i e R&D,
𝑠0
, which, condi ional on he pa ame e es ima es, is he only
emaining andom quan i y:
E[Va (𝑠𝑡∣
𝜽,𝑋)∣𝑋]=E[ 𝜅𝑠2𝑡⋅Va ( 𝑠0∣
𝜽,𝑋)∣𝑋] (69)
=E[ 𝜅𝑠2𝑡∣𝑋]Va ( 𝑠0|𝑋)+Co [ 𝜅𝑠2𝑡,Va ( 𝑠0∣
𝜽,𝑋)∣𝑋]. (70)
Unde s a iona i y o
𝑠𝑡
and a consis en eco e y p o ided by
𝑠𝑡
, bo h
Va ( 𝑠0∣𝑋)
and
Va ( 𝑠0∣
𝜽,𝑋)can be consis en ly es ima ed by

𝜎2𝑠 =1
𝑇𝑇
∑
𝑡=1(𝑠𝑡)2(71)
=1
𝑇𝑇
∑
𝑡=1{(Δln𝑆𝑡− 𝛼𝐿Δln 𝐿𝑡)2[𝑇−𝑡
∑
𝑙=0 𝜅𝑠2𝑙]}. (72)
Then, app oxima ing
𝜅𝑠2𝑡
using a second-o de expansion and applying he Del a me hod o
app oxima e he co a iance e m,

E[Va (𝑠𝑡∣
𝜽,𝑋)∣𝑋]=( 𝜅𝑠2𝑡+𝑡(2𝑡−1) 𝜅𝑠2(𝑡−1)⋅
Va ( 𝜅𝑠|𝑋))
𝜎2𝑠
+∇𝜅2𝑡( 𝜅𝑠)
Va ( 𝜅𝑠|𝑋)∇
𝜎2𝑠( 𝜅𝑠), (73)
whe e
∇𝜅2𝑡( 𝜅𝑠)=2𝑡⋅ 𝜅𝑠2𝑡−1 (74)
∇
𝜎2𝑠( 𝜅𝑠)=1
𝑇𝑇
∑
𝑡=1{(Δln𝑆𝑡− 𝛼𝐿Δln 𝐿𝑡)2[𝑇−𝑡
∑
𝑙=0𝑙⋅ 𝜅𝑠2𝑙−1]}. (75)
The o al a iance o he e ec i e R&D eco e y is hus quan i ied by

Va [𝑠𝑡∣𝑋]= 
Va (E[𝑠𝑡∣
𝜽,𝑋]∣𝑋)+ 
E[Va (𝑠𝑡∣
𝜽,𝑋)∣𝑋]. (76)
A.4 The cash- low-based asse p icing amewo k
Acco ding o he a gumen p esen ed in he main ex , Bansal e al. (2005) adop he simple
c oss-sec ional p icing condi ion
E𝑡[𝑅𝑖𝑡+1]−𝑅𝑓
𝑡=𝜆𝑥𝛽𝑖
𝑥(77)
as a easonable app oxima ion o he heo e ical long- un isk p icing equa ion. The p icing
equa ion in
(27)
hen ollows om applying he Campbell (1996) decomposi ion, which
exp esses unexpec ed e u ns as he sum o news abou u u e cash- low g ow h and discoun
47
a es:
ln𝑅𝑖𝑡+1−E𝑡[ln𝑅𝑖𝑡+1]≈𝛿𝑖
𝐷,𝑡+1−𝛿𝑖
𝑅,𝑡+1 (78)
whe e
𝛿𝑖
𝐷,𝑡 ={E𝑡−E𝑡−1}[∞
∑
𝑗=0 𝜅𝑗Δln𝐷𝑖𝑡+𝑗], 𝛿𝑖
𝑅,𝑡 ={E𝑡−E𝑡−1}[∞
∑
𝑗=1 𝜅𝑗ln𝑅𝑖𝑡+𝑗]. (79)
This decomposi ion implies ha any e u n be a can be app oxima ed as he di e ence
be ween a di idend be a, 𝛽𝑖
𝑥,𝐷, and a discoun - a e be a, 𝛽𝑖
𝑥,𝑅:27
𝛽𝑖
𝑥=Co [𝑅𝑖𝑡,𝜀𝑥,𝑡]
Va [𝜀𝑥,𝑡]≈Co [𝛿𝑖
𝐷,𝑡,𝜀𝑥,𝑡]
Va [𝜀𝑥,𝑡]−Co [𝛿𝑖
𝑅,𝑡,𝜀𝑥,𝑡]
Va [𝜀𝑥,𝑡]=𝛽𝑖
𝑥,𝐷−𝛽𝑖
𝑥,𝑅,(80)
and ocusing on he di idend be a alone isola es he componen o isk s emming om asse s’
undamen als, abs ac ing om ha a ising h ough he discoun - a e channel.
The be a es ima es om (28) a e asymp o ically equi alen o hose om
1
𝐻𝐻
∑
𝑙=1Δln 𝐷𝑖𝑡+𝑙 =
𝛽𝑖
0,𝐷+
𝛽𝑖
𝑥,𝐷𝜀𝑠,𝑡+ 𝑢𝑖𝑡+1.(81)
The la e o mula ion makes he in e p e a ion o he sensi i i y as he ‘long-las ing impac
on cash- low g ow h’ mo e explici , he eby be e cla i ying he mapping be ween he
sensi i i y es ima es and he heo e ical pa ame e
𝛽𝑖
𝑥,𝐷
om
(80)
. Howe e , as illus a ed
by Hod ick (1992), he o me o e s in e en ial ad an ages in small samples. The p emium
𝜆𝑥is hen es ima ed by eg essing he di idend-be as on he asse s’ e u ns.
B De ails on he da a
B.1 Mac oeconomic da a
This sec ion p esen s he mac oeconomic da a, showing he se ies and desc ip i e s a is ics.
27
The app oxima ion ollows om bo h
(78)
and
𝑅𝑖
𝑡≈1+ln 𝑅𝑖
𝑡
. See Campbell and Vuol eenaho (2004) o
a sys ema ic applica ion.
48
Figu e 7: Raw mac oeconomic da a.
𝑠
deno es he panel showing da a used o es ima e e ec i e R&D,
while BS and LN indica e he espec i e ac o se s.
Table 7: Desc ip i e s a is ics: e ec i e R&D da a. N. Obs. is he numbe o obse a ions.
𝑡𝑡
and
𝑡𝑡2
a e he ime end and squa ed ime end coe icien s (wi h HAC s anda d e o s); ADF and KPSS
a e s a iona i y es s in le els; AR(1) is he coe icien om an AR(1) i .
P i . R&D To . R&D Raw TFP Adj. TFP To . Empl. N.F. Empl.
𝑡𝑡 0.017*** 0.019** 0.018*** 0.018*** 0.018*** 0.017***
𝑡𝑡20.000 0.000 0.000*** 0.000 0.000*** 0.000***
AR(1) 1.000*** 1.000*** 1.000*** 1.000*** 1.000*** 1.000***
ADF −1.482 −2.027** −2.346** −2.891*** −1.353 −0.948
KPSS 2.004*** 1.924*** 1.976*** 1.978*** 2.001*** 2.011***
N. Obs. 314 314 314 314 310 310
* p <0.1, ** p <0.05, *** p <0.01
49
Table 8: Desc ip i e s a is ics: BS ac o s. N. Obs. is he numbe o obse a ions.
𝑡𝑡
and
𝑡𝑡2
a e he ime end and squa ed ime end coe icien s (wi h
HAC s anda d e o s); ADF and KPSS a e s a iona i y es s in le els; AR(1) is he coe icien om an AR(1) i .
CAPE 10Y yield 3M yield 3Y yield 5Y yield In . Vol Co p. P o i s N.F. Liq. Asse s
𝑡𝑡 −0.001 −0.004*−0.003 −0.004*−0.004*0.002 0.002 0.005*
𝑡𝑡20.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
AR(1) −0.334*** −0.053 −0.187*** −0.117*−0.102*−0.297*** 0.031 0.068
ADF −13.638*** −11.188*** −13.206*** −11.844*** −11.734*** −15.514*** −10.569*** −10.688***
KPSS 0.078*0.184*0.099*0.144*0.170*0.259*0.052*0.097*
N. Obs. 270 270 270 270 270 270 270 269
* p <0.1, ** p <0.05, *** p <0.01
Table 9: Desc ip i e s a is ics: LN ac o s. N. Obs. is he numbe o obse a ions.
𝑡𝑡
and
𝑡𝑡2
a e he ime end and squa ed ime end coe icien s (wi h
HAC s anda d e o s); ADF and KPSS a e s a iona i y es s in le els; AR(1) is he coe icien om an AR(1) i .
1 2 3 4 5 6 7 8 9
𝑡𝑡 0.005 0.005 0.002 −0.004 −0.001 0.006 −0.001 0.017*** 0.003
𝑡𝑡20.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000*** 0.000
AR(1) 0.725*** 0.360*** 0.749*** 0.426*** 0.703*** 0.620*** 0.138** 0.513*** 0.329***
ADF −6.467*** −7.252*** −4.577*** −6.929*** −4.911*** −5.758*** −10.204*** −7.415*** −9.526***
KPSS 0.324*0.514** 0.525** 0.373*0.224*0.278*0.190*0.603** 0.062*
N. Obs. 262 262 262 262 262 262 262 262 262
* p <0.1, ** p <0.05, *** p <0.01
50
B.2 Financial da a
Cash- low g ow h a es
Fi s , a measu e
ℎ𝑖,𝑡
o capi al gain is compu ed o each s ock by adjus ing CRSP ex-di idend
e u ns RETX o sha e epu chases as
ℎ𝑖,𝑡 =(𝑃𝑖,𝑡+1
𝑃𝑖,𝑡 )⋅min(𝑛𝑖,𝑡+1
𝑛𝑖,𝑡 ,1). (82)
Fo each po olio, hese s ock-le el measu es a e agg ega ed using ma ke -capi aliza ion
weigh s o ob ain a po olio capi al gain se ies
ℎ𝑝,𝑡
. F om each po olio se ies, he alue o one
dolla in es ed a he beginning o he sample is compu ed ecu si ely as
𝑉𝑝,𝑡+1 =ℎ𝑝,𝑡+1𝑉𝑝,𝑡
wi h
𝑉𝑝,0 =1
. Payou s a e hen gi en by
𝐷𝑝,𝑡+1 =𝑦𝑝,𝑡+1𝑉𝑝,𝑡
, whe e
𝑦𝑝,𝑡+1
is he po olio
di idend yield, ob ained om
𝑅𝑝,𝑡 =ℎ𝑝,𝑡+𝑦𝑝,𝑡
. Capi al gains a e less han p opo ional
o p ice app ecia ion when equi alen sha es ou s anding decline, ypically due o sha e
epu chases, which a e a o m o payou no eco ded in di idend da a. Qua e ly di idend
se ies a e ob ained by summing mon hly alues and de la ing hem wi h he implici p ice
de la o o nondu able and se ices consump ion, cons uc ed as in Bu eau o Economic
Analysis (2024). The se ies a e log- ans o med and, ollowing Bansal e al. (2005), de-
seasoned wi h a 4-qua e olling mean o emo e esidual seasonali y. Finally, cash- low
g ow h a es a e compu ed as i s di e ences o he log- ans o med, de-seasoned eal
qua e ly payou s.
Po olio Fo ma ion
To mi iga e liquidi y conce ns, he sample is es ic ed o common s ocks wi h ma ke
capi aliza ion abo e he 1s pe cen ile o he mon hly NYSE dis ibu ion and sha e p ices
abo e $2, emo ing less han 0.4% o o al ma ke capi aliza ion a any poin in ime.
Fi ms mus also ha e a leas wel e consecu i e mon hly obse a ions o en e he inal
sample. Po olio cons uc ion ollows s anda d me hodology: po olios a e o med a he
end o June, alue-weigh ed, and held un il he ollowing June. So ing a iables a e Ma ke
capi aliza ion (Size), Book- o-Ma ke (BM), Momen um (Mom), i m-speci ic R&D in ensi y
(RD), Tu no e (To), P o i abili y (P o ), Le e age (L g), Asse g ow h (AG), Tobin’s Q
(TQ), and Indus y (Ind). Mos a iables a e so ed using a 2
×
3 amewo k, whe e s ocks
a e i s spli by whe he ma ke capi aliza ion is abo e o below he NYSE median and
hen di ided in o e ciles wi hin each size g oup. Excep ions a e Size, BM, and Mom, which
ollow uni a ia e so s as in Bansal e al. (2005), and Indus y, which is di ec ly classi ied.
All po olios exhibi pa e ns consis en wi h es ablished li e a u e.
Size po olios All i ms a e assigned o quin iles based on ma ke capi aliza ion ela i e
o NYSE b eakpoin s. Bo h e u ns and cash- low g ow h dec ease wi h size.
51

BM po olios All non- inancial i ms (SIC ou side 6000–6999) a e assigned o quin iles
based on book equi y in iscal yea
𝑡−1
o ma ke capi aliza ion a end o calenda yea
𝑡−1
, ela i e o NYSE b eakpoin s. Bo h e u ns and cash- low g ow h inc ease wi h he
B/M a io.
Mom po olios All i ms a e assigned o quin iles based on cumula i e e u ns om
mon h 𝑡−12 o mon h 𝑡−1. Bo h e u ns and cash- lows inc ease wi h momen um.
RD po olios See Chan e al. (2001) and Lin (2011). All non- inancial and non-u ili y
i ms (SIC ou side 4000–4999, 6000–6999) a e anked by R&D expendi u es om he p e ious
iscal yea o ma ke capi aliza ion a end o calenda yea
𝑡−1
. Re u ns and cash- low
g ow h inc ease wi h wi h R&D in ensi y in bo h small and big po olio.
To po olios See Haugen and Bake (1996). All i ms a e assigned o quin iles based on
he sales- o-asse s a io om he p e ious iscal yea . Re u ns and cash- low g ow h inc ease
wi h u no e o bo h small and big po olios.
P o po olios See Fama and F ench (2015), Hou e al. (2015), and No y-Ma x (2013).
All non- inancial and non-u ili y i ms a e so ed on g oss p o i s o e asse s. Re u ns and
cash- low g ow h inc ease wi h p o i abili y o bo h small and big po olios, wi h s onge
e ec s among small po olios.
L g po olios See Bhanda i (1988). All non- inancial i ms a e so ed on deb - o-asse s
a io, used ins ead o deb - o-ma ke capi aliza ion o consis ency wi h co po a e inance
li e a u e (e.g. Rajan and Zingales 1995) and wi h o he so s ela ed o inancing capabili ies
in his s udy. Re u ns and cash- low g ow h inc ease wi h le e age among small po olios,
while cash- low g ow h dec eases among big po olios, wi h e u ns showing no clea pa e n.
AG po olios See Coope e al. (2008) and Hou e al. (2015). All non- inancial and
non-u ili y i ms a e so ed on asse g ow h ( i s di e ence o asse s o e lagged alue). Bo h
cash- low g ow h and e u ns dec ease wi h asse g ow h o small and big po olios.
TQ po olios See Hou e al. (2015). All i ms a e so ed on Tobin’s Q, de ined as (asse s
- book equi y + ma ke capi aliza ion) / asse s as in Chung and P ui (1994). Cash- low
g ow h and e u ns clea ly dec ease wi h Tobin’s Q o small po olios, wi h no e iden
pa e n o big po olios.
52
C Addi ional ables and igu es
C.1 Tables
This subsec ion p esen s supplemen a y ables on co ela ions, R&D measu es, and es asse
po olio s a is ics.
Table 10: Co ela ion among e o co ec ion e ms om he speci ica ions in Table 1.
Baseline 𝑆: To . R&D 𝑍: Raw TFP 𝑄: N.F. Empl.
𝑆: To . R&D 0.880 . . .
𝑍: Raw TFP 0.855 0.721 . .
𝑄: N.F. Empl. 0.997 0.858 0.859 .
Es . Me h.: IM 0.511 0.504 0.377 0.503
Table 11: Co ela ion among es ima es o 𝑠𝑡 om he speci ica ions in Table 2.
Speci ica ion Baseline 𝑆: To . R&D 𝑍: Raw TFP 𝑄: N.F. Empl.
BS LN BS LN BS LN BS LN
Baseline-LN 1.000 . . . . . . .
𝑆: To . R&D-BS 0.777 0.777 . . . . . .
𝑆: To . R&D-LN 0.806 0.806 0.996 .....
𝑍: Raw TFP-BS 0.858 0.859 0.640 0.641 ....
𝑍: Raw TFP-LN 0.853 0.853 0.644 0.636 0.994 ...
𝑄: N.F. Empl.-BS 0.998 0.998 0.752 0.781 0.868 0.862 . .
𝑄: N.F. Empl.-LN 0.998 0.998 0.752 0.781 0.868 0.861 1.000 .
Table 12: Co ela ion among es ima es o s uc u al shocks om he VAR speci ica ions in Table 3.
Baseline 𝑆: To . R&D 𝑍: Raw TFP 𝑄: N.F. Empl.
𝑆: To . R&D 0.690 . . .
𝑍: Raw TFP 0.915 0.630 . .
𝑄: N.F. Empl. 0.991 0.699 0.901 .
𝑠 0.926 0.646 0.837 0.929
* p <0.1, ** p <0.05, *** p <0.01
53
Table 13: Summa y s a is ics o he (upda ed) Kung and Schmid (2015) R&D in ensi y measu e.
Column 1 epo s yea ly R&D expendi u e (
𝑆
, NSF) and R&D s ock (
𝐼
, BLS) o 1963-2020 (yea ly
obse a ions om 1986 onwa d); columns 2-3 show da a om he baseline speci ica ions in he main
analysis.
𝜉
co esponds o commonly used labo -sha e alues, ollowing he o iginal pape .
𝑇
deno es
he numbe o obse a ions;
𝜎
, he s anda d de ia ion;
𝑡𝑡
and
𝑡𝑡2
, linea and quad a ic ime ends;
ADF and KPSS, espec i e s a iona i y es s; AR(1), he i s -o de au o eg essi e coe icien ; and
HL, he implied hal -li e based on AR(1).
𝑠𝑡:(ln𝑆𝑡−ln𝐼𝑡) (ln𝑆𝑡−1
𝜉ln𝑍𝑡)
1−𝜉:− 0.35 0.3
𝑇 62 314 314
𝜎 0.321 0.864 0.835
𝑡𝑡 −0.043 0.01 0.01
𝑡𝑡20.000 −0.00 −0.00
ADF −2.82∗3.80 3.72
KPSS 1.29∗∗∗ 0.82∗∗∗ 0.79∗∗∗
AR(1) 0.995∗∗∗ 1.000∗∗∗ 1.000∗∗∗
(0.063) (0.000) (0.000)
HL low 40.2 ∞ ∞
HL high ∞ ∞ ∞
∗∗∗𝑝<0.01,∗∗𝑝<0.05,∗𝑝<0.1
Table 14: G ange causali y F- es
𝑝
- alues om a VAR o e ec i e R&D (
𝑠
) and he agg ega ed
inno a ion measu e o Kogan e al. (2017) (KPSS), bo h included as s uc u al shocks o hogonal
o p oduc i i y g ow h, es ima ed om a i s -s ep bi a ia e VAR wi h p oduc i i y g ow h (as
in Table 3). VAR lags a e selec ed by Hannan–Quinn in o ma ion c i e ion (max 10);
𝑘
deno es
second-s ep VAR lags. Resul s a e shown using s anda d, HC, and HAC co a iance es ima o s.
Baseline 𝑆: To . R&D 𝑍: Raw TFP 𝑄: N.F. Empl. 𝑠
𝑘 1 1 2 1 1
N. Obs. 298 298 298 298 216
GC(𝑠) p. . 0.072 0.926 0.095 0.074 0.073
GC(KPSS) p. . 0.786 0.395 0.667 0.662 0.154
GC(𝑠) HC p. . 0.069 0.934 0.131 0.072 0.066
GC(KPSS) HC p. . 0.838 0.436 0.699 0.745 0.345
GC(𝑠) HAC p. . 0.011 0.921 0.013 0.008 0.031
GC(KPSS) HAC p. . 0.824 0.497 0.636 0.719 0.288
* p <0.1, ** p <0.05, *** p <0.01
54
Table 15: Desc ip i e s a is ics o he legacy pool es asse s used in Sec ion 3.3. Qua e ly e u ns
and cash- low g ow h a es a e epo ed om 1967 Q1 o 2022 Q1; summa y s a is ics include mean
and s anda d de ia ion.
Po olio CF G ow h Mean CF G ow h SD Re u ns Mean Re u ns SD
Size(1) 134.68 604.09 2.47 13.20
Size(2) 68.34 373.78 2.31 11.83
Size(3) 36.33 193.21 2.18 10.77
Size(4) 29.38 161.65 2.14 10.06
Size(5) 6.38 36.26 1.59 8.34
BM(1) 3.09 16.54 1.72 9.76
BM(2) 2.31 21.74 1.78 8.86
BM(3) 1.49 17.17 1.67 8.22
BM(4) 3.47 36.58 1.96 8.65
BM(5) 8.70 69.61 2.41 9.37
Mom(1) 1.11 20.96 1.47 12.61
Mom(2) 7.44 74.50 1.75 9.31
Mom(3) 8.34 58.26 1.78 8.47
Mom(4) 17.24 107.76 1.85 8.77
Mom(5) 34.09 280.68 1.98 11.66
55
Figu e 11:
𝑠𝑡
om all speci ica ions es ed. Bands show 95% con idence in e als assuming no mali y, wi h a iance compu ed as in
(64)
. Shaded a eas
indica e NBER ecessions.
62

Figu e 12: S uc u al shocks om VAR es ima ions; selec ed esul s a e epo ed in Table 3. Shaded a eas indica e NBER ecessions.
63
Figu e 13: P incipal componen sc ee plo s o he es asse s in Sec ion 3.3. The le y-axis shows
he a iance explained by each ac o (solid line), and he igh y-axis shows he cumula i e a iance
explained (do ed line). Ve ical g een lines indica e he op imal numbe o ac o s acco ding o Alessi
e al. (2010), while he e ical blue line ma ks he minimum op imal numbe o ac o s as in Bai and
Ng (2002). The ho izon al ed line co esponds o he ecip ocal o he numbe o es asse s.
0
5
10
15
20
0
25
50
75
100
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Componen (s)
%
Cumula i e %
Cumula i e Va iance Fac o Va iance
Figu e 14: C oss-sec ional a e age e u ns: a iance explained by he p incipal componen s used as
isk ac o s in Sec ion 3.3. The le y-axis shows he a iance explained by each ac o (solid line), and
he igh y-axis shows he cumula i e a iance explained (do ed line). Ve ical g een lines indica e
he op imal numbe o ac o s acco ding o Alessi e al. (2010), while he e ical blue line ma ks he
minimum op imal numbe o ac o s as in Bai and Ng (2002). The ho izon al ed line co esponds o
he ecip ocal o he numbe o es asse s.
64
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