Ridge regularized estimation of VAR models for inference

Balla in, Gio anni
A icle — Published Ve sion
Ridge egula ized es ima ion o VAR models o in e ence
Jou nal o Time Se ies Analysis
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Sugges ed Ci a ion: Balla in, Gio anni (2024) : Ridge egula ized es ima ion o VAR models o
in e ence, Jou nal o Time Se ies Analysis, ISSN 1467-9892, John Wiley & Sons, L d, Ox o d, UK, Vol.
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h ps://doi.o g/10.1111/j sa.12737
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JOURNAL OF TIME SERIES ANALYSIS
J. Time Se . Anal. 46: 235–257 (2025)
Published online 18 Feb ua y 2024 in Wiley Online Lib a y
(wileyonlinelib a y.com) DOI: 10.1111/j sa.12737
SPECIAL ISSUE ARTICLE
RIDGE REGULARIZED ESTIMATION OF VAR MODELS FOR INFERENCE
GIOVANNI BALLARIN
Depa men o Economics, Uni e si y o Mannheim, Mannheim, Ge many
Ridge eg ession is a popula me hod o dense leas squa es egula iza ion. In his a icle, idge eg ession is s udied in he
con ex o VAR model es ima ion and in e ence. The implica ions o aniso opic penaliza ion a e discussed, and a compa ison
is made wi h Bayesian idge- ype es ima o s. The asymp o ic dis ibu ion and he p ope ies o c oss- alida ion echniques a e
analyzed. Finally, he es ima ion o impulse esponse unc ions is e alua ed wi h Mon e Ca lo simula ions and idge eg ession
is compa ed wi h a numbe o simila and compe ing me hods.
Recei ed 17 July 2023; Accep ed 30 Janua y 2024
Keywo ds: Impulse esponses; in e ence; idge egula iza ion; ec o au o eg ession.
JEL. C32; C51; C52.
MOS subjec classi ica ion: 60G10; 60G25; 62F12; 62J07; 62M10; 62M20; 62P20.
1. INTRODUCTION
While he idea o using idge eg ession o ec o au o eg essi e model es ima ion da es back o Hamil on (1994),
he e seems o be no comple e analysis o i s p ope ies and asymp o ic heo y in he li e a u e. This a icle ills his
gap by analyzing he geome ic and dis ibu ional p ope ies o idge in a VAR es ima ion amewo k, discussing
i s compa ison o well-known Bayesian app oaches and de i ing he alidi y o c oss- alida ion as a selec ion
p ocedu e o he idge penal y.
Fi s , I show ha he sh inkage induced by he idge es ima o , while in ui i e in he se ing o an iso opic
penal y, p oduces complex e ec s when es ima ing a VAR model wi h a mo e lexible penaliza ion scheme. This
implies ha he bene i s o he bias- a iance ade-o (Has ie, 2020) may be ha d o gauge a p io i. I p o ide
a ac able example whe e idge can yield es ima es ha ha e highe au o eg essi e dependence han he leas
squa es solu ion. To be e unde s and how di e en idge penaliza ion s a egies can be designed, I also make a
compa ison wi h Bayesian VAR es ima o s commonly used in mac oeconome ic p ac ice.
Second, I gene alize he analysis o Fu and Knigh (2000) and p o e he consis ency and asymp o ic no mali y
o he idge es ima o , a esul ha seems o be missing in he li e a u e. Fo s anda d in e ence, he idge penal y
should ei he be negligible in he limi o i s cen e ing con e ge in p obabili y o he ue pa ame e ec o . In bo h
hese cases, he e is no asymp o ic bias and no educ ion in a iance. Al e na i ely, in se ings whe e a esea che
is willing o assume ha a subse o he VAR pa ame e s ea u es small coe icien s, one can achie e an asymp o ic
educ ion o a iance by co ec ly uning he idge penal y ma ix. I u he de i e he p ope ies o c oss- alida ion,
which is a popula app oach in p ac ical applica ions o une penalized es ima o s (Has ie e al.,2009; Be gmei
e al.,2018). Mo e speci ically, I show ha c oss- alida ion is able o selec penal ies ha a e asymp o ically alid
∗Co espondence o: Gio anni Balla in, Depa men o Economics, Uni e si y o Mannheim, L7, 3-5, Mannheim, 68131, Ge many.
Email: gio [email p o ec ed]
© 2024 The Au ho s. Jou nal o Time Se ies Analysis published by John Wiley & Sons L d.
This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion-NonComme cial License, which pe mi s use,
dis ibu ion and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed and is no used o comme cial pu poses.
236 G. BALLARIN
o in e ence. In passing, I also p o e ha , in an au o eg essi e se up, he ime dependence o eg esso s has an
exponen ially small e ec on in-sample p edic ion e o e alua ion.
Las ly, I use Mon e Ca lo simula ions o s udy he pe o mance o he di e en idge app oaches discussed,
ocusing on impulse esponse in e ence. I conside wo exe cises: one is based on a h ee- a iable VARMA(1,1)
da a gene a ing p ocess om Kilian and Kim (2011); he o he is a VAR(5) model es ima ed in le els om a se o
se en mac oeconomic se ies, ollowing Giannone e al. (2015). The inding is ha idge can lead o imp o emen s
o e un egula ized me hods in impulse esponse con idence in e al leng h, while Bayesian es ima o s show he
bes o e all pe o mance due o he unde lying lexibili y o hei p io s.
1.1. Rela ed Li e a u e
This a icle does no discuss he high-dimensional se ing, whe e he numbe o eg esso s g ows oge he wi h he
sample size. Some impo an wo k has been done in his di ec ion al eady. Dob iban and Wage (2018)de i ean
explici exp ession o he p edic i e isk o idge eg ession assuming a high-dimensional andom e ec s model.
O he wo ks in his ein a e Liu and Dob iban (2020); Pa il e al. (2021) and Has ie e al. (2022), which a e mos ly
ocused on penal y selec ion by c oss- alida ion, as well as s uc u al ea u es o idge. Gene ally speaking, he
complexi y o analyzing idge eg ession in high dimensions is a challenge o p ecisely unde s anding i s p ac ical
implica ions. As I show below, in he con ex o ini e-dimensional VARs, asymp o ic in e ence demands ha
he idge penal y becomes asymp o ically negligible a app op ia e a es. Thus, a challenge is unde s anding in
wha way high-dimensional ime se ies p oblems would bene i om he use o idge. This ques ion is beyond
he scope o his a icle.
In he ime se ies o ecas ing li e a u e, idge eg ession is commonly used o p edic ion. I p o ide a pa ial lis
o con ibu ions in his di ec ion. Inoue and Kilian (2008) use idge egula iza ion o o ecas ing U.S. consume
p ice in la ion and a gue ha i compa es a o ably wi h bagging echniques; De Mol e al. (2008) use a Bayesian
VAR wi h pos e io mean equi alen o a idge eg ession in o ecas ing; Ghosh e al. (2019) again s udy he
Bayesian idge, his ime, howe e , in he high-dimensional con ex ; Goule Coulombe e al. (2022), Fuleky (2020),
Babii e al. (2021), and Medei os e al. (2021) compa e LASSO, idge and o he machine lea ning echniques
o o ecas ing wi h la ge economic da ase s. Fuleky (2020) gi es a ex book ea men o penalized ime se ies
es ima ion, including idge, bu does no discuss in e ence. The idge penal y is conside ed wi hin a mo e gene al
mixed 𝓁1-𝓁2penaliza ion se ing in Smeekes and Wijle (2018), who s udy he pe o mance and obus ness o
penalized es ima es o cons uc ing o ecas s.
Rega ding in e ence, Li e al. (2024) p o ided a gene al explo a ion o sh inkage p ocedu es in he con ex
o s uc u al impulse esponse es ima ion. Ve y ecen ly, Ca alie e e al. (2023) sugges ed a me hodology o
in e ence on idge- ype es ima o s ha elies on boo s apping. Finally, sh inkage o au o eg essi e models o
cons ained submodels was discussed by Hansen (2016b) in a mo e gene al se ing.
Finally, a ious es ima ion p oblems can ei he be cas as o augmen ed wi h idge- ype eg essions. Goule
Coulombe (2023) shows ha he es ima ion o VARs wi h ime- a ying pa ame e s can be w i en as idge eg es-
sion. Plagbo g-Mølle (2016) and Ba nichon and B ownlees (2019) bo h use idge o de i e smoo hed local
p ojec ion impulse esponse unc ions.
1.2. Ou line
Sec ion 2p o ides a discussion o he idge penal y and he idge VAR es ima o . In Sec ion 3I deal wi h he p op-
e ies o idge-induced sh inkage in he au o eg essi e coe icien s. I discuss he connec ions be ween equen is
and Bayesian idge o VAR models wi hin Sec ion 4. Sec ion 5de elops he asymp o ic heo y and in e ence
esul in he case whe e he e is no asymp o ic sh inkage. This includes s udying he p ope y o c oss- alida ion
unde dependence. Sec ion 6p o ides in e ence and CV esul s in a se ing whe e some sh inkage o a subse
o pa ame e s is possible. Sec ion 7p esen s Mon e Ca lo simula ions ocused on impulse esponse es ima ion.
Sec ion 8concludes. Finally, he Da a S1 Supplemen a y Appendix con ains mo e de ailed de i a ions, as well as
all p oo s, addi ional ables and u he in o ma ion on simula ions.
wileyonlinelib a y.com/jou nal/j sa © 2024 The Au ho s. J. Time Se . Anal. 46: 235–257 (2025)
Jou nal o Time Se ies Analysis published by John Wiley & Sons L d. DOI: 10.1111/j sa.12737
RIDGE REGULARIZED ESTIMATION OF VAR MODELS 237
1.3. No a ion
De ine R+ o be he se o s ic ly posi i e eal numbe s. Vec o s ∈RNand ma ices A∈RN×Ma e always
deno ed wi h lowe and uppe case le e s espec i ely. Th oughou , I will use IM o ep esen he iden i y ma ix
o dimension M. Fo any ec o ∈RN,‖ ‖is he Euclidean no m. Fo any ma ix A∈RN×M,‖A‖is he spec al
no m unless s a ed o he wise; ‖A‖max =maxi,j|||aij|||is he maximal en y no m; ‖A‖F=( {A′A})−1∕2is he
F obenius no m; ec(⋅)is he ec o iza ion ope a o and ⊗is he K onecke p oduc (Lü kepohl, 2005). I a ec o
ep esen s a ec o ized ma ix, hen i will be w i en in bold, ha is, o A∈RN×MI w i e ec(A)=a∈RNM.
Le Λ=diag{𝜆1,…,𝜆K2p},𝜆i>0 o alli=1,…,K2p. To gi e he pa ial o de ing o diagonal posi i e
semi-de ini e penaliza ion ma ices, le Λ1=diag{𝜆1,j}K2p
j=1and Λ2=diag{𝜆2,j}K2p
j=1. I w i e Λ1≺Λ2i 𝜆1,i<𝜆
2,i
o all i=1,…,K2p;Λ1⪯Λ
2i 𝜆1,i≤𝜆2,i o all iand ∃j∈1,…,K2psuch ha 𝜆1,j<𝜆
2,j. Symbols P
−−→and
d
−−→a e used o indica e con e gence in p obabili y and con e gence in dis ibu ion espec i ely.
2. RIDGE REGULARIZED VAR ESTIMATION
Le y =(y1 ,…,yK )′be a K-dimensional ec o au o eg essi e p ocess wi h lag leng h p≥1 and pa ame iza ion
y =𝜈 +A1y −1+A2y −2+···+Apy −p+u ,(1)
whe e u =(u1 ,…,uK )′is addi i e noise such ha u a e i.i.d. wi h E[ui ]=0andVa [u ]=Σ
u,and𝜈
is a de e minis ic end. Fo simplici y, in he emainde I shall assume ha 𝜈 =0so ha y has no end
componen – equi alen ly, y is a de ended se ies.
Fo a gi en sample size Tde ine Y=(y1,…,yT)∈RK×T,z =(y′
,y′
−1,…,y′
−p+1)′∈RKp,Z=
(z0,…,zT−1)∈RKp×T,B=(A1,…,Ap)∈RK×Kp,U=(u1,…,uT)∈RK×T, and ec o ized coun e -
pa s y= ec(Y),𝜷= ec(B)and u= ec(U). Acco dingly, Y=BZ +Uand y=(Z′⊗IK)𝜷+u,
whe e Σu=IK⊗Σu. Impo an ly, h oughou his a icle, I will assume ha he c oss-sec ional dimension K
emains ixed.
Ridge egula iza ion is a modi ica ion o he leas squa es objec i e by he addi ion o a e m dependen on
he Euclidean no m o he coe icien ec o . The iso opic Ridge- egula ized Leas Squa es (RLS) es ima o is
he e o e de ined as

𝜷R(𝜆)∶=a g min
𝜷
1
T‖‖y−(Z′⊗IK)𝜷‖‖2+𝜆‖𝜷‖2,
whe e 𝜆>0 is he scala egula iza ion pa ame e o egula ize . When 𝜆‖𝜷‖2is eplaced wi h quad a ic o m
𝜷′Λ𝜷 o a posi i e de ini e ma ix Λ he abo e is o en e med Tikhono egula iza ion. To a oid con usion, I
shall also e e o i as ‘ idge’, since in wha ollows Λwill always be assumed diagonal. As Λdoes no , in gene al,
penalize all coe icien s equally, i can be used o cons uc an aniso opic idge es ima o . By sol ing he no mal
equa ions (see Supplemen a y Appendix A.1), he RLS es ima o wi h posi i e semi-de ini e egula iza ion ma ix
Λ∈RK2p×K2pisshown obe

𝜷R(Λ) = (ZZ′
T⊗IK+Λ
)−1(Z⊗IK)y
T.
When a cen e ing ec o 𝜷0≠0isincludedinpenal y(𝜷−𝜷0)′Λ(𝜷−𝜷0), he RLS es ima o becomes

𝜷R(Λ,𝜷0)=(ZZ′
T⊗IK+Λ
)−1((Z⊗IK)y
T+Λ𝜷0).(2)
J. Time Se . Anal. 46: 235–257 (2025) © 2024 The Au ho s. wileyonlinelib a y.com/jou nal/j sa
DOI: 10.1111/j sa.12737 Jou nal o Time Se ies Analysis published by John Wiley & Sons L d.
238 G. BALLARIN
In he con ex o mul i- a ia e es ima ion, one has o make a u he dis inc ion be ween wo ela ed ypes o
idge es ima o s. I le 
BR(Λ,𝜷0)be he de- ec o ized coe icien es ima o ob ained om eshaping 
𝜷R(Λ,𝜷0) o a
K×Kp ma ix. Bu one can also conside he ma ix RLS es ima o 
BR
ma (ΛKp,B0)gi en by

BR
ma (ΛKp,B0)=T−1(Y+B0ΛKp)Z′(T−1ZZ′+Λ
Kp)−1,
whe e ΛKp =diag{𝜆1,…,𝜆Kp}and B0is a cen e ing ma ix. The dis inc ion is impo an because he ec o ized
and ma ix RLS es ima o s in gene al need no coincide. As discussed in Supplemen a y Appendix A.2,
BR(Λ,𝜷0)
allows o mo e gene al penal y s uc u es compa ed o 
BR
ma (ΛKp,B0). I, he e o e, ocus on he o me a he han
he la e .
Rema k 1. Equa ion (2) implies ha 
𝜷R(Λ,𝜷0)and, he e o e, 
𝜷R(Λ) p o ide simul aneous es ima ion o all
he coe icien s in 𝜷. Howe e , by analogy wi h o dina y leas squa es (OLS) VAR es ima ion, one may also
conside an equa ion-by-equa ion idge eg ession (ebe-RLS) scheme. Fo k=1,…,K,le yk=(Z′⊗
IK)𝜷k+ukbe he au o eg essi e equa ion o he k h se ies o y . Then, we can de ine he k h equa ion RLS
es ima o o be

𝜷R
k(Λ,𝜷0)=(ZZ′
T⊗IK+Λ
k)−1((Z⊗IK)yk
T+Λ𝜷0k),
whe e 0 ⪯Λ
kand 𝜷0ka e he k h equa ion egula ize and cen e ing espec i ely. No ice ha he ebe-RLS
app oach allows, by cons uc ion, o penalize he es ima es o one componen di e en ly han o ano he , and
he wo can be independen ly chosen. This p o ides a highe deg ee o eedom han he one a o ded by, o
example, he aniso opic lag-adap ed scheme p oposed in Sec ion 3.2 o he Bayesian schemes o Sec ion 4.
Howe e , implemen ing ebe-RLS in applica ions inhe en ly implies ha da a-d i en uning o Λkwill be signi i-
can ly mo e compu a ionally in ensi e – wi h cos s g owing linea ly in K. Due o his highe complexi y, in bo h
heo e ical de i a ion and simula ions below, I will ocus on s udying he p ope ies o he simul aneous RLS
es ima o .
Rema k 2. Fu he ega ding ebe-RLS, ano he way o app oach es ima ion is h ough he ecu si e o m o he
VAR model. Le Σu=P−1DP−1′,whe eP−1is an uni iangula ma ix and Da diagonal ma ix, so ha we may
w i e
Y=GZ −
PY +D−1∕2E,
whe e G=PB,
P=P−IKand noise e m Ehas iden i y co a iance ma ix. Es ima ion can now be pe o med in
ebe-RLS ashion, and ma ices P,Band Da e eco e ed (Hausman, 1983). No ice, howe e , ha in his ame-
wo k he o de ing o a iables plays a ole, since i also de e mines he decomposi ion o Σu. Indeed, e en i a
penaliza ion scheme is ixed, pe mu ing he en ies may yield di e en penalized es ima es o P,Band D,so ha
bo h slope and co a iance pa ame e es ima es a e di e en , implying (s uc u al) IRF es ima es will also di e .
Howe e , no e ha his issue is somewha mi o ed in a ecu si e shock iden i ica ion app oach: a e es ima ion,

Σuis Cholesky decomposed o iden i y he shocks’ o a ion, and he o de ing o a iables is key and mus be
economically jus i ied.
3. SHRINKAGE
He e, I discuss bo h he iso opic idge penal y, i.e., he ‘s anda d’ idge app oach, and an aniso opic penal y
ha is be e adap ed o he VAR se ing. An impo an esul is ha , e en in simple se ups wi h only wo a i-
ables, he sh inkage induced by idge can ei he inc ease o educe bias, as well as he s abili y o au o eg essi e
es ima es.
wileyonlinelib a y.com/jou nal/j sa © 2024 The Au ho s. J. Time Se . Anal. 46: 235–257 (2025)
Jou nal o Time Se ies Analysis published by John Wiley & Sons L d. DOI: 10.1111/j sa.12737

RIDGE REGULARIZED ESTIMATION OF VAR MODELS 239
Th oughou his sec ion, I conside ixed design ma ices and he ocus will be on he geome ic p ope ies o
idge.
3.1. Iso opic Penal y
The mos common way o pe o m a idge eg ession is h ough iso opic egula iza ion, ha is, Λ=𝜆I o
some scala 𝜆≥0. Iso opic idge has been ex ensi ely s udied, see o example, he comp ehensi e e iew o
Has ie (2020). Wi h ega d o sh inkage, an iso opic idge penal y can be eadily s udied.
P oposi ion 1. Le Z∈RM×T,Y∈RT o T>Mbe eg ession ma ices. Fo 𝜆•>𝜆>0 and iso opic RLS
es ima o 
𝛽R(𝜆)∶=(T−1ZZ′+𝜆IM)−1(T−1ZY)i holds
‖‖‖
𝛽R(𝜆•)‖‖‖<‖‖‖
𝛽R(𝜆)‖‖‖.
P oo . Using he ull singula - alue decomposi ion (SVD), decompose T−1∕2Z=UDV′∈RM×Twhe e Uis
M×Mo hogonal, Dis M×Tdiagonal and Vis T×To hogonal. W i e

𝛽R(𝜆•)=(T−1ZZ′+𝜆•IM)−1(T−1ZY)
=(UDV′VDU′+𝜆•IM)−1UDV′(T−1∕2Y)
=U(D2+𝜆•IM)−1DV′(T−1∕2Y)
=U(D2+𝜆•IM)−1(D2+𝜆IM)(D2+𝜆IM)−1DV′(T−1∕2Y)
=[U(D2+𝜆•IM)−1(D2+𝜆IM)U′]
𝛽R(𝜆).
Since D2=diag{𝜎2
j}M
j=1, he e m wi hin b acke s is Udiag{(𝜎2
j+𝜆)∕(𝜎2
j+𝜆•)}M
j=1U′. Mo eo e , because he
spec al no m is uni a y in a ian and 𝜆1>𝜆
2, i ollows ha
‖‖U(D2+𝜆•IM)−1(D2+𝜆IM)U′‖‖=‖‖‖diag{(𝜎2
j+𝜆)∕(𝜎2
j+𝜆•)}M
j=1‖‖‖<1.
Finally, by he sub-mul iplica i e p ope y i holds
‖‖‖
𝛽R(𝜆•)‖‖‖≤‖‖U(D2+𝜆1IM)−1(D2+𝜆IM)U′‖‖⋅‖‖‖
𝛽R(𝜆)‖‖‖<‖‖‖
𝛽R(𝜆)‖‖‖,
as claimed. ◾
P oposi ion 1and i s p oo expose he main ing edien s o idge eg ession. F om he SVD o T−1∕2Zused abo e,
i is clea ha idge egula iza ion ac s uni o mly along he o hogonal di ec ions ha a e he columns o V.The
imp o emen in he condi ioning o he in e se comes om all diagonal ac o s [(D2+𝜆•IM)−1D]j=𝜎j∕(𝜎2
j+𝜆•)
being well-de ined, e en when 𝜎j=0 (as is he case in sys ems wi h collinea eg esso s).
Howe e , di ec ly applying iso opic idge o ec o au o eg essi e models is no necessa ily he mos e ec i e
es ima ion app oach. S able VAR models show decay in he absolu e size o coe icien s o e lags. Thus, i is
easonable o choose a mo e gene al idge penal y ha can accommoda e lag decay.
3.2. Lag-Adap ed Penal y
I now conside a di e en o m o Λ ha is o in e es when applying idge speci ically o a VAR model. De ine
amily (p)o lag-adap ed idge penal y ma ices as
(p)={diag{𝜆1,…,𝜆p}⊗IK2|𝜆i∈R+,i=1,…,p},
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240 G. BALLARIN
whe e each 𝜆iin ui i ely implies a di e en penal y o he elemen s o each coe icien ma ix Ai,i=1,…,p.1
The amily (p)allows imposing a idge penal y ha is cohe en wi h he lag dimension o an au o eg essi e model.
I is pa ame ized by pdis inc penal y ac o s, meaning ha he penaliza ion is aniso opic.
P oposi ion 2. Le Z∈RKp×T,y∈RKT o T>Kp be mul i- a ia e VAR eg ession ma ices. Gi en subse
⊆{1,…,p}o ca dinali y s=||, o Λ(p)∈(p)de ine 
𝜷R(Λ(p))[]as he ec o o sK2coe icien es ima es
loca ed a indexes 1 +K2(j−1),…,K2j o j∈.Le c={1,…,p}⧵be he complemen o .
a. I 𝜆1≥𝜆2, hen‖‖‖
𝜷R(𝜆1IK2p)[]‖‖‖≤‖‖‖
𝜷R(𝜆2IK2p)[]‖‖‖ o any ⊂{1,…,K2p}. The inequali y is s ic when
𝜆1>𝜆
2.
b. Le 
𝜷LS
[]be he leas squa es es ima o o he au o eg essi e model wi h only he lags indexed by included
and ze os as coe icien s o he lags indexed by c. Simila ly, le Λ(p)
[]be he subse o diagonal elemen s in
Λ(p)penalizing he lags in .Then
lim
Λ(p)
[]→0
Λ(p)
[c]→∞

𝜷R(Λ(p))=
𝜷LS
[],
whe e Λ(p)
[]→0andΛ(p)
[c]→∞a e o be in ended as he elemen -wise con e gence.
P oposi ion 2shows ha he limi ing geome y o a lag-adap ed idge es ima o is hus iden ical o ha o a
leas squa es eg ession un on he subse speci ied by . By con olling he size o coe icien s {𝜆1,…,𝜆p}
i is he e o e possible o ob ain pseudo-model-selec ion. Howe e , in he nex sec ion, I show ha aniso opic
penaliza ion p oduces complex e ec s on he model’s coe icien es ima es.
3.3. Illus a ion o Aniso opic Penaliza ion
He e, I aim o illus a e he e ec s o a lag-adap ed idge penal y on VAR coe icien s es ima es using a pa icula
example. This u he helps mo i a e and con ex ualize he esul s o he simula ion exe cises p o ided in Sec ion 7.
Mo e gene ally, be o e mo ing on o he discussion o mo e sophis ica ed o ms o idge eg ession, i is impo an
o gain some in ui ion eg ading he p ope ies o aniso opic penaliza ion, which I highligh wi h he help o a
simple bi a ia e VAR(2) model.
No e ha , since idge ope a es along p incipal componen s, he e is no immedia e ela ionship be ween a speci ic
subse o he es ima ed coe icien s and a gi en diagonal block o Λ(p). Wi h ega d o au o eg essi e modeling,
h ee e ec s a e o in e es : he sh inkage o coe icien ma ices Ai ela i e o he choice o 𝜆i; he en i y o he
bias in oduced by sh inkage, and he impac o sh inkage on he pe sis ence o he es ima ed model.
To showcase hese e ec s, I conside he VAR(2) model
y =A1y −1+A2y −2+u ,u ∼i.i.d. (0,Σu),
whe e
A1=[0.80.1
−0.10.7],A2=[0.1−0.2
−0.10.1],Σu=[0.30
05
].
1No e ha wi h a lag-adap ed penal y i is also possible o di ec ly use he ma ix idge es ima o since he penal y o 
𝜷Ris gi en by
diag{𝜆1,…,𝜆p}⊗IK2=(diag{𝜆1,…,𝜆p}⊗IK)⊗IK, see Supplemen a y Appendix A.2. Impo an ly, his kind o s uc u e is minimal in
e ms o modeling he ela i e size o coe icien s wi hin each coe icien ma ix Ai. I economic heo y o in ui ion p o ides in o ma ion abou
he e ec s o one speci ic a iable and lag on ano he – say, he con empo aneous e ec o he i s se ies on he second se ies is ze o – mo e
s uc u e can be in eg a ed in o he idge penal y ma ix. This would mean, howe e , ha di e en idge es ima o o ms a e no equi alen .
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RIDGE REGULARIZED ESTIMATION OF VAR MODELS 241
Figu e 1. Sh inkage o coe icien s es ima e in F obenius no m (a); bias induced by sh inkage (b); change in s abili y
o es ima ed VAR model a di e en le els o penaliza ion, measu ed by he absolu e alue o he la ges companion
o m eigen alue (c)
A single sample o leng h T=200 is d awn, demeaned and used o es ima e coe icien s A1and A2. The VAR(2)
model is i ed using he lag-adap ed idge es ima o 
BR(Λ(2)),whe eΛ(2)=diag{𝜆1,𝜆2}⊗I2. No e ha 
BR(Λ(2))
can be pa i ioned in o es ima es ÂR
1(Λ(2))and ÂR
2(Λ(2)) o he espec i e pa ame e ma ices.
3.3.1. Sh inkage
To illus a e sh inkage, I conside he es ic ed case o 𝜆1∈[10−2,106]and 𝜆2=0. The idge es ima o is
compu ed o a ying 𝜆1o e a loga i hmically spaced g id. Figu e 1(a) shows ha ‖‖‖
BR(Λ(2))‖‖‖F≈‖‖‖
BLS‖‖‖F o
𝜆1≈0, bu as he penal y inc eases ‖‖‖ÂR
1(Λ(2))‖‖‖Fdec eases while ‖‖‖ÂR
2(Λ(2))‖‖‖Fg ows. The esul ing beha io o
‖‖‖
BR(Λ(2))‖‖‖Fis non-mono onic in 𝜆1, al hough indeed ‖‖‖
BR(Λ(2))‖‖‖F<‖‖‖
BLS‖‖‖Fin he limi 𝜆1→∞. This e ec is
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242 G. BALLARIN
due o he model selec ion p ope ies o lag-adap ed idge, and he esul ing omi ed a iable bias. The e o e, in
p ac ice, i is no gene ally ue ha aniso opic idge induces mono onic sh inkage o es ima es.
3.4. Bias
Since idge bias is ha d o s udy heo e ically, I use a simula ion wi h he same se up o Figu e 1(a), his ime wi h
𝜆1,𝜆2∈[10−2,104]. The g id is loga i hmic wi h 150 poin s. Figu e 1(b) p esen s a le el plo o he sup-no m
idge bias ‖‖‖
BR(Λ(2))−B‖‖‖∞gi en mul iple combina ions o 𝜆1and 𝜆2. While he e can be gains compa ed o he
leas squa es es ima o 
BLS, hey a e modes . Mo eo e , le el cu es o he bias su ace show ha gains concen a e
in a e y hin egion o he pa ame e space. Consequen ly, one may imagine ha , in p ac ice, any (da a-d i en)
idge penal y selec ion c i e ion is unlikely o yield bias imp o emen o e leas squa es. Ye , in la ge VAR models
wi h many lags, he educ ion in a iance o he idge es ima o o en yields imp o emen s o e un- egula ized
p ocedu es (Li e al.,2024). Howe e , he bias- a iance ade-o in idge is no a ee-lunch when pe o ming
in e ence. P a (1961) showed ha i is no possible o p oduce a es (equi alen ly, a CI p ocedu e) which is alid
uni o mly o e he pa ame e space and yields meaning ully smalle con idence in e als han any o he alid
me hod.
3.5. S abili y
To s udy he s abili y o idge VAR es ima es, I euse he esul s o he bias simula ion abo e. Le Abe he com-
panion ma ix o [A1,A2],and 
AR he companion ma ix o es ima es [ÂR
1(Λ(2)),ÂR
2(Λ(2))]. Fo all combina ions
(𝜆1,𝜆2), I compu e he la ges eigen alue 𝜔1(
AR)o 
AR. No e ha i |𝜔1(
A)|<1, hen he es ima ed VAR(2) is
s able (Lü kepohl, 2005). Figu e 1(c) p esen s he le el se s o he su ace o maximal eigen alue moduli, and o
compa ison |𝜔1(
BLS)|is shown a he o igin.2While along he main diagonal he e is a clea dec ease in |𝜔1(
AR)|
as iso opic penaliza ion inc eases, when 𝜆1is la ge and 𝜆2≪1 (o ice e sa) he maximal eigen alue inc eases
ins ead. The e o e, an es ima e o a VAR model ob ained wi h aniso opic idge may be close o uni oo han
he leas squa es es ima e.
4. BAYESIAN AND FREQUENTIST RIDGE
So a , I ha e discussed s anda d idge penaliza ion schemes. He e, I s udy he pos e io mean o Bayesian VAR
(BVAR) p io s commonly applied in he mac oeconome ics li e a u e. I show ha such pos e io s a e in ac
speci ic GLS o mula ions o he idge es ima o . This compa ison highligh s ha idge can be seen as a way o
embed p io knowledge in o he leas squa es es ima ion p ocedu e by means o cen e ing and escaling coe icien
es ima es.
4.1. Li e man–Minneso a P io s
In Bayesian ime se ies modeling, he so-called Minneso a o Li e man p io has ound g ea success
(Li e man, 1986). Fo s a iona y p ocesses which one belie es o ha e easonably small dependence, a ze o-mean
no mal p io can be pu on he VAR pa ame e s, wi h non-ze o p io a iance. Assuming ha he co a iance ma ix
o e o s Σuis known, he Li e man–Minneso a has pos e io mean
𝜷|Σu=[V−1
𝜷+(ZZ′⊗Σ−1
u)]−1(Z⊗Σ−1
u)y,(3)
2I Λ(2)→0, hen by con inui y o eigen alues i ollows ha |𝜔1(
AR)|→|𝜔1(
ALS)|, see Supplemen a y Appendix A.3.
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RIDGE REGULARIZED ESTIMATION OF VAR MODELS 249
(i) 𝜷=(𝜷′
1,𝜷′
2)′whe e 𝜷1∈RK2(p−n)and 𝜷2=T−(1∕2+𝛿)b2 o 𝛿>0, b2∈RK2n ixed.
(ii) Λ=diag{(L′
1,L′
2)′}whe e L1∈RK2(p−n)
+and L2∈RK2n
+.
(iii) L1=oP(T−1∕2)and L2
P
−−→L2as T→∞.
(i ) 𝜷0=0.
Le ΓΛ=Γ+Λwhe e Λ≽0isgi enby(8). Then, esul s (a)-(c) hold and
(d′′)√T(
𝜷R(Λ,𝜷0)−𝜷)d
−−→(0,Γ−1
ΛΓΓ
−1
Λ⊗Σu).
I is easy o see ha indeed he e m Γ−1
ΛΓΓ
−1
Λin Theo em 6is weakly smalle han Γ−1in he posi i e-de ini e
sense. No e ha
Γ−1
ΛΓΓ
−1
Λ≺Γ−1⟺(Γ + Λ)−1Γ≺Γ−1(Γ + Λ)
⟺IK2p−(Γ+Λ)−1Λ≺IK2p+Γ
−1Λ
⟺0⪯((Γ+Λ)−1+Γ
−1)Λ.
The las inequali y is ue by de ini ion o Λ. Sh inkage gains a e concen a ed a he componen s ha ha e non-ze o
asymp o ic sh inkage, i.e. hose penalized by L2.
Rema k 6. A key poin in he applica ion o Theo em 6is iden i ica ion o 𝜷1and 𝜷2. In p ac ice, one may hen
p oceed in wo ways. As discussed in Sec ion 4, one can see he idge app oach as a equen is ‘coun e pa ’
o implemen ing a Bayesian p io . The e o e, he esea che may spli 𝜷in o subse s o small and la ge pa ame-
e s based on economic in ui ion, domain knowledge o p elimina y in o ma ion. Al e na i ely, in he ollowing
sec ion, I show ha c oss- alida ion is able o au oma ically une Λapp op ia ely.
Finally, i is immedia e o gene alize he a gumen o Theo em 6 o he case whe e 𝜷is no spli in o subse s
based on he ela i e size o coe icien s, bu a he a non-ze o, pa ially consis en cen e ing sequence 𝜷0is used.
Co olla y 2. Conside he se up o Theo em 6, whe e now assump ions (i) and (i ) a e eplaced by
(i′)𝜷=(𝜷′
1,𝜷′
2)′whe e 𝜷1∈RK2(p−n)and 𝜷2∈RK2na e ixed.
(i ′)𝜷0=(𝜷′
01,𝜷′
02)′whe e 𝜷01 ∈RK2(p−n)is such ha 𝜷01 ≠𝜷1,and𝜷02 =𝜷2+T−(1∕2+𝛿)b2 o 𝛿>0,
b2∈RK2n ixed.
Then, esul s (a)-(c) and (d′′) s ill hold.
6.1. C oss- alida ion wi h Pa i ioned Coe icien s
One can use he same app oach applied o de i e Theo em 5 o show ha c oss- alida ing he RLS es ima o wi h
E (
𝜷R
⧫(Λ)) is also asymp o ically alid unde pa i ioning.
Co olla y 3. Conside he se up o Theo em 6and assume ha he assump ions o Theo em 5a e me . I holds
[Λ1,⧫0
0Λ2,⧫]∶= a g min
Λ∈𝜆
E (
𝜷R
⧫(Λ))=[op(T−1∕2)0
0oP(1)]
Mo eo e , any Λ2,⧫such ha 0 ⪯Λ
2,⧫⪯𝜆Iis asymp o ically alid.
In heo y, one would like o be able o quan i y he gains ob ained in he asymp o ic sh inkage se up o Theo em 6
compa ed o he s anda d se ing o Theo ems 1and 2, pa icula ly when using c oss- alida ion. Un o una ely, i is
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250 G. BALLARIN
in gene al ha d o s udy he c oss- alida ion e o loss e en in se ups wi hou dependence. S ephenson e al. (2021),
in ac , show ha he idge lea e-one-ou CV loss is no gene ally con ex. This sugges s ha s udying he beha io
o CV when penalizing wi h a diagonal aniso opic Λcan be a e y complex ask in a ini e sample se up.
7. SIMULATIONS
To s udy he pe o mance o idge- egula ized es ima o s, I now pe o m simula ion exe cises ocused on impulse
esponse unc ions (IRFs). Th oughou he expe imen s I will conside s uc u al impulse esponses, and I assume
ha iden i ica ion can be ob ained in a ecu si e way (Kilian and Lü kepohl, 2017), which is a widely used app oach
o s uc u al shock iden i ica ion in mac oeconome ics.
I conside wo se ups:
1. The h ee- a iable VARMA(1,1) design o Kilian and Kim (2011), ep esen ing a small-scale mac o model.
I e m his se up ‘A’.
2. A VAR(5) model in le els, using he model speci ica ion o Giannone e al. (2015) wi h he da ase o
Hansen (2016b) consis ing o K=7 a iables in le els.8I e m his se up ‘B’. Fo he ease o exposi ion,
in he discussion I will abula e esul s only o h ee a iables – eal GDP, in es men and ede al unds
a e – bu comple e ables can be ound in Supplemen a y Appendix D.5.
The speci ica ion o Kilian and Kim (2011) has al eady been ex ensi ely used in he li e a u e as a benchma k o
gauge he basic p ope ies o in e ence me hods. On he o he hand, he es ima ion ask o Giannone e al. (2015)
in ol es mo e a iables and a highe deg ee o pe sis ence. This se ing is use ul o e alua e he e ec s o idge
sh inkage when applied o ealis ic mac oeconomic ques ions. I is also a sui able es bench o compa e Bayesian
me hods wi h equen is idge.
7.1. Es ima o s
Fo equen is me hods, I include bo h 
𝜷Rand 
𝜷RGLS idge es ima o s as well as he local p ojec ion es ima o
o Jo dà (2005). Fo Bayesian me hods, I implemen bo h he Minneso a p io app oach o Ba´
nbu a e al. (2010)
wi h s a iona y p io and he hie a chical p io BVAR o Giannone e al. (2015).9The ull lis o me hod I conside
is gi en in Table I. To make me hods compa able, I ha e ex ended he idge es ima o s o include an in e cep in
he eg ession. A p ecise discussion ega ding he uning o penal ies and hype pa ame e s o all me hods can be
ound in Supplemen a y Appendix D.
7.2. Poin wise MSE
The i s wo simula ion designs explo e he MSE pe o mance o idge- ype es ima o s e sus al e na i es. Le
𝜃km(h)be he ho izon hs uc u al IRF o a iable kgi en a uni shock om a iable m. To compu e he MSE o
each k,de ine
MSEk(h)∶=
K
∑
m=1
E[(
𝜃km(h)−𝜃km(h))2],
which is he o al MSE o he k h a iable o e all possible s uc u al shocks. In simula ions, I use B eplica ions
o es ima e he expec a ion. All MSEs a e no malized by he mean squa ed e o o he leas squa es es ima o .
8The da ase is supplied by he au ho a h ps://use s.ssc.wisc.edu/~bhansen/p ogs/ a .h ml. While he da a p o ided by Hansen (2016b)
includes eleases un il 2016, I do no include mo e ecen qua e ly da a since his is a simula ion exe cise. Mo eo e , due o he e ec s o he
COVID-19 global pandemic, an ex ended sample would likely only add da a eleased un il Q4 2019 due o o e whelming conce ns o a b eak
poin .
9To es ima e hie a chical p io BVARs I ely on he o iginal MATLAB implemen a ion p o ided by Giannone e al. (2015) on he au ho s’
websi e a h p:// acul y.wcas.no hwes e n.edu/gep575/GLP eplica ionWeb.zip.
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RIDGE REGULARIZED ESTIMATION OF VAR MODELS 251
Table I. Lis o es ima ion me hods
Type Name Desc ip ion
F equen is LS Leas squa es es ima o
RIDGE Ridge es ima o , CV penal y
RIDGE-GLS GLS idge es ima o , CV penal y
RIDGE-AS Ridge es ima o wi h asymp o ic sh inkage, CV penal y
LP Local p ojec ions wi h Newey-Wes co a iance es ima e
Bayesian BVAR-CV Li e man-Minneso a Bayesian VAR, CV igh ness p io
H-BVAR Hie a chical Bayesian VAR o Giannone e al. (2015)
Table II. MSE ela i e o OLS – Se up A
Va iable Me hod h=1h=4h=8h=12 h=16 h=20 h=24
In es men RIDGE 0.97 0.74 0.64 0.64 0.65 0.63 0.60
g ow h RIDGE-GLS 5.16 0.89 0.55 0.47 0.44 0.41 0.38
LP 1.00 1.05 1.13 1.52 2.15 3.20 4.87
BVAR-CV 1.55 0.84 0.70 0.70 0.71 0.70 0.66
H-BVAR 1.80 0.66 0.53 0.52 0.54 0.53 0.50
De la o RIDGE 0.93 0.78 0.69 0.68 0.67 0.64 0.59
RIDGE-GLS 2.43 0.83 0.59 0.52 0.48 0.44 0.40
LP 1.00 1.05 1.13 1.44 1.99 2.90 4.47
BVAR-CV 1.03 0.89 0.74 0.73 0.73 0.70 0.66
H-BVAR 1.01 0.70 0.58 0.56 0.55 0.53 0.50
Pape a e RIDGE 0.94 0.76 0.66 0.66 0.66 0.64 0.60
RIDGE-GLS 1.80 0.87 0.59 0.52 0.47 0.43 0.39
LP 1.00 1.05 1.13 1.46 1.99 2.86 4.31
BVAR-CV 0.87 0.87 0.74 0.73 0.73 0.71 0.66
H-BVAR 0.81 0.69 0.57 0.55 0.56 0.54 0.51
7.2.1. Se up A
A ime se ies o leng h T=200 is gene a ed a numbe B=10,000 o imes o eplica ion. All VAR es ima o s
a e compu ed using p=10 lags, while LPs include q=10 eg ession lags. Table II shows ela i e MSEs o his
design. I is impo an o no ice ha , in his si ua ion, GLS idge has ema kably low pe o mance a ho izon h=1
compa ed o o he me hods. The p ima y issue is ha Σu ea u es s ong co ela ion be ween componen s, and hus
he diagonal lag-adap ed s uc u e does no sh ink along he app op ia e di ec ions. This is much less p ominen as
he ho izon inc eases due o he ac ha impulse esponses e en ually decay o ze o, since he unde lying VARMA
DGP is s a iona y. While he e is no clea anking, he MSE o he baseline idge VAR es ima o is in be ween
hose o he BVAR and hie a chical BVAR app oaches. The deg ading quali y o local p ojec ion es ima es a e
mainly due o he smalle samples a ailable in eg essions a each inc easing ho izon (Kilian and Kim, 2011).
This beha io is one o he p ime easons behind he de elopmen o LP sh inkage es ima o s, like ha p oposed
in Plagbo g-Mølle (2016) o he SLP es ima o o Ba nichon and B ownlees (2019).
7.2.2. Se up B
Using he da a o Hansen (2016b), I es ima e and simula e a s a iona y bu highly pe sis en VAR(5) model using
he same sample size and numbe o eplica ions as Se up A. Fo all me hods, p=5 lags a e used, so ha VAR
es ima o s a e co ec ly speci ied. The esul s can be ound in Table III. In his se up, unlike in he p e ious expe -
imen , one can clea ly no ice ha impulse esponses compu ed ia c oss- alida ed idge show inc easing MSE as
ho izon hg ows. The e a e wo main easons behind his beha io . Fi s , he chosen se up ea u es a e y pe sis en
da a gene a ing p ocess, as he la ges oo o he unde lying VAR model is 0.9945. This means ha he ue IRFs
J. Time Se . Anal. 46: 235–257 (2025) © 2024 The Au ho s. wileyonlinelib a y.com/jou nal/j sa
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252 G. BALLARIN
Table III. MSE ela i e o OLS – Se up B
Va iable Me hod h=1h=4h=8h=12 h=16 h=20 h=24
Real GDP RIDGE 1.11 1.08 1.16 1.06 0.90 0.89 0.94
RIDGE-GLS 1.16 1.00 0.99 1.00 0.93 0.93 0.95
LP 1.00 1.14 1.37 1.52 1.72 1.98 2.24
BVAR-CV 0.90 0.87 1.04 1.01 0.92 0.92 0.98
H-BVAR 0.83 0.62 0.78 0.73 0.62 0.62 0.68
In es men RIDGE 1.49 1.27 1.17 0.99 0.70 0.73 1.61
RIDGE-GLS 1.34 1.14 1.02 1.02 0.86 0.82 0.86
LP 1.00 1.15 1.40 1.63 2.03 2.76 3.59
BVAR-CV 1.51 1.01 0.97 0.97 0.93 1.08 1.24
H-BVAR 1.06 0.68 0.69 0.66 0.63 0.87 1.14
Fed unds RIDGE 2.17 1.21 0.96 0.93 1.03 4.00 53.18
a e RIDGE-GLS 1.21 1.04 0.90 0.93 0.90 0.88 0.91
LP 1.00 1.18 1.51 1.71 1.97 2.44 2.99
BVAR-CV 0.92 0.94 0.91 0.90 0.86 0.87 0.92
H-BVAR 0.75 0.77 1.32 1.38 1.25 1.15 1.20
e e o ze o only o e long ho izons, while lag-adap ed idge es ima es yields models wi h lowe pe sis ence
and hus la e impulse esponses. Second, he da ase om Hansen (2016b) is no no malized, and he included
se ies ha e ma kedly he e ogenous a iances. Since GLS idge sh inks along co a iance- o a ed da a, sh inkage
is adjus ed acco ding o each se ies a iance, unlike ha baseline idge es ima o 
𝜷R. The MSE o he Fed Fund
Ra e impulse esponses shows ha he poin wise di e ence be ween baseline and GLS idge can be se e e o
long ho izon IRFs when he DGP is highly pe sis en . On sho ho izons, Bayesian es ima o s pe o m on pa o
be e han baseline leas squa es es ima es, while a longe ho izons di e ences a e less s a k. I is, howe e , clea
ha he hie a chical p io BVAR o Giannone e al. (2015) shows he o e all bes esul s. As in he p e ious se up,
local p ojec ions show deg ading pe o mance a la ge ho izons.
Rema k 7. The compa ison be ween me hods in bo h Se up A and Se up B is la gely consis en wi h he indings
o Li e al. (2024), who make ex ensi e compu a ional simula ions by simula ing om syn he ic DGPs. They
p o ide a comp ehensi e ea men o he ques ion o which model – VAR o LP – is bes sui ed o IRF in e ence
in a gi en scena io in e ms o bias- a iance ade-o . They show ha a key balance o bias s. a iance exis s
be ween LP and VAR es ima es o impulse esponses: LPs end o ha e low bias, due o hei lexibili y, bu hey
also ea u e la ge a iance a highe ho izons. Thei esul s allow one o be e unde s and he ade-o s a play in
Tables II and III. In pa icula , i is clea ha idge sh inkage is bene icial a sho ho izons only i he penaliza ion
scheme is well-adap ed o he DGP a hand. O he wise, as is he case o RIDGE and RIDGE-GLS me hods, he
induced bias can be such ha idge MSEs su pass ha o OLS es ima es. One also inds ha he medium and long
ho izons MSE gains o e LPs a e mo e p onounced in cases o mode a e dependence, bu in he case o he Fede al
Funds Ra e IRFs in Se up B ze o-cen e ed RIDGE es ima es ho oughly mis ake long- e m beha io .
7.3. Con idence In e als
I now y and e alua e whe he idge sh inkage has a nega i e impac on in e ence. The e ha e also been ecen
con ibu ions di ec ly aimed a s udying sh inkage e ec s. Using he same simula ion se ups as in he p e ious
sec ion, I in es iga e co e age and size p ope ies o poin wise CIs cons uc ed using he me hods in Table I.All
con idence in e als a e cons uc ed wi h nominal 90% le el co e age.
In his se o simula ions, I swap GLS idge o he asymp o ic sh inkage idge es ima o , 
𝜷R
as, see Sec ion 6,
since he la e allows o a pa ially non-negligible penaliza ion in he limi . To implemen 
𝜷R
as, one needs o
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Jou nal o Time Se ies Analysis published by John Wiley & Sons L d. DOI: 10.1111/j sa.12737
RIDGE REGULARIZED ESTIMATION OF VAR MODELS 253
Table IV. Impulse esponse in e ence – Se upA–CIco e age
Va iable Me hod h=1h=4h=8h=12 h=16 h=20 h=24
In es men LS 0.88 0.88 0.87 0.88 0.91 0.93 0.94
g ow h RIDGE 0.90 0.92 0.94 0.93 0.94 0.95 0.95
RIDGE-AS 0.90 0.92 0.88 0.88 0.88 0.89 0.89
LP 0.88 0.97 0.99 0.99 0.99 0.99 0.99
BVAR-CV 0.77 0.88 0.88 0.90 0.92 0.94 0.96
H-BVAR 0.72 0.89 0.89 0.92 0.93 0.95 0.96
De la o LS 0.88 0.87 0.86 0.88 0.91 0.92 0.94
RIDGE 0.91 0.92 0.93 0.92 0.93 0.94 0.95
RIDGE-AS 0.91 0.91 0.88 0.88 0.87 0.87 0.88
LP 0.88 0.97 0.99 0.99 0.99 0.99 1.00
BVAR-CV 0.80 0.86 0.88 0.91 0.93 0.94 0.96
H-BVAR 0.84 0.88 0.90 0.92 0.94 0.95 0.97
Pape a e LS 0.87 0.86 0.86 0.88 0.90 0.92 0.94
RIDGE 0.90 0.91 0.93 0.93 0.93 0.94 0.95
RIDGE-AS 0.89 0.90 0.89 0.88 0.88 0.88 0.88
LP 0.87 0.97 0.99 0.99 0.99 0.99 0.99
BVAR-CV 0.82 0.84 0.87 0.90 0.92 0.93 0.95
H-BVAR 0.88 0.88 0.90 0.92 0.93 0.95 0.96
choose a pa i ion o 𝜷which iden i ies asymp o ically negligible coe icien . To do his, I spli 𝜷by lag and
penalize all coe icien s wi h lag o de s g ea e han a gi en h eshold p, such ha 1 <p<p.Inse upA,I
choose p=6, while in se up B I se p=3. In Bayesian me hods, including he c oss- alida ed Minneso a
BVAR, I cons uc high-p obabili y in e als by d awing om he pos e io . Compa ison be ween equen is CIs
and Bayesian pos e io densi ies is no gene ally alid, because hey a e no analogous concep s. The e o e, he
discussion below is in ended o highligh di e ences in s uc u e be ween idge app oaches.
7.3.1. Se up A
Simula ions wi h he DGP o Kilian and Kim (2011), p esen ed in Tables IV and V, highligh some o he ad an-
ages o applying idge when pe o ming in e ence. Focusing on es ima o 
𝜷R, i is clea ha CI co e age is in
ac highe han he in e als ob ained by leas squa es es ima ion in all si ua ions. A impac , idge CIs a e la ge
han he LS baseline, bu hey sh ink as ho izons inc ease. Thus, is IRFs e e ela i ely quickly o ze o, idge can
e ec i ely educe leng h while p ese ing co e age. As discussed in Sec ion 3, hese gains a e inhe en ly local o
he DGP – sh inkage o ze o a deep lags embodies co ec p io knowledge o a weakly pe sis en p ocess. Fo
Bayesian es ima o s, one can no e ha quan ile in e als a small ho izons end o be sho e compa ed o leas
squa es and idge me hods.
7.3.2. Se up B
The e ec s o idge sh inkage on a DGP wi h high pe sis ence a e much mo e se e e, as shown in Tables VI and
VII. Focusing on equen is idge, one can obse e ha close o impac (h=1) idge has simila o e en highe
co e age han o he me hods o eal GDP10 Howe e , as he IRF ho izon g ows, sh inkage o en leads o se e e
unde co e age, wi h asymp o ic sh inkage es ima o 
𝜷R
as gi ing he wo s esul s. In compa ison, Bayesian me hods
a e much mo e eliable a all ho izons, al hough he only es ima o ha can consis en ly imp o e on he benchma k
leas squa es VAR CIs is he hie a chical p io BVAR o Giannone e al. (2015). The eason behind his is simple
enough: he implemen a ion o he Minneso a-p io BVAR I ha e used he e has a whi e noise p io on all a iables,
which in his case is a om he u h. Indeed, Ba´
nbu a e al. (2010) implemen he same BVAR by uning he
10 This also is he case wi h consump ion and compensa ion, see also Tables 9 and 10 in Supplemen a y Appendix D.5.
J. Time Se . Anal. 46: 235–257 (2025) © 2024 The Au ho s. wileyonlinelib a y.com/jou nal/j sa
DOI: 10.1111/j sa.12737 Jou nal o Time Se ies Analysis published by John Wiley & Sons L d.
254 G. BALLARIN
Table V. Impulse esponse in e ence – Se upA–CIleng h
Va iable Me hod h=1h=4h=8h=12 h=16 h=20 h=24
In es men LS 2.99 5.11 5.78 5.35 4.79 4.17 3.56
g ow h RIDGE 3.13 5.20 5.82 5.17 4.48 3.78 3.09
RIDGE-AS 3.11 5.15 4.84 4.33 3.70 3.06 2.48
LP 2.99 7.50 10.97 12.89 13.99 14.55 14.70
BVAR-CV 2.84 4.48 4.70 4.38 3.99 3.56 3.11
H-BVAR 2.71 4.20 4.50 4.29 3.96 3.56 3.13
De la o LS 1.19 1.92 2.23 2.14 1.94 1.71 1.46
RIDGE 1.24 1.97 2.25 2.09 1.84 1.54 1.26
RIDGE-AS 1.24 1.95 1.95 1.78 1.52 1.25 1.01
LP 1.19 3.03 4.56 5.42 5.90 6.14 6.21
BVAR-CV 1.03 1.69 1.87 1.80 1.67 1.50 1.31
H-BVAR 1.01 1.64 1.83 1.79 1.67 1.51 1.33
Pape a e LS 0.97 1.42 1.64 1.57 1.44 1.27 1.09
RIDGE 1.01 1.44 1.65 1.53 1.36 1.16 0.95
RIDGE-AS 1.01 1.43 1.42 1.31 1.13 0.94 0.77
LP 0.97 2.19 3.28 3.90 4.26 4.43 4.48
BVAR-CV 0.84 1.22 1.35 1.30 1.21 1.09 0.96
H-BVAR 0.85 1.21 1.34 1.31 1.22 1.10 0.97
Table VI. Impulse esponse in e ence – Se up B: CI co e age
Va iable Me hod h=1h=4h=8h=12 h=16 h=20 h=24
Real GDP LS 0.87 0.81 0.75 0.72 0.71 0.72 0.73
RIDGE 0.90 0.79 0.66 0.62 0.65 0.68 0.68
RIDGE-AS 0.89 0.72 0.61 0.58 0.61 0.65 0.65
LP 0.87 0.93 0.94 0.94 0.93 0.93 0.91
BVAR-CV 0.70 0.71 0.63 0.64 0.71 0.75 0.76
H-BVAR 0.84 0.86 0.76 0.76 0.83 0.88 0.88
In es men LS 0.87 0.82 0.76 0.73 0.75 0.82 0.87
RIDGE 0.85 0.79 0.65 0.62 0.73 0.80 0.81
RIDGE-AS 0.82 0.69 0.59 0.57 0.68 0.77 0.77
LP 0.87 0.94 0.94 0.95 0.94 0.94 0.94
BVAR-CV 0.70 0.73 0.67 0.71 0.77 0.81 0.83
H-BVAR 0.80 0.86 0.81 0.82 0.87 0.88 0.88
Fed unds LS 0.85 0.83 0.80 0.78 0.77 0.79 0.80
a e RIDGE 0.79 0.77 0.74 0.68 0.68 0.72 0.72
RIDGE-AS 0.78 0.66 0.68 0.64 0.64 0.68 0.69
LP 0.85 0.94 0.96 0.96 0.95 0.94 0.93
BVAR-CV 0.76 0.72 0.76 0.77 0.77 0.81 0.83
H-BVAR 0.87 0.86 0.74 0.73 0.78 0.84 0.87
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RIDGE REGULARIZED ESTIMATION OF VAR MODELS 255
Table VII. Impulse esponse in e ence – Se up B: CI leng h ( escaled ×100)
Va iable Me hod h=1h=4h=8h=12 h=16 h=20 h=24
Real GDP LS 0.71 1.56 2.07 2.31 2.32 2.24 2.15
RIDGE 0.79 1.56 1.85 1.95 1.92 1.85 1.77
RIDGE-AS 0.74 1.31 1.65 1.76 1.75 1.70 1.64
LP 0.71 2.42 4.21 5.40 5.90 5.91 5.70
BVAR-CV 0.53 1.23 1.74 2.00 2.10 2.13 2.15
H-BVAR 0.58 1.36 1.87 2.16 2.32 2.44 2.55
In es men LS 3.38 6.65 7.89 7.89 7.31 6.69 6.18
RIDGE 3.79 6.81 6.93 6.46 5.79 5.19 4.73
RIDGE-AS 3.59 5.57 6.11 5.77 5.21 4.72 4.34
LP 3.37 10.16 16.00 18.85 19.06 18.22 17.23
BVAR-CV 2.64 5.26 6.59 6.91 6.78 6.57 6.38
H-BVAR 2.89 5.74 7.08 7.54 7.63 7.60 7.58
Fed unds LS 0.25 0.39 0.43 0.43 0.41 0.38 0.35
a e RIDGE 0.29 0.39 0.37 0.36 0.33 0.30 0.29
RIDGE-AS 0.27 0.31 0.33 0.32 0.30 0.28 0.27
LP 0.25 0.59 0.88 1.01 1.05 1.03 0.98
BVAR-CV 0.21 0.31 0.36 0.37 0.36 0.35 0.34
H-BVAR 0.23 0.36 0.42 0.44 0.45 0.45 0.46
p io o a andom walk o e y pe sis en a iables in hei applica ions. In his sense, he c oss- alida ed BVAR
conside ed – which is assumed cen e ed a ze o – is eally he lip-side o idge es ima o s. The e o e, he addi ion
o a p io on he mean o he au o eg essi e pa ame e s as done by Giannone e al. (2015) is a key elemen o
pe o m sh inkage in high pe sis ence se ups in a way ha does no sys ema ically unde mine asymp o ic in e ence
on impulse esponses.
8. CONCLUSION
In his a icle, I ha e s udied idge eg ession and i s applica ion o ec o au o eg essi e model es ima ion in
de ail. This appea s o be he i s wo k ha p o ides a ho ough analysis o idge penaliza ion in he con ex o
ime se ies da a, including geome ic as well as asymp o ic p ope ies. I ha e also de i ed esul s on he alidi y
o c oss- alida ion as a me hod o selec he penal y in ensi y in p ac ice, and I ha e shown ha CV p oduces
asymp o ically alid penaliza ion a es. Finally, I ha e compa ed bo h equen is and Bayesian idge o mula ion
in simula ions aimed a quan i ying he applicabili y o idge o impulse esponse in e ence.
The key akeaway o his a icle is ha idge penaliza ion is a use ul app oach o VAR es ima ion as long
as he chosen penal y s uc u e is well-adap ed o he model’s s uc u e. Bayesian idge pos e io s a e espe-
cially lexible, wi h hie a chical p io s also allowing sh inkage owa d non-ze o coe icien ec o s. Howe e , i
is impo an o no e ha he Bayesian app oach also pe mi s he esea che o speci y unin o ma i e p io s, so
ha he in luence o he p io s’ hype pa ame e s is less p onounced. This is no he case in equen is idge,
c . including an explici non-ze o cen e ing ec o . Howe e , p io knowledge o a p e-es ima ion p ocedu e
may be a ailable o he esea che , so ha idge can be e ec i ely implemen ed wi hou he need o implemen
aBVAR.
To conclude, he e a e s ill a enues o esea ch ega ding idge which would be in e es ing o de elop. Fi s and
o emos , he high-dimensional se up, o which, howe e , i seems non- i ial o ind a domain o applicabili y, as
discussed in he in oduc ion. Second, a mo e in-dep h analysis o c oss- alida ion, especially in he mul i- a ia e
case, would be ex emely aluable. Mo eo e , bo h he la e and o me opics should be join ly add essed in he
con ex o mild c oss-sec ional dimension g ow h, i.e., K→∞such ha K∕T→𝜌∈(0,1), which is compa able
o ac o model se ups.
J. Time Se . Anal. 46: 235–257 (2025) © 2024 The Au ho s. wileyonlinelib a y.com/jou nal/j sa
DOI: 10.1111/j sa.12737 Jou nal o Time Se ies Analysis published by John Wiley & Sons L d.
256 G. BALLARIN
ACKNOWLEDGEMENTS
I am g a e ul o he commen s and sugges ions om Lyudmila G igo ye a, So Jin Lee, Thomasz Olma, Oli e
P äu i and Mikkel Plagbo g-Mølle , and he semina pa icipan s a he Uni e si y o Mannheim, he HKMe ics
Wo kshop and he Young Resea che s Wo kshop on Big and Sma Da a Analysis in Finance. I am especially
hank ul o Claudia Noack o poin ing ou an impo an e o in a p e ious e sion o his a icle, as well as
Jonas K ampe and Ca s en T enkle o hei insigh ul discussions which helped de elop his a icle signi ican ly.
Las ly, I wish o hank Pe e C. B. Phillips, A sushi Inoue and many o he colleagues o he sugges ion o con-
side adding a o mal analysis o c oss- alida ion in he a icle. The au ho acknowledges suppo by he s a e o
Baden–Wü embe g h ough bwHPC. Open Access unding enabled and o ganized by P ojek DEAL.
CONFLICT OF INTEREST STATEMENT
The au ho epo s ha he e a e no compe ing in e es s o decla e.
DATA AVAILABILITY STATEMENT
The da a ha suppo he indings o his s udy a e openly a ailable in Gi Hub a h ps://gi hub.com/giob1994
/ idge_ a .
SUPPORTING INFORMATION
Addi ional Suppo ing In o ma ion may be ound online in he suppo ing in o ma ion ab o his a icle.
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DOI: 10.1111/j sa.12737 Jou nal o Time Se ies Analysis published by John Wiley & Sons L d.