Structural identification of vector autoregressions

Tony YatesLectures to MSc Time Series

students, Bristol, Spring 2014

Overview

• The algebra of the identification problem in VARs.

• Cholesky-factoring; timing restrictions.• Long run impact restrictions.• (Max share restrictions).• Sign restrictions.• Identification through heteroskedasticity.

Some useful sources

• Luktepohl, Hamilton.• Kilian survey.• Wouter den Haan lecture notes• Karl Whelan lecture notes• Many more!

Why bother with structural identification

• Empirical form of business cycle accounting, which is important for informing policy. Eg if RBC claim that tech shocks dominant true, maybe no need for stabilisation policy?

• Not needed for forecasting.• Needed for estimation of our economic models,

eg impulse response function matching.• Hence needed to understand appropriate policy

design. Eg identify a policy shock.

Lucas(1980) on why identifying shocks is important

Methods and problems in business cycle theory, JMCB

Reduced form vs structural model for Y

Yt A1Yt 1 A2Yt 2 A3Yt 3 . . .ApYt p et

Eu tut u IK

We are estimating this VAR(p), in the vector Y....

..in order to learn about this structural model, with different coefficients, and driven by structural shocks

Structural shocks mutually uncorrelated, and normalised so that vcov matrix is identity=dimension of Y

B0Yt B1Yt 1 . . .BpYt p ut

I BLYt ut

Structural vs reduced form VARs

Once we have estimated the reduced form VAR for Y, if only we knew the B_0, we could recover the sructural shocks, and ALL the coefficients of the structrual model.If only!!Structural identification is about trying to find B_0.Long and controversial story.We will tell it chronologically.

B0 1B0Yt B0

1B1Yt 1 . . .B0 1BpYt p B0

1u t

Yt B0 1B1Yt 1 . . .B0

1BpYt p B0 1u t

A i B0 1Bi,e t B0

1ut

Idealised factorisation of the reduced form vcov matrix

e B0 1Eu tut

B0 1

e B0 1 uB0

1 B0 1B0

1

LHS known; RHS elements unknown. System of nonlinear equations.Need to restrict B_0 so that have same number of unknowns as equations.Sigma_e is symmetric, as it’s a vcov matrix, hence has K(K+1)/2 independent elements only.B_0 not necessarily symmetricReferred to as ‘order condition’ for identification.

Simple illustration of the identification problem in 2 dimensions

Yt y1y2

t

AYt 1 e1e2

t

b0,11 b0,12

b0,21 b0,22

y1y2

t

b1,11 b1,12

b1,21 b1,22

y1y2

t 1

u1

u2t

y1y2

t

b1,11 b1,12

b1,21 b1,22

1b1,11 b1,12

b1,21 b1,22

y1y2

t 1

b1,11 b1,12

b1,21 b1,22

1u1

u2t

We estimate this 2 variable reduced form VAR(1) to learn about this 2 variable structural VAR.

Here we invert the coefficient matrix B0 so that the structural VAR has the same form as the reduced form VAR.Which allows us to see relation between rf and structural errors….

Identification problem in 2 dimensions, ctd…

e Ee1e2

e1e2

e,11 e,21

e,21 e,22

b1,11 b1,12

b1,21 b1,22

1b1,11 b1,12

b1,21 b1,22

1

b1,112 b1,21

2

b1,212 b1,22

2

First line: we compute the vcov matrix of reduced form errors which we see has only 3 separate elements.Second line: we note that this is equal to inv(B0)*inv(B0)’.Problem: this is a 2*2 with 4 independent unknown elements.We have only 3 knowns to find these 4 unknowns.This is the identification problem in VARs/SVARs.

Cholesky identification

e PP,P chol e

P B0 1

Yt

x t

t

it

p11 0 0

p21 p22 0

p31 p32 p33

B1Yt 1 . . .p11 0 0

p21 p22 0

p31 p32 p33

ux

u

uit

P is lower triangularUsed eg to identify a monetary policy shockAssumes strict causal chain in the VARGDP and inflation don’t react within period to a monetary policy shock.

For those studying DSGE models.Third equation looks like a central bank reaction function.But it isn’t! Coefficients of the central bank reaction function will show up in all of the VARs reduced form equations.See, eg, Canova comment on Benati/Surico paper.

Recovering the monetary policy shock and structural coefficients with Cholesky identification

exe

e it

p11 0 0

p21 p22 0

p31 p32 p33

ux

u

u it

eit p31uxt p32u t p33uit

eit p31uxt p32u tp33

u it

A i B0 1Bi

B1 A1

p11 0 0

p21 p22 0

p31 p32 p33

1

Finishing the 3d monetary policy shock

ext p11uxt

uxt ext/p11

e t p21uxt p22u t

p21ext/p11 p22u t

e t p21ext/p11p22

u t

e it p 31u xt p 32u tp 33

uitOur expression for the mon pol shock u_it was in terms of some unknowns.

Here we can get rid of one of them, u_xt.

Here we get rid of the second, u_pit, also using expression for u_xt

And we are done. u_it now entirely in terms of knowns.

Impulse responses to the structural shocks

e B0 1u u B0e

Yhirf AhB0e

We can use the reduced form VAR estimates to compute the response to structural shocks.Our structural shocks are functions of the reduced form shocks, a function of our assumed value for B_0.So we then feed these through the reduced form VAR in the normal way, taking successively higher and higher powers of the A matrix.

Problems with recursive structure implied by Cholesky identification

• Is it economically plausible? Quarterly data. Does output or inflation really not respond to interest rates within the quarter?

• Assumed causal ordering means we can’t use the VAR to find out about causality.

• Common practice to check for robustness to alternative causal orderings, but this is rarely done comprehensively; and [so Kilian says] is nonsensical!

Recap on causal assumptions in Cholesky identification of mon pol shock

exe

eit

p11 0 0

p21 p22 0

p31 p32 p33

ux

u

uit

Neither e_x nor e_pi are functions of the structural shock u_it.

Implies output gap and inflation do not respond [within the period] to a monetary policy shock.

Is this really plausible?

Remember the NK DSGE model.

t E t 1 x t u t

x t Ex t 1 it E t 1 uxt

it t xx t u it

If we trace the timing of the effects of the monetary policy shock, we can see that it affects inflation and the output gap straight away.Of course, the NK model could be a load of nonsense!

More problems with recursive identification.

• Sims and the ‘price puzzle’.• Would find that recursively identified, contract

mon pol shock leads to increase in prices, contrary to theory.

• Concluded that had omitted variable cb responding to, eg commodity prices.

• CP up means prices up, despite central bank rate increase.

Recursive ID problems: omitted variables

• Omitted variable bias leads us to enlarge the VAR.

• But there is a cost: imprecision in the coefficients. And the search for more restrictions.

• Solutions: Bayesian shrinkage and factor modelling to reduce dimensionality.

• In time we will cover both of these.

What are these mp shocks anyway?

• Why would policymakers induce shocks?• Are they really conducting experiments for their own

edification [a control literature involving Sargent, Cogley shows there is a benefit]. Alan Blinder says not.

• Shocks are just misspecifications by econometrician• Shocks are policymakers’ real time measurement error?• Shocks are policymakers’ [or our] model mis-

specification• Same goes, of course, for fiscal policy shocks.

Alternative, less intensive measures of identifying (eg) mp shocks

• Romer and Romer (1989) narrative measures of monetary policy (and fiscal policy) shocks.

• Rudebusch (1998): gap between actual Fed Funds Rate and expectations implied by Fed Funds futures.

Romer and Romer’s narrative shock definition

Problem: they are including movements in rates prompted by concerns about inflation. So if we think i=a*pi+b*y+shock, they are not identifying m p shocks

Romer and Romer(1989), quoting Friedman and Schwartz (1964)

Rudebusch, and Sims on Rudebusch

• Rudebusch: FFFs produce better forecasts than VARs. His surprise a better measure of policy ‘shocks’.

• Sims (1996): 1. False premise. FFR shocks confound surprises due to non policy shocks with those due to shocks. 2. Not true that if shock measures badly correlated, estimated effects are very different.

Famous applications

• Rotemburg and Woodford (1998).– Early application. Estimated DSGE model by fitting

IRFs of monetary policy shock using MDE.– Later pointed out that in DSGE model EVERYTHING

responds within the period to monetary policy.• Christiano, Eichenbaum and Evans (2005)– Same exercise. But DSGE model is consistent with

timing assumption.

Minimum distance estimation

mde

arg min Y YA, e

DSGE model is defined by a vector of parameters, PHI.

We take as our PHI_hat the value of these parameters that makes the impulse response in the DSGE model as close as possible to the identified, estimated IRF in the VAR.

Some costs and benefits wrt eg MLE estimation of DSGE model.

Cost: partial information, means bad identification.Benefit: MLE only consistent if model well-specified.

Identification using long run restrictions

• Attractive because much agreement on certain long run restrictions. So these are ‘credible’ in Sims language. At least among members of the RBC/DSGE cult [like me]!

• Famous early applications are Blanchard and Quah (1989), and Gali (1999). More later.

• Technique sparked famous arguments between Chari et al and VAR proponents, notably Christiano and coauthors.

Egs of restrictions uncontroversial, at least in DSGE/RBC-land

• Nominal shocks should have no long run impact on real variables.

• Corrollary: only real shocks should have a long run impact on real variables.

• Real shocks like technology should be neutral on inflation in the long run.

• Only inflation regime changes should affect inflation in the long run.

Deriving long run restrictions for an SVAR

Yt A t 1 B0 1ut

impact period: B0 1

1 period after: AB0 1

2 periods after:A2B0 1

n periods after:AnB0 1

Reduced form VAR(1) with an expression in terms of structural shocks and the unkown structural impact matrix substiuted in place of the RF shock

This is how we would compute the IRF to the structural shock, if only we knew the structural impat matrix....

You can see this defines an infinite sequence. Effect on level of something=0, implies sum of effects on difference=0.

Deriving long run restrictions

D B0 1 AB0

1 A2B0 1 . . . AnB0

1

B0 1I A A2 . . .An

D B0 1I A 1

i 0

a i 11 a

Factor the long run IRF in terms of something you might recognise from high school as the expression for an infinite geometric series, or its matrix equivalent.

DD I A 1B0 1B0

1I A 1

I A 1 eI A 1

Some algebra and we spot that DD’ involves something we can estimate from the data, the vcov of the RF residuals!

Long run restrictions and the Cholesky factor, again

D cholI A 1 eI A 1

D B0 1I A 1

B0 1 I AD

DD I A 1B0 1B0

1I A 1

I A 1 eI A 1

DD’ which we know to be related to the magic B0 which we are trying to find……we also know to be related entirely to things we do know.

Gali’s (AER, 1999) search for technology shocks

Yt logyt/h t

loght

Only the tech shock (which we say comes first, has a long run effect on the first variable, (the change in) output per hour

D d11 0

d21 d22

Celebrated applications of LR restrictions

• Blanchard-Quah. Tech shock is the only thing to affect output in the long run. Q: how important are tech shocks in driving the business cycle?

• Gali (1999). Tech shock is only thing driving labour productivity. Appears to cause hours to fall, not rise. Consistent with sticky price model, not RBC models. Hours tend to comove positively with business cycle. Suggests tech shocks not dominant driver of business cycle.

• Christiano-Eichenbaum-Vigfussen (). Reexamination of Gali. Conclusion depends on definition of hours worked used in VAR

Identification of VARs using sign restrictions

e PP,P chol e

e PCC P,C Givens

Cholesky factorisation of rf varcov can be expanded with product of any orthonormal matrixSo, parameterising with the angle theta, we choose all those that satisfy certain sign restrictions

Sign is the ‘signum’ function; S is a selector matrix with ones for restricted elements, 0s otherwise. C is our givens matrix. A is our estimated rf impact matrix.

1 0 0 0

0 c s 0

0 s c 0

0 0 0 1

,c cos , s sin

signS.PCAh R

Signum function and the selector matrix

sign0 0.05

2 3

0 1

1 1

An example of what the signum function does to a matrix. Turns things into 0s, 1s or -1s.Just a way to record in an algorithm whether things are positive, negative or zero.Could do it differently, with more if, then else statements.

S.2 6

0 1

1 0

1 0.

2 6

0 1

2 0

0 0

An example of doing element by element multiplication with a selector matrix.

Verbal description of sign restrictions

• Take cholesky factor of vcov matrix P• Multiply by some Givens, C(theta)• Check signs of elements of interest.• If they agree with your restrictions, store and

keep.• If not, move on.• At the end, plot ALL the IRFs, and or

summarise them somehow.

i=cb rate, pi=inflation, x=output, ur=unemployment rate

Mp shock in first column: contraction raises rate, lowers pi, lowers output, lowers unemployment.

Tech shock in second column: reduction has ambiguous effect on i, increases pi, lowers output, ambiguous effect on ue

Demand shock in third column: cb raises rates to fight it, inflation and output increase anyway, ue falls.

Fourth shock unidentified. A dustbin containing many things we don’t need to worry about.

Yt

it t

x t

urt

AYt 1 et

R

1 0 1 0

1 1 1 0

1 1 1 0

1 0 1 0

S

1 0 1 0

1 1 1 0

1 1 1 0

1 0 1 0

C

1 0 0 0

0 c s 0

0 s c 0

0 0 0 1

Alternative way to do sign restrictions using the QR decomposition

1.LKK,L ij NID0,1

2.L QR,QQ IK3.C Q

4.Proceed as before

Any square matrix can be decomposed into a product of an orthonormal matrix and something else.Matlab will calculate this for you in a flash.Note equivalence which seems not to be widely understood. Derives from fact that all orthonormal matrices can be shown to be product of Givens matrices.Personal preference: use Givens method. Systematic way to explore the space.Random number generators for step 1 in computers are not random, they are pseudo random. Don’t know if this matters much.

Sign restrictions at different or multiple horizons

signSPC Ah RChoose h the horizon, then keep IRF if it satisifies this condition

signSPC Ah 1 R, signSPC Ah 2 R

For multiple horizons, keep if the condition above applied for multiple horizons holds!

Warning: spanning the entire space of possible impulse responses

nn 1/2A givens indexed by one angle only not enough to guarantee to find all possible IRFs that satisfy the sign restriction, except in the 2 variable case [eg n=2, 2*(1)/2=1]

More generally, we need to search across orthonormal matrices formed by products of Givens matrices.

i 1

nn 1/2

C i

Reporting distributions from sign restrictions

1.For each h compute: IRFh 1/n i 1

n

Yh, i

irf

2. Find arg min h Yh, i

irf IRFh2

3. Plot Yh,irf

Sign restrictions generate arbitrary numbers of impulse response functions.

How to report them? Can just plot the whole damned lot.

People used to report moments at each h eg the median.

Fry and Pagan suggested above. Find the single SVAR corresponding to single angle that is closest to this median.

Sign restrictions: examples

• Giraitis, Kapetanios, Yates.• Actually a TVP study, but don’t worry about

that until later on in the lecture series.• Sign restrictions to identify various shocks.• Studies time variation in the IRFs.

Sign restrictions in Giraitis, Kapetanios, Yates

c i y h w/p r

monetary policy - - - -

technology -

labour supply - -

demand

Identification using Mountford and Uhlig’s penalty method

w_h are weights; V records not just signs, but magnitudes.Why would you do this? Isn’t it just assuming the answer?Well, one motivation is to use one implication of the model to test another, that you don’t impose.Another is that some rotations may satisfy the sign restriction literally, but you want the IRF not just to clear the zero line by a tiny amount.

1. draw rotation matrixC

2. compute K h 1

H

whSPCAh VhSPCAh Vh

3. Rank according to K

Using the ‘max share’ criterion in a VAR to identify news shocks

• Barsky-Sims (2010): identified news shocks for technology.

• Long history of identifying technology shocks, measuring impact, quantifying contribution to business cycles

• RE assumption suggests agents may also react to news about future events.

• Failure to account for this may mislead us in properly measuring and quantifying effects of technology shocks.

Other work on news shocks

• Beaudry-Portier: news shocks in VARs: news causes hours to increase.

• Jaimovich and Rebello: news shocks in RBC model, neutralising the wealth effect so that hours don’t fall.

• Schmitt-Grohe and Uribe (2012): multiple news shocks in large RBC style model

• Christiano, Motto, Rostagno and Pinter, Theodoridis and Yates: risk news shocks in DSGE model and VAR respectively

News shocks: algebrayt BLu t

u t A tAA

y t h E t 1yt h 0

h

BAQ t h

Q Q IN, KK 1/2 1

Q Q 1 Q 2 . . . Q KK 1/2 1

A chol

Note annoying change of notation; matches Barsky-Sims. Sorry. Good for the soul though.This slide is about writing down the expression for the forecast error up to horizon h.

News shocks identification, ctd...

i,jh e i 0

h B AQ e je jQ AB e i

e i 0h B B e i

tfp t tfp t 1 u tfp,t u tfp,t 1news

1,1h 1,2h 1

The share of the FEV up to horizon h, of variable i, accounted for by shock j...

Example news process for tfp. Could be more general than this.

In an ideal setting, share of current and news shock to tfp that accounts for tfp should be 1. tfp is exogenous after all!

News shocks: the max share criterion

max

i,jh

subject to

A 1, j 0, j 1

signSA 22 F

By choice of the K.(K-1)/2 vector of angles w, maximise the share of tfp forecast error variance up to horizon h, accounted for by the news shock to tfp, subject to the restriction that the news shock is orthogonal to tfp today. [if it weren’t, it wouldn’t be news, it would be a contemp. Shock to tfp]

In Pinter, Theodoridis and Yates, in our search for ‘risk news’ shocks, we impose a sign restrictions condition too.

Comments on news shock ID

• Method proposed originally as an alternative to LR restrictions that does not depend on uncovering zero frequency events in finite samples: see work by Faust and Diebold et al.

• Note practical contradiction. Proxy for shock is supposed to be exogenous, but is included as an endogenous variable.

• Monte carlo tests in economic laboratory show that it works in ‘theory’.

Application : ‘risk news shock’ in Pinter, Theodoridis and Yates

• Looks for the same shock as in CMR(2013), AER.• Risk shocks are fluctuations in the variance of

idiosyncratic returns to entrepreneurs.• Literally, in CMR, these guys build capital goods from

capital inputs, selling them to sticky-price producers.• They borrow from banks at a spread related to their

cross-sectional risk.• Higher risk means more defaults, means bigger

spread to compensate.

Pinter, Theodoridis, Yates

• VAR study allows us to drop lots of contestable assumptions in the full information estimation of CMR’s DSGE model.

• Cost: we have to assume we can observe risk.• Do this using the VIX, and using cross section

of stock returns computed from a US panel.

Risk proxies

‘F’: Sign and zero restrictions in the VAR

t t 1

news tech net w mpol news tech net w mpol

risk 0 0 0 0

spread

GDP growth - - - -

C growth

I growth

hours

r wage growth

inflation - - - - - -

policy rate - - - -

net worth growth - - - -

IRF to a risk news shock: VIX vs CSR

FEVD contributions (VIX)

Headlines from application

• PTY get a much smaller contribution of the risk and risk news shocks to the business cycle than CMR. 20% contribution in total, compared to 60% in CMR.

Rigobon: identification through heteroskedasticity

• If you can persuasively argue there was a change in the structural shock variances

• And at the same time the VARs coefficients were constant

• Then you can use the change to generate more equations to find your unknowns and get identification.

• In the limit, this is like event study analysis.

Similar variance of demand and supply shocks generates cloud of dots, unable to see shape of either curve of course.

Increase in variance of supply shocks starts to trace out shape of demand curve helping us to ‘identify’ demand coefficients.In the limit : event study; no demand shocks at all.

The identification problem: Rigobon’s simple demand-supply example

p t q t tq t p t t

P p1 ,p2 . . .pT ,

Q q1 ,q2 . . .qT

, , 2 ,

2

11 2

2 2

2 2 2

2

2 2

2 2

2

A simultaneous demand and supply system.

All we observe is time series on prices P and quantities Q

We try to estimate four unknowns, the two slopes, and the two shock variances

However all we can estimate consistently is the reduced form vcov matrix. Its symmetry means we only have 3 different elements, so this is 3 equations in 4 unknowns.

Rigobon’s heteroskedasticity solution to the identification problem

s 11 2

2 ,s2 ,s

2 2 ,s2 ,s

2

2 ,s2 ,s

2 ,s2 ,s

2, s 1,2

There are TWO values for s, so we have TWO reduced form vcov matrices. This means two lots of 3 equations=6.Slope coefficients are unchanged, so they provide 2 unknowns as before.Structural variances change, so there are now double 2=4 unknowns.Making 6 equations in 6 unknowns.Luktepohl has applied this to VARs.