Topics in Time Series Econometrics Structural VAR

Domenico Giannone, Université Libre de Bruxelles and CEPR

Trend stationary processes

yt = Tt + Ct

Trend (deterministic): Tt = α + δt

Cycle (stationary process): Ct = ψ(L)et = et + ψ1et−1 + ψ2et−2 + ... et ∼WN(0, σ2)

• Shocks have only temporary effects

∂Ct+h ∂et

= ψh → 0 as h→∞

yt+h|t − yt+h|t−1 = ψhet ⇒ All deviation from the deterministic trend are temporary

Remark: ŷt+h|t = δ(t + h) + ψhet + ψh−1et−1 + ψh−2et−2 + ...

Difference stationary processes

yt = yt−1 + δ + zt

The changes are stationary zt = ψ(L)et et ∼WN(0, σ2)

yt = y0 + δt + (z1 + ...+ zt )

• Shocks have permanent effects

yt+h|t−yt+h|t−1 = ∂yt+h ∂et

= 1+ψ1+...+ψh → ψ(1) as h→∞

lim h→∞

yt+h|t − yt+h|t−1 = ψ(1)et

long run multipliers: ψ(1) 6= 0 Shocks move the long run⇒ Stochastic trend

Stochastic or deterministic trend?

Nelson and Plosser, 1986, Journal of Monetary Economics

The authors cannot reject the hypothesis that most of the macroeconomic time series for the US are non stationary stochastic processes with no tendency to return to a deterministic path.

⇒ Shocks that drive the long run trend might also be driving business cycle fluctuations (Real Business Cycles)

This findings cast doubts on the importance of demand/monetary shocks as the main sources of business cycles fluctuations

⇒ debate on the relative importance of demand and supply shocks as driving forces for business cycle fluctuations

Identification of Aggregate Demand and Supply Shocks

Blanchard and Quah, 1989 American Economic Review

∆yt = 100× log GDPt−GDPt−1GDPt−1 : quarterly growth rate of GDP

URt : unemployment rate( ∆yt URt

) =

( µy µUR

) +

( B11(L) B12(L) B21(L) B22(L)

)( uSt uDt

) Identification

(i) (

eyt eUt

) =

( B11,0 B12,0 B21,0 B22,0

)( uSt uDt

)

(ii) (

uSt uDt

) ∼ WN

[( 0 0

) ,

( 1 0 0 1

)] (iii) limh→∞

∂yt+h uDt

= ∂∆ytuDt + ∂∆yt+1uDt

+ ∂∆yt+2uDt + ... = B12(1) = 0

From the forecasting to the structural model: step 1

( ∆yt URt

) =

( µy µUR

) +

( ψ11(L) ψ12(L) ψ21(L) ψ22(L)

)( eyt eut

) (

eyt eut

) ∼ WN

[( 0 0

) ,

( σ21 σ12 σ21 σ

2 2

)] (

∆yt URt

) =

( µy µUR

) +

( ψ̃11(L) ψ̃12(L) ψ̃21(L) ψ̃22(L)

)( ẽyt ẽut

) (

ẽyt ẽut

) ∼ WN

[( 0 0

) ,

( 1 0 0 1

)] (

eyt eUt

) =

( ψ11,0 0 ψ21,0 ψ22,0

)( ẽyt ẽUt

) ( ψ̃11,0 0 ψ̃21,0 ψ̃22,0

)( ψ̃11,0 ψ̃21,0

0 ψ̃22,0

) =

( σ21 σ1,2 σ21 σ

2 2

)

From the forecasting to the structural model: step 2

( ∆yt URt

) =

( µy µUR

) +

( ψ̃11(L) ψ̃12(L) ψ̃21(L) ψ̃22(L)

)( ẽyt ẽut

) (

ẽyt ẽut

) ∼ WN

[( 0 0

) ,

( 1 0 0 1

)] (

∆yt URt

) =

( µy µUR

) +

( B11(L) B12(L) B21(L) B22(L)

)( uSt uDt

) (

ẽyt ẽUt

) =

( R11 R12 R21 R22

)( uSt uDt

) (

B11(L) B12(L) B21(L) B22(L)

) =

( ψ̃11(L) ψ̃12(L) ψ̃21(L) ψ̃22(L)

)( R11 R12 R21 R22

) (

R11 R12 R21 R22

)( R11 R21 R12 R22

) =

( 1 0 0 1

)

Rotation matrix

( R11 R12 R21 R22

) =

( cos θ sin θ − sin θ cos θ

) θ ∈ [−π, π]

Choose θ such that B12(1) = 0 end than you get the identified responses(

B11(L) B12(L) B21(L) B22(L)

) =

( ψ̃11(L) ψ̃12(L) ψ̃21(L) ψ̃22(L)

)( R11 R12 R21 R22

) ( ψ̃11(L) ψ̃12(L) ψ̃21(L) ψ̃22(L)

) =

( ψ11(L) ψ12(L) ψ21(L) ψ22(L)

)( σ21 σ12 σ21 σ

2 2

)1/2

Summary of the SVAR estimation: step 1

The Goal Estimation on the structural model

yt = µ+ B(L)ut ut ∼WN(0, Iq)

The basic insight is that under rational expectations the structural shocks are forecast errors B0ut = yt − proj{yt |It−1}

It−1: Information set of the Economic Agents when they take expectations at time t − 1.

If we are able to approximate the information set of the economic agents using observable variables we can estimate the forecast errors and track back the structural shocks (with the help of some economic theory)

Usually: span{yt−1, yt−2, ...} =Mt−1 ⊆ It−1 = span{ut−1,ut−2, ...}

SVAR strategy: assume It−1 ≈Mt−1

Summary of the SVAR estimation: step 2

If It−1 ≈Mt−1 then the forecasts of the economic agents and the forecasts of the econometrician coincide.

proj{yt |It−1} ≈ proj{yt |Mt−1}

hence

et = yt − proj{yt |Mt−1} = yt − proj{yt |It−1} = B0ut

where

proj{yt |Mt−1} = c + ∞∑

s=1

Asyt−s : VAR(∞)

Wold: by inverting the VAR(∞) we obtain the MA(∞)

yt = µ+ Ψ(L)et et ∼WN(0,Σ)

Summary of the SVAR estimation: step 3

yt = µ+ B(L)ut ut ∼WN(0, I) yt = µ+ Ψ(L)et et ∼WN(0,Σ)

B0ut = et

• yt = µ+ Ψ(L)B0ut = µ+ B(L)ut ⇔ B(L) = Ψ(L)B0 • Σ = E[ete′t ] = E[B0utu

′ tB ′ 0]⇔ B0B′0 = Σ

B0 = Σ1/2R and B(L) = Ψ(L)Σ1/2R

where R is a rotation matrix ( R−1 = R′) which depends on n2 − n(n − 1)/2 unknown which have to be obtained by using economic restrictions

yt = µ+ Ψ(L) Σ1/2 In︷︸︸︷

RR′ Σ−1/2︸ ︷︷ ︸ In

et == µ+

B(L)︷ ︸︸ ︷ Ψ(L)Σ1/2R

ut︷ ︸︸ ︷ R′Σ−1/2et

Summary of the SVAR estimation: step 3

Rotation Matrices Definition: RR′ = I ⇔ R−1 = R′

Two dimensions: ( cos θ sin θ − sin θ cos θ

)

Three dimensions:

 cos θ1 sin θ1 0− sin θ1 cos θ1 0 0 0 1

 cos θ2 0 sin θ20 1 0 − sin θ2 0 cos θ2

 1 0 00 cos θ3 sin θ3 0 − sin θ3 cos θ3



n dimensions...

Summary of the SVAR estimation: step 4

In the Wold decomposition we have VAR(∞), which is impossible to estimate with a finite number of observations. Solution, limit the number of lags

Mt−1 ≈Mpt−1 = span{yt−1, yt−2, ..., yt−p}

=⇒ et = yt −

( c +

p∑ s=1

Asyt−s

) and et ⊥ yt−s for s = 1,2, ...,p Finite order Vector Autoregressive Model: VAR(p)

Summary of the SVAR estimation: step 5 Inverting the VAR(p) to the MA(∞): Wold Let us first rewrite the VAR(p) into a VAR(1) form:

yt = c + p∑

s=1

Asyt−s + et

Define



yt yt−1 yt−2

.

.

. yt−p+1

 ︸ ︷︷ ︸

Yt

=



c 0n×n 0n×n

.

.

. 0n×n

+ 

A1 A − 2 . . . Ap−1 Ap In 0n×n . . . 0n×n 0n×n

0n×n In . . . 0n×n 0n×n . . .

.

.

. . . .

.

.

. . . .

0n×n 0n×n . . . In 0n×n

 ︸ ︷︷ ︸

A



yt−1 yt−2 yt−3

.

.

. yt−p

 ︸ ︷︷ ︸

Yt−1

+



et 0n×n 0n×n

.

.

. 0n×n

 ︸ ︷︷ ︸

Et

Yt = C + AYt−1 + Et

define J = (In 0n×n . . . 0n×n︸ ︷︷ ︸ p times

) =⇒ yt = JYt and et = JEt

Summary of the SVAR estimation: step 5 Inverting the VAR(p) to the MA(∞): Wold

Yt = C + AYt−1 + Et solve it backward

Yt = (Inp − A)−1C + ∞∑ j=1

AjEt =⇒ yt = J

( ∞∑ s=0

Aj )

C + ∞∑ j=1

JAjJ ′et

Equating term by term with the wold decomposition

Ψj = JAjJ ′

Ψ(1) = J(Inp − A)−1J ′

µ = J

( ∞∑ s=0

Aj )

C = J(Inp − A)−1C = Ψ(1)c

SVAR: estimation and identification

• Estimate VAR(p): ĉ, Â1, ..., Âp, Σ̂ • Derive the coefficients of the Wold representation

Ψ̂1, Ψ̂2, ...

• Compute Σ̂1/2, i.e. from the Cholesky decomposition Σ̂ • derive the impulses Ψ̂jΣ̂1/2R for all possible rotation

matrices R • find the rotation matrix R∗ for which the impulses satisfy

your “preferred” economic restrictions:

B(L) = Ψ̂jΣ̂1/2R∗

• derive structural shocks

B(L)ût = R∗′Σ̂1/2êt

Widely used identification schemes

• on the contemporaneous multipliers B0: e.g. monetary policy shocks, fiscal shocks

• on the lung run multipliers B(1) supply/demand, te