Structural VAR Models and Applications Laurent Ferrara1 (with the help of Jean-Paul Renne) 1 University

of Paris West

M2 EIPMC 2012

Overview of the presentation

1. Structural Vector Auto-Regressions I I

Definition Estimation

2. Impulse responses functions (IRF) I I

Concept General IRF

3. Applications 3.1 Smets-Tsataronis (1995) 3.2 Kilian (AER, 2009)

Vector Auto-Regressions: Causal mechanisms I

Assume that the GDP growth gt is affected by some real shocks ur ,t following gt = −0.3(it−1 − πt ) + ur ,t where it denotes the nominal interest rate and πt denotes the inflation rate.

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Besides, assume that we have it πt

= 0.9it−1 + 1.5πt + ump,t = 0.9πt−1 + 0.2gt−1 + un,t

where ump,t and un,t are respectively some monetary-policy and cost-push shocks. I

The strucural shocks ut are uncorrelated (i.e., the covariance matrix of ut , denoted with Ωu is diagonal) and ut is serially uncorrelated (i.e. Cov (ut−k , ut ) = 0 for any t and k > 0).

Vector Auto-Regressions: Causal mechanisms

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The “structural” model reads   gt = −0.3(it−1 − πt ) + ur ,t it = 0.9it−1 + 1.5πt + ump,t   πt = 0.9πt−1 + 0.2gt−1 + un,t .

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To get it in a “reduced-form”, let us substitute πt in the right-hand sides of the first two equations:   gt = 0.06gt−1 − 0.3it−1 + 0.27πt−1 + 0.3un,t + ur ,t it = 0.9it−1 + 1.35πt−1 + 0.3gt−1 + ump,t + 1.5un,t   πt = 0.9πt−1 + 0.2gt−1 + un,t .

Vector Auto-Regressions:Impulse responses I

In matrix form        gt 0.06 −0.3 0.27 gt−1 εg ,t  it  =  0.3 0.9 1.35   it−1  +  εi,t  πt 0.2 0 0.9 πt−1 επ,t 

      εg ,t 1 0 0.3 u r ,t u r ,t with  εi,t  =  0 1 1.5   ump,t  = B  ump,t . επ,t 0 0 1 un,t un,t I

With the procedure described above, one only gets an estimate of Ωε where   εg ,t Ωε = Var  εi,t  . επ,t

Vector Auto-Regressions:Impulse responses

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Note however that we must have Ωε = BΩu B 0 where Ωu is diagonal positive.

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In addition, given the “structural” framework, one knows that B is an upper-triangular matrix.

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The Choleshy decomposition can therefore be used to get the B matrix.

Vector Auto-Regressions:Impulse responses

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Whereas the VAR model is able to capture efficiently the interactions between the different variables, it does not allow to reveal the underlying causal mecanisms since two different causal schemes can correspond to the same reduced forms.

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By taking into account certain economic relationships, a Structural VAR model (SVAR) makes it possible to identify structural shocks while letting play the interactions between the different variables (see Gali, 1992 or Gerlach and Smets 1995).

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Formally, let assume that the residuals εt are some linear combinations of the structural shocks ut , that is: εt = But .

Vector Auto-Regressions:Impulse responses

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How it has been shown previously, a SVAR is based on a structural model that draws from a theoretical framework.

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As a starting point, we always have Ωε = BΩu B 0 that provides us with n(n + 1)/2 restrictions to recover the B matrix.

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Consequently, to get the B matrix, one have to impose n(n − 1)/2 additional restrictions.

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There exist two kinds of restrictions that can be easily implemented in a SVAR: short-run and long-run restrictions: I

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a short-run restriction prevents a structural shock from affecting an endogenous variable contemporaneously; a long-run restriction prevents a structural shock from affecting an endogenous variable in a cumulative way.

Vector Auto-Regressions:Impulse responses I

Concretely, the short-run restrictions consists in setting to zero some entries of B.

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The long-run restrictions require additional computations to be applied. More precisely, one needs to implement the computation of the cumulative effect of one of the structural shocks ut on one of the endogenous variable.

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Assume that we have the (reduced-form) VAR yt = c + Φ1 yt−1 + . . . Φp yt−p + εt .

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As was shown previously, one can always write a VAR(p) as a VAR(1), by stacking yt , yt−1 , . . . yt−p+1 in a vector yt∗ . Consequently, let us consider only the VAR(1) case: yt = c + Φyt−1 + εt .

Vector Auto-Regressions:Impulse responses I

Once more, let us consider the Wold’s form of yt : yt

= c + εt + Φ (c + εt−1 ) + . . . + Φk (c + εt−k ) + . . . = c + But + Φ (c + But−1 ) + . . . + Φk (c + But−k ) + . . .

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Consequently, the cumulated effect of the first structural shock u1,t on the endogenous variables is obtained by computing   γ  0    (B + ΦB + . . . + Φk B + . . .)  .   ..  0 if the initial shock of u1,t is of magnitude γ.

Vector Auto-Regressions:Impulse responses

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In this context, consider the following long-run restriction: the j th structural shock does not affect,in a cumulative way, the i th endogenous variable.

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Then, denoting with Θ the matrix (I + Φ + . . . + Φk + . . .)B, it comes that the entry (i, j) of Θ must be equal to zero.

A simple SVAR

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It can be noted that short-run restrictions are simpler to implement than long-run one.

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There are particular cases in which some well-known matrix decompostion can be used to easily estimate some specific SVAR.

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Imagine a context in which you can argue that there exists an “ordering” of the shocks: I

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A first shock (say, ε1,t ) can affect instantaneously (i.e., in t) only one of the endogenous variable (say, y1,t ); A second shock (say, ε2,t ) can affect instantaneously (i.e., in t) the first two endogenous variables (say, y1,t and y2,t ); ...

Cholesky decomposition: Illustration I

Dedola and Lippi (“The Monetary Transmission Mechanism: Evidence from the Industries of Five OECD Countries”, 2005) estimate 5 structural VAR for the US, the UK, Germany, France and Italy to analyse the monetary-policy transmission mechanisms.

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They estimate an SVAR over the period 1975-1997, using 5 lags in VAR.

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The shock-identification scheme is based on Cholesky decompositions, the ordering of the endogenous variables being: the industrial production, the consumer price index, a commodity price index, the short-term rate, a monetary aggregate and (except for the US).

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This ordering implies that monetary policy reacts to the shocks affecting the first three variables but that the latter react to monetary policy with a one-period lag.

Responses of Macro-variables to a monetary policy shock Figure 1

Responses of the main macro variables to a monetary policy shock !±"#$%&'(&)(#*))+)#,&'($-

Responses of Macro-variables to a monetary policy shock

Note: The boxes in each column show the response of the VAR variables to a shock to the short term interest rate (equal to

Illustration: Blanchard and Quah (1989)

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The article “The Dynamics Effects of Aggregate Demand and Supply Disturbances” (AER, 1989) implements long-run restrictions in a small-sized VAR.

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Two variables are considered GDP and unemployment.

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Consequently, the VAR is affected by two types of shocks.

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The authors want to identify supply shocks (that xan have a permanent effect on output) and demand shocks (that can not have a permanent effect on output).

Illustration: Blanchard and Quah (1989) I

The motivation of the authors regarding their long-run restrictions can be obtained from a traditional Keynesian view of fluctuations.

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The authors propose a variant of a model from Stanly and Fisher (1977) Yt

= Mt − Pt + a.θt

(1)

Yt

= Nt + θ t

(2)

Pt

= Wt − θt  = W | Et−1 Nt = N

(3)

Wt I

(4)

To close the model, the aithors assume the following dynamics for the money supply and the productivity Mt

= Mt−1 + εdt

θt

= θt−1 + εst

Illustration: Blanchard and Quah (1989) I

In this context, it can be shown that 4gnpt ut

= εdt − εdt−1 + a.(εst − εst−1 ) + εst = −εdt − aεst

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Then, it appears that the demand shocks have no long-run impact on output. Besides, neither shocks have a long-run impact on unemployment.

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The endogenous variable is ( 4gnpt denotes the logarithm of GNP.

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It is assumed to be stationary. Therefore, neither disturbances has a long-run effect on unemployment or the rate of change in output.

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The long-run restriction implies that the demand shocks also have no long-run effect on the output level gnp itself.

ut ) where gdpt

Illustration: Blanchard and Quah (1989)

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Estimation data: quarterly, from 1950:2 to 1987:4.

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8 lags.

Dynamic effects of demand disturbances

THE AMERICAN ECONOMIC RE VIEW

mpositions only

amic effects of nces.

nd and Supply

mand and supd in Figures 1 igures 1 and 2 of output and the horizontal s. Figures 3-6 , but now with ds around the

a hump-shaped loyment. Their r quarters. The

SEPTEMBER 1989

1.40

1.20 1.000.80 -0.600.40

0.20 0.00

,,,

-0.20 0

10

20

30

40

-0.40

-0.60 FIGURE

1. RESPONSE TO DEMAND,-= -

1.00

=

UNEMPLOYMENT

OUTPUT,

, but now with ds around the

a hump-shaped loyment. Their r quarters. The line to vanish The responses are mirror imn to this aspect cussing the ef-

allest when the allowing for a unemployment ecays the most change in the is allowed, the changes in untively unimporemand distur-

consistent with amic effects of and unemploy-

FIGURE

1. RESPONSE TO DEMAND,-=

OUTPUT,

Dynamic effects of supply disturbances = UNEMPLOYMENT 1.00 0.80 0.60 -/ 0.400.20 0.00 0 -0.20 -

10

20

l+q 30

ii

i 40

-0.40

-0.60FIGURE

2.

RESPONSE TO SUPPLY, = UNEMPLOYMENT

OUTPUT,

and wages leads the economy back to equi-

and output devia7.40 n. At the peak rets an implied coef-Vector 7.20 Auto-Regressions:Impulse responses four, higher in ab7.00 's coefficient. That 1980 1970 1960 1950 coefficient is higher Thedemand output gap isFIGURE the component of GNP ABSENT that is explained 7. OUTPUT FLUCTUATIONS than for DEMAND Suphat we expect. demand shocks. y to affect the relad employment, and 0.10 ttle or no change in 0.08

ns of Demand and rbances.

amic effects of each next step is to assess n to fluctuations in ent. We do this in ormal, and entails a orical time-series of of output to the usiness cycles. The e decompositions of ent in demand and arious horizons.

0.06 0.04 0.02

v

0.007 -0.04 -0.06 -0.08 -0.10 1950

1960

1970

1980

FIGURE 8. OUTPUT FLUCTUATIONSDUE TO DEMAND

components of unemployment are stationary. The time-series for these components are

only by

Variance decomposition

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The k quarter-ahead forecast error in output is defined as the difference between the actual value of output and its VAR-based forecast.

Vector Auto-Regressions:Impulse responses

666

THE AMERICAN ECONOMIC RE VIEW

SEPTEMBER 1989

TABLE 2-VARIANCE DECOMPOSITION OF OUTPUT AND UNEMPLOYMENT (CHANGE IN OUTPUT GROWTH AT 1973/1974; UNEMPLOYMENT DETRENDED)

Percentage of Variance Due to Demand: Horizon (Quarters) 1 2 3 4 8 12 40

Output

Unemployment

99.0 (76.9,99.7) 99.6 (78.4,99.9) 99.0 (76.0,99.6) 97.9 (71.0,98.9) 81.7 (46.3,87.0) 67.6 (30.9,73.9) 39.3 (7.5,39.3)

51.9 (35.8,77.6) 63.9 (41.8,80.3) 73.8 (46.2,85.6) 80.2 (49.7,89.5) 87.3 (53.6,92.9) 86.2 (52.9,92.1) 85.6 (52.6,91.6)

TABLE 2A-VARIANCE DECOMPOSITION OF OUTPUT AND UNEMPLOYMENT (No DUMMY BREAK, TIME TREND IN UNEMPLOYMENT)

Smets and Tsatsaronis (1997)

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“Why does the yield curve predict economic activity? ” BIS Working Paper No.49.

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Objective: Investigating why the slope of the yield curve predicts future economic activity in Germany and the United States.

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Methodology: A structural VAR is used to identify aggregate supply, aggregate demand, monetary policy and inflation scare shocks and to analyse their effects on the real, nominal and term premium components of the term spread and on output.

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Findings: In both countries demand and monetary-policy shocks contribute to the covariance between output growth and the lagged term spread, while inflation scares do not.

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Yield-curve slope (10yr-3mth) vs. Output gap, Euro area data, Source: OECD

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