New Keynesian Macroeconomics - LMU

New Keynesian MacroeconomicsChapter 3: VAR Models

Prof. Dr. Kai Carstensen

ifo Institute for Economic Research and LMU Munich

Prof. Dr. Kai Carstensen (LMU Munich) New Keynesian Macroeconomics, Ch. 3 1 / 47

The identification problem in macroeconomics


The identification problem in macroeconomics

What are the effects of monetary policy on output and inflation?

→ Problem: Identify exogenous changes in monetary policy

Traditional Cowles commission approach:

Backward-looking models with exogeneity restrictions, assume that monetary policy instruments(money, interest rate) are exogenous, while all other macroeconomic variables, including outputand inflation, are endogenous

Great critiques:

Lucas (1976): backward-looking models do not take expectations explicitly into account, andidentified parameters are hence a mixture of deep parameters reflecting e.g. household’spreferences or technology and expectational parameters, which are not stable across policyregimes → development of dynamic stochastic general equilibrium (DSGE) models

Sims (1980): no variable can be seen as exogenous in a world of forward-looking agents whosebehavior depends on the solution of an intertemporal optimization model → development ofvector autoregressive (VAR) models with endogenous monetary policy


The VAR model

Reference: H. Lutkepohl (1993) Introduction to Multiple Time Series Analysis, Springer.


Example (1)

Consider the following example:

xt = −a012mt + a1

11xt−1 + a112mt−1 + d11ε1t + d12ε2t (1)

mt = −a021xt + a1

21xt−1 + a122mt−1 + d21ε1t + d22ε2t (2)

where xt denotes output and mt is a monetary aggregate.

Let us assume that the structural shocks to output and money, ε1t and ε2t , are uncorrelated (andnormally distributed). Then these shocks occur independently from each other. Hence, a ceterisparibus interpretation of each shock is possible.

Note that the parameters of these two equations are not estimable. This can be seen easily whenyou remember how OLS works. You need predetermined or exogenous variables that can be usedto generate normal equations/orthogonality restrictions of the type X ′u = 0. In each of the aboveequations, there are only two predetermined variables on the right-hand side of each equation(xt−1 and mt−1) which can be used to estimate two parameters. This leaves three parametersundetermined.


Example (2)

To make more general statements, let us write the system of equations

xt = −a012mt + a1

11xt−1 + a112mt−1 + d11ε1t + d12ε2t

mt = −a021xt + a1

21xt−1 + a122mt−1 + d21ε1t + d22ε2t

in matrix form as

(1 a0

12a0

21 1

)(xtmt

)=

(a1

11 a112

a121 a1

22

)(xt−1

mt−1

)+

(d11 d12

d21 d22

)(ε1t

ε2t

)or simply as

A0yt = A1yt−1 + Dεt . (3)


The general VAR(p) model

In general, the VAR(p) model in structural form can be written as

A0yt = A1yt−1 + ...+ Apyt−p + Dεt (4)

or in more compact form as

A (L) yt = Dεt

where A (L) = A0 − A1L− A2L2 − ...− ApLp and yt is a n × 1 vector of variables.

The lag operator simply means Liyt = yt−i .

The structural shocks are assumed to be mutually uncorrelated and their variances are normalized

to unity. Hence, the covariance matrix is E[εtε

′t

]= I , where I is the identity matrix.


The reduced form (1)

The reduced form of the example VAR is given by

yt = A−10 A1yt−1 + A−1

0 Dεt

yt = B1yt−1 + ut

where B1 = A−10 A1 and ut = A−1

0 Dεt .

In general, the VAR(p) model in reduced form is:

yt = B1yt−1 + ...+ Bpyt−p + ut

or

B (L) yt = ut

where Bi = A−10 Ai . The disturbances ut are, in general, contemporaneously correlated with

covariance matrix E[utu

′t

]= Σ.


The reduced form (2)

Note that the VAR(p) model in reduced form

yt = B1yt−1 + ...+ Bpyt−p + ut

is just identified. For each equation there as many predetermined right-hand side variables asparameters in B1, . . . ,Bp . Hence, they can be estimated by OLS using a sample of observationst = 1, . . . ,T plus p pre-sample observations t = −p + 1, . . . , 0 for the lags.

The covariance matrix Σ can be estimated from the OLS residuals as

Σ =1

T

T∑t=1

ut u′t .


Identification of the structural form parameters (1)

The question now is how to recover the structural parameters, which the researcher is interestedin, from the reduced form parameters. This is only possible if the structural form parameters arerestricted such that there are at most as many structural as reduced form parameters.

To this end, observe that the number of parameters in the reduced form matrices B1, . . . ,Bp

equals the number of parameters in the structural form matrices A1, . . . ,Ap . From the definition

of the reduced form, we know that Bi = A−10 Ai and thus Ai = A0Bi . If A0 was known, it would

thus be straightforward to calculate estimates of A1, . . . ,Ap from the OLS estimates ofB1, . . . ,Bp .

Consequently, all we need is to find an estimate for A0 and D.



How can we estimate A0 and D?

Remember that

ut = A−10 Dεt and thus εt = D−1A0ut ,

E[εtε

′t

]= I and E

[utu

′t

]= Σ.

This implies

I = E[εtε

′t

]= E

[D−1A0utu

′tA

′0

(D

′)−1]

= D−1A0 E[utu

′t

]︸︷︷︸

Σ

A′0

(D

′)−1.

Thus, I = D−1A0ΣA′0

(D

′)−1

or alternatively

Σ = A−10 DD

′ (A

′0

)−1. (5)

This is a set of non-linear equations that, in principle, allow to calculate the elements in A0 andD from Σ.



Consider again the example in Eqs. (1) and (2). We have

Σ =

(σ11 σ12

σ12 σ22

)The matrices A0 and D contain 6 unknown parameters that shall be uncovered from the 3 distinctelements in Σ, namely from σ11, σ12 and σ22. Hence, we need 3 additional identifying restrictions.

To this end, let us assume that the output shock does not contemporaneously affect mt and thatthe monetary shock does not contemporaneously affect output: d12 = d21 = 0. Moreover, assumethat changes in the monetary aggregate have only a delayed impact on output: a0

12 = 0.

The system given by Eqs. (1) and (2) then reduces to

xt = a111xt−1 + a1

12mt−1 + d11ε1t

mt = −a021xt + a1

21xt−1 + a122mt−1 + d22ε2t



To solve for the parameters of the structural VAR model, plug these restrictions in Eq. (5) andobtain

(σ11 σ12

σ12 σ22

)=

(1 0a0

21 1

)−1 (d11 00 d22

)(d11 00 d22

)′ (1 a0

210 1

)−1

which implies that

(σ11 σ12

σ12 σ22

)=

(d2

11 −a021d

211

−a021d

211

(a0

21

)2d2

11 + d222

)

and thus d211 = σ11, a0

21 = −σ12σ11

and d222 = σ22 −

σ212σ11

.


Identification in VAR models — Summary

In VAR models, the classical identification problem boils down to the question how torecover the structural form parameters form the reduced form.

In principle, all structural VAR (SVAR) models use the same procedure to solve this problem.

First, they use an appropriate number of restrictions on the elements in A0 and D to recover

them from the reduced form covariance matrix, i.e., to solve Σ = A−10 DD

′(A

′0

)−1.

Second, given an estimate of A0, the remaining parameters in A1, . . . ,Ap are calculated fromthe reduced form according to Ai = A0Bi .

However, there are many possible ways to restrict the elements in A0 and D. Thus there aremany different SVAR models.


Specific identification schemes — Overview

Approaches to identify the structural parameters, i.e., to restrict the elements in A0 and D:

Choleski decomposition of Σ (=recursiveness assumption on the contemporaneousrelationships between shocks and variables)

General restrictions on the contemporaneous relationships between shocks and variables

Sign restrictions on the contemporaneous and medium-run relationships between shocks andvariables

Restrictions on the long-run relationships between shocks and variables

Restrictions on the long-run relationships between shocks and variables, taking cointegrationrestrictions into account


Specific identification schemes — Cholesky decomposition

Sims (1980) proposes a Cholesky decomposition of Σ to identify the VAR model. The Choleskydecomposition imposes a recursive structure on the model, implying the ordering of the variablesin yt matters.

A0 =

1 0 · · · 0

a021 1

. . ....

.... . .

. . . 0a0n1 · · · a0

n,n−1 1

, D =

d11 0 · · · 0

0 d22

. . ....

.... . .

. . . 00 · · · 0 dnn

or alternatively

A0 =

a0

11 0 · · · 0

a021 a0

22

. . ....

.... . .

. . . 0a0n1 · · · a0

n,n−1 a0nn

, D =

1 0 · · · 0

0 1. . .

......

. . .. . . 0

0 · · · 0 1

A0 and D contain n (n + 1) /2 free elements and hence as many free elements as in Σ. Thus, themodel is just identified.


Specific identification schemes — General contemporaneous restrictions

The Cholesky decomposition has the virtue of simplicity but is completely atheoretical. We canalternatively achieve identification by directly imposing economically meaningful restrictions onA0 and D. This gives rise to structural VAR models (SVARs) in a stricter sense.

Consider the general model

A0yt = A1yt−1 + ...+ Apyt−p + Dεt .

The partial, direct relation between the variables yt and the shocks εt is

A0yt = Dεt

To identify the model, we again need at least n (n + 1) /2 identifying restrictions, which can bederived, e.g., from economic models, timing constraints, or the institutional setting.

Examples for this and other approaches will be given below.


Estimating VAR models — Likelihood function

Maximum Likelihood (ML) estimation:

Suppose that

ut ∼ N (0,Σ) , t = 1, ...,T

The multivariate (log) likelihood function is then given by

lnL = c −T

2log |Σ| −

1

2

T∑t=1

u′tΣ−1ut

Replace ut by the VAR model and obtain

lnL = c −T

2log |Σ| −

1

2

T∑t=1

(yt − B1yt−1 + ...+ Bpyt−p)′

Σ−1 (yt − B1yt−1 + ...+ Bpyt−p)


Estimating VAR models — Estimate of B1, ...,Bp

First step: estimate the reduced form parameters B1, ...,Bp by OLS and replace ut by theestimated first step residuals ut to obtain the concentrated likelihood function

lnL = c −T

2log |Σ| −

1

2

T∑t=1

u′tΣ−1ut (6)

that is now independent of the reduced form parameters. Simplify Eq. (6) to

lnL = c −T

2log |Σ| −

T

2tr(

Σ−1Σ)

where tr denotes the trace of a matrix (= sum of all elements on the main diagonal) and

Σ = ut u′t/T is the estimated covariance matrix of the reduced form residuals.


Estimating VAR models — Estimate of A0 and D

Second step: estimate the structural parameters in A0 and D using the relation

Σ = A−10 DD

′(A

′0

)−1. We have that

lnL = c −T

2log |A−1

0 DD′ (

A′0

)−1| −

T

2tr

[(A−1

0 DD′ (

A′0

)−1)−1

Σ

]

Maximizing lnL with respect to A0 and D for a given Σ leads to a unique solution for A0 and D ifa sufficiently large number of restrictions is considered.

If the model is just identified, we can directly calculate A0 and D from Σ: Σ = A−10 DD

′(A

′0

)−1,

which is not possible if the model is overidentified.

Given the estimate of A0, the remaining parameters in A1, . . . ,Ap are calculated according to

Ai = A0Bi .


Estimating VAR models — Test for overidentifying restrictions

In case the overidentifying restrictions are correct, deviations of A−10 DD

′(A

′0)−1 from Σ are

completely random. This can be used to test for the validity of the overidentifying restrictions.Since we use a maximum likelihood approach, it is straightforward to implement a likelihood ratiotest.

Under H0 : Σ = A−10 DD

′(A

′0)−1, the most efficient estimator of the unknown covariance matrix

Σ is the restricted estimator Σ0 = A−10 DD

′(A

′0)−1. The likelihood function then is

lnL0 = c −T

2log |Σ0| −

T

2tr(

Σ−10 Σ

)= c −

T

2log |A−1

0 DD′(A

′0)−1| −

T

2tr

[(A−1

0 DD′(A

′0)−1

)−1Σ

].

Under H1, the model is just identified. Thus, the best estimate of the unknown covariance matrixΣ is the unrestricted reduced form estimator Σ1 = Σ. The likelihood function then is

lnL1 = c −T

2log |Σ1| −

T

2tr(

Σ−11 Σ

)= c −

T

2log |Σ| −

T

2tr (In) = c −

T

2log |Σ| −

nT

2.

The test statistic LR = 2 (lnL1 − lnL0) is asymptotically χ2 distributed with the degrees offreedom equal to the number of overidentifying restrictions.


Interpreting VAR models — Impulse response functions (1)

Impulse response functions (IRFs):

The IRFs build on the vector moving average presentation (VMA) of the VAR model:

yt = A (L)−1 Dεt = B (L)−1 A−10 Dεt

=∞∑i=0

Ci εt−i = C (L) εt

which requires the invertibility of the VAR polynomial. The matrices Ci are recursively calculatedas follows

Ci = CiA−10 D with C0 = I for i = 0

and Ci =i∑

k=1

Ci−kBk for i = 1, 2, ...



The VMA matrices Ci represent the partial effect of εt on yt+i since

yt+i = C0εt+i + ...+ Ci εt + ...

and hence

∂yt+i

∂ε′t

= Ci =

c i11 · · · c i1n...

. . ....

c in1 · · · c inn

meaning that c ikl is the partial effect of the l th structural shock ε1,t on the kth variable after i

periods (yk,t+i ). The sequence (c0kl , c

1kl , c

2kl , ...) is the IRF of the kth variable after a structural

shock l .



Even though it is possible to derive the asymptotic distribution of the confidence intervals, inpractice simulation methods are used instead.

For example, we could use the following algorithm.

Estimate the parameters A0, ...,Ap and D of the VAR model and obtain A0, ..., Ap and D.

Under the assumption that εt ∼ N (0, I ), we can simulate a series of artificial shocksε1, ..., εT using a random number generator. Using this series in combination with theestimated parameters allows us to generate a sequence of artificial data:

yt = A−10 A1yt−1 + ...+ A−1

0 Ap yt−p + A−10 D εt , t = 1, ...,T .

Re-estimate the VAR model, now using the artificial data series, and again calculate andsave the IRFs.

Repeat the last two steps a large number of times, e.g. 1000 times. On the basis of thedrawings kept, calculate, e.g., the 2.5% and 97.5% quantiles.


Interpreting VAR models — Forecast error variance decomposition (1)

Forecast error variance decomposition (FEVD):

Consider again the VMA representation of the VAR model:

yt+h = C0εt+h + C1εt+h−1 + C2εt+h−2 + ... (7)

The h-step-ahead forecast for yt+h is given by

Et [yt+h] = Et [C0εt+h + C1εt+h−1 + C2εt+h−2 + ...] = Chεt + Ch−1εt−1 + ... (8)

since Et [εt+i ] = 0 for i > 0.

The h-step-ahed forecast error is the difference between Eqs. (7) and (8):

yt+h − Et [yt+h] = C0εt+h + C1εt+h−1 + C2εt+h−2 + ...+ Ch−1εt+1



The covariance of the h-step-ahead forecast error can be calculated as follows

Σh = Et

[(yt+h − Et [yt+h]) (yt+h − Et [yt+h])

′]= C0Et

[εt+hε

′t+h

]C

′0 + C1Et

[εt+h−1ε

′t+h−1

]C

′1 + ...+ Ch−1Et

[εt+1ε

′t+1

]C

′h−1

= C0C′0 + C1C

′1 + ...+ Ch−1C

′h−1

since Et

[εt+i ε

′t+i

]= I for i > 0. Given that the shocks are uncorrelated, we can decompose the

total forecast arror variance into the contribution of the respective shocks. For h = 1 we have forexample

Σ1 = C0C′0 =

∑n

i=1

(c0

1i

)2 · · ·∑n

i=1 c01ic

0ni

.... . .

...∑ni=1 c

01ic

0ni · · ·

∑ni=1

(c0ni

)2



The forecast error variance of the first variable y1t is the first diagonal element of Σ1, i.e.

n∑i=1

(c0

1i

)2=(c0

11

)2+(c0

12

)2+ ...+

(c0

1n

)2

where(c0

11

)2denotes the variance that is due to the first shock,

(c0

12

)2the variance that is due to

the second shock, and so on. Thus,

(c0

11

)2/

n∑i=1

(c0

1i

)2

is for example the 1-step-ahead forecast error variance share for the variable y1t which is due tothe first shock ε1t .

To obtain confidence intervals, we could again use simulation techniques.


Identification strategies for VAR models used in the literature


Cholesky decomposition (1)

Cholesky decomposition (Christiano, Eichenbaum, Evans, 2000):

Consider a VAR(p) model in reduced form:

yt = B1yt−1 + ...+ Bpyt−p + ut (9)

where yt is a n× 1 vector of variables and ut is a n× 1 vector of residuals with a n× n covariance

matrix E[utu

′t

]= Σ. We can obtain consistent estimates of the n × n coefficient matrices Bi ,

i = 1, ..., p, and of Σ by running OLS equation by equation on Eq. (9).

But we cannot directly compute the impulse responses for yt to a monetary policy shock sinceeach element of the residual vector ut reflects the impact of all fundamental shocks and there isno good reason to presume that any element of ut corresponds to a particular shock, say amonetary policy shock εmt . This is refelected by

ut = A−10 Dεt (10)

where εt is a n × 1 vector of fundamental shocks, including, among others, the monetary policyshock εmt . In what follows, we assume that D = In and define A0 ≡ A−1

0 .



To identify a monetary policy shock, we assume a recursive model structure of the following form:

yt =

x1,t

FFRt

x2t

where FFRt is the Federal Funds Rate or monetary policy instrument, x1t is a n1 × 1 vector ofvariables that do not contemporaneously respond to a monetary policy shock, x2t is a n2 × 1vector of variables that instantaneously respond to a monetary policy shock and n1 + n2 + 1 = n.

The recursiveness assumption imposes the following restrictions on the system (forn1 = 2, n2 = 1):

ut =

a0

11 0 0 0a0

21 a022 0 0

a031 a0

32 a033 0

a041 a0

42 a043 a0

44

−1

ε1tε2tεmtε4t

=

a0

11 0 0 0a0

21 a022 0 0

a031 a0

32 a033 0

a041 a0

42 a043 a0

44

ε1tε2tεmtε4t

,

where we use fact that the inverse of a lower triangular matrix is again lower triangular. Thisidentification scheme allows us to interpret the shock associated with the interest rate equation asa monetary policy shock εmt .



In practice, we can directly compute A0 (and thus A−10 ) using the unique lower Cholesky factor of

the estimated covariance matrix Σ, such that A0A′0 = Σ.

Example: Let yt contain output, inflation, the Federal Funds Rate and a monetary aggregate(e.g. M1) in that order. The recursiveness assumption implies that output and inflation respondwith a lag of one period only to the monetary policy shock, while the monetary aggregateresponds within the period.


The sign restriction approach (1)

Sign restrictions (Uhlig, 2005):

The Cholesky factorization above uses only one particular decomposition of the covariance matrixto identify the monetary policy shock: A0A′0 = Σ. However, if there exists a n × n orthonormalmatrix Q such that QQ′ = I , the impulse matrix

A0Q

is also feasible because it continues to hold that

A0QQ′A′0 = A0A′0 = Σ.

Hence, it is possible to construct infinitely many impact matrices and thus impulse responsefunctions. This is just another way to describe the identification problem: without restrictions, itis impossible to infer the structural from the reduced form model.

Quite often economic theory does not offer strict parametric restrictions (such as recursiveness)that allow to pin down the structural parameters. However, it might be possible to derive signrestrictions from theory. For example, one could assume that the policy interest rate should moveupwards on impact after a contractionary monetary policy shock. Sign restrictions of this kindcan be used to identify the impact matrix.



Sign restrictions can be imposed as follows.

1 Estimate the reduced form VAR and calculate the Cholesky factor A0 such that A0A′0 = Σ.

2 Construct a large number of Q matrices that are in some sense representative of theinfinitely large set of possible Q matrices that satisfy QQ′ = I .

3 From each Q matrix construct the corresponding impact matrix A0Q and impulse responsefunctions.

4 Check whether the impulse responses fulfils your sign restrictions. (The restrictions mayapply not only to the impact matrix but to any impulse horizon.) If yes, keep this draw. Ifnot, discard it.

5 For all feasible draws, show the results: either the full set of IRF’s or, e.g., the mediantogether with the upper and lower bound of all IRF’s or the 10 and 90 percent quantiles.

From this it is obvious that sign restrictions typically cannot identify a single structural model buta range of models. This is certainly a drawback!



Figure: Some impulse responses for GDP—together with upper and lower bounds—after a monetary policyshock calculated from models that satisfy certain sign restrictions (Uhlig, 2005, Fig. 2, upper left panel)



How to find the Q’s (cf. Fry and Pagan, 2009):

In high dimensional systems, one may draw the elements of a n × n matrix W from the standardnormal distribution and apply the so-called QR factorization to W which yields an orthonormalmatrix Q. Do this as many times as desired.

In a two-dimensional system, it is instructive to proceed in a more straightforward way. Note thatany rotation matrix Rα that rotates points in an Euclidian coordinate system by the angle αsatisfies RαR′α = I . Hence, we can construct a set of Rα matrices for a fine grid of anglesα = 0, . . . , 2π.

This can be achieved using the specific form

Rα =

[cos α −sinαsinα cos α

]For example, α = π/2 yields a clockwise rotation by 90 degrees.



Figure: Rotation of the Cholesky matrix

−2 0 2−2

−1

0

1

2Columns of the Cholesky factor

−2 0 2−2

−1

0

1

2Rotation by 0.5 π

−2 0 2−2

−1

0

1

2Rotation by π

−2 0 2−2

−1

0

1

2Rotation by 1.5 π

Covariance matrix Σ =

[3.0625 1.31251.3125 2.8125

], Cholesky factor A0 =

[1.75 00.75 1.5

]Prof. Dr. Kai Carstensen (LMU Munich) New Keynesian Macroeconomics, Ch. 3 36 / 47


Uhlig (2005) uses a Bayesian approach to estimate the structural VAR with sign restrictions. Thisis straightforward because sign restriction may be interpreted as priori knowledge that–in aBayesian sense–can be combined with observations to yield sharper a posteriori knowledge.Consider the same example as for the Cholesky approach above: Let yt contain output, inflation,the Federal Funds Rate and a monetary aggregate. Consistent with a wide class of DSGE modelsone may impose the restrictions that a monetary policy shock raises the Federal Funds Rates butdecreases output, inflation and the monetary aggregate.

The advantage of this approach is not only that it allows to use sign restriction readily availablefrom theory. It also makes public the otherwise informal and typically unreported search formodels that researchers find “reasonable”.

The disadvantages are also obvious. Using sign restrictions typically cannot identify a singlestructural model but a range of models. There is no guarantee that, e.g., there is any singlemodel that behaves like the median of this range (which is typically reported). Also note thatusing only weak restrictions may result in including models which do not make sense. Inparticular, if only restrictions concerning one shock are used, there are two problems (see Fry andPagan, 2009): First, the shock of interest need not be uncorrelated with the remaining(unidentified) shocks. Second, it is not ruled out that one of the remaining shocks satisfies thesame restrictions as the shock of interest which makes the interpretation unclear.


Short-run restrictions (1)

Short-run restrictions (Bernanke and Mihov, 1998):

In this identification scheme some a priori information is used to impose restrictions on theelements of matrices A0 and D, different from the Cholesky ordering.

Sometimes, as in this example, only identification of one shock (the monetary policy shock) isachieved while the other shocks are left unrestricted up to the assumption that they areuncorrelated with the shock of interest.

Bernanke and Mihov include real GDP (GDP), consumer prices (P), commodity prices (Pcm), theFederal Funds Rate (FF), the quantity of total bank reserves (TR) and the amount ofnon-borrowed reserves (NBR). Their maximum sample is 1965M1 to 1996M12.

To identify the monetary policy shock, they make the following assumptions: (a) Thefundamental shocks are uncorrelated; (b) the macro variables do not simultaneously respond tomonetary variables (reaction lag), but not vice versa; and (c) the monetary block of the modelreflects the operational procedure implemented by the monetary policy authority.



To understand the operational model, remember that banks need to hold a certain fraction oftheir deposits as reserves. In the US this is currently roughly 10 percent. Reserves are either vaultcash (that can be directly used to pay out customers) or the banks’ balances on its FederalReserve account. These balances can come from

open market operations of the Fed (in particular, purchases and sales of US Treasury andfederal agency securities by the Fed),

the discount window of the Fed (borrowed reserves: against collateral, at the discount rate)or

unsecured lending from other banks that have more balances on their Federal Reserveaccount than necessary. This is done on the Fed funds market. The interest rate forovernight lending is called the Fed funds rate.

Hence, non-borrowed reserves is total reserves minus borrowed reserves (borrowed at the discountwindow).



The contemporaneous relations among the Federal Funds Rate and the reserve aggregates arederived from a specific model of the reserve market:

uTR = −αuFF + νD

uBR = β(uFF − uDISC ) + νB

uNBR = φDνD + φBνB + νS

The first equation states that an innovation in banks’ demand for total reserves (TR) dependsnegatively on the innovation in the Fed funds rate (the cost of reserves), and on the demanddisturbance νD .

The second equation states that an innovation in the demand for borrowed reserves (BR) dependson the difference between the innovation in the discount rate (the cost of borrowed reserves) andthe innovation in the Fed funds rate (the rate at which borrowed reserves can be relent), and on aborrowing disturbance νB . As discount rate changes very infrequently, in the following it isassumed that its innovations are always zero.

The supply of non-borrowed reserves in the last equation reflects the behavior of the Fed, whereνS is the monetary policy shock of interest. It is assumed that the Fed reacts to demand andborrowing shocks within the period.



Combining the market for reserves with the macro variables, we can explicitly write A0ut = Dεt as

1 0 0 0 0 0a0

21 1 0 0 0 0a0

31 a032 1 0 0 0

a041 a0

42 a043 1 − 1

β1β

a051 a0

52 a053 α 1 0

a061 a0

62 a063 0 0 1

uGDPt

uPt

uPcmt

uFFt

uTRt

uNBRt

=

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 − 1

β 0 0

0 0 0 0 1 0

0 0 0 φB φD 1

νNP1t

νNP2t

νNP3t

νBt

νDt

νSt

Note that

VAR residuals from the first three equations, describing the non-policy part of the system,are orthogonalized simply by assuming a recursive (Cholesky) structure for the correspondingblock of the A0 matrix, yielding non-policy shocks without any structural interpretation.

The number of restrictions imposed on A0 and D are still not sufficient to identify themonetary policy shock: there are 16 structural but only 15 reduced form (sub-diagonal)parameters.

Here the model is written assuming that the covariance matrix of the structural disturbancesis diagonal but the diagonal elements are unrestricted. To be fully in line with our notation(structural covariance matrix = identity matrix), one would have to add 6 parameters on themain diagonal of A0.



As it stands, the model is underidentified and needs at least one more restriction. This can bederived from additional assumptions regarding the behavior of the Fed.

At the time of the analysis, this behavior was far from obvious: “The Federal Reserve’s objectivefor open market operations has varied over the years. During the 1980s, the focus graduallyshifted toward attaining a specified level of the federal funds rate, a process that was largelycomplete by the end of the decade. Beginning in 1994, the FOMC began announcing changes inits policy stance, and in 1995 it began to explicitly state its target level for the federal fundsrate.”(Cited from www.federalreserve.gov/monetarypolicy/openmarket.htm)



Bernanke and Mihov examine several different such assumptions. For example, the Fed may havetargeted the Fed funds rate, non-borrowed reserves or borrowed reserves. In the following, wefocus on one assumption, namely that the Fed targets the Federal funds rate. This means thatthe Fed completely offsets demand and borrowing shocks to the Fed funds rate: φD = 1 andφB = −1. Consequently, the innovation in the Fed funds rate reflects solely the monetary policyshocks. Now the monetary policy block reads as:

uTRt

uNBRt

uFFt

=

1 αβ+α

0

1 1 −10 − 1

β+α0

νDtνStνBt

The model is now over-identified. Note that this identification scheme is equivalent to a Choleskyidentification with the ordering Federal Funds Rate, total reserves and non-borrowed reserves plusadditional within and cross equation restrictions.


Long-run restrictions (1)

Long-run restrictions (Blanchard and Quah, 1989):

In contrast to the previous identification schemes, we now restrict the long-run behavior of thesystem to identify a shock. For example, we may assume that in the long-run demand shocks εdthave zero impact on output, while supply shocks εst may have long-run effects.

In a stationary VAR, the partial effect of a shock in period t on the variables in period t + iapproaches zero as i →∞: C∞ = 0.

Therefore, the long-run effect of a shock is defined equivalently as either

the cumulative effect on yt , yt+1, yt+2, . . . of a single on-off unit shock εt in period t or

as the long-run (steady state) effect on y of a permanent change in ε (=a series of on-offshocks in t, t + 1, t + 2, . . .).



Consider the first definition (the second one is treated in the tutorial). For simplicity assume thatA0 = In but leave D completely unrestricted. The long-run or cumulative effect (LRE) of a singleon-off unit shock εt is

LRE =∞∑i=0

∂yt+i

∂ε′t=∞∑i=0

Ci = C0 + C1 + · · ·+ C∞ = C(1).

Use the definition C(L) = A(L)−1D to write

C(1) = A(1)−1D =

(I −

p∑i=1

Ai

)−1

D.

Note that for A(1) to be invertible the VAR must be specified in stationary variables.

Due to the assumption A0 = I , the coefficients in Ai are readily obtained from the reduced form(and thus identified). The long-run restrictions are then chosen such that certain elements inC(1) are zero.



Blanchard and Quah (1989) include two variables: the unemployment rate UNt and the logdifference (≈ growth rate) of output ∆LYt . Neglecting deterministics, the VAR is specified asfollows:(

∆LYt

UNt

)= A1

(∆LYt−1

UNt−1

)+ ...+ A8

(∆LYt−8

UNt−8

)+

(d11 d12

d21 d22

)(εstεdt

)For identification, one long-run restriction on D is needed (three parameters in D are identified byrequiring the structural shocks to be uncorrelated). To impose the identifying restrictions considerthe long-run effect matrix

LRE =

(I −

8∑i=1

Ai

)−1 (d11 d12

d21 d22

)=

(k11 k12

k21 k22

)(d11 d12

d21 d22

)

The demand shock εdt (ordered second) is identified by imposing that its long-run impact on thelevel of output (which is equivalent to the cumulative effect on the log changes) is zero:

k11d12 + k12d22 = 0

Note the difference between this methodology and the Cholesky decomposition, which wouldsimply restrict d12 in the instantaneous impact matrix to zero.


Long-run with cointegration (1)

Long-run with cointegration (King, Plosser, Stock and Watson, 1991):

Consider the three variables model yt = (qt , ct , it)′, where qt denotes (log) output, ct is (log)

consumption and it stands for (log) investment. Assume that all three variables are I (1), i.e.,non-stationary, and that the ratios (ct − qt) and (it − qt) are stationary, i.e., the variables arecointegrated with cointegration vectors (1,−1). Hence, there is only one permanent innovation(the number of variables less the number of cointegrating vectors). Call this the productivityshock εat . In addition, there are two transitory shocks, ε1

t and ε2t that are not of specific interest.

Similar to Blanchard and Quah (1989), we can write the long-run solution of the model (instationary variables) as follows

∆qtct − qtit − qt

=

k11 k12 k13

k21 k22 k23

k31 k32 k33

d11 d12 d13

d21 d22 d23

d31 d32 d33

εatε1tε2t

,

where the k’s can be estimated from the reduced form. We assume that the two transitoryshocks have no long-run effects on output, implying that k11d12 + k12d22 + k13d32 = 0 andk11d13 + k12d23 + k13d33 = 0. Since we need three restrictions in total to identify the system, weimpose that the second transitory shock ε2

t has no contemporaneous impact on the consumptionto output ratio (ct − qt). One may set d23 = 0.


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