An IV framework for combining sign and long-run parametric restrictions in SVARs

Journal of Macroeconomics 61 (2019) 103125

Contents lists available at ScienceDirect

Journal of Macroeconomics journal homepage: www.elsevier.com/locate/jmacro

An IV framework for combining sign and long-run parametric restrictions in SVARs

T

Lance A. Fishera, Hyeon-seung Huhb,

⁎

a b

Department of Economics, Macquarie University, Sydney 2109, Australia School of Economics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 03722, Republic of Korea

ARTICLE INFO

ABSTRACT

JEL classification: C32 C36 C51 E30

This paper derives the sufficient conditions on the coefficients in a system of structural equations which imply the long-run exclusion restrictions on the impulse responses which are used to identify the model. A recent contribution shows a method to impose the sufficient conditions on the structural equations and to estimate them by instrumental variables (IV). This method has the advantage that it can be combined with a new method for sign restrictions which can be cast in an IV framework. This paper shows that the sufficient conditions which imply the two long-run exclusion restrictions in a SVAR taken from the literature imply other long-run exclusion restrictions as well which are not part of the identifying assumptions. In this case, this method is not suitable. This paper shows how to impose the two long-run exclusion restrictions directly on the structural equations of the model on each draw in sign restrictions which utilizes the IV method.

Keywords: Structural vector-autoregression Sign restrictions Long-run parametric restrictions Instrumental variables Generated coefficients Algorithms

1. Introduction The method in Fisher, Huh and Pagan (FHP, 2016) for imposing zero long-run identifying restrictions in SVARs, which draws on the earlier instrumental variables (IV) approach of Shapiro and Watson (1988), is particularly suitable for combining long-run zero restrictions with the sign restrictions methodology of Ouliaris and Pagan (OP, 2016) in an IV setting. In fact, OP (2016, p. 614) suggest the method in FHP as one way to combine the two long-run zero restrictions on the responses with their sign restriction methodology in the context of their small macro model. The method in FHP shows how to impose the sufficient conditions on the coefficients in a system of structural equations which imply the long-run zero restrictions on the responses which are used to identify the model. However, the sufficient conditions in some SVARs can imply other long-run zero restrictions on the responses as well which may not be justifiable on economic grounds. In such circumstances, the method in FHP is not suitable and another method is required which can be easily combined with the sign restrictions approach of OP, which utilizes IV estimation. This paper develops such a method in the context of Peersman's SVAR (2005). The paper also considers a variation of Peersman's model for which the method in FHP is suitable. Traditionally, SVARs are identified by imposing zero restrictions on the contemporaneous and/or long-run impulse responses of the variables to certain shocks. An early example is Galí (1992) who identified the structural shocks by a combination of contemporaneous and long-run zero restrictions on the responses. Peersman (2005) also used both types of exclusion restrictions for the case of full parametric identification of his model. Fisher et al. (2000) extended the identification that utilizes both contemporaneous

⁎

Corresponding author. E-mail addresses: [email protected] (L.A. Fisher), [email protected] (H.-s. Huh).

https://doi.org/10.1016/j.jmacro.2019.103125 Received 23 April 2018; Received in revised form 27 April 2019; Accepted 4 May 2019 Available online 05 May 2019 0164-0704/ © 2019 Elsevier Inc. All rights reserved.

Journal of Macroeconomics 61 (2019) 103125

L.A. Fisher and H.-s. Huh

and long-run zero restrictions to cointegrated SVARs.1 The methods of these papers produce a system of non-linear equations in the unrestricted coefficients which are solved for using a non-linear equation solver. In an important paper, Rubio-Ramírez et al. (2010) formulate the problem of imposing contemporaneous and long-run zero restrictions to exactly identify a SVAR as one of finding a rotation matrix Q, which satisfies the exclusion restrictions. This produces a system of linear equations in the unrestricted coefficients which can easily be solved for using matrix methods. In the same paper, they establish the rank conditions for identification of SVARs. Sign restrictions on the impulse responses have recently become a popular way of identifying the shocks in SVARs. The sign restrictions approach was first introduced by Faust (1998), Canova and De Nicoló (2002) and Uhlig (2005). This approach involves rotating an initial set of orthogonal shocks in a SVAR by applying an orthogonal rotation matrix Q to form a new set of orthogonal shocks from which impulse responses are obtained. If the responses satisfy the sign restrictions they are retained and, if not, they are discarded. In order to consider the responses from all possible SVARs that are consistent with the reduced-form VAR, the rotation matrix Q on each draw needs to come from the space of all possible orthogonal matrices Q. Canova and De Nicoló (2002), and Peersman (2005) construct Q by way of Givens rotation matrices while Rubio-Ramírez et al. (2010) obtain Q from the QR decomposition of a matrix, which may be computed using the Householder transformation. Baumeister and Hamilton (2015) point out that the method of generating the rotation matrix Q will affect the distribution of the impulse responses before the sign restrictions are applied. In the OP (2016) approach to sign restrictions, the above diagonal elements in the matrix of contemporaneous coefficients in the SVAR (the A0 matrix in Eq. (1) below) are drawn randomly and the elements along the principal diagonal are normalized to unity. For each draw of the above diagonal elements, the structural equations of the model are estimated by instrumental variables (IV) estimation. Impulse responses are then obtained and judged by the sign restrictions. Fisher and Huh (2016) apply this method to SVARs of small open economies. This method, rather than sampling from the space of all possible orthogonal Q matrices, samples from the set of all possible values for the elements of A0, which is typically the set of all real numbers. The Baumeister and Hamilton (2015) observation applies here as well as the method by which the above diagonal elements of A0 are generated will affect the distribution of the impulse responses before the application of signs. There is an extensive literature on combining parametric restrictions with sign restrictions when the latter are implemented by rotating the initial set of orthogonal shocks with the matrix Q. Baumeister and Benati (2013) impose a single contemporaneous zero restriction on a response in combination with sign restrictions utilizing the Givens rotation matrix. Haberis and Sokol (2014) extend their methodology to multiple contemporaneous zero restrictions with signs. Benati and Lubik (2014) and Benati (2015) impose longrun exclusion restrictions on the responses in combination with sign restrictions using the Householder transformation i.e. by obtaining Q from a QR decomposition. A more general algorithm for combining contemporaneous exclusion restrictions, long-run exclusion restrictions and sign restrictions which utilizes the Householder transformation has been developed by Arias et al. (2018). It is an extension of the algorithm in Rubio-Ramírez et al. (2010) to the case of where there are fewer zero parametric restrictions than needed for exact identification and sign restrictions are added to these to complete the identification. A detailed exposition of both algorithms is provided in Kilian and Lütkepohl (2017). Binning (2013) developed an algorithm which is similar to that of Arias et al. (2018). OP (2016) provide two examples of combining exclusion restrictions on the responses with their sign restrictions approach. The first combines a contemporaneous zero restriction on an impulse response with their signs approach in a model of the interest rate, inflation and the output gap. They present an algorithm, which estimates the structural equations by IV to implement their signs approach given the contemporaneous restriction. The second combines long-run zero restrictions on the responses with their signs approach in a model of the interest rate, inflation and the log level of output. In this model, output is an I(1) variable. They impose two long-run zero restrictions, namely, that the shocks associated with the interest rate and inflation equations have a zero long-run effect on output and combine these two parametric restrictions with their sign restriction approach to fully identify the shocks. They suggest that the two long-run zero restrictions can be imposed on the SVAR using the method in FHP (2016). In this paper, we show that the method in FHP is suitable to use in this case because the sufficient conditions on the coefficients in the structural equations which imply the two long-run zero restrictions on the responses have no other implications. However, this is not necessarily always the case. In this paper, this is demonstrated for the SVAR of Peersman (2005). Peersman's SVAR consists of four variables which are the oil price, output, consumer prices and the interest rate. The variables are I(1), except for the interest rate which is I(0). He separates the shocks first by using parametric restrictions alone and then by using sign restrictions alone where the signs method is implemented by way of a Givens rotation matrix. In the parametric identification, he utilizes two long-run and four contemporaneous exclusion restrictions. The two long-run restrictions are that the shocks associated with the interest rate and consumer price equations have a zero long-run effect on output. In this paper, we take Peersman's model and data and combine his two long-run zero restrictions with sign restrictions utilizing the approach of OP. We first establish the sufficient conditions on the coefficients in the structural equations which will imply that the response of output to the shocks associated with the interest rate and consumer prices is zero in the long-run. We show that these sufficient conditions imply that some of the other structural shocks have a zero long-run impact on some of the other variables as well and that such restrictions on the

1 An alternative approach which considers identification with contemporaneous and long-run exclusion restrictions together is developed by Lastrapes (1998) in a Bayesian framework. As an example, consider two structural models where the first (Model 1) is identified by contemporaneous zero restrictions, while the second (Model 2) is identified by long-run zero restrictions. This approach calculates the posterior mean of the impulse responses which is the weighted average of the responses at each horizon from Model 1 and Model 2, where the weights are the posterior probability of each model.

2

Journal of Macroeconomics 61 (2019) 103125

L.A. Fisher and H.-s. Huh

model are difficult to justify on economic grounds. While the method in FHP (2016) shows how to impose the sufficient conditions on the structural model, one would not want to proceed with this method as the sufficient conditions also imply zero long-run impacts of some of the shocks that can't be justified by standard macroeconomic reasoning. In cases such as this, another approach which can easily be combined with the sign restrictions approach of OP is required. In this paper, we develop a method to directly impose the two long-run zero restrictions on the responses in Peersman's model, and which is combined with sign restrictions following the approach of OP. This method involves finding the values for certain contemporaneous coefficients in the structural equations which enforce the long-run exclusion restrictions on each draw in the sign restrictions approach of OP. This method is not as general as the approach of Arias et al. (2018), which can implement zero parametric and sign restrictions in SVARs of a general nature. The method here is more restrictive because in some SVARs it may not be possible to solve for the values of the contemporaneous coefficients to enforce the long-run restrictions on each draw. However, this is not the case in Peersman's SVAR which is considered here. The structure of the paper is as follows. Section 2 describes the econometric methods in a three variable setting and discusses their application to the small macro model of OP (2016). Section 3 considers Peersman's model. It shows that the method in FHP for imposing the two long-run zero restrictions on the responses is not suitable and develops a direct method to impose them in combination with the sign restrictions methodology of OP. Section 4 presents the results from this method. Section 5 considers a variation of Peersman's model for which the method in FHP is suitable and presents the results under the long-run zero restrictions and signs. Section 6 provides concluding remarks. 2. Econometric methods 2.1. The structural model Consider the following SVAR with p lags,

A0 z t = A1 z t

1

+ A2 z t

2

+

+ Ap z t

p

+

(1)

t

where zt is an n × 1 vector of variables and ɛt is the n × 1 vector of structural shocks. The structural shocks have diagonal covariance matrix and each of the variables can be either I(0) or I(1). The n × n matrix A0 is the matrix of the contemporaneous coefficients. It has typical element [ aij0], and, when i = j, aij0 = 1 i.e. the particular equation is normalized on a dependent variable. The coefficients on the lagged variables are given by the n × n matrices Ak , k = 1, …, p. The typical element of Ak is [aijk ]. We have excluded deterministic terms from Eq. (1) as that can be done without a loss of generality for what follows. Eq. (1) can be written as:

A (L) z t =

(2)

t

Ap L p . The structural moving average representation is where L is the lag operator i.e. Lz t = z t 1, and A (L) = A0 A1 L A2 L2 z t = A (L) 1 t = C (L) t where C (L) = A (L) 1 is the matrix of the responses of the variables at horizon L to the structural shocks in ɛt. It follows that C (L) A (L) = I , and that: (3)

C (1) A (1) = I

Ap ]. From where C(1) is the matrix of the cumulative impacts of the structural shocks in the long-run and A (1) = [A0 A1 A2 Eq. (3), C (1) = [A (1)] 1 . The key methods of the paper can be demonstrated in the context of a three variable SVAR of arbitrary lag length p, to which we now turn. 2.2. The three variable SVAR Suppose that z t = [ y1t y2t y3t ] , where yit = yit yit 1 . We are assuming that y1t is an I(0) variable and that y2t and y3t are both I(1) variables.2 The I(1) variables enter the structural model in first difference form. We emphasize that the shock associated with the I(0) variable can have a permanent effect on either or both of the I(1) variables, in which case it is a permanent shock. The shock associated with the I(0) variable is a transitory shock only if it is restricted to have a zero long-run effect on both of the I(1) variables. In other words, the shock associated with y1t is a transitory shock only if identifying assumptions are imposed which restrict the shock to have a zero long-run effect on both y2t and y3t. Writing Eq. (3) out in full for the three variable model gives:

c11 (1) c12 (1) c13 (1) c21 (1) c22 (1) c23 (1) c31 (1) c32 (1) c33 (1)

a11 (1) a12 (1) a13 (1) 1 0 0 a21 (1) a22 (1) a23 (1) = 0 1 0 0 0 1 a31 (1) a32 (1) a33 (1)

(4)

where aij (1) = We now consider the question of how to impose one zero restriction in the long-run impact matrix of the structural shocks i.e. in C(1) in this framework. Suppose, for argument sake, that the restriction is c31 (1) = 0. This restriction says that the shock associated with y1t, the I(0) variable, has a zero long-run effect on y3t, the second I(1) variable. The shock associated with the I(0) variable is a

aij0

aij1

aij2

aijp .

2 We assume the two I(1) variables are not cointegrated. If they are, the vector of variables in zt can be specified as y1t, the I(0) error-correction variable, and Δy3t. The case of two I(0) variables and one I(1) variable is considered below.

3

Journal of Macroeconomics 61 (2019) 103125

L.A. Fisher and H.-s. Huh

permanent shock because it can have a non-zero long-run effect on y2t, the other I(1) variable. In this case, there are three permanent shocks and two I(1) variables since the shock associated with the I(0) variable is a permanent shock. For it to be a transitory shock, it must have a zero long-run effect on both of the I(1) variables and, for this to be the case, two restrictions are required, namely, c31 (1) = 0 and c21 (1) = 0. The fact that y1t is I(0) does not imply that the shock associated with it is transitory; it is only transitory if these two long-run restrictions are imposed on it. Our focus at the moment is with one restriction, namely, c31 (1) = 0. Now, from Eq. (4) and the expression for the inverse of a 3 × 3 matrix, we obtain3:

c31 (1) = a21 (1) a32 (1)

(5)

a22 (1) a31 (1)

From Eq. (5), a sufficient condition for c31 (1) = 0 is that a31 (1) = 0 and either a32 (1) = 0 or a21 (1) = 0. Let us consider each of these sufficient conditions in turn.4 Case 1: a31 (1) = 0 and a32 (1) = 0. In the same manner, we obtain the following expression:

c32 (1) = a12 (1) a31 (1)

(6)

a11 (1) a32 (1)

From Eq. (6), these sufficient conditions also make c32 (1) = 0, i.e. they restrict not only the shock associated with y1t but also the shock associated with y2t to have a zero long-run effect on y3t. We summarize the findings here as: Result 1: a31 (1) = 0 and a32 (1) = 0 are sufficient for c31 (1) = 0 and c32 (1) = 0. The sufficient conditions mean that the first differences of y1t and Δy2t appear as right-hand side variables in the equation for Δy3t. For instance, when p = 2, we have: 0 0 1 y3t = a31 y1t + (a31 + a31 ) y1t

1

0 + a32

2y 2t

0 1 + (a32 + a32 )

2y 2t 1

1 + a33 y3t

1

2 + a33 y3t

2

+

(7)

3t

The formulation of Eq. (7) corresponds to what is found in Shapiro and Watson (1988). In a structural equation of theirs, all of the other variables appear on the right-hand side in first difference form. This equation was for the variable upon which all the other shocks had a zero long-run effect on it. Case 1 is an instance of this. Now consider the other case. Case 2: a31 (1) = 0 and a21 (1) = 0. The expression for c21(1) is:

c21 (1) = a23 (1) a31 (1)

(8)

a21 (1) a33 (1)

It can be seen from Eq. (8) that these sufficient conditions also make c21 (1) = 0. They restrict the shock associated with y1t to have a zero long-run effect on y3t and on y2t as well. In this case, the shock associated with the I(0) variable is transitory because it is restricted to have a zero long-run effect on both of the I(1) variables. We summarize the findings here as: Result 2: a31 (1) = 0 and a21 (1) = 0 are sufficient for c31 (1) = 0 and c21 (1) = 0. The sufficient conditions here mean that the first difference of y1t appears as a right hand side variable in the equations for Δy2t and Δy3t. For the two lag case, we have: 0 0 1 y2t = a21 y1t + (a21 + a21 ) y1t

1

1 + a22 y2t

0 0 1 y3t = a31 y1t + (a31 + a31 ) y1t

1

0 1 + a32 y2t + a32 y2t

1

2 + a22 y2t 1

2

0 1 + a23 y3t + a23 y3t

2 + a32 y2t

2

1 + a33 y3t

1

2 + a23 y3t

2

+

2t

(9)

1

2 + a33 y3t

2

+

3t

(10)

This is a demonstration of the general result that if the shocks associated with the I(0) variables are all transitory, i.e. have a zero long-run effect on all of the I(1) variables, then in the structural equations for the I(1) variables, all of the I(0) variables appear in first difference form. A generalization of this result to cointegated systems is found in Fisher et al. (2016). We now illustrate these methods in the context of the small macro model of Ouliaris and Pagan (2016, p. 614). Here z t = [ y1t y2t y3t ] , where y1t is the policy interest rate, y2t is inflation and y3t is the log level of GDP. The interest rate and inflation are I(0) variables and GDP is an I(1) variable. Ouliaris and Pagan restrict the shocks associated with the interest rate and inflation to be transitory, that is, they restrict these shocks to have a zero long-run effect on GDP. These two long-run parametric restrictions are c31 (1) = 0 and c32 (1) = 0. We know from Result 1 that these two zero long-run impacts of the shocks are delivered by the restrictions a31 (1) = 0 and a32 (1) = 0. Ouliaris and Pagan impose these restrictions on the structural coefficients in the equation for GDP to obtain Eq. (12) on p. 614 of their paper. This equation is analogous to Eq. (7) above. They then explain how to combine sign restrictions with the two long-run zero parametric restrictions in this model. This case is straightforward because the two long-run zero restrictions are easily imposed by Result 1. A case which is not straightforward occurs when the shock associated with interest rate is restricted to have a zero long-run effect on GDP but the shock associated with inflation is allowed to have a permanent effect on GDP. We abstract from the economic rationale of this case as it is only for illustration. In this case there is one long-run parametric restriction which is c31 (1) = 0. The difficulty is that this restriction cannot be imposed in isolation using the framework above. It can be imposed using Result 1 or Result 2 but in each case that will imply another zero long-run effect of a shock. In the next sub-section, we will develop a method to impose the long-run restriction c31 (1) = 0 in isolation i.e. directly on the structural model. A development of this method is required to impose only the two long-run zero restrictions on the impacts of the shocks in Peersman's model in combination with sign 3 This and other like expressions in the paper are obtained from MATLAB. It reports the results for the inverse of the matrix A as B = inv(A)*|A| which means that the right hand side of Eq. (5) and similar expressions in the paper are to be divided by the determinant of A(1) but we ignore that as it has no bearing on the results. 4 Note that, in consideration of the sufficient conditions, a22(1) cannot be zero by construction.

4

Journal of Macroeconomics 61 (2019) 103125

L.A. Fisher and H.-s. Huh

restrictions. 2.3. A direct method The reduced-form VAR underlying the SVAR is: (11)

D (L) z t = et

Dp where D (L) = I D1 L D2 and et is the n × 1 vector of reduced-form errors which has covariance matrix Ω. The coefficients on the lagged variables are given by the n × n matrices Dk, k = 1, , p with typical element [dijk ]. The reduced-form moving average representation is z t = D (L) 1et = B (L) et where B (L) = D (L) 1 is the matrix of the responses of the variables at horizon L to the reduced-form shocks in et. It follows that: L2

L p,

(12)

C (1) A0 = B (1)

where B(1) is the matrix of the cumulative impacts of the VAR errors on the variables in the long-run, the elements of which are denoted bij(1). Estimates of its elements are obtained from the estimated reduced-form and are denoted b^ij (1). The method we develop 0 to impose the long-run restriction that c31 (1) = 0, involves working with Eq. (12) to obtain an estimate of a31 which will enforce the p 0 1 2 a31, by itself will not deliver the long-run restriction. We have already established that setting a31 (1) = 0 i.e. a31 = a31 a31 long-run restriction as it is not sufficient (recall Results 1 and 2). In Eq. (12), multiply the third row of C(1), under the restriction c31 (1) = 0, with the first and third columns of A0 to obtain, respectively, the equations: 0 a21 c32 (1)

(13)

0 a31 c33 (1) = b31 (1)

(14)

0 a23 c32 (1) + c33 (1) = b33 (1) 5

Two further restrictions are required for exact identification. They are selected as and (14) to be solved recursively to obtain: 0 a31 =

b31 (1) b33 (1)

0 a21

= 0 and

0 a23

= 0 and they allow Eqs. (13)

(15)

This is analogous to an expression derived in Fry and Pagan (2005, Eq. (9), p.10) and in Levtchenkova et al. (1998, p.512). The estimation of the SVAR proceeds as follows. Suppose z t = [ y1t y2t y3t ] , and note the case for y2t being an I(0) variable is 0 analogous. First, estimate the value of a31 as: 0 a˜31 =

b^31 (1) b^ (1)

(16)

33

The equation for Δy2t is the first to estimate as the two contemporaneous restrictions are placed on it. We regress Δy2t on the lags of all 0 y1t on the rightof the variables and obtain the estimated residuals ^2t . We then estimate the third equation by regressing y3t a˜31 hand side variables using ^2t as the instrument for Δy2t, and obtain ^3t . The equation for y1t is estimated last using ^2t and ^3t as instruments for Δy2t and Δy3t, respectively. This method delivers c31 (1) = 0 as part of the solution to the system of equations shown by 0 Eq. (4), evaluated at the SVAR's estimated coefficients and the value of a˜31 given in Eq. (16). 3. Peersman's SVAR The methods we develop in this paper are demonstrated in the four variable model of Peersman (2005). The four variables are the logs of the oil price (ot), output (yt), and consumer prices (pt), and the level of the short-term interest rate (it). On the basis of unit root tests, Peersman finds that the oil price, output and consumer prices are I(1) variables and that the interest rate is an I(0) variable. Accordingly, the vector of variables in his SVAR is specified as z t = ( ot yt pt it ) . Peersman identifies the structural shocks, first by utilizing six parametric zero restrictions, which exactly identify the model, and then by utilizing sign restrictions on the impulse responses. Under each method, the shocks are identified as either an oil price (OP) shock, an aggregate supply (AS) shock, an aggregate demand (AD) shock or a monetary policy (MP) shock. To identify the shocks parametrically, Peersman utilizes two longrun zero restrictions and four contemporaneous zero restrictions so that the shock associated with the oil price (ɛ1t) is an OP shock, the shock associated with output (ɛ2t) is the AS shock, the shock associated with consumer prices (ɛ3t) is the AD shock and the shock associated with the interest rate (ɛ4t) is the MP shock. Three restrictions are placed on the MP shock, namely, that it has a zero longrun and contemporaneous effect on output and a zero contemporaneous effect on the oil price. Two restrictions are placed on the AD shock, namely, that it has a zero long-run effect on output and a zero contemporaneous effect on the oil price. Finally, one restriction is placed on the AS shock, namely, that it has a zero contemporaneous effect on the oil price.6 Peersman compares the impulse 5 In a SVAR of n variables, n (n 1)/2 parametric restrictions are required to identify the structural model. This is the order condition. It is a necessary but not sufficient condition for identification. 6 The restrictions that the MP, AD and AS shocks have no immediate impact on the oil price is imposed by excluding the contemporaneous values of the interest rate, consumer prices and output from the structural equation for the oil price, which is what Peersman did.

5

Journal of Macroeconomics 61 (2019) 103125

L.A. Fisher and H.-s. Huh

responses from the parametric identification to those from a full sign restrictions methodology. In this paper, we show how to combine his two long-run parametric restrictions with the sign restrictions method of Ouliaris and Pagan (2016). 3.1. The two long-run restrictions: sufficient conditions The two long-run restrictions, namely, that the AD and MP shocks have a zero long-run effect on output are that c23 (1) = 0 and c24 (1) = 0. From C (1) A (1) = I , C (1) = A (1) 1 and from the expression for the inverse of a 4 × 4 matrix, we obtain:

c23 (1) = a11 (1) a24 (1) a43 (1) + a13 (1) a21 (1) a44 (1) + a14 (1) a23 (1) a41 (1)

a11 (1) a23 (1) a44 (1)

a13 (1) a24 (1) a41 (1) (17)

a14 (1) a21 (1) a43 (1) c24 (1) = a11 (1) a23 (1) a34 (1) + a13 (1) a24 (1) a31 (1) + a14 (1) a21 (1) a33 (1)

a11 (1) a24 (1) a33 (1)

a13 (1) a21 (1) a34 (1) (18)

a14 (1) a23 (1) a31 (1) Eqs. (17) and (18) establish the following proposition. Proposition. Sufficient conditions for c23 (1) = 0 and c24 (1) = 0 are: a23 (1) = 0 and a24 (1) = 0 and either a21 (1) = 0 or a13 (1) = 0 and a14 (1) = 0.□ Consider first the sufficient condition (SC) in the first part of the proposition. SC1. a23 (1) = 0 and a24 (1) = 0 and a21 (1) = 0. From the following expression,

c21 (1) = a21 (1) a34 (1) a43 (1) + a23 (1) a31 (1) a44 (1) + a24 (1) a33 (1) a41 (1)

a21 (1) a33 (1) a44 (1)

a23 (1) a34 (1) a41 (1) (19)

a24 (1) a31 (1) a43 (1)

it can be seen that the sufficient condition SC1 also makes c21 (1) = 0. The sufficient condition SC1, which delivers the two long-run restrictions of Peersman also imposes a further long-run restriction, which is that the oil price shock has a zero long-run effect on output. While Peersman's two long-run restrictions are justifiable on the grounds of nominal neutrality, the long-run restriction on the oil price shock may not be plausible particularly if the source of the shock originates from supply side disturbances in the oil market. In that case, we would expect an oil price shock to potentially have a long-run effect on output. We summarize the results here as: Corollary 1. a23 (1) = 0 and a24 (1) = 0 and a21 (1) = 0 are sufficient for c23 (1) = 0 and c24 (1) = 0 and c21 (1) = 0. □ Consider next the sufficient condition in the second part of the proposition. SC2. a23 (1) = 0 and a24 (1) = 0 and a13 (1) = 0 and a14 (1) = 0. From the following expressions

c13 (1) = a12 (1) a23 (1) a44 (1) + a13 (1) a24 (1) a42 (1) + a14 (1) a22 (1) a43 (1)

a12 (1) a24 (1) a43 (1)

a13 (1) a22 (1) a44 (1) (20)

a14 (1) a23 (1) a42 (1) c14 (1) = a12 (1) a24 (1) a33 (1) + a13 (1) a22 (1) a34 (1) + a14 (1) a23 (1) a32 (1)

a12 (1) a23 (1) a34 (1)

a13 (1) a24 (1) a32 (1) (21)

a14 (1) a22 (1) a33 (1)

it can be seen that the sufficient condition SC2 also makes c13 (1) = 0 and c14 (1) = 0. The sufficient condition SC2 which delivers Peersman's two long-run restrictions also imposes two other long-run restrictions, namely that the AD and MP shocks have a zero long-run effect on the oil price. These restrictions do not appear to be plausible as the level of the oil price is likely to be permanently affected by AD and MP shocks. We summarize the results here as: Corollary 2. a23 (1) = 0 and a24 (1) = 0 and a13 (1) = 0 and a14 (1) = 0 are sufficient for c23 (1) = 0 and c24 (1) = 0 and c13 (1) = 0 and c14 (1) = 0. □ We now develop a method to directly impose Peersman's two long-run restrictions in conjunction with the sign restriction methodology of Ouliaris and Pagan. 3.2. Combining the two long-run restrictions with signs Recall that Peersman's two long-run restrictions are c23 (1) = 0 and c24 (1) = 0. We know from the proposition that the restrictions a23 (1) = 0 and a24 (1) = 0 are not by themselves sufficient to deliver the two long-run restrictions. We also know that the sufficient conditions of the proposition deliver these two long-run restrictions but others as well. Accordingly, we develop a method which 0 0 involves working with Eq. (12) to obtain estimates for a23 and a24 which will enforce the two long-run restrictions, and which can be combined with the Ouliaris and Pagan sign restrictions methodology. We begin with Eq. (12), and specify: 6

Journal of Macroeconomics 61 (2019) 103125

L.A. Fisher and H.-s. Huh

1 0 a21 0 a31 0 a41

A0 =

0 a¯12

0 a¯13

0 a¯14

1

0 a23

0 a24

0 a32 0 a42

1

0 a¯ 34

0 a43

1

(22)

The Ouliaris and Pagan approach to sign restrictions involves assigning values to a set of coefficients in A0. In this application, four of the coefficients are assigned values and they are designated with a “−“ above them in Eq. (22). The method finds the values for the 0 0 coefficients a23 and a24 which will enforce the two long-run restrictions. Because four coefficients are generated and two coefficients are restricted in Eq. (22) the model is exactly identified. The four coefficients are generated as: 0 a¯12 =

1

1

| 1|

0 , a¯13 =

2

1

| 2|

0 , a¯ 14 =

3

1

| 3|

0 , a¯ 34 =

4

1

| 4|

(23)

where i , i = 1, …, 4 are drawn from a uniform probability density function over (−1, 1) and |.| denotes the absolute value. We now derive the values for the two restricted coefficients in A0. In Eq. (12), multiply the second row of C(1), for c23 (1) = 0 and c24 (1) = 0, with the second, third and fourth columns of A0 in Eq. (22) to obtain, respectively, the equations: 0 c21 (1) a¯ 12 + c22 (1) = b22 (1)

(24)

0 c21 (1) a¯ 13

0 c22 (1) a23

= b23 (1)

(25)

0 c21 (1) a¯ 14

0 c22 (1) a24 = b24 (1)

(26)

Solve Eq. (24) for c22(1) and substitute that expression into Eqs. (25) and (26) to obtain, respectively: 0 a23 =

0 b23 (1) + c21 (1) a¯13 0 b22 (1) + c21 (1) a¯12

(27)

0 a24 =

0 c21 (1) a¯14 0 c21 (1) a¯12

(28)

b24 (1) + b22 (1) +

From the reduced-form VAR, we can obtain an estimate of each bij(1) i.e. we have b^ij (1). All that remains is to find an estimate of c21(1), since the generated coefficients are known. Denote the elements of A0 1 as a0ij . It follows from Eq. (12) that:

a011 c21 (1) = [ b21 (1) b22 (1) b23 (1) b24 (1)]

a021 a031 a041

(29)

and it can be seen that once estimates for a0i1, i = 1, 2, 3, 4 are found, an estimate of c21(1) is found. From the relationship between the reduced-form errors and the structural shocks, given by et = A0 1 t , it follows that:

eit = a0i1

1t

+

it ,

it

= a0i2

2t

+ a0i3

3t

+ a0i 4

4t ,

i = 1, 2, 3, 4.

(30)

Note that the residuals ɛ1t can be found because the coefficients on all of the contemporaneous variables in the oil price equation are generated so that the equation can be estimated consistently. Estimation of Eq. (30) by OLS will produce consistent estimates of a0i1 because ηit is uncorrelated with ɛ1t since the structural shocks are orthogonal. These are used in Eq. (29) to give an estimate of 0 0 . c21(1) which is substituted into Eqs. (27) and (28) to obtain an estimate of a23 and a24 3.3. The sign restriction algorithm The algorithm begins by obtaining a value for each of the generated coefficients in expression (23) by taking a draw of four θi coefficients as described there. Then 0 0 0 yt a¯ 13 pt a¯ 14 it on the lags of the variables and on the de(i) Estimate the oil price equation by a regression of ot a¯ 12 terministic terms. Compute ^1t . i1 (ii) Regress e^it on ^1t , i = 1, 2, 3, 4 to obtain a^0 which, from Eq. (29), produces c^21 (1). Place this into Eqs. (27) and (28) to obtain the 0 0 0 0 . estimates of a23 and a24 , which we denote as a˜ 23 and a˜ 24 0 0 pt a˜ 24 it on Δot, the lagged variables and deterministic terms, using (iii) Estimate the output equation by a regression of yt a˜ 23 ^1t as the instrument for Δot. Compute ^2t . 0 it on Δot, Δyt and the remaining right-hand side terms, using ^1t and (iv) Estimate the consumer price equation by regressing pt a¯ 34 ^2t as the instruments for Δot and Δyt, respectively. Compute ^3t . (v) Estimate the interest rate equation by regressing it on the variables, their lags and on the deterministic terms, using ^1t , ^2t and ^3t as the instruments for Δot, Δyt and Δpt, respectively. Compute ^4t .

7

Journal of Macroeconomics 61 (2019) 103125

L.A. Fisher and H.-s. Huh

0 0 , and estimates all of the other coefficients in the structural model. Substituting A^ (1) For each draw, the algorithm finds a˜ 23 and a˜ 24 into C (1) A (1) = I and solving for C(1) produces c^23 (1) = 0 and c^24 (1) = 0 as part of the solution. The impulse responses of the variables (in their levels) to each orthogonal shock in ɛt are calculated where the size of the shock is one standard error. The impulse responses are judged for either acceptance or rejection by the sign restrictions. The algorithm is repeated for another draw, and once a predetermined number of sets of impulse responses are accepted, no further draws are made and the algorithm terminates. We now turn to the empirical implementation of the algorithm using Peersman's original data set.

4. Empirical results The structural equations are estimated with Peersman's data for the United States (US). The data covers the period 1980:Q1 to 2002:Q2 and is obtained from the Journal of Applied Econometrics data archive.7 In accordance with Peersman's specification, we estimate each structural equation with three lags, and with a constant and a time trend. For a draw of the four generated coefficients, the algorithm estimates the structural equations and obtains the structural shocks. The impulse responses are calculated for the levels of the series and the set of responses are either accepted or rejected on the basis of the sign restrictions. The algorithm is repeated until 1000 sets of responses are retained. Given the two long-run zero parametric restrictions, the sign restrictions on the impulse responses which are required to separate the shocks are given in Table 1 below. In the table, the designation “≥0” denotes a non-negative response so that the variable does not fall in response to the shock while “≤0” denotes a non-positive response so that the variable does not rise in response to the shock. Peersman did not sign restrict the response of oil prices to the AS shock and this is shown by “?”, which denotes an unrestricted response.8 The two long-run zero restrictions are indicated by LR = 0 and these replace Peersman's sign restrictions on the response of output to the AD and MP shocks. The two long-run zero restrictions separate the AD and MP shocks from the AS and OP shocks. Put differently, ɛ3t and ɛ4t can only be either AD or MP shocks because they have a zero long-run effect on output while ɛ1t and ɛ2t can only be either AS or OP shocks. The sign restrictions separate the AD and MP shocks from each other. However, the sign restrictions can't separate the AS shock from the OP shock. Peersman separated them by utilizing a size restriction, namely, the shock with the largest contemporaneous effect on the oil price was treated as the oil price shock. On a successful draw, the responses satisfy the sign (and size) restrictions for either ɛ1t and ɛ2t to be the OP or AS shock, and for either ɛ3t and ɛ4t to be the AD or MP shock. If neither occurs, the draw is unsuccessful, and all the impulse responses are discarded. The sign restrictions shown in the table are those utilized by Peersman and are standard. For example, they rule out a price puzzle so that in response to an MP shock which raises the interest rate, the oil price and consumer prices cannot rise. Following Peersman, we apply the sign restrictions to the impulse responses of output and consumer prices for four quarters and to the responses of the oil price and the interest rate for one quarter. In sign restrictions, the accepted responses are arranged into ascending order at each horizon and the median response is found. The medians are connected point-wise across horizons to form the median impulse response. We find the 16th and 84th percentile responses in this way as well. It is important to note that the median (and any percentile) response from one horizon to the next is most likely to come from a different model i.e. from a different draw of the θi parameters. Fry and Pagan (2011) refer to this as the multiple models problem, and propose a metric to construct the ‘median-target’ response which corresponds to a single model.9 Both the median and median-target responses are reported in the figures that follow together with the 16th and 84th percentile responses with the region between them shaded. Fig. 1 shows the results under sign restrictions combined with the two long-run parametric restrictions. The two long-run restrictions are apparent in the results as the median and percentile responses of output to the AD and MP shocks all converge to zero by 24 quarters. The median responses under full sign identification, reported by Peersman (2005, Fig. 2(a), p.193), show that both of these shocks have a small long-run effect on output. Here they are restricted to have a zero long-run effect which is evident in the responses of output shown in the figure. In response to a positive oil price shock, output falls steadily to a lower long-run level. Consumer prices rise steadily to a higher long-run level and following this, the interest rate rises before returning to its level prior to the shock. The impacts of the oil price shock suggest that it can be interpreted as a negative AS supply shock as its effects on output and consumer prices are opposite to those seen for the positive AS shock in the figure. In response to the AS shock, output rises steadily to a higher long-run level while consumer prices fall steadily to a lower long-run level. The similar though opposite output and price patterns suggest that oil price shocks are also a reflection of shifts in the aggregate supply curve. Note that the median impact response of oil prices to the AS shock is positive, while the 16th percentile response is very close to zero. This suggests that oil prices rise somewhat on impact even though consumer prices fall (see fn. 8). The oil price steadily rises to a higher long-run level in response to the AD shock. Consumer prices rise steadily for five quarters 7

The URL for the Journal of Applied Econometrics data archive is: http://qed.econ.queensu.ca/jae/. The response is unrestricted because there are two possible effects of the AS shock on oil prices each with a different sign. On the one hand, a positive AS shock would be expected to increase the demand for oil because output rises, thereby raising the oil price. On the other hand, the AS shock reduces consumer prices and among them is the oil price. 9 This metric finds the particular draw of the θi parameters that minimises the distance between the accepted impulse responses and the median responses for all of the shocks. The median-target responses are the responses produced by this particular draw of the parameters i.e. the mediantarget responses come from the single model which corresponds to this draw. 8

8

Journal of Macroeconomics 61 (2019) 103125

L.A. Fisher and H.-s. Huh

Table 1 Sign restrictions. Shock\Variable

Oil price

Output

Consumer prices

Interest rate

OP AS AD MP

≥0 ? ≥0 ≤0

≤0 ≥0 LR = 0 LR = 0

≥0 ≤0 ≥0 ≤0

≥0 ≤0 ≥0 ≥0

Fig. 1. Sign and parametric identification of Peersman's model: Two long-run zero restrictions.

and then decline gradually to a higher long-run level. The oil price response is considerably larger than the consumer price response at all horizons. Output increases on impact and in the subsequent quarter, before declining smoothly to zero as required by the longrun parametric restriction. In response to the MP shock, the oil price falls and converges to a lower long-run level, which is considerably below the impact fall. Consumer prices steadily fall and converge to a lower long-run level. The long-run fall in the oil price is considerably greater than the long-run fall in consumer prices. Over all horizons, there is no oil price or consumer price puzzle in the responses.10 The maximum fall in output to the MP shock occurs after three quarters. Subsequently, output gradually returns to its level prior to the shock as required by the long-run parametric restriction. The 84th percentile response lies below the zero axis at all horizons so there is no evidence for an output puzzle either. The results here are virtually no different to those obtained by Peersman under full sign identification of the shocks. This may not be surprising as he finds only a small positive long-run effect of the AD and MP shocks on output. Finally, we note that the algorithm's success rate was 6.23%.11 5. Peersman's SVAR: a variation Fisher et al. (2016) reconsider Peersman's SVAR by replacing the price of oil with the relative price of oil, defined as t = ot pt , which they treat as an I(1) variable. They identify the model by parametric restrictions alone. They impose four long-run restrictions which are that the AD and MP shocks have a zero long-run effect on the relative price of oil and on output. The four long-run restrictions are consistent with standard macroeconomic models because nominal neutrality implies that relative prices and output are unaffected in the long-run by aggregate demand and monetary policy shocks. The two contemporaneous restrictions they impose are that the AS and AD shocks have a zero contemporaneous effect on the relative price of oil. Here we reconsider their variation of Peersman's model by combining the four long-run zero restrictions i.e. the four nominal 10 We follow convention and classify rises in prices and output in response to an MP shock which raises the interest rate as puzzles. The puzzles can only arise at horizons for which the sign restrictions do not rule them out i.e. at horizons for which the sign restrictions are not applied. 11 The success rate is the number of accepted sets of responses divided by the number of times the algorithm was utilized to obtain them i.e. by the number of draws. In this paper, all of the algorithms terminate once one thousand sets of responses are accepted.

9

Journal of Macroeconomics 61 (2019) 103125

L.A. Fisher and H.-s. Huh

neutrality restrictions with sign restrictions to separate the shocks. The four long-run restrictions are c13 (1) = 0, c14 (1) = 0, c23 (1) = 0 and c24 (1) = 0. In this case, combining the long-run zero restrictions with sign restrictions is straightforward, unlike in Peersman's original model, because Corollary 2 shows that these restrictions (and no more) are implied by a13 (1) = 0, a14 (1) = 0, a23 (1) = 0 and a24 (1) = 0. These conditions on the coefficients in the structural equations for the relative price of oil and output are shown (for a one lag model) by Eqs. (32) and (33), respectively, in Fisher et al. (2016, p. 903). However, they did not prove that these conditions are sufficient for the long-run neutrality restrictions which is what we have shown here.12 Under these conditions, the relative oil price and output equations (for three lags and ignoring deterministic terms), become, respectively13: t

0 0 = a12 yt + a13 2

+

0 yt = a21

+

i=0

t

i a13

0 + a23 2 i=0

i a23

2p t

0 1 + a14 it + a11

2p t 2

2p t

+

2 i=0

i a14

0 1 + a24 it + a21

2p t 2

+

2 i=0

i a24

1 + a12 yt

t 1

it

t 1

2

3 + a11

1 + a22 yt

it

2

1

3 + a21

+

t 3

1

+

t 3

1 i=0

i a13

3 + a12 yt

1 i=0

2p t 1

3

i a23

3 + a22 yt

+

+

1 i=0

i a14

it

1

2 + a11

t 2

2 + a12 yt

2

1t

2p t 1

3

+

2t

(31)

+

1 i=0

i a24

it

1

2 + a21

t 2

2 + a22 yt

2

(32)

0 0 . The There are four long-run zero restrictions which means that only two coefficients in A0 are generated and they are a¯12 and a¯ 34 sign restrictions algorithm begins by obtaining a value for each of the generated coefficients using the method described previously. Then: 0 yt on the right hand side variables (and de(i) Estimate the relative oil price equation, Eq. (31), by a regression of t a¯ 12 2 terministic terms) using as instruments pt 1 for Δ pt, and it 1 for Δit, respectively. Compute ^1t . (ii) Estimate the output equation, Eq. (32), by a regression of Δyt on the right-hand side variables (and deterministic terms) using as instruments ^1t for Δςt, pt 1 for Δ2pt and it 1 for Δit. Compute ^2t . (iii) Estimate the consumer price equation (with relative oil prices) as before and compute ^3t . (iv) Estimate the interest rate equation (with relative oil prices) as before and compute ^4t .

Under this identification, the sign restrictions on the response of the relative oil price to the AD and MP shocks in Table 1 are replaced by LR = 0. In this case, ɛ3t and ɛ4t can only be either AD or MP shocks, and they are separated as such by the sign restrictions on the response of consumer prices and the interest rate. Similarly, ɛ1t and ɛ2t can only be either OP or AS shocks, and they are separated as before by the size restriction on the relative oil price response. The number of quarters over which the sign restrictions are applied to the responses is the same as previously and the algorithm stops at 1000 acceptances as before. Fig. 2 shows the results under sign restrictions combined with the four long-run neutrality restrictions. The response of the oil price itself to a shock (shown in the last row) is the sum of the relative oil price and consumer price response to that shock. The success rate of the algorithm is 5.11%. The responses of the relative oil price and output to the AD and MP shocks converge to zero by 24 quarters, which shows that the long-run restrictions are satisfied. The short-run response of the oil price to these shocks is not consistent with standard economic models. In response to the AD shock, the oil price falls over short and intermediate horizons while consumer prices, output and the interest rate rise. We would expect the oil price to rise. In response to the MP shock, the oil price rises so there is an oil price puzzle, though there is no consumer price puzzle. The median response shows that output falls following the MP shock. However, the 84th percentile response shows a slight output puzzle one quarter after impact of the MP shock. By comparison, under the full parametric identification of FHP, there is an output and consumer price puzzle but not an oil price puzzle (FHP, 2016, Fig. 1, p. 902). Our findings collaborate those of FHP, who find that at least one puzzle emerges under the four long-run neutrality restrictions. 6. Concluding remarks This paper establishes the sufficient conditions on the coefficients in the structural equations which imply the AD and MP shocks have a zero long-run effect on output in Peersman's model. The method in FHP shows how to impose the sufficient conditions on the structural equations and to estimate the system by IV. In this case, however, the use of the method in FHP is not suitable because the sufficient conditions imply other long-run zero responses, which are not part of Peersman's identifying assumptions. Accordingly, this paper develops a direct method to impose Peersman's two long-run zero restrictions on the responses on output in conjunction with 12 They conjectured that these conditions would deliver the four long-run neutrality restrictions because they constitute a block of zeros in the top right-hand corner of C(1), the long-run impact of the structural shocks, giving it a block recursive structure. Their results, shown in Fig. 1 on p. 902 of their paper, confirmed their conjecture. 13 These equations are analogous to Eqs. (34) and (35), respectively, in Fisher et al. (2016, p.904). However, there they are written for a one lag model.

10

Journal of Macroeconomics 61 (2019) 103125

L.A. Fisher and H.-s. Huh

Fig. 2. Sign and parametric identification of Peersman's model with relative oil prices: Four long-run restrictions.

sign restrictions following the approach of OP. It finds the values for the contemporaneous coefficient on consumer prices and the interest rate in the structural equation for output on each draw in sign restrictions which enforces the long-run effect of the AD and MP shocks on output to be zero. Under this method, the responses are similar to those in Peersman under full sign restrictions, except here all the responses of output to the AD and MP shocks converge to zero by the two long-run restrictions. We also considered a variation of Peersman's model where use of the method in FHP is suitable because the sufficient conditions imply the long-run identifying assumptions that are used to identify the model and have no other implications. The method can be generalized to other SVARs that utilize various long-run restrictions provided the values of the relevant contemporaneous coefficients in the structural equations can be solved for recursively from a system of equations, analogous to Eqs. (24) to (26). In our experience, this may not be the case when more than one long-run restriction is imposed on a shock. As such, the suitability of the method will depend on the application. Acknowledgements We are grateful to Adrian Pagan for helping us formulate the ideas that are contained in Sections 2 and 3 of the paper. Any errors or admissions are our own. We thank the Editor, William Lastrapes, for very helpful and considered advice and two anonymous referees for helpful comments. Thanks also to seminar participants at Korea University for their comments. The first author also thanks Yonsei university for its generous hospitality. Funding The research by the first author was supported by a Macquarie University OSP grant. The research by the second author is supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF2018S1A5A2A01033502). Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.jmacro.2019.103125. References Arias, J.E., Rubio-Ramírez, J.F., Waggoner, D.F., 2018. Inference based on structural vector autoregressions identified with sign and zero restrictions: theory and applications. Econometrica 86 (2), 685–720. Baumeister, C., Benati, L., 2013. Unconventional monetary policy and the great recession: estimating the macroeconomic effects of a spread compression at the zero lower bound. Int. J. Cent. Bank. 9 (2), 165–212. Baumeister, C., Hamilton, J.D., 2015. Sign restrictions, structural vector autoregressions, and useful prior information. Econometrica 83 (5), 1963–1999.

11

Journal of Macroeconomics 61 (2019) 103125

L.A. Fisher and H.-s. Huh

Benati, L., 2015. The long-run Phillips curve: a structural VAR investigation. J. Monet. Econ. 76, 15–28. Benati, L., Lubik, T.A., 2014. Sales, inventories, and real interest rates: a century of stylized facts. J. Appl. Econom. 29 (7), 1210–1222. Binning, A., 2013. Underidentified SVAR Models: A Framework for Combining Short and Long-Run Restrictions with Sign-Restrictions. Norges Bank Working paper. Canova, F., De Nicoló, G., 2002. Monetary disturbances matter for business fluctuations in the G-7. J. Monet. Econ. 49 (6), 1131–1159. Faust, J., 1998. The robustness of identified VAR conclusions about money. In: Carnegie-Rochester Conference on Public Policy. 49. pp. 207–244. Fisher, L.A., Huh, H-S., 2016. Monetary policy and exchange rates: further evidence using a new method for implementing sign restrictions. J. Macroecon. 49, 177–191. Fisher, L.A., Huh, H-S., Pagan, A.R., 2016. Econometric methods for modelling systems with a mixture of I(1) and I(0) variables. J. Appl. Econom. 31 (5), 892–911. Fisher, L.A., Huh, H-S., Summers, P.M., 2000. Structural identification of permanent shocks in VEC models: a generalization. J. Macroecon. 22 (1), 53–68. Fry, R., Pagan, A.R., 2005. Some Issues in Using VARs for Macroeconometric Research. Centre for Applied Macroeconomic Analysis, Australian National University CAMA Working Paper 19/2005. Fry, R., Pagan, A.R., 2011. Sign restrictions in structural vector autoregressions: a critical review. J. Econ. Lit. 49 (4), 938–960. Galí, J., 1992. How well does the IS-LM model fit postwar U.S. data. Q. J. Econ. 107 (2), 709–738. Haberis, A., Sokol, A., 2014. A Procedure For Combining Zero And Sign Restrictions in a VAR-Identification Scheme. Centre for Macroeconomics CFM discussion paper series, CFM-DP2014-10. Kilian, L., Lütkepohl, H., 2017. Structural Vector Autoregressive Analysis. Cambridge University Press, Cambridge, United Kingdom. Lastrapes, W.D., 1998. The dynamic effects of money: combining short-run and long-run identifying restrictions using Bayesian techniques. Rev. Econ. Stat. 80 (4), 588–599. Levtchenkova, S., Pagan, A.R., Robertson, J.C., 1998. Shocking stories. J. Econ. Surv. 12 (5), 507–532. Ouliaris, S., Pagan, A.R., 2016. A method for working with sign restrictions in structural equation modelling. Oxf. Bull. Econ. Stat. 78 (5), 605–622. Peersman, G., 2005. What caused the early millennium slowdown? Evidence based on vector autoregressions. J. Appl. Econom. 20 (2), 185–207. Rubio-Ramírez, J.F., Waggoner, D.F., Zha, T., 2010. Structural vector-autoregressions: theory of identification and algorithms for inference. Rev. Econ. Stud. 77 (2), 665–696. Shapiro, M.D., Watson, M.W., 1988. Sources of business cycle fluctuations. In: Fischer, S. (Ed.), NBER Macroeconomics Annual 3. pp. 111–148. Uhlig, H., 2005. What are the effects of monetary policy on output? Results from an agnostic identification procedure. J. Monet. Econ. 52 (2), 381–419.

12