A Program for the Identification of Structural VAR-Models - CiteSeerX

A Program for the Identification of Structural VAR-Models Caesar Lack and Carlos Lenz Abstract This paper presents a menu-driven RATS-program which allows to identify structural shocks in vector-autoregressive (VAR) models. Identification is achieved by imposing short-run or long-run restrictions (or a combination of both) on the structural form of a model. The only requirement is that the matrix of restrictions consisting of rows of the restricted matrices of short-run and long-run effects can be written as an upper triangular matrix.

1

Introduction

When introducing the vector-autoregressive (VAR) methodology in 1980, Sims raised several objections to traditional macroeconometric models. He pointed out that for reasons of identification, exclusion restrictions were routinely imposed and the decision whether a variable should be regarded as exogenous with respect to the system was made rather arbitrarily. Identification was therefore often achieved without solid economic or econometric arguments. Starting from the notion that it is not obvious which variables are exogenous and which variables are endogenous, Sims’ basic idea was therefore to treat them all as endogenous and first estimate an unrestricted model in a reduced form. No prior knowledge is used except to decide which variables should enter the system. After estimation by OLS (which is consistent and, under normality of the error terms, efficient), structural shocks are identified by assuming that contemporaneous interactions among variables are recursive, i.e. by imposing a certain ordering of the variables. In terms of the moving average (MA) representation, the structural shocks do not affect preceding variables simultaneously.

1

The structural form of the model can then be conveniently summarized by the impulse response functions and the variance decomposition. The impulse response function describes the in-sample effect of a typical shock to the system and can be used to economically interpret the behaviour of the system. The variance decomposition assesses the importance of different shocks by determining the relative share of variance that each structural shock contributes to the total variance of each variable. Cooley and LeRoy (1985) criticized the VAR methodology because of its “atheoretical” identification scheme. They argued that Sims did not explicitly justify the identification restrictions and claimed that a model identified by this arbitrary procedure cannot be interpreted as a structural model, because a different variable ordering yields different structural parameters. As an alternative to the recursive identification scheme, Bernanke (1986), Blanchard and Watson (1986) and Sims (1986) introduced non-recursive restrictions on the contemporaneous interactions among variables for identification. As economic theory often does not provide enough meaningful contemporaneous restrictions, the search for additional identifying restrictions led Blanchard and Quah (1989), Shapiro and Watson (1988), King et al. (1992) and Gal´ı (1992) to introduce restrictions on the system’s long-run properties, which are usually based on neutrality postulates. The objective of this paper is to present a RATS program which allows to identify structural VAR models and to give an overview of its functionality. The program allows to implement all the above mentioned identification strategies in a user friendly and straightforward way. In addition, the paper deals in some detail with the algorithm which is used to identify structural shocks. It is crucial to understand this algorithm in order to make full use of the program’s capabilities. The remainder of this paper is organized as follows: Section 2 makes some general remarks about the identification of structural VAR models, section 3 presents the identification algorithm, and section 4 gives a quick tour around the program using a concrete example. The last section summarizes the paper and the properties of the program.

2

Identification of VARs

In this section we will concentrate on the identification process and leave aside topics such as the determination of the correct lag length, detrending of variables, stationarity issues and cointegrating relations between the variables. We assume that the variables entering the system are all stationary after appropriate transformation. In particular, we do not allow non-stationary 2

variables to enter the system in levels. Even though this has some benefits in the case of cointegrating relations between the variables, long-run identifying restrictions cannot be implemented if the VAR is estimated in levels. If there are cointegration relations, these can be exploited by directly introducing the stationary linear combination of the cointegrated variables as a single variable in the system The starting point for the analysis of a structural VAR is the estimation of a reduced form VAR including sufficient lags in order to describe the underlying dynamics. For a meaningful interpretation of the dynamics of the system, it has to be identified. That is, the reduced form model with correlated innovations has to be transformed into a structural form with uncorrelated, economically interpretable shocks. We assume that the model contains n variables (excluding the constant term). For identification purposes, at least n2 independent restrictions on parameters of the structural form are needed to exactly identify the system. Structural shocks are supposed to be mutually uncorrelated, therefore the variance-covariance matrix of the structural shocks is required to be diagonal. Without loss of generality, the standard deviations of the structural shocks are normalized to 1, i.e. the variance-covariance matrix of the structural shocks is set to the identity matrix, which yields n(n + 1)/2 restrictions. Consequently, additional n(n−1)/2 restrictions are needed. These remaining restrictions can now be imposed either on the contemporaneous or on the long-run properties of the system. Sims (1980) placed the remaining restrictions on the contemporaneous relationship of the variables: he assumed the matrix of contemporaneous effects of structural shocks on the variables to be lower triangular, which yields exactly the needed n(n − 1)/2 additional constraints. Restrictions on the contemporaneous effects in the structural MA representation will be called “short-run restrictions” throughout the following text. Short-run restrictions of this lower-triangular type are also implemented in most econometric packages to achieve identification of VAR models. The results of this type of identification scheme obviously depend on the ordering of the variables, especially if there are large off-diagonal elements in the variance-covariance matrix of the innovations of the reduced-form. Apart from the lower triangular type of short-run restrictions, there are other ways to identify a VAR model. The first obvious choice is to impose non-triangular restrictions on the matrix of contemporaneous effects. Another possibility is to place triangular or non-triangular restrictions on the matrix of the contemporaneous structural autoregressive (AR) representation. As long as the restricted matrix of contemporaneous interactions of the structural MA or the structural AR form has full rank, any set of 3

restrictions is feasible. As an alternative to restrictions on the short-run properties of the system, restrictions can also be imposed on the long-run properties of the system, as economic theory may tell us something about the long-run effects of structural shocks on certain variables. These restrictions can again be of a triangular or of a non-triangular form (as long as the restricted matrix has full rank) and will be called “long-run restrictions” throughout the following text. A typical example are shocks to nominal variables which are restricted to have no effect on real variables in the long run. Finally, the n(n − 1)/2 restrictions can be obtained by imposing both short-run and long-run restrictions. Any VAR which is not identified by using only triangular short-run restrictions will be called a “structural VAR” throughout the following text. We will now explain in more detail how short-run and long-run restrictions can be used for identification. Assume that xt is a vector containing n economic variables. Note that xt is assumed to be covariance stationary (after appropriate treatment of the variables). Suppose that the VAR representation of the reduced-form model can be written as: E(t 
t ) = Ω,

D(L)xt = t

(1)

where D(L) = D0 + D1 L + D2 L2 + ... + Dp Lp and L is the lag-operator with Li xt = xt−i . As (1) is a reduced form, D0 is equal to the identity matrix I. The pth order VAR process in (1) can be taken to be the true data generating process for xt or a finite order linear approximation to an underlying infinite order linear or nonlinear model. The covariance matrix of the residuals t is, in general, nondiagonal. Therefore, the shocks in t cannot be the structural innovations which are assumed to be uncorrelated with each other. If the matrix polynomial D(L) has all its roots greater than one in modulus, it is invertible and there exists an infinite order MA-representation. This Wold representation will be written as: xt = C(L)t ,

(2)

where C(L) = D(L)−1 . Now suppose that the VAR representation of the structural form can be written as: E(ut u
t ) = I.

B(L)xt = ut

(3)

Without loss of generality, the covariance matrix of the structural shocks ut is normalized to I. If the matrix polynomial D(L) is invertible, so is the matrix polynomial B(L) and one can write the structural M A(∞) representation as: xt = A(L)ut . 4

(4)

Note that A(L) = B(L)−1 . The structural MA representation in (4) is also called the final form of an economic model because the endogenous variables xt are expressed as distributed lags of the exogenous variables, given by the elements of ut . However, the exogenous structural shocks ut are not directly observed. Rather, the elements of ut are indirectly observed through their effects on the elements of xt . We can obtain the structural shocks ut by first estimating the reduced form VAR (1) and transforming the reduced form residuals. From (2) and (4) we have: (5) A(L)ut = C(L)t . Let subscripts indicate the matrix of coefficients at the corresponding lag. As C0 = I and (5) must hold for all t, we have: A0 u t =  t .

(6)

Squaring both sides and taking expectations yields: A0 A
0 = Ω.

(7)

A(L)ut = C(L)A0 ut ,

(8)

Ai = Ci A0 .

(9)

Combining (5) and (6) we find:

which implies: Note that knowledge of A0 is sufficient for the full identification of the structural system. When A0 is known, all structural coefficients of the lag polynomial A(L) and the structural innovations ut are easily calculated from the estimated reduced form VAR using (6) and (9). The identification of the structural VAR is achieved through two sets of restrictions. First, equation (7) places n(n+1)/2 restrictions on the elements of A0 . Second, additional n(n−1)/2 additional restrictions are needed to fully determine A0 . In the structural VAR literature these restrictions are usually taken from economic theory and are intended to represent some meaningful short-run or long-run relationship between the variables and the structural shocks. Short-run restrictions are imposed directly on A0 which describes the contemporaneous reaction of the variables to structural innovations. Sims (1980) restricted the upper triangle of A0 to be zero. Note that in this case, A0 is simply the Choleski-decomposition of Ω, which always exists as Ω is a symmetric positive definite matrix of full rank. Using the Choleski-decomposition 5

imposes an ordering where structural shocks contemporaneously affect only succeeding variables in a prespecified ordering. But short-run restrictions need not be of a triangular form. As long as condition (7) is met, restrictions can be put on any element of A0 . It is also obvious from (7) that the matrix A0 has to be of full rank. As the matrix of the contemporaneous structural AR representation B0 is simply the inverse of A0 , restrictions can also be placed on B0 , which also must have full rank. Long-run restrictions on the coefficients of structural vector MA representation A(1) can also be used for identification. From (9) we have C(1)A0 = A(1),

(10)

where C(1) and A(1) represent the cumulated effects of innovations. Again, C(1) can be obtained from the estimation of the reduced form system. Therefore, restrictions on A(1) can be used to identify A0 . Shapiro and Watson (1988) and Blanchard and Quah (1989) are early examples which used longrun restrictions in order to identify structural VARs. In some cases, it might also be useful to place identifying restrictions on B(1). This can easily be done as A(1) is the inverse of B(1). Finally, it is also possible to use a combination of short-run and long-run restrictions for identification purposes. Gal´ı (1992) pioneered this rather flexible approach. His identification strategy will be considered in more detail in section 4 where it will serve as an example to illustrate the use of the program.

3

The Identification Algorithm

This section explains in detail the functioning of the identification procedure which stands at the heart of the program presented in the next section. The starting point is equation (7) which implies: ˜ = A0 , ΩS

(11)

˜ is the Choleski-decomposition of Ω and S an arbitrary orthogonal where Ω matrix. Note that orthogonality of S implies S
= S −1 . From (4) and (3) we −1 have B(L) = A(L)−1 , which implies B0 = A−1 0 and B(1) = A(1) . Together with (9) and (11) this implies: ˜
)−1 S = B0
(Ω ˜ C(1)ΩS = A(1)
−1 ˜
(C(1)Ω) S = B(1)
. 6

(12) (13) (14)

Furthermore, knowledge of S is equivalent to knowledge of A0 and therefore to the identification of the structural form. Let n denote the number of variables in the system. Our procedure solves for S if the restrictions can be written in the following form:     H  

˜ Ω ˜ (Ω
)−1 ˜ C(1)Ω
 ˜
−1 (C(1)Ω)





· r12 · · · · · · r1n .. . · r23 · · · r2n .. .. .. .. . . . . . . . · · · · · · · · rn−1,n

      S =     

    ,  

(15)

where H is an (n − 1) × 4n selection matrix whose function is to choose ˜ and [(C(1)Ω) ˜
]−1 , representing ˜ (Ω ˜
)−1 , C(1)Ω, the appropriate rows of Ω, restrictions on A0 , B0 , A(1), and B(1), respectively. Note that by setting the elements of H appropriately, it is not only possible to restrict individual elements of A0 , B0 , A(1), and B(1), but also to impose inter- or intra-equational constraints. The upper triangle of the right-hand matrix contains the restrictions put on the rows of A0 , B0
, A(1), and B(1)
, respectively. For the identification algorithm to work, it is crucial for these restrictions to be ordered in this triangular fashion. In addition, depending on the data used, the rij cannot adopt arbitrary large values, otherwise no solution on the set of real numbers will exist. This is clear intuitively, as one cannot enforce structural shocks with a variance set equal to 1 to produce arbitrary large short-run or long-run effects on given economic time series. In order to solve for S, we first define:     Z =H  






     −1 
˜ (C(1)Ω)

   R=  

˜ Ω ˜
)−1 (Ω ˜ C(1)Ω




· r12 · · · · · · r1n .. . · r23 · · · r2n .. .. .. . . .. . . . . . · · · · · · · · rn−1,n



    .  

˜ n [S1 . . . Sn ] = [R1 . . . Rn ] where subThis can also be written as ZZ scripts indicate the corresponding column of a matrix and Z˜ contains the first n − 1 columns of Z. Now we can solve recursively for each column of S, starting with the last column: 




˜ n Sn = Z˜   ZZ 

s1n .. . sn−1,n

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   + Zn snn = Rn . 

Letting Sˆn = [s1n , s2n , . . . , sn−1,n ]
and solving for Sˆn yields Sˆn = Z˜ −1 Rn − Z˜ −1 Zn snn .

(16)

Using the orthogonality of S, which implies Sˆn
Sˆn + s2nn = 1, we can write:

1 − s2nn = Z˜ −1 Rn − Z˜ −1 Zn snn






Z˜ −1 Rn − Z˜ −1 Zn snn ,

(17)

which can be solved for snn and, inserted into (16), yields Sn . After having calculated the nth column of S, the same procedure can be applied iteratively to calculate the columns n − i, i = 1, . . . , n − 1 of S. Note that in order to be able to apply this iterative procedure, after each iteration
and the last i elements of the n − ith row of Z must be replaced by Sn−i+1 Rn−i must be set equal to 0. As a final remark note that S is unique up to a change of signs of each column. The signs are set such that the long-run effect of a structural shock on the respective variable is positive (i.e. the diagonal elements of A(1) are all positive), which is just a normalization.

4

Example

The handling of the program is best explained by applying it to an economic model. The model is the one proposed by Gal´ı (1992). It is appropriate to show the capabilities of the program as it identifies the structural shocks using a combination of short and long-run restrictions.

4.1

User Input Section

Before running the program, several adaptations have to be made in the user input section of the main program SVAR.PRG. 1. The directories of the source files and the data files must be specified correctly. If necessary, the data have to be transformed appropriately to obtain stationary data. All series which are imported from a data file or generated in the user input section can then be used in the program. 2. The calendar date and the parameter start must be set to the same value, otherwise the program does not work. They denote the start of the sample period. The parameter end denotes the end of the sample period. If some of the variables are being differenced in the program, the parameter ordiff has to be set to the largest difference taken. This variable is needed to determine the actual start of the estimation period allowing for the number of lags included in the VAR. 8

3. If long-run restrictions are used, the parameter steps denotes the time span after which the variables are forced to comply with the long-run restrictions. Impulse response functions and variance decompositions are also calculated for steps intervals in advance. 4. The parameter ndraws denotes the number of bootstrapping simulations used for calculating confidence intervals for the impulse response functions and confidencelevel denotes their significance level. The only function of the main program SVAR.PRG is to administer the drop-down menus. All the calculations are done in procedures. Every menu point has its procedures stored in a separate source file. There are the following source files: • MENU.SRC: a small procedure for the drop-down menus • IDENTIFY.SRC: main procedure for identification • LAGLENGTH.SRC: calculates information criteria to determine the optimal lag length • OPTION.SRC: allows to set the confidence level and the number of bootstrap simulations • READNLAGS.SRC: determines the number of lags • READVARIABLES.SRC: selects the variables • STRUCRES.SRC: prints the structural residuals • UNIT.SRC: contains the unit-root tests • VARDEC.SRC: performs the variance decomposition

4.2

Defining Restrictions and Ordering the Structural Shocks

The program allows for identification by restricting the matrices A0 , A(1), B0
and B(1)
. Restrictions may be imposed on one or several of these matrices. A necessary condition for the program to work is that the matrix of restrictions R can be written in a triangular form. This triangular form can often be attained by reordering the variables and the structural shocks. We demonstrate this process for Gal´ı’s model. He estimates an augmented IS-LM model containing four variables: real output (∆y), the nominal interest rate (∆i), 9

the real interest rate (i−∆p), and real balances (∆m−∆p). The variables are assumed to be driven by a vector of four structural shocks: [us , ums , umd , uIS ]
representing aggregate supply shocks, money supply shocks, money demand shocks, and spending shocks, respectively. The matrix A0 has 16 elements, the same number of restrictions is therefore needed to identify the structural shocks. The orthogonality assumption which implies that the covariance matrix of structural shocks is diagonal, yields 10 restrictions. A second set of restrictions sorts out the supply shocks from the three aggregate demand disturbances by constraining the latter to have no long-run effect on the level of GNP. This yields three additional restrictions. A third set of restrictions rules out contemporaneous effects of money demand and money supply shocks on output, which yields two more restrictions. Finally, one additional restriction is needed, and Gal´ı offers three alternatives, which will be presented in some detail below. The variables and shocks have to be ordered such that a triangular restriction matrix results. Note that the constraint of the three demand shocks having no long-run effect on GNP imposes three zero restrictions on a row of A(1). This implies that ∆y is the first variable and us is the first shock. The restriction that money supply and money demand shocks have no contemporaneous effect on output imposes two zero restrictions on a row of A0 . To obtain a triangular restriction matrix, this leads to uIS being the second shock. This (yet incomplete) set of restrictions imposes the following ordering of variables and shocks: Variables: [∆y,

. . . ]


Shocks: [us , uIS ,

. . ]
.

According to each of the three proposed alternatives for the final restriction, a suitable ordering of the remaining three variables and two shocks has to be found. The first alternative proposes that contemporaneous GNP does not enter the money supply rule, which is a constraint on B0 . This restriction requires the money supply shock to come last and therefore the money demand shock to be the third shock. This fixes the ordering of all shocks, but not of all variables: Variables: [∆y,

. . . ]


Shocks: [us , uIS , umd , ums ]
.

Therefore, we have the following restrictions on the matrices A0 , A(1) and

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B0 :    

A(1) = 

· · · ·

0 · · ·

0 · · ·

0 · · ·





   

A0 = 

  

· · · ·

· · · ·

0 · · ·

0 · · ·





   

B0 = 

  

· · · 0

· · · ·

· · · ·

· · · ·

   . 

Using the first row of A(1) as the first row of R, the first row of A0 as the second row of R and the first row of B0
as the third and last row of R, yields a triangular restriction matrix R. The second alternative proposes that there is contemporaneous homogeneity in money demand. This constrains the coefficient of i − ∆p in the money demand equation to be zero, which requires the money demand shock to come last and therefore the money supply shock to be the third shock. That is, we have the following ordering of the shocks and variables: . . . ]


Variables: [∆y,

Shocks: [us , uIS , ums , umd ]


The first variable has to be ∆y, the ordering of the remaining three variables is in principle arbitrary. Due to a peculiarity of the algorithm used for identification, the row containing only one restriction (i.e. the last row of the restriction matrix R) must not be the last row of B0
. It is therefore not possible that the variable i − ∆p comes last, it has to come second or third. The matrices A0 and A(1) still look the same as in the previous case, but B0 can now have two forms, depending on whether i − ∆p is ordered second or third:  

 B0 =  

· · · ·

· · · 0

· · · ·

· · · ·





   

 or: B0 = 

 

· · · ·

· · · ·

· · · 0

· · · ·

   . 

Using the first row of A(1) as the first row of R, the first row of A0 as the second row of R and the second or third row of B0
as the third row of R, again yields a triangular restriction matrix R. The third alternative restriction says that contemporaneous prices do not enter the money supply rule and therefore imposes the restriction that the coefficients of i − ∆p and ∆m − ∆p in the money supply equation add up to zero. This intraequational constraint refers to two elements of B0 and can be imposed by appropriately setting the elements of the selection matrix H.

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In particular, the selection matrix is set equal to:  

 H= 

0 1 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 0 0 1 −1 0 0 0 0

1 0 0 0

0 0 0 0

0 0 0 0



0 0 0 0

  . 

Together with the following ordering of variables and shocks: Variables: [∆y, ∆m − ∆p, i − ∆p, ∆i]
this implies that matrices:  · 0  · ·  A(1) =   · · · ·

Shocks: [us , uIS , umd , ums ]

the desired restrictions are imposed on the corresponding 0 · · ·

0 · · ·





   

 A0 = 

 

· · · ·

· · · ·

0 · · ·

0 · · ·





   

 B0 = 

 

· · · ·

· · · · · · z −z

· · · ·

   . 

Here z is an arbitrary real number. The user interface offers the possibility to manipulate H directly and therefore to impose intra-equational and interequational constraints. Although the restrictions used by Gal´ı imply that the matrix of restrictions R has only zeroes in its upper triangle, the program allows also for non-zero restrictions. If the imposed restrictions yield a solution which is not in the set of real numbers, the program warns the user and stops its calculations. If the calculation of point estimates yields a real solution but during the process of bootstrapping, a non-real solution arises, the results are displayed without confidence bands.

4.3

Running the Program

This subsection explains how to handle the user interface. Define a separate output window and run the program. A new menu will appear:

12

Select the first point Choose Variables and all the variables defined in the user input section or contained in the data file will appear in a window. The variables now have to be selected and ordered in such a way that the restriction matrix R is triangular. After choosing the last variable, press cancel. Then select Choose Number of Lags to determine the number of lags to include in the estimation of the restricted system. Finally, select Decompose VAR. The results of the estimation of the reduced form will appear in the output window. The user can now choose between Choleski (a standard triangular Choleski-decomposition where identification is achieved by imposing a triangular structure on A0 ) or Structural (restrictions on A0 , A(1), B0
and B(1)).

Next, select Structural. A triangular restriction matrix R appears, containing zeros as default values for the restrictions. Only zero-restrictions are used by Gal´ı, therefore nothing has to be changed here and the window can be closed. If non-zero restrictions are imposed, do not forget to press enter after inserting any number and confirm that the changes in R are saved when closing the window. The next step consists of selecting the appropriate rows of the restricted matrices, i.e. to define the selection matrix H. Remember that in the present case, the first row of the restriction matrix R corresponds to the first row of A(1), the second row of R corresponds to the first row of A0 and the last row of R corresponds to the fourth row of B0
. This is told to the program as follows:

13

If the selection matrix H only selects single rows (as is the case when using the first two alternatives of Gal´ı’s final restriction), a menu-driven procedure helps the user in correctly specifying H, therefore select Automatic in this box. If inter- or intra-equational constraints are imposed (as it is the case when applying Gal´ı’s third alternative), H has to be modified manually and one would select Manual. As the first row of R, choose A(1) and 1 (meaning the first row of A(1)).

As the second row of R, select A0 and 1 (meaning the first row of A0 ).

14

And for the last row of R, select B0
and 4 (meaning the fourth row of B0
).

Short-run and long-run matrices of the structural and reduced form A0 , A(1), B0 and B(1) are then displayed and confidence intervals are calculated by bootstrapping, which may take some time. Impulse response functions for the variables and the accumulated variables are shown graphically and numerically. The confidence level of the error-bands and the number of bootstrapping simulations can be changed in Options. By selecting Structural Innovations, the structural shocks are calculated, shown on the screen and saved to a file. Variance Decomposition shows the variance decomposition of the level of variables and the cumulated variables. The identification progress can be repeated with a different number of lags, other variables and other identifying restrictions.

5

Summary

In this paper we have presented a menu-driven RATS-program which allows to identify structural shocks in VAR models and the underlying identification algorithm. The RATS program including all source files is printed in 15

the appendix or can be downloaded from the Internet.1 It is a convenient, easy-to-use and fast tool which allows to identify VAR models using shortrun and long-run restrictions simultaneously. The only requirement is that the identifying restrictions imposed on the contemporaneous or long-run effects of the structural shocks can be written as a triangular matrix. Even if the restriction matrix is initially not triangular for a certain ordering of the structural variables and shocks, it might very well be triangular for a different ordering. In addition to the identification algorithm, the program offers the following features: • Graphical and numerical output of the impulse response functions of the (stationary) input variables and of the integrated variables. • Graphical and numerical output of the variance decompositions for the (stationary) input variables and for the integrated variables. • Graphical and numerical output of structural innovations. • Calculation of confidence intervals by bootstrapping at different levels of significance and with different numbers of simulations. • Unlimited number of input variables. The identification procedure is embedded in a menu-driven RATS-environment which also incorporates all necessary tools for the specification of a VAR such as: • Augmented Dickey-Fuller and Phillips-Perron unit root tests. • Unit root test by Kwiatkowski, Phillips, Schmidt and Shin (1992). • Different information criteria (AIC, FPE, Schwarz, Hannan-Quinn) as well as Ljung-Box Q-statistics for determining the optimal number of lags to include in the estimation of the VAR.

1

http://www.unibas.ch/wwz/makro/svar

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References Bayoumi, T. and B. Eichengreen (1993). “Shocking Aspects of European Monetary Unification”, in F. Torres and F. Giavazzi (eds.), Adjustment and Growth in the European Monetary Union. Cambridge University Press, Cambridge (UK). Bernanke, B. (1986). “Alternative Explanations of the Money-Income Correlation”, Carnegie Rochester Conference in Public Policy, 25, 49-100. Blanchard, O. J. and M. W. Watson (1986). “Are Business Cycles All Alike?” in R. Gordon, ed., The American Business Cycle - Continuity and Change, NBER Studies in Business Cycles, Vol. 25, Chicago, University of Chicago Press. Blanchard, O. J. and D. Quah (1989). “The Dynamic Effects of Aggregate Demand and Supply Disturbances”, The American Economic Review, 79, 655-673. Cooley, T. F. and S. F. Leroy (1985). “Atheoretical Macroeconometrics. A Critique.” Journal of Monetary Economics, 16 , 283-308. Gal´ı, J. (1992). “How Well Does the IS-LM Model Fit Postwar U.S. Data?” Quarterly Journal of Economics, CVII, 709-735. Kwiatkowski D., P.C.B. Phillips, P. Schmidt and Y. Shin (1992). “Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root”, Journal of Econometrics 54, 159-178. Shapiro, M.D. and M.W. Watson (1988). “Sources of Business Cycle Fluctuations”, in S. Fischer (ed.), NBER Macroeconomics Annual 1988. The MIT Press, Cambridge (Mass.) and London. Sims, C.A. (1980). “Macroeconomics and Reality”, Econometrica, Vol 48, Jan. 1980, 1-48.

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