This paper introduces asymmetric impulse response functions and asymmetric variance decompositions. It is shown how the underlying variables can be transformed into cumulative positive and negative changes in order to estimate the impulses to an asymmetric innovation. An application is provided to demonstrate how the propagation mechanism of these asymmetric impulses and responses operates.VAR modelling, Asymmetric Impulses, Fiscal Policy

Hatemi-J, Abdulnasser

English

Research Papers in Economics

1 
 
Asymmetric Generalized Impulse Responses and Variance 
Decompositions with an Application  
 
Abdulnasser Hatemi-J 
UAE University 
E-mail: AHatemi@uaeu.ac.ae 
 
 
 
Abstract 
This  paper  introduces  asymmetric  impulse  response  functions  and  asymmetric  variance 
decompositions. It is shown how the underlying variables can be transformed into cumulative 
positive and negative changes in order to estimate the impulses to an asymmetric innovation. 
An  application  is  provided  to  demonstrate  how  the  propagation  mechanism  of  these 
asymmetric impulses and responses operates.  
 
Running title: Asymmetric Generalized Impulses  
JEL Classification: C32, H21  
Keywords: VAR modelling, Asymmetric Impulses, Fiscal Policy 
 
1.  Introduction 
 
Since the pioneer  work by Sims  (1980), the impulse response functions  and variance 
decompositions are regularly used to capture the dynamic interaction between the variables of 
interest. These calculations are produced by transforming the vector autoregressive (VAR) 
model  into  its  vector  moving  average  representation.  Sims  suggested  using  the  Cholesky 
decomposition in order to identify the underlying chocks. However, this approach is sensitive 
to the order in which the variables enter the model. Koop et. al., (1996) and Pesaran and Shin 
(2008) have introduced generalized impulse response functions, which are not sensitive to the 
way  the  variables  are  ordered  in  the  model.  Nonetheless,  in  all  previous  approaches  on 
generating impulses it has been assumed that the response to a negative shock is similar as the 
response to a positive shock in absolute terms. There are logical reasons to believe that this 
might not be the case and in reality it matters, in terms of the absolute size of the response, 
whether the shock is positive or negative. This is likely to be the case even in situations in 
which the absolute size of the shocks is the same. It is a well-established fact that economic 
actors respond differently to a bad news compared to a good news. For example, if the profit 2 
 
of a company increases by 5% it will lead to different consequences compared to the case in 
which when the profit decreases by 5%. It is surely much easier in a boom market to expand 
and employ more people compared to a bust market potential layoffs. It is easier to hire 
people than firing them for legal and other pertinent reasons. However, the standard impulse 
response analyses do not account for this potential asymmetric effect. Another reason for the 
existence  of  asymmetric  effects  is  the  fact  that  imperfect  information  prevails  in  many 
circumstances as is indicated by the seminal contributions of Akerlof (1970), Spense (1973) 
and  Stiglitz  (1974).  By  relying  on  these  facts,  we  conclude  that  allowing  for  potential 
asymmetry in estimating the impulse responses and variance decompositions is important. 
Thus, the aim of this paper is to introduce asymmetric generalized impulse response (AGIR) 
functions  and  asymmetric  variance  decompositions  (AVD).  We  demonstrate  how  the 
underlying variables can be fragmented into positive and negative components in order to 
generate the AGIR functions as well as the AVD. An application is provided to evaluate the 
impact of contractionary as well as expansionary fiscal policy on economic performance in 
Sweden.  
The rest of the article is organised as follows. Section 2 introduces the AGIR functions 
and the AVD. Section 3 provides an application. The last section offers ending remarks.  
 
2.  Asymmetric Generalised Impulses 
 
One operational approach to construct the positive shocks as well as negative shocks of the 
underlying variables is provided by Granger and Yoon (2002), which was used for conducting 
hidden cointegration analysis. Based on their idea we suggest the calculation of asymmetric 
impulse responses and variance decompositions. For simplicity, assume that we are interested 
in the dynamic interaction between two integrated variables X1t and X2t. However, the results 
can be generalised to higher dimensions. By using the recursive method, we can express each 
variable as  
,
1
1 0 , 1 1 1 1 1     


t
r
r t t t X X X         (1) 
and 
,
1
2 0 , 2 2 1 2 2     


t
r
r t t t X X X         (2) 
 
for t =1,2,…T. The values X1,0 and X2,0 represent initial values. The denotations ε1r and ε2r 
signify  the  error  terms.  The  underlying  shocks  can  be  defined  as   , 0   , max : 1 1 r r      3 
 
 , 0   , max : 2 2 r r         0   , min : 1 1 r r       and    0   , min : 2 2 r r     .  These  results  lead  to  having 
  1 1 1
    r r r    and      r r r 2 2 2      . This in turn means  
,
1 1
1 1 0 , 1 1 1 1 1       
 
 

t
r
t
r
r r t t t X X X         (3) 
and  
      
 
 

t
r
t
r
r r t t t X X X
1 1
2 2 0 , 2 2 1 2 2    .     (4) 
 
These results can be used to obtain the cumulative representation of the positive and negative 
shocks  of  each  variable  in  the  form  of    , :
1
1 1  

  t
r
r t X    , :
1
1 1  

  t
r
r t X     

  t
r
r t X
1
2 2 :    and 
 

  t
r
r t X
1
2 2 :  . These values can be utilized to estimate the asymmetric impulses and variance 
decompositions. Since truly exogenous variables are rare in reality, the VAR model that treats 
all  variables  in  the  model  endogenously  can  be  used.  Assume  the  we  are  interested  in 
capturing the dynamic interaction between cumulative negative and positive shocks, i.e., the 
vector  ) , ( 2 1
    t t t X X X . Then, the following VAR(k) model can be estimated: 
 
 



      t k t k t t u X B X B B X  1 1 0       (5) 
 
Where B0 is 2×1 vector, Bs (s=1, …, k.) is an 2×2 matrix, and  
t u  is an 2×1 vector of error 
terms.  The  exogeneity  problem  is  resolved  since  past  values  of  
t X   are  exogenous  in 
determining  
t X . The optimal  lag order, k, can be chosen by minimizing an information 
criterion. The forecasting performance in this model is strong since past values of  
t X  are 
expected to be the best information set in determining  
t X  itself.
1 This VAR model can be 
used to trace out the effect of a shock in any variable within the system on other variables or 
                                                 
1  A  VAR  system  allows  testing  of  economic  relationship  by  testing  whether  variables  within  
t X   are 
methodically linked so they do not wander too far from each other (cointegration testing) and by testing whether 
past values of one variable improves the forecasting capability of another variable after that other variable’s past 
values  have  already  been  taken  into  account  (Granger  causality  testing).  However,  conducting  this  kind  of 
analysis is beyond the scope of this paper. The interested reader is referred to Granger and Yoo (2002) and 
Hatemi-J (2010, 2011).  4 
 
itself. To estimate the impulses we present the VAR model (eq. 5) in the moving average 
representation form as follows: 
 
T.   ,   1,   for t                 ,
0 0
      






i t
i
i
i
i t u A C X       (6) 
 
where the 2×2 coefficient matrixes (Ai) are obtained recursively as the following: 
 
   ,   2,    1,   for        , 2 2 1 1           i A B A B A B A k i k i i i     (7) 
 
with  2 0 I A    and  0   , 0    i Ai ,  and  0 B A C i i  .  The  asymmetric  generalized  impulse 
response of the effect of a standard error shock in the jth equation at time t on  
n t X  is defined 
as: 
 
, ) ( 5 . 0
j n jj e A n AGIR      for  , , 2 , 1 , 0   n       (8) 
 
where  is the variance-covariance matrix in the VAR model (   . 2   , 1 , ,    j i ij  ) and ej is a 
2×1 selection vector with its jth element equal to one and zero for all other elements. The 
asymmetric forecast error variance decomposition, denoted by    n AVDij , can be calculated as 
 
 
, ) (
0
' '
0
2 ' 1
 
 




n
l i l l i
n
l j l i ii
ij
e A A e
e A e
n AVD

 ij= 1, 2.       (9) 
 
An algorithm written in Gauss is used to obtain the cumulative positive and negative changes 
for each variable based on equations (3) and (4). The algorithm is available on request. After 
creating the cumulative positive and negative changes for each variable, it is straight forward 
to estimate the asymmetric impulses and variance decompositions as presented by equations 
(8)  and  (9)  via  a  number  of  well-known  econometric  packages  that  are  available  on  the 
market. It should be mentioned that these estimations can also be conducted by transforming 
the VAR model into its vector error correction representation first. 
  5 
 
3.  An Application 
 
The procedure suggested in this paper is applied to investigate the potentially asymmetric 
relationship between government spending and GDP in Sweden during the period 1993-2010 
on quarterly basis. The variables are used at constant prices. The source of the data is the 
Swedish  statistical  bureau.  The  results  for  asymmetric  impulses  combined  with  the  95% 
confidence intervals are presented in Figures 1-3. 
 
Figure 1. The asymmetric Generalised Responses for Original Data. 
-.01
.00
.01
.02
.03
1 2 3 4 5 6 7 8 9 10
Response of GPD to GPD
-.01
.00
.01
.02
.03
1 2 3 4 5 6 7 8 9 10
Response of GPD to Spending
-.01
.00
.01
.02
.03
1 2 3 4 5 6 7 8 9 10
Response of Spending to GPD
-.01
.00
.01
.02
.03
1 2 3 4 5 6 7 8 9 10
Response of Spending to Spesning
Response to Generalized One S.D. Innovations ± 2 S.E.
 
 
Figure 1 presents the impulses for the original data. As is shown, government spending has a 
positive impact on economic performance. However, it is  statistically significant for short 
period of time only. The response of spending to economic output is not significant. These 
results  are  also  supported  by  the  asymmetric  variance  decompositions   to  some  extent. 
Because more than 87% of variation in the forecast error in GDP is explained by itself and 
only less than 13% is explained by the forecast error in government spending (S) after 10 
questers,. The corresponding values for S are 81.5% by itself and 18.55% by GDP.  
Next we produce the asymmetric impulses. Figure 2 demonstrates the response for the 
variables represented in cumulative positive changes. As can be clearly seen the cumulative 
positive changes of spending significantly react to an impulse in neither cumulative positive 
changes of GDP nor the revers. The estimated asymmetric variance decompositions for this 
particular case revealed the following. Around 93% of variation in the forecast error in GDP
+ 6 
 
is  explained by itself and while around 7% is explained by the forecast  error in  S
+. The 
comparable values for S
+ are 89% by itself and 11% by GDP
+. 
 
Figure 2. The Asymmetric Generalised Responses for Cumulative Positive Shocks. 
-.002
.000
.002
.004
.006
.008
1 2 3 4 5 6 7 8 9 10
Response of GDP+ to GDP+
-.002
.000
.002
.004
.006
.008
1 2 3 4 5 6 7 8 9 10
Response of GDP+ to Spending+
-.005
.000
.005
.010
.015
.020
.025
1 2 3 4 5 6 7 8 9 10
Response of Spending+ to GDP+
-.005
.000
.005
.010
.015
.020
.025
1 2 3 4 5 6 7 8 9 10
Response of Spending+ to  Spending+
Response to Generalized One S.D. Innovations ± 2 S.E.
 
 
Figure  3  illustrates  the  response  for  the  variables  in the cumulative  negative format.  It  is 
evident from these estimations that the cumulative negative changes of spending significantly 
react to an impulse in neither the cumulative negative changes of GDP nor the reverse. This 
conclusion is also confirmed by the asymmetric variance decompositions. More than 97% of 
variation in the forecast error in GDP
- is explained by an innovation in itself and only less 
than 3% is explained by an innovation in the forecast error in S
-. The analogous values for S
- 
are 95% by itself and 5% by GDP
-.  
 7 
 
Figure 3. The Asymmetric Generalised Responses for Cumulative Negative Shocks. 
-.005
.000
.005
.010
.015
.020
.025
1 2 3 4 5 6 7 8 9 10
Response of GDP- to GDP-
-.005
.000
.005
.010
.015
.020
.025
1 2 3 4 5 6 7 8 9 10
Response of GDP- to Spending-
-.02
-.01
.00
.01
.02
.03
1 2 3 4 5 6 7 8 9 10
Response of Spending- to GDP-
-.02
-.01
.00
.01
.02
.03
1 2 3 4 5 6 7 8 9 10
Response of Spending- to Spending-
Response to Generalized One S.D. Innovations ± 2 S.E.
 
 
 
4.  Concluding Remarks 
 
This article introduces a procedure for calculating the asymmetric impulse response functions 
and asymmetric variance decompositions. Allowing for asymmetry in these estimates might 
be important in order to figure out whether positive shocks have the same absolute impact as 
the negative shocks or not. This can be achieved by estimating the impulses and responses to 
innovations  based  on  the  cumulative  values  of  the  positive  and  negative  changes  of  the 
underlying variables.  
An application is provided to investigate the dynamic relationship between government 
spending and GDP in Sweden. The estimations indicate that an innovation in the government 
spending  does  not  lead  to  a  significant  response  in  the  GDP.  This  seems  to  be  the  case 
regardless if the asymmetric property is taken into account in estimating the impulses or not. 
The  asymmetric  variance  decompositions  also  provide  support  for  this  conclusion  to  an 
extent. These results can be interpreted as empirical evidence that the conducted fiscal policy, 
regardless if it is contractionary or expansionary, is not a major factor behind the successful 
economic performance in Sweden. 
 8 
 
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Hatemi-J  A.  (2010)  Asymmetric  Causality  Tests  with  an  Application,  Empirical 
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Asymmetric Causality Tests with an Application,

Generalized Impulse Response Analysis in Linear Multivariate Models,

Hidden Panel Cointegration, MPRA working paper series,

Impulse Response Analysis in Nonlinear Multivariate Models,

Incentives and Risk Sharing in Sharecropping,

Job Market Signalling,

The Market for Lemons: Quality Uncertainty and the Market Mechanism,

Asymmetric generalized impulse responses and variance decompositions with an application

Asymmetric generalized impulse responses and variance decompositions with an application

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