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Fatih Chellai: Regime-Dependent Effects of Public Spending in Algeria: A Structural VAR and
Markov-Switching Approach
Theoretical aspects of the SVAR model
The Vector Autoregression (VAR) model is an econometric tool widely used to model
dynamic relationships between several time series. A reduced VAR model is written
as:
Y = A + AY + A Y ++ A Y + u
t 0 1 t− 1 2 t− 2 p t p− t (1)
WhereY is a vector of n endogenous variables at time t , A is a vector of constants,
0
t
Ai (i = 1,2, , p ) are matrices of coefficients(n n ) , and ut is a vector of white noise
residuals with variance-covariance matrix u = [ E u u ] .
t t
The residuals u are uncorrelated in time but may be correlated with each other at time t
t
The SVAR (Structural Vector Autoregressive) model extends the VAR model by
imposing restrictions to identify the underlying structural shocks affecting system
variables. Unlike reduced shocks u , structural shocks are uncorrelated and have unit
t
t
variance. The relationship between reduced shocks and structural shocks is given by:
u = B
t t (2)
where B is a matrix (n n ) of coefficients that captures contemporaneous relationships
between variables.
The variance-covariance matrix of the reduced residuals is then u = BB , where B is
the Cholesky decomposition if B is lower triangular, or another form if different
restrictions are imposed.
The SVAR model can be expressed in its structural form as:
A Y = A Y + A Y ++ A Y + B
s
s
s
s
0 t 1 t− 1 2 t− 2 p t p− t (3)
s
where A is a matrix (n n ) of contemporaneous relations, A = A A for
s
s
i
i
0
0
i = 1,2, , p , and B represents structural shocks.
t
The structural shocks are uncorrelated and have unit variance: E t t ] I= n
[
t
s
The aim of SVAR analysis is to estimate the matrix A (or B ) to identify structural
0
shocks and analyze their effects.
Identifying the SVAR model
s
Identifying the matrix A (or B ) requires the imposition of restrictions. These restrictions
0
can be of two types:
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