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Unformatted text preview: 1. What is a VAR model? What are the stationarity conditions for a VAR model? What are the advantages/disadvantages of using a VAR model to analyze economic data? AS. A vector autoregressive model can be interpreted as an unconstrained reduced form of a dynamic simultaneous equations model. It expresses a vector of endogenous variables as linear functions of heir own and each other's lagged values. Contemporaneous and lagged exogenous variables may also be included in the system. This modeling technique was introduced by Sims (1980). A VAR (p) model can be represented as Y t = φ 1 Y t-1 + φ 2 Y t-2 + ... + φ p Y t-p + ε t , where Y t is an n by 1 vector (of endogenous variables), the φ i 's are n by n (constant) coefficient matrices, ε t is an n by 1 vector of disturbances. It is assumed the E ( ε t ) = 0 ∀ t, E ( ε s , ε t ) = Σ for s = t and = 0 for s ≠ t. Thus, a VAR (p) implicitly assume the p lags of Y t are sufficient to summarize the dynamic interactions of elements of Y t . A VAR(p) process, Y t , is stationary if 1. E(Y t ) = μ , ∀ t. 2. V(Y it ) < ∞ , ∀ i. 3. COV(Y t , Y t+k ) = Γ k , which depends on k but not on t. Remarks: 1. A necessary condition of stationarity is the polynomial defined by the determinant DET(I - φ 1 B - φ 2 B 2- ... - φ p B P ) has all its root outside the (complex) unit circle. 2. In general Γ k ≠ Γ-k . However, it is true that Γ k = Γ ’-k 3. Stationary univariate ARMA (p, q) models have a stationary VAR representation. In economics, most variables are jointly determined in the equilibrium and, in this sense, they are endogenous. An advantage of VAR analysis is that it treats all the variables in the system as jointly endogenous. Each variable is allowed to depend on the lagged values of all the variables (not just its own lagged values) in the system. Thus, the VAR approach makes it possible to study the possible dynamic interactions between economic variables. As pointed out above, a vector autoregressive model can be interpreted as an unconstrained reduced form equation, it is impossible to recover the underlying structural parameters without imposing restrictions on the VAR model. The use of Choleski decomposition is one way to recover the structural parameters. Another way to recover the structural parameters is to impose restrictions derived from economic theory; for example, long run restrictions such as neutrality of money. 2. a. Use a bivariate VAR(1) model to discuss and explain i) impulse response analysis, and ii) forecast error variance decomposition. AS. Impulse Response Analysis Given its multivariate nature, the VAR model can be used to analyze the effect of a shock on each of the variables in the system. To simplify the presentation, consider a bivariate VAR(1) process Y t = φ Y t-1 + ε t . Suppose Y t = 0 and ε t = 0 for t < 0. At t = 0, = 0 for t < 0....
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This note was uploaded on 09/29/2010 for the course ECON Econometri taught by Professor Fairlie during the Winter '09 term at University of California, Santa Cruz.
- Winter '09