TimeSeriesBook.pdf

This discrepancy between the dimensions of the domain

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This discrepancy between the dimensions of the domain and the range space of g is known as the identification problem . To put it in another way, there are only n ( n + 1) / 2 (nonlinear) equations for n (2 n - 1) unknowns. 7 To overcome the identification problem, we have to bring in additional information. A customary approach is to impose a priori assumptions on the structural parameters. The Implicit Function Theorem tells us that we need 3 n ( n - 1) / 2 = n (2 n - 1) - n ( n + 1) / 2 (15.6) such restrictions, so-called identifying restrictions, to be able to invert the function g . Note that this is only a necessary condition and that the identifi- cation problem becomes more severe as the dimension of the VAR increases because the number of restrictions grows at a rate proportional to n 2 . This result can also be obtained by noting that the function g in equa- tion (15.5) is invariant to the following transformation h : h : ( A, B, Ω) -→ ( RA, RB 1 / 2 Q - 1 / 2 , D D - 1 ) 6 As usual, we concentrate on the first two moments only. 7 Note also that our discussion of the identification problem focuses on local identifi- cation, i.e. the invertibility of g in an open neighborhood of Σ. See Rothenberg (1971) and Rubio-Ram´ ırez et al. (2010) for details on the distinction between local and global identification.
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284 CHAPTER 15. INTERPRETATION OF VAR MODELS where R , Q and D are arbitrary invertible matrices such that R respects the normalization of A and B , Q is an orthogonal matrix, and D is a diagonal matrix. It can be verified that ( g h )( A, B, Ω) = g ( A, B, Ω) . The dimensions of the matrices R , Q , and D are n 2 - 2 n , n ( n - 1) / 2, and n , respectively. Summing up gives 3 n ( n - 1) / 2 = n 2 - 2 n + n ( n - 1) / 2 + n degrees of freedom as before. The applied economics literature proposed several identification schemes: (i) Short-run restrictions (among many others, Sims, 1980b; Blanchard, 1989; Blanchard and Watson, 1986; Christiano et al., 1999) (ii) Long-run restrictions (Blanchard and Quah, 1989; Gal´ ı, 1992) (iii) Sign restrictions: this method restricts the set of possible impulse re- sponse functions (see Section 15.4.1) and can be seen as complementary to the other identification schemes (Faust, 1998; Uhlig, 2005; Fry and Pagan, 2011; Kilian and Murphy, 2012). (iv) Identification through heteroskedasticity (Rigobon, 2003) (v) Restrictions derived from a dynamic stochastic general equilibrium (DSGE) model. Typically, the identification issue is overcome by im- posing a priori restrictions via a Bayesian approach (Negro and Schorfheide (2004) among many others). (vi) Identification using information on global versus idiosyncratic shocks in the context of multi-country or multi-region VAR models (Canova and Ciccarelli, 2008; Dees et al., 2007) (vii) Instead of identifying all parameters, researchers may be interested in identifying only one equation or a subset of equations. This case is known as partial identification . The schemes presented above can be extended in a straightforward manner to the partial identification case.
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