A systematic framework for analyzing the dynamic effects of permanent and transitory shocks☆
Introduction
In this paper, we propose a simple and coherent framework for isolating the permanent and the transitory shocks from a system of integrated variables, making explicit the relationship between the common trends and the innovations underlying the reduced form model. We show how dynamic impulse response functions can be constructed to trace out the propagating mechanism of the permanent and the transitory shocks. The analysis is conducted using a VECM (Vector Error Correction Model), i.e. a vector autoregression (VAR) that incorporates cointegration restrictions. The procedure consists of two steps. The first step distinguishes innovations that have permanent effects from those that have transitory effects only. This is accomplished by a transformation of the residuals using information that are readily available from the VECM. The second step uses Choleski decomposition to obtain a set of permanent and transitory shocks that are mutually orthogonal. The advantage of the method is its simplicity, since it operates in much the same way a stationary VAR is used to do impulse response analysis. The only difference is that instead of applying Choleski decomposition to the residuals of the VAR, we apply it to a set of transformed residuals.
In conventional VAR analysis, the identified shocks are usually viewed as innovations to the variables in the system. Because identification is typically based on a priori assumptions, it is only in a limited sense the data reveal the source of the shock.1 As forcefully argued in Cochrane (1994b), after decades of analysis, we still know very little and perhaps will never know enough about the origin of shocks. A key feature of cointegrated systems is that the variables move together at low frequencies. Given this coherence, our ability to identify the source of the shock is even more limited. But we can exploit the low frequency comovements to identify shocks according to whether their effects are permanent or transitory. Accordingly, in our analysis, the shocks are distinguished by their degree of persistence, rather than their origin. However, in general, the low frequency movements alone are not sufficient to identify permanent and transitory shocks that are mutually uncorrelated. We show how this can be achieved in a simple framework, and in the process, clarify the limits of cointegration in identifying the permanent and transitory shocks.
The plan of this paper is as follows. The econometric framework used to isolate the permanent and transitory shocks is presented in the next section. We then put into context our decomposition with related work in the literature. Section 3 presents simulated and empirical examples. Pitfalls of analysis on permanent and transitory components are discussed in Section 4. A conclusion completes the analysis.
Section snippets
The econometric framework
The objective of this section is to present a framework which systematically isolates the permanent and the transitory shocks from a VECM.
Simulations and examples
To see that the P–T decomposition functions well in practice, we provide two simple simulated examples. We assume that the rank of the cointegrating matrix is known to focus on orthogonalization issues. We use reduced rank regressions with two lags to obtain .7
Two caveats
The procedure outlined above necessitates estimates of r, α, and γ⊥. One can use the reduced rank analysis of Johansen (1988) to obtain r and α. An alternative is to use the common trend statistic of Stock and Watson (1988) to determine r, and then estimate α by fully efficient estimators. Which method (or combination of methods) to use is at the user's discretion. In this section, we stress the strong dependence of any P–T decomposition on precise and consistent estimation of these parameters.
Conclusion
Vector autoregressions is a valuable framework for dynamic economic analyses. When some variables share common stochastic trends, the system of variables is bind together by cointegrating restrictions. This paper shows that information on these linear relationships can be used to decompose shocks into permanent and transitory components. The analysis (i) presents the two steps necessary to obtain a permanent–transitory decomposition, and (ii) clarifies that cointegration restrictions are used
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This paper was presented at the University of British Columbia, Boston College, University of California (Riverside), University of Toronto, and the winter meeting of the Econometric Society in San Francisco. We thank an anonymous referee for helpful comments. We also thank the seminar participants for helpful discussion and Paul Beaudry and Pierre Perron for comments on an earlier draft. The first author thanks the Foundation of Caja Madrid and the Spanish Secretary of Education (PB 950298) for financial support. The second author acknowledges grants from the Social Science and Humanities Research Council of Canada (SSHRC) and the Fonds de la Formation de Chercheurs et l'Aide à la Recherche du Québec (FCAR).