ON THE ASYMPTOTIC DISTRIBUTION OF IMPULSE RESPONSE FUNCTIONS WITH LONG-RUN RESTRICTIONS

Abstract

This paper adopts a two-step technique to estimate vector error correction models and provides the asymptotic distribution of the impulse response functions of such a system. The method combines two popular tools in econometrics, namely, vector autoregressive cointegration analysis in the first step and vector autoregression analysis in the second. The proposed model structure is very general in the sense that all just-identifying or overidentifying schemes that can be expressed as linear restrictions on either the contemporaneous or long-run impact of the shocks are allowed for. The long-run restrictions complicate the derivation of the asymptotic distribution of the parameter estimates as these restrictions are a function of the reduced form parameters. Consequently, the asymptotic distribution involves an extra partial derivative. This paper adopts a two-step procedure to estimate vector error correction models and provides the asymptotic distribution of the impulse response functions of such a system even if those contain long-run restrictions. The method combines two popular tools in econometrics, namely, vector autoregressive cointegration analysis and vector autoregressions. Vector autoregression (VAR) analysis has become a popular tool in empirical macroeconomics and finance. An important element in these models is the identification of independent shocks. Originally (Sims, 1980) a Choleski decomposition of the covariance matrix of the residuals was used for this purpose, thereby implicitly assuming a recursive causal ordering. Later on, structural VAR models were developed, in which the identifying restrictions are explicitly derived from theory. These restrictions can involve either contemporaneous relationships (Blanchard and Watson, 1986; Bernanke, 1986;