Widely used modified least squares estimators for estimation and inference in cointegrating regressions are discussed. The standard case with cointegration in the I(1) setting is examined and some relevant extensions are sketched. These include cointegration analysis with panel data as well as nonlinear cointegrating relationships. Extensions to higher order (co)integration, seasonal (co)integration and fractional (co)integration are very briefly mentioned. Recent developments and some avenues for future research are discussed.
Helmut Herwartz and Alexander Lange
Unlike traditional first order asymptotic approximations, the bootstrap is a simulation method to solve inferential issues in statistics and econometrics conditional on the available sample information (e.g. constructing confidence intervals, generating critical values for test statistics). Even though econometric theory yet provides sophisticated central limit theory covering various data characteristics, bootstrap approaches are of particular appeal if establishing asymptotic pivotalness of (econometric) diagnostics is infeasible or requires rather complex assessments of estimation uncertainty. Moreover, empirical macroeconomic analysis is typically constrained by short- to medium-sized time windows of sample information, and convergence of macroeconometric model estimates toward their asymptotic limits is often slow. Consistent bootstrap schemes have the potential to improve empirical significance levels in macroeconometric analysis and, moreover, could avoid explicit assessments of estimation uncertainty. In addition, as time-varying (co)variance structures and unmodeled serial correlation patterns are frequently diagnosed in macroeconometric analysis, more advanced bootstrap techniques (e.g., wild bootstrap, moving-block bootstrap) have been developed to account for nonpivotalness as a results of such data characteristics.
The cointegrated VAR approach combines differences of variables with cointegration among them and by doing so allows the user to study both long-run and short-run effects in the same model. The CVAR describes an economic system where variables have been pushed away from long-run equilibria by exogenous shocks (the pushing forces) and where short-run adjustments forces pull them back toward long-run equilibria (the pulling forces). In this model framework, basic assumptions underlying a theory model can be translated into testable hypotheses on the order of integration and cointegration of key variables and their relationships. The set of hypotheses describes the empirical regularities we would expect to see in the data if the long-run properties of a theory model are empirically relevant.