Structural vector autoregressions (SVARs) represent a prominent class of time series models used for macroeconomic analysis. The model consists of a set of multivariate linear autoregressive equations characterizing the joint dynamics of economic variables. The residuals of these equations are combinations of the underlying structural economic shocks, assumed to be orthogonal to each other. Using a minimal set of restrictions, these relations can be estimated—the so-called shock identification—and the variables can be expressed as linear functions of current and past structural shocks. The coefficients of these equations, called impulse response functions, represent the dynamic response of model variables to shocks. Several ways of identifying structural shocks have been proposed in the literature: short-run restrictions, long-run restrictions, and sign restrictions, to mention a few. SVAR models have been extensively employed to study the transmission mechanisms of macroeconomic shocks and test economic theories. Special attention has been paid to monetary and fiscal policy shocks as well as other nonpolicy shocks like technology and financial shocks. In recent years, many advances have been made both in terms of theory and empirical strategies. Several works have contributed to extend the standard model in order to incorporate new features like large information sets, nonlinearities, and time-varying coefficients. New strategies to identify structural shocks have been designed, and new methods to do inference have been introduced.
Zoë Fannon and Bent Nielsen
Outcomes of interest often depend on the age, period, or cohort of the individual observed, where cohort and age add up to period. An example is consumption: consumption patterns change over the lifecycle (age) but are also affected by the availability of products at different times (period) and by birth-cohort-specific habits and preferences (cohort). Age-period-cohort (APC) models are additive models where the predictor is a sum of three time effects, which are functions of age, period, and cohort, respectively. Variations of these models are available for data aggregated over age, period, and cohort, and for data drawn from repeated cross-sections, where the time effects can be combined with individual covariates. The age, period, and cohort time effects are intertwined. Inclusion of an indicator variable for each level of age, period, and cohort results in perfect collinearity, which is referred to as “the age-period-cohort identification problem.” Estimation can be done by dropping some indicator variables. However, dropping indicators has adverse consequences such as the time effects are not individually interpretable and inference becomes complicated. These consequences are avoided by instead decomposing the time effects into linear and non-linear components and noting that the identification problem relates to the linear components, whereas the non-linear components are identifiable. Thus, confusion is avoided by keeping the identifiable non-linear components of the time effects and the unidentifiable linear components apart. A variety of hypotheses of practical interest can be expressed in terms of the non-linear components.
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.