Multivariate, classic econometric models

The convention of using a multiple time series model with constant parameters and assuming that the indicators in the model are hit with shocks of equal sizes over time may not always be realistic in practice, especially for longer periods of time. The Time-varying Bayesian VAR model can ease these assumptions and produce a more flexible model and is sometimes used in cases where the time period is a bit longer or when the economy is subject to policy changes.

The Minnesota BVAR is a Bayesian VAR model with a prior developed by Litterman and Sims at the University of Minnesota. Similar to how a penalized model shrinks the parameters towards zero, the Minnesota prior shrinks them towards a random walk. The prior also specifies a larger variance for shorter lags, implying a prior belief that shorter lags have a larger impact than longer.

Bayesian statistics, in contrast to classical statistics, use probability more widely to model uncertainty. A Bayesian model use informative priors, i.e. a probability distribution that express one's beliefs of a certain variable, to create a benchmark model and then shrink the parameter uncertainty to get more accurate forecasts. In macroeconomics the steady state model has been honored for the ability to incorporate economic theory by the prior. An example of a prior is inflation that, in the long-term, should center around 2%, which can be incorporated into the model by the prior. A criticism of the Bayesian model is the self-confirming effect you may get when setting a too tight prior, confirming your own belief rather than building an accurate model.

Auto-Regressive Distributed Lag was the standard model before the VAR model was invented. Compared to the VAR, it’s a less complex model, where the variables are not seen as interrelated. The main variable that are forecasted depends on the indicators, but the indicators do not depend on other indicators or the main variable.

In the statistical analysis of time series, Auto-Regressive–Moving-Average (ARMA) models provide a description of the relationships between the variables in terms of the two factors: autoregression (AR) and moving average (MA). The AR part involves regressing the variable on its own lagged (i.e. past) values. The MA part involves modeling the error term as a linear combination of error terms occurring contemporaneously and at various times in the past. VARMA is the VAR (multivariate) version of the ARMA model.

Vector Error Correction Models are especially useful for data sets with long-run relationships (also called cointegration). VECMs are however useful for estimating both short-term and long-term effects of one-time series on another. The term error-correction relates to the fact that the last period’s deviation from a long-run equilibrium, the error, influences its short-run dynamics. These models estimate, besides the long-run relationships between variables, also directly the speed at which a dependent variable returns to equilibrium after a change in other variables.

Vector Auto Regression is a model that captures the linear relations among multiple time series. VAR models generalize the univariate autoregressive model (AR model) by allowing for multiple variables. All variables in a VAR enter the model in the same way: each variable has an equation explaining its evolution based on its own lagged values, the lagged values of the other model variables, and an error term. The calculations find the best common lag length for all variables in all equations (vectors).