Forecast accuracy, coefficient bias and Bayesian vector autoregressions
Introduction
For more than two decades, vector autoregressive models (VARs) have been used for macroeconomic modelling and forecasting. The original Litterman [10] model contained six variables, and six lags of each variable in each equation, making the demands on data extreme. This led Litterman to suggest that more accurate forecasting models might be produced by combining the evidence contained in the data with priors specifying that each time series is a random walk. The resultant Bayesian VAR (BVAR), based on Theil and Goldberger’s [13] mixed estimation procedure, has become a standard and successful method of forecasting macroeconomic time series.
Although the 20 years that have elapsed since the pioneering days of VAR modelling resulted in the availability of significantly longer time series, the need for methods such as BVARs to reduce the demands of modelling on data have not diminished. Researchers have begun to develop larger models [8] and structural changes often prevent earlier data being used.
Recently, Abadir et al. [1] demonstrated that the bias in parameter estimates of VARs with I (1) data increases with the inclusion of extraneous I (1) variables. As a result, two major questions are posed for researchers. Should one block-segment models to reduce this parameter bias? Should the estimated number of unit roots be imposed upon such models before estimation?
Given that the equations from classical econometric models often contain only three or four variables in an equation, it is reasonable to ask whether it is better to build VARs with, say, 10 equations, or a number of inter-linked VARs, each of only three or four variables. Since any additional variables, over and above those needed to capture the macroeconomic transmission mechanism, increase the bias in the estimated coefficients, this is a significant, and as yet, largely unexplored question [11].
Clements and Hendry [6] conducted a Monte Carlo study and reported that if the true number of unit roots is unknown, it is preferable to err on the side of too many cointegrating equations rather than too few, whilst Brandner and Kunst [4] concluded to the contrary.
The second purpose of this paper is to illuminate this Clements–Hendry and Kunst–Brandner contradiction. The forecasting performance of vector error correction (VEC) models based on an estimated number of unit roots is compared to the extremes of a VAR in levels and a VAR in differences (DVAR), and all three are compared to a BVAR specification.
Forecasts from VAR models with biased estimates produce forecasts that are unconditionally unbiased [7], but this bias significantly contributes to forecast mean squared error (MSE) and bias in a conditional sense. By directly comparing the forecast performance of BVARs with VECs in a Monte Carlo experiment, some light can be shed on whether BVARs are as successful as they are because of bias-correction or parameter reduction.
Section snippets
Monte Carlo experiment
The core of the data generating process (DGP) for this experiment is a 4-equation cointegrated system with one common trend (when |φ|<1)where each ξit is a normally and independently distributed, mutually uncorrelated, zero mean, unit variance disturbance term.
In a second DGP, the core model is augmented by six extraneous and mutually uncorrelated random walks. From [1] it is known that the coefficient bias in unrestricted VARs increases
Forecast accuracy
One result, which is not reported, was particularly clear. Initially, each of the models was also estimated with an arbitrary lag length of four [12]. There was extremely strong support for using an information criterion to select the lag length over arbitrarily setting it to four in both VAR and DVAR variants. In some cases the MSFE more than halved when the lag length was estimated. Since the BVAR method is designed to be less susceptible to inefficiency due to over-specifying the lag length,
Coefficient bias
These experiments also shed some light on the degree of coefficient bias present in the eight cases considered. For simplicity and clarity, only the cases of the unrestricted VAR and the BVAR(M) will be considered with no moving average terms (θ=0) in the DGP. In this case, the DGP can also be written asusing obvious notation where Y is T×n and n is the number of equations. When there is no cointegration A=In, and when cointegration is present (φ=0)for the 4-equation model, where
Conclusions
It is common for researchers to estimate VAR forecasting models with up to 10 equations using nonstationary data and relatively short samples of say 75–100 observations. A set of experiments has been devised in an attempt to mimic some of the problems encountered in this literature by embedding a 4-equation VAR with three cointegrating equations within a 10-equation VAR that contains six extraneous random walks.
For example, the four cointegrated variables could be two domestic and two foreign
Acknowledgements
I gratefully acknowledge funding from the Australian Research Council (A79700597) and thank Stuart Nolan and Tommy Ng for research assistance.
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