Averaging forecasts from VARs with uncertain instabilities

A body of recent work suggests commonly–used VAR models of output, inflation, and interest rates may be prone to instabilities. In the face of such instabilities, a variety of estimation or forecasting methods might be used to improve the accuracy of forecasts from a VAR. These methods include using different approaches to lag selection, different observation windows for estimation, (over-) differencing, intercept correction, stochastically time–varying parameters, break dating, discounted least squares, Bayesian shrinkage, and detrending of inflation and interest rates. Although each individual method could be useful, the uncertainty inherent in any single representation of instability could mean that combining forecasts from the entire range of VAR estimates will further improve forecast accuracy. Focusing on models of U.S. output, prices, and interest rates, this paper examines the effectiveness of combination in improving VAR forecasts made with real–time data. The combinations include simple averages, medians, trimmed means, and a number of weighted combinations, based on: Bates-Granger regressions, factor model estimates, regressions involving just forecast quartiles, Bayesian model averaging, and predictive least squares–based weighting. Our goal is to identify those approaches that, in real time, yield the most accurate forecasts of these variables. We use forecasts from simple univariate time series models and the Survey of Professional Forecasters as benchmarks.Economic forecasting ; Vector autoregression

Combining forecasts from nested models

Motivated by the common finding that linear autoregressive models often forecast better than models that incorporate additional information, this paper presents analytical, Monte Carlo, and empirical evidence on the effectiveness of combining forecasts from nested models. In our analytics, the unrestricted model is true, but a subset of the coefficients are treated as being local-to-zero. This approach captures the practical reality that the predictive content of variables of interest is often low. We derive MSE-minimizing weights for combining the restricted and unrestricted forecasts. Monte Carlo and empirical analyses verify the practical e effectiveness of our combination approach.Econometric models ; Economic forecasting

Averaging forecasts from VARs with uncertain instabilities

Recent work suggests VAR models of output, inflation, and interest rates may be prone to instabilities. In the face of such instabilities, a variety of estimation or forecasting methods might be used to improve the accuracy of forecasts from a VAR. The uncertainty inherent in any single representation of instability could mean that combining forecasts from a range of approaches will improve forecast accuracy. Focusing on models of U.S. output, prices, and interest rates, this paper examines the effectiveness of combining various models of instability in improving VAR forecasts made with real-time data.Econometric models ; Economic forecasting

Tests of Equal Forecast Accuracy and Encompassing for Nested Models

We examine the asymptotic and finite-sample properties of tests for equal forecast accuracy and encompassing applied to 1-step ahead forecasts from nested parametric models. We first derive the asymptotic distributions of two standard tests and one new test of encompassing. Tables of asymptotically valid critical values are provided. Monte Carlo methods are then used to evaluate the size and power of the tests of equal forecast accuracy and encompassing. The simulations indicate that post-sample tests can be reasonably well sized. Of the post-sample tests considered, the encompassing test proposed in this paper is the most powerful. We conclude with an empirical application regarding the predictive content of unemployment for inflation.

Nested forecast model comparisons: a new approach to testing equal accuracy

This paper develops bootstrap methods for testing whether, in a finite sample, competing out-of-sample forecasts from nested models are equally accurate. Most prior work on forecast tests for nested models has focused on a null hypothesis of equal accuracy in population basically, whether coefficients on the extra variables in the larger, nesting model are zero. We instead use an asymptotic approximation that treats the coefficients as non-zero but small, such that, in a finite sample, forecasts from the small model are expected to be as accurate as forecasts from the large model. Under that approximation, we derive the limiting distributions of pairwise tests of equal mean square error, and develop bootstrap methods for estimating critical values. Monte Carlo experiments show that our proposed procedures have good size and power properties for the null of equal finite-sample forecast accuracy. We illustrate the use of the procedures with applications to forecasting stock returns and inflation.

Tests of equal predictive ability with real-time data

This paper examines the asymptotic and finite-sample properties of tests of equal forecast accuracy applied to direct, multi-step predictions from both non-nested and nested linear regression models. In contrast to earlier work in the literature, our asymptotics take account of the real-time, revised nature of the data. Monte Carlo simulations indicate that our asymptotic approximations yield reasonable size and power properties in most circumstances. The paper concludes with an examination of the real-time predictive content of various measures of economic activity for inflation.Economic forecasting ; Real-time data

In-sample tests of predictive ability: a new approach

This paper presents analytical, Monte Carlo, and empirical evidence linking in-sample tests of predictive content and out-of-sample forecast accuracy. Our approach focuses on the negative effect that finite-sample estimation error has on forecast accuracy despite the presence of significant population-level predictive content. Specifically, we derive simple-to-use in-sample tests that test not only whether a particular variable has predictive content but also whether this content is estimated precisely enough to improve forecast accuracy. Our tests are asymptotically non-central chi-square or non-central normal. We provide a convenient bootstrap method for computing the relevant critical values. In the Monte Carlo and empirical analysis, we compare the effectiveness of our testing procedure with more common testing procedures.

In-sample tests of predictive ability: a new approach

This paper presents analytical, Monte Carlo, and empirical evidence linking in-sample tests of predictive content and out-of-sample forecast accuracy. Our approach focuses on the negative effect that finite-sample estimation error has on forecast accuracy despite the presence of significant population-level predictive content. Specifically, we derive simple-to-use in-sample tests that test not only whether a particular variable has predictive content but also whether this content is estimated precisely enough to improve forecast accuracy. Our tests are asymptotically non-central chi-square or non-central normal. We provide a convenient bootstrap method for computing the relevant critical values. In the Monte Carlo and empirical analysis, we compare the effectiveness of our testing procedure with more common testing procedures.Economic forecasting

Advances in forecast evaluation

This paper surveys recent developments in the evaluation of point forecasts. Taking West's (2006) survey as a starting point, we briefly cover the state of the literature as of the time of West's writing. We then focus on recent developments, including advancements in the evaluation of forecasts at the population level (based on true, unknown model coefficients), the evaluation of forecasts in the finite sample (based on estimated model coefficients), and the evaluation of conditional versus unconditional forecasts. We present original results in a few subject areas: the optimization of power in determining the split of a sample into in-sample and out-of-sample portions; whether the accuracy of inference in evaluation of multi-step forecasts can be improved with judicious choice of HAC estimator (it can); and the extension of West's (1996) theory results for population-level, unconditional forecast evaluation to the case of conditional forecast evaluation.Forecasting