On forecasting cointegrated seasonal time series

We analyze periodic and seasonal cointegration models for bivariate quarterly observed time series in an empirical forecasting study. We include both single equation and multiple equation methods for those two classes of models. A VAR model in first differences, with and without cointegration restrictions, and a VAR model in annual differences are also included in the analysis, where they serve as benchmark models. Our empirical results indicate that the VAR model in first differences without cointegration is best if one-step ahead forecasts are considered. For longer forecast horizons however, the VAR model in annual differences is better. When comparing periodic versus seasonal cointegration models, we find that the seasonal cointegration models tend to yield better forecasts. Finally, there is no clear indication that multiple equations methods improve on single equation methods.     

The COVID-19 shock and challenges for inflation modelling

We document the impact of COVID-19 on inflation modelling within a vector autoregression (VAR) model and provide guidance for forecasting euro area inflation during the pandemic. We show that estimated parameters are strongly affected, leading to different and sometimes implausible projections. As a solution, we propose to augment the VAR by allowing the residuals to have a fat-tailed distribution instead of a Gaussian one. This also outperforms with respect to unconditional forecasts. Yet, what brings sizeable forecast gains during the pandemic is adding meaningful off-model information, such as that entailed in the Survey of Professional Forecasters. The fat-tailed VAR loses part, but not all of its relative advantage compared to the Gaussian version when producing conditional inflation forecasts in a real-time setup. It is the joint fat-tailed errors and multi-equation modelling that manage to robustify models against extreme observations; in a single-equation model the same solution is less effective.     

The performance of alternative forecasting methods for SETAR models

We compare a number of methods that have been proposed in the literature for obtaining h-step ahead minimum mean square error forecasts for self-exciting threshold autoregressive (SETAR) models. These forecasts are compared to those from an AR model. The comparison of forecasting methods is made using Monte Carlo simulation. The Monte-Carlo method of calculating SETAR forecasts is generally at least as good as that of the other methods we consider. An exception is when the disturbances in the SETAR model come from a highly asymmetric distribution, when a Bootstrap method is to be preferred.An empirical application calculates multi-period forecasts from a SETAR model of US gross national product using a number of the forecasting methods. We find that whether there are improvements in forecast performance relative to a linear AR model depends on the historical epoch we select, and whether forecasts are evaluated conditional on the regime the process was in at the time the forecast was made.     

Can the random walk model be beaten in out-of-sample density forecasts? Evidence from intraday foreign exchange rates

It has been documented that random walk outperforms most economic structural and time series models in out-of-sample forecasts of the conditional mean dynamics of exchange rates. In this paper, we study whether random walk has similar dominance in out-of-sample forecasts of the conditional probability density of exchange rates given that the probability density forecasts are often needed in many applications in economics and finance. We first develop a nonparametric portmanteau test for optimal density forecasts of univariate time series models in an out-of-sample setting and provide simulation evidence on its finite sample performance. Then we conduct a comprehensive empirical analysis on the out-of-sample performances of a wide variety of nonlinear time series models in forecasting the intraday probability densities of two major exchange rates—Euro/Dollar and Yen/Dollar. It is found that some sophisticated time series models that capture time-varying higher order conditional moments, such as Markov regime-switching models, have better density forecasts for exchange rates than random walk or modified random walk with GARCH and Student-t innovations. This finding dramatically differs from that on mean forecasts and suggests that sophisticated time series models could be useful in out-of-sample applications involving the probability density.     

An empirical investigation of direct and iterated multistep conditional forecasts

When constructing unconditional point forecasts, both direct and iterated multistep (DMS and IMS) approaches are common. However, in the context of producing conditional forecasts, IMS approaches based on vector autoregressions are far more common than simpler DMS models. This is despite the fact that there are theoretical reasons to believe that DMS models are more robust to misspecification than are IMS models. In the context of unconditional forecasts, Marcellino et al. (Journal of Econometrics, 2006, 135, 499–526) investigate the empirical relevance of these theories. In this paper, we extend that work to conditional forecasts. We do so based on linear bivariate and trivariate models estimated using a large dataset of macroeconomic time series. Over comparable samples, our results reinforce those in Marcellino et al.: the IMS approach is typically a bit better than DMS with significant improvements only at longer horizons. In contrast, when we focus on the Great Moderation sample we find a marked improvement in the DMS approach relative to IMS. The distinction is particularly clear when we forecast nominal rather than real variables where the relative gains can be substantial.     

Forecasting combination and encompassing tests

In this paper we demonstrate that forecast encompassing tests are valuable tools in getting an insight into why competing forecasts may be combined to produce a composite forecast which is superior to the individual forecasts. We also argue that results from forecast encompassing tests are potentially useful in model specification. We illustrate this using forecasts of quarterly UK consumption expenditure data from three classes of models: ARIMA, DHSY and VAR models.     

Do precious metal prices help in forecasting South African inflation?

In this paper we test whether the key metals prices of gold and platinum significantly improve inflation forecasts for the South African economy. We also test whether controlling for conditional correlations in a dynamic setup, using bivariate Bayesian-Dynamic Conditional Correlation (B-DCC) models, improves inflation forecasts. To achieve this we compare out-of-sample forecast estimates of the B-DCC model to Random Walk, Autoregressive and Bayesian VAR models. We find that for both the BVAR and BDCC models, improving point forecasts of the Autoregressive model of inflation remains an elusive exercise. This, we argue, is of less importance relative to the more informative density forecasts. For this we find improved forecasts of inflation for the B-DCC models at all forecasting horizons tested. We thus conclude that including metals price series as inputs to inflation models leads to improved density forecasts, while controlling for the dynamic relationship between the included price series and inflation similarly leads to significantly improved density forecasts.     

On the forecasting performance of a small-scale DSGE model

Dynamic stochastic general equilibrium (DSGE) models have recently become standard tools for policy analysis. Nevertheless, their forecasting properties have still barely been explored. In this article, we address this problem by examining the quality of forecasts of the key U.S. economic variables: the three-month Treasury bill yield, the GDP growth rate and GDP price index inflation, from a small-size DSGE model, trivariate vector autoregression (VAR) models and the Philadelphia Fed Survey of Professional Forecasters (SPF). The ex post forecast errors are evaluated on the basis of the data from the period 1994–2006. We apply the Philadelphia Fed “Real-Time Data Set for Macroeconomists” to ensure that the data used in estimating the DSGE and VAR models was comparable to the information available to the SPF.Overall, the results are mixed. When comparing the root mean squared errors for some forecast horizons, it appears that the DSGE model outperforms the other methods in forecasting the GDP growth rate. However, this characteristic turned out to be statistically insignificant. Most of the SPF's forecasts of GDP price index inflation and the short-term interest rate are better than those from the DSGE and VAR models.     

Efficient forecast tests for conditional policy forecasts

Central Banks regularly make forecasts, such as the Fed’s Greenbook forecast, that are conditioned on hypothetical paths for the policy interest rate. While there are good public policy reasons to evaluate the quality of such forecasts, up until now, the most common approach has been to ignore their conditional nature and apply standard forecast efficiency tests. In this paper we derive tests for the efficiency of conditional forecasts. Intuitively, these tests involve implicit estimates of the degree to which the conditioning path is counterfactual and the magnitude of the policy feedback over the forecast horizon. We apply the tests to the Greenbook forecast and the Bank of England’s inflation report forecast, finding some evidence of forecast inefficiency. Nonetheless, we argue that the conditional nature of the forecasts made by central banks represents a substantial impediment to the analysis of their quality—stronger assumptions are needed and forecast inefficiency may go undetected for longer than would be the case if central banks were instead to report unconditional forecasts.     

Evaluating the accuracy of value-at-risk forecasts: New multilevel tests

We propose independence and conditional coverage tests which are aimed at evaluating the accuracy of Value-at-Risk (VaR) forecasts from the same model at different confidence levels. The proposed procedures are multilevel tests, i.e., joint tests of several quantiles corresponding to different confidence levels. In a comprehensive Monte Carlo exercise, we document the superiority of the proposed tests with respect to existing multilevel tests. In an empirical application, we illustrate the implementation of the tests using several VaR models and daily data for 15 MSCI world indices.     

Using low frequency information for predicting high frequency variables

We analyze ways of incorporating low frequency information into models for the prediction of high frequency variables. In doing so, we consider the two existing versions of the mixed frequency VAR, with a focus on the forecasts for the high frequency variables. Furthermore, we introduce new models, namely the reverse unrestricted MIDAS (RU-MIDAS) and reverse MIDAS (R-MIDAS), which can be used for producing forecasts of high frequency variables that also incorporate low frequency information. We then conduct several empirical applications for assessing the relevance of quarterly survey data for forecasting a set of monthly macroeconomic indicators. Overall, it turns out that low frequency information is important, particularly when it has just been released.     

Forecasting with vector autoregressive (VAR) models subject to business cycle restrictions

In the last decade VAR models have become a widely-used tool for forecasting macroeconomic time series. To improve the out-of-sample forecasting accuracy of these models, Bayesian random-walk prior restrictions are often imposed on VAR model parameters. This paper focuses on whether placing an alternative type of restriction on the parameters of unrestricted VAR models improves the out-of-sample forecasting performance of these models. The type of restriction analyzed here is based on the business cycle characteristics of U.S. macroeconomic data, and in particular, requires that the dynamic behavior of the restricted VAR model mimic the business cycle characteristics of historical data. The question posed in this paper is: would a VAR model, estimated subject to the restriction that the cyclical characteristics of simulated data from the model “match up” with the business cycle characteristics of U.S. data, generate more accurate out-of-sample forecasts than unrestricted or Bayesian VAR models?     

Prediction regions for interval-valued time series

We approximate probabilistic forecasts for interval-valued time series by offering alternative approaches. After fitting a possibly non-Gaussian bivariate vector autoregression (VAR) model to the center/log-range system, we transform prediction regions (analytical and bootstrap) for this system into regions for center/range and upper/lower bounds systems. Monte Carlo simulations show that bootstrap methods are preferred according to several new metrics. For daily S&P 500 low/high returns, we build joint conditional prediction regions of the return level and volatility. We illustrate the usefulness of obtaining bootstrap forecasts regions for low/high returns by developing a trading strategy and showing its profitability when compared to using point forecasts.     

Forecasting volatility via stock return,range, trading volume and spillover effects: The case of Brazil

For the purpose of developing alternative approach for forecasting volatility, we consider heterogeneous VAR (HVAR) model which accommodates the market effects of different horizons, namely, daily, weekly and monthly effects, and examine the interdependence of stock markets in Brazil and the US, based on information of daily return, range and trading volume. To compare with the new approach, we also work with the univariate and multivariate GARCH models with asymmetric effects, trading volumes and fat-tails. The heteroskedasticity-corrected Granger causality tests based on the HVAR show the strong evidence of such spillover effects. We assess the value-at-risk thresholds for Brazil, based on the out-of-sample forecasts of the HVAR model, finding the new approach works satisfactory for the periods including the global financial crisis, without assuming heavy-tailed conditional distributions.     

DCC- and DECO-HEAVY: Multivariate GARCH models based on realized variances and correlations

This paper introduces the scalar DCC-HEAVY and DECO-HEAVY models for conditional variances and correlations of daily returns based on measures of realized variances and correlations built from intraday data. Formulas for multi-step forecasts of conditional variances and correlations are provided. Asymmetric versions of the models are developed. An empirical study shows that in terms of forecasts the scalar HEAVY models outperform the scalar BEKK-HEAVY model based on realized covariances and the scalar BEKK, DCC, and DECO multivariate GARCH models based exclusively on daily data.     

Combining survey long-run forecasts and nowcasts with BVAR forecasts using relative entropy

This paper constructs hybrid forecasts that combine forecasts from vector autoregressive (VAR) model(s) with both short- and long-term expectations from surveys. Specifically, we use the relative entropy to tilt one-step-ahead and long-horizon VAR forecasts to match the nowcasts and long-horizon forecasts from the Survey of Professional Forecasters. We consider a variety of VAR models, ranging from simple fixed-parameter to time-varying parameters. The results across models indicate meaningful gains in multi-horizon forecast accuracy relative to model forecasts that do not incorporate long-term survey conditions. Accuracy improvements are achieved for a range of variables, including those that are not tilted directly but are affected through spillover effects from tilted variables. The accuracy gains for hybrid inflation forecasts from simple VARs are substantial, statistically significant, and competitive to time-varying VARs, univariate benchmarks, and survey forecasts. We view our proposal as an indirect approach to accommodating structural change and moving end points.     

The Incremental Predictive Information Associated with Using Theoretical New Keynesian DSGE Models vs. Simple Linear Econometric Models*

In this paper we construct output gap and inflation predictions using a variety of dynamic stochastic general equilibrium (DSGE) sticky price models. Predictive density accuracy tests related to the test discussed in Corradi and Swanson [Journal of Econometrics (2005a), forthcoming] as well as predictive accuracy tests due to Diebold and Mariano [Journal of Business and Economic Statistics (1995) , Vol. 13, pp. 253–263]; and West [Econometrica (1996) , Vol. 64, pp. 1067–1084] are used to compare the alternative models. A number of simple time‐series prediction models (such as autoregressive and vector autoregressive (VAR) models) are additionally used as strawman models. Given that DSGE model restrictions are routinely nested within VAR models, the addition of our strawman models allows us to indirectly assess the usefulness of imposing theoretical restrictions implied by DSGE models on unrestricted econometric models. With respect to predictive density evaluation, our results suggest that the standard sticky price model discussed in Calvo [Journal of Monetary Economics (1983), Vol. XII, pp. 383–398] is not outperformed by the same model augmented either with information or indexation, when used to predict the output gap. On the other hand, there are clear gains to using the more recent models when predicting inflation. Results based on mean square forecast error analysis are less clear‐cut, although the standard sticky price model fares best at our longest forecast horizon of 3 years, it performs relatively poorly at shorter horizons. When the strawman time‐series models are added to the picture, we find that the DSGE models still fare very well, often outperforming our forecast competitions, suggesting that theoretical macroeconomic restrictions yield useful additional information for forming macroeconomic forecasts.     

A Reconciliation of Some Paradoxical Empirical Results on the Expectations Model of the Term Structure

In this paper we attempt to reconcile contradictory empirical results for the expectations model of the term structure which are found when it is tested by a variety of methods based on single-equation and VAR models. Using monthly data for one-month and three-month interest rates, we show that the expectations hypothesis is rejected for the US and UK term structures on the basis of some popular tests. However, tests based on VAR models or on IV regressions of yield spreads on future short rate changes provide no evidence against the expectations model with a random component in the term premium.     

Testing for Misspecification in Binary Response Models with Competing Distributions*

In empirical studies, the probit and logit models are often used without checks for their competing distributional specifications. It is also rare for econometric tests to be focused on this issue. Santos Silva [Journal of Applied Econometrics (2001 ), Vol. 16, pp. 577–597] is an important recent exception. By using the conditional moment test principle, we discuss a wide class of non‐nested tests that can easily be applied to detect the competing distributions for the binary response models. This class of tests includes the test of Santos Silva (2001 ) for the same task as a particular example and provides other useful alternatives. We also compare the performance of these tests by a Monte Carlo simulation.     

Bayesian VAR forecasts,survey information,and structural change in the euro area

We incorporate external information extracted from the European Central Bank’s Survey of Professional Forecasters into the predictions of a Bayesian VAR using entropic tilting and soft conditioning. The resulting conditional forecasts significantly improve the plain BVAR point and density forecasts. Importantly, we do not restrict the forecasts at a specific quarterly horizon but their possible paths over several horizons jointly since the survey information comes in the form of one- and two-year-ahead expectations. As well as improving the accuracy of the variable that we target, the spillover effects on “other-than-targeted” variables are relevant in size and are statistically significant. We document that the baseline BVAR exhibits an upward bias for GDP growth after the financial crisis, and our results provide evidence that survey forecasts can help mitigate the effects of structural breaks on the forecasting performance of a popular macroeconometric model.