Ffuse prior and posterior detailed in such studies as Kadiyala and Karlsson (1997). .4/A. Carriero, T. E. Clark and M. MarcellinoAt any given forecast origin, estimation is quite fast, because the forecasting model is a single equation. The model with stochastic volatility is estimated with a Metropolis-within-Gibbs NS-018 biological activity algorithm, used in such studies as Clark (2011) and Carriero et al. (2012). The posterior mean and variance of the coefficient vector are given by ? = ?T t=-1 Xm,t yt + -1 , tT t=.7/?-1 = -1 +-1 Xm,t Xm,t , t.8/where we again omit the m-index from the parameters for notational simplicity. In presenting our results, we focus on forecasts that are obtained by estimating the forecasting models with a recursive scheme: the estimation sample expands as forecasting moves forwards in time. A rolling scheme, under which the size of the estimation sample remains fixed over time but the first observation moves forwards in time, is in general less efficient but can be more robust in the presence of changes in regression parameters and (for density forecasts) error variances. Hence, for the BMF models with constant volatility we also report results based on a rolling estimation scheme. However, as we shall show below, rolling window estimation of the model is not sufficient to obtain point and density forecasts that are as good as those obtained with the (recursive) BMFSV specification (with a gap that is particularly large for density forecasts). In all cases, we obtain forecast distributions by sampling as appropriate from the posterior distribution. For example, in the case of the BMFSV model, for each set of draws of parameters, we (a) simulate volatility for the quarter being forecast by using the random-walk structure of log-volatility, (b) draw shocks to the variable with variance equal to the draw of volatility and (c) use the structure of the model to obtain a draw of the future value (i.e. forecast) of the variable. We then form point forecasts as means of the draws of simulated forecasts and density forecasts from the simulated distribution of forecasts. Conditionally on the model, the posterior distribution reflects all sources of uncertainty (latent states, parameters and shocks over the forecast interval).4.Competing nowcastsWe compare our BMF and BMFSV nowcasts with those generated from AR models and with survey-based forecasts (which pool many predictions, based on timely information). These are typically tough benchmarks in forecast competitions. The results in Carriero et al. (2013) also indicate that MIDAS and unrestricted MIDAS specifications can produce relatively good nowcasts of GDP growth. However, these models are primarily designed for use with small sets of indicators (possibly with a high frequency mismatch) and point forecasts; using large sets of indicators and allowing stochastic volatility to obtain reliable density forecasts is feasible but relatively difficult. As a result, we abstract from MIDAS and unrestricted MIDAS forecasts in the comparison.Realtime Nowcasting4.1. Auto-regressive models In our forecast evaluation, in light of evidence in other studies of the difficulty of Sch66336 site beating simple AR models for GDP growth, we include forecasts from AR(2) models. The models take the same basic forms given in expressions (1) and (2), with Xm,t defined to include just a constant and two lags of GDP growth. In keeping with our realtime set-up, we generate four different AR-based forecasts of GDP growth in eac.Ffuse prior and posterior detailed in such studies as Kadiyala and Karlsson (1997). .4/A. Carriero, T. E. Clark and M. MarcellinoAt any given forecast origin, estimation is quite fast, because the forecasting model is a single equation. The model with stochastic volatility is estimated with a Metropolis-within-Gibbs algorithm, used in such studies as Clark (2011) and Carriero et al. (2012). The posterior mean and variance of the coefficient vector are given by ? = ?T t=-1 Xm,t yt + -1 , tT t=.7/?-1 = -1 +-1 Xm,t Xm,t , t.8/where we again omit the m-index from the parameters for notational simplicity. In presenting our results, we focus on forecasts that are obtained by estimating the forecasting models with a recursive scheme: the estimation sample expands as forecasting moves forwards in time. A rolling scheme, under which the size of the estimation sample remains fixed over time but the first observation moves forwards in time, is in general less efficient but can be more robust in the presence of changes in regression parameters and (for density forecasts) error variances. Hence, for the BMF models with constant volatility we also report results based on a rolling estimation scheme. However, as we shall show below, rolling window estimation of the model is not sufficient to obtain point and density forecasts that are as good as those obtained with the (recursive) BMFSV specification (with a gap that is particularly large for density forecasts). In all cases, we obtain forecast distributions by sampling as appropriate from the posterior distribution. For example, in the case of the BMFSV model, for each set of draws of parameters, we (a) simulate volatility for the quarter being forecast by using the random-walk structure of log-volatility, (b) draw shocks to the variable with variance equal to the draw of volatility and (c) use the structure of the model to obtain a draw of the future value (i.e. forecast) of the variable. We then form point forecasts as means of the draws of simulated forecasts and density forecasts from the simulated distribution of forecasts. Conditionally on the model, the posterior distribution reflects all sources of uncertainty (latent states, parameters and shocks over the forecast interval).4.Competing nowcastsWe compare our BMF and BMFSV nowcasts with those generated from AR models and with survey-based forecasts (which pool many predictions, based on timely information). These are typically tough benchmarks in forecast competitions. The results in Carriero et al. (2013) also indicate that MIDAS and unrestricted MIDAS specifications can produce relatively good nowcasts of GDP growth. However, these models are primarily designed for use with small sets of indicators (possibly with a high frequency mismatch) and point forecasts; using large sets of indicators and allowing stochastic volatility to obtain reliable density forecasts is feasible but relatively difficult. As a result, we abstract from MIDAS and unrestricted MIDAS forecasts in the comparison.Realtime Nowcasting4.1. Auto-regressive models In our forecast evaluation, in light of evidence in other studies of the difficulty of beating simple AR models for GDP growth, we include forecasts from AR(2) models. The models take the same basic forms given in expressions (1) and (2), with Xm,t defined to include just a constant and two lags of GDP growth. In keeping with our realtime set-up, we generate four different AR-based forecasts of GDP growth in eac.