We carry out a pseudo out-of-sample density forecasting study for US GDP with an autoregressive benchmark and alternatives to the benchmark that include both oil prices and stochastic volatility. The alternatives to the benchmark produce superior density forecasts. This comparative density performance appears to be driven more by stochastic volatility than by oil prices, and it primarily occurs outside of the great recession. We use our density forecasts to compute a recession risk indicator around the great recession. The alternative model with the real price of oil generates the earliest strong signal of a recession; but it surprisingly indicates reduced recession immediately after the Lehman Brothers bankruptcy. Use of the “net oil-price increase” nonlinear transformation of oil prices does lead to warnings of highly elevated risk during the Great Recession.
We thank our anonymous referee, Lutz Kilian, and participants at the BI CAMP Workshop on “Forecasting and Analysing Oil Prices,” the 22nd Annual Symposium of the Society for Nonlinear Dynamics and Econometrics, and the 2nd International Workshop on “Financial Markets and Nonlinear Dynamics” for suggestions and comments.
A. Appendix: Model estimation with stochastic volatility
We estimate the models incorporating stochastic volatility with a five-step Gibbs sampling algorithm. Let Xt denote the collection of right-hand side variables of the either the AR-SV or ADL-SV model (for the ADL-SV model, this includes both the AR lag terms and lags of the oil price measure), and let B denote a vector containing the intercept α, the AR coefficients βi, i=0, …, p–1, and, for the ADL-SV model, the oil price measure coefficients δi, i=0, …, p–1, defined in the paper’s equations (2) or (3).
Step 1: Draw the coefficients B conditional on the history of λt, and σ2.
The vector of coefficients is sampled from a conditional posterior distribution that is multivariate normal with mean and variance based on the prior mean μB and variance ΩB. The posterior mean and variance are:
Step 2: Draw the elements of the states for the mixture distribution used to approximate the χ2 distribution under the Kim, Shephard, and Chib (1998) algorithm, conditional on B, the history of λt, and σ2.
Step 3: Draw the elements of the variance λt conditional on B, σ2, and the mixture states.
Step 4: Draw the variance σ2, conditional on B and the history of λt.
The sampling of σ2, the variance of innovations to the log variances, is based on an inverse-γ priors and posteriors. The scale matrix of the posterior distribution is the sum of the prior mean × the prior degrees of freedom and where denotes the vector of innovations to the posterior draw of the volatilities in step 3.
Step 5: Draw the posterior distributions of forecasts accounting for the uncertainty in all parameters and shocks occurring over the forecast horizon.
From a forecast origin of period t, for each retained draw of the time series of λt up through t+h, B, and σ2, we: (1) draw innovations to log volatilities for periods t+1 through t+h from a normal distribution with variance matrix σ2 and use the random walk model of logλt+h to compute λt+1, …, λt+h; (2) draw innovations to Δyt+h from a normal distribution with variance λt+h, and use the autoregressive structure of the model, and oil price measures for the ADL-SV model, along with the time series of coefficients B to obtain draws of Δyt+h. The draws of Δyt+h are used to compute the forecast statistics of interest.
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