Combining Large Numbers of Density Predictions with Bayesian Predictive Synthesis

Tony Chernis

doi:10.1515/snde-2022-0108

Article

Combining Large Numbers of Density Predictions with Bayesian Predictive Synthesis

Tony Chernis

Published/Copyright: December 18, 2023

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 28 Issue 2

Abstract

Bayesian Predictive Synthesis is a flexible method of combining density predictions. The flexibility comes from the ability to choose an arbitrary synthesis function to combine predictions. I study choice of synthesis function when combining large numbers of predictions – a common occurrence in macroeconomics. Estimating combination weights with many predictions is difficult, so I consider shrinkage priors and factor modelling techniques to address this problem. These techniques provide an interesting contrast between the sparse weights implied by shrinkage priors and dense weights of factor modelling techniques. I find that the sparse weights of shrinkage priors perform well across exercises.

Keywords: density prediction; forecast combination; econometric and statistical methods

Corresponding author: Tony Chernis, Bank of Canada, K1A 0G9, Ottawa, Canada, E-mail: tchernis@bankofcanada.ca

Appendix A: Technical Appendix

A.1 MCMC Algorithm

This section describes the Markov Chain Monte Carlo (MCMC) algorithm used to estimate the forecast combinations. It largely follows McAlinn and West (2019) for the BPS steps, Cadonna, Frühwirth-Schnatter, and Knaus (2020) for the global-local shrinkage priors combinations, and Lopes and West (2004) for the factor model combinations. The MCMC follows a two-component block Gibbs sampler: one component samples the synthesis function parameters, and the second samples from the expert forecast distributions or the agent states. As such, I discuss the estimation of each synthesis function separately, followed by details on sampling the agent states.

A.2 Global-local Shrinkage Combinations

This section describes the estimation of the global-local shrinkage synthesis functions. Knaus et al. (2021) provide an R package and the vignette is an excellent overview of the estimation and priors of these models. More details are available in Cadonna, Frühwirth-Schnatter, and Knaus (2020) and Bitto and Frühwirth-Schnatter (2019). I first describe the model, followed by the priors, and then describe the MCMC algorithm.

Starting with the centered parameterization of the synthesis function, for t = 1, …, T, we have that

(7) y t = x t β t + ϵ t β t = β t − 1 + u t ϵ t ∼ N 0 , σ t 2 u t ∼ N ( 0 , Q )

where y _t is a univariate response variable and x _t = (x _t0, x _t1, …, x _td) is a d-dimensional row vector containing the regressors at time t, with x _t1 corresponding to the intercept.

For simplicity, I assume here that Q = Diag(θ ₁, …, θ _d) is a diagonal matrix, implying that the state innovations are conditionally independent. Moreover, I assume the initial value follows a normal distribution (i.e. β 0 ∼ N d ( β , Q ) ), with initial mean β = (β ₁, …, β _d). Model (7) can be rewritten equivalently in the non-centered parametrization as

(8) y t = x t β + x t D i a g ( θ 1 , … , θ d ) β t ̃ + ϵ t , ϵ t ∼ N 0 , σ t 2 β t ̃ = β ̃ t − 1 + u t ̃ , u t ̃ ∼ N d ( 0 , I d )

with β ̃ 0 ∼ N d ( 0 , I d ) , where I _d is the d-dimensional identity matrix. Furthermore, the model can accommodate stochastic volatility or constant volatility. In the former case, the log-volatility h t = log ⁡ σ t 2 follows a random-walk. More specifically,

(9) h t | h t − 1 , σ η 2 ∼ N h t − 1 , σ η 2 ,

with initial state h 0 ∼ N a 0 , b 0 .

A.2.1 Shrinkage Priors on Variances and Model Parameters

This section describes the priors used in the previously discussed synthesis function. The triple gamma prior can be represented as a conditionally normal distribution, where the component specific variance is itself a compound probability distribution resulting from two gamma distributions. This results in independent normal-gamma-gamma (NGG) priors (Cadonna, Frühwirth-Schnatter, and Knaus 2020), both on the standard deviations of the innovations, that is the θ j ’s, and on the means of the initial value β _j, for j = 1, …, d. Note that, in the case of the standard deviations, this can equivalently be seen as a triple gamma prior on the innovation variances θ _j, for j = 1, …, d. In the constant parameterizations, I place an NGG prior on the β _j using the centered parameterization:

(10) θ j | ξ j 2 ∼ N 0 , ξ j 2 , ξ j 2 | a ξ , κ j 2 ∼ G a ξ , a ξ κ j 2 2 , κ j 2 | c ξ , κ B 2 ∼ G c ξ , c ξ κ B 2

(11) β j | τ j 2 ∼ N 0 , τ j 2 , τ j 2 | a τ , λ j 2 ∼ N a τ , a τ λ j 2 2 λ j 2 | c τ , λ B 2 ∼ N c τ , c τ λ B 2 .

Letting c ^ξ and c ^τ go to infinity results in a normal-gamma (NG) prior (Brown and Griffin 2010) on the θ j ’s and β _j’s. It also has a representation as a conditionally normal distribution, with the component specific variance following a gamma distribution; that is

(12) θ j | ξ j 2 ∼ N 0 , ξ j 2 , ξ j 2 | a ξ , κ B 2 ∼ G a ξ , a ξ κ B 2 2 ,

(13) β j | τ j 2 ∼ N 0 , τ j 2 , τ j 2 | a τ , λ B 2 ∼ G a τ , a τ λ B 2 2 .

The parameters a ^ξ, a ^τ, c ^ξ, c ^τ, κ B 2 , and λ B 2 can be learned from the data through appropriate prior distributions. Results from Cadonna, Frühwirth-Schnatter, and Knaus (2020) motivate the use of different distributions for these parameters under the NGG and NG prior. In the NGG case, the scaled global shrinkage parameters conditionally follow F distributions, depending on their respective pole and tail parameters:

(14) κ B 2 2 | a ξ , c ξ ∼ F ( 2 a ξ , 2 c ξ ) , λ B 2 2 | a τ , c τ ∼ F ( 2 a τ , 2 c τ ) .

The scaled tail and pole parameters, in turn, follow beta distributions:

(15) 2 a ξ ∼ B α a ξ , β a ξ , 2 c ξ ∼ B α c ξ , β c ξ ,

(16) 2 a τ ∼ B α a τ , β a τ , 2 c τ ∼ B α c τ , β c τ .

These priors are chosen as they imply a uniform prior on a suitably defined model size; see Cadonna, Frühwirth-Schnatter, and Knaus (2020) for details. In the NG case, the global shrinkage parameters follow independent gamma distributions:

(17) κ B 2 ∼ G ( d 1 , d 2 ) , λ B 2 ∼ G ( e 1 , e 2 ) .

In order to learn the pole parameters in the NG case, I generalize the approach taken in Bitto and Frühwirth-Schnatter (2019) and place the following gamma distributions as priors:

(18) a ξ ∼ G α a ξ , α a ξ β a ξ , a τ ∼ G α a τ , α a τ β a τ ,

which correspond to the exponential priors used in Bitto and Frühwirth-Schnatter (2019) when α a ξ = 1 and α a τ = 1 . The parameters α a ξ and α a τ act as degrees of freedom and allow the prior to be bounded away from zero.

In the constant parameter case, I employ a hierarchical prior, where the scale of an inverse gamma prior for σ ² follows a gamma distribution; that is,

(19) σ 2 | C 0 ∼ G − 1 ( c 0 , C 0 ) , C 0 ∼ G ( c 0 + g 0 , G 0 + σ − 2 − 1 ) ,

with hyperparameters c ₀, g ₀, and G ₀.

In the case of stochastic volatility, the priors on the parameters σ η 2 in Equation (9) are,

(20) σ η 2 ∼ G − 1 ( ν , S h ) , h 0 ∼ N ( a 0 , b 0 )

with hyperparameters ν, S _h, a ₀ and b ₀.

A.2.2 MCMC Sampling Algorithm

This next section describes the MCMC Gibbs sampling algorithm with Metropolis–Hastings steps to obtain draws from the posterior distribution of the global-local shrinkage prior synthesis function parameters. This is meant to be an overview of the algorithm; for more details, please refer to Cadonna, Frühwirth-Schnatter, and Knaus (2020) and Bitto and Frühwirth-Schnatter (2019).

Algorithm 1

Gibbs Sampling Algorithm

If in TVP specification, sample the latent states β ̃ = ( β ̃ 0 , … , β ̃ T ) in the non-centered parametrization from a multivariate normal distribution using precision sampling (Chan and Jeliazkov 2009). Otherwise skip.
If in TVP specification, sample jointly β ₁, …, β _d, and θ 1 , … , θ d in the non-centered parametrization from a multivariate normal distribution. Otherwise, sample β ₁, …, β _d, in the centered parameterization from a multivariate normal distribution.
If in TVP specification, perform an ancillarity-sufficiency interweaving step and redraw each β ₁, …, β _d from a normal distribution and each θ ₁, …, θ _d from a generalized inverse Gaussian distribution using the MATLAB implementation (Hartkopf 2022) of Hörmann and Leydold (2014). Otherwise skip.
Sample (where required) the prior variances ξ 1 2 , … ξ d 2 and τ 1 2 , … τ d 2 and the component specific hyper-parameters. Sample the pole, tail, and global shrinkage parameters. In the NGG case, this is done by employing steps (b)–(f) from Algorithm 1 in Cadonna, Frühwirth-Schnatter, and Knaus (2020). In the NG case, use steps (d) and (e) from Algorithm 1 in Bitto and Frühwirth-Schnatter (2019).
Sample the error variance σ ² from an inverse gamma distribution in the homoscedastic case or, in the SV case, sample the volatility of the volatility σ η 2 and the log-volatilities h = (h ₀, …, h _T).

Step 4 presents a fork in the algorithm, as different parameterizations are used in the NGG and NG case, to improve mixing. For details on the exact parameterization used in the NGG case, see Cadonna, Frühwirth-Schnatter, and Knaus (2020). One key feature of the algorithm is the joint sampling of the time-varying parameters β ̃ t , for t = 0, …, T in step 1 of Algorithm 1. I employ the procedure described in Chan and Jeliazkov (2009) and McCausland, Miller, and Pelletier (2011) from Rue and Held (2005), which exploits the sparse, block tri-diagonal structure of the precision matrix of the full conditional distribution of β ̃ = ( β 0 ̃ , … , β T ̃ ) , to speed up computations.

Step 3, as described in Bitto and Frühwirth-Schnatter (2019), makes use of the ancillarity-sufficiency interweaving strategy (ASIS) introduced by Yu and Meng (2011). ASIS is well known to improve mixing by sampling certain parameters both in the centered and non-centered parameterization.

A.3 Factor Model Combinations

The second synthesis function considered in this paper is a Bayesian Factor Model similar to that of Lopes and West (2004), and Lopes (2014) provides an overview of Modern Bayesian Factor Analysis. Please refer to those references for detailed discussion on the methods. Here I provide a brief overview of the model and estimation technique.^[11]

(21) y t = F t ′ γ t + ϵ t γ t = γ t − 1 + u t x t = Λ f t + ν t

(22) ϵ t ∼ N 0 , σ t 2 u t ∼ N ( 0 , θ ) ν t ∼ N ( 0 , R )

where f _t is a k × 1 vector of factors, F _t = (1, f _t), γ _t is k + 1 vector of coefficients, Λ is a J × k vector of loadings, and R is a diagonal covariance matrix with elements σ ν J 2 . In order to derive combination weights, I need to identify the factors. This is done by the following restriction f t ′ f t = I J and by restricting the first k elements of the loadings matrix to be positive block lower diagonal. This is a common identification scheme used to fix indeterminacy in the estimation of the factors.

To complete model specifications, I need priors for Λ, R, σ t 2 , and θ. The factor loadings have independent priors Λ i j ∼ N ( 0 , C 0 ) when i ≠ j and Λ i j ∼ N ( 0 , C 0 ) 1 ( Λ i i > 0 ) for the upper-diagonal elements of positive loadings i = 1, …, k. Each of the prior variances are independent and modeled as σ ν J 2 ∼ I G ( ν / 2 , ν s 2 / 2 ) , similarly θ ∼ I G ν θ / 2 , ν θ s θ 2 / 2 . Initial conditions for the γ _t are γ 0 ∼ N ( 0 , P 0 ) , where P 0 ∼ I G ( ν P , ( ν P − 1 ) × c P ) .

With the model specified, the next section provides a sketch of the MCMC routine. Interested readers can refer to Lopes and West (2004).

Algorithm 2

Gibbs Sampling Algorithm

Sample f _t from independent normal distributions for every t, namely, f t ∼ N ( I k + Λ ′ R − 1 Λ − 1 Λ ′ R − 1 x t , I k + Λ ′ R − 1 Λ − 1 ) .
Sample Λ for i = 1 , … , k Λ i ∼ N ( m i , C i ) 1 ( Λ i i > 0 ) where m i = C i C 0 − 1 μ 0 1 i + σ ν i 2 F i x i and C i − 1 = C 0 − 1 I i + σ ν i 2 F i ′ F i .
Sample Λ for i = k + 1 , … , J Λ i ∼ N ( m i , C i ) 1 ( Λ i i > 0 ) where m i = C i C 0 − 1 μ 0 1 k + σ ν i 2 F ′ x i and C i − 1 = C 0 − 1 I k + σ ν i 2 F ′ F .
Sample σ ν i 2 ∼ I G ( ( ν + T ) / 2 , ( ν s 2 + d i ) / 2 ) where d i = x i − F Λ ′ ′ x i − F Λ ′ .
If in TVP specification, sample the latent states γ ₁, …, γ _d, jointly from a multivariate normal distribution using the precision sampler of Chan and Jeliazkov (2009). Otherwise, sample γ = (γ ₀, …, γ _T) from a multivariate normal distribution.
Sample the error variance σ ² from an inverse gamma distribution in the homoscedastic case or, in the SV case, sample the volatility of the volatility σ ² and the log-volatilities h = (h ₀, …, h _T).

A.4 Sampling the Agent States

After estimating the synthesis function parameters, the next step in BPS is to draw x _1:t from p ( x 1 : t | Φ 1 : t , y 1 : t , H 1 : t ) where Φ is the model parameters, y _t is the target variable, and H 1 : t is the set of agent densities. As shown in McAlinn and West (2019), the x _t, draws from agent densities, are conditionally independent over t with time t conditionals:

(23) p ( x t | Φ t , y t , H t ) ∝ N y t | X t ′ β t , ϵ t ∏ j = 1 : J h t j ( x t j ) with X t = ( 1 , x t 1 , … , x t J ) ′

If the agents provide normal forecast densities, then 23 yields a multivariate normal distribution for x _t. The posterior distribution for each x _t is:

(24) p ( x t | Φ t , y t , H t ) = N h t + b t c t , H t − b t b t ′ g t

where c t = y t − β t 0 − h t ′ β t , 1 : J , g t = σ t 2 + β t , 1 : J ′ H t β t , 1 : J , and b _t = H _t β _t,1:J/g _t. Unfortunately, the applications in this paper do not have analytical forms; instead, histograms represent the agent densities. With no analytical form, I use a Block Metropolis–Hastings step with 24 as a proposal distribution. Since the number of agent densities can be large, I break the MH step into blocks of five experts that are sampled at a time.

There are a few details for Bayesian Factor Model combinations that warrant explanation. First, the model has to be re-parameterized in terms of the x _t so that I can use the proposal distribution from 24 in the MH step. The model is straightforward to re-parameterize with the following steps:

(25) y t = x t ′ γ t + ϵ t x t = Λ f t + ν t

(26) f t = ( Λ ′ Λ ) − 1 Λ ′ x t − ( Λ ′ Λ ) − 1 Λ ′ ν t where, Ω = ( Λ ′ Λ ) − 1 Λ ′

(27) y t = x t ′ Ω ′ γ t − ν t ′ Ω ′ γ t + ϵ t → y t = x t ′ γ t * + ϵ t *

(28) where, ϵ t * = − ν t ′ Ω ′ γ t + ϵ t and γ t * = Ω ′ γ t

Now that the model has been re-parameterized, I can use the equation (24) in the MH step by substituting in β t = γ t * , and error variance ϵ t * ∼ N 0 , γ t ′ Ω R Ω ′ γ t + σ t 2 .

The second issue that the data (x _t) used to estimate Bayesian Factor Models is standardized to be mean 0 and variance 1. Since the agents provide forecast distributions, I calculate mean and variance used to standardized draws from the agent densities using the marginal density of each expert over all T (h(x)_1:T). Each x _t draw is standardized during each MCMC iteration.

Appendix B: Calibration Appendix

This section assesses the calibration of the BPS predictions. Calibration (also referred to as absolute accuracy) is achieved when a predictive density properly characterizes the probability of the events that it is predicting. For example, events predicted to occur with a 20 percent probability should be observed in the data roughly 20 percent of the time. More formally, calibration refers to the statistical consistency between the predictive distributions and the observations of the data they are predicting (Gneiting and Raftery 2007). I assess calibration with a test based off of the probability integral transforms (PITs) (Diebold, Gunther, and Tay 1998) as proposed in Knüppel (2015). In general, I find little evidence to suggest that the predictions from any of the synthesis functions are not calibrated.

Figures 13 and 14 show results from the nowcasting application and the SPF forecasting application. For the most part in the nowcasting applications, the factor model combinations show little evidence of being uncalibrated. However, the shrinkage approaches have slightly different results. The LASSO synthesis function does not appear to produce calibrated predictions, and the test rejects calibration for the time-varying double gamma specification at most horizons. In contrast, the constant parameter specifications produce calibrated predictions at most horizons, the exception being the shortest horizons where calibration is rejected at the 10 percent level. The SPF application has more straightforward results – there is little evidence to suggest the BPS predictions are uncalibrated from any synthesis function. In only two cases is the null hypothesis rejected at the 10 percent level.

Figure 13:

Knüpple test for probabilistic calibration: nowcasting application. Results from the Knüpple test for probabilistic calibration. Null hypothesis is for calibration and values in the table correspond to p-values. Red shading corresponds to rejection of calibration at 5 percent level and yellow at 10 percent level.

Figure 14:

Knüpple test for probabilistic calibration: survey forecast application. Results from the Knüpple test for probabilistic calibration. Null hypothesis is for calibration and values in the table correspond to p-values. Red shading corresponds to rejection of calibration at 5 percent level and yellow at 10 percent level.

Appendix C: MCMC Convergence Appendix

In this section, I assess the convergence of the MCMC algorithms. This is done using the Gelman-Rubin diagnostic (Rubin et al. 2015) and implemented through the MATLAB MCMC Diagnostics Toolbox (Vehtari and Särkkä 2014). The Gelman-Rubin diagnostic compares within-chain variance to across-chain variance to estimate a potential scale reduction factor (R), which can be used to assess convergence of the MCMC chain. As a rule of thumb, values below 1.1 suggest convergence. The diagnostic is performed with five chains and on four specifications of BPS using the SPF “tall” dataset. I focus on the constant and time-varying versions of the triple gamma and one factor synthesis functions. This is because the other shrinkage priors are special cases of the triple gamma prior and reduce to simpler versions of the sampler. Results for other shrinkage are available upon request. Since the number of parameters, state variables, and hyper-parameters sampled can number in the thousands, I report box plots of the potential scale reduction factor in Figure 15. These results show reasonable convergence of the MCMC algorithms.

Figure 15:

Box plots of potential scale reduction factors. The above shows the potential scale reduction factors (R) from the Gelman-Rubin diagnostic with 5 chains on the SPF ‘Tall’ dataset. Values less than 1.1 provide evidence that the MCMC has converged.

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/snde-2022-0108).

Received: 2022-11-28

Accepted: 2023-11-01

Published Online: 2023-12-18

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Supplementary Material Details

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https://doi.org/10.1515/snde-2022-0108

Keywords for this article

density prediction; forecast combination; econometric and statistical methods