A Flexible Mixed-Frequency Vector Autoregression with a Steady-State Prior

2.1 State-Space Representation of the Mixed-Frequency Model

To cope with mixed observed frequencies of the data, we assume the system to be evolving at the highest available frequency. This assumption frames the problem of frequency mismatch as a missing data problem. By doing so, the approach naturally lends itself to a state-space representation of the system in which the underlying monthly series of the quarterly variables become the latent states of the system. Because we have a mix of monthly and quarterly frequencies in our empirical application, we will in the following proceed with the presentation of the model from this perspective. It should, however, be noted that other compositions of frequencies are viable within the same framework.

The VAR model at the core of the analysis is specified for the high-frequency and partially missing variables. More specifcally, a VAR(p) for the n × 1 vector z_t is employed such that

where Π(L) = (I_n−Π₁L−Π₂L²−⋯−Π_pL^p) is a p-th order invertible lag polynomial, d_t is an m × 1 vector of deterministic components and Φ is an n × m matrix of parameters. The time index t is here monthly. We let the error term u_t be heteroskedastic and return to the specifics thereof in Section 2.2.

The model in (1) is a conventional VAR specification, but, in the spirit of Villani (2009), we instead employ the mean-adjusted form as

where Ψ=[Π(L)]−1Φ it can be readily confirmed that E(zt|Π,Ψ,Σ)=Ψdt:=μt, and thus μt is the unconditional mean—steady state—of the process. The steady-state representation (2) requires an explicit prior on the steady state parameters.

However, common practice is to use (1) with a loose prior on Φ, which implicitly defines an intricate (but loose) prior on Ψ and, subsequently, μt. We argue that in many applications, the parametrization in (2) is more convenient as it allows for a more natural elicitation of prior beliefs. In what follows, we will extend the work of Villani (2009) such that (2) may still constitute a viable option in the presence of mixed frequencies.

Next, we partition the high-frequency underlying process z_t as, zt=(z′m,t,z′q,t)′ where z_m,t represents the n_m monthly and z_q,t the n_q quarterly variables. Recall that the time t here takes the highest frequency, i. e. monthly. A ubiquitous problem with macroeconomic data is that what is observed varies between months such that z_t is not always fully observed.

To distinguish between the underlying process and actual observations, we denote the latter by y_t. A consequence of all variables not being observed at every time point t is that the dimension n_t of y_t is not always equal to n = n_m + n_q. The observed data in y_t are generally supposed to be some linear aggregate of Zt=(z′t,⋯,z′t−p+1)′ such that

where M_q,t and Λ_q are deterministic selection and aggregation matrices, respectively.

We let M_q,t be the n_q identity matrix Inq if all quarterly variables are observed at time t so that yq,t=(0Λq)Zt. In the remaining periods, M_q,t is an empty matrix such that y_t = y_m,t. More complicated observational structures can easily be accommodated in the same manner; instead of being empty or a full I_n matrix, M_t can have rows that correspond to unobserved variables omitted. This idea allows for seamlessly handling missing data for a subset of the monthly variables at the end of the sample.

The aggregation matrix Λ_q represents the assumed aggregation scheme of unobserved high-frequency latent observations z_q,t into occasionally observed low-frequency observations y_q,t. To make the presentation simpler, we can write the bottom block of ΛZ_t as

where Λ_qq collects the columns of Λ_q that correspond to quarterly variables in Z_t.

Schorfheide and Song (2015), working with log-levels of the data, used the intra-quarterly average yq,t*=13(zq,t*+zq,t−1*+zq,t−2*), where yq,t* denotes the observed quarterly log-levels and the latent monthly log-levels. Because we use log-differenced data, we instead follow Mariano and Murasawa (2003, 2010. By taking the quarterly difference of yq,t* to construct our observed growth rates, we obtain

Finally, the expression can be written as

Because the set of weights in (4)(6) sum to three, we define our latent variable of interest to be zq,t=3Δzq,t*, i. e. the latent month-on-month growth rate scaled to be commensurate in scale with the quarterly level.

Equations (2) and (3) form a state-space model that can be used for estimation of the model. Schorfheide and Song (2015) suggested an efficient compact formulation of the employed state-space model that is statistically equivalent but computationally more convenient. The compact treatment is based on the observation that the set of monthly variables included in the model are observed for all time points except for a handful at the end of the sample, known as a ragged edge (Bańbura, Giannone, and Reichlin 2011). The treatment proposed by Schorfheide and Song (2015) is to let the monthly variables enter the model as exogenous for t = 1, …, T_b, where T_b denotes the final time period where the monthly variables are all observed. By this approach, the monthly variables are excluded from the state equation. The state dimension is thereby reduced from np to n_q(p + 1), which substantially improves the computational efficiency.

Let

denote the mean-adjusted data. The state-space model is thereafter formulated in terms of y˜t and z˜t=zt−Ψdt, leading to the model

where Π_i,j, i, j ∈ {m, q} refer to the submatrices of regression parameters relating the j frequency variables to the conditional mean of the i frequency variables. The errors are the corresponding partitions of ut=(u′m,t,u′q,t)′ and are consequently correlated. Finally, Y˜m,t−1 stacks the mean-adjusted monthly variables as Y˜m,t−1=(y˜′m,t−1,…,y˜′m,t−p)′ and Z˜q,t=(z˜′q,t,…,z˜′q,t−p)′.

The above state-space model remains valid as long as t ≤ T_b, implying that all of the monthly series are observed. To deal with ragged edges and unbalanced monthly data for t = T_b + 1, we follow Ankargren and Jonéus (2020) and adaptively add the monthly series with missing data as appropriate. Contrary to Schorfheide and Song (2015), we thereby avoid use of the full companion form altogether.

2.2 Extending the Basic Steady-State Model

The standard Bayesian VAR (BVAR) with the steady-state prior typically produces good forecasts and is for this reason used by e.g. Sveriges Riksbank as one of its main forecasting models (see Iversen et al. 2016). However, recent work in the VAR literature demonstrates that allowing for more flexibility may be beneficial. Particularly, letting the error covariance matrix in the model vary over time by incorporating stochastic volatility often improves the predictive ability as demonstrated by e.g. Clark (2011); Clark and Ravazzolo (2015); Carriero et al. (2016). Moreover, studies such as Bańbura, Giannone, and Reichlin (2010); Giannone, Lenza, and Primiceri (2015); Koop (2013) have shown that medium-sized models including 10–20 variables often outperform smaller models when forecasting. The caveat, however, when extending the size of the model under the use of the steady-state prior is that the researcher must set a prior mean and variance for the unconditional mean for each variable in the model. For key variables such as inflation, gross domestic product (GDP) growth and unemployment this task is relatively effortless, but it can be more challenging when the previous literature does not offer any guidance on reasonable prior specifications. To simplify the process of specifying the steady-state prior, Louzis (2019) developed a hierarchical prior for the steady-state prior that effectively relieves the researcher from eliciting the prior variances of the steady-state parameters. Instead, only prior means are required. Providing a sensible prior for the unconditional mean is generally much simpler than quantifying the uncertainty of one’s specification. We next briefly describe the stochastic volatility and hierarchical steady-state prior specifications that we extend our basic model with.

2.3 Prior Distributions

We use a standard normal inverse Wishart prior for the VAR coefficients and error covariance (Π, Σ). Thus, we have a priori

where Π = (Π₁, …, Π_p). The main diagonal of the prior covariance matrix for the regression parameters, Ω̲Π, is set in the Minnesota-style fashion

where λ1 is the overall tightness and λ2 determines the lag decay rate; the inclusion of s_r adjusts for differences in measurement scale of the variables. For a more thorough exposition of the normal inverse Wishart prior, the reader is referred to Karlsson (2013).

While Σ describes the fixed covariance structure, the time-varying volatility in the model is governed by the latent volatility f_t. For the two parameters associated with its evolution, (ϕ,σ2), we use a normal distribution truncated to the stationary region for ϕ, and an inverse gamma prior for σ2:

As discussed in Section 2.2, the priors for the steady-state parameters are normal conditional on the local shrinkage parameters. Instead of fixing the top-level hyperparameters ϕψ and λψ, Huber and Feldkircher (2019) proceeded with an additional hierarchy by specifying priors for ϕψ and λψ. We follow their suggestion and obtain the following hierarchical prior specification for the steady-state parameters:

2.4 Posterior Sampling

To estimate the model and produce forecasts, we employ Markov Chain Monte Carlo (MCMC). The MCMC algorithm consists of multiple Gibbs sampling steps, which we describe next. We relegate some of the details to Appendix A.

3.1 Data

Our dataset consists of 13 key macroeconomic variables for the United States. The dataset we use largely parallels that of Carriero et al. (2016); Louzis (2019) with the exception that we use consumer price index (CPI) inflation as the sole measure of inflation. The data consist of 10 monthly and three quarterly variables and ranges over the period 1980M01–2018M12. Most of the included variables are available with real-time vintages in the ALFRED database. For variables not available in ALFRED, we turn to FRED and FRED-MD (McCracken and Ng 2016). A summary of the data is provided in Table 1.

We follow Louzis (2019) and transform the raw series to growth levels. For our monthly variables, we use month-on-month growth rates, whereas the three quarterly variables are computed as quarter-on-quarter rates. All growth rates are annualized. The final two columns of Table 1, μ¯ψ,j and ωψ,j, display the prior means and prior standard deviations of the unconditional means of the variables. The values are drawn from Louzis (2019), but are also in line with e.g. Clark (2011); Österholm (2012).

We use real-time data where available throughout the forecasting exercise. To obtain a realistic pattern of available observations, we first consider the information set available on the 10th day of every month. Figure 1 displays the publication pattern during 2005–2018 and shows the number of months that has passed since the last available publication.

Figure 1:

Publication Delays. The color of each box represents the number of months since the last available observation. The delay is computed for the 10th day of the corresponding month; a zero-month delay implies that the observation for the preceding month is available.

Figure 1 shows a characteristic pattern for real-time forecasting of macroeconomic data. Data for financial and select real and nominal variables are already available for the previous month, whereas the previous month’s outcomes for some of the monthly variables are unknown. The pattern of availability displayed shows that consumption and inflation are available with a one-month delay at every month except for a handful of occasions. Similarly, non-farm employment, hours, unemployment and the federal funds rate are typically available with a zero- month delay with the exception of a few months. In the final dataset that we use in our forecasting exercise, we make adjustments to the publication delays in order to obtain a more uniform dataset. The adjustments change the publication structure in the vintages so that the aforementioned variables have the same delay in all vintages, i. e. consumption and inflation are always observed wtih a delay of one month, whereas non-farm employment, hours, unemployment and the federal funds rate are always observed without any delay. Consequently, at every month that we make our forecasts observations are available for the preceding month for six of the monthly variables, whereas four still lack data.

3.2 Implementation Details

The mixed-frequency models that we estimate use p = 12 lags following e.g. Bańbura et al. (2010). The overall tightness in the prior distribution for the regression parameters is set to λ1=0.2 and the lag decay used is λ2=1. We use 15,000 draws in the MCMC procedure and discard the first 5000.

For the hierarchical steady-state prior, we let c₀ = c₁ = 0.01 in line with Huber and Feldkircher (2019); Louzis (2019). To set the scale of the proposal distribution for ϕψ, we employ the adaptive scaling procedure discussed by Roberts and Rosenthal (2009). We use a batch size of 100 and check every 100 iterations if the fraction of acceptances within the most recent batch exceeds 0.44. If it does, we increase s by δ(k)=min(0.01,k−1/2), where k denotes the batch number. If the fraction of acceptances was less than 0.44, s is instead decreased by δ(k).

For the parameters of the log-volatility process, we let the prior mean and standard deviation for ϕ be μ¯ϕ=0.9 and Ω¯ϕ=0.1, respectively. The prior mean and degrees of freedom of σ2 are σ¯2=0.01 and d¯=4.

4.1 Forecasting Setup

The quarterly steady-state Bayesian VAR model has been used in several previous studies, see for example Adolfson et al. (2007); Österholm (2008); Villani (2009); Clark (2011); Ankargren, Bjellerup, and Shahnazarian (2017). The model is employed both for policy purposes and for forecasting and is implemented in the Matlab toolbox BEAR developed at the European Central Bank (Dieppe, Legrand, and van Roye 2016). Our empirical application targets this audience, and our main interest lies in seeing whether the components we add to the model—mixed frequencies, stochastic volatility and hierarchical steady states—improve upon the benchmark model of Villani (2009) estimated on single-frequency data. The forecasting results are also compared to models using Minnesota-style normal inverse Wishart priors, i. e. without use of the steady-state component. A summary of the models that we include in the forecast evaluation is presented in Table 2.

The benchmark model is the steady-state model estimated on single-frequency data. Depending on whether it serves as benchmark for quarterly or monthly variables, we include either the full set of variables (aggregated to the quarterly frequency) or the 10 monthly variables. The quarterly VAR uses p = 4, whereas for the monthly VAR p = 12.

We use a recursive forecasting scheme to evaluate the forecasting performance of the considered models. Beginning in January 2005, we estimate the models and make forecasts and then recursively add months to the set of data used for estimation. The benchmark models use the balanced data, whereas the mixed-frequency models automatically handle the ragged edges.

The forecasting ability of the models is evaluated with respect to both point and density forecasts. For point forecasts, we consider the root mean squared errors (RMSE). For density forecasts, we compute univariate and multivariate log predictive density scores (LPDS). We do so by fitting a normal density to the draws from the predictive distribution following e.g. Adolfson et al. (2007); Carriero et al. (2015). That is, we compute

where m denotes the model, s denotes the set of variables the LPDS is computed for, n_s is the dimension of s, h is the forecast horizon, and y¯ and V are the mean and covariance of the draws from the relevant predictive distribution. For fixed (m, s, h), we compute the summary LPDS by averaging over the evaluation period. We calculate the LPDS jointly but separately for the monthly and quarterly variables, and univariately for all variables.

What vintage the forecasts should be evaluated with respect to is an important question when using real-time data. Two alternatives are commonly employed in the literature. The first, as used by e.g. Romer and Romer (2000) and Clark (2011), is to use the second available vintage. This choice can be justified by acknowledging that revisions that occur after longer periods of time may be unforeseeable and more structural in nature by relating to e.g. definitions, methods of measurement, etc. The second available estimate therefore provides a less noisy estimate than the initial available value, yet is produced in the same environment as the forecaster is active. The second common approach for evaluation, as followed by e.g. Schorfheide and Song (2015), is to use the most recent vintage. For whatever reason revisions may have taken place, the currently available data provide the best estimates of e.g. inflation and output in previous years. We follow the latter approach and use the most recent vintage for evaluating the forecasts, but for transparency provide the main results of the evaluation using the second available vintage in Appendix D.

4.2 In-Sample Estimation

As a preliminary analysis, we begin by estimating the mixed-frequency VAR model using the steady-state (SS) and steady-state normal-gamma (SSNG) priors to see whether the obtained steady-state posteriors differ. Because the long-term forecasts are largely determined by the steady-state posterior, seeing whether differences are present is of direct importance for forecasts beyond the immediate short term. Figure 2 displays kernel density estimates of the posterior distributions from the mixed-frequency model with CSV. As a point of reference, the figure includes the prior distribution detailed in Table 1.

Figure 2:

Steady States. Kernel density estimates of the posterior distribution of the steady states (unconditional means).

As expected, the posteriors in Figure 2 are for the most part similar. The modes of the posteriors are close to perfectly aligned for variables such as bond spread, inflation, residential investment and GDP. For others—e.g., hours, the federal funds rate and industrial production—the SSNG posteriors deviate more from both the priors and the SS posteriors.

While the steady states are of central importance for the levels of the forecasts, the precision thereof is highly influenced by the CSV factor. Figure 3 displays the mean of ft together with 90% bands for the SS-CSV and SSNG-CSV models.

Figure 3:

Volatility. THe lines display the posterior means of ft and the bands how the 90 % posterior intervals.

Figure 3 shows that there is little difference between the estimated volatility factors in the two steady-state models. Peaks of volatility are aligned and reach the same levels, while the level of the factor in the SSNG model is slightly higher in normal times. Both display the entrance into the Great Moderation in the beginning of the 1980s with heightened volatility again around the recent financial crisis. The interpretation of the level of the factor is that the time-invariant elements in the error covariance matrix Σ have been scaled by f_t, which roughly amounts to an amplification by a factor of 4–6 during the recent financial crisis and a compression of around 0.5–0.75 in recent years. This feature has a direct effect on the width of the predictive distribution.

4.3 Forecast Evaluation

In this section, we present the main results of the forecast evaluation. For space considerations, the presentation includes the results from the joint evaluations as well as the univariate results for the three quarterly variables and the three monthly variables that are typically of primary interest: the inflation, federal funds and unemployment rates.