2.1 An adaptive hierarchical steady-state prior

Prior elicitation with respect to Equation 3 means that the researcher has to choose each of the elements of θ_μ, θ_μ,j, and then decide with regard to the prior informativeness by setting the diagonal elements of Ω_μ, ω_μ,j,j  =  1,…,n. According to the literature, the steady-state prior distribution should be reasonably informative, translating to reasonably small values for , to reap the benefits of the mean-adjusted VAR model in terms of forecasting performance and avoid convergence problems of the MCMC algorithm (Villani, 2005, 2009).

In general, prior tightness for each of the n variables in the VAR model depends on the uncertainty of the researcher regarding the selected level of the steady-state prior mean. For variables with much a priori information available—for example, long-term survey forecasts—a researcher may feel much more comfortable with a very tight prior—that is, very small ω_μ,j. Otherwise, in cases where it is difficult to formulate a prior opinion on the unconditional mean and as a result the steady-state prior mean differs largely from its true value, then a very informative prior may lead to inaccurate steady-state inference. Therefore, we seek for a prior distribution adaptive enough to place a lot of mass on θ_μ,js but at the same time have heavy tails to let the data speak.

An eligible candidate is the NG prior distribution of Griffin and Brown (2010, 2017), defined in hierarchical form as

(4)

where is the gamma distribution and ϕ_μ and λ_μ are hyperparameters, with their own hyperprior distributions playing an important role in adaptive shrinkage.

More specifically, as Griffin and Brown (2010) show, the hyperparameter λ_μ controls for the overall (or global) prior informativeness, pushing μ_j toward the prespecified values θ_μ,j ∀j. Larger values of λ_μ imply a smaller variance for the marginal prior on μ_j, obtained by integrating out ω_μ,j. On the contrary, the excess kurtosis of the marginal prior depends on the hyperparameter ϕ_μ; as ϕ_μ decreases, the prior places more mass on θ_μ,j but at the same time becomes more fat tailed. This means that, even when we use a tight prior with large λ_μ and small ϕ_μ, the prior is flexible enough to use the information content of the data if they strongly suggest different steady-state prior mean locations. Another advantageous property of the NG prior is that it allows for an idiosyncratic level of tightness for each μ_j irrespective of the size of the steady-state coefficient. Following Huber and Feldkircher (2017), we define an exponential hyperprior distribution centered on unity for ϕ_μ:

(5)

For the hyperparameter λ_μ we assume a gamma hyperprior distribution:

(6)

with hyperparameters being set to a low value, indicating a very tight prior on μ; that is, c₀  =  c₁  =  0.01. 12

2.2 Time-varying steady states

In this section, we introduce more flexibility into the standard SS-VAR model by allowing the unconditional mean of the model to change over time, capturing possible structural breaks in the steady state. In particular, we define a VAR model with time-varying steady-states (TVSS-VAR) as

(7)

(8)

where the dynamics of the steady-state parameters, μ_t, are modeled as a driftless random walk, a standard approach in the time-varying parameter VAR literature (see, e.g., Primiceri, 2005), and the error terms {u_t,η_t} are assumed to be independent with each other. Here, the prior information regarding the steady-state parameters is incorporated into the model via the prior on the initial conditions; that is:

(9)

We can also extend the TVSS-VAR to adopt the NG prior on the initial conditions and thus combining the hierarchical shrinkage towards economic information with the time variation of the steady states. The full details of such a model are provided in Appendix B.

The model is completed by assuming that Q is a priori distributed as

(10)

where is the inverse Wishart distribution with scale matrix, S_Q and degrees of freedom d_Q. The prior on Q plays an important role in empirical applications since it governs the degree of time variation of μ_t. Usually the literature prefers a tight prior to avoid implausible behavior in terms of the time evolution of the time-varying parameter and optimize the forecasting performance of the model (see, e.g., D'Agostino et al., 2013). 13

3.1 Asymmetric priors and stochastic volatility

The natural conjugate N-IW prior facilitates the estimation of large-scale VARs with fat tails and heteroskedastic errors but imposes, possibly undesirable, restrictions on the structure of the volatility and the prior of the VAR coefficients. 14 Carriero et al. (2016b) proposes a simple triangularization which enables an equation-by-equation estimation of the model even in cases of asymmetry in either the likelihood or the priors. It is worth noting that the introduction of asymmetry necessitates the vectorization of the model and increases exponentially the computational complexity. We define a steady-state VAR with a flexible prior on VAR coefficients and flexible stochastic volatility structure as

(19)

(20)

Here, we assume that H_t is a diagonal matrix, with h_j,t, j  =  1,…,n, being the jth diagonal element. The dynamics of h_j,t are specified as a geometric random walk:

(21)

where ε_t is the vector that collects the innovations ε_j,t. The model is completed by specifying the prior distributions on the rest of the coefficients. Specifically:

(22)

(23)

(24)

Obviously, the prior on VAR coefficients in Equation 22 is not conditional on the variance of the VAR coefficients, meaning that we can introduce asymmetry in the prior across equations as in the traditional Minnesota prior of Litterman (1986).

The decomposition in Equation 20 implies that we can write the jth generic Equation 19 as

(25)

where , . Given that we have estimated all previous j − 1 equations, the terms on the left-hand side of Equation 25 can be replaced with their estimations and the model can be estimated equation by equation, alleviating substantially the computational burden (for more details see Carriero et al., 2016b).

4.1 Posterior distributions related to the NG steady-state prior

The hierarchical formation of the NG prior requires to sample ω_μ,j , λ_μ and ϕ_μ from their own posterior distributions (see the Technical Appendix for the related derivations). More specifically, ω_μ,j, ∀j has a generalized inverse Gaussian (GIG) conditional posterior:

(26)

Next, we draw λ_μ from a gamma conditional posterior of the form

(27)

with shape and rate parameters nϕ_μ + c₀ and , respectively. The conditional posterior distribution of ϕ_μ has no closed form and, thus, we rely on a random-walk Metropolis–Hastings (RWMS) step. Particularly, the proposed values for ϕ_μ are given by , where is the last accepted draw, s_ϕ is a scaling factor, and z is a standard normal random variable. The proposed value, , is accepted with probability

(28)

The scaling factor s_ϕ is calibrated to achieve an acceptance rate of approximately 30%. As soon as we have the posterior draws for ω_μ,js we form the diagonal matrix Ω_μ and we draw μ as in Villani (2009).

4.2 Estimation of the time-varying steady-state parameters

Regarding the TVSS-VAR model and the estimation of μ_t, we rewrite the model in Equations 7 and 8 in a state-space form. In particular, we show that the TVSS-VAR can be written as

(29)

where μ_t is specified as in Equation 8, y_B,t≡y_t − B(L)y_t, Z_t≡I_n⊗D_tU, , , and 1_p is a 1 × p vector of ones (a proof of this result is provided in Louzis, 2016a). Equations 29 and 8 form the observation and transition equations respectively, and conditional on all other parameters we can use standard Bayesian techniques for TVP-VARs to estimate the series of μ_t (see, e.g., (Primiceri, 2005). In particular, assuming a prior on initial conditions as in Equation 9, we draw μ_t ∀t from its conditional posterior distribution using the Carter and Kohn (1994) (CK) algorithm. Next, conditional on the dth draw, , we calculate the demeaned variables , and then we can apply standard Bayesian estimation for the rest of the parameters.

6.1 Competing models

The main scope of the paper is to investigate whether the proposed steady-state VAR extensions improve the forecasting ability of the VAR models relative to the standard benchmarks. Therefore, except for the stationary steady-state VAR specifications using variables in growth rates, we also consider VAR models with variables in log levels, a typical approach in macroeconomic forecasting literature (see, e.g., Carriero, Clark, & Marcellino, 2015; Giannone et al., 2015). A list of competing models, along with a short description of the alternative specifications and their priors, is presented in Table 2.

The benchmark model is a standard BVAR model in growth rates with a natural conjugate N-IW prior (see Equations 15 and 16). The VAR models using variables in levels (L) are also enriched with the routinely used sum-of-coefficients and dummy-initial-observation priors (Doan, Litterman, & Sims, 1984; Sims, 1993). The former, also known as no-cointegration prior, is imposed on the sum of coefficients of the model lags and assumes that each variable has a separate stochastic trend, as opposed to common stochastic trends implied by cointegration. The use of the sum-of-coefficients is standard in macroeconomic forecasting since it improves the predicting ability of the models by correcting for the deterministic overfitting in VARs. 14On the other hand, the sum-of-coefficients prior cancels out cointegration in the limit (when the prior is too tight), which is an undesirable property for models using variables in levels. The dummy initial observation, also known as single-unit-root prior, alleviates this shortcoming and assumes that no-change forecast provides a good description of the dynamics of the model. Thus, depending on the tightness, the prior either forces the variables towards the unconditional mean of the model, suggesting stationarity, or assumes that there is an unspecified number of unit roots without drift; both extremes are consistent with cointegration. For consistency and comparability reasons we also augment the VARs in levels with fat tails (suffix “-t”) and common stochastic volatility combined with fat tails (suffix “-CSV-t”), as described in Section 3.

All three categories of steady-state models—that is, the standard steady-state model (SS), the hierarchical steady-state model with the NG prior (HSS), and the time-varying steady-state model (TVSS)—are combined with fat tails and common stochastic volatility for the error terms. For the steady-state models we also consider the original asymmetric Minnesota prior for the dynamic VAR coefficients combined with stochastic volatility (suffix “-SV”) and estimated as described in Section 3.1.

Generally, the aforementioned choice of competing models allows us (i) to investigate the forecasting performance of the HSS and TVSS models relative to standard benchmarks in the literature—that is, the SS and L models—revealing the value added of the proposed steady-state specifications in terms of forecasting, and (ii) to examine whether fat tails and stochastic volatility improve the forecasting ability across alternative specifications with informative steady states—that is, L, SS, HSS and TVSS—thus providing additional support to the use of large-scale VAR models with nonconstant volatility and non-Gaussian innovations.

6.2 Data

We examine the estimation and forecasting properties of the proposed specifications using a large dataset of 14 macroeconomic time series from the US economy. Table 3 presents the details on the data set along with the transformations of the variables for the stationary and nonstationary models and the corresponding sources.

Table 3. The dataset

	Transformation
Variable	Steady-state VARs	VARs in log levels	Source
Real GDP			RTDSM
Consumption			RTDSM
Business fixed inv. (BFI)			RTDSM
Resind. inv.			RTDSM
Ind. prod.			RTDSM
Cap. util.	None	None	RTDSM
Employment			RTDSM
Hours			RTDSM
Unempl. rate	None	None	RTDSM
GDP deflator			RTDSM
PCE deflator			RTDSM
Fed funds rate	None	None	FRED
Term spread	None	None	FRED
Real stock prices			FRED

• Notes.
• 1. “Term spread” is the difference between the 10-year treasury bond yield and the fed funds rate.
• 2. “Real stock prices” is the S&P 500 stock index deflated with the GDP deflator.
• 3. “RTDSM” is the Real Time Data Set for Macroeconomists database of the Federal Reserve Bank of Philadelphia; “FRED” is the Federal Reserve Economic Data maintained by the Federal Reserve Bank of St. Louis.

In brief, the dataset includes quarterly variables from the real sector of the economy, such as the real gross domestic product (GDP), consumption, investments, employment, capacity utilization, industrial production, etc.; prices (GDP and consumption deflators); and monetary and financial variables (i.e., the short-term interest rate, the term spread, and a stock index). All variables are expressed in annualized log differences or log levels except for those already expressed in annualized percentages, such as the unemployment rate, capacity utilization, or the interest rate. For variables available only at the monthly frequency we take the average value within the quarter.

More specifically, we rely on the real-time dataset for macroeconomists (RTDSM) obtained form the website of the Federal Reserve Bank of Philadelphia, and we use exactly the same 14 variables as in Carriero et al. (2016a), employing real-time vintages from 1965:Q1 to 2016:Q2. The quarterly values of the variables in each vintage reflect the information available at the middle of each quarter, while the t vintage contains data until t − 1 (Croushore & Stark, 2001). For instance, the available observations in the 1985:Q1 real-time vintage run through 1984:Q4. Following the literature, for the variables with immaterial or no revisions, such as the unemployment rate or the financial variables, we depart from the real-time approach and we use the last available vintage: 2016:Q2 in this paper. 11

6.3 Specification of the priors

In this section we concentrate on the specification of the priors with respect to the steady-state and time-varying steady-state parameters. For the prior elicitation of the rest of the parameters we follow closely the contributions of Carriero et al. (2016a, 2016b) and Chan (2018); full details are provided in Supporting Information Appendix D.

As regards the prior mean on the steady-state coefficients, θ_μ, we follow the recent literature and set the steady-state prior mean across all steady-state specifications—that is, SS, HSS, TVSS—according to the second column of Table 4 (see, e.g., Clark, 2011; Jarocinski & Smets, 2008; Österholm, 2012).

Table 4. Specification of the steady-state prior

	θμ,j (%)	(%)
GDP	3	0.5
Cons.	3	0.7
BFI	3	1.5
Resind. inv.	3	1.5
IP	3	0.7
Cap. util.	80	0.7
Empl.	3	0.5
Hours	3	0.5
UR	6	1
GDP def.	2	0.5
PCE def.	2	0.5
FFR	5	0.7
Spread	1	1
S&P 500	0	2

For the standard SS models we also have to elicit the diagonal elements of the prior covariance matrix, Ω_μ. A typical but somewhat restrictive strategy in larger VAR models is to suppose a common prior variance across all variables (see, e.g., (Wright, 2013). To add more flexibility to the SS models we choose to specify a different prior standard deviation for each variable, presented in the third column of Table 4.

The hyperparameters related to the NG steady-state prior employed by the HSS models are specified in Section 2.1, while for the TVSS we also set the prior mean, and covariance matrix, , of the initial conditions according to Table 4. For the degrees of freedom, d_Q, and the scale matrix, S_Q, in Equation 10, we follow a standard approach in the literature (see, e.g., (Primiceri, 2005) and we use a training sample for prior elicitation. In particular, we set d_Q  =  T^∗ and S_Q=k_Q·T^∗·Γ_Q, where T^∗  =  40 is the number of observations in the training sample and Γ_Q is a diagonal matrix the main diagonal of which contains the sample variance of the variables estimated over the presample. The hyperparameter k_Q controls for the degree of time variation of the steady states and should be sensibly specified to avoid implausible behavior, hence we set k_Q  =  0.005.

6.4 In-sample estimation and MCMC convergence

In general, throughout this paper, we estimate all constant volatility models using 6,000 draws after discarding the first 5,000 draws used for initial convergence (burn-in period). The estimation of all models encompassing stochastic volatility is also based on 1,000 retained draws, but now they are obtained from a total of 10,000 draws with 5,000 draws of burn-in period and a thinning of 5; that is, we keep one every five draws. The lag length for all models using variables in growth rates is set equal to 4 (p  =  4), while for those using variables in log levels we use 5 lags (p  =  5).

First, we evaluate the convergence of the proposed MCMC algorithms by implementing the inefficiency factors (IFs) metric of Primiceri (2005) . We present IF results for the most flexible models (i.e., HSS-CSV-t and TVSS-CSV-t) using the last vintage data of the US economy. More specifically, Figure 1 presents the box-plots of the inefficiency factors corresponding to the posterior draws of the parameters. Box-plots summarize visually the distribution of the inefficiency factors, with the middle line in the box being the median of the distribution, while the upper and lower lines are the 75 and 25 percentiles respectively. The tails in each box-plot are the maximum and minimum. The empirical evidence reveals that all maximum values are well below the threshold value of 20 (see Primiceri, 2005), meaning that the convergence of the Gibbs sampler is more than satisfactory and the proposed sampler can produce posterior draws that are not highly correlated.

Next, we present in Figure 2 the steady-state prior distribution implied by the standard SS and hierarchical HSS models. 12For the SS model we simulate from a normal distribution with first and second moments given in Table 4, while for the HSS model we simulate from the marginal distribution of the steady-state parameters using a normal distribution with first moments given in Table 4 and second moments using the posterior estimates of ω_μ,js. The histograms in Figure 2 reveal a typical shape of the NG prior, placing most of its mass around the predetermined mean values but at the same time allowing for fatter tails compared to an informative normal prior, thus giving the model enough space to draw information from the likelihood.

In Figures 3-6 we also present the posterior median of the steady-state parameter for four selected variables routinely examined in similar forecasting applications—that is, the real GDP growth rate, the unemployment rate, the GDP deflator, and the federal funds rate (FFR)—using the SS-CSV-t, HSS-CSV-t, and TVSS-CSV-t models. 13The steady-state estimates produced by the SS and HSS models (dotted and dashed lines, respectively) are very close to each other for the GDP and the FFR, while the HSS model produces lower estimates for the remaining two variables. More specifically, the SS model estimates the steady-state level for the unemployment rate (GDP deflator) at 5.36% (2.49%) as opposed to 5.12% (2.22%) of the HSS model. The intriguing point here is to examine whether these differences in estimates between the alternative methods have a significant effect on forecasting accuracy.

Estimation results regarding the TVSS model are, overall, in line with the stylized facts for the US economy. Particularly, the steady-state estimates for the GDP deflator and especially for the FFR have both an upward trend during the 1970s and the Great Inflation period and they de-escalate in the 1980s during the Great Moderation period. This empirical evidence aligns also with Chan and Koop (2014), who find a structural break on the steady state of inflation and the interest rates during the 1970s. Another interesting point is that during the Great recession (2007–2009) the steady-state level for all variables fall well below (above for the unemployment rate) the constant steady-state level (see the dashed and dotted straight lines). Finally, excluding GDP, all other variables seem to recover towards the end of the sample, probably pointing at a lower growth rate regime for the US economy.

6.5 Forecasting analysis and evaluation

We evaluate the alternative VAR specifications listed in Table 2 in terms of out-of-sample point and density forecasting. The out-of-sample evaluation period is from 1985:Q1 to 2015:Q4 and requires real-time data vintages from 1985:Q1 to 2016:Q2. Moreover, we follow the standard practice in the literature and we choose the second available estimate of the real-time variables as the observed value in the forecasting evaluation (see, e.g., the discussion in ; Clark, 2011; Carriero et al., 2016a and references therein). Following Chan (2018), we proceed with a recursive estimation of all models generating h-step-ahead iterated forecasts with h  =  1, 2, 5, 9 and 13, which correspond to current quarter nowcasts, 1-quarter-ahead forecasts, 1-, 2-, and 3-year-ahead forecasts, due to reporting lags.

We evaluate the forecasting performance of the various competing models, implementing the average root mean square error (RMSE) for point forecasts and the robust continuous ranked probability score (CRPS) for evaluation of the density forecasts (Gneiting & Raftery, 2007). 14 Compared to other density evaluation measures, such as the log score, the CRPS is less sensitive to outliers and is able to reward more efficiently values from the predictive density that are near to but not identical to the realized outcome. 15 In particular, the CRPS favors density forecasts with both high sharpness (i.e., concentration) and small distance between the mean of the predictive density and the observed value. The CRPS metric for the jth variable at time t is defined as

(33)

where F is the cumulative distribution function associated with the predictive density f, denotes the observed value, while and are independent random draws from the posterior predictive density. We follow Panagiotelis and Smith (2008) and we compute Equation 33 using the posterior draws from the MCMC output, where is obtained by independently resampling from the posterior predictive density without replacement. It is also worth noting that the aforementioned definition of the CRPS metric implies that the lower the values of the CRPS, the more accurate the predictive density is.

The overall forecasting performance of the models across all variables is evaluated using the multivariate counterparts of the evaluation metrics discussed above. In particular, we use the weighted mean squared error (WMSE), which for the tth observation is defined as

(34)

where ε_t + h is an n × 1 vector of h-step-ahead forecast errors and W is an n × n diagonal matrix with the inverse of the variances of the series on the main diagonal. W accounts for the different volatility and predictability of the various endogenous variables of the model (Carriero, Kapetanios, & Marcellino, 2011). The multivariate CRPS (MCRPS) at time t is given by

(35)

where || is the Euclidean norm.

Following the standard practice in the related literature, we present the results concerning both point and density metrics in relation to the benchmark VAR model. Thus we facilitate comparisons between the models and the benchmark, but also among the various competing models, since values of the relative metrics below one indicate that the corresponding model outperforms the benchmark and vice versa.

We also provide a rough gauge of whether the improvement in the forecasting accuracy relative to the benchmark is significant. To that end, we employ the Diebold and Mariano (1995) t-statistic for equal MSE and CRPS, both compared against normal critical values (see also ; Amisano & Giacomini, 2007). Following the literature, we choose the Diebold and Mariano (1995) test because it is considered a conservative test for nesting models in finite samples, in a sense that its size tends to be below the nominal size (Clark & McCracken, 2011, 2015). Therefore, we consider one-sided tests (we reject the null in favor of the benchmark and not the alternative model in favor of the benchmark), because most of the models can be viewed as nesting the benchmark (see, e.g., Carriero et al., 2016a). In addition, t-statistics are robust to serial correlation using a rectangular kernel with h − 1 lags for the variances, which are also adjusted according to Harvey, Leybourne, and Newbold (1997) in order to alleviate size distortions related to small samples.

6.6 Forecasting results

This section discusses the forecasting results generated by the various competing models using the real-time dataset for the US economy. Table 5 presents the overall point and density forecasting results using the WMSE and MCRPS metrics, respectively. Regarding the point forecasts the picture is clear cut. All models outperform the benchmark across all horizons, and the more flexible HSS-CSV-t model ranks first for horizons that are equal to or greater than 1 year—that is, for h  =  5, 9, and 13, while the L-CSV-t model ranks first for current and next-quarter forecasts (h  =  1 and 2). A closer look at panel A of Table 5 also reveals that the HSS-CSV-t outperforms the standard SS-type models across almost all horizons, with forecasting gains ranging between 0.3% for h  =  2 to 3.6% for h  =  13, highlighting the importance of the NG steady-state priors in forecasting applications. The TVSS type models are usually the second-best-performing class of models across all horizons and generally produce better short-term forecasts (h  =  1 and 2) compared to the HSS-type models. 16Finally, in line with Carriero et al. (2016a, 2016b) and Chan (2018), among others, we find that, overall, the models that incorporate stochastic volatility and/or fat tails produce better forecasts compared to the models with constant volatility and Gaussian errors.

Turning to panel B of Table 5 and the joint density forecasting results, we see that the overall picture is slightly different compared to the point forecasting results presented in panel A. Now, only two classes of models outrank the benchmark across all forecasting horizons: the HSS- and SS-type models augmented with stochastic volatility and/or fat tails—that is, HSS-t, HSS-CSV-t, SS-t, and SS-CSV-t. Generally, the HSS-type models are again the best-performing models for h ≥ 9, with the HSS-t model ranking first for forecasting horizons of 2 and 3 years, while the L-CSV-t, TVSS-CSV-t, and SS-t models outrank their counterparts for h  =  1, 2 and 5, respectively. The stochastic volatility and/or the t-distributed error terms improve materially the forecasting performance across models especially for shorter-term forecasting horizons, a result that is also supported by the empirical findings of Carriero et al. (2016a). In general, the empirical evidence presented so far suggests that for forecasting horizons exceeding 1 year steady-state priors play a crucial role in terms of density forecasting, while for nowcasts and shorter-terms horizons fat tails and stochastic volatility of either kind are the key factors that improve forecasting accuracy.

In Tables 6-9 we also provide point and density forecasting results for the real GDP, the unemployment rate, the GDP deflator, and the Fed Funds rate (the results for the rest of the variables are provided in the Supporting Information because of space considerations). Starting from GDP, it is evident that the HSS-type models augmented with stochastic volatility and/or t distributed innovations are usually the best-performing models for h ≥ 5 across evaluation metrics. In general, the results for GDP growth, but also for consumption, business fixed investment, residential investment, the spread and partially for hours and the S&P 500 are, on average, in line with the multivariate results presented in Table 5, with the HSS class prevailing over its rivals for longer-term forecasts. 12

By contrast, for the nominal variables GDP deflator and FFR, the TVSS-type models are typically the best performers across all forecasting horizons (with the exception of the point forecasts for the GDP deflator). This also holds true for some of the real variables of the model such as industrial production, capacity utilization, and employment, and partly for residential investment (for h  =  1, 2, and 5), where the TVSS models usually rank among the best models and never forecast poorly. To shed more light on this interesting empirical finding we should go back to the figures presenting the steady-state estimates (see Figures 3-6 in Section 6.4 and Supporting Information Figures F.2–F.11). A more detailed look at the relevant figures reveals that the time-varying steady-state estimates for the above-mentioned variables share a common characteristic: They fall consistently well below the constant steady-state estimates after 2000, probably capturing a structural break on the unconditional mean of these variables. See, for instance, the steady-state estimates for the capacity utilization, which after 2000 fluctuates around 76% as opposed to 80% estimated by the constant steady-state models; or the steady-state of FFR, which declines constantly after 2000 and stabilizes slightly below 2% after 2012. This evidence underlines the empirical relevance of the proposed time-varying specification in cases where the unconditional mean of the variable has undergone a significant structural change.

Finally, the results for the unemployment rate presented in Table 7 reveal that generally steady-state models do not consistently outperform the benchmark except for the HSS-SV model, which outranks the benchmark across all horizons. The models in log levels typically rank first, with the exception of the 3 years forecasting horizon, where the HSS-SV model beats its counterparts.

For robustness we also present results concerning an out-of-sample period ending in 2007:Q4 and excluding the global financial crisis. The joint forecasting results are presented in Table 10 and are qualitatively similar to those for the full out-of-sample period. The only exception is for the short-term point forecasts, where now the TVSS-SV model ranks first. The results for each variable separately for the normal out-of-sample period 1985:Q1–2007:Q4 are presented in the Supporting Information.

Overall, we provide empirical evidence that VAR models with NG steady-state priors can materially improve forecasting quality upon standard steady-state models or models using variables in log levels for forecasting horizons greater than 1 year. The models using a time-varying steady-state specification produce superior forecasts across all forecasting horizons in the case of structural changes in the unconditional mean of the process. Finally, the performance of the proposed models is further improved when we account for the stochastic volatility and/or fat tails of the error terms.