Note: Due to a small coding error in the original modelling, this post has been edited since its initial release. Results and discussion remain qualitatively the same, while some differences in the impulse responses emerged. Historical decompositions remain practically the same.

Although practical algorithms for Gibbs sampling have been available for some time (see Villani, 2005) and included in toolboxes like BEAR, informative priors on the BVAR steady state have not become a staple in the macroeconometric zeitgeist. This is somewhat odd. The usual Minnesota-type structure upon which most BVARs build upon imposes prior information on the dynamic coefficients – ie. short-run co-movements of endogenous variables -, while leaving the the prior uninformative for exogenous coefficients such as the constant and trend components – ie. the steady state.

Despite being undoubtedly clever and insightful, the Minnesota-setup is in essence built upon statistical, not economic intuition: if it looks like a Random Walk, assume that. If not, assume it as an IID series. From this point of view stance on exogenous components is hard to take and thus the prior is set as diffuse.

Economic intuition runs often the opposite direction however. If anything, economists tend to have heterogeneous views on the short-run co-movements of macroeconomic variables while being remarkably unanimous on the long-run dynamics. For example, by how much do headline HICP and oil prices co-move within a given month? Does the other lead the other and by how much? How persistent are changes in the headline HICP? How many months does it take for an ECB rate increase to be fully in consumer prices? Most economists would exercise caution of varying degree in answering these questions. This said, what is the long-run path of headline HICP? If any economist gave an answer more than 0.1 percentage points away from 2% year-on-year, I’d consider eating my hat.

Thus, if anything, imposing informative priors on the steady state and diffuse priors on the dynamic coefficients – ie. the exact opposite of most Minnesota-based prior structures – would seem more justified. Of course, in practice these shouldn’t be seen as mutually exclusive but complementary. Like alluded to earlier, in most cases the Minnesota-setup is strongly justified making it preferable to a diffuse setup as available information can be taken into account in posterior inference.

So why aren’t informative priors on the steady state the norm? The most probable explanation seems to be that in most applications the data drives the posterior steady state to the right ballpark anyway despite the prior being diffuse. At the same time, adding an informative prior excludes conjugate solutions making parametrisation and estimation (supposedly) somewhat more awkward.

At least one situation where the above does not obviously apply comes to mind however. Suppose the available data length is short while the period covered includes a series of major shocks to the economy driving variables away from their steady states. Then, the risk of the likelihood over- or underestimating the steady state due to stochastic shocks becomes non-trivial. Thus, if prior information on these steady states is available, including them obviously improves posterior inference.

Let’s now turn into a concrete example. Suppose one would like to use a BVAR to disentangle the drivers of the volatile 2020-2024 period on the euro area. This period includes such events as the pandemic and related economic downturn, the following inflation surge and the full-scale invasion of Ukraine by Russia with all its consequences. Furthermore, suppose one is not content on using industrial production as the monthly proxy for economic activity, as it tends to be more volatile than GDP, and more importantly the close co-movement arguably broke down during the lockdowns as services were closed for prolonged periods of time while heavy stimulus spending supported consumption as bored consumers sought to use their money elsewhere. One could thus add the production of services into the system. However, this is available only from 2015 onward. Thus, especially for HICP, the high inflation period of 2021-2023 covers a large part of the data making it a prior quite probable that this will drive the unconditional mean of the likelihood up (which as we will soon see is indeed the case).

For illustration, let’s estimate two similar VARs for the euro area for the periods 2015M01-2024M01. For both, endogenous variables will include industrial production excluding construction (IP), services production excluding trade, financial and insurance activities (SERV), headline HICP index, European Brent Oil Spot price and the Shadow Rate for the euro area estimated by Krippner. All variables except the shadow rate are in log-differences and the models include six lags and a constant. Both models are estimated with a normal-diffuse prior. Hyperparameters for both are as follows (for details on parametrisation see the working paper on the BEAR toolbox): overall parameter shrinkage is set to 0.1, cross-variable shrinkage to 2 and shrinkage for growing lags to 1. A Gibbs sampler with a burn-in size of 2000 and 1000 subsequent iterations is used. For both models sign restrictions are used to identify following shocks: an aggregate demand shock, an ad hoc consumption switch shock between IP and SERV, aggregate supply, energy supply and a monetary shock. Restrictions are as follows:

	Demand	Switch	Supply	Energy	Monetary
IP	+	–	–	–	–
SERV	+	+	–	–	–
HICP	+		+		–
Oil	+			+	–
Rate	+				+

Now the only difference between these models is the fact that the other is non-informative regarding the steady state (hyperparameter for exogenous parameter shrinkage is set to 10 000) while the other is informative. As in Villani (2005), the prior is set by specifying a subjective 95%-probability interval from which the variance and mean of the Gaussian prior are calculated. These intervals are as follows:

IP	SERV	HICP	Oil	Rate
0% – 4%	0% – 4%	1.5% – 2.2%	-1% – 3%	-1% – 4%

The prior steady state for the first four variables are expressed here in year-on-year growth rates and inputted to the model by dividing them by 12. The steady state for the shadow rate could be seen as the nominal natural rate. Due to uncertainty regarding the estimation of the real natural rate, the role of inflation in nominalising it and the estimation of the shadow rate under the ZLB, a less informative stance on it is warranted. As in Hernández de Cos (2025), the real natural rate during this period is assumed to be anything between -2% and 2%. In nominal terms let’s add 1% to 2% from inflation (persistently low inflation before 2021) and we get a nominal natural rate of -1% to 4%.

So how

As can be seen above, the impulse responses produces by the two models are practically the same. This is to be expected, as the constant terms do not (directly) affect the impulse responses. Thus, even with a short data span, impulse responses are likely to converge to reasonable estimates regardless of the prior on the steady-state.

Differences start to emerge when historical decompositions are considered. The diffuse prior puts the steady state of the headline HICP around 2.5%-3% in year-on-year terms (the differenced series has been cumulated and then converted to year-on-year differences which in logarithms correspond to percentage changes). The steady state of the shadow rate on the other hand clearly exhibits a stochastic trend. In both cases it seem likely that the low inflation and low interest environment followed by high inflation and high interest rates simply pushes the likelihood to estimate a stochastic trend due to it statistically fitting the short data length. With the informative prior it is a different story: HICP steady state is strongly anchored at close but below 2% year-on-year while the shadow rate steady state seems to converge to a stable value.

Two observations from the decomposition for the shadow rate hint toward the steady-state prior yielding more plausible estimates. First, the diffuse prior would indicate that the nominal natural rate turned positive amid the chaotic first year of the COVID pandemic while the informative prior yields a converging result. Although the exact interpretation of the VAR steady-state in regards of the natural rate and its evolution over time can be debated, the fact that the nominal rate rises during the pandemic while inflation and economic activity are going down hints toward the steady-state simply reflecting a statistical fit being driven by the likelihood. I.e. the model simply reflects the fact that the rate is first low and then high. Second, in the decomposition from with an informative prior monetary policy before the inflation episode of 2021-2023 is mainly driven by demand, which seems to be in line with theoretical and institutional aspects.

So what does this model say about the inflation drivers of 2021-2023? According to our informative prior model, at its peak in September 2023 aggregate demand and shifts back to services contributed around 3 percentage points, aggregate supply and energy supply disruptions around 3 percentage points and monetary policy whoppingly around 2 percentage points.

Whether the differences between the two models presented above are large enough to sway against diffuse prior use in general is left to the reader. However, the evidence shown here should make a case for the fact that when short and volatile data samples are used, fairly uncontroversial priors on the BVAR steady state can be enough to make a difference on the implications of the model.