# Theory coherent shrinkage of Time-Varying Parameters in VARs111I would like
to thank Frank Schorfheide and Andrea Carriero for invaluable guidance and
support. I also thank participants at the NBER-NSF SBIES Conference at the
Federal Reserve of Philadelphia.

Andrea Renzetti222Department of Economics, Alma Mater Studiorium Università di
Bologna, Piazza Scaravilli 2, 40126 Bologna, Italy

Latest Draft Here

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###### Abstract

Time-varying parameters Vector Autoregressive (TVP-VAR) models are frequently
used in economics to capture evolving relationships among the macroeconomic
variables. However, TVP-VARs have the tendency of overfitting the data,
resulting in inaccurate forecasts and imprecise estimates of typical objects
of interests such as the impulse response functions. This paper introduces a
Theory Coherent Time-Varying Parameters Vector Autoregressive Model (TC-TVP-
VAR), which leverages on an arbitrary theoretical framework derived by an
underlying economic theory to form a prior for the time varying parameters.
This “theory coherent” shrinkage prior significantly improves inference
precision and forecast accuracy over the standard TVP-VAR. Furthermore, the
TC-TVP-VAR can be used to perform indirect posterior inference on the deep
parameters of the underlying economic theory. The paper reveals that using the
classical 3-equation New Keynesian block to form a prior for the TVP-VAR
substantially enhances forecast accuracy of output growth and of the inflation
rate in a standard model of monetary policy. Additionally, the paper shows
that the TC-TVP-VAR can be used to address the inferential challenges during
the Zero Lower Bound (ZLB) period.

_J.E.L Classification Code: C32, C53_

_Keywords:_ Time Varying Parameters VARs, Bayesian Econometrics, DSGE-VARs

##  1 Introduction

Over the past four decades vector autoregressive models have become the
leading tool for description, forecasting, structural inference and policy
analysis of macroeconomic data [34, 38]. A natural progression in the
literature was to allow for time-varying parameters to capture changes in the
complex dynamic interrelationship among the variables in the system [13, 32,
14]. On one side, this class of models known as Time Varying Parameters VARs
(TVP-VARs) can be flexible enough to fit many different forms of structural
instabilities and evolving nonlinear relationships among the macroeconomic
variables. On the other side, due to the growing number of parameters, they
can easily become too flexible with adverse consequences on the precision of
inference and on the reliability of the forecasts.

In this paper I propose to use economic theory to sharpen inference in TVP-
VARs. The approach consists in exploiting prior information coming from a more
tightly parameterized model derived by an underlying economic theory about the
macroeconomic variables in the system. This prior information provides a set
of economically grounded (fuzzy) restrictions which are incorporated into a
shrinkage prior for the TVP-VAR. The resulting model, that I label Theory
Coherent TVP-VAR (TC-TVP-VAR), is a flexible statistical model for the data
that leverages on economic theory to enhance inference about the time varying
parameters. In the TC-TVP-VAR, two crucial hyper-parameters govern the
behavior of the time varying coefficients: the first one determines their
intrinsic time variation, while the other one determines their degree of
theory coherence stemming from the amount of shrinkage towards the cross-
equation restrictions implied by the economic theory. Both hyper-parameters
can be optimally tuned by maximizing the marginal data density of the TC-TVP-
VAR which is available in closed form. Moreover, the TC-TVP-VAR can also be
used as a tool for learning about the deep parameters from the underlying
economic theory. As a matter of fact, the deep parameters from the economic
theory are another set of hyper-parameters of the model that are indirectly
estimated by projecting the TVP-VAR estimates onto the restrictions implied by
the model from the economic theory. Thus, in this approach, learning about the
deep parameters from the economic theory happens indirectly through learning
about the TVP-VAR parameters.

In the paper I show that incorporating the restrictions implied by the
economic theory into a prior for the coefficients of TVP-VARs can be
beneficial to improve forecast accuracy and to obtain more precise estimates
of typical objects of interest such as the impulse response functions. In
particular, I find that encoding the restrictions from a conventional three
equation New Keynesian model into a prior for the parameters of a trivariate
TVP-VAR for output growth, inflation rate and the interest rate improves both
point and density forecast accuracy of both output growth and the inflation
rate at all the horizons considered (one quarter ahead, two quarters ahead and
one year ahead). Then, I exploit the TC-TVP-VAR to investigate whether the US
economy’s performance was affected by a binding zero lower bound (ZLB)
constraint as predicted by a standard New Keynesian model. Indeed, according
to a standard New-Keynesian model, the economy is expected to exhibit a
different response to demand and supply shocks when the ZLB constraint is in
effect. However, and more importantly, the short length of the ZLB period in
the US makes the standard TVP-VAR unfit to detect the change in the responses
predicted by the NK model [4]. In other words, whether or not there was a
change in the response of the economy during the ZLB as predicted by a
standard NK model, cannot be directly inferred by using a standard TVP-VAR.
Based on a simulation study, I show that the TC-TVP-VAR can in principle be
used to solve this inferential problem. In particular, I exploit the time
varying restriction functions implied by a medium scale NK model that accounts
for forward guidance and the ZLB period to parametrize the shrinkage prior in
the TC-TVP-VAR. I show that this approach allows to estimate more precisely
the response of the economy to macroeconomic shocks inside and outside the ZLB
period, solving the inferential problems of the standard TVP-VAR. Finally,
estimating the model on US data, I find that there are convincing evidences
supporting a change in the response of the economy during the ZLB period
similar to the one predicted by the NK model.

Related Literature An increasing number of studies has recently focused on the
issue of mitigating complexity and over-parametrization in TVP-VARs. One
strand of literature has focused on identifying fixed versus time varying
coefficients, by concentrating on the variance selection problem in the
generic state equations of each of the TVP-VARs’ coefficients. [23, 3, 29, 5].
This literature has produced shrinkage priors for variances aimed at
“automatically” reducing time-varying coefficients to static ones if the model
is overfitting.333Similarly, from a frequentist perspective, [16] showed that
time varying parameters can be framed as ridge regressions problems and used
cross validation to tune the optimal amount of time variation in each of the
state equations of the coefficients of the TVP-VAR. While treating the
coefficients of the model as independent stochastic processes and just
focusing on the problem of tuning the optimal amount of time variation of the
single coefficients, this strand of literature typically entirely neglects co-
movement and correlation among the coefficients. However, in macroeconomic
applications, the high degree of co-movement in the parameters is an empirical
regularity. This fact was already found and stressed by [14] in one of the
papers that introduced TVP-VARs in the field. In the same paper, the authors
envisaged that the reduced-form parameters should move in a highly structured
way because of the cross-equation restrictions suggesting that “a formal
treatment of cross-equation restrictions with parameter drift is a priority
for future work” (p. 274). More in line with these considerations, a smaller
number of studies [39, 37, 10] proposed to use a factor structure to model the
time variation of the parameters. Despite being compatible with the idea of
the coefficients varying in a highly structured way because of the cross-
equation restrictions associated with macroeconomic equations, this approach
is purely statistical and abstracts from any macroeconomic theory disciplining
the behavior of the coefficients.

This paper fills this gap in the literature by showing how to exploit prior
information grounded on the basis of an economic theory to state a priori a
plausible correlation structure among the time varying parameters of the
model. In this aspect, the contribution conceptually borrows from ideas from
the seminal work of [28] and operationally from the insights in [20] which
show how to exploit the non-linear cross equation restrictions implied by a
DSGE to form a prior for the parameters of a constant parameters VAR model.
Extending the framework of [20] to TVP-VARs is important at least for two
reasons. First, because in macroeconomic applications the assumption of
constant coefficients is often restrictive. Indeed, instabilities in the
autoregressive coefficients of VARs used to model the dynamics of key
macroeconomic indicators such as output and inflation have been widely
documented in the literature [13, 32, 17]. Especially in this setting, as the
model becomes more flexible, additional shrinkage can be particularly
beneficial to reduce overfitting. Second, because economic theories themselves
might imply time varying moments for the data that result in time-varying
restriction functions for the coefficients. For example, macroeconomic
theories assuming rational expectations extended so as to allow some
parameters to vary according to Markov process with given transition
probabilities, lead to state space representation with time varying
coefficients [21]. At the same time, solutions for linear stochastic rational
expectations models in the face of a finite sequence of anticipated structural
changes lead to state space representation with time varying coefficients
[7].444Another notable case within the rational expectations framework are
models log-linearized around time varying trend for inflation [15, 2, 1].
Likewise, outside the rational expectation framework, macroeconomic theories
that assume learning can lead to state space representation with time varying
coefficients [31]. In all those cases, the proposed TC-TVP-VAR can be used
both to incorporate the implied time varying restrictions into a prior for the
time varying coefficients of the VAR and to estimate the deep parameters of
the underlying economic model. In this sense, the paper is also related to the
strand of literature that exploits an auxiliary flexible statistical model for
the data to make indirect inference on the deep parameters of a structural
model from the economic theory (see for example [24] [22]).555In the TC-TVP-
VAR the structural parameters from the underlying economic theory are
estimated by implicitly minimizing the weighted discrepancy between the
unrestricted TVP-VAR estimates and the restriction functions. This approach
can be thought as a Bayesian version of [36] as in [20]. Finally, the paper is
more broadly related to the econometric literature showing that moment
conditions from economic theory can successfully be exploited for forecasting
macroeconomic and financial variables ([25, 8] most notably).

Outline The paper is organized as follows. In section 2 I introduce the TC-
TVP-VAR, presenting an analytical derivation of the proposed theory coherent
prior and a simulation from the prior to showcase its main properties. In
addition, I discuss the fit-complexity trade-off linked to the calibration of
the hyper-parameters determining the intrinsic persistence of the time
varying-coefficients and the degree of shrinkage towards the restrictions from
the theory. I conclude section 2 by presenting an MCMC sampler used for
estimation of the parameters of the TC-TVP-VAR. Afterwards, in section 3 I
exploit the well known 3-equations New Keynesian (NK) block to form a prior
for the parameters of a trivariate TVP-VAR for GDP growth, inflation and the
interest rate for the US economy. I compare the forecasts from a TC-TVP-VAR
that encodes the restrictions from the NK model as a prior for the time
varying coefficients to the forecasts from a standard TVP-VAR, showing that
the former outperforms the latter both in terms of point and density forecast
accuracy. Then, in section 4 I conduct a simulation study and show that a
standard TVP-VAR struggles to detect the change in the response of the economy
to macroeconomic shocks during the ZLB period as predicted by a standard NK
model. I show that instead the TC-TVP-VAR can in principle be used to solve
this inferential problem. Finally, I estimate a TC-TVP-VAR on US data to
investigate whether the US economy was indeed affected by a binding ZLB.
Section 5 concludes.

##  2 Theory coherent TVP-VAR

The construction of a theory coherent prior builds on the idea that an
economic theory implies restrictions on the parameters of the TVP-VAR.
Intuitively we can find out these restrictions by specifying a prior
distribution on the deep parameters from the theory, simulating the data from
the theory and then estimating a TVP-VAR on the simulated data. Imposing a
prior on the deep parameters from the theory will then induce a prior on the
parameters of the TVP-VAR encoding the restrictions imposed by the economic
theory. Based on this idea, in this section I derive analytically a prior for
a TVP-VAR which is theory coherent, in the sense that it centers the time
varying coefficients on the cross equation restrictions implied by an
underlying economic theory about the variables in the system.

Notation Before moving on, I introduce some notations conventions used in the
paper. Scalars are in lowercase and normal weight. Vectors are in lowercase
and in bold. Matrices are in uppercase and bold.

###  2.1 Construction of the prior

A TVP-VAR is given by:

$\underbrace{\bm{y}^{\prime}_{t}}_{\text{$1\times
N$}}=\underbrace{\bm{x^{\prime}}_{t}}_{\text{$1\times
k$}}\underbrace{\bm{\Phi}_{t}}_{\text{$k\times
N$}}+\underbrace{\bm{u_{t}}^{\prime}}_{\text{$1\times N$}}\hskip
85.35826pt\bm{u_{t}}\sim\mathcal{N}(\bm{0}_{N\times 1},\bm{\Sigma_{u}})$ (1)
$vec(\bm{\Phi_{t}})=vec(\bm{\Phi_{t-1}})+\bm{\eta}_{t}\hskip
56.9055pt\bm{\eta}_{t}\sim\mathcal{N}(\bm{0},\bm{\Omega})$ (2)

where $\bm{y_{t}}$ is an $N$ dimensional random vector observed for
$t=1,\ldots,T$ periods,
$\bm{x_{t}}=[1,\bm{y}_{t-1}^{\prime},\ldots,\bm{y}_{t-p}^{\prime}]^{\prime}$,
$p$ is the lag order and $k=1+Np$ is the number of coefficients in each
equation of the VAR. For convenience, we can rewrite the TVP-VAR in (1) in
“static” compact form as

$\underbrace{\bm{Y}}_{\text{$T\times N$}}=\underbrace{\bm{X}}_{\text{$T\times
Tk$}}\underbrace{\bm{\Phi}}_{\text{$Tk\times
N$}}+\underbrace{\bm{U}}_{\text{$T\times N$}}\hskip 85.35826pt\bm{U}\sim
MVN(\bm{0},\bm{\Sigma_{u}},\bm{I_{T}})$ (3)

and the state equations in (2) as

$\bm{H}_{Tk}\bm{\Phi}=\bm{\Phi_{00}}+\bm{\eta}$ (4)

which graphically is

$\underbrace{\begin{bmatrix}\bm{I}_{k}&\bm{0}&\ldots&\bm{0}\\\
-\bm{I}_{k}&\bm{I}_{k}&\ldots&\bm{0}\\\ \bm{0}&-\bm{I}_{k}&\ddots&\bm{0}\\\
\vdots&\vdots&\vdots&\vdots\\\ \bm{0}&\bm{0}&-\bm{I}_{k}&\bm{I}_{k}\\\
\end{bmatrix}}_{\text{$\bm{H_{Tk}}$}}\underbrace{\begin{bmatrix}\bm{\Phi_{1}}\\\
\bm{\Phi_{2}}\\\ \bm{\Phi_{3}}\\\ \vdots\\\ \bm{\Phi_{T}}\\\
\end{bmatrix}}_{\text{$\bm{\Phi}$}}=\underbrace{\begin{bmatrix}\bm{\Phi_{0}}\\\
\bm{0}\\\ \bm{0}\\\ \vdots\\\ \bm{0}\\\
\end{bmatrix}}_{\text{$\bm{\Phi_{00}}$}}+\underbrace{\begin{bmatrix}\bm{\eta}_{1}\\\
\bm{\eta}_{2}\\\ \bm{\eta}_{3}\\\ \vdots\\\ \bm{\eta}_{T}\\\
\end{bmatrix}}_{\text{$\bm{\eta}$}}$

In this representation, the matrix $\bm{\Phi}$ stores the matrices with the
time varying coefficients one on the top of the other, for all the time
periods $t=1,\ldots,T$. I assume that an hyperparameter $\lambda$ governs the
serial correlation of the coefficients stored in the matrices
$\bm{\Phi_{1}},\ldots,\bm{\Phi_{T}}$. In particular, I assume that the
variance covariance matrix $\bm{\Omega}$ has the Kronecker structure
$\bm{\Omega}=\bm{\Sigma_{u}}\otimes(\lambda^{2}\bm{I_{k}})^{-1}$. This
assumption implies that the variance in the state equation of the coefficients
is proportional to the variance of the innovations of the equation to which
the coefficients appertain $\sigma^{2}_{ii}$. In practice this means that the
generic state equation of the coefficient attached to the $j^{th}$ regressor
in the $i^{th}$ equation reads as follows

$\phi_{jt}^{(i)}=\phi_{jt-1}^{(i)}+\eta_{jt}^{(i)}\hskip
56.9055pt\eta_{jt}^{(i)}\sim\mathcal{N}\left(0,\frac{\sigma^{2}_{ii}}{\lambda^{2}}\right)$
(5)

with $i=1,\ldots,N$ and $j=1,\ldots,k$. As $\lambda\rightarrow 0$ the variance
of the Normal prior centering a coefficient at time $t$ on the realization at
time $t-1$ increases, meaning that this prior becomes more and more diffuse
and the coefficients of two consecutive time periods are not forced a priori
to be close to each other. Conversely as $\lambda\rightarrow\infty$ the prior
centering the coefficient at time $t$ on the realization at time $t-1$ becomes
more and more tight. Clearly the calibration of $\lambda$ involves a fit
complexity trade off since by letting $\lambda\rightarrow\infty$ we allow the
coefficient at time $t$ to be arbitrary distant from the coefficient at $t-1$
up to the point that we can almost fit the data perfectly in sample.666This
point will be covered more in detail in Section 2.4. Exploiting representation
(4) and the Kronecker structure of $\bm{\Omega}$, we can conclude that
equation (4) implies a joint normal prior for $\bm{\Phi}$, that is

$vec(\bm{\Phi}|\bm{\Sigma}_{u},\lambda)\sim\mathcal{N}(vec\underbrace{(\bm{H_{Tk}}^{-1}\bm{\Phi_{00}}}_{\text{$\underline{\Phi_{0}}$}}),\bm{\Sigma}_{u}\otimes(\underbrace{\lambda^{2}\bm{H}_{Tk}^{\prime}\bm{H}_{Tk})^{-1}}_{\text{$\underline{\Psi}(\lambda)$}})$
(6)

since $|\bm{H}_{Tk}|=1$, $\bm{H}_{Tk}$ is always invertible and the
determinant of Jacobian matrix of the linear transformation cancels
out.777Notice that this is true also if we consider an AR(1) dynamics for the
time varying coefficients. In that case the elements off the main diagonal of
the band matrix $H_{Tk}$ would store the AR coefficients of the state
equations. Combining this prior for the time varying parameters with an
Inverse-Wishart prior for $\bm{\Sigma}_{u}$

$p(\bm{\Sigma_{u}})\sim\mathcal{IW}(\underline{\textbf{S}},\underline{\nu})$
(7)

leads to

$p(\bm{\Phi,\Sigma_{u}}|\lambda)\sim\mathcal{NIW}$ (8)

Equation (8) states a prior for the time varying parameters and the variance
covariance matrix of the innovations of the TVP-VAR which is conditional on
$\lambda$, the crucial hyper-parameter determining the amount of time
variation of the coefficients in the state equations (2). 888To be precise
this prior is also conditional on $\bm{\Phi_{0}}$, $\bm{\underline{S}}$ and
$\underline{\nu}$. However, for the purpose of the paper I will consider
$\bm{\Phi_{0}}$, $\bm{\underline{S}}$ and $\underline{\nu}$ as fixed.
Importantly, this prior is conjugate to the Gaussian likelihood. We exploit
the conjugacy to update equation (8) with observations simulated from the
theory and obtain a Normal-Inverse-Wishart prior which is theory coherent,
meaning that it is centered on the restrictions from the economic
theory.999Note that equivalently we can consider a flat prior for
$\bm{\Sigma_{u}}$, that is:
$p(\bm{\Sigma_{u}})\propto\bm{\Sigma_{u}}^{-\frac{N+1}{2}}$ (9) In the
specific, updating (8) with observations from the theory we get

$p(\bm{\Phi,\Sigma_{u}}|\lambda,\bm{\theta},\gamma)=c(\lambda,\bm{\theta},\gamma)^{-1}p(\bm{\Phi,\Sigma_{u}}|\lambda)p(\bm{Y^{*}(\theta)}|\gamma,\bm{\Phi},\bm{\Sigma_{u}})$
(10)

where $p(\bm{Y(\theta)}|\bm{\Phi},\bm{\Sigma_{u}},\gamma)$ is the likelihood
of $\gamma$ simulated samples of $\bm{Y(\theta)}$ from the theory and

$c(\lambda,\bm{\theta},\gamma)=\int_{-\infty}^{\infty}p(\bm{\Phi,\Sigma_{u}}|\lambda)p(\bm{Y^{*}(\theta)}|\gamma,\bm{\Phi},\bm{\Sigma_{u}})d\bm{\Phi}d\bm{\Sigma_{u}}$
(11)

is an integrating constant which ensures that the prior density is proper and
integrates to one.101010In the appendix A.1.3 I report details on the
integrating constant. Equation (10) makes clear that the theory-coherent prior
for $(\bm{\Phi},\bm{\Sigma_{u}})$ is obtained estimating a TVP-VAR on the
simulated data from the theory. As a matter of fact, equation (10) is just the
posterior distribution of $(\bm{\Phi},\bm{\Sigma_{u}})$ obtained by updating
the hierarchical prior $p(\bm{\Phi,\Sigma_{u}}|\lambda)$ with the likelihood
of $\gamma$ simulated samples of observations generated by the model from the
theory. As in [20], in $p(\bm{Y(\theta)}|\bm{\Phi},\bm{\Sigma_{u}})$ I replace
the sample moments $\bm{Y(\theta)^{\prime}Y(\theta)}$,
$\bm{Y(\theta)^{\prime}X(\theta)}$, and $\bm{X(\theta)^{\prime}X(\theta)}$ by
their expected values, obtaining 111111The notation $Y(\bm{\theta})$ makes
clear that we are conditioning on the vector $\bm{\theta}$.

$\displaystyle p(\bm{Y(\theta)}|\gamma,\bm{\Phi},\bm{\Sigma_{u}})$
$\displaystyle=(2\pi)^{-\frac{\gamma TN}{2}}|\bm{\Sigma_{u}}|^{-\frac{\gamma
T}{2}}exp\left[-\frac{1}{2}tr\left((\bm{\Sigma_{u}}^{-1})(\gamma\bm{\Gamma_{yy}(\theta)}-\gamma\bm{\Phi^{\prime}\Gamma_{xy}(\theta))}-\gamma\bm{\Gamma_{yx}(\theta)\Phi}+\gamma\bm{\Phi^{\prime}\Gamma_{xx}(\theta)\Phi})\right)\right]$
(12)

This step is meant to avoid stochastic simulation of the artificial
observations from the economic theory and it let us write the likelihood of
the artificial observations directly as a function of the vector of deep
parameters from the economic theory $\bm{\theta}$.121212Details on
$\bm{\Gamma_{xx}}$, $\bm{\Gamma_{yy}}$, $\bm{\Gamma_{xy}}$ are reported in the
appendix A.1.2. It is easy to show that (10) is also Normal-Inverse-Wishart
since it is de-facto obtained as the posterior distribution that combines a
Normal-Inverse Wishart prior for $(\bm{\Phi},\bm{\Sigma_{u}})$ with a Gaussian
likelihood for the simulated data from the theory. Therefore the theory
coherent prior in equation (10) takes the form

$\displaystyle p(\bm{\Sigma}_{u})$
$\displaystyle\sim\mathcal{IW}\left({\underline{\underline{\bm{S}}}},\underline{\underline{\nu}}\right)$
(13) $\displaystyle p(vec(\bm{\Phi})|\bm{\Sigma}_{u})$
$\displaystyle\sim\mathcal{N}(vec(\bm{\underline{\underline{\Phi}}}),\bm{\Sigma}_{u}\otimes\bm{\underline{\underline{\Psi}}})$
$\underline{\underline{\bm{S}}}=\underline{\bm{S}}+\gamma\bm{\Gamma_{yy}(\theta)}+\bm{\underline{\Phi_{0}}}^{\prime}(\lambda^{2}\bm{H_{Tk}}^{\prime}\bm{H_{Tk}})\bm{\underline{\Phi_{0}}}-\bm{\underline{\underline{\Phi}}}^{\prime}\bm{\underline{\underline{\Psi}}}^{-1}\bm{\underline{\underline{\Phi}}}$
(14) $\underline{\underline{\nu}}=\underline{\nu}+T\gamma$ (15)
$vec(\bm{\underline{\underline{\Phi}}})=vec(\left(\gamma\bm{\Gamma_{xx}(\theta)}+\lambda^{2}\bm{H_{Tk}^{\prime}H_{Tk}}\right)^{-1}(\gamma\bm{\Gamma_{xy}(\theta)}+\lambda^{2}\bm{H}_{Tk}^{\prime}\bm{H}_{Tk}\bm{\underline{\Phi_{0}}}))$
(16)
$\bm{\underline{\underline{\Psi}}}=\left(\gamma\bm{\Gamma_{xx}(\theta)}+\lambda^{2}\bm{H_{Tk}^{\prime}H_{Tk}}\right)^{-1}$
(17)

This prior for $\bm{\Phi}$ encompasses two different pieces of information
about the time varying coefficients. The first piece of information is their
intrinsic persistence determined by the shrinkage hyperparameter $\lambda$.
The second instead, is the degree of theory coherence determined by the
shrinkage hyperparameter $\gamma$ which defines the tightness of the Normal
prior around the restriction function defined by the economic theory

$\bm{\Phi(\theta)^{*}}=\bm{\Gamma_{xx}(\theta)^{-1}\Gamma_{xy}(\theta)}$ (18)

which implies $\bm{\Phi}_{t}=\bm{\Gamma_{xx,t}}^{-1}\bm{\Gamma_{xy,t}}$ for
$t=1,\ldots,T$ where
$\bm{\Gamma_{xx,t}}\equiv\operatorname*{\mathbb{E}}[\bm{x_{t}x_{t}^{\prime}}|\bm{\theta}]$
and
$\bm{\Gamma_{xy,t}}\equiv\operatorname*{\mathbb{E}}[\bm{x_{t}y_{t}^{\prime}}|\bm{\theta}]$.
A couple of considerations are worthwhile. First, the assumption of a unique
hyperparameter $\lambda$ determining the amount of time variation of the time
varying coefficients can be easily replaced by assuming $\lambda_{j}$’s hyper-
parameters with $j=1,\ldots,k$ such that $\bm{\Omega}$ can be factorized as

$\bm{\Omega}=\bm{\Sigma}\otimes(\bm{\Lambda_{k}}^{\prime}\bm{\Lambda_{k}})^{-1}$
(19)

where $\bm{\Lambda_{k}}=diag(\lambda_{1},\ldots,\lambda_{k})$ and the Kroneker
structure of the Normal-Inverse-Wishart prior (6) is preserved. Keeping the
assumption of a Kronecker structure allows to have a closed form expression
for the likelihood of $\gamma,\lambda$ and $\bm{\theta}$ marginally of the
time varying parameters $\bm{\Phi_{1},},\ldots,\bm{\Phi_{T}}$ and of
$\bm{\Sigma_{u}}$. In practice this allows to make inference on
$\bm{\Phi},\bm{\Sigma}_{u}$, the shrinking hyper-parameters $\gamma$ and
$\lambda$ and also on the deep parameters from the theory $\bm{\theta}$
jointly as in [20]. Second, the population moments encoded in the TVP-VAR
$\bm{\Gamma_{xx}}$, $\bm{\Gamma_{xy}}$, $\bm{\Gamma_{yy}}$, which are assumed
to be available in closed form as function of the deep parameters from the
theory $\bm{\theta}$ might not be available in some applications. If it is the
case, these moments can be obtained by stochastic simulation as in [30].
Furthermore, and more importantly, the moments stored in the matrices
$\bm{\Gamma_{xx}}$, $\bm{\Gamma_{xy}}$, $\bm{\Gamma_{yy}}$ can be either
constant or time varying conditionally on the deep parameters from the theory.
That is, the prior in (13) can accommodate theories which imply both constant
and time-varying restrictions. This is the main innovation to the DSGE-VAR
toolkit developed by [20]. As a last note, the joint prior (13) implies that
the conditional prior $p(vec(\bm{\Phi_{t}}|vec(\bm{\Phi_{t-1}}))$ is still
Normal and the prior for $vec(\bm{\Phi_{t}})$ is still a Markov process,
meaning that:131313The full derivation of the conditional form of the prior as
a function of the elements of the precision matrix
$\bm{\underline{\underline{\Psi}}}^{-1}$ and the matrix of means
$\bm{\underline{\underline{\Phi}}}$ can be found in the appendix A.1.4.

$p(vec(\bm{\Phi_{t}})|vec(\bm{\Phi_{t-1}}),vec(\bm{\Phi_{t-2}}),\ldots,vec(\bm{\Phi_{1}}))=p(vec(\bm{\Phi_{t}})|vec(\bm{\Phi_{t-1}}))$
(20)

###  2.2 Simulation of the prior

To showcase the properties of the proposed theory coherent prior, figure 1
shows random draws for the time series of a generic $\phi_{jt}^{(i)}$
coefficient of the TVP-VAR for $t=1,\ldots,250$ from the prior. The dashed
black line represents the restriction function $(\ref{restriction_function})$
defined by the economic theory for that coefficient, that I label
$\phi_{jt}^{(i)}(\bm{\theta})^{*}$. Just for illustrative purposes I consider
the case of a constant restriction function, however having time varying
moments stored in the matrices $\bm{\Gamma_{xx}}$ and $\bm{\Gamma_{xy}}$
conditionally on the deep parameters of the theory, would directly imply a
time varying restriction function. The first row shows five draws from the
theory coherent prior for different values of $\lambda$ conditioning on a
positive $\gamma$, meaning that for a given degree of theory coherence I
change the value of hyper-parameter which determines the persistence of the
coefficient. When $\lambda=0$, the draws for the coefficient $\phi_{jt}^{(i)}$
are independent for $t=1,\ldots,T$ and centered on the restriction function
coming from the the theory. Hence, nothing prevents a coefficient at time $t$
to be arbitrary distant from a coefficient at time $t-1$ except the fact both
coefficients are centered, with a precision determined by $\gamma$, on the
restriction function defined by the theory. As $\lambda$ increases the time
series of the coefficient becomes more and more persistent up to the point
that when $\lambda$ is very big, the coefficient becomes almost constant over
time. Instead, in the second row, I let the degree of theory coherence change
conditionally on a given intrinsic persistence of the time varying
coefficients. In other words, I show random draws for $\phi_{jt}^{(i)}$ for
different values of $\gamma$ conditionally on $\lambda>0$. When $\gamma=0$,
the coefficients are just random walks with variance equal to
$\frac{\sigma_{ii}}{\lambda^{2}}$, and they are completely unrelated to the
restriction function implied by the economic theory
$\phi_{jt}^{(i)}(\bm{\theta})^{*}$.141414Note that the draws fore the time
varying coefficients were initiated near the restriction functions just for
visualization purposes. As $\gamma$ increases the draws for the coefficient
are centered on the restriction function defined by the theory with an
increasing precision and on the limit, with $\gamma\rightarrow\infty$, they go
to $\phi_{jt}^{(i)}(\bm{\theta})^{*}$.151515The coefficients would have a
degenerate distribution, with point mass on the restriction function.
Importantly, the hyperparameter $\gamma$ is contemporaneously shrinking all
the time varying coefficients $\phi_{t,j}^{(i)}$ in the equations of the VAR
towards the restrictions implied by the economic theory. This reflects the
idea that it is economic theory that a priori postulates a plausible
correlation structure among the coefficients of VAR model through the
restriction function in (18).

Figure 1:  Each subfigure presents five draws from the theory coherent prior
for a generic coefficient of the TVP-VAR for different values of the
hyperparameters $\lambda$ (shrinkage towards constant) and $\gamma$ (theory
coherent shrinkage). The first row, presents draws for $\lambda=0$,
$\lambda>0$ and a big $\lambda$, conditionally on $\gamma>0$. The second row
presents draws for $\gamma=0$, $\gamma>0$ and a big $\gamma$, conditionally on
$\lambda>0$. The dashed black line represents the restriction functions coming
from the theory.

###  2.3 Conditional posterior of $\Phi$ and $\Sigma_{u}$

It is straightforward to derive the posterior distribution of
$(\bm{\Phi},\bm{\Sigma}_{u})$ conditional on the hyper-parameters $\lambda$,
$\gamma$ and on the deep parameters from the theory $\bm{\theta}$ since this
is just a Normal-Inverse-Wishart distribution obtained by updating a Gaussian
likelihood with the Normal-Inverse-Wishart prior (13), namely

$\displaystyle
p(vec(\bm{\Phi})|\bm{\Sigma}_{u},\gamma,\lambda,\bm{\theta},\bm{Y})$
$\displaystyle\sim\mathcal{N}(vec(\bm{\tilde{\Phi}}),\bm{\Sigma}_{u}\otimes\bm{\tilde{\Psi}})$
(21) $\displaystyle p(\bm{\Sigma}_{u}|\gamma,\lambda,\bm{\theta},\bm{Y})$
$\displaystyle\sim\mathcal{IW}\left({\bm{\tilde{S}}},\tilde{\nu}\right)$
$vec(\bm{\tilde{\Phi}})=vec(\left(\bm{X^{\prime}X}+\gamma\bm{\Gamma_{xx}(\theta)}+\lambda^{2}\bm{H_{Tk}^{\prime}H_{Tk}}\right)^{-1}(\bm{X^{\prime}Y}+\gamma\bm{\Gamma_{xy}(\theta)}+\lambda^{2}\bm{H}_{Tk}^{\prime}\bm{H}_{Tk}\bm{\underline{\Phi_{0}}}))$
(22)
$\bm{\tilde{\Psi}}=\left(\bm{X^{\prime}X}+\gamma\bm{\Gamma_{xx}(\theta)}+\lambda^{2}\bm{H_{Tk}^{\prime}H_{Tk}}\right)^{-1}$
(23)
$\tilde{\bm{S}}=\bm{Y^{\prime}Y}+\underline{\underline{\bm{S}}}+\bm{\underline{\underline{\Phi}}}^{\prime}\bm{\underline{\underline{\Psi}}}^{-1}\bm{\underline{\underline{\Phi}}}-\bm{\tilde{\Phi}}^{\prime}\bm{\tilde{\Psi}}^{-1}\bm{{\tilde{\Phi}}}$
(24) $\tilde{\nu}=\underline{\underline{\nu}}+T$ (25)

Equation (22) makes clear that thanks to the conjugacy of the prior the
formula for the mean of the conditional posterior of $\bm{\Phi}$ boils to a
standard OLS regression formula based on a sample augmented with a set of
dummy observations that determines the degree of persistence on the time
varying coefficients and another set of dummy observations that centers the
coefficients on the restriction function implied by the theory. While the
first set of dummy observations shapes the correlation of the time varying
coefficients over time by imposing a set of linear restrictions as a function
of the hyper-parameter $\lambda$, the second set of dummy observations induces
at each time period a correlation structure implied by the non-linear cross
equation restrictions coming from the theory, as a function of the deep
parameters from the theory $\bm{\theta}$ and of the shrinking parameter
$\gamma$.161616In the appendix A.1.1 I show that thanks to the Kronecker
structure of the prior (6) the time variation of the coefficients can be
modelled by dummy observations. It is worth to remark that from a
computational perspective equation (22) makes the estimation of the time
varying coefficients very efficient since it allows to draw all the history of
the time varying coefficients in all the equations of the VAR in a single
step, avoiding forward filtering and backward smoothing algorithms à la [9].
171717Indeed, as in the precision sampler by [11], we can draw all the latent
states from $t=1,\ldots,T$ in a single step and thanks to the Kronecker
structure of the posterior, we can do it for all the $N$ equations of the TVP-
VAR jointly. More in general, the Kronecker structure (19) coupled with the
precision sampler by [11] can be exploited to estimate medium to large scale
TVP-VARs.

###  2.4 $\lambda,\gamma$ and fit complexity trade off

Tuning of the optimal degree of theory coherence and the intrinsic amount of
time variation of the coefficients is a delicate matter, since it clearly
involves a fit-complexity trade off. As a matter of fact, very low values of
both $\gamma$ and $\lambda$ imply a priori that the coefficients in two
consecutive time periods can potentially be very different from each others
and distant from the restriction functions defined by the theory.181818When
both $\gamma=0$ and $\lambda=0$ the model is left totally unrestricted, with
the prior variance covariance of the coefficients being equal to infinity.
Clearly, in this case there are more parameters than you can feasibly estimate
with a flat prior meaning that the conditional posterior of $\bm{\Phi}$ cannot
be computed due to the non-invertibility of $\bm{X^{\prime}X}$ (this can be
seen from equation (21)). Intuitively, this model will fit the data very well
in sample but will perform badly for forecasting out-of-sample. Indeed,
decreasing $\gamma$ and $\lambda$ will in general increase in-sample fit of
the model at the expense of out-of-sample accuracy. Based on this argument, I
recommend to base the optimal choice of both the hyper-parameters on the
maximization of the marginal likelihood of the model or equivalently on the
maximization of the posterior of the hyper-parameters $\lambda$ and $\gamma$
under a flat prior for these hyper-parameters. This translates into the
maximizing the one-step-ahead out-of-sample forecasting ability of the model.
Indeed, the log-marginal likelihood (or Bayesian evidence) can be interpreted
as the sum of the one step-ahead predictive scores, since it equivalent to the
scoring rule of the form

$S(\bm{Y})=\sum_{t=1}^{T}s(\bm{y}_{t}|\bm{y}_{t-1})=\sum_{t=1}^{T}log(p(\bm{y}_{t}|\bm{y}_{t-1}))$
(26)

An attractive feature of the TC-TVP-VAR is that the marginal likelihood
$p(\bm{Y}|\gamma,\lambda,\bm{\theta})$ obtained by integrating out
$\bm{\Phi},\bm{\Sigma}_{u}$ from the conditional posterior
$p(\bm{\Phi,\Sigma_{u}}|\lambda,\bm{\theta},\gamma,\bm{Y})$ is available in
closed form and it is equal to

$p(\bm{Y}|\lambda,\bm{\theta},\gamma)=\left(\pi\right)^{-\frac{TN}{2}}\frac{\Gamma_{N}\left(\frac{\tilde{{\nu}}}{2}\right)|\tilde{{S}}|^{-\frac{\tilde{\nu}}{2}}|\tilde{{\bm{\Psi}}}|^{\frac{N}{2}}}{\Gamma_{N}\left(\frac{\underline{\nu}}{2}\right)|\underline{\underline{S}}|^{-\frac{\underline{\underline{\nu}}}{2}}|\bm{\underline{\underline{\Psi}}}|^{\frac{N}{2}}}$
(27)

As a consequence, calibrating $\gamma$ and $\lambda$ to maximize (27)
corresponds to finding $\gamma$ and $\lambda$ maximizing the one-step-ahead
out-of-sample forecasting ability of the model. This strategy of estimating
hyper-parameters by maximizing the marginal likelihood is an empirical Bayes
method which has a clear frequentist interpretation. In what follows, and in
particular in the estimation algorithm detailed in the next section 2.5 I will
regard $\gamma$ and $\lambda$ as random variables and perform full posterior
inference on the hyper-parameters, but analogously maximizing the posterior of
the hyper-parameters will correspond to maximizing the one-step-ahead out of
sample forecastability of the model. Following the same steps as in [26], we
can rewrite equation (27) as

$p(\bm{Y}|\lambda,\bm{\theta},\gamma)\propto|\bm{(V_{\varepsilon}^{post})^{-1}V^{prior}_{\varepsilon}}|^{\frac{T+\underline{\underline{\nu}}}{2}}\prod_{t=1}^{T}|\bm{V_{t|t-1}}|^{-\frac{1}{2}}$
(28)

where $\bm{V_{\varepsilon}^{post}}$ and $\bm{V^{prior}_{\varepsilon}}$ are the
posterior and prior means (or modes) of the residual variance, while
$\bm{V_{t|t-1}}$ is equal to the variance (conditional on $\bm{\Sigma_{u}}$)
of the one-step-ahead forecast of $\bm{y}$, averaged across all possible
a-priori realizations of $\bm{\Sigma_{u}}$. Complete formulas of these objects
are reported in the appendix A.1.5 and are the analog of the formulas reported
in [26]. Equation (28) makes clear that the marginal likelihood involves two
terms: a reward for model fit,
$|\bm{(V_{\varepsilon}^{post})^{-1}V^{prior}_{\varepsilon}}|^{\frac{T+\underline{\underline{\nu}}}{2}}$
and a penalty term for model complexity
$\prod_{t=1}^{T}|\bm{V_{t|t-1}}|^{-\frac{1}{2}}$. Figure 2 plots the model fit
term and the penalty term of the marginal likelihood as a function of the two
hyper-parameters $\gamma$ and $\lambda$ conditionally on a set of deep
parameters from the theory $\bm{\theta}$. As $\lambda$ decreases the tightness
of the prior centering each coefficient on its realization in the previous
period eases and in sample model fit improves, meaning that $\bm{V}^{post}$
decreases. However, at the same time, as $\lambda$ decreases $\bm{V_{t|t-1}}$
increases, since the variance (conditional on $\bm{\Sigma}$) of the one-step-
ahead forecast of $\bm{y}$ increases. Analogously, as $\gamma$ decreases the
restriction functions from the theory become less and less binding, meaning
that the model will fit better in sample, namely $\bm{V}^{post}$ decreases,
but the variance of the one step ahead forecast error $\bm{V_{t|t-1}}$ will
increase.

Figure 2:  The figure shows the model fit term (on the left) and the penalty
term (on the right) of the marginal likelihood (27) as a function of the
hyper-parameters $\lambda$ (persistency) and $\gamma$ (theory coherence).

###  2.5 Estimation

In this section I describe a posterior simulation sampler to make inference on
the TVP-VAR parameters $\bm{\Phi}$ and $\bm{\Sigma}_{u}$ and on the shrinkage
hyper-parameters $\lambda$ and $\gamma$ together with the deep parameters from
the theory $\bm{\theta}$ .191919For the purpose of the paper we fix
$\bm{\Phi}_{0}=\bm{\hat{\Phi}_{0}}$ and
$\bm{\underline{S}}=\bm{\hat{S}}=diag(\hat{s}_{1}^{2},\ldots,\hat{s}_{N}^{2})$
and $\nu=N+2$. Alternatively inference on these hyper-parameters can be made
by extending the Random Walk Metropolis step in the MCMC sampler described
below. We can simulate draws from the posterior
$p(\bm{\Phi,\Sigma_{u}},\lambda,\bm{\theta},\gamma|\bm{Y})$ exploiting the
factorization:

$p(\bm{\Phi,\Sigma_{u}},\lambda,\bm{\theta},\gamma|\bm{Y})=p(\bm{\Phi,\Sigma_{u}}|\lambda,\bm{\theta},\gamma,\bm{Y})p(\lambda,\bm{\theta},\gamma|\bm{Y})$
(29)

This factorization makes clear that we can sample from the posterior of
$\bm{\Phi},\bm{\Sigma_{u}},\bm{\theta},\gamma,\lambda$ building a MCMC
algorithm that iterates the following two steps:

  1. 1.

Draw from $p(\lambda,\bm{\theta},\gamma|\bm{Y})\propto
p(\bm{Y}|\lambda,\bm{\theta},\gamma)p(\lambda,\bm{\theta},\gamma)$.

  2. 2.

Draw from $p(\bm{\Phi,\Sigma_{u}}|\lambda,\bm{\theta},\gamma,\bm{Y})$

In the first step, estimation of both the shrinkage parameters $\gamma$ and
$\lambda$ and of deep parameters from the theory $\bm{\theta}$ happens by
posterior evaluation of

$p(\lambda,\bm{\theta},\gamma|Y)\propto
p(\bm{Y}|\lambda,\bm{\theta},\gamma)p(\bm{\theta})p(\gamma)p(\lambda)$ (30)

where $p(\gamma)$ and $p(\lambda)$ are the priors for the shrinkage hyper-
parameters 202020For both $\gamma$ and $\lambda$ I consider the Uniform prior
$\gamma\sim\mathcal{U}(0,b_{\gamma})$ and
$\lambda\sim\mathcal{U}(0,b_{\lambda})$ where $b_{\gamma}=10^{10}$ and
$b_{\lambda}=10^{10}$ while $p(\bm{\theta})$ is the prior of the deep
parameters from the theory. Hence, learning about the structural parameters
happens implicitly by projecting the VAR estimates onto the restrictions
implied by the model from the theory. More precisely, the estimates of the
deep parameters minimizes the weighted discrepancy between the TVP-VAR
unrestricted estimates and the restriction function (18). This approach can be
thought as a Bayesian version of [36] and was pioneered by [20]. In
particular, the TVP-VAR is used to summarize the statistical properties of
both the observed data and the theory-simulated data and an estimate of the
deep parameters from the theory is obtained by matching as close as possible
TVP-VAR parameters from observed data and from the simulated data.212121See
Proposition 1 and Proposition 2 in [20] for further details on this.  To
sample from $p(\lambda,\bm{\theta},\gamma|\bm{Y})$ I consider a two blocks
random walk metropolis algorithm. This is basically a Gibbs Sampler where in
the first block, I draw the hyper-parameters $\gamma$ and $\lambda$
conditionally on $\bm{\theta}$ while in the second block I draw $\bm{\theta}$
conditionally on the shrinking hyper-parameters $\gamma$ and $\lambda$. Step
2, instead, consists just on Monte Carlo draws from the posterior of
$(\bm{\Sigma_{u}},\bm{\Phi})$ conditional on $\lambda,\gamma$ and
$\bm{\theta}$ which is the Normal-Inverse-Wishart distribution in (21) that
is:

  1. 2.1)

$p(\bm{\Sigma_{u}}|\bm{\Phi},\lambda,\bm{\theta},\gamma,\bm{Y})\sim\mathcal{IW}\left({\bm{\tilde{S}}},\tilde{\nu}\right)$

  2. 2.2)

$p(vec(\bm{\Phi})|\bm{\Sigma_{u}},\lambda,\bm{\theta},\gamma,\bm{Y})\sim\mathcal{N}(\bm{\tilde{\Phi}},\bm{\Sigma}_{u}\otimes\bm{\tilde{\Psi}})$

In this approach, the time varying parameters and the variance covariance
matrix are drawn in a single step from their Normal-Inverse-Wishart
conditional posterior distribution. It is worth to mention that in the case we
allow for regressor specific shrinkage hyper-parameters $\lambda_{j}$ for
$j=1,\ldots,k$ nothing changes conceptually, since the algorithm should just
be adapted to draw all the $\lambda_{j}$’s hyper-parameters together with
$\gamma$ in the second block of the random walk metropolis in Step 1.
222222Formulas for the marginal likelihood and the conditional posterior
distribution of $\bm{\Phi}$ and $\bm{\Sigma}_{u}$ can be found in the appendix
A.1.6. As a last note, to let the estimated path of time varying coefficients
$\bm{\Phi_{1}},\ldots,\bm{\Phi_{T}}$ and the estimate of the hyper-parameter
$\lambda$ less affected by the choice of the initial condition $\bm{\Phi_{0}}$
I assume that in equation (4)
$\bm{\eta_{1}}\sim\mathcal{N}(\bm{0},\bm{\Sigma_{u}}\otimes
5(\sum_{t=1}^{T_{pre}}\bm{x}_{t}\bm{x}_{t}^{\prime})^{-1})$ such that
$\bm{\Phi_{1}}\sim\mathcal{N}(\bm{\hat{\Phi}_{0}},\bm{\Sigma_{u}}\otimes
5(\sum_{t=1}^{T_{pre}}\bm{x}_{t}\bm{x}_{t}^{\prime})^{-1})$ where $T_{pre}$ is
the size of a pre-sample of observations while $\bm{\hat{\Phi}_{0}}=0$.

##  3 Forecasting with the TC-TVP-VAR

In this section I consider the problem of forecasting the rate of growth of
GDP, the inflation rate and the Fed Fund rate using a trivariate TVP-VAR
model. In the specific, I estimate a trivariate TVP-VAR for the US economy
using data from 1970 up to 2019 and I compare the forecast accuracy of a
standard TVP-VAR model to the forecasts from a TC-TVP-VAR.

###  3.1 Small scale New Keynesian model for the US economy

In the TC-TVP-VAR I exploit the New Keynesian model in [20] to parametrize the
theory coherent prior. The conceptual framework commonly denoted as the
3-equation New Keynesian model constitutes the nucleus of Michael Woodford’s
book “Interest and Prices” [40] and underpins most of modern monetary
macroeconomics models.232323See [20] for the more in depth details on the New
Keynesian Model. More specifically, the structural model is composed by an IS
curve (31), a New Keynesian Phillips curve (32), a monetary policy rule (33)
and two equations that describe the dynamics for the log-deviation from the
steady state of technological process (34) and government spending (35),
namely

$\hat{y_{t}}=E_{t}[\hat{y}_{t+1}]-\frac{1}{\tau}\left(\hat{R}_{t}-E_{t}[\hat{\pi}_{t+1}]\right)+(1-\rho_{g})\hat{g}_{t}+\frac{\rho_{z}}{\tau}\hat{z}_{t}\\\
$ (31)
$\hat{\pi}_{t}=\frac{\gamma}{r^{*}}E_{t}[\hat{\pi}_{t+1}]+\kappa(\hat{y}_{t}-\hat{g}_{t})\\\
$ (32)
$\hat{R}_{t}=\rho_{R}\hat{R}_{t-1}+(1-\rho_{R})(\psi_{1}\hat{\pi}_{t}+\psi_{2}\hat{y}_{t})+\varepsilon_{R,t}\hskip
19.91684pt\varepsilon_{R,t}\sim\mathcal{N}(0,\sigma_{R})\\\ $ (33)
$\hat{z}_{t}=\rho_{z}\hat{z}_{t-1}+\varepsilon_{z,t}\hskip
119.50148pt\varepsilon_{z,t}\sim\mathcal{N}(0,\sigma_{z})\\\ $ (34)
$\hat{g}_{t}=\rho_{g}\hat{g}_{t-1}+\varepsilon_{g}\hskip
128.0374pt\varepsilon_{g,t}\sim\mathcal{N}(0,\sigma_{g})\\\ $ (35)

The population moments needed to parametrize the prior are derived from the
state-space representation of the New Keynesian model obtained by solving the
system of non-linear rational expectation equations. In particular, the non-
linear rational expectation equations are solved using the method based on
matrix eigenvalue decomposition by [33] leading to a solution which has the
form

$\bm{s_{t}}=\mathcal{T}(\bm{\theta})\bm{s_{t-1}}+\mathcal{R}(\bm{\theta})\bm{\epsilon}_{t}$
(36)

that is complemented with the set of observation equations

$YGR_{t}=ln(\gamma)+\Delta\hat{y_{t}}+\hat{z}_{t}$ (37)
$INFL_{t}=ln(\pi^{*})+\hat{\pi}_{t}$ (38)
$ln(INT_{t})=4ln(r^{*})+4ln(\pi^{*})+4\hat{R}_{t}$ (39)

which relate the unobservable latent states in (36) to the observed time
series of output growth, inflation rate and the Fed Fund rate. Equations
(36),(37),(38),(39) are used to derive the population moments used to
parametrize the prior. The population moments $\bm{\Gamma_{xx}}(\bm{\theta})$,
$\bm{\Gamma_{xy}}(\bm{\theta})$, $\bm{\Gamma_{yy}}(\bm{\theta})$ are derived
conditioning on $\bm{\theta}$, the vector of the deep parameters of the NK
model

$\bm{\theta}=[ln(\gamma),ln(\pi^{*}),ln(r^{*}),\kappa,\tau,\psi_{1},\psi_{2},\rho_{R},\rho_{g},\rho_{z},\sigma_{R}^{2},\sigma_{g}^{2},\sigma_{z}^{2}]$
(40)

###  3.2 Forecast comparison

Table 1 shows the comparison of the forecasts from a standard TVP-VAR model
and the TC-TVP-VAR model. The forecasting exercise is designed such that I
compute the recursive one quarter, two quarters, and one year ahead forecasts
starting from 1985-Q1 up to 2019-Q4.242424Details on the data can be found in
the appendix A.2.1 To compare relative point forecast accuracy, in the table I
report the Root Mean Squared Error (RMSE) while for evaluating density
forecast accuracy I report the average Cumulative Ranked Probability Scores
(CRPS). In the table I also include the results concerning the forecasts from
a constant parameters VAR with flat prior and a constant parameters Bayesian
VAR (BVAR) with Minnesota type of prior.252525Appendix A.2.2 the reports
details on the competing forecasting models. Overall, the TC-TVP-VAR provides
the most accurate point and density forecasts for both output growth and
inflation rate, outperforming the standard TVP-VAR model at all the horizons
considered. For forecasting output growth, the standard TVP-VAR performs
poorly relative to the TC-TVP-VAR, but also to the constant parameters BVAR
with Minnesota prior, suggesting that the model tends to fit some noise in the
time series of output growth. In line with the previous forecasting
literature, allowing for time variation of the parameters of the VAR is
important for obtaining accurate forecasts of the inflation rate, as the
standard TVP-VAR outperforms both the VAR with flat prior and the BVAR with
the Minnesota prior. However, economic shrinkage is helpful to obtain more
reliable point and density forecasts when modelling the time variation of the
coefficients. Indeed the TC-TVP-VAR outperforms the standard TVP-VAR at all
the horizons. As a caveat, the standard Minnesota prior centering the
autoregressive coefficients on a random walk process outperforms the other
competitors, including the TC-TVP-VAR, for forecasting the Fed Fund Rate. This
result is consistent with the results of the forecasting exercise in [20]
which use the same small scale NK model to parametrize a prior for a constant
parameters VAR.

Table 1: Point and density forecast accuracy 1985Q1 - 2019Q4

| GDP growth | Inflation rate | Interest Rate
---|---|---|---
| RMSE | CRPS | RMSE | CRPS | RMSE | CRPS
a) One quarter ahead |  |  |  |  |  |
VAR flat prior | 0.4417 | 0.3703 | 0.3451 | 0.2877 | $\mathbf{0.0402}$ | 0.0954
B-VAR min | 0.3422 | 0.3584 | 0.2854 | 0.2637 | 0.0403 | $\mathbf{0.0936}$
Standard TVP-VAR | 0.4283 | 0.3853 | 0.3362 | 0.2682 | 0.0451 | 0.1096
TC-TVP-VAR | $\mathbf{0.3027}$ | $\mathbf{0.3485}$ | $\mathbf{0.2715}$ | 0.2461 | 0.0608 | 0.1312
b) Two quarters ahead |  |  |  |  |  |
VAR flat prior | 0.5045 | 0.3885 | 0.4771 | 0.3423 | 0.1384 | 0.1784
B-VAR min | 0.3990 | 0.3909 | 0.3819 | 0.3110 | $\mathbf{0.1303}$ | $\mathbf{0.1709}$
Standard TVP-VAR | 0.5131 | 0.4151 | 0.3362 | 0.2911 | 0.1548 | 0.2037
TC-TVP-VAR | $\mathbf{0.3340}$ | $\mathbf{0.3668}$ | $\mathbf{0.3183}$ | $\mathbf{0.2722}$ | 0.1689 | 0.2096
c) One year ahead |  |  |  |  |  |
VAR flat prior | 0.4278 | $\mathbf{0.3728}$ | 0.4965 | 0.3675 | 0.4475 | 0.3385
B-VAR min | 0.4037 | 0.4065 | 0.4099 | 0.3393 | $\mathbf{0.4142}$ | $\mathbf{0.3229}$
Standard TVP-VAR | 0.6472 | 0.4775 | 0.2855 | 0.3039 | 0.5665 | 0.3854
TC-TVP-VAR | $\mathbf{0.3617}$ | 0.3887 | $\mathbf{0.2758}$ | $\mathbf{0.2802}$ | 0.5156 | 0.3593

Notes: The Table reports the Root Mean Squared Error (RMSE) and the average

Cumulative Ranked Probability Scores (CRPS). In bold the best model according
to

each forecast metric.

##  4 Response analysis at the ZLB with the TC-TVP-VAR

TVP-VAR are extensively used in applied research not only to make forecasts,
but also to infer changes in the response of the economy to macroeconomic
shocks. In this section, I show that the proposed shrinkage prior can be
useful also to enhance inference on the impulse response functions estimated
from a TVP-VAR. Recent studies in empirical macroeconomics have used TVP-VARs
to assess whether the US economy’s performance was affected by a binding ZLB
constraint [18, 4]. As a matter of fact, according to a standard New-Keynesian
model, the economy is expected to exhibit different responses when the ZLB
constraint is in effect. For example, the model predicts a distinct response
of output and inflation following both demand and supply shocks when the
conventional stabilizing monetary policy response to aggregate shocks is
constrained as a consequence of the Federal Funds Rate hitting the ZLB. Figure
3 makes this point, by showing the responses to a pure demand shock
(preference shock) in the [35] model version considered in [19].262626In the
model, the ZLB period is treated as in [19], and more details on the model
will be explained in the next subsection. As expected, when monetary policy is
constrained by the policy rate hitting the zero lower bound, the response of
both output and inflation to the preference shock is magnified on impact and
the effect of the shock dies down much more slowly over time.

Figure 3: The two plots show the cumulative response of output growth (on the
left) and the response of the inflation rate (on the right) to a preference
shock inside and outside the ZLB period in the New Keynesian model.

Despite the sharp difference in the propagation of the shocks in the economy
inside and outside the zero lower bound period foreseen by a standard NK
model, empirical evidences investigating this issue are mixed and many studies
do not find substantial evidence supporting a different response of the US
economy during the zero lower bound period. For example, [18], support the
irrelevance hypothesis i.e. the hypothesis that the economy’s performance has
not been affected by a binding ZLB constraint, embracing the view that
unconventional monetary policy have been effective at getting around the zero
lower bound (ZLB) constraint. [4] recently showed that, given the short length
of the zero-lower-bound period, inference based on a standard TVP-VAR where
the time varying parameters are regarded as slow moving stochastic processes,
doesn’t allow to capture the changing relationship among the macroeconomic
variables during the ZLB period predicted by the New Keynesian model. In what
follows, based on a simulation study, I also find that a standard TVP-VAR
struggles to detect the change in the responses of the economy in the ZLB
period generated by a NK model. I show that the TC-TVP-VAR can in principle be
used to solve this inferential issue and therefore to recover the distinct
response of the economy during the zero lower bound period.

###  4.1 Medium scale NK model accounting for ZLB and forward guidance

The New Keynesian model is the version of the [35] model considered in [19].
The model features price and wage stickiness, investment adjustment costs,
habit formation in consumption and 7 shocks being monetary policy shocks,
technology shocks, price mark-up shocks, wage mark-up shocks, preference
shocks, fiscal policy shock and shocks to the marginal efficiency of capital.
In the model the monetary policy rule accounts for ZLB period and forward
guidance.272727The log-linearized equilibrium conditions of the model can be
found in the appendix A.3 The model is labelled ”SW” in [19] and assumes a
constant inflation target. More specifically, when accounting for the zero
lower bound and forward guidance the solution of the model implies a state
state space representation which exhibits time varying coefficients. The
solution method follows the approach developed by [7] for linear stochastic
rational expectations models in the face of a finite sequence of anticipated
structural changes. In particular, it is assumed that at a given period $t$,
agents expect the nominal interest rate to be at the ZLB for $\bar{H}$
periods. That is, the monetary policy rule

$R_{t}=\rho_{R}R_{t-1}+(1-\rho_{R})\left(\psi_{1}(\pi_{t}-\pi_{t}^{*})+\psi_{2}(y_{t}-y_{t}^{f})\right)+\psi_{3}\left((y_{t}-y_{t}^{f})-(y_{t-1}-y_{t-1}^{f})\right)+r_{t}^{m}$
(41)

becomes

$R_{\tau}=-R_{*}\hskip 28.45274pt$ (42)

for $\tau=t,\ldots,t+\bar{H}$ while it is determined by (41) for
$\tau>t+\bar{H}$. This implies that the equilibrium conditions

$\bm{\Gamma}_{2,\tau}E_{\tau}[\bm{{s}}_{\tau+1}]+\bm{\Gamma}_{0,\tau}\bm{{s}}_{\tau}=\bm{\Gamma}_{c,\tau}+\bm{\Gamma}_{1,\tau}\bm{{s}}_{\tau-1}+\bm{\Psi_{\tau}}\bm{\varepsilon}_{\tau}$
(43)

differ over time depending on whether $\tau\leq t+\bar{H}$, with the row
corresponding to the policy rule differing across $\tau$. Following [7] the
solution for the linear rational expectations model under anticipated
structural variations takes the form

$\bm{s_{\tau}}=\mathcal{C}_{\tau}^{(\tau,\bar{H})}+\mathcal{T}_{\tau}^{(\tau,\bar{H})}\bm{s_{\tau-1}}+\mathcal{R}_{\tau}^{(\tau,\bar{H})}\bm{\epsilon_{\tau}}$
(44)

where $\mathcal{C}_{\tau}^{(\tau,\bar{H})}$
$\mathcal{T}_{\tau}^{(\tau,\bar{H})}$ and
$\mathcal{R}_{\tau}^{(\tau,\bar{H})}$ are computed by recursion

$\mathcal{C}_{\tau}^{(\tau,\bar{H})}=(\bm{\Gamma}_{2,\tau}\mathcal{T}_{\tau+1}^{(\tau,\bar{H})}+\bm{\Gamma}_{0,\tau})^{-1}(\bm{\Gamma}_{c,\tau}-\bm{\Gamma}_{2,\tau}\mathcal{C}_{\tau+1}^{(\tau,\bar{H})})$
(45)
$\mathcal{T}_{\tau}^{(\tau,\bar{H})}=(\bm{\Gamma}_{2,\tau}\mathcal{T}_{\tau+1}^{(\tau,\bar{H})}+\bm{\Gamma}_{0,\tau})^{-1}\bm{\Gamma}_{1,\tau}$
(46)
$\mathcal{R}_{\tau}^{(\tau,\bar{H})}=(\bm{\Gamma}_{2,\tau}\mathcal{T}_{\tau+1}^{(\tau,\bar{H})}+\bm{\Gamma}_{0,\tau})^{-1}\bm{\Psi_{\tau}}$
(47)

starting from $\mathcal{T}_{t+\bar{H}+1}^{(\tau,\bar{H})}=\mathcal{T}$ ,
$\mathcal{C}_{t+1+\bar{H}}^{(\tau,\bar{H})}=0$ and where the superscript
$(t,\bar{H})$ is used to indicate that the solution is obtained under the
assumption that the announcement of zero interest rates for a duration of
$\bar{H}$ periods was made in period $t$. 282828Following [19] to measure the
number of quarters $\bar{H}$ that the Federal Funds Rate is expected to remain
at the ZLB I exploit information based on the overnight index swap (OIS)
rates. In particular I identify the ZLB period as the quarters in which the
OIS rate is lower then 0.35. This classification leads to the same ZLB period
considered in [4] and [18] namely 2009Q1-2015Q3 (28 quarters). Following [12]
I assume that the number of quarters such that the policy rate is expected to
stay fixed, is at most equal to four. The matrices
($\mathcal{C}_{t}^{(\tau,\bar{H})},\mathcal{T}_{t}^{(\tau,\bar{H})},\mathcal{R}_{t}^{(\tau,\bar{H})})$
parameterize the transition equation of time varying coefficients state space
model

$\bm{y}_{t}=\mathcal{D}+\mathcal{B}\bm{s_{t}}+\bm{v}_{t}$ (48)
$\bm{s}_{t}=\mathcal{C}_{t}+\mathcal{T}_{t}\bm{s_{t-1}}+\mathcal{R}_{t}\bm{\epsilon}_{t}$
(49)

where the constant matrices $\mathcal{D}$ $\mathcal{B}$ and time varying
matrices $\mathcal{C}_{t}$, $\mathcal{T}_{t}$, $\mathcal{R}_{t}$ depend on the
structural parameters $\bm{\theta}$ while $\bm{v}_{t}$ is a measurement error.
292929Details for the observation equations are in the appendix A.3 together
with the list of the deep structural parameters $\bm{\theta}$ (appendix
LABEL:secpriors_dsge_medium).

###  4.2 Simulation study

As anticipated above and shown in figure 3, the NK model predicts a distinct
response of the economy to a demand shock inside and outside the ZLB period.
Conditioning on the vector of deep parameters of the NK model, I simulate data
from the state space representation (48) and (49). 303030Parameters are
calibrated according to the posterior mode in [19] In the simulation I
consider artificial samples with $T=139$ mimicking quarterly observations for
the period 1985Q1-2019Q3. The length of the ZLB period is 28 quarters,
covering the period 2009Q1-2015Q3. I estimate a standard TVP-VAR model [11] on
the simulated data to understand whether the model is able to recover the
change in the response of the economy during the ZLB period generated by the
NK model.313131The standard TVP-VAR model is the model in [11]. To estimate
the model I use the MATLAB codes kindly made available by Joshua Chan on his
personal website. Figure 4 shows the estimated responses of output and
inflation to a preference shock obtained from a standard TVP-VAR. The figure
plots the responses of output growth (cumulative) and the inflation rate to a
one standard deviation preference shock in two reference dates, one outside
and the other one inside the ZLB period. In order to identify the shocks in
the structural TVP-VAR I exploit the true impact matrix of the NK model given
by

$\frac{\partial\bm{y_{t}}}{\partial\bm{\epsilon}^{\prime}_{t}}=\underbrace{\mathcal{B}(\bm{\theta})\bm{\mathcal{R}_{t}}(\bm{\theta})\bm{\Omega}(\bm{\theta})}_{\text{$\bm{A_{0,t}}(\bm{\theta})^{NK}$}}$
(50)

where
$\bm{\Omega}(\bm{\theta})=diag(\sigma_{g}^{2},\sigma_{b}^{2},\sigma_{\mu}^{2},\sigma_{z}^{2},\sigma_{\lambda_{f}}^{2},\sigma_{\lambda_{w}}^{2},\sigma_{r}^{2})$
is the diagonal matrix containing the volatility of the structural shocks. As
figure 4 shows, despite conditioning on the correct identification scheme to
identify the structural shocks, inference based on a standard TVP-VAR
struggles to provide convincing evidences supporting a distinct response of
both output growth and the inflation rate to a preference shock inside and
outside the ZLB period. As a matter of fact, the estimates of the impulse
responses are so imprecise that the model doesn’t allow to detect any change
in the response of the economy inside and outside the ZLB. 323232As shown in
the appendix in figure 7, this result is not driven by this specific simulated
sample of artificial observations.

Figure 4: Estimated $90^{th}-10^{th}$ credible sets of the cumulative response
of output growth (on the left) and of the response of the inflation rate (on
the right) to a preference shock inside (shaded grey) and outside (shaded red)
the ZLB period by a standard TVP-VAR on data simulated from the NK model.

Figure 5, instead, shows the responses obtained by estimating a TC-TVP-VAR
that exploits the NK model as a prior for the time varying coefficients. The
moment matrices $\bm{\Gamma_{xx}}$, $\bm{\Gamma_{xy}}$, $\bm{\Gamma_{yy}}$
used to parametrize the Normal-Inverse-Wishart prior are derived from the
equations of the state representation (48) and (49). Clearly, due to the time
variation of the coefficients in the state space representation, the moments
of the structural model are time varying with
$\operatorname*{\mathbb{E}}[\bm{x_{t}x_{t}^{\prime}}|\bm{\theta}]$,
$\operatorname*{\mathbb{E}}[\bm{x_{t}y_{t}^{\prime}}|\bm{\theta}]$ and
$\operatorname*{\mathbb{E}}[\bm{y_{t}y_{t}^{\prime}}|\bm{\theta}]$ changing
over time. This, in turns, implies time varying restriction functions for the
coefficients of the TVP-VAR encoded in
$\bm{\Phi(\theta)^{*}}=\bm{\Gamma_{xx}(\theta)^{-1}\Gamma_{xy}(\theta)}$.
Since the time varying restrictions incorporate the change in the response of
the economy foreseen by the NK model, the responses of output growth and the
inflation rate to the preference shock are more precisely estimated as the
time varying coefficients are shrinked towards these restrictions. This is
shown in figure 5, which plots the responses of output growth and of the
inflation rate to a preference shock for different fixed values of $\gamma$,
that is for a different amount of shrinkage towards the restrictions implied
by the NK model.333333Also in this case, in the TC-TVP-VAR we condition on the
correct identification scheme for the structural shocks. As the figure shows,
a sufficiently high $\gamma$ is needed to detect a precise and distinct
response of the economy inside and outside the ZLB period as predicted by the
NK model. As $\gamma$ increases and gets big enough, we get more and more
precise estimates of the time varying coefficients. This allows to detect the
different propagation of the shocks inside and outside the ZLB period, as
predicted by the New Keynesian model. Letting $\gamma\rightarrow\infty$ the
estimated model is a restricted TVP-VAR in which the time varying coefficients
exactly satisfy
$\bm{\Phi(\theta)^{*}}=\bm{\Gamma_{xx}(\theta)^{-1}\Gamma_{xy}(\theta)}$. As
expected, the estimated responses from this model almost exactly resemble the
responses from the NK model (panel (d)).

(a)

(b)

(c)

(d)

Figure 5: Estimated $10^{th}-90^{th}$ credible sets of the cumulative
responses of output growth (left) and the response of the inflation rate
(right) to a preference shock in the TC-TVP-VAR fixing the hyper-parameter
$\gamma$. The figures reports also the true responses. Data are simulated from
the NK model. Horizons are in quarters. a) $\gamma=10$ b) $\gamma=100$ c)
$\gamma=300$ d) $\gamma=\infty$.

###  4.3 Estimating the TC-TVP-VAR on US data

When analyzing real data, it is reasonable to think of the NK model just as an
approximate (most likely misspecified) tightly parameterized representation of
the true data generating process. In other words, we do not expect the
restrictions implied by the NK model to hold exactly. However, when encoded
into a prior, these restrictions might prove to be useful to get more precise
estimates of the time varying coefficients and of nonlinear functions of these
coefficients, such as the impulse response functions. In particular, in our
context, shrinkage might turn out to be particularly useful to detect the
change in the response of the economy during the ZLB period predicted by the
NK model. In order to analyze to what extent the data support the changing
behavior during the ZLB, my approach consists in exploiting the New Keynesian
model as a prior for the TC-TVP-VAR. The extent to which the predictions from
the New Keynesian model will be supported by the data will depend on the
estimated posterior distribution of $\gamma$, the hyper-parameter governing
the degree of shrinkage of the parameters towards the restriction implied by
the New Keynesian model. If the posterior distribution of $\gamma$ is
concentrated relatively far from zero, the coefficients of the TVP-VAR will be
shrinked towards the restrictions implied by the NK model and the estimated
responses from the TC-TVP-VAR will resemble the predictions of the NK model.
This would happen in practice if the restrictions from the NK model find
enough support on the data. Conversely, if the restrictions from the NK are
deemed implausible by the data, the posterior distribution of $\gamma$ will be
concentrated around zero and the estimates of the time varying coefficients
will not reflect the restrictions coming from the prior. The estimated model
is a 7-variable TC-TVP-VAR for the US economy including output growth,
consumption growth, investment growth, real wage growth, hours worked,
inflation and the Fed Fund rate and it is estimated over the sample
1985Q1-2019Q3.343434Details on the variables and their transformation is
available in the appendix A.3. In order to identify the shocks in the
structural TVP-VAR, I exploit the impact matrix of the NK model. The deep
parameters of the NK model are treated as unknown and estimated along with the
other parameters of the model, which implies that on impact there is
uncertainty on the effect of the shocks on the variables of the system. Figure
6 shows the estimated responses from the TC-TVP-VAR where the hyper-parameter
$\gamma$ is estimated along with the other parameters of the model. The
posterior distribution of $\gamma$ is estimated to be concentrated far from
zero, meaning that the time varying restriction functions from the NK model
find sufficient support on the data. This, in turns, is reflected on the
estimates of the $10^{th}-90^{th}$ credible sets which provide some evidences
supporting a distinct responses of the economy inside and outside the ZLB
period. As predicted by the NK model, the estimates suggest that when monetary
policy is constrained by the nominal rate hitting the ZLB, the preference
shock is reabsorbed at a much slower pace by the economy. Following the
positive demand shock both output growth and the inflation rate increase, with
the effect on output growth being more precisely estimated at shorter
horizons. The effect on output peaks after almost two quarters outside the ZLB
period, while about after five quarters inside the ZLB period. As for
inflation, the effect peaks after one quarter outside the ZLB period and after
two quarters inside the ZLB period. After the peak, the effect of the
preference shock on both output and inflation dies at a much slower pace
inside the ZLB period, as foreseen by the NK model.

Figure 6:  Estimated $90^{th}-10^{th}$ credible sets of the cumulative
response of output growth (on the left) and of the response of the inflation
rate (on the right) to a preference shock inside (shaded grey) and outside
(shaded red) the ZLB period from the TC-TVP-VAR on US data.

##  5 Conclusion

TVP-VAR are flexible statistical models used both for prediction and policy
analysis in macroeconomics. Despite their flexibility allows to capture
changes in the dynamic relationship among the macroeconomic variables, these
models can easily become too flexible with the risk of over-fitting the data.
This translate into poor forecasting performances and imprecise inference on
the time-varying parameters and of typical objects of interests such as the
impulse responses. On the other side, models from the economic theory
typically provide a more tightly parameterized representation of the
macroeconomy and therefore have the opposite tendency of fitting the data
rather poorly. This paper exploits the restrictions implied by economic theory
to formulate a prior for the parameters of TVP-VARs so as to enhance inference
within this class of models. It does so by introducing a shrinkage prior that
centers the time varying coefficients at each time period on the cross
equation restrictions implied by an underlying economic theory about the
variables in the system. The paper shows that “economic shrinkage” can be
successfully used to obtain more accurate forecasts and more precise estimates
of typical objects of interests such as the impulse responses.

Future research For future research the prior proposed in this paper could be
modified to accommodate for multiple competing theories à la [30]. As well,
the model could be extended to embed stochastic volatility, adapting the
algorithm in [6]. More in general, restrictions from economic theory could
successfully be used to reduce overfitting and sharpen inference in non-
parametric VARs and Gaussian Procesess VARs [27].

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##  Appendix A Appendix

###  A.1 Theory Coherent TVP-VAR

####  A.1.1 Time Varying Parameters by dummy observations

Starting from:

$\underbrace{\bm{y}^{\prime}_{t}}_{\text{$1\times
N$}}=\underbrace{\bm{x}^{\prime}_{t}}_{\text{$1\times
k$}}\underbrace{\bm{\Phi}_{t}}_{\text{$k\times
N$}}+\underbrace{\bm{u_{t}}}_{\text{$1\times N$}}\hskip
85.35826pt\bm{u_{t}}\sim\mathcal{N}(\bm{0}_{1\times N},\bm{\Sigma_{u}})$

we can write the TVP-VAR in static compact form as:

$\underbrace{\bm{Y}}_{\text{$T\times N$}}=\underbrace{\bm{X}}_{\text{$T\times
Tk$}}\underbrace{\bm{\Phi}}_{\text{$Tk\times
N$}}+\underbrace{\bm{U}}_{\text{$T\times N$}}\hskip 85.35826pt\bm{U}\sim
MVN(\bm{0},\bm{I_{T}},\bm{\Sigma_{u}})$

Suppose we want to specify independent RW stochastic processes for all the
coefficients in $\bm{\Phi}$ as:

$\phi_{jt}^{(i)}=\phi_{jt-1}^{(i)}+\eta_{jt}^{(i)}\hskip
56.9055pt\eta_{j}^{(i)}\sim\mathcal{N}\left(0,\frac{\Sigma_{ii}}{\lambda_{j}^{2}}\right)$
(51)

for $t=1,\ldots,T$ $i=1,\ldots,N$ and $j=1,\ldots,k$. Defining
$\bm{\Lambda_{k}}=diag(\lambda_{1},\ldots,\lambda_{k})$ we can use a set of
dummy observations to model the time variation of the coefficients. The dummy
observations imply the linear fuzzy restrictions in (51) on the coefficients
in $\bm{\Phi}$, in the specific we can write:

$\underbrace{\begin{bmatrix}\bm{\Lambda}_{k}\bm{\Phi_{0}}\\\ \bm{0}\\\
\bm{0}\\\ \vdots\\\ \bm{0}\\\
\end{bmatrix}}_{\text{$\bm{Y^{*}}$}}=\underbrace{\begin{bmatrix}\bm{\Lambda_{k}}&\bm{0}&\ldots&\bm{0}\\\
0&\bm{\Lambda_{k}}&\ldots&\bm{0}\\\ \bm{0}&0&\ldots&\bm{0}\\\
\vdots&\vdots&\vdots&\vdots\\\ \bm{0}&\bm{0}&0&\bm{\Lambda_{k}}\\\
\end{bmatrix}\begin{bmatrix}\bm{I}_{k}&\bm{0}&\ldots&\bm{0}\\\
-\bm{I}_{k}&\bm{I}_{k}&\ldots&\bm{0}\\\ \bm{0}&-\bm{I}_{k}&\ldots&\bm{0}\\\
\vdots&\vdots&\vdots&\vdots\\\ \bm{0}&\bm{0}&-\bm{I}_{k}&\bm{I}_{k}\\\
\end{bmatrix}}_{\text{$\bm{X}^{*}$}}\begin{bmatrix}\bm{\Phi_{1}}\\\
\bm{\Phi_{2}}\\\ \bm{\Phi_{3}}\\\ \vdots\\\ \bm{\Phi_{T}}\\\
\end{bmatrix}+\underbrace{\begin{bmatrix}\bm{u}^{*}_{1}\\\ \bm{u}^{*}_{2}\\\
\bm{u}^{*}_{3}\\\ \vdots\\\ \bm{u}^{*}_{T}\\\
\end{bmatrix}}_{\text{$\bm{U}^{*}$}}$ (52)

$\bm{Y}^{*}=\bm{X}^{*}\bm{\Phi}+\bm{U}^{*}$ (53)

This is just another way of writing:

$\underbrace{\begin{bmatrix}\bm{I}_{k}&\bm{0}&\ldots&\bm{0}\\\
-\bm{I}_{k}&\bm{I}_{k}&\ldots&\bm{0}\\\ \bm{0}&-\bm{I}_{k}&\ldots&\bm{0}\\\
\vdots&\vdots&\vdots&\vdots\\\ \bm{0}&\bm{0}&-\bm{I}_{k}&\bm{I}_{k}\\\
\end{bmatrix}}_{\text{$\bm{H_{Tk}}$}}\underbrace{\begin{bmatrix}\bm{\Phi_{1}}\\\
\bm{\Phi_{2}}\\\ \bm{\Phi_{3}}\\\ \vdots\\\ \bm{\Phi_{T}}\\\
\end{bmatrix}}_{\text{$\bm{\Phi}$}}=\underbrace{\begin{bmatrix}\bm{\Phi_{0}}\\\
\bm{0}\\\ \bm{0}\\\ \vdots\\\ \bm{0}\\\
\end{bmatrix}}_{\text{$\bm{\Phi_{00}}$}}-\underbrace{\begin{bmatrix}\bm{\Lambda_{k}^{-1}}&\bm{0}&\ldots&\bm{0}\\\
0&\bm{\Lambda_{k}^{-1}}&\ldots&\bm{0}\\\ \bm{0}&0&\ldots&\bm{0}\\\
\vdots&\vdots&\vdots&\vdots\\\ \bm{0}&\bm{0}&0&\bm{\Lambda_{k}^{-1}}\\\
\end{bmatrix}}_{\text{$\bm{\Lambda}^{-1}$}}\begin{bmatrix}\bm{u}^{*}_{1}\\\
\bm{u}^{*}_{2}\\\ \bm{u}^{*}_{3}\\\ \vdots\\\ \bm{u}^{*}_{T}\\\ \end{bmatrix}$
(54)

Since the generic $t^{th}$ block of dimension $k\times N$ reads as follows :

$\bm{\Phi_{t}}=\bm{\Phi}_{t-1}+\bm{\Lambda_{k}}^{-1}\bm{u_{t}}^{*}\hskip
56.9055ptvec(\bm{u}_{t})\sim\mathcal{N}(0,\bm{\Sigma_{u}}\otimes\bm{I}_{k})$
(55)

defining $\bm{\eta_{t}}=\bm{\Lambda_{k}}^{-1}\bm{u_{t}}$ we obtain:

$vec(\bm{\eta_{t}})\sim\mathcal{N}(0,\bm{\Sigma_{u}}\otimes(\bm{\Lambda_{k}^{\prime}\Lambda_{k})}^{-1})$
(56)

that is equivalent to the dynamic linear model :

$\underbrace{\bm{y}^{\prime}_{t}}_{\text{$1\times
N$}}=\underbrace{\bm{x}^{\prime}_{t}}_{\text{$1\times
k$}}\underbrace{\bm{\Phi}_{t}}_{\text{$k\times
N$}}+\underbrace{\bm{u_{t}}^{\prime}}_{\text{$1\times N$}}\hskip
85.35826pt\bm{u_{t}}\sim\mathcal{N}(\bm{0}_{N\times 1},\bm{\Sigma_{u}})$ (57)

$vec(\bm{\Phi_{t}})=vec(\bm{\Phi_{t-1}})+\bm{\eta_{t}}\hskip
56.9055pt\bm{\eta_{t}}\sim\mathcal{N}(0,\bm{\Sigma}\otimes(\bm{\Lambda_{k}}^{\prime}\bm{\Lambda_{k}})^{-1})$
(58)

where $\bm{\Omega}=(\bm{\Lambda_{k}^{\prime}\Lambda_{k})}^{-1}$

####  A.1.2 Population moments

The likelihood of the observations simulated from the model of the economic
theory $\bm{Y(\theta)}$ is:

$\displaystyle p(\bm{Y(\theta)}|\bm{\Phi},\bm{\Sigma_{u}})$
$\displaystyle=(2\pi)^{-\frac{TN}{2}}|\bm{\Sigma_{u}}|^{-\frac{T}{2}}exp\left[-\frac{1}{2}tr\left((\bm{\Sigma_{u}}^{-1})(\bm{Y(\theta)^{\prime}Y(\theta)}-\bm{\Phi^{\prime}X(\theta)^{\prime}Y(\theta))}-\bm{Y(\theta)^{\prime}X(\theta)\Phi}+\bm{\Phi^{\prime}X(\theta)^{\prime}X(\theta)\Phi})\right)\right]$
(59)

Considering $\gamma$ replications of $Y(\bm{\theta})$ we get

$\displaystyle p(\bm{Y(\theta)}|\bm{\Phi},\bm{\Sigma_{u}})$
$\displaystyle=(2\pi)^{-\frac{\gamma TN}{2}}|\bm{\Sigma_{u}}|^{-\frac{\gamma
T}{2}}exp\left[-\frac{1}{2}tr\left((\bm{\Sigma_{u}}^{-1})(\bm{\gamma
Y(\theta)^{\prime}Y(\theta)}-\gamma\bm{\Phi^{\prime}X(\theta)^{\prime}Y(\theta))}-\gamma\bm{Y(\theta)^{\prime}X(\theta)\Phi}+\gamma\bm{\Phi^{\prime}X(\theta)^{\prime}X(\theta)\Phi})\right)\right]$
(60)

Next, following the approach of [20] we replace the sample moments
$\bm{X(\theta)^{\prime}X(\theta)}$, $\bm{X(\theta)^{\prime}Y(\theta)}$,
$\bm{Y(\theta)^{\prime}Y(\theta)}$ by their expected values. Taking
expectations conditionally on $\bm{\theta}$ we define:

$\bm{\Gamma_{xx}(\theta)}\equiv\operatorname*{\mathbb{E}}[\bm{X^{\prime}X}|\bm{\theta}]=\begin{bmatrix}\bm{\Gamma_{xx,1}(\theta)}&\bm{0}_{k}&\ldots&\bm{0}_{k}\\\
\bm{0}_{k}&\bm{\Gamma_{xx,2}(\theta)}&\ldots&\bm{0}_{k}\\\
\vdots&\bm{0}_{k}&\ddots&\bm{0}_{k}\\\
\bm{0}_{k}&\ldots&\bm{0}_{k}&\bm{\Gamma_{xx,T}(\theta)}\\\ \end{bmatrix}$ (61)
$\bm{\Gamma_{xy}(\theta)}\equiv\operatorname*{\mathbb{E}}[\bm{X^{\prime}X}|\bm{\theta}]=\begin{bmatrix}\bm{\Gamma_{xy,1}(\theta)}\\\
\bm{\Gamma_{xy,2}(\theta)}\\\ \ldots\\\
\bm{\Gamma_{xy,T}(\theta)}\end{bmatrix}$ (62)
$\operatorname*{\mathbb{E}}[\bm{Y^{\prime}Y}|\bm{\theta}]\equiv\bm{\Gamma_{yy}(\theta)}=\sum_{t=1}^{T}\bm{\Gamma_{yy,t}(\theta)}$
(63)

where
$\bm{\Gamma_{xx,t}}=\operatorname*{\mathbb{E}}[\bm{x_{t}x_{t}^{\prime}}|\bm{\theta}]$,
$\bm{\Gamma_{xy,t}}=\operatorname*{\mathbb{E}}[\bm{x_{t}y_{t}^{\prime}}|\bm{\theta}]$
and
$\bm{\Gamma_{yy,t}}\equiv\operatorname*{\mathbb{E}}[\bm{y_{t}y_{t}^{\prime}}|\bm{\theta}]$
for $t=1,\ldots,T$. Substituting in the likelihood we get

$\displaystyle p(\bm{Y(\theta)}^{*}|\gamma,\bm{\Phi},\bm{\Sigma_{u}})$
$\displaystyle=(2\pi)^{-\frac{\gamma TN}{2}}|\bm{\Sigma_{u}}|^{-\frac{\gamma
T}{2}}exp\left[-\frac{1}{2}tr\left((\bm{\Sigma_{u}}^{-1})(\gamma\bm{\Gamma_{yy}(\theta)}-\gamma\bm{\Phi^{\prime}\Gamma_{xy}(\theta))}-\gamma\bm{\Gamma_{yx}(\theta)\Phi}+\gamma\bm{\Phi^{\prime}\Gamma_{xx}(\theta)\Phi})\right)\right]$
(64)

####  A.1.3 Integrating constant of the theory coherent prior

The integrating constant of the Normal-Inverse-Wishart prior

$c(\lambda,\bm{\theta},\gamma)=\int_{-\infty}^{\infty}p(\bm{\Phi,\Sigma_{u}}|\lambda)p(\bm{Y(\theta)}|\gamma,\bm{\Phi},\bm{\Sigma_{u}})d\bm{\Phi}d\bm{\Sigma_{u}}$
(65)

is given by:

$c(\bm{\lambda,\theta},\gamma)=\left(\pi\right)^{-\frac{\gamma
N}{2}}\frac{\Gamma_{N}\left(\frac{\underline{\underline{\nu}}}{2}\right)|\underline{\underline{S}}|^{\frac{-\underline{\underline{\nu}}}{2}}|\underline{\underline{\bm{\Psi}}}|^{\frac{N}{2}}}{\Gamma_{N}\left(\frac{\underline{\nu}}{2}\right)|\underline{S}|^{-\frac{\underline{\nu}}{2}}|\bm{\underline{\Psi}}|^{\frac{N}{2}}}=\left(\pi\right)^{-\frac{\gamma
N}{2}}\frac{\Gamma_{N}\left(\frac{\underline{\underline{\nu}}}{2}\right)|\underline{\underline{S}}|^{\frac{-\underline{\underline{\nu}}}{2}}|\underline{\underline{\bm{\Psi}}}|^{\frac{N}{2}}}{\Gamma_{N}\left(\frac{\underline{\nu}}{2}\right)|\underline{S}|^{-\frac{\underline{\nu}}{2}}|\bm{\underline{\Psi}}|^{\frac{N}{2}}}$
(66)

where the definitions for $\underline{\underline{\bm{S}}}$,
$\bm{\underline{\underline{\Psi}}}$,
$\bm{\underline{\underline{\Phi}}}$,$\bm{\underline{\Psi}}$,
$\underline{\bm{S}}$, $\bm{\underline{\Psi}}$, are given above in the text.

####  A.1.4 Conditional distribution of theory coherent prior

The joint normal prior (6) for $\bm{\Phi_{1}},\ldots,\bm{\Phi_{T}}$ implies
that the conditional prior $p(vec(\bm{\Phi_{t}})|vec(\bm{\Phi_{t-1}}))$ is
still normal and that the prior for $\bm{\Phi}_{t}$ is still a Markov process.
Defining $\bm{\Psi}_{i,j}$ the (i,j) block of the variance covariance matrix
$\underline{\underline{\bm{\Psi}}}$, from standard results for the conditional
distribution of jointly normal random variables we have:

$\small
p(vec(\bm{\Phi_{t}})|vec(\bm{\Phi_{t-1}}))\sim\mathcal{N}\left(\bm{\mu}_{t}+\bm{\Psi}_{t,t-1}(\bm{\Psi}_{t-1,t-1})^{-1}(\bm{\Phi_{t-1}}-\bm{\mu}_{t-1}),\bm{\Sigma_{u}}\otimes(\bm{\Psi}_{t,t}-\bm{\Psi}_{t,t-1}(\bm{\Psi}_{t-1,t-1})^{-1}\bm{\Psi}_{t-1,t})\right)$
(67)

where $\bm{\mu}_{t}$ is the vectorization of the $t^{th}$ block of dimension
$k\times N$ of

$\bm{\underline{\underline{\Phi}}}=\left(\gamma\bm{\Gamma_{xx}(\theta)}+\lambda^{2}\bm{H_{Tk}^{\prime}H_{Tk}}\right)^{-1}(\gamma\bm{\Gamma_{xy}(\theta)}+\lambda^{2}\bm{H}_{Tk}^{\prime}\bm{H}_{Tk}\bm{\underline{\Phi_{0}}})$
(68)

From inspection of the precision matrix
$\bm{P}=\underline{\underline{\bm{\Psi}}}^{-1}$ it is also easy to see that
the theory coherent prior implies that $p(vec(\bm{\Phi_{t}}))$ is still a
Markov process. As a matter of fact the precision matrix

$\bm{P}=\begin{bmatrix}\gamma\bm{\Gamma_{xx,1}(\theta)}+2\bm{\Lambda}_{k}&-\bm{\Lambda}_{k}&\ldots&\bm{0}_{k}\\\
-\bm{\Lambda}_{k}&\gamma\bm{\Gamma_{xx,2}(\theta)}+2\bm{\Lambda}_{k}&\ldots&\bm{0}_{k}\\\
\vdots&-\bm{\Lambda}_{k}&\ddots&-\bm{\Lambda}_{k}\\\
\bm{0}_{k}&\ldots&-\bm{\Lambda}_{k}&\gamma\bm{\Gamma_{xx,T}(\theta)}+\bm{\Lambda}_{k}\\\
\end{bmatrix}$ (69)

is block tri-diagonal matrix, meaning that

$p(vec(\bm{\Phi_{t}})|vec(\bm{\Phi_{t-1}}),vec(\bm{\Phi_{t-2}}),\ldots,vec(\bm{\Phi_{1}}))=p(vec(\bm{\Phi_{t}})|vec(\bm{\Phi_{t-1}}))$
(70)

To derive the conditional distribution (70) as a function of the elements of
the precision matrix $\bm{\underline{\underline{\Psi}}}^{-1}$ and the mean
$\underline{\underline{\bm{\Phi}}}$ consider the Cholesky root $\mathbf{L}$
such that $\mathbf{P}=\mathbf{L}\mathbf{L}^{\top}$. The matrix $\mathbf{L}$ is
block tri-diagonal, with blocks of dimension $k\times k$

$\mathbf{L}=\begin{bmatrix}\mathbf{L}_{1,1}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\\
\mathbf{L}_{2,1}&\mathbf{L}_{2,2}&\mathbf{0}&\ddots&\ddots&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&\ddots&\ddots&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\mathbf{0}&\mathbf{L}_{T-1,T-1}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{L}_{T,T-1}&\mathbf{L}_{T,T}\end{bmatrix}.$
(71)

Considering the three first blocks we get

$\displaystyle\mathbf{L}_{1,1}\mathbf{L}_{1,1}^{\top}$
$\displaystyle=\mathbf{P}_{1,1},$ (72)
$\displaystyle\mathbf{L}_{2,1}\mathbf{L}_{1,1}^{\top}$
$\displaystyle=\mathbf{P}_{2,1},$
$\displaystyle\mathbf{L}_{2,1}\mathbf{L}_{2,1}^{\top}+\mathbf{L}_{2,2}\mathbf{L}_{2,2}^{\top}$
$\displaystyle=\mathbf{P}_{2,2}.$

We can notice that $\mathbf{L}_{1,1}$ is the Cholesky root of
$\mathbf{P}_{1,1}$, and that
$\mathbf{L}_{2,1}=\mathbf{P}_{2,1}\mathbf{L}_{1,1}^{-\top}$ while
$\mathbf{L}_{2,2}$ is the Cholesky root of
$\mathbf{P}_{2,2}-\mathbf{L}_{2,1}\mathbf{L}_{2,1}^{\top}$. Therefore more in
general

$\left\\{\begin{aligned}
\mathbf{L}_{t,t-1}&=\mathbf{P}_{t,t-1}\mathbf{L}_{t-1,t-1}^{-\top},\\\
\mathbf{L}_{t,t}&=chol(\mathbf{P}_{t,t}-\mathbf{L}_{t,t-1}\mathbf{L}_{t,t-1}^{\top})\end{aligned}\right.\qquad
t=2,\dots,\,T$ (73)

Now define $\bm{E}=\mathbf{L}^{\top}(\bm{\phi_{.}}-\bm{\mu_{.}})$ with block
components $\bm{\varepsilon}_{t}$, where $\bm{\phi_{.}}$ is a generic column
of $\bm{\Phi}$ and $\bm{\mu}_{.}$ is a generic column of the $Tk\times N$
matrix of means $\bm{\underline{\underline{\Phi}}}$. Then

$Cov(\bm{E})=\mathbf{L}^{\top}Cov(\bm{\phi_{.}}-\bm{\mu_{.}})\mathbf{L}=\mathbf{L}^{\top}\mathbf{P}^{-1}\mathbf{L}=\mathbf{L}^{\top}\mathbf{L}^{-\top}\mathbf{L}^{-1}\mathbf{L}=\mathbf{I}_{Tk}$
(74)

which implies that the block components $\bm{\varepsilon}_{t}$ are
independent. By back substitution

$\displaystyle\bm{\varepsilon}_{T}$
$\displaystyle=\mathbf{L}_{T,T}^{\top}(\bm{\phi_{.T}}-\bm{\mu}_{.T})$ (75)
$\displaystyle\bm{\varepsilon}_{t}$
$\displaystyle=\mathbf{L}_{t,t}^{\top}(\bm{\phi_{.t}}-\bm{\mu}_{.t})+\mathbf{L}_{t+1,t}^{\top}(\bm{\phi_{.t+1}}-\bm{\mu}_{.t+1})\qquad
t=T-1,\,T-2,\,\dots,\,1.$

Hence we get an auto-regression

$(\bm{\phi_{.t+1}}-\bm{\mu}_{.t+1})=\mathbf{L}_{t+1,t}^{-\top}\mathbf{L}_{t,t}^{\top}(\bm{\phi_{.t}}-\bm{\mu}_{.t})+\mathbf{L}_{t+1,t}^{-\top}\bm{\varepsilon}_{t},\qquad
t=T-1,\,T-2,\,\dots,\,1$ (76)

$(\bm{\phi_{.t+1}}-\bm{\mu}_{.t+1})=\bm{A}_{t+1}(\bm{\phi_{.t}}-\bm{\mu}_{.t})+\mathbf{V}_{t+1}\bm{\varepsilon}_{t},\qquad
t=T-1,\,T-2,\,\dots,\,1$ (77)

with $\bm{A}_{t+1}:=-\mathbf{L}_{t+1,t}^{-\top}\mathbf{L}^{\top}_{t,t}$ and
$\mathbf{V}_{t+1}:=\mathbf{L}_{t+1,t}^{-\top}$.

####  A.1.5 Marginal likelihood and fit-complexity trade off

The marginal likelihood is given by:

$p(\bm{y}|\lambda,\bm{\theta},\gamma)=\left(\pi\right)^{-\frac{TN}{2}}\frac{\Gamma_{N}\left(\frac{\tilde{{\nu}}}{2}\right)|\tilde{{S}}|^{-\frac{\tilde{\nu}}{2}}|\tilde{{\bm{\Psi}}}|^{\frac{N}{2}}}{\Gamma_{N}\left(\frac{\underline{\nu}}{2}\right)|\underline{\underline{S}}|^{-\frac{\underline{\underline{\nu}}}{2}}|\bm{\underline{\underline{\Psi}}}|^{\frac{N}{2}}}$
(78)

Following the same steps as in [26] it can be re-written as :

$p(\bm{Y}|\lambda,\bm{\theta},\gamma)=const|\bm{(V_{\varepsilon}^{post})^{-1}V^{prior}_{\varepsilon}}|^{\frac{T+\underline{\underline{\nu}}}{2}}\prod_{t=1}^{T}|\bm{V_{t|t-1}}|^{-\frac{1}{2}}$
(79)

where

$vec(\bm{\hat{\varepsilon}_{t}})=vec(\bm{Y-X\tilde{\Phi}})$ (80)

$\bm{V}_{\varepsilon}^{prior}=E[\bm{\Sigma}]=\frac{\bm{\underline{\underline{S}}}}{\underline{\underline{\nu}}-N-1}$
(81)
$\bm{V}_{\varepsilon}^{post}=E[\bm{\Sigma}|\bm{Y}]=\frac{\bm{\underline{\underline{S}}}+\bm{\hat{\varepsilon}^{\prime}\hat{\varepsilon}}+(\bm{\tilde{\Phi}}-\bm{\underline{\underline{\Phi}}})^{\prime}\bm{\underline{\underline{\Psi}}}^{-1}(\bm{\tilde{\Phi}}-\bm{\underline{\underline{\Phi}}})}{T+\underline{\underline{\nu}}-N-1}$
(82)
$\bm{V}_{t|t-1}=\frac{\bm{{\underline{\underline{S}}}}}{\underline{\underline{\nu}}-N-1}\bm{\otimes}\left(1+\bm{X_{t}^{\prime}(X_{t-1}^{\prime}X_{t-1}+\underline{\underline{\Psi}}^{-1})^{-1}X_{t}}\right)$
(83)
$const=\left(\frac{1}{\pi}\right)^{\frac{NT}{2}}\frac{\Gamma_{N}\left(\frac{T+\underline{\underline{\nu}}}{2}\right)}{\Gamma_{N}\left(\frac{\underline{\underline{\nu}}}{2}\right)}\frac{(T+\underline{\underline{\nu}}-N-1)^{\underline{\underline{\nu}}/2}}{(\underline{\underline{\nu}}-N-1)^{T+{\underline{\underline{\nu}}}/2}}$
(84)

####  A.1.6 Formulas with distinct $\lambda_{j}$ for $j=1,\ldots,K$

In the case we consider regressor specific hyper-parameters $\lambda_{j}$ with
$j=1,\ldots,K$, defining
$\bm{\Lambda_{k}}=diag(\lambda_{1},\ldots,\lambda_{k})$ the theory coherent
prior becomes:

$p(vec(\bm{\Phi})|\bm{\Sigma}_{u},\bm{\Lambda},\bm{\theta},\gamma)\sim\mathcal{N}(vec(\bm{\underline{\underline{\Phi}}}),\bm{\Sigma}_{u}\otimes\bm{\underline{\underline{\Psi}}})$
(85)
$vec(\bm{\underline{\underline{\Phi}}})=vec(\left(\gamma\bm{\Gamma_{xx}(\theta)}+\bm{H_{Tk}^{\prime}H_{Tk}(I_{T}\otimes\Lambda_{k}^{\prime}\Lambda_{k}})\right)^{-1}(\gamma\bm{\Gamma_{xy}(\theta)}+\bm{H}_{Tk}^{\prime}\bm{H}_{Tk}(\bm{I_{T}\otimes\Lambda_{k}^{\prime}\Lambda_{k}})\bm{\underline{\Phi_{0}}}))$
(86)
$\bm{\underline{\underline{\Psi}}}=\left((\gamma\bm{\Gamma_{xx}(\theta)}+\bm{H_{Tk}^{\prime}H_{Tk}(I_{T}\otimes\Lambda_{k}^{\prime}\Lambda_{k}})\right)^{-1}$
(87)

$p(\bm{\Sigma}_{u}|\bm{\Lambda},\bm{\theta},\gamma)\sim\mathcal{IW}\left({\underline{\underline{\bm{S}}}},\underline{\underline{\nu}}\right)$
(88)

$\underline{\underline{\bm{S}}}=\underline{S}+\gamma\bm{\Gamma_{yy}(\theta)}+\bm{\underline{\Phi_{0}}}^{\prime}(\bm{H_{Tk}}^{\prime}\bm{H_{Tk}})(\bm{I_{T}}\otimes\bm{\Lambda^{\prime}\Lambda})\bm{\underline{\Phi_{0}}}-\bm{\underline{\underline{\Phi}}}^{\prime}\bm{\underline{\underline{\Psi}}}^{-1}\bm{\underline{\underline{\Phi}}}$
(89) $\underline{\underline{\nu}}=\underline{\nu}+\gamma$ (90)

The formula of the conditional posterior of $\bm{\Phi}$ and $\bm{\Sigma_{u}}$
becomes

$\displaystyle
p(vec(\bm{\Phi})|\bm{\Sigma}_{u},\bm{\Lambda},\bm{\theta},\gamma,\bm{Y})$
$\displaystyle\sim\mathcal{N}(vec(\bm{\tilde{\Phi}}),\bm{\Sigma}_{u}\otimes\bm{\tilde{\Psi}})$
(91) $\displaystyle p(\bm{\Sigma}_{u}|\bm{\Lambda},\bm{\theta},\gamma,\bm{Y})$
$\displaystyle\sim\mathcal{IW}\left({\bm{\tilde{S}}},\tilde{\nu}\right)$
$\small
vec(\bm{\tilde{\Phi}})=vec(\left(\bm{X^{\prime}X}+\gamma\bm{\Gamma_{xx}(\theta)}+\bm{H_{Tk}^{\prime}H_{Tk}(I_{T}\otimes\Lambda_{k}^{\prime}\Lambda_{k}})\right)^{-1}(\bm{X^{\prime}Y}+\gamma\bm{\Gamma_{xy}(\theta)}+\bm{H_{Tk}^{\prime}H_{Tk}(I_{T}\otimes\Lambda_{k}^{\prime}\Lambda_{k}})\bm{\underline{\Phi_{0}}}))$
(92)
$\bm{\tilde{\Psi}}=\left(\bm{X^{\prime}X}+\gamma\bm{\Gamma_{xx}(\theta)}+\bm{H_{Tk}^{\prime}H_{Tk}(I_{T}\otimes\Lambda_{k}^{\prime}\Lambda_{k}})\right)^{-1}$
(93)
$\tilde{\bm{S}}=\bm{Y^{\prime}Y}+\underline{\underline{\bm{S}}}+\bm{\underline{\underline{\Phi}}}^{\prime}\bm{\underline{\underline{\Psi}}}^{-1}\bm{\underline{\underline{\Phi}}}-\bm{\tilde{\Phi}}^{\prime}\bm{\tilde{\Psi}}^{-1}\bm{{\tilde{\Phi}}}$
(94) $\tilde{\nu}=\underline{\underline{\nu}}+T$ (95)

The marginal likelihood is just (27), updated with the new definitions of
$\underline{\underline{\nu}}$, $\underline{\underline{\bm{S}}}$,
$\bm{\underline{\underline{\Psi}}}$,
$\bm{\underline{\underline{\Phi}}}$,$\bm{\tilde{\Psi}}$, $\tilde{\bm{S}}$,
$\bm{\tilde{\Psi}}$.

###  A.2 Small scale New Keynesian model for the forecasting exercise

####  A.2.1 Data

The data for the out of sample forecasting exercise in Section 3 are taken
from the FRED-QD Dataset from the Federal bank of St. Louis. The series IDs of
the time series are GDPC1, CPIAUCSL, FEDFUNDS. Quarterly GDP growth and
quarterly inflation rate are obtained transforming the series according to
$\Delta\%y_{t}=100\left(\frac{y_{t}-y_{t-1}}{y_{t-1}}\right)$. As for the Fed
Funds interest rate it is transformed by taking the logarithm.

####  A.2.2 Competing models in the forecasting exercise:

The competing models in the out of sample forecasting exercise in Section 3
are

  * •

A constant parameters VAR with flat prior.

  * •

A constant parameters VAR with Normal Inverse-Wishart prior.

  * •

A TVP-VAR model.

The VAR with Normal Inverse-Wishart prior is given by:

$\bm{Y}=\bm{X\Pi}+\bm{U}\hskip 56.9055pt\bm{U}\sim
MVN(\bm{0},\bm{\Sigma}_{u},\bm{I_{T}})$ (96)

where $\bm{Y}$ is $T\times N$, $\bm{X}$ is $T\times k$ with $k=Np+1$,
$\bm{\Pi}$ is $k\times n$ , $\bm{U}$ is $T\times N$ and $MVN$ stands for the
matricvariate normal. The prior for the autoregressive coefficients and the
variance covariance matrix is:

$vec(\bm{\Pi})\sim\mathcal{N}(vec(\bm{\uline{\mu_{\Pi}}}),\bm{\Sigma}_{u}\otimes\bm{\uline{\Omega_{\Pi}}})$
(97) $\bm{\Sigma}\sim\mathcal{IW}(\bm{S_{0}},v_{0})$ (98)

where $vec(\bm{\Pi})$ centers the non-stationary series on a random walk
process and the stationary ones on a white noise process and
$\bm{\uline{\Omega_{\Pi}}}$ is a diagonal matrix with first element element
equal to

$\omega_{1}=100\\\ $ (99)

and

$\omega_{s}=\frac{\theta_{1}}{\sigma_{i}^{2}l^{2}}\\\ $ (100)

for the other elements $s=2,\ldots,Np$ with $\theta_{1}=0.1$ and
$\sigma_{i}^{2}$ is set equal to $\hat{\sigma}_{i}^{2}$ being the estimated
variance of the $i^{th}$ variable in a VAR model using a pre-sample of
observations. $l$ is the lag order of the variable associated to that
variable. $\bm{S_{0}}$ is set equal to $\bm{\hat{S}_{0}}$ the variance
covariance matrix estimated variance from a VAR model on a pre-sample of
observations and $v_{0}=N+2$. The standard TVP-VAR model and the corresponding
choice of the prior distributions follows [11]. The model is given by:

$\underbrace{\bm{y}^{\prime}_{t}}_{\text{$1\times
N$}}=\underbrace{\bm{x^{\prime}}_{t}}_{\text{$1\times
k$}}\underbrace{\bm{\Phi}_{t}}_{\text{$k\times
N$}}+\underbrace{\bm{u_{t}}^{\prime}}_{\text{$1\times N$}}\hskip
85.35826pt\bm{u_{t}}\sim\mathcal{N}(\bm{0}_{N\times 1},\bm{\Sigma_{u}})$ (101)
$vec(\bm{\Phi_{t}})=vec(\bm{\Phi_{t-1}})+\bm{\eta}_{t}\hskip
56.9055pt\bm{\eta}_{t}\sim\mathcal{N}(\bm{0},\bm{\Omega})$ (102)

where $\bm{\Omega}=diag(\omega_{1},\ldots,\omega_{Nk})$ 353535Note that in
this model the variances in the state equation of the time varying
coefficients are not constrained to be proportional across equations (the
structure of $\bm{\Omega}$ is not constrained to be a Kronecker product
$\bm{\Omega}=\bm{\Sigma}_{u}\otimes\bm{\Lambda_{k}}$).

$\omega_{i}\sim\mathcal{IG}\left(\nu,s^{2}\right)$ (103)

where $\nu=3$ $s^{2}=0.005$ and the prior for the variance covariance matrix
is set as in the constant parameter BVAR with Normal-Inverse-Wishart prior.

####  A.2.3 Prior for the DSGE parameters

Table 2: Prior distribution for the parameters of the New Keynesian model

Parameter | Prior distribution | Mean | Standard Deviation
---|---|---|---
$ln(\gamma)$ | Normal | 0.500 | 0.250
$ln(\pi^{*})$ | Normal | 1.000 | 0.500
$ln(r^{*})$ | Gamma | 0.500 | 0.250
$\kappa$ | Gamma | 0.300 | 0.150
$\tau$ | Gamma | 2.000 | 0.500
$\psi_{1}$ | Gamma | 1.500 | 0.250
$\psi_{2}$ | Gamma | 0.500 | 0.200
$\rho_{R}$ | Beta | 0.500 | 0.250
$\rho_{g}$ | Beta | 0.800 | 0.100
$\rho_{z}$ | Beta | 0.300 | 0.100
$\sigma_{R}$ | Inverse Gamma | 0.251 | 0.139
$\sigma_{g}$ | Inverse Gamma | 0.630 | 0.323
$\sigma_{z}$ | Inverse Gamma | 0.875 | 0.430

The table reports the details on the prior distribution of the parameters of
the New Keynesian model.

####  A.2.4 Posterior estimate for the DSGE parameters and IRFs from the TC-
TVP-VAR

Table 3: Posterior estimates of the parameters of the New Keynesian model

Parameter | Estimate
---|---
$ln(\gamma)$ | 0.979
| [0.734 - 1.230]
$ln(\pi^{*})$ | 0.770
| [0.547 - 1.156]
$ln(r^{*})$ | 0.227
| [0.105 - 0.427]
$\kappa$ | 0.886
| [0.619 - 1.123]
$\tau$ | 1.153
| [0.748 - 1.709]
$\psi_{1}$ | 1.534
| [1.250 - 1.895]
$\psi_{2}$ | 0.139
| [0.032 - 0.284]
$\rho_{R}$ | 0.849
| [0.761 - 0.897]
$\rho_{g}$ | 0.794
| [0.643 - 0.908]
$\rho_{z}$ | 0.209
| [0.121 - 0.314]
$\sigma_{R}$ | 0.080
| [0.068 - 0.095]
$\sigma_{g}$ | 0.345
| [0.269 - 0.449 ]
$\sigma_{z}$ | 0.416
| [0.340 - 0.505]

The table reports the estimated posterior median and $10^{th}$ and $90^{th}$
credible sets
(in square brackets) of the parameters of the New Keynesian model.

###  A.3 Medium scale New Keynesian model

The model is taken from [19] and it is a version of the popular medium scale
New Keynesian model in [35]. The set of log-linearized equilibrium conditions
of the model is

$\scriptsize
c_{t}=-\frac{(1-he^{-\gamma})}{\sigma_{c}(1+he^{-\gamma})}(R_{t}-E_{t}[\pi_{t+1}]+b_{t})+\frac{(he^{-\gamma})}{(1+he^{-\gamma})}(c_{t-1}-z_{t})+\frac{1}{(1+he^{-\gamma})}E_{t}[c_{t+1}+z_{t+1}]+\frac{(\sigma_{c}-1)}{\sigma_{c}(1+he^{-\gamma})}\frac{w_{*}l_{*}}{c_{*}}(i_{t}-E_{t}[l_{t+1}])$
(104)
$q_{t}^{k}=S^{\prime\prime}e^{2\gamma}(1+\bar{\beta})\left(i_{t}-\frac{1}{1+\bar{\beta}}(i_{t-1}-z_{t})-\frac{\bar{\beta}}{1+\bar{\beta}}E_{t}[i_{t+1}+z_{t+1}]-\mu_{t}\right)$
(105)
$\bar{k}_{t}=\left(1-\frac{i_{*}}{k_{*}}\right)(\bar{k}_{t-1}-z_{t})+\frac{i_{*}}{k_{*}}i_{t}+\frac{i_{*}}{k_{*}}S^{\prime\prime}e^{2\gamma}(1+\bar{\beta})\mu_{t}$
(106)
$\frac{r_{*}^{k}}{r_{*}^{k}+(1-\delta)}E_{t}[r_{t+1}^{k}]+\frac{1-\delta}{r_{*}^{k}+(1-\delta)}E_{t}[q_{t+1}^{k}]-q_{t}^{k}=R_{t}+b_{t}-E_{t}[\pi_{t+1}]$
(107) $k_{t}=u_{t}-z_{t}+\bar{k}_{t-1}$ (108)
$\frac{1-\psi}{\psi}r_{t}^{k}=u_{t}$ (109) $mc_{t}=w_{t}+\alpha l_{t}-\alpha
k_{t}$ (110) $k_{t}=w_{t}-r_{t}^{k}+l_{t}$ (111) $y_{t}=\Phi_{p}(\alpha
k_{t}+(1-\alpha)l_{t})+\mathcal{I}\\{\rho_{z}<1\\}(\Phi_{p}-1)\frac{1}{1-\alpha}\tilde{z}_{t}$
(112)
$y_{t}=g_{t}+\frac{c_{*}}{y_{*}}c_{t}+\frac{i_{*}}{y_{*}}i_{t}+\frac{r_{*}^{k}k_{*}}{y_{*}}u_{t}-\mathcal{I}\\{\rho_{z}<1\\}\frac{1}{1-\alpha}\tilde{z}_{t}$
(113)
$g_{t}=\rho_{g}g_{t-1}+\sigma_{g}\varepsilon_{g,t}+\eta_{gz}\sigma_{z}\varepsilon_{z,t}$
(114) $\pi_{t}=\kappa
mc_{t}\frac{\iota_{p}}{1+\iota_{p}\bar{\beta}}\pi_{t-1}+\frac{\bar{\beta}}{1+\iota_{p}\bar{\beta}}E_{t}[\pi_{t+1}]+\lambda_{f,t}$
(115) $\scriptsize
w_{t}=\frac{(1-\zeta_{w}\bar{\beta})(1-\zeta_{w})}{(1+\bar{\beta})\zeta_{w}((\lambda_{w}-1)\epsilon_{w}+1)}(w^{h}-w_{t})-\frac{1+\iota_{w}\bar{\beta}}{(1+\bar{\beta})}\pi_{t}+\frac{1}{1+\bar{\beta}}(w_{t-1}-z_{t}-\iota_{w}\pi_{t-1})+\frac{\bar{\beta}}{1+\bar{\beta}}E_{t}[w_{t+1}+z_{t+1}+\pi_{t+1}]+\lambda_{w,t}$
(116)
$w_{t}^{h}=\frac{1}{1-he^{-\gamma}}(c_{t}-he^{-\gamma}c_{t-1}+he^{-\gamma}z_{t})+\nu_{l}l_{t}$
(117)
$\lambda_{f,t}=\rho_{\lambda_{f}}\lambda_{f,t-1}+\sigma_{\lambda_{f}}\varepsilon_{\lambda_{f},t}-\eta_{\lambda_{f}}\sigma_{\lambda_{f}}\varepsilon_{\lambda_{f},t-1}$
(118)
$\lambda_{w,t}=\rho_{\lambda_{w}}\lambda_{w,t-1}+\sigma_{\lambda_{w}}\varepsilon_{\lambda_{w},t}-\eta_{\lambda_{w}}\sigma_{\lambda_{w}}\varepsilon_{\lambda_{w},t-1}$
(119)
$R_{t}=\rho_{R}R_{t-1}+(1-\rho_{R})\left(\psi_{1}(\pi_{t}-\pi_{t}^{*})+\psi_{2}(y_{t}-y_{t}^{f})\right)+\psi_{3}\left((y_{t}-y_{t}^{f})-(y_{t-1}-y_{t-1}^{f})\right)+r_{t}^{m}$
(120) $E_{t}[\tilde{R}_{t+1}^{k}-R_{t}]=b_{t}$ (121)
$\tilde{R}_{t+1}^{k}-\pi_{t}=\frac{r_{*}^{k}}{r_{*}^{k}+(1-\delta)}r_{t}^{k}+\frac{(1-\delta)}{r_{*}^{k}+(1-\delta)}q_{t}^{k}-q_{t-1}^{k}$
(122) $b_{t}=\rho_{b}b_{t-1}+\sigma_{b}\varepsilon_{b,t}$ (123)
$\mu_{t}=\rho_{\mu}\mu_{t-1}+\sigma_{\mu}\varepsilon_{\mu,t}$ (124)
$\tilde{z}_{t}=\rho_{z}\tilde{z}_{t-1}+\sigma_{z}\varepsilon_{z,t}$ (125)
$r_{t}^{m}=\rho_{r}\tilde{z}_{t-1}+\sigma_{r}\varepsilon_{r,t}$ (126)

The observation equations are:

$\text{Output growth}=\gamma+100(y_{t}-y_{t-1}+z_{t})$ (127)
$\text{Consumption growth}=\gamma+100(c_{t}-c_{t-1}+z_{t})$ (128)
$\text{Investment growth}=\gamma+100(i_{t}-i_{t-1}+z_{t})$ (129) $\text{Real
wage growth}=\gamma+100(w_{t}-w_{t-1}+z_{t})$ (130) $\text{Hours
worked}=\bar{l}+100l_{t}$ (131) $\text{Inflation}=\pi_{*}+100\pi_{t}$ (132)
$\text{FFR}=R_{*}+100(R_{t})$ (133)

Data sources and transformations are as in [19] to which I refer for further
details. During the ZLB period, the observation equation for the interest rate
includes a measurement error, namely:

$\text{FFR}=R_{*}-R_{*}+v_{r,t}$ (134)

with $\operatorname*{\mathbb{E}}[v_{r,t}]=0$ and $var(v_{r,t})=0.001$. This is
to account for the fact that the Fed Fund Rate was not exactly equal to zero,
remaining slightly above zero in the ZLB. As well, this condition ensures
invertibility of $\bm{\Gamma_{xx,t}}$ during the ZLB period.

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Figure 7: Estimated $10^{th}$ and $90^{th}$ credible intervals of responses of
output growth (cumulative) and inflation rate inside (in shaded red) and
outside (in shaded grey) the ZLB period from a standard TVP-VAR on distinct
simulated samples from the NK model. The dotted line plots are the responses
inside (in red) and outside (in black) the ZLB period in the NK model.