Long Run Risk in Stationary Structural Vector Autoregressive Models

C., GOURIEROUX111Toronto University, Toulouse School of Economics, and CREST
e-mail<EMAIL_ADDRESS>and J., JASIAK 222York University,
Canada, e-mail<EMAIL_ADDRESS>
The first author gratefully acknowledges financial support from the ACPR chair
”Regulation and Systemic Risk”, and the ERC DYSMOIA. The second author thanks
the Natural Sciences and Engineering Council of Canada (NSERC).

February, 16, 2022

Abstract

This paper introduces a local-to-unity/small sigma process for a stationary
time series with strong persistence and non-negligible long run risk. This
process represents the stationary long run component in an unobserved short-
and long-run components model involving different time scales. More
specifically, the short run component evolves in the calendar time and the
long run component evolves in an ultra long time scale. We develop the methods
of estimation and long run prediction for the univariate and multivariate
Structural VAR (SVAR) models with unobserved components and reveal the
impossibility to consistently estimate some of the long run parameters. The
approach is illustrated by a Monte-Carlo study and an application to
macroeconomic data.

Keywords: Structural VAR, Ultra Long Run Process, Identification,
Impossibility Theorem, Autocorrelation Function, Ultra Long Run Prediction,
Required Capital, Prudential Principle.

##  1 Introduction

The macroeconomic and financial models used in practice provide reliable
predictions at short horizons of 1 to 5 years. Recently, there has been a
growing interest in providing long run predictions at horizons of 10 to 50
years, in the context of transition to low carbon economy, climate risk and
rare extreme events, for evaluating of necessary behavioral and technical
changes. In the financial sector as well, the long run predictions may soon
become mandatory for prudential supervision in banks and insurance companies.

The long-run predictions are difficult to compute for several reasons. The
standard prediction models with short lags and the associated statistical
inference methods are inadequate for long horizons. Moreover, long run
predictions at horizons of 50 or 100 years are difficult to compute from
macroeconomic or financial time series observed over periods shorter or equal
to the prediction horizon of interest. Therefore, the long run predictions
remain mainly model based. As such, they can be improved so that:

i) The estimation methods account for the long run component, even though it
is difficult to detect over the sampling period.

ii) The long run predictions produce reasonable outcomes in the sense that
point forecasts should take values from the set of admissible values of the
predicted variable.

iii) The prediction errors are not underestimated due to the selected dynamic
model and its estimation method.

This paper examines the feasibility of such improvements in the class of
dynamic stochastic linear models with long run properties already known in the
literature to some extent. The Structural Vector Autoregressive (SVAR) models
are stochastic linear systems which are second-order identifiable under a set
of parameter restrictions. The SVAR models are used in macroeconomics,
monetary economics and macro-finance for the analysis of multivariate economic
processes and prediction of future shock effects through impulse response
functions [see Hurwicz (1962) for the introduction of the term ”structural”
and Sims (1980), (2002) for SVAR applications]. Two types of SVAR models can
be distinguished, which are the stationary (regular) SVAR models and
cointegrated SVAR models with nonstationary unit root components. While both
types of SVAR models are efficient instruments of short run analysis, they
have limitations in application to the long run and ultra long run analysis.
More specifically,

i) the presence of nonstationary features, such as nonstationary unit roots,
implies explosive patterns in trajectories, which are incompatible with the
behavior of variables such as the growth rate of per capita real GDP,
productivity, real food expenditure per capita, interest rates, real exchange
rates, and some commodity prices, or spot-forward spreads. 111See for instance
the discussion on the mean reverting feature of consumption in Beeler,
Campbell (2012) and the analysis of spot-forward spreads in Gospodinov (2009),
Gospodinov et al. (2021). In general, variables transformed to stationarity by
differencing or expressed as rates of growth are not explosive; ii) the
identification restrictions imposed on the long run behaviour, i.e. the so-
called long run identification restrictions [Blanchard, Quah (1989),
Christiano et al. (2006)] can affect the long-run predictions and the long run
patterns of impulse response functions [Sims (1986)] as well as their signs
[Uhlig (2005)]; iii) the long run risk is diversified away and often
disregarded in stationary SVAR models [see e.g. Bansal, Yaron (2004), Bansal,
Kiku, Yaron (2010), Croce, Lettau, Ludvigson (2015) for recent papers on the
long run analysis with regular SVAR models estimated as stationary processes].

Among the macroeconomic time series, those considered stationary often display
persistence up to high lags, in the sample autocorrelation functions.
Therefore, a large body of literature use the local-to-unity model linking the
aforementioned two types of SVAR dynamics and develop associated inference
methods [see e.g. Chan, Wei (1987), Phillips (1987), Magdalinos, Phillips
(2007), and the recent survey by Muller, Watson (2020)]. However, the standard
local-to-unity models may not be suitable for stationary time series (see the
discussion in Appendix 1).

The aim of our paper is to propose an alternative approach by considering a
stationary VAR model with a stationary short run (SR) component and a
stationary multivariate ultra long run (ULR) component with asymptotic unit
root and close to zero sigma,

For illustration, let us consider the framework of stationary Gaussian
processes (where strict stationarity and second-order stationarity are
equivalent). This simplified framework is not only convenient for comparing
our analysis with the large body of literature on structural SVAR models, but
also for introducing and interpreting the notion of an ultra long run process
in a stationary time series seen as a sum of a regular and a singular
components. There exist two generic representation theorems of a time series
seen the Wold representation theorem for weakly stationary processes [Wold
(1938)] and the Volterra representation theorem for strictly stationary
processes [Rozanov (1967), Priestley (1988)]. These two representations
coincide for the Gaussian stationary processes. According to the Wold
representation, any (multivariate) stationary Gaussian process $y(t)$ can be
written as:

$y(t)=\sum_{j=0}^{\infty}A_{j}\epsilon_{t-j}+Z_{t},$ (1.1)

where $dim\,y=\dim\epsilon=dim\;Z\equiv n$, the $(n\times n)$ matrices $A_{j}$
are such that $\sum_{j=0}^{\infty}A_{j}A_{j}^{\prime}$ exists and is finite,
$(\epsilon_{t})$ is a Gaussian white noise, $\epsilon_{t}\sim IIN(0,Id)$. The
moving average component is the regular component. The component $Z_{t}$ is
the singular component of process $(y(t))$ and is measurable with respect to
$F_{-\infty}=\cap_{t}F_{t}$ where $F_{t}$ is the $\sigma$-algebra generated by
the current and lagged values of $y$. Following Doob (1944), this singular
component is also referred to the deterministic component of the process. This
explains why in practice, $Z_{t}$ is often disregarded and replaced by a
deterministic constant:

$y(t)=\sum_{j=0}^{\infty}A_{j}\epsilon_{t-j}+m.$ (1.2)

The term ”deterministic” is misleading, because the stationary singular
component can be constant over time while being stochastic, so that $Z_{t}=Z$
is independent of the noise and normally distributed: $Z\sim N(m,\Sigma)$,
where $\Sigma$ does not necessarily degenerate to zero. The fact that $Z_{t}$
is measurable with respect to $F_{-\infty}$ means that it is only influenced
by the infinitely distant past and has a long run interpretation. In
particular, the singular component is such that:

$E[Z_{t+h}|F_{t}]=Z_{t}=Z,\;\mbox{for any }h,t.$

Thus $Z_{t}$ is a stationary martingale, which is a particular stationary unit
root process 222Since $Z_{t+h}=Z_{t}=Z$, the theoretical prediction error is
equal to zero. This explains the terminology ”purely predictable” process also
used for this singular component. This other terminology is also misleading
since in practice we observe $y$ from an initial date only, and, if $F_{0,t}$
is the $\sigma$-algebra generated by $y(0),\ldots,y(t)$, then
$E(Z_{t+h}|F_{0,t})$ is different from $Z_{t}$.. The singular component can be
approximated by a moving average:

$Z_{T}(t)=\sum_{j=0}^{\infty}A_{Tj}^{*}\epsilon^{*}_{t-j},$

where $(\epsilon^{*}_{t})$ is another Gaussian white noise, independent of
$(\epsilon_{t})$ and
$\displaystyle\sum_{j=0}^{\infty}A_{Tj}^{*}A_{Tj}^{*^{\prime}}$ is finite.
This approximation leads to the triangular array (doubly indexed by $t$ and
$T$):

$\displaystyle y_{T}(t)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}A_{j}\epsilon_{t-j}+\sum_{j=0}^{\infty}A_{Tj}^{*}\epsilon^{*}_{t-j}$
(1.3) $\displaystyle=$
$\displaystyle\sum_{j=0}^{\infty}A_{j}\epsilon_{t-j}+Z_{T}(t),$

which approximates the representation (1.1) of $y(t)$.

The stationary moving average approximation of the singular component
$Z_{t}=Z$ is slowly varying over time and has a negligible effect on the
dynamics of $y(t)$ in the short run. As shown later in the text, it has a
significant impact in the long run dynamic, and is therefore interpreted as
the Ultra Long Run (ULR) component. Moreover, process
$\left\\{y_{T}(t)\right\\}$ remains stationary, but becomes non-ergodic.

More generally, in the VAR framework we can write the n-dimensional process
$y_{T}(t)$ as a sum of the short run (SR) component $y_{s}(t)$, which for ease
of exposition is assumed to follow a VAR(1) process, and an ULR component:

$y_{T}(t)=y_{s}(t)+Ay_{l}(t/T),$ (1.4)

where $A$ is a $(n,L)$ matrix of coefficients of full column rank L and :

$y_{s}(t)=\Phi y_{s}(t-1)+\Omega^{1/2}\epsilon_{t},$ (1.5)

where the eigenvalues of $\Phi$ are of modulus strictly less than 1 to ensure
the stationarity of the SR process $y_{s}$, the $(n,n)$ matrix $\Omega$ is
positive definite and $\epsilon_{t}$ is a multivariate strong white noise
process with second-order moments, but not necessarily Gaussian. The component
$[y_{l}(\tau)]$ follows a L-dimensional (continuous time) multivariate
Ornstein-Uhlenbeck process:

$dy_{l}(\tau)=-\Theta y_{l}(\tau)d\tau+SdW_{\tau},$ (1.6)

where $\Theta$ is of dimension $L\times L$ and such that the eigenvalues of
matrix 333$\exp(-\Theta)$ is defined as :
$\exp(-\Theta)=\sum^{\infty}_{j=0}[(-1)^{j}\Theta^{j}/j!].$ $\exp(-\Theta)$
are of modulus strictly less than one to ensure the stationarity of process
$y_{l}$. Process $(W_{\tau})$ is a L-dimensional Brownian motion independent
of the noise in the short run component and $S$ is a positive definite matrix
of dimension $(L,L)$. The ULR component is deduced from the continuous time
underlying process $[y_{l}(\tau)]$ by applying the time deformation and time
discretization. More specifically, the short run component evolves over
calendar time, whereas the long run component evolves over time measured on a
scale with an ultra large time unit $T$ that tends to infinity with $T$. In
particular, for any date $t$, $y_{l}(t/T)$ will tend to $y_{l}(0)$, which is
constant over time and random. This limit is a singular process. Equations
(1.4)-(1.6) impose a special structure on the VAR(1) model, turning it into a
Structural VAR where the SR and ULR shocks can be identified.

The process (1.4)-(1.6) is an example of a local-to-unity model with
behavioral foundations [see the discussion in Sims (1988), Section 2]. Indeed,
its dynamics is compatible with a dynamic equilibrium of two types of
individuals (consumers, investors), who operate at two different frequencies
and have different information sets. By considering these two frequencies
jointly, we do not support the ”belief that data collection and modelling
strategy usually lead economists to use data corresponding to a ”fine” time
unit relative to the phenomena they are studying” [Sims (1991), p429].

In the approximate Wold representation (1.3) of (1.4)-(1.6) the number of
underlying shocks is equal to $n+L=dim\epsilon_{t}+dimW_{\tau}$, which is
larger than the dimension of $[y_{T}(t)]$. These two types of shocks are
measured on different time scales. The number of ULR processes $L$ is assumed
to be less that $n$ (the dimension of $Z$) by imposing the condition of full
column rank on matrix $A$. The $(n,L)$ matrix $A$ is introduced to identify
the components of the array that are not sensitive to long run shocks. More
precisely, for any direction $\gamma$ such that $\gamma^{\prime}A=0$, the
combination $\gamma^{\prime}y_{T}(t)=\gamma^{\prime}y_{s}(t)$, no longer
depends on the ULR component and $T$. This allows us for considering the ULR
co-movement relationships (i.e. the $\gamma^{\prime}s)$.

The paper is organized as follows. Section 2 discusses the second-order
properties of the model such as the theoretical autocovariances and spectral
density of the (triangular) array. We show how the theoretical autocovariances
and spectra depend on $T$, derive their limits when $T$ tends to infinity and
reveal a lack of uniformity in their convergences. As a consequence, new
identification issues arise for statistical inference.

Statistical inference is discussed in Section 3. We focus on estimation
methods based on the first- and second-order sample moments. Under the
triangular array representation $[y_{T}(t)],$ the sample autocovariances can
have asymptotic properties which are very different from the standard
properties. Therefore, the sample autocovariances and autocorrelations may not
be fully informative about the parameters of interest. We show that different
types of autocorrelation functions (ACF) have to be considered jointly. These
are:

i) the standard ACF evaluated at multiples of large lags;

ii) short run ACF evaluated at short lags over short episodes of time;

iii) long run ACF at large lags computed from historical averages of the
observed process $[y_{T}(t)]$.

We show that, from rolling averages and different ACF’s, it is possible to
estimate consistently (under the standard structural identification
restrictions for SVAR):

i) the parameters $\Phi$ and $\Omega$ driving the short run dynamics;

ii) the levels $[y_{l}(\tau),\tau\in(0,1)]$ of the underlying long run
continuous time

model;

iii) the matrix A of factor sensitivities;

iv) the volatility of the ULR component, i.e. the matrix $SS^{\prime}$.

The ”estimators” in i)-iv) converge at different rates. In particular the ULR
parameters converge at slower rates, which reveals weak identification issues.
However, the main identification issue is the impossibility to estimate
consistently the drift $\Theta$ in the Ornstein-Uhlenbeck (OU) dynamics (the
impossibility theorem). This implies that, even with an infinite number of
observations, the long run estimation risk cannot be diversified away. Thus,
the evaluation of the long run estimation risk is crucial for the long run
predictions and impulse responses. Section 4 highlights the importance of
taking into account the estimation risk when computing the long run prediction
intervals and the Value-at-Risk. We also discuss the prudential principle and
the limitation of Bayesian approaches in this respect. Section 5 illustrates
the proposed methodology by Monte-Carlo studies and provides an empirical
application to a set of macroeconomic series. It compares the approach
introduced in this paper with the treatment of long run risk in the applied
macroeconomic literature. Section 6 concludes. A comparison with alternative
local-to-unity models is provided in Appendix 1. The Law of Large Numbers
(LLN) for triangular arrays used for deriving the asymptotic results is given
in Appendix 2, and the Central Limit Theorems (CLT) in Appendix 3. Additional
figures are provided in Appendix 4.

##  2 Dynamic properties

###  2.1 The VAR representation

A time discretized multivariate Ornstein-Uhlenbeck process is a Gaussian
VAR(1) process. Hence, the triangular array model (1.4) - (1.6) can be
rewritten as [see Meucci (2010)]:

$y_{T}(t)=y_{s}(t)+Ay_{l,T}(t),$ (2.1)

where :

$y_{s}(t)=\Phi y_{s}(t-1)+\Omega^{1/2}\epsilon_{t},$ (2.2)

$y_{l,T}(t)=exp(-\Theta/T)y_{l,T}(t-1)+\Sigma_{T}^{1/2}\epsilon_{t}^{*},$
(2.3)

with :

$vec(\Sigma_{T})=[\Theta\oplus\Theta]^{-1}\\{Id-
exp[-(\Theta\oplus\Theta)/T]\\}vec(SS^{\prime}),$ (2.4)

and the Kronecker sum defined by:

$\Theta\oplus\Theta=Id\otimes\Theta+\Theta\otimes Id,$ (2.5)

with $\otimes$ the Kronecker product.

System (2.1) -(2.4) shows that the process $[(y_{T}(t)]$ for any given $T$ is
the sum of two independent VAR(1) processes 444When the unobserved components
are integrated out, the process $[y_{T}(t)]$ follows a weak VARMA model with
orders depending on $L$ and on the assumptions on $\Phi$ and $\Theta$. This
interpretation is not used later on.. These two processes are stationary, when
their initial values are drawn in a stationary distribution such as, for
example, N(0, $\Sigma$), with
$vec\Sigma=[\Theta\oplus\Theta]^{-1}vec(SS^{\prime})$ for the ULR component.

The VAR dynamics of the ULR component is such that the autoregressive
coefficient $exp(-\Theta/T)$ tends to $Id$, when $T$ tends to infinity while
at the same time the volatility $\Sigma_{T}$ given in (2.4) tends to 0.
Therefore it is a sequence of multivariate stationary processes, with an
asymptotic multivariate unit root and close to zero sigma. They are examined
in the framework of a local-to-unity and small sigma analysis with respective
rates of convergence to Id and 0 being of equal order 555Since
$\exp(-\Theta/T)\sim Id-\Theta/T,\exp[-\Theta\oplus\Theta/T]\sim
Id-(\Theta\oplus\Theta)/T.$ $1/T$ [see Kadane (1971) for the introduction of
small sigma asymptotics to econometrics]. This approach differs from the
local-to-unity analysis with fixed sigma that leads to nonstationary processes
at the limit [see e.g. Stock (1991), p 437, Muller, Watson (2016), p 1723, in
the univariate case]. To clarify this difference with the literature (we
elaborate on this point in Appendix 1), let us consider below a univariate
process:

$y_{T}(t)=y_{s}(t)+y_{l,T}(t),$ (2.6)

where :

$y_{s}(t)=\phi y_{s}(t-1)+\eta\varepsilon_{t},\eta>0,$ (2.7)

$y_{l,T}(t)=exp(-\theta/T)y_{l,T}(t-1)+[\frac{s^{2}}{2\theta}[1-\exp(-2\theta/T)]]^{1/2}\epsilon_{T,t}^{*}.$
(2.8)

For a given $T$, the variance of the innovation of the ULR component is
proportional to $1-\rho_{T}^{2}$, where $\rho_{T}=exp(-\theta/T)$ is the
autoregressive coefficient. Therefore, the autoregressive coefficient and the
volatility tend to 1 and 0, respectively, in a constrained manner that is
consistent with the time deformation interpretation.

An array with such a dynamic might be interpreted as a local level model. Let
us recall the standard definition of a local level model introduced by Harvey
(1989) [see also Durbin, Koopman (2002)]. In our notation, a local level model
is a time series model such that:

$y(t)=y_{s}(t)+y_{l}(t),$

where $y_{s}(t)=\eta\epsilon_{t}$, $y_{l}(t)=y_{l}(t-1)+s\epsilon_{t}^{*}$.
Like our model, the local level involves a decomposition with a short term
component, here a noise process, and a long run component, which is a random
walk. The main difference between the standard local level and our model is
the specification of the long run component. The random walk representing the
long-run component in the standard local level model is nonstationary and has
”explosive” trajectories in the long run unlike the ULR process (see also the
discussion in Appendix 1 of the different local-to-unity models proposed in
the literature).

As shown in the next Section, the ULR model allows for a stochastically
varying ”local level” without an explosive pattern. The ULR model also differs
from the model with a local level smoothly varying in time in a deterministic
pattern [Dalhaus (2012), Dalhaus et al. (2017)].

###  2.2 Autocovariances and Spectrum

Let us now consider the theoretical autocovariances and spectrum. For ease of
exposition, we consider the univariate model (2.6)-(2.8) and known parameter
values.

For a fixed $T$, the autocovariance of process $y_{T}(t)$ defined by
(2.6)-(2.8) at lag $h$ is given by:

$\gamma_{T}(h)=\eta^{2}\varphi^{h}+\frac{s^{2}}{2\theta}[1-exp(-\frac{2h\theta}{T})],\;h\geq
0.$ (2.9)

We have the following proposition:

Proposition 1

i) If $h$ is fixed, $T\rightarrow\infty$,
$\gamma_{T}(h)\rightarrow\eta^{2}\varphi^{h}.$

ii) If $h=h_{T}\rightarrow\infty$, when $T\rightarrow\infty$,
$\gamma_{T}(h_{T})\sim\frac{s^{2}}{2\theta}[1-exp(-\frac{2h_{T}}{T})]$.

In particular, if $h_{T}\rightarrow\infty$,
$T\rightarrow\infty,h_{T}/T\rightarrow 0$, then $\gamma_{T}(h_{T})\rightarrow
0$.

If $h_{T}\rightarrow\infty$, $T\rightarrow\infty,h_{T}/T\rightarrow c$, then
$\gamma_{T}(h_{T})\rightarrow\frac{s^{2}}{2\theta}[1-exp(-2c\theta)]$.

The asymptotic behaviour of the sequence of autocovariances is complex as it
is different at short and ULR lags. For the short lag (resp. long lag) only
the short run (resp. long run) component matters. Moreover, the convergence of
this autocovariance function to a limit is not uniform. This causes
identification issues discussed in the next Section.

Similarly, let us consider the spectrum at a given date $T$ given by :

$S_{T}(w)=\frac{\eta^{2}}{2\pi}\frac{1}{1+\varphi^{2}-2\varphi
cosw}+\frac{s^{2}}{2\pi\theta}\frac{1-exp(-2\theta/T)}{1+exp(-2\theta/T)-2exp(-\theta/T)cosw}.$
(2.10)

We get the following limiting behaviour:

Proposition 2

i) For $w$ fixed, $T\rightarrow\infty$,
$S_{T}(w)\rightarrow\frac{\eta^{2}}{2\pi}\displaystyle\frac{1}{1+\varphi^{2}-2\varphi\cos
w}$, if $w\neq 0$,

$S_{T}(w)\rightarrow\infty$ , if $w=0$.

ii) If $w_{T}\rightarrow 0^{+}$, $T\rightarrow\infty$,

$\displaystyle S_{T}(w_{T})$ $\displaystyle\sim$
$\displaystyle\frac{\eta^{2}}{2\pi}\frac{1}{(1-\varphi)^{2}}+\frac{s^{2}}{2\pi\theta}\frac{1-exp(-2\theta/T)}{1+exp(-2\theta/T)-2exp(-\theta/T)cosw_{T}}$
$\displaystyle\sim\frac{\eta^{2}}{2\pi}\frac{1}{(1-\varphi)^{2}}+\frac{2\theta/T}{w_{T}^{2}}.$

In particular, if $Tw^{2}_{T}\approx\lambda$, then
$S_{T}(w_{T})\approx\displaystyle\frac{\eta^{2}}{2\pi}\displaystyle\frac{1}{(1-\varphi)^{2}}+\displaystyle\frac{2\theta}{\lambda}$.

The lack of uniform convergence of the sequence of spectra is analogous to the
lack of uniform convergence of the sequences of autocovariances discussed
above. For a fixed $w$, $w\neq 0$, the limit depends on the short run dynamics
only. When $w_{T}$ tends to 0 at an appropriate rate $1/\sqrt{T}$, the limit
involves both the short- and long-run parameters, $\varphi$ and $\theta$,
respectively.

##  3 Statistical Inference

So far, we have discussed the asymptotic behaviours of the theoretical
autocovariances and spectrum in Section 2.2. This Section examines the
asymptotic properties of the sample autocovariances and sample spectrum
computed from the observations on $y_{T}(t),t=1,...,T,$ corresponding to a
triangular array.

The standard sample autocovariances are based on global averages that conceal
the local mean effect due to the ULR component. Therefore, we distinguish the
three following types of sample ACF:

i) the standard ACF evaluated at long lags from observations that are
sufficiently far apart;

ii) the short run ACF computed at short lags from short time series of
observations;

iii) the long run ACF based on short run averages of observations.

As shown below, these ACF’s have distinct behaviours and interpretations,
explaining the long run predictability puzzle reported in Fama, French (1988),
(1989), or the debate on the predictability of consumption growth from log
price/dividend ratio [Becler, Campbell (2012), Section 4.1]. This shows also
that the standard asymptotic properties of moment estimators have to be
modified in models with long run component [see e.g. Bansal, Kiku, Yaron
(2012), (2016) for the use of GMM in models with long run risk].

###  3.1 ACF Evaluated at Distant Lags

Let us denote the multivariate sample autocovariance by:

$\hat{\Gamma}_{T}(h)=\frac{1}{T-h}\sum_{t=h+1}^{T}y_{t}y_{t-h}^{\prime}-\frac{1}{T-h}\sum_{t=h+1}^{T}\;y_{t}\;\frac{1}{T-h}\sum_{t=h+1}^{T}y_{t-h}^{\prime}.$
(3.1)

From Proposition 1 ii) given for the one-dimensional case, we expect a rather
regular behaviour of the multivariate sample autocovariances at long lags
$h_{T}=cT$ for any given $c,c\in(0,1).$

To interpret this result, let us first consider the sample mean. We have:

$\frac{1}{T-h_{T}}\sum_{t=h_{T}+1}^{T}y_{t}=\frac{1}{T-cT}\sum_{t=cT+1}^{T}y_{s}(t)+\frac{A}{T-cT}\sum_{t=cT+1}^{T}y_{l}(t/T).$

The first term tends to 0 by the standard Law of Large Numbers applied to the
short run stationary component. The second term is a Riemann sum that tends to
the associated stochastic integral, by applying the Stroock, Varadhan theory
of diffusion approximation [Stroock, Varadhan (1979), Section 11]. In brief,
we have asymptotically:

$\frac{1}{T-cT}\sum_{t=cT+1}^{T}y_{t}\rightarrow\frac{A}{1-c}\int_{c}^{1}y_{l}(v)dv,$
(3.2)

where $\rightarrow$ denotes the convergence in probability (and also the weak
convergence of processes indexed by $c$).

The sample mean does not converge to the theoretical mean of process
$[y_{T}(t)]$ equal to 0. A similar derivation can be done for the cross-
product (see Appendix 2). This result is summarized in the Proposition below.

Proposition 3

For large $T$,

$\displaystyle\hat{\Gamma}_{T}(cT)$ $\displaystyle=$
$\displaystyle\frac{A}{1-c}\int_{c}^{1}y_{l}(v)y_{l}^{\prime}(v-c)dv\;A^{\prime}$
$\displaystyle-\left[\frac{A}{1-c}\int_{c}^{1}y_{l}(v)dv\right]\left[\frac{A}{1-c}\int_{c}^{1}y_{l}(v-c)dv\right]^{\prime}+o_{p}(1),$

where $o_{P}(1)$ is negligible in probability.

Proposition 3 reveals a non-standard asymptotic behavior of sample
autocovariances evaluated at long lags. When $h$ increases, these sample
autocovariances do not converge to constant values, such as their theoretical
analogues (i.e. $\frac{s^{2}}{2\theta}[1-exp(-2\theta)]$ in the univariate
case, see Proposition 1). The sample autocovariances at large lags still
converge, but to a stochastic limit that depends on the trajectory of the ULR
component.

###  3.2 Short Run Sample Autocovariances

For a fixed $h$, the sample autocovariance $\hat{\Gamma}_{T}(h)$ does not
provide a good approximation of $\Gamma_{T}(h),$ which is the autocovariance
function of the short run component (which follows from Proposition 1, i)).
Indeed, $\hat{\Gamma}_{T}(h)$ is a global measure that is impacted by the
local level effect due to the ULR component. This effect can be adjusted for
as follows. Let us consider a subset of observations corresponding to the
interval $(cT,cT+H_{T}),$ where $H_{T}\rightarrow\infty,H_{T}/T\rightarrow 0$,
and compute the sample autocovariances over this interval:

$\hat{\Gamma}_{c,T}(h)=\frac{1}{H_{T}}\sum_{t=cT+1}^{cT+H_{T}}y_{t}\;y_{t-h}^{\prime}-\frac{1}{H_{T}}\sum_{t=cT+1}^{cT+H_{T}}y_{t}\;\frac{1}{H_{T}}\sum_{t=cT+1}^{cT+H_{T}}y_{t-h}^{\prime}.$
(3.3)

Because $H_{T}<<T$ and the ULR component is close to the purely predictable
component, we have:

$\displaystyle y_{t}$ $\displaystyle=$ $\displaystyle y_{s}(t)+Ay_{l}(t/T)$
$\displaystyle\approx$ $\displaystyle y_{s}(t)+Ay_{l}(c)+o_{p}(1),$

for $t\in(cT+1,cT+H_{T})$. The process corresponds to the short run component
plus a stochastic ”local mean” equal to $Ay_{l}(c)$. Then the standard LLN can
be applied to the process conditional on $[y_{l}(\tau)]$.

Proposition 4

For any fixed $h,h\leq H_{T}$, any $c,c\in(0,1),$ and any $H_{T},$ with
$H_{T}\rightarrow\infty,H_{T}/T\rightarrow 0$, we have :
$\hat{\Gamma}_{c,T}(h)\rightarrow\Gamma_{s}(h),$ where $\Gamma_{s}(h)$ is the
theoretical autocovariance function of the short run component.

The above result can be used to estimate consistently some parameters of
interest. Let us consider a grid $c_{1}\leq...\leq c_{K_{T}}$ of $[0,1]$ .

Then, for each $c_{k}$ we compute the sample mean:

$\displaystyle\hat{m}_{T}(c_{k})$ $\displaystyle=$
$\displaystyle\frac{1}{H_{T}}\sum_{t=c_{k}T+1}^{c_{k}T+H_{T}}y_{t}$
$\displaystyle\approx$
$\displaystyle\frac{1}{H_{T}}\sum_{t=c_{k}T+1}^{c_{k}T+H_{T}}[y_{s}(t)+Ay_{l}(c_{k})]$
$\displaystyle\approx$ $\displaystyle Ay_{l}(c_{k}),\mbox{by the LLN applied
to the short run component.}$

Corollary 1:

The sample means $\hat{m}_{T}(c_{k})$ are consistent estimates (predictors) of
the values of the underlying URL process, up to the multiplicative matrix $A$.
By standard arguments, the sample means converge at speed $1/\sqrt{H_{T}}$.

Then several estimates of the autocovariances of the short run component
$\hat{\Gamma}_{c_{k},T}(h),k=1,...,K$ can be computed and used to estimate
$\Gamma_{s}(h)$ by:

$\hat{\Gamma}_{T,s}(h)=\frac{1}{K}\sum_{k=1}^{K}\hat{\Gamma}_{c_{k},T}(h),h\leq
H_{T}.$ (3.5)

Corollary 2:

The autocovariances $\Gamma_{s}(h),h\leq H_{T},$ can be consistently estimated
by $\hat{\Gamma}_{T,s}(h),h\leq H_{T}$. They converge at speed
$1/\sqrt{H_{T}K}$.

In a structural SVAR model, under the usual identification restrictions on
$\Phi,\Omega$, consistent estimates of the identifiable short run parameters
can be obtained from the moment conditions:

$\displaystyle\Gamma_{s}(1)=\Phi\Gamma_{s}(0),\;\;\;\Gamma_{s}(0)=\sum^{\infty}_{j=0}\Phi^{j}\Omega\Phi^{j^{\prime}},$
(3.6)

after replacing $\Gamma_{s}(h),\;h=0,1,...$ by $\hat{\Gamma}_{T,s}(h)$ These
estimators converge at speed $1/\sqrt{H_{T}K}$.

Let us now discuss the estimation of matrix $A$. A multivariate factor model
is commonly defined up to an invertible transformation. In the linear
framework, for any $(L,L)$ invertible matrix $Q$, we get the same $L$-variate
model with $A$ replaced by $AQ^{-1}$ and $y_{l}$ replaced by $Qy_{l}$ that is
an O.U. process as well. Under the identification restriction on $A$ which is
imposed to solve this multiplicity, we get an estimate $\hat{A}_{T}$ of $A$ by
using the principal component analysis (PCA). Next, by applying a pseudo
inverse of $\hat{A}_{T}$, which is
$\hat{A}^{+}_{T}=(\hat{A}^{\prime}_{T}\hat{A}_{T})^{-1}\hat{A}^{\prime}_{T},$
approximations $\hat{y}_{lT}(c_{k})=\hat{A}^{+}_{T}\hat{m}_{T}(c_{k})$ of
$y_{l}(c_{k})$ are obtained. This leads to the following corollary: 666If the
identification restriction on $C$ is $C^{\prime}C=Id_{L}$, the pseudo inverse
becomes $\hat{C}^{+}_{T}=\hat{C}^{\prime}_{T}$.

Corollary 3:

The matrix $A$ and the values $y_{l}(c_{k})$ are consistently estimated with
the convergence rate of estimators of $1/\sqrt{H_{T}}$.

In particular,

Corollary 4:

For $T=\infty$, the long run component $y_{l}(c),c\in[0,1],$ is filtered
without errors. Since

$y_{s}(t)=y_{t}-y_{l}(t/T)\approx y_{t}-\hat{y}_{lT}(c),$

for $t\in(cT,cT+H_{T})$, the short run component is known as well in any
neighbourhood of a date $cT$.

Corollary 4 shows that asymptotically, the underlying SR and ULR components
can be detected. Therefore, in this model involving two time scales, the
identification of the short and long run dynamics at different ”frequencies”
is possible without introducing any additional structural identifying
restrictions. This result is analogous to the asymptotic detection of smooth
deterministic local level [Dalhaus (2012), Dalhaus et al. (2017)], although in
a context of stochastic local levels. Corollary 4 is also compatible with the
remark of Beeler, Campbell (2012) in a critic of the Bansal et al. (2012) long
run modelling approach : ”Predictable variations in long run (component) might
exists in the data, but might be masked by temporary fluctuations that are
omitted from the long-run risk model. If this is the case, however, economic
agents must perceive those variations in order for them to influence asset
prices”.

It is also possible to derive the asymptotic distribution of the difference
between the local sample mean $\hat{m}_{T}(c_{k})$ and its stochastic limit
$m(c_{k})=Ay_{l}(c_{k})$ (see Appendix 3).

Proposition 5 : Let us assume that $H_{T}\rightarrow\infty$ and
$H_{T}^{2}/T\rightarrow 0$, when $T\rightarrow\infty$. Then we have :

$\sqrt{H_{T}}[\hat{m}_{T}(c_{k})-Ay_{l}(c_{k}),k=1,\ldots,K]$

tends in distribution to $N(0,Id\otimes\Sigma_{\infty}),$ with :

$\Sigma_{\infty}=\sum^{+\infty}_{h=-\infty}\;\mbox{cov}\;[y_{s}(t),y_{s}(t-h)].$

This asymptotic result can be used to derive the asymptotic distributions of
$\sqrt{H_{T}}[\hat{A}_{T}-A]$ and of
$\sqrt{H_{T}}[\hat{y}_{lT}(c_{k})-y_{l}(c_{k})]$.

Proposition 5 shows that the condition $H_{T}^{2}/T\rightarrow 0,$ if
$T\rightarrow\infty$, eliminates the effect of the smooth varying stochastic
level, when deriving these asymptotic distributions. If
$H_{T}^{2}/T\rightarrow\gamma>0$, say, an additional term should have to be
included in the asymptotic variance formula.

###  3.3 Long Run ACF

The dynamics of the ULR component can be analysed by means of the proxies
$\hat{y}_{lT}(c)$ of $y_{l}(c)$. Asymptotically, this is as if the underlying
continuous time O.U. process were observed on the interval $[0,1]$. Then, a
maximum likelihood method, that is an OLS estimator in our Gaussian case, can
be applied after substituting the proxies $\hat{y}_{lT}(c)$ for the true value
$y_{l}(c)$ over the grid of $(c_{1},\ldots,c_{K})$. Let us examine the
consequences of disregarding the first step estimation error, that is the fact
that $\hat{y}_{Tl}(c)\neq y_{l}(c)$ and computing the ML estimator on a
regularly spaced grid which is getting finer with $T$. The following well-
known result is commonly used in the high frequency data literature.

Proposition 6

Let us consider a regular grid $c_{1}<\ldots<c_{K_{T}}$ of [0,1], and assume
observable $y_{l}(c_{1}),\ldots,y_{l}(c_{K_{T}})$, then if
$K_{T}\rightarrow\infty$,

i) The volatility parameter $S$ can be estimated consistently with a
convergence rate $1/\sqrt{K_{T}}$.

ii) The ”drift” parameter $\Theta$ cannot be consistently estimated.

The last statement follows from the impossibility theorem for continuous time
models [see Banon (1978), Jiang, Knight (1997)]. Although the drift parameter
can be estimated, the distribution of its estimator does not degenerate to a
point mass at the true value even when $T\rightarrow\infty$. The estimation
risk on some of the long run parameters persists indefinitely, and it cannot
be diversified away given the observations on a single time series. Below, we
show that finite sample techniques can be used to address this issue.

###  3.4 Summary of the Estimation Approach

Under the identification restriction on the short run structural VAR and the
identification restriction on the long run factor (and $A$), the parameters
can be estimated along the following steps:

Step 1 : Compute the short run sample autocovariances at lags 0 and 1 from
formula (3.5) and apply a method of moment estimation based on the moment
conditions (3.6) under the short run identification restrictions on the short
run SVAR. This step provides consistent estimates of short run parameters
$\Phi,\Omega$.

Step 2 : Compute the short run local means at different dates
$c_{k}T,k=1,\ldots,K$, from formula (3.4). This step provides approximations
$\hat{m}_{T}(c_{k})$ of $Ay_{l}(c_{k}),k=1,\ldots,K.$

Step 3 : Concatenate the vector of $K$ summary statistics
$\hat{m}_{T}(c_{k}),k=1,\ldots,K$, each of dimension $n$ to perform a
principal component analysis (PCA) of the matrix :
$\displaystyle\sum^{K}_{k=1}[(\hat{m}_{T}(c_{k})\hat{m}^{\prime}_{T}(c_{k})]\equiv\hat{M}_{T}.$.
Keep only the principal directions associated with the statistically
significant eigenvalues of $\hat{M}_{T}$. Under the identification restriction
$A^{\prime}A=Id_{L}$ for$A$, this step provides :

i) an estimate $\hat{L}_{T}$ of the number of underlying long run components;

ii) an estimate of the ”factor sensitivity” matrix $A,\hat{A}_{T}$, say;

iii) the approximations of the underlying long run component values at the
$c^{\prime}_{k}s,k=1,\ldots,K$:

$\hat{y}_{lT}(c_{k})=\hat{A}^{\prime}_{T}\hat{m}_{T}(c_{k}),k=1,\ldots,K,$

where $\hat{A}^{\prime}_{T}$ is a pseudo-inverse of $\hat{A}_{T}.$

Step 4 : Apply the maximum likelihood estimation based on the joint
distribution of $y_{l}(c_{k}),k=1,\ldots,K,$ with the pseudo-observations
$\hat{y}_{lT}(c_{k}),k=1,\ldots,K,$ instead of the unobserved
$y_{l}(c_{k}),k=1,\ldots,K$, to get estimates $\hat{\Theta}_{T},\hat{S}_{T}$
of the long run parameters in (1.6).

Step 5 : This approach can be completed by a residual plot that compares the
observed values $y$ to fitted values $\hat{y}$ computed on another grid
$\gamma_{1},\ldots,\gamma_{K^{*}},$ i.e. the values $y(\gamma_{k}T)$ to the
values :

$\begin{array}[]{lcl}\hat{y}_{T}(\gamma_{k}T)&=&\hat{\Phi}_{T}^{(\gamma_{k}-\gamma_{k-1})T}\hat{y}_{s}(\gamma_{k-1}T)-\hat{y}_{lT}(\gamma_{k})\\\
\\\
&=&\hat{\Phi}_{T}^{(\gamma_{k}-\gamma_{k-1})T}[y(\gamma_{k-1}T)-\hat{y}_{lT}(\gamma_{k-1})]+\hat{y}_{lT}(\gamma_{k}).\end{array}$

Since the emphasis of this paper is on structural dynamic modelling rather
than the estimation in a narrow sense, a more detailed discussion of the
asymptotic theory addressing the first step estimation errors on
$y_{l}(c_{k})$ and of the appropriate choices of $H_{T}$ and the size $K_{T}$
of a regular grid on $[0,1]$, when $T$ tends to infinity is out of the scope
of this paper. This would require a functional limit theorem for triangular
arrays [see Gourieroux et al. (2022)].

The distinct rates of convergence of the consistent estimators mentioned in
the subsection above, and the impossibility theorem which implies that
$\hat{\Theta}_{T}$ is inconsistent, complicate the efficiency analysis 777The
block decomposition introduced in the proposed estimation approach is a time
series analogue of the granularity theory for panel data with a dynamic common
factor [Gagliardini, Gourieroux (2014)]. We refer to this literature on panel
data for the difficulty in defining the notion of efficient estimation when
the rates of convergence of estimators are different.. It follows from the
impossibility theorem that when $T\rightarrow\infty$ the accuracy of the
consistent approximations of the long run component values
$\hat{y}_{lT}(c_{k}),k=1,\ldots,K_{T}$, that converge at a speed of order
$1/\sqrt{H_{T}},$ can be disregarded and the accuracy of
$\hat{\Theta}_{T},\hat{S}_{T}$, can be negligible when $K_{T}/H_{T}\rightarrow
0$. Then the finite sample confidence intervals (CI) of any known scalar
nonlinear transformation of $\hat{\Theta}_{T},\hat{S}_{T}$ can be derived by
the test inversion [see Kendall, Stuart (1967), Chap. 20, for a description of
this technique, Stock (1991), Section 2.3, or Pesavento, Rossi (2006), for an
application in a univariate standard local-to-unity model, Carpenter (1999),
Dufour (2006) for nuisance parameters and bootstrap extensions].

The estimation steps 1-4 can be related to the literature on mixed frequency
data because of the apparent resemblance of combining data measured on one
time scale with averaged data measured on a time scale with larger units [see
e.g. Deistler et al. (2016), Ghysels (2016)]. Our approach differs from this
literature in the following respects:

i) The observations at mixed frequencies (MF) are generated by the same time
series, while we consider two different latent processes. In our approach, the
observations at different frequencies are used as a tool for identifying the
short and long run dynamics, rather than to interpolate the missing
observations.

ii) The MF literature considers fixed frequencies, i.e. two fixed time scales.
In our framework, we consider an ultra long time scale with a unit that tends
to infinity with the number of observations. Therefore, the impossibility
theorem and its consequences for the long run prediction intervals do not
concern the MF literature.

iv) Similarly to the MF literature, our estimation approach is based on the
Yule-Walker equations at different frequencies to disentangle the short and
long run parameters [see e.g. Chen, Zadrozny (1998)].

##  4 Long Run Predictions

Let us now consider the long run predictions 888The prediction problem is
different from the problem of testing the unit root hypothesis and comparing
the power functions of the alternative test procedures.. Our objective is to
find the intervals for predictions at large leads as a fixed function of the
sample size with a good coverage rate. We assume that the underlying
parametric model is well specified.

###  4.1 Information Sets

Corollary 4 implying that the long and short run components can be detected
and identified asymptotically simplifies the analysis of predictions, which
depend on different information sets. The predictions are conditioned on the
current and lagged values of the observed process
$\underline{y_{t}}=(y_{t},y_{t-1},...)$. As only shocks to the ULR component
are needed to analyse the systematic long run risk 999Such a practice is
facilitated in our framework since the SR and ULR components are independent.
This independence assumption and the two time scales provide the structural
aspect to the VAR allowing for separate analysis of shocks on the SR and ULR
components, respectively., it is preferable to work with the information set
$(\underline{y_{s}(t)},\underline{y_{l}(t/T)}).$ From Corollary 4 these two
information sets are equivalent when $T$ is large.

This result is easy to interpret in the context of the Wold/Volterra
decomposition. Let $(F_{t})$ denote the sequence of $\sigma$-algebras
generated by process $(y_{t})$. The purely predictable component $Z$ is
measurable with respect to the $\sigma$-algebra $F_{-\infty}=\cap_{t}F_{t}$
and is stochastic if $F_{-\infty}$ does not degenerate to
$({\O},\mathcal{Y})$. Thus this component belongs to the $\sigma$-algebra
generated by process $(y_{t})$. Even if the observations are available only
since an initial date $t=0$ rather than since $-\infty$, the purely
predictable component $Z$ can be recovered asymptotically, if
$T\rightarrow\infty$

###  4.2 Theoretical Predictions

Let us consider the predictive distribution
$l(y(T+H_{T})|\underline{y_{(T)}})$ of $y$ at horizon $H_{T}$, that is, the
conditional distribution of $y(T+H_{T})$ given $\underline{y_{(T)}}$. Given
the equivalence of the two information sets, the independence between the
short and long run components and the Markov assumption on these components,
the predictive distribution is:

$l(y(T+H_{T})|\underline{y(T)})\approx
l_{s}(y_{s}(T+H_{T})|y_{s}(T))*l_{l}[Ay_{l,T}(T+H_{T})|y_{l,T}(T)],$ (4.1)

where $*$ denotes the convolution $l_{s},l_{l}$ the (conditional)
distributions of SR and ULR components respectively.

Two cases can be distinguished:

i) For short run predictions, with $H_{T}/T\rightarrow 0$, when
$T\rightarrow\infty$, we have :

$\displaystyle l(y(T+H_{T})|\underline{y(T)})$ $\displaystyle\simeq$
$\displaystyle l_{s}(y_{s}(T+H_{T})|y_{s}(T))*\delta_{Ay_{l,T}(t)}.$ (4.2)
$\displaystyle=$ $\displaystyle l_{s}(y(T+H_{T})-Ay_{l,T}(T)|y_{s}(T)),$ (4.3)

where $\delta_{Ay_{l,T}(T)}$ denotes the point mass at $Ay_{l,T}(T)$.

ii) For long run predictions, with $H_{T}/T\rightarrow\gamma,\gamma>0$, when
$T\rightarrow\infty$, by the ergodicity of the short run component and the
fact that it has zero mean :

$\displaystyle l(y(T+H_{T})|\underline{y(T)})$ $\displaystyle\simeq$
$\displaystyle l_{l}[Ay_{l,T}(T+H_{T})|y_{l,T}(T)]$ (4.4) $\displaystyle=$
$\displaystyle l_{l}[Ay_{l}(1+\gamma)|y_{l}(1)].$ (4.5)

Hence, at short horizons, only the short run dynamics and short run component
matter, up to a known drift. At long horizons, only the long run dynamics and
long run component matter.

###  4.3 Sample-Based Predictions

The theoretical predictive distributions derived in Section 4.2 depend on the
unobserved factor values and on the unknown short and long run parameters. In
practice, they need to be replaced by the estimated predictive distributions.
This replacement is not innocuous, even for large T, because of the
impossibility theorem implying the impossibility to approximate $\Theta$
consistently. There is a long run estimation risk that has to be accounted for
in the prediction interval computation. In applied research, it would concern
the computation of the Value-at-Risk (VaR), the required capital (in a
financial application of VaR), or deriving the impulse response functions
[compare to Gospodinov (2004), Pesavento, Rossi (2006)].

In this section we focus on a combination $X_{T+H_{T}}=a^{\prime}Y(T+H_{T})$
of variables to be predicted at large horizons $H_{T}=cT$ and on one-sided
prediction intervals. The one-sided prediction intervals are essential for the
computation of the VaR and required capital at horizon $H_{T}$ under the
current financial regulation. The approach below is easily extended to two
sided prediction intervals. We denote the set of all model parameters by
$\beta$.

####  4.3.1 The standard approach

Before we describe the proposed method, let us recall the standard prediction
interval which can be computed when a consistent estimator $\hat{\beta}_{T}$
of $\beta$ is available.

i) For a given risk level $\alpha$, a theoretical conditional quantile $q$
satisfies the following equality:

$P_{\beta}[X_{T+H_{T}}<q(1-\alpha,\beta,\underline{Y(T)})|\underline{Y(T)}]=1-\alpha,\forall\beta.$
(4.6)

In our Markov framework, it is asymptotically equivalent to :

$P_{\beta}[a^{\prime}Y_{s}(T+cT)+a^{\prime}Y_{l}(T+cT)<q(1-\alpha;\beta,Y_{s}(T),Y_{l}(T))|Y_{s}(T),Y_{l}(T)]=1-\alpha,\forall\beta,$

or, by the independence between the SR and ULR components and the ergodicity
of the SR component :

$P_{\beta}[a^{\prime}Y_{s}(T+cT)+a^{\prime}Y_{l}(T+cT)<q(1-\alpha;\beta,Y_{l}(T))|Y_{l}(T)]=1-\alpha,\forall\beta.$
(4.7)

The only parameters appearing in this expression are the long run parameters
$\Theta,S$, and the parameters of the stationary distribution of the short run
component (since processes $Y_{s}$ and $Y_{l}$ are independent and $Y_{s}(T)$
no longer appears in the conditioning set).

ii) Since the conditional quantile depends on the unknown parameter $\beta$
and on the unobserved long run component, it is common to replace them by
their consistent approximations in the expression of the conditional quantile,
that is $q(1-\alpha,\hat{\beta}_{T},\hat{Y}_{l}(T)),$ and to introduce the
associated estimated prediction interval. Then for large $T$, we have :

$P_{\beta}[a^{\prime}X_{T+H_{T}}<q(1-\alpha;\hat{\beta}_{T},\hat{Y}_{l}(T))|Y_{l}(T)]\simeq
1-\alpha,\forall\beta.$ (4.8)

iii) Even if a consistent estimator $\hat{\beta}_{T}$ is available, in finite
sample the estimation risk can exist. It is commonly evaluated by a bootstrap
equivalent of a Monte-Carlo experiment based on simulated trajectories with
the true parameter value replaced by its estimate $\hat{\beta}_{T}$ [see
Kilian (1999)].

####  4.3.2 The approach adjusted for finite sample

As shown in Section 3.3, parameter $\Theta$ cannot be consistently estimated,
which follows from the impossibility theorem. This implies that the standard
approach mentioned above, including possibly the bootstrap procedure, is no
longer valid. Yet, in Section 3.4 we have shown that when
$K_{T}/H_{T}\rightarrow 0,$ confidence sets for $\Theta$ , which are valid in
finite sample, can be obtained. Then, the remaining parameters, for which
consistent estimators are available, can be replaced by their estimates in the
prediction interval formula without entailing significant errors at large $T$.

Let $I_{T}(1-\alpha_{1};\underline{Y(T)})$ denote a confidence set for
parameter $\beta$ at confidence level $1-\alpha_{1}$. It satisfies :

$P_{\beta}[\beta\in
I_{T}(1-\alpha_{1};\underline{Y(T)})]=1-\alpha_{1},\forall\beta.$ (4.9)

Then, a one-sided sample-based prediction interval at confidence level
$1-\alpha$ is obtained by fixing the upper bound of the interval as :

$Q(\alpha,\alpha_{1};\underline{Y(T)})=\max_{\beta\in
I_{T}(1-\alpha_{1};\underline{Y_{T}})}q(1-\alpha+\alpha_{1},\beta,\hat{Y}_{l}(T)),$
(4.10)

for any choice of $\alpha_{1}\in[0,\alpha]$. Indeed we have :

$P_{\beta}[a^{\prime}X_{T+H_{T}}<Q(\alpha,\alpha_{1};\underline{Y(T)})|\underline{Y(T)}]\geq
1-\alpha,\forall\beta,\alpha_{1},$ (4.11)

by applying the Bonferroni inequality.

This sample-based prediction interval depends on the confidence set chosen for
the estimation of $\beta$ and in particular on the level $\alpha_{1}$. Then,
we can also optimize it with respect to $\alpha_{1}$. In terms of financial
regulation, this corresponds to searching for the minimum required capital
compatible with level $1-\alpha$. Thus, the one-sided sample-based prediction
interval has an upper bound defined by a min-max procedure :

$Q^{*}(\alpha;\underline{Y(T)})=\min_{0<\alpha_{1}<\alpha}\max_{\beta\in
I_{T}(1-\alpha_{1};\underline{Y_{T}})}q(1-\alpha+\alpha_{1};\beta,\hat{Y}_{l}(T)).$
(4.12)

Let $\alpha^{*}_{1T},\beta^{*}_{T}$ denote the solutions of the min-max
optimization (4.12). These solutions depend on $\underline{Y(T)}$ and are
fixed endogenously.

The consequences of the additional estimation risk, as compared with the
standard computation of Section 4.3.1 are examined below. We have :

$\displaystyle Q^{*}(\alpha;\underline{Y(T)})$ $\displaystyle=$ $\displaystyle
q(1-\alpha;\hat{\beta}_{T},\hat{Y}_{l}(T))+\\{q(1-\alpha+\alpha^{*}_{1T};\beta^{*}_{T},\hat{Y}_{l}(T))-q(1-\alpha;\hat{\beta}^{*}_{T},\hat{Y}_{l}(T))\\}$
(4.13) $\displaystyle=$ $\displaystyle
q(1-\alpha;\hat{\beta}_{T},\hat{Y}_{l}(T))+\\{q(1-\alpha;\beta^{*}_{T},\hat{Y}_{l}(T))-q(1-\alpha;\hat{\beta}_{T},\hat{Y}_{l}(T))\\}$
$\displaystyle+$
$\displaystyle\\{q(1-\alpha+\alpha^{*}_{1T};\beta^{*}_{T},\hat{Y}_{l}(T))-q(1-\alpha;\beta^{*}_{T},\hat{Y}_{l}(T))\\}.$

Formula (4.13) has three terms with the following interpretations:

$\bullet$ the first term is the estimated quantile, that is, the estimated
theoretical measure of risk.

$\bullet$ the next two terms represent the estimation risk. The first among
them is due to the replacement of approximation $\hat{\beta}_{T}$ by an
extreme $\beta^{*}_{T}$ of the confidence set. In the univariate case
illustrated below, it is either a lower or upper bound of the confidence
interval. The second term is due to adjusting of confidence levels for the
lack of accuracy. It increases the confidence interval from $1-\alpha$ to
$1-\alpha+\alpha^{*}_{1T},$ making it more conservative in the light of the
definition of a theoretical Value-at Risk (VaR) [Robinson (1977)].

Let us now illustrate the above approach in a univariate framework.

Example 1: In the univariate model (2.6)-(2.8) the impossibility theorem
concerns the scalar parameter $\theta$. All other parameters $\eta,\varphi,s$,
as well as the value of the underlying long run component at $T$ can be
replaced by consistent estimators and approximations. Therefore, we focus on
the theoretical quantile as a function of parameter $\theta$. It is easy to
check that for predicting $y(T+H_{T})$ at time $T$ and a large horizon
$H_{T}=\gamma T$, the corresponding quantile is:

$q(\theta,\alpha)=\exp(-\theta\gamma)y_{l}(T)+\Psi^{-1}(1-\alpha)\sqrt{\eta^{2}+\displaystyle\frac{s^{2}}{2\theta}[1-\exp(-2\theta\gamma)]},$

where $\Psi$ denotes the cumulative distribution function of the standard
normal.

Let us now consider a confidence interval for $\theta$ at level $1-\alpha_{1}$
:

$I_{T}[1-\alpha_{1};\underline{Y(T)}]=[\theta_{L,T}(\alpha_{1}),\theta_{U,T}(\alpha_{1})],\;\mbox{say}.$

Since function $q(\theta;\alpha)$ is not necessarily increasing or decreasing
in $\theta$, we get :

$Q(\alpha,\alpha_{1};\underline{Y(T)})=\max[q(\theta_{L,T}(\alpha_{1}),\alpha-\alpha_{1}),q(\theta_{U,T}(\alpha_{1}),\alpha-\alpha_{1})].$

In the last optimization with respect to $\alpha_{1},0<\alpha_{1}<\alpha$, the
optimum will not be reached in practice for $\alpha_{1}=0$, or
$\alpha_{1}=\alpha$. Indeed for $\alpha_{1}=0$, the confidence interval is
usually of infinite length, whereas for $\alpha_{1}=\alpha$, that is
$\alpha-\alpha_{1}=0,$ $q(\theta,\alpha-\alpha_{1})$ is infinite.

####  4.3.3 Bayesian approach and the prudential principle

The approach developed in Section 4.3.2 assumes a well-specified parametric
dynamics for the SR and ULR components. The same assumption underlies the
derivation of the one-sided prediction intervals in a Bayesian framework [see
e.g. Muller, Watson (2016), (2020), Schorfheide, Song, Yaron (2018) for such
approaches). Let us introduce a prior $\pi(\beta)$. Then a Bayesian one-sided
prediction interval at level $1-\alpha$ can be defined by a quantile
$Q^{B}(\alpha;\underline{Y_{T}})$ such that :

$P[a^{\prime}X_{T+H}<Q^{B}(\alpha;\underline{Y(T)})|\underline{Y(T)}]=1-\alpha,$
(4.14)

where the conditional probability includes the integration with respect to the
”stochastic” parameter $\beta$. More precisely, we get :

$\displaystyle\int
P_{\beta}[a^{\prime}X_{T+H}<Q^{B}(\alpha;\underline{Y(T)})|\underline{Y(T)}]\pi(\beta|\underline{Y(T)})d\beta=1-\alpha,$
(4.15)

where $\pi(\beta|\underline{Y(T)})$ denotes the posterior distribution of
$\beta$. As already noted in the literature [see [Robinson (1977), Muller,
Norets (2016), Muller, Watson (2020), Section 7.4, for confidence sets versus
credible sets], this condition does not ensure that the condition :

$P_{\beta}[a^{\prime}X_{T+H}<Q^{B}(\alpha;\underline{Y(T)})|\underline{Y(T)}]\geq
1-\alpha,\forall\beta,$

holds uniformly in $\beta$, which is a common characteristic of a prediction
interval. This coverage condition is only satisfied in average for a specific
posterior (i.e. indirectly for a specific prior). In particular, for financial
regulation, this Bayesian practice is not prudential, as in general it implies
an undervaluation of required capital and is very sensitive to the choice of
the prior 101010Except in the special case when
$P_{\beta}[\alpha^{\prime}X_{T+H}<Q^{B}(\alpha;\underline{Y(T)}|\underline{Y(T)}]$
is independent of $\beta$. In a simple framework, this can arise if the
statistic used to construct the prediction interval has some invariance
property with respect to the nonidentifiable parameters. But such invariant
procedures are difficult to construct for a multivariate $\Theta$.. As a
consequence, it is not permitted by the current prudential financial
supervision.

##  5 Illustration

To illustrate the properties of the SVAR model with a stationary ULR
component, we perform a Monte-Carlo study to show the behaviour of various
summary statistics. Next we apply the proposed methodology to a set of
macroeconomic time series.

###  5.1 A Monte-Carlo study

####  5.1.1 The model

We consider a bivariate model $h=2$ with one long run component $L=1.$ The
short run parameters are :

$\Phi=\left(\begin{array}[]{cc}0.3&0\\\
0&0.7\end{array}\right),\Omega^{1/2}=\left(\begin{array}[]{cc}1&1\\\
0&2\end{array}\right),$

with a triangular $\Omega^{1/2}$ matrix to get directional short run causality
between these two components. The matrix of ”factor sensitivities” is set to :
$A=(1,1)^{\prime}$ and the long run parameters for the univariate long run
component to $\theta$ such that $\exp(-10\;\theta)=0.4$ and $s=1.$

The simulated series are of length $T=7200,$ which corresponds to about 20
years of 360 days, if the basic time unit is one day. We fix the long run grid
to $c_{1}=1/20,c_{2}=2/20,\ldots,c_{20}=20/20$ corresponding to multiples of
years, and the length of the short run episode to $H_{T}=60$, which is
equivalent to two months.

Figure 1 provides a simulated path of process $y(t)=(y_{1t},y_{2t})^{\prime}$.
For each series the effect of the stationary ULR component is masked by the
variation of the short run component. We also observe rather parallel smooth
waves created by the single latent ULR factor.

[Figure 1 : Simulated Trajectory of $y(t)$]

Figure 2 displays the evolutions of the averages
$\hat{m}_{1}(c_{k}),\hat{m}_{2}(c_{k}),k=1,\ldots,20$ obtained with a
bandwidth of 60.

[Figure 2 : Trajectory of $\hat{m}(c_{k})$]

We observe parallel evolutions of these averages revealing an underlying
single long run component.

####  5.1.2 The sample autocorrelation functions

Let us now consider the different types of ACF introduced in Section 3. These
are the standard joint ACF computed from $y_{t},t=1,\ldots,T,$ up to lag
$H=1000$ (Figure 3) and the standard joint ACF computed from
$y_{t},t=1,\ldots,H_{T}=60$, up to lag $H=15$ (Figure 4). The main difference
between them is how the autocorrelation is demeaned, i.e. globally in Figure
3, and locally in Figure 4.

Figure 5 provides the joint ACF computed from 20 observations of the averages
$\hat{m}(c_{k}),k=1,\ldots,K=20$, up to lag 10 of the long run time unit. It
is given for bandwidths of 60 and 80, respectively.

These figures show the importance of considering jointly the different ACF’s.
The standard one in Figure 3 reveals the persistence, that is, a slow decay of
the auto- and cross-correlations. However, this persistence effect conceals
the important features concerning the short and long run components,
respectively. By diminishing the length of the time series in Figure 4, and
then adjusting for the mean, we reveal the dynamics of the short run
component. Next by averaging and changing the time scale in Figure 5, we
reveal the dynamics of the long run component.

[Figure 3 : The Standard ACF]

[Figure 4 : The Local ACF ]

[Figure 5 : ACF of Averages]

####  5.1.3 Finite sample properties of the ML estimator of $\theta$.

Let us also provide information on the lack of consistency and finite sample
distribution of the ML estimator of $\theta$ computed from observations on
process $y_{l}$ over a grid $c_{k}=k/K$, of $[0,1].$

We consider $K=25$ used in the application of Section 5.2, and the Gaussian
AR(1) model :

$\tilde{y}(k)=\exp(-\theta/K)\tilde{y}(k-1)+\sqrt{1-\exp(-2\theta/K)}\tilde{\varepsilon}(k),k=1,\ldots,K,$
(5.1)

where the $\tilde{\varepsilon}(k)^{\prime}s$ are IIN(0,1). Without loss of
generality the scale parameter is set equal to 1, since parameter $\theta$ is
scale invariant. Parameter $\theta$ is estimated by the maximum likelihood
conditional on $y(0)$ where $y(0)$ is drawn in $N(0,1)$.

For each level $\alpha,\alpha=0.05,0.1,0.5,0.9.0.95$, we derive the one sided
$\alpha$\- percentiles interval for $\hat{\rho}_{K}=\exp(-\hat{\theta}/K)$ as
a function of $\rho=\exp(-\theta/K)$. This provides belts that can be inverted
to get the $1-\alpha$ confidence intervals for $\rho$ from an estimate
$\hat{\rho}_{K}$. The sudden change of slope in the 0.95 quantile occurs when
$\rho$ is too large, It is due to the finite sample properties of the
estimator close to a unit root [see Stock (1991), Figure 2, in the pure unit
root case].

By comparing the estimate with the 50% quantile, we get information on the
skewness of the finite sample distribution of the estimator of $\rho$.

[Insert Figure 6: Confidence Belts]

###  5.2 Application to Economic Time Series

####  5.2.1 The series

As an empirical exercise and to facilitate the comparison with the recent
literature [Muller, Watson (2016), (2020)], we consider the same five
quarterly US macro-series as those examined by Muller, Watson in their papers.
The series obtained from the Federal Reserve Bank of St Louis FRED database
are the growth rates of total factor productivity (TFP), per-capita GDP,
consumption, investment and labor compensation. They are not annualized and
cover the period 1948 Q2-2019 Q2 prior to the COVID pandemic. The evolutions
of these series are displayed in Figure a.1, which corresponds to Figure 2 in
Muller, Watson (2020). We give in Figure a.2 in Appendix 4 the associated ACF
and cross-ACF.

The standard tools of time series analysis reveal no persistence, i.e. no
remaining ”unit root” features in the growth rates. However, in the framework
of a model with SR and ULR components, the standard ACF provides no clear
information about the ULR. The ACF at short lags reveals only the persistence
of the short run component. The autocorrelations at very long lags are close
to zero due to the large variance of the SR component relatively to the
variance of the underlying ULR component.

Following the standard approach, ARMA models are fitted to the series without
taking into account the underlying structure represented as the sum of a SR
and an ULR components. In particular, they are estimated by OLS-type
estimators that optimize the SR prediction performance at horizon 1, whereas
we are interested in the ULR prediction performance. Nevertheless we estimate
an ARMA model for each series and then compute the standard prediction
intervals at a date corresponding to $T=275$. Consistently with our proposed
ULR approach, we choose the ultra-long horizon equal to $H=200$ quarters, i.e.
50 years 111111Different ULR horizons corresponding to other fractions of $T$
can also be considered, such as $H=120$, i.e. $30$ years.. The standard
prediction intervals are given in the first column of Table 2.

####  5.2.2 Filtering the ULR component

To filter out the potential ULR component, we partition the time scale
measured in quarters into non-overlapping intervals of length 11 quarters.
Then, the raw data are averaged over each interval and their average is
assigned to the middle date of each interval. We get new series of length 25
on a deformed time scale of 21 quarters. Although this filtering approach
follows the same idea as the low frequency approach of Muller, Watson (2016),
(2020), it differs from their approach with respect to the following. Muller,
Watson base their analysis on projections on a fixed number of low frequency
periodic functions of time, with frequencies such as $2T/j,j=1,\ldots,14$ [see
Muller, Watson (2020), Section 3]. This approach uses all observations to
construct the summary statistics. Therefore, Muller and Watson’s approach is a
global-in-time approach while ours is local-in-time. A drawback of the global-
in-time approach is easy to point out in the context of a model with two
additive components, which are a ULR component and a white noise as the SR
component. It is well-known that the spectrum of white noise process is
characterized by positive equal weights at each frequency. Hence, the global-
in-time projection approach does not eliminate the effect of the noise, and as
a consequence does not focus on the ULR component only. From a purely
theoretical perspective, this approach shows a lack of coherency with respect
to the Wold representation. While the modelling assumes a regular infinite
order moving average process, the projection is intended to detect singular
components (which are stochastic combinations of deterministic periodic
functions of time). Our approach distinguishes between the regular and
singular (i.e. SR and ULR) components and attenuates the effects of the SR
component in order to filter out the ULR component. The ACF and cross-ACF of
the series of filtered components are provided in Figure a.3 in Appendix 4.
The AR(1) models are estimated from each filtered serie. The values of
estimated autoregressive coefficients in the ULR time scale and the calendar
time scale measured in quarters are given in Table 1.

Table 1 : Autogressive Coefficients for the ULR Component

Series | tfp | gdp | c | inv | lab.
---|---|---|---|---|---
ULR Scale | $(-)0.275$ | $(-)0.139$ | $(-)0.025$ | $0.546$ | $0.160$
quarter scale | $0.889$ | $0.836$ | $0.714$ | $0.947$ | $0.847$

Due to time deformation, the small values of autregressive coefficients in the
ULR scale becomes much larger and closer to 1 in the calendar time scale
measured in quarters.

Next, each estimated AR(1) model is used to predict at the same ULR horizon as
described in Section 5.2.1 corresponding to about $H=18$ in the ULR time
scale. 121212Some filtered ULR series show negative autocorrelation
coefficients. They seem to be due to the special cleaning treatment of the
series in the database. Since the prediction formulas depend on $\rho^{2}$
only, the sign has no impact on the prediction intervals. We provide the plug-
in theoretical prediction intervals in the second column of Table 2, and the
intervals adjusted for estimation risk (see Section 4.3.2) in the third
column. The differences between columns 2 and 3 show the importance of taking
into account the estimation risk in ULR predictions due to the significant
increase of $10\%$ to $200\%$ in the length of the prediction formula in the
third column.

These results can be compared with the prediction intervals computed from the
series of raw data. The prediction intervals based on raw data are shorter
than the adjusted filtered ULR prediction intervals. In addition, they are
biased, since they assume a constant mean instead of a slowly varying mean.
Our results cannot be directly compared to the prediction intervals derived
for the same series in Muller, Watson as they derived prediction intervals for
the change averages between $T$ and $T+H$ at $90\%$. Because of the averaging
and using critical values associated with lower confidence levels, Muller,
Watson report shorter prediction intervals.

Table 2 : Prediction Intervals at $95\%$

Series | Raw Data | Filtered ULR | Filtered ULR
---|---|---|---
|  | (unadjusted) | (adjusted for
|  |  | estimation risk)
tfp | $(0.781,1.634)$ | $(0.437,1.436)$ | $(0.108,1.763)$
gdp | $(0.0065,0.0090)$ | $(0.0054,0.0081)$ | $(0.0048,0.0087)$
c | $(0.0069,0.0089)$ | $(0.0070,0.0094)$ | $(0.0067,0.0096)$
inv | $(0.0065,0.0085)$ | $(-0.0017,0.0113)$ | $(-0.0136,0.0231)$
lab | $(0.0065,0.0091)$ | $(0.0076,0.0111)$ | $(0.0069,0.0119)$

##  6 Concluding Remarks

The aim of this paper is to introduce stationary long run components for
stationary SVAR models and to derive long run prediction intervals for this
model. It follows from the impossibility theorem that the long run predictions
have to account for a significant long run estimation risk that makes the
prediction intervals significantly wider.

The ULR framework is an alternative to other local-to-unity models introduced
in the literature, where the long run component leads to explosive evolutions,
with trajectories that either explode in the ultra long run, or take extreme
values with high probabilities (see Appendix 1). This questions the validity
of long run predictions based on these models for a variety of economic or
financial variables, which are not explosive and take values from limited
sets.

As shown in this paper, by applying well-chosen block decomposition of
observations, it is possible to separate the statistical inference on short
and long run parameters as well as to identify the short and long run
components of the process. The main message of this paper is related to the
impossibility theorem, and conveys the impossibility to consistently estimate
some long run parameters. This implies that estimation risk is significant and
has to be taken into account in the long run prediction intervals, as well as
in the VaR and required capital computation methods. While the proposed
approach is clearly model-based, from a prudential perspective the long run
estimation risk should not be artificially underestimated by either using an
over-scaled local-to-unity model (see Appendix 1), or a Bayesian prediction
approach that does not ensure the uniform coverage of the prediction interval
(Section 4.3.3). More generally, prediction methods that appear misleadingly
precise because of underestimating the long run risk, need to be avoided.

The analysis presented in the paper can be extended to additive decompositions
of short and long run component with non- Gaussian distributions, to avoid the
standard identification issues in the second-order analysis of SVAR [see
Gourieroux, Monfort, Renne (2017)]. It can also be extended to nonlinear
dynamic models with a non-additive decomposition into the short and long
components with independent nonlinear dynamics of those components [see
Gourieroux et al. (2021)].

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Figure 1 : Simulated Trajectory of Bivariate Process $y(t)$

Figure 2 : Trajectory of $\hat{m}(c_{k})$

Figure 3 : The Standard ACF

Figure 4 : The Local ACF

Figure 5 : ACF of Averages

bandwidth = 60

bandwidth= 80

Figure 6: Quantiles of Estimated $\rho$ at 0.05, 0.1, 0.5, 0.9 and 0.95

Appendix 1

Local-To-Unity (LTU) Models

This appendix provides a brief survey of the different types of LTU models
introduced in the literature to compare their ability of representing the
dynamics of time series featuring ”persistence” in sample autocorrelations and
a significant long run risk. The condition on the extremes is :
$\sup_{t}P[|y_{t}|>A]\equiv\pi(A)<1$, say. The above condition eliminates
explosive trends as well as explosive variances. For expository purpose, we
consider the univariate (AR(1)) Gaussian process, where $(\varepsilon_{t})$
denotes a Gaussian white noise.

Initially the aim of the LTU models was to approximate a unit root model such
that $Corr(y_{t},y_{t-1})=1.$ Recall that there exists two types of unit root
Gaussian AR(1) models:

i) The random walk

$y_{t}=y_{t-1}+\sigma\varepsilon_{t},\sigma>0,$

where $y_{0}$ is either deterministic, or stochastic, independent of
$\varepsilon_{t},t>0$. This process is such that
$y_{t}=y_{0}+\Sigma^{t}_{\tau=1}\varepsilon_{\tau}$. It is nonstationary and
”explosive” with $\sup_{t}P(|y_{t}|>A)=1.$

Such models have for instance be applied to time varying parameters (mean and
log-variances) to analyze the long run GDP growth in Antolin-Diaz et al.
(2017).

ii) The singular process

$y_{t}=y_{t-1},$

with a stochastic $y_{0}$. This is a stationary process such that
$\sup_{t}P[|y_{t}|>A]=P(|y_{0}|>A)=\pi(A)<1.$ It has no explosive features.

Let us now consider local-to-unity models, which are written for triangular
arrays, $y_{T}(t)$. Thus the condition on the extremes is expected to be
uniform in $T$:

$\sup_{T}\sup_{t\leq T}P_{T}[|y_{T}(t)|>A]=\pi(A)<1.$ (a.1)

a) Local-to-Unity with 0 initial value [Chan, Wei (1987), Phillips (1987)]

$y_{T}(t)=\exp(-c/T)y_{T}(t-1)+\sigma\varepsilon_{t}$, with $y_{T}(0)=0,c>0.$

The distribution of $y_{T}(t)$ is
$N\left(0,\sigma\displaystyle\frac{1-\exp(-2ct/T)}{1-\exp(-2c/T)}\right)$.

In particular $y_{T}(\gamma T)\sim
N\left[0,\sigma^{2}\displaystyle\frac{1-\exp(-2c\gamma)}{1-\exp(2c/T)}\right],\gamma>0$.

At dates equal to multiples of the number of observations, we see that
$P_{T}[|y_{T}(\gamma T)|>A]\rightarrow 1$, for $T\rightarrow\infty$. The
condition on the extremes is not satisfied.

b) Stationary Local-to-Unity [Elliott (1999), top of p820, Elliott, Muller
(2003)].

$y_{T}(t)=\exp(-c/T)y_{T}(t-1)+\sigma\varepsilon_{t},$

where the initial value $y_{T}(0)$ is drawn in the unconditional distribution
$N(0,\displaystyle\frac{\sigma^{2}}{1-\exp(-2c/T)})$.

When $T\rightarrow\infty$, the unconditional distribution of $y_{T}(t)$ tends
to $N(0,\infty)$, and the condition on the extremes is not satisfied.

c) Scaled Local-to-unity with $1/T$ scale effect [Muller, Watson (2016),
footnote 9, Muller, Watson (2020), p11].

$\left\\{\begin{array}[]{lcl}y_{T}(t)&=&\exp(-c/T)y_{T}(t-1)+\displaystyle\frac{\sigma}{T}\varepsilon_{t},\\\
\\\ y_{T}(0)&=&0.\end{array}\right.$

In this case, we get :

$y_{T}(\gamma T)\sim
N(0,\displaystyle\frac{\sigma^{2}}{T^{2}}\displaystyle\frac{1-\exp(-2c\gamma)}{1-\exp(-2c/T)})\approx
N(0,0),$

when $T$ tends to infinity. The scale effect is too strong leading to a
deterministic constant zero at long horizons and an artificial total
diversification of the long run risk.

d) Scaled Random Walk [Muller, Watson (2020), p11]

$\left\\{\begin{array}[]{lcl}y_{T}(t)&=&y_{T}(t-1)+\displaystyle\frac{\sigma}{T}\varepsilon_{t},\\\
\\\ y_{T}(0)&=&0.\end{array}\right.$

Similarly as for the scaled LTU in c) we have :

$y_{T}(\gamma T)\sim N(0,\displaystyle\frac{\sigma^{2}}{T^{2}}\gamma
T)=N(0,\displaystyle\frac{\sigma^{2}\gamma}{T})\approx N(0,0),$

when $T$ tends to infinity. The scale effect is too strong.

e) The Ultra Long Run (ULR)

This dynamic model bridges the gap between the standard LTU models a), b) and
the scaled LTU in c). It is defined as :

$y_{T}(t)=\exp(-c/T)y_{T}(t-1)+\sigma\sqrt{1-\exp(-2c/T)}\varepsilon_{t}.$

For each fixed $T$, we get a stationary process as in example b), but due to
the choice of the variance, the unconditional distribution $N(0,\sigma^{2})$
no longer depends on $T$. Therefore the uniform condition for the extreme
becomes:

$\sup_{T}\sup_{t\leq T}P_{T}[|y_{T}(t)|<A]=P(\sigma\mathcal{U}<A)=\pi(A)<1,$
where $\mathcal{U}\sim N[0,1]$, and is satisfied.

When $T\rightarrow\infty$, the dynamic is close to the ”dampered unit root”
process :
$y_{T}(t)=(1-\displaystyle\frac{c}{T})y_{T}(t-1)+\lambda/\sqrt{T}\varepsilon_{t}$,
considered in Gospodinov (2009), Gospodinov, Maynard, Pesavento (2021).

f) Other time deformation models

The ULR model can be extended to other time deformations, for instance
corresponding to a large time unit of $T^{d},d>0$. It is easy to show that the
ULR model corresponding to $d=1$ is a limiting case. Indeed, by taking
appropriate averages of the observations, we observe asymptotically the
underlying OU process on $(0,\infty),$ if $d<1$. Therefore parameter $c$ (in
the notation of this appendix, $\theta$ in the notation of the text) can be
estimated consistently. In some sense the ULR model is the first prudential
model in this class of time deformed autoregressive models.

Appendix 2

Asymptotic Behaviour of
$\displaystyle\frac{1}{T-cT}\displaystyle\sum^{T}_{t=cT+1}[y_{T}(t)y^{\prime}_{T}(t-cT)]$

We have :

$\begin{array}[]{lcl}&&\displaystyle\frac{1}{T-cT}\displaystyle\sum^{T}_{t=cT+1}[y_{T}(t)y^{\prime}_{T}(t-cT)]\\\
\\\
&=&\displaystyle\frac{1}{T-cT}\displaystyle\sum^{T}_{t=cT+1}[y_{s}(t)y^{\prime}_{s}(t-cT)]+A\displaystyle\frac{1}{T-cT}\displaystyle\sum^{T}_{t=cT+1}[y_{lT}(t)y^{\prime}_{lT}(t-cT)]A^{\prime}\\\
\\\
&+&\displaystyle\frac{1}{T-cT}\displaystyle\sum^{T}_{t=cT+1}[y_{s}(t)y^{\prime}_{lT}(t-cT)]A^{\prime}+\displaystyle\frac{A}{T-cT}\displaystyle\sum^{T}_{t=cT+1}[y_{lT}(t)y^{\prime}_{s}(t-cT)].\end{array}$

The first term of the right hand side of the equality tends to zero in
probability by the standard Law of Large Numbers for stationary geometrically
mixing Gaussian processes.

The second term tends to the corresponding stochastic integral by applying the
convergence of Riemann sums [Stroock, Varadhan (1979), Section 11,]

With respect to the discussion on asymptotic behaviour of sample means, the
two remaining terms representing the cross-effects of the short and long run
components are new. We have to check if they are negligible. This is a
consequence of the weak Law of Large Numbers for mixingales [De Jong (1998)].

Since the weak LLN for a triangular array is usually discussed in the
univariate process, we consider below the case $n=L=1$ for expository purpose.
The approach is easily extended to matrices by applying the same reasoning
element par element. Let us consider the fourth term and denote :

$X_{Tt}=y_{l,T}(t)y_{s}(t-cT)=y_{l}(t/T)y_{s}(t-cT),$

and
$\mathcal{F}_{Tt}=\left(\underline{y_{l}(t/T)},\underline{y_{s}(t-cT)}\right)$
an infinite array of $\sigma$-fields.

a) $L_{1}$-mixingale condition

Let us check that $(X_{Tt},\mathcal{F}_{Tt})$ satisfies the condition of a
$L_{1}-$mixingale array [Andrews (1988), De Jong (1998), Definition 1].

i) By construction $X_{Tt}$ is measurable with respect to the increasing
sequence of $\sigma$-fields $\mathcal{F}_{Tt},t$ varying. Therefore condition
(2) in De Jong (1998), definition 1, is trivially satisfied [see Andrews
(1988), p460, remark b)].

ii) Let us now consider condition (1) in De Jong definition 1. We have :

$\begin{array}[]{l}||E[X_{Tt}|\mathcal{F}_{T,t-m}]||_{1}\\\ \\\
=||E[y_{l}(t/T)|y_{l}\left(\displaystyle\frac{t-m}{T}\right)]E[y_{s}(t-cT)|y_{s}(t-m-
cT)]||_{1},\end{array}$

by the independence between the SR and ULR components and their Markov
properties. Then,

$\begin{array}[]{l}||E[X_{Tt}|\mathcal{F}_{T,t-m}]||_{1}\\\ \\\
=||\exp(-\theta m/T)y_{l}(\displaystyle\frac{t-m}{T})\varphi^{m}y_{s}(t-m-
cT)||_{1}\\\ \\\ \leq||y_{l}(\displaystyle\frac{t-m}{T})y_{s}(t-m-
cT)||_{1}\varphi^{m}\equiv K\psi(m),\end{array}$

where $K$ is a constant by the independence of the SR and ULR components and
their stationarity, and where $\psi(m)=\varphi^{m}$ tends to zero when $m$
tends to infinity. Thus condition (2) is satisfied.

b) Weak LLN

There exist different versions of weak LLN for $L_{1}$-mixingales. The
standard version in Andrews (1988), Theorem 1, is not applicable in our
framework due to a too strong condition of uniform integrability of $X_{Tt}$.
However, we can apply De Jong (1995), Theorem 1 [see also De Jong (1998), p
214]. In his notations, the conditions are satisfied with $B_{T}=1$ constant.
Indeed we have :

$\begin{array}[]{l}\lim_{K\rightarrow\infty}\lim\sup_{T\rightarrow\infty}\displaystyle\frac{1}{T}\displaystyle\sum^{T}_{t=1}||X_{Tt}1\hskip-3.0pt\mbox{l}_{[X_{Tt}]>K}||_{1}\\\
\\\
=\lim_{K\rightarrow\infty}||X_{Tt}1\hskip-3.0pt\mbox{l}_{[X_{Tt}|>K}||_{1}\;\mbox{by
stationarity,}\\\ \\\ =0\;,\mbox{since the expectation
of}\;X_{Tt}\;\mbox{exists.}\end{array}$

Appendix 3

Asymptotic Behaviour of Local Means

The local mean at date $cT$ is defined by [see eq. (3.4)] :

$\begin{array}[]{lcl}\hat{m}_{T}(c)&=&\displaystyle\frac{1}{H_{T}}\sum^{cT+H_{T}}_{t=cT+1}y_{T}(t)\\\
\\\
&=&\displaystyle\frac{1}{H_{T}}\sum^{cT+H_{T}}_{t=cT+1}y_{s}(t)+\displaystyle\frac{1}{H_{T}}\sum^{cT+H_{T}}_{t=cT+1}Ay_{l}(t/T).\end{array}$

By the independence assumption the two components of the right hand side can
be analyzed separately.

a) Analysis of the SR component

Since the SR process is stationary, the sample average
$\hat{m}_{sT}(c)=\displaystyle\frac{1}{H_{T}}\sum^{cT+H_{T}}_{t=cT+1}y_{s}(t)$
has the same distribution as the average
$\displaystyle\frac{1}{H_{T}}\sum^{H_{T}}_{t=1}y_{s}(t)$. Therefore under the
assumption of geometric ergodicity of $y_{s}$ (that is the modulus of the
eigenvalues of $\Phi$ strictly smaller than 1) and the existence of its
second-order moments, we have :

i) $\hat{m}_{sT}(c)\rightarrow 0$, if $H_{T}\rightarrow\infty$,

ii) $\sqrt{H_{T}}\hat{m}_{sT}(c)\rightarrow\Sigma^{1/2}_{\infty}Z_{s}(c)$,

where $Z_{s}(c)\sim N(0,Id),$ and
$\Sigma_{\infty}=\sum^{+\infty}_{h=-\infty}\;\mbox{cov}\;[y_{s}(t),y_{s}(t-h)].$

iii) Moreover, for different levels $c_{k},k=1,\ldots,K$, the limiting
$Z_{s}(c_{k}),k=1,\ldots,K$ can be chosen independent, if $H_{T}/T\rightarrow
0$.

b) Analysis of the second component

Let us consider the difference :

$\Delta_{T}(c)=\displaystyle\frac{1}{H_{T}}\sum^{cT+H_{T}}_{t=cT+1}y_{l}(t/T)-y_{l}(c).$

Since $(y_{l})$ is a Gaussian process, $\Delta_{T}(c)$ is also a Gaussian
process. We have :

$\begin{array}[]{lcl}\sqrt{H_{T}}\Delta_{T}(c)&=&\displaystyle\frac{1}{\sqrt{H_{T}}}\sum^{cT+H_{T}}_{t=cT+1}[y_{l}(t/T)-y_{l}(c)]\\\
\\\
&=&\displaystyle\frac{1}{\sqrt{H_{T}}}\sum^{H_{T}}_{h=1}[y_{l}(c+h/T)-y_{l}(c)].\end{array}$

The Ornstein-Uhlenbeck process has a closed form expression with respect to
its value $y_{l}(c)$ at $c$ and the future values of the Brownian motion as :

$y_{l}(c+h/T)=\exp(-\Theta h/T)y_{l}(c)+\exp(-\Theta
h/T)\displaystyle\int^{c+h/T}_{c}\exp(\Theta u)SdW_{u}.$

Then the asymptotic behaviour of $\sqrt{H_{T}}\Delta_{T}(c)$ depends on the
independent effects of $y_{l}(c)$ and the future of the Brownian motion
$W_{c+u}$.

i) Effect of $y_{l}(c)$

The component of $\Delta_{T}(c)$ depending on $y_{l}(c)$ is :

$\begin{array}[]{lcl}\Delta_{1T}(c)&=&\displaystyle\frac{1}{H_{T}}\sum^{H_{T}}_{h=1}[\exp(-\Theta
h/T)-Id]y_{l}(c)\\\ \\\
&\simeq&-\displaystyle\frac{1}{H_{T}}\sum^{H_{T}}_{h=1}[\Theta
h/T]y_{l}(c),\;\mbox{if}\;H_{T}/T\rightarrow
0,\;\mbox{when}\;T\rightarrow\infty,\\\ \\\
&\simeq&-\displaystyle\frac{\Theta}{TH_{T}}\sum^{H_{T}}_{h=1}hy_{l}(c)\\\ \\\
&\simeq&-\displaystyle\frac{H_{T}}{2T}\Theta y_{l}(c).\end{array}$

Therefore,
$\sqrt{H_{T}}\Delta_{1T}(c)\sim-\displaystyle\frac{H_{T}^{3/2}}{2T}\Theta
y_{l}(c),$ and this term is negligible if $H_{T}/T^{2/3}\rightarrow 0$, if
$T\rightarrow\infty$.

iii) Effect of $W_{c+u}$

Let us now denote $\Delta_{2,T}(c)$ the component associated with the future
of the Brownian motion. We have :

$\begin{array}[]{lcl}\Delta_{2T}(c)&=&\displaystyle\frac{1}{H_{T}}\sum^{H_{T}}_{h=1}\\{\exp[-\Theta
h/T]\displaystyle\int^{c+h/T}_{c}\exp(\Theta u)SdW_{u}\\}\\\ \\\
&=&\displaystyle\frac{1}{H_{T}}\sum^{H_{T}}_{h=1}\sum^{h}_{k=1}\\{\exp(-\Theta
h/T)\displaystyle\int^{c+k/T}_{c+(k-1)/T}\exp(\Theta u)SdW_{u}\\}\\\ \\\
&=&\displaystyle\frac{1}{H_{T}}\sum^{H_{T}}_{k=1}\\{[\sum^{H_{T}}_{h=k}\exp(-\Theta
h/T)]\displaystyle\int^{c+k/T}_{c+(k-1)/T}\exp(\Theta u)SdW_{u}\\}\\\ \\\
&=&\displaystyle\frac{1}{H_{T}}[Id-\exp(-\Theta/T)]^{-1}\\\ \\\
&&\sum^{H_{T}}_{k=1}\\{(Id-\exp[-\Theta(H_{T}-k)/T])\exp(-\Theta
k/T)\displaystyle\int^{c+k/T}_{c+(k-1)/T}\exp(\Theta u)SdW_{u}\\}.\end{array}$

For expository purpose, i.e. to avoid the matrix notations, let us consider
the one-dimensional case. $\Delta_{2T}(c)$ is zero-mean with variance :

$\begin{array}[]{lcl}V[\Delta_{2T}(c)]&=&\displaystyle\frac{s^{2}}{H^{2}_{T}}\displaystyle\frac{1}{[1-\exp(-\theta/T)]^{2}}\sum^{H_{T}}_{k=1}\\{(1-\exp[-\theta(H_{T}-k)/T])^{2}\exp(-2\theta
k/T)\\\ &&\displaystyle\int^{c+k/T}_{c+(k-1)/T}\exp(2\theta u)du\\}\\\ \\\
&=&\displaystyle\frac{s^{2}}{H^{2}_{T}}\displaystyle\frac{\exp(2\theta
c)}{2\theta[1-\exp(-\theta/T)]^{2}}\sum^{H_{T}}_{k=1}[1-\exp(-\theta(H_{T}-k)/T)]^{2}[1-\exp(-2\theta/T)]\\\
\\\ &=&\displaystyle\frac{s^{2}}{H^{2}_{T}}\displaystyle\frac{\exp(2\theta
c)}{2\theta}\displaystyle\frac{1-\exp(-2\theta/T)}{[1-\exp(-\theta/T)]^{2}}\sum^{H_{T}}_{k=1}(1-\exp(-\theta(H_{T}-k)/T])^{2}\\\
\\\ &=&\displaystyle\frac{s^{2}}{H^{2}_{T}}\displaystyle\frac{\exp(2\theta
c)}{2\theta}\displaystyle\frac{1-\exp(-2\theta/T)}{[1-\exp(-\theta/T)]^{2}}\\\
\\\ &&\\{H_{T}-2\displaystyle\frac{1-\exp(-\theta
H_{T}/T)}{1-\exp(-\theta/T)}+\displaystyle\frac{1-\exp(-2\theta
H_{T}/T)}{1-\exp(-2\theta/T)}\\}.\;\;\;(*)\end{array}$

Let us now expand the term within brackets when
$T\rightarrow\infty,H_{T}\rightarrow\infty$, $H_{T}/T\rightarrow 0.$ We get
the equivalent expression :

$\begin{array}[]{l}H_{T}-\frac{2T}{\theta}[1-exp(-\frac{\theta
H_{T}}{T})]+\frac{T}{2\theta}[1-exp(-\frac{2\theta H_{T}}{T})]\\\ \\\
\approx\frac{1}{3}\frac{\theta^{2}H_{T}^{3}}{T^{2}}.\end{array}$

Therefore, by introducing this expansion in equation $(*)$ we get :

$\displaystyle V[\sqrt{H_{T}}\Delta_{2T}(c)]$ $\displaystyle\approx$
$\displaystyle s^{2}\displaystyle\frac{\exp(2\theta
c)}{2\theta}\displaystyle\frac{1-\exp(-2\theta/T)}{[1-\exp(-\theta/T)]^{2}}\frac{1}{3}\frac{\theta^{2}H_{T}^{2}}{T^{2}}$
$\displaystyle\approx$ $\displaystyle\frac{1}{3}s^{2}\exp(2\theta
c)\frac{H_{T}^{2}}{T}.$

Appendix 4

Additional Figures

We provide in this appendix additional figures for illustrating the results of
Section 5.2.

Figure a.1 : Growth Rates of TFP, Per-Capita GDP, Consumption, Investment and
Labor Compensation in US

Figure a.2 : ACF and Cross-ACF of the Raw Data

Figure a.3 : ACF and Cross-ACF of Filtered ULR component.

(a) TFP rate

(b) GDP

(c) Consumption

(d) Investment

(e) Labor Compensation

Figure a.1 : The series

Figure a.2: ACF and Cross-ACF of the Raw Data

Figure a.3: ACF and Cross-ACF of Filtered ULR Components