identified SVARs are interpreted as achieving identification through internal
instruments. Consequently, in these models, structural shocks are the
interventions of interest, and thus the goal is to estimate the dynamic causal
effect of these shocks on macroeconomic outcomes. In contrast, modern
microeconometric identification stategies rely heavily on external sources of
variation that provide quasi-experiemnts to identify causal effects. Such
external variation might be found, for example, in firm-specific
characteristics that introduce as-if randomness in the variable of interest
(the treatment). The use of such external instruments in microeconometrics has
proven highly productive and has yielded compelling estimates of causal
effects. Therefore, the use of external instruments has opened a new and
rapidly growing research avenue in macroeconometrics, in which credible
identification is obtained using as-if random variation in the shock of
interest that is distinct from - external to - the macroeconomic shocks
hitting the economy (Stock and Watson, (2018)).

###  5.1. Causal Effects and IV Regression

Following the paper of Stock and Watson, (2018), a starting point for
formulating the theory on identification and estimation of dynamic causal
effects in macroeconomics, is that the expected difference in outcomes between
the treatment and control groups in a randomized experiment with a binary
treatment is the average treatment effect. Roughly speaking, if a binary
treatment $X$ is randomly assigned, then all other determinants of $Y$ are
independent of $X$, which implies that the (average) treatment effect is

$\displaystyle\mathbb{E}\big{[}Y|X=1\big{]}-\mathbb{E}\big{[}Y|X=0\big{]}$
(5.1)

In particular, in the linear model $Y=\alpha+\beta X+u$, where $\beta$ is the
treatment effect, random assignment implies that
$\mathbb{E}\left(u|X\right)=0$ so that the population regression coefficient
is the treatment effect. Furthermore, if randomization is conditional on
covariates $W$, then the treatment effect for an individual with covariates
$W=w$ is estimated by the outcome of a random experiment on a group of
subjects with the same value of $W$, that is, it is

$\displaystyle\mathbb{E}\big{[}Y|X=1,W=w\big{]}-\mathbb{E}\big{[}Y|X=0,W=w\big{]}$
(5.2)

######  Example 22.

Usually the path of observed macroeconomic variables arising from current and
past shocks and measurement error are collected into the $(m\times 1)$,
$\varepsilon_{t}$ error vector $\varepsilon_{t}$. Therefore, the $(n\times 1)$
vector of macroeconomic variables $Y_{t}$ can be written in terms of current
and past innovation terms

$\displaystyle Y_{t}=\Theta(L)\varepsilon_{t},$ (5.3)

where $\Theta(L)=\Theta_{0}+\Theta_{1}L+\Theta_{2}L^{2}+...$, where
$\Theta_{h}$ is an $(n\times m)$ matrix of coefficients.

Then the variance matrix of the error terms
$\boldsymbol{\Sigma}=\mathbb{E}[\boldsymbol{\epsilon}_{t}\boldsymbol{\epsilon}_{t}]$
is assumed to be positive-definite to ensure the existence of a non ill-
conditioned covariance matrix. Moreover these disturbance terms (shocks) are
assumed to be mutually uncorrelated. The expression
$Y_{t}=\Theta(L)\varepsilon_{t}$, corresponds to the structural moving average
representation of $Y_{t}$. Specifically, the coefficients of $\Theta(L)$ are
the structural impulse response functions, which are the dynamic causal
effects of the shocks (see, also Bacchiocchi et al., (2018)).

Under the null of invertability, the following SVAR representation applies:

$\displaystyle\boldsymbol{A}(L)\boldsymbol{Y}_{t}=\boldsymbol{\Theta}(L)\boldsymbol{\varepsilon}_{t},$
(5.4)

Therefore, under invertability, the structural moving average is
$\boldsymbol{Y}_{t}=\boldsymbol{C}(L)\boldsymbol{\Theta}_{0}\boldsymbol{\varepsilon}_{t}$,
where $\boldsymbol{C}(L)=\boldsymbol{A}(L)^{-1}$, such that
$\boldsymbol{\Theta}(L)=\boldsymbol{C}(L)\boldsymbol{\Theta}_{0}$. The null
and alternative hypotheses are then

$\displaystyle
H_{0}:\boldsymbol{C}_{h}\boldsymbol{\Theta}_{0,1}=\boldsymbol{\Theta}_{h,1}\ \
\ \text{against}\ \ \
H_{1}:\boldsymbol{C}_{h}\boldsymbol{\Theta}_{0,1}\neq\boldsymbol{\Theta}_{h,1},\
\ \text{for some}\ h.$ (5.5)

Recall that the SVAR can be also written in state-space form as below:

$\displaystyle\boldsymbol{Y}_{t}$
$\displaystyle=\boldsymbol{\mathcal{B}}\boldsymbol{X}_{t}$ (5.6)
$\displaystyle\boldsymbol{X}_{t}$
$\displaystyle=\boldsymbol{A}\boldsymbol{X}_{t-1}+\boldsymbol{G}\boldsymbol{\varepsilon}_{t},$
(5.7)

where
$\boldsymbol{X}_{t}=\big{(}\boldsymbol{Y}_{t}^{\top},\boldsymbol{Y}_{t-1}^{\top},....,\boldsymbol{Y}_{t-p+1}^{\top}\big{)}^{\top}$,
such that $\boldsymbol{A}$ is the companion matrix and
$\boldsymbol{\mathcal{B}}=\big{(}\boldsymbol{I}_{n}\
\boldsymbol{0}\cdots\boldsymbol{0}\big{)}$ is a selection matrix. Then, the
local projection regression equation is written as below:

$\displaystyle\boldsymbol{Y}_{t+h}=\boldsymbol{\Theta}_{h,1}\boldsymbol{Y}_{1,t}+\boldsymbol{\Gamma}_{h}\boldsymbol{W}_{t}+\boldsymbol{u}_{t+h},$
(5.8)

######  Remark 15.

The study of Stock and Watson, (2018) verifies important insights regarding
the identification of dynamic causal effects. In particular, under the
assumption of Gaussian errors, every invertible model has multiple
observationally equivalent non-invertible representations, which imply that to
identify a unique representation some external information regarding the
system is required. If we assume that the structural shocks are independent
and non-Gaussian then using information from higher-order restrictions the
causal structure of the system can be identified. In practice, external
instruments can be employed to estimate dynamic causal effects directly
without using an indirect VAR identification step.

######  Example 23 (Causal effects of Lockdown Policies on Health and Macro
Outcomes).

We present the study of Arias et al., (2023) who consider causal impact of
pandemic-induced lockdowns and other nonpharmaceutical policy interventions
(NPIs), on health and macroeconomic outcomes. However, identification
assumptions are needed to access causality. In the first step, using the
Bayesian approach we can estimate an epidiomiological model with time
variation in the parameters controlling an infectious disease’s dynamics. In
particular, time variation in the parameters of the model allows to: (i)
capture changes in the behaviour of individuals as they respond to public
health conditions and (ii) to include shifts in the transmission and clinical
outcomes of the pandemic.

We write our SVAR model such that

$\displaystyle\boldsymbol{y}_{t}^{\prime}\otimes\boldsymbol{A}_{0}=\boldsymbol{x}_{t}^{\prime}\otimes\boldsymbol{A}_{+}+\varepsilon_{t}^{\prime},\
\ 1\leq t\leq T$ (5.9)

  * •

$\boldsymbol{y}_{t}$ is an $(n\times 1)$ vector of endogenous variables,

  * •

$\boldsymbol{x}_{t}^{\prime}=\big{[}\boldsymbol{y}_{t}^{\prime},...,\boldsymbol{y}_{t-p}^{\prime},\boldsymbol{z}_{t},\boldsymbol{1}\big{]}$,
where $\boldsymbol{z}_{t}$ is a $(z\times 1)$ vector of exogenous variables,

  * •

$\boldsymbol{\varepsilon}_{t}$ is an $(n\times 1)$ vector of structural shocks
and,

  * •

$\boldsymbol{A}_{0}$ is an $(n\times n)$ invertible matrix of parameters,

  * •

$\boldsymbol{A}_{+}$ is an $(np+z+1)\times n$ matrix of parameters and $p$ is
the lag length and $T$ is the sample size.

In addition, we assume that the vector $\boldsymbol{\varepsilon}_{t}$,
conditional on past information and the initial conditions
$\boldsymbol{y}_{0},...,\boldsymbol{y}_{1-p}$, is Gaussian with mean zero and
covariance matrix $\boldsymbol{I}_{n}$. Without loss of generality, we assume
that the first equation of the SVAR characterizes the policy rule. This
implies that

$\displaystyle\boldsymbol{y}_{t}^{\prime}\boldsymbol{\alpha}_{0,1}=\boldsymbol{x}_{t}^{\prime}\boldsymbol{\alpha}_{+,1}+\boldsymbol{\varepsilon}_{1t},\
\ 1\leq t\leq T$ (5.10)

is the policy equation such that

  * •

$\boldsymbol{\varepsilon}_{1t}$ denotes the first entry of the vector
$\boldsymbol{\varepsilon}_{t}$.

  * •

$\boldsymbol{\alpha}_{+,1}$ denotes the first column of $\boldsymbol{A}_{+}$
for $\ell\in\left\\{0,...,p\right\\}$ and $\alpha_{s,ij}$ denotes the $(i,j)$
entry of $\boldsymbol{A}_{s}$, where $s\in\left\\{0,+\right\\}$, and describes
the systematic component of the policy rule.

Restricting the systematic component of the policy rule is equivalent to
restricting $\alpha_{s,ij}$ and identifying a policy shock that we call the
stringency shock.

######  Example 24 (ECB monetary policy and bank default risk (Soenen and
Vander Vennet, (2022))).

In this example, the primary focus is the impact of ECB monetary policy on
bank risk. There is an ongoing debate on the effect of accommodative monetary
policy on bank risk taking and financial stability in general. Specifically,
one concern relates to the potential increase in risk taking and the possible
under-pricing of risk. The relevant question is whether or not monetary policy
causes excessive risk taking by banks, since this could hamper financial
stability which can be reflected in higher bank CDS spreads. Next, regarding
the choice of the variable of interest: we cannot use the policy rate because
of the zero lower bound constraint and, similarly, we cannot use the ECB
balance sheet because some important monetary policy measures did not affect
the balance sheet.

Therefore, when assessing the causal impact of monetary policy, we decide to
employ a structural VAR because incorporating a broad set of financial market
indicators allows us not only to identify actual ECB monetary policy
decisions, but also to capture anticipation effects and instances in which
financial markets judge that monetary policy actions were insufficient, given
the prevailing market conditions. In other words, we can estimate a time
series of exogeneous monetary policy shocks by modelling a set of relevant
financial market variables in a structural VAR model which is given by the
following econometric specification:

$\displaystyle\boldsymbol{y}_{t}=\boldsymbol{A}_{1}\boldsymbol{y}_{t-1}+...+\boldsymbol{A}_{p}\boldsymbol{y}_{t-p}+\boldsymbol{R}\boldsymbol{v}_{t},$
(5.11)

where $\boldsymbol{y}_{t}$ is an $n-$dimensional vector of endogenous
variables, $\boldsymbol{v}_{t}$ is an $n-$dimensional vector of orthogonal
structural innovations with mean zero and
$\left\\{\boldsymbol{A}_{1},...,\boldsymbol{A}_{p}\right\\}$ and
$\boldsymbol{R}$ are $(n\times n)$ time-invariant parameter matrices. The
reduced-form residuals corresponding to this structural model are given by
$\boldsymbol{\varepsilon}_{t}=\boldsymbol{R}\boldsymbol{v}_{t}$.

An existing macroeconomic puzzle that involve the optimal economic policy
decision-making with respect to macroprudential restructuring and debt
collection is the aspect of the tax policy conducted over the Business Cycle.
In particular, it is well known that government spending has typically been
procyclical in developing economics but exhibit acyclical or countercyclical
behaviour in industrial economies. Moreover, an interesting avenue of further
research is the counterfactual distribution of the cyclical behaviour of tax
rates, as opposed to tax revenues which are endogenous to the business cycle
and hence cannot shed light on the cyclicality of tax policy (see, Vegh and
Vuletin, (2015)).

Therefore it is of paramount importance to investigate important theoretical
and empirical driven questions regarding the identification and estimation of
dynamic treatment effects focusing on the VAT behaviour of firms in relation
to imposed by the government tax rates. Governmental taxation is considered to
be the pillar of fiscal policy and decisions on the optimal tax policy are
commonly questions by economists in relation to their effectiveness in various
macroeconomic outcomes, such as output fluctuations as well as structural
changes in agent’s behaviour over the business cycle. Our main interest is to
derive the asymptotic behaviour of the dynamic treatment effect estimator
across a panel of economics which have undergone a period of fiscal adaption
to new tax policies. In other words, we are interested to obtain the
asymptotic behaviour of these estimators based on the trajectory of the
control against the experimental groups in relation to a common taxation
policy as well as to obtain counterfactual outcomes when a control group has
adapted an updated fiscal policy across the business cycle. In this direction,
we are interested to incorporate tax policy variables such as taxation
revenues which are often considered as policy outcomes. Furthermore, due to
the fact that these variables might be positive correlated during the business
cycle, we consider suitable instrumental variables that can ensure consistent
model estimation and inference procedures. In this direction, a suitable
instrumental variable to be employed as a taxation policy variable is the VAT
rates across economic during the business cycle. A relevant novel tax database
which can be utilized has been created by Vegh and Vuletin, (2015) which
combines both corporate and personal income tax rates as well as data on
value-added taxes. A relevant study which discusses the effects of global tax
reform is proposed by Gomez Cram and Olbert, (2023).

###  5.2. Main Assumptions for Identification and Estimation of Causal Effects

In particular, the parameter of interest is the ATE
$:=\mathbb{E}\left[Y_{1}-Y_{0}\right]$. Moreover, the Conditional Average
Treatment Effect (CATE) defined by

$\displaystyle\tau(x):=\mathbb{E}\left[Y_{1}-Y_{0}|X=x\right],$ (5.12)

captures how the average treatment effect changes across sub-populations with
covariate value $X=x$. Furthermore, under the unconfoundness assumption
$\tau(x)$ can be nonparametrically identified by

$\displaystyle\tau(x):=\mathbb{E}\left[Y|D=1,X=x\right]-\mathbb{E}\left[Y|D=0,X=x\right]$
(5.13)

######  Assumption 3.

Suppose that the following conditions hold

  * (i)

$\big{(}Y(1),Y(0)\big{)}\ \textit{independent}\ D|X$.

  * (ii)

$X$ has a compact support $\mathsf{Supp}(X)$ and there exists $c>0$ such that
$c<\Pi(x)<1-c$ for all $x\in\mathsf{Supp}(X)$.

######  Remark 16.

Notice that the above assumptions are fairly standard in the causal inference
literature and we assume that they hold throughout the remaining of the paper
unless otherwise states. In particular, Assumption 1 (i), corresponds to the
unconfoundedness condition while Assumption 1 (ii) corresponds to the common
support condition for the set of covariates in the model. Furthermore, as one
of the main interest of our paper is the aspect of covariate shift, we will
introduce related conditions.

##### Causal Effect Inference with Time Series Data

Consider the $k-$step ahead forecast of $Y_{t}$ given the information
available up to time $t^{*}$ which is given by the following expression

$\displaystyle
Y_{t^{*}+k|t^{*}}(0):=\widehat{\mathbb{E}}\big{[}Y_{t^{*}+k}(0)|\mathcal{F}_{t^{*}}\big{]}\equiv
X_{t^{*}+k}^{\prime}\widehat{\beta}+\widehat{\mathbb{E}}\big{[}Z_{t^{*}+k}|\mathcal{F}_{t^{*}}\big{]}.$
(5.14)

Notice that by definition, $Y_{t^{*}+k}(0)$ is the potential outcome expected
at time $t^{*}+k$ in case the intervention does not occur. Therefore, an
estimator of the point causal effect can be defined as

$\displaystyle\widehat{\tau}_{t^{*}+k}:=Y_{t^{*}+k}(1)-Y_{t^{*}+k|t^{*}}(0)=\tau_{t^{*}+k}+X_{t^{*}+k}\big{(}\beta-\widehat{\beta}\big{)}+Z_{t^{*}+k}-\widehat{\mathbb{E}}\big{[}Z_{t^{*}+k}|\mathcal{F}_{t^{*}}\big{]}$

###  5.3. Counterfactual Analysis

Let $D\in\left\\{0,1\right\\}$ be a binary treatment indicator and $Y_{d}$ be
the potential outcomes for $d=0,1$ in the status quo environment (see, also
Hsu et al., (2022)). In particular, $Y_{1}$ is the outcome if an individual is
exogenously assigned to the treatement $(D=1)$ and $Y_{0}$ is the outcome in
the absence of treatement $(D=0)$. Then, the actual observed outcome is

$\displaystyle Y=DY_{1}+(1-D)Y_{0}$ (5.15)

We observe a $d-$dimensional vector of pretreatment covariates
$X=\left(X_{1},...,X_{k}\right)$ in the status quo environment and
$X^{\star}=\left(X^{\star}_{1},...,X^{\star}_{k}\right)$ in the counterfactual
environment which is of the same dimension as the set of covariates $X$.

  * (i)

$X$ and $X^{\star}$ are statistically independent

  * (ii)

$X^{\star}$ is a deterministic transformation of $X$, such that
$X^{\star}=\pi(X)$ for some known function $\pi$.

The corresponding treatment indicator, outcome, and potential outcomes in the
counterfactual environment are denoted as $D^{*},Y^{*}$ and $Y_{d}^{*}$ for
$d=0,1$, respectively with

$\displaystyle Y^{\star}=D^{\star}Y^{\star}_{1}+(1-D^{\star})Y^{\star}_{0}.$
(5.16)

However, notice that since the treatment has not been implemented in the
counterfactual environment yet, neither $D^{*}$ nor $Y^{*}$ is observed in our
model (see, Hsu et al., (2022)). We adopt the former approach such that a
closed-form solution exists

$\displaystyle\widehat{\delta}^{*}=\frac{1}{n}\sum_{j=1}^{n}\left[\widehat{\mathbb{E}}\left(Y_{1}|X=X_{j}^{*}\right)-\widehat{\mathbb{E}}\left(Y_{0}|X=X_{j}^{*}\right)\right]$
(5.17)

where $\widehat{\mathbb{E}}\left(Y_{d}|X=X_{j}^{*}\right)$ is the Nadaraya-
Watson estimator such that

$\displaystyle\widehat{\mathbb{E}}\left(Y_{d}|X=X_{j}^{*}\right)=\frac{\displaystyle\sum_{i=1}^{n}Y_{i}\mathbf{1}\left\\{D_{i}=d\right\\}K_{x,h}\left(X_{i}-x\right)}{\displaystyle\sum_{i=1}^{n}\mathbf{1}\left\\{D_{i}=d\right\\}K_{x,h}\left(X_{i}-x\right)}$
(5.18)

The asymptotic properties for $\widehat{\delta}^{*}$ can be derived under
weaker regularity conditions stated below. Furthermore based on regularity
conditions it can be proved that

$\displaystyle\sqrt{n}\left(\widehat{\delta}^{*}-\delta^{*}\right)\overset{d}{\to}\mathcal{N}\left(0,\sigma^{2}_{\delta^{*}}\right).$
(5.19)

Therefore, compared to the semiparametric efficiency bound of the ATE
estimator we have that

$\displaystyle\mathbb{E}\left\\{\frac{\mathsf{Var}(Y_{1}|X)}{p(X)}+\frac{\mathsf{Var}(Y_{0}|X)}{1-p(X)}+\mathbb{E}\big{[}Y_{1}-Y_{0}|X\big{]}-\mathbb{E}\big{[}\left(Y_{1}-Y_{0}\right)\big{]}^{2}\right\\}.$
(5.20)

###  5.4. Time Series Experiments and Causal Effects

Time series causal effects are defined as a comparison between the potential
outcomes at a fixed point in time. Thus, the primary object of interest is the
temporal average of these causal effects. Therefore, the only source of
randomness in our formulation is the randomization of the treatment. Once we
begin the experiment, the randomization reveals a particular outcome path;
however, this does not change the set of potential outcomes, and therefore,
does not alter what we would have observed had we administered a different
treatment. Thus, in this sense, we are treating the potential outcomes as
predetermined, and the only source of randomness comes from which path we
observe.

######  Definition 6 (General Causal Effects).

For paths $w_{1:t}$ and $w_{1:t}^{\prime}$, the $t-$th causal effect is given
by

$\displaystyle\tau(w_{1:t},w_{1:t}^{\prime})=\mathbb{E}\big{[}Y_{it}(w_{1:t})-Y_{it}(w_{1:t}^{\prime})\big{]}.$
(5.21)

Then, the temporal average treatment effect of the paths $w_{1:T}$ and
$w_{1:T}^{\prime}$ is

$\displaystyle\tau(w_{1:t},w_{1:t}^{\prime})=\frac{1}{T}\sum_{t=1}^{T}\tau_{t}\left(w_{1:t},w_{1:T}^{\prime}\right).$
(5.22)

Moreover, we define the potential outcomes just intervening on treatment the
last $j$ periods as

$\displaystyle Y_{it}(x_{t-j:t})=Y_{it}\big{(}X_{i,1:t-j-1},x_{t-j:t}\big{)}$
(5.23)

In other words, this "marginal" potential outcome represents the potential or
counterfactual level of regime A in country $i$, if we let welfare spending
run its natural course up to $t-j-1$ and just set the last $j$ lags of
spending to $x_{t-j:t}$. Thus, we can define an important quantity of
interest, that is, the contemporaneous effect of treatment (CET) of $X_{it}$
on $Y_{it}$ such that:

$\displaystyle\tau_{c}(t)$
$\displaystyle:=\mathbb{E}\big{[}Y_{it}\big{(}X_{i,1:t-1},1\big{)}-Y_{it}\big{(}X_{i,1:t-1},0\big{)}\big{]},$
$\displaystyle:=\mathbb{E}\big{[}Y_{it}(1)-Y_{it}(0)\big{]}.$

Furthermore, following Bojinov and Shephard, (2019) specifically in
longitudinal studies and survey studies with a follow-up wave of cross-
sectional datasets, we are particulary interested to investigate the treatment
effect of survey participants as well as their membership in certain
endogenously generated groups. This implies that we are modelling the presence
of time-varying exposure to dynamic causal estimands. Therefore, to construct
a feasible structural econometric framework we consider the treatment and
potential outcome path with time-varying effects. At each time step,
$t=1,...,T$, we assume that each individual (sampling unit), is exposed to
either treatment, $W_{t}=1$, or control, $W_{t}=0$, and then we measure the
corresponding outcome variable of interest.

Although we focus on binary treatments, our results can be generalized to
multiple treatments as well. Thus, the random "treatment path" is given by

$\displaystyle W_{1:t}=\big{(}W_{1},...,W_{t}\big{)}.$ (5.24)

Then, the potential path for the treatment path $w_{1:t}$ is given by

$\displaystyle
Y_{1:t}(w_{1:t})\big{\\{}Y_{1}(w_{1}),Y_{2}(w_{1:2}),...,Y_{2}(w_{1:t})\big{\\}}$
(5.25)

####  5.4.1. Null Hypothesis of Temporal Causal Effects

Following the framework proposed by Bojinov and Shephard, (2019), to access
whether the treatment has a statistically significant effect, we consider the
sharp null of no temporal causal effects

$\displaystyle H_{0}:Y_{t}(w_{1:t})=Y_{t}(w_{1:t}^{\prime}),\ \ \ \text{for
all}\ w_{1:t}=w_{1:t}^{\prime},\ \ \ t=1,2,...,T.$ (5.26)

In particular, this will null hypothesis will be tested against a portmanteu
alternative. Thus, invoking the sharp null hypothesis implies that
$Y_{t}(w_{1:t}^{\text{obs}})=Y_{t}(w_{1:t}^{\prime})$ for all
$w_{1:t}^{\prime}$. Moreover, the particular formulation of the null
hypothesis means "the null of no temporal causality at lag $p\geq 0$ of the
treatment on the outcome" holds, such that $H_{0,p}:\tau_{t,p}=0$, for all
$t=1,2,...,T$.

In the case where $p=0$, then $\widehat{\tau}_{t,0}$ estimation error can be
written as below

$\displaystyle
u_{t,0}=\left(\frac{\boldsymbol{1}\left\\{W_{t}=1\right\\}}{p_{t}(1)}-\frac{\boldsymbol{1}\left\\{W_{t}=0\right\\}}{p_{t}(0)}\right)Y_{t}(w_{1:t}^{\text{obs}})$
(5.27)

and under the sharp null hypothesis we obtain that

$\displaystyle\mathbb{E}^{R}\big{[}u_{t,0}|\mathcal{F}_{T,t-1}\big{]}=0\ \ \
\text{and}\ \ \
Var\big{[}u_{t,0}|\mathcal{F}_{T,t-1}\big{]}=\frac{Y_{t}(w_{1:t}^{\text{obs}})}{p_{t}(1)p_{t}(0)}.$
(5.28)

In particular, whenever $p_{t}(1)=p_{t}(0)$ the estimator of the upper bound
to the variance for the contemporaneous causal effect, is equal to the
variance under the sharp null of no treatment effect. Then, the $p$ lagged
error variance $Var^{R}\big{(}u_{t-p,p}|\mathcal{F}_{T,t-p-1}\big{)}$ equals
to

$\displaystyle\nu_{p}^{2}=Y_{t}(w_{1:t}^{\text{obs}})^{2}\frac{1}{2^{2p}}\sum_{w\in\left\\{0,1\right\\}^{p}}\left(\frac{1}{p_{t}(1,w)}+\frac{1}{p_{t}(0,w)}\right).$
(5.29)

We begin by rewriting the within-unit matching estimator as below

$\displaystyle\hat{\tau}_{\text{match}}=\frac{1}{\frac{1}{N}\sum_{i=1}^{N}C_{i}}.\frac{1}{N}\sum_{i=1}^{N}C_{i}\left\\{\frac{\displaystyle\sum_{t=1}^{T}X_{it}Y_{it}}{\displaystyle\sum_{t=1}^{T}X_{it}}-\frac{\displaystyle\sum_{t=1}^{T}X_{it}Y_{it}}{\displaystyle\sum_{t=1}^{T}(1-X_{it})}\right\\}.$
(5.30)

Consider the conditional independence property which implies that
$\big{\\{}Y_{it}(1),Y_{it}(0)\big{\\}}\perp\boldsymbol{X}_{i}|\boldsymbol{U}_{i}$.
By the law of iterated expectations this implies that

$\displaystyle\mathbb{E}\big{[}Y_{it}(x)|C_{i}=1\big{]}=\mathbb{E}\bigg{[}\mathbb{E}\big{[}Y_{it}(x)|\boldsymbol{U}_{i},C_{i}=1\big{]}\big{|}C_{i}=1\bigg{]}$
(5.31)

for $x=0,1$.

###  5.5. Bias-Reduced Doubly Robust Estimation

According to Vermeulen and Vansteelandt, (2015) the proposed approach may more
generally lend itself better to small-sample inference. For instance, suppose
that interest lies in the marginal causal effect $\tau=\mathbb{E}[Y(1)-Y(0)]$.
Because the proposed estimation strategy does not require acknowledging the
uncertainty of the estimated nuisance parameters (up to first order), we
foresee that it may potentially lend itself better to randomization inference.
How such randomization inference could be accomplished and how it performs in
small to large samples is indeed relevant questions for further investigation.

####  5.5.1. Marginal Treatment Effects

Consider i.i.d data
$\left\\{\boldsymbol{Z}_{i}=\left(Y_{i},A_{i},\boldsymbol{X}_{i}\right),i=1,...,n\right\\}$,
where $Y_{i}$ is the outcome of interest, $A_{i}$ is a dichotomous treatment
taking values zero and one and $\boldsymbol{X}_{i}$ is a sufficient set of
covariates to control for confounding of the treatment effect, in the sence
that $Y(a)\perp A|\boldsymbol{X}$ for $a\in\left\\{0,1\right\\}$.
Specifically, $Y(a)$ denotes the counterfactual outcome for treatment level
$a\in\left\\{0,1\right\\}$, which is linked to the observed data through the
consistency assumption (i.e., $Y(a)=Y$ iff $A=a$).

######  Assumption 4 (Covariates-Treatment Independence).

Let $X_{j,t}$ be a vector of covariates that are predictive of the outcome of
unit $j$, for all $t\in\left\\{t^{*}+1,...,T\right\\}$ such covariates are not
affected by the intervention, such that, $X_{j,t}(1)=X_{j,t}(0)$.

######  Definition 7.

Let $t^{*}$ be the time of the intervention and define with
$k\in\left\\{1,...,K\right\\}$ such that $t^{*}+K=T$. For any $k$, the point,
cumulative and temporal average causal effects on unit $j$ at time $t^{*}+k$
are defined, respectively, such that

$\displaystyle\tau_{j,t^{*}+k}(1;0)$
$\displaystyle=Y_{j,t^{*}+k}(1)-Y_{j,t^{*}+k}(0)$ (5.32)
$\displaystyle\Delta_{j,t^{*}+k}(1;0)$
$\displaystyle=\sum_{h=1}^{k}\tau_{j,t^{*}+k}(1;0)$ (5.33)

######  Assumption 5 (Selection on observables).

The following conditional independence assumption ensures that model selection
is conditioned on observables

$\displaystyle Y_{t,1}(d),Y_{t,2}(d),...,\perp D_{t}|z_{t},\ \ \ \text{for
all}\ \ d\ \text{and}\ \ \psi\in\Psi_{t}.$ (5.34)

Notice that the selection on observables assumption says that policies are
independent of potential outcomes after appropriate conditioning. Moreover,
the conditioning set includes the unknown error vector of the model,
$\varepsilon_{t}$, such that
$Y_{t,1}(d),Y_{t,2}(d),...,\perp\varepsilon_{t}|z_{t}$. This is because,
conditional on the set of instruments $z_{t}$, randomness in $D_{t}$ is due
exclusively due to randomness induced from the unknown $\varepsilon_{t}$
vector. Therefore, in order to convert the above assumption into a testable
hypothesis which has an identifiable parameter space, we consider the sharp
null hypothesis such that $Y_{t,j}(d^{\prime})=Y_{t,j}(d)=Y_{t+j}$.

Then, if we substitute the observed with the potential outcomes we obtain the
following testable conditional independence assumption:

$\displaystyle Y_{t+1},...,Y_{t+j},...,\perp D_{t}|z_{t},$ (5.35)

Notice that $D_{t}$ is not necessarily a binary treatment status but can be a
set of different taxation policies which implies that we have a continuum of
treatment status.

######  Definition 8 (Granger noncausality).

Under suitable regularity conditions it holds that

$\displaystyle Y_{t+1}\perp D_{t},\bar{D}_{t+1}|\bar{Y}_{t}.$ (5.36)

The following theorem of Sims and Granger causality holds.

######  Theorem 3.

Let $\chi_{t}$ be a stochastic process defined on a probability space
$\big{(}\Omega,\mathcal{F},\mathbb{P}\big{)}$, assuming that conditional
probability measures $\mathbb{P}\big{(}Y_{t+1},D_{t}|z_{t}\big{)}$ are well
defined for all $t$ expect possibly on a set of measure zero.

######  Remark 17.

A relevant example is presented by Menchetti et al., (2023) who consider
estimating the effect of policy interventions in observational time series
settings in the absence of untreated units, by combining counterfactual
outcomes and ARIMA models for prediction purposes. Specifically, when focusing
on observational time series data, commonly used methodologies for policy
evaluation within the Rubin Causal Model (RCM) framework include Differences-
in-Differences (DiD) and synthetic control methods. However, a main drawback
of these two approaches is that they require the presence of controls that did
not experience the treatment, and therefore finding untreated units is a
challenging task (Menchetti et al.,, 2023). Moreover, the particular method is
not guaranteed to reveal relevant causal effects especially due to the
presence of unobserved factors. For example, if our main goal is to estimate
the causal effect of a new intervention policy such as due to the adoption of
new tax rates, as well as its indirect impact on a common fiscal policy across
similar economies (e.g., VAT policy), then a causal time series modelling
approach can be employed to obtain the causal estimands of interest. However,
one may find that these causal effects have no negative or positive impact on
the spending trajectory of similar economies that did not adopt new fiscal
policies, suggesting that unobserved factors may drive spending decision
making more than taxation policy. Other relevant counterfactual frameworks can
be constructed (see, Kilian and Lewis, (2011)), around issues from the
international finance literature such as the modelling sovereign default risk
(e.g., see Hatchondo et al., (2016)).

###  5.6. A Panel Data regression model for Dynamic Causal Effects

Following the framework proposed by Bojinov et al., (2021) we consider a
design-based framework which provides a generalization of the finite
population literature in cross-sectional causal inference and time series
experiments to panel experiments.

######  Definition 9.

For $p<T$ and $\boldsymbol{w},\tilde{\boldsymbol{w}}\in\mathcal{W}^{p+1}$,

  * (i)

the time$-t$ lag$-p$ average dynamic causal effect is

$\displaystyle\bar{\tau}_{.t}(\boldsymbol{w},\tilde{\boldsymbol{w}};p):=\frac{1}{N}\sum_{i=1}^{N}\tau_{i,t}(\boldsymbol{w},\tilde{\boldsymbol{w}};p).$
(5.37)

  * (ii)

the unit$-i$ lag$-p$ average dynamic causal effect is

$\displaystyle\bar{\tau}_{i.}(\boldsymbol{w},\tilde{\boldsymbol{w}};p):=\frac{1}{T-p}\sum_{t=p+1}^{T}\tau_{i,t}(\boldsymbol{w},\tilde{\boldsymbol{w}};p).$
(5.38)

  * (iii)

the total lag$-p$ average dynamic causal effect is

$\displaystyle\tau(\boldsymbol{w},\tilde{\boldsymbol{w}};p):=\frac{1}{N(T-p)}\sum_{t=p+1}^{T}\sum_{i=1}^{N}\tau_{i,t}(\boldsymbol{w},\tilde{\boldsymbol{w}};p)$
(5.39)

####  5.6.1. Simulation Design

We generate the potential outcomes for the panel experiment using an
autoregressive model

$\displaystyle
Y_{i,t}=\phi_{i,t,1}Y_{i,t-1}(w_{i,1:t-1})+...+\phi_{i,t,t-1}Y_{i,1}(w_{i,1})+\beta_{i,t,0}w_{i,t}+...+\beta_{i,t,t-1}w_{i,1}+\epsilon_{i,t},\
\ \ \forall\ t>1,$ (5.40)

where

$\displaystyle Y_{i,1}(w_{i,1})=\beta_{i,1,0}w_{i,1}+\epsilon_{i,1}$ (5.41)

with

$\displaystyle\phi_{i,t,s}=\begin{cases}\phi,&\text{for}\ s=1\\\
0,&\text{for}\ s>1.\end{cases}\ \ \ \ \text{and}\ \ \ \
\beta_{i,t,s}\begin{cases}\beta,&\text{for}\ s=0\\\ 0,&\text{for}\
s>0.\end{cases}$ (5.42)

We vary the choice of $\phi$, which governs the persistence of the process,
and $\beta$, which governs the size of the contemporaneous causal effects.
Moreover, we vary the probability of treatment $p_{i,t-p}(w)=p(w)$ as well as
the distribution of the errors $\epsilon_{i,t}$, which we either sample from a
standard normal or Cauchy distribution.

######  Example 25 (see, Goes, (2016)).

Consider the following panel SVAR(1) model defined as below

$\displaystyle\boldsymbol{B}\boldsymbol{y}_{i,t}=\boldsymbol{f}_{i}+\boldsymbol{A}(L)\boldsymbol{y}_{i,t-1}+\boldsymbol{e}_{i,t},\
\ \ i\in\left\\{1,...,N\right\\},\ \ t\in\left\\{1,...,T\right\\}$ (5.43)

where $y_{i,t}\equiv[c_{i,t},k_{i,t}]^{\prime}$ is a bi-dimensional vector of
stacked endogenous variables, such that $c_{i,t}$ is the log of GDP per capita
and $k_{i,t}$ is the proxy for institutional quality, $\boldsymbol{f}_{i}$ is
a diagonal matrix of time-invariant individual-specific intercepts. Moreover,
$\boldsymbol{A}(L)=\sum_{j=0}^{p}\boldsymbol{A}_{j}L^{j}$ is a polynomial of
lagged coefficients, $\boldsymbol{A}_{j}$ is a matrix of coefficients, and
$\boldsymbol{e}_{i,t}$ is a vector of stacked residuals, and $\boldsymbol{B}$
is a matrix of contemporaneous coefficients. However, since
$\boldsymbol{f}_{i}$ is correlated to the error terms, estimation through OLS
leads to biased coefficients. As proposed by Baltagi, (2008), a strategy to
obtain consistent parameters and eliminate individual fixed-effects when $N$
is large and $T$ is fixed, is to apply first-differencing and use lagged
instruments. We consider the GMM/IV technique using a system of $m=2$
equations.

Each equation in the system has the first difference of an endogenous variable
on the left hand side, $p$ lagged first differences of all $m$ endogenous
variables on the right hand side, and no constant.

$\displaystyle\Delta y_{1,i,t}$
$\displaystyle=\sum_{j=1}^{p}{\color[rgb]{0,0,1}\gamma_{11}^{j}}\Delta
y_{1,i,t-j}+...+\sum_{j=1}^{p}{\color[rgb]{0,0,1}\gamma_{1m}^{j}}\Delta
y_{m,i,t-j}+e_{1,i,t}$ (5.44) $\displaystyle\vdots\ \ $ $\displaystyle=\ \
\vdots$ (5.45) $\displaystyle\Delta y_{m,i,t}$
$\displaystyle=\sum_{j=1}^{p}{\color[rgb]{0,0,1}\gamma_{m1}^{j}}\Delta
y_{1,i,t-j}+...+\sum_{j=1}^{p}{\color[rgb]{0,0,1}\gamma_{mm}^{j}}\Delta
y_{m,i,t-j}+e_{m,i,t}$ (5.46)

Moreover, the model has an equivalent vector moving average $(VMA)$
representation which implies that the Panel SVAR model can be formulated as
follows

$\displaystyle\boldsymbol{B}\boldsymbol{y}_{i,t}=\boldsymbol{\Phi}(L)\boldsymbol{e}_{i,t},\
\ \
\boldsymbol{\Phi}(L):=\sum_{j=0}^{\infty}\boldsymbol{\Phi}_{j}L^{j}\equiv\sum_{j=0}^{\infty}\boldsymbol{A}_{1}^{j}L^{j}$
(5.47)

is a polynomial of reduced-form responses to stochastic innovations and
$\boldsymbol{\Phi}_{0}=\boldsymbol{A}_{1}^{0}\equiv\boldsymbol{I}_{m}$.

Thus, to recover the $\boldsymbol{B}$ matrix and ensure robust identification,
we first retrieve the variance-covariance matrix
$\boldsymbol{\Sigma}_{e}=\mathbb{E}\left[\boldsymbol{e}_{i,t}\boldsymbol{e}_{i,t}^{\prime}\right]$.
Since, it holds that
$\boldsymbol{B}^{-1}\boldsymbol{e}_{i,t}=\boldsymbol{u}_{i,t}$, then
$\boldsymbol{\Sigma}_{e}=\mathbb{E}\left[\boldsymbol{B}\boldsymbol{u}_{i,t}\boldsymbol{u}_{i,t}^{\prime}\boldsymbol{B}^{\prime}\right]$.
The structural shocks of the model are assumed to be uncorrelated, such that,
$\boldsymbol{u}_{i,t}\boldsymbol{u}_{i,t}^{\prime}=\boldsymbol{I}_{m}$, we
identify the matrix $\boldsymbol{B}$ by decomposing the variance-covariance
matrix into two triangular matrices. Therefore, to identify the model we
impose one restriction in order to orthogonalize the contemporaneous
responses. In particular, using the Cholesky ordering, and based on the
variables of interest, institutional quality is set to have no contemporaneous
effect on GDP per capita while the latter is allowed to contemporaneously
impact the former. Overall, the study of Goes, (2016) investigates the
relation between institutions and economic growth using a structural panel
VAR(1) model. Our next task is to recover the IRs from the coefficient
matrices. Specifically, we take the following VMA representation of the Panel
SVAR such that

$\displaystyle\boldsymbol{B}\boldsymbol{M}(L)=\boldsymbol{e}_{i,t}.$ (5.48)

######  Example 26.

We briefly discuss the econometric specification proposed by Huber et al.,
(2023) which is suitable for dealing with cross-country heterogeneity in panel
VARs using finite mixture models. Specifically, a global econometric
identification and estimation model (see, also Feldkircher et al., (2020)) is

$\displaystyle\boldsymbol{y}_{it}=\boldsymbol{\beta}_{i}+\boldsymbol{A}_{i1}\boldsymbol{y}_{it-1}+...+\boldsymbol{A}_{ip}\boldsymbol{y}_{it-p}+\boldsymbol{B}_{i1}\boldsymbol{y}_{-it-1}+...+\boldsymbol{B}_{ip}\boldsymbol{y}_{-it-p}+\boldsymbol{\varepsilon}_{it}.$
(5.49)

Under the Gaussian "state-of-the-world" one can impose Gaussian assumptions
regarding the error covariances across countries. Thus, we can stack the
country-specific errors $\boldsymbol{\varepsilon}_{it}$ in a $K-$dimensional
vector
$\boldsymbol{\varepsilon}_{t}=\left(\boldsymbol{\varepsilon}_{1t}^{\prime},...,\boldsymbol{\varepsilon}_{Nt}^{\prime}\right)^{\prime}$
and assume that
$\boldsymbol{\varepsilon}_{t}\sim\mathcal{N}\big{(}\boldsymbol{0},\boldsymbol{\Sigma}_{t}\big{)}$,
where $\boldsymbol{\Sigma}_{t}$ is a $(K\times K)$ dimensional covariance
matrix. Furthermore, according to Huber et al., (2023), there are three
important dimensions of model uncertainty which have been identified in the
literature as summarized below:

  * •

The first source of model uncertainty is concerned with modelling
contemporaneous relations across the shocks in the system (static
interdependancies).

  * •

The second source of model uncertainty focuses on the question whether
coefficients associated with lagged domestic variables are homogeneous across
countries (homogeneity restrictions). In particular, if such domestic
coefficients are similar, then the so-called homogeneity restrictions might be
imposed, effectively introducing the same set of coefficients for various
countries and thus reducing the number of free parameters.

  * •

The third source of model uncertainty deals with the question whether to allow
for lagged dependencies between countries (dynamic interdependencies).
Therefore, if we are interested to capture the international effects of
climate shocks in globalized markets then capturing cross-country
interdependencies on the macroeconomic level is crucial.

  * •

From the empirical contribution perspective one can focus on how climate
shocks impact country-specific macroeconomic fundamentals. Then the empirical
analysis of the novel framework proposed by demonstrate that climate shocks
are have a substantial impact on short-term interest rates and inflation,
which is the primary conventional tool and target variable of central banks,
and to a lesser degree on output and exchange rates (see, Huber et al.,
(2023)).

######  Example 27 (Climate Econometric Model see, Phillips et al., (2020)).

Consider the econometric model which relates local temperature $(T_{i})$ at
time $t+1$ to local temperature and surface radiation $(R_{i})$, as well as
global factors, denote with $\lambda_{t}$, at time $t$. Consider the
econometric specification below:

$\displaystyle T_{i,t+1}$
$\displaystyle=\alpha_{0i}+\alpha_{1}T_{i,1}+\alpha_{2}R_{i,t}+\lambda_{t}+u_{it+1},\
\ \ i\in\left\\{1,...,N\right\\},t\in\left\\{1,...,n\right\\}.$ (5.50)
$\displaystyle\lambda_{t}$
$\displaystyle=\beta_{0}+\beta_{1}\bar{T}_{t}+\beta_{2}\bar{R}_{t}+\beta_{3}\mathsf{ln}(C_{t})$
(5.51)

The above expression can be interpreted as an energy balancing relationship
that captures the global linkage between temperature, radiation and greenhouse
gas atmospheric influences. In other words, the dynamic panel regression
equation chatacterizes the dynamic adjustement mechanism of station level
temperature $T_{it+1}$ as an autoregression on past temperature, $T_{it}$,
radiation $R_{it}$, and the time specific global energy effects embodied in
$\lambda_{t}$.

###  5.7. The Augmented Synthetic Control Method

The synthetic control method (SCM) is a popular approach for estimating the
impact of a treatment on a single unit in panel data settings (see, Ben-
Michael et al., (2021)). In other words is a weighted average of control units
that balances the treated units pre-treatment outcomes and other covariates as
closely as possible. We define the treated potential outcome as
$Y_{it}=Y_{it}(0)+\tau_{it}$, where the treatment effects $\tau_{it}$ are
fixed parameters. Since the first unit is treated, the key estimand of
interest is

$\displaystyle\tau=\tau_{1T}=Y_{1T}(1)-Y_{1T}(0).$ (5.52)

Then, the observed outcomes are as below

$\displaystyle Y_{it}=\begin{cases}Y_{it}(0)&\ \text{if}\ W_{i}=0\ \text{or}\
t\leq T_{0},\\\ Y_{it}(1)&\ \text{if}\ W_{i}=1\ \text{or}\
t>T_{0}.\end{cases}$ (5.53)

######  Remark 18.

Notice that pre-treatment outcomes serve as covariates in SCM, we use
$X_{it}$, for $t\leq T_{0}$, to represent pre-treatment outcomes.

  * •

We are interested on a potential outcome system as a foundational framework
for analyzing dynamic causal effects on outcomes in observational time series
settings. We consider settings in which there is a single unit observed over
time. At each time period $t\geq 1$, the unit receives a vector of assignments
$W_{t}$, and an associated vector of outcomes $Y_{t}$ are generated. The
outcomes are causally related to the assignments through a potential outcome
process, which is a stochastic process that describes what would be observed
along counterfactual assignment paths.

  * •

A dynamic causal effect is generally defined as the comparison of the
potential outcome process along different assignment paths at a fixed point in
time.

######  Assumption 6 (Assignment and Potential Outcome).

The assignment process $\left\\{W_{t}\right\\}_{t\geq 1}$ satisfies

$\displaystyle W_{t}\in\mathcal{W}:=\times_{k=1}^{d_{w}}\mathcal{W}_{k}$
(5.54)

The potential outcome process, is for any deterministic sequence
$\left\\{w_{s}\right\\}_{s\geq 1}$ with $w_{s}\in\mathcal{W}$ for all $s\geq
1$, $\big{\\{}Y_{t}\big{(}\left\\{w_{s}\right\\}_{s\geq
1}\big{)}\big{\\}}_{t\geq 1}$, where the time$-t$ potential outcomes satisfies
$Y_{t}\big{(}\left\\{w_{s}\right\\}_{s\geq
1}\big{)}\in\mathcal{Y}\subset\mathbb{R}^{d_{y}}$.

######  Remark 19.

The simplest case is when the assignment is scalar and binary
$\mathcal{W}=\left\\{0,1\right\\}$, in which case $W_{t}=1$ corresponds to
"treatment" and $W_{t}=0$ is "control". Moreover, the potential outcome
$Y_{t}\big{(}\left\\{w_{s}\right\\}_{s\geq 1}\big{)}$ may depend on future
assignments $\left\\{w_{s}\right\\}_{s\geq t+1}$. The next assumption
considers the dependence structure we allow in the proposed econometric
framework, restricting the potential outcome to only depend on past and
contemporaneous assignments.

######  Assumption 7 (Non-anticipating Potential Outcomes).

For each $t\geq 1$, and all deterministic $\left\\{w_{t}\right\\}_{t\geq 1}$
and $\left\\{w_{t}^{\prime}\right\\}_{t\geq 1}$ with
$w_{t},w_{t}^{\prime}\in\mathcal{W}$,

$\displaystyle Y_{t}\big{(}w_{1:t},\left\\{w_{s}\right\\}_{s\geq
t+1}\big{)}=Y_{t}\big{(}w_{1:t},\left\\{w_{s}^{\prime}\right\\}_{s\geq
t+1}\big{)}\ \text{almost surely}.$ (5.55)

######  Remark 20.

The above assumption is a stochastic process analogue of non-interference.
Furthermore, it still allows for rich dependence on past and contemporaneous
assignments. Under Assumption 2, we drop references to the future assignmnets
in the potential outcome process and write

$\displaystyle\big{\\{}Y_{t}\big{(}\left\\{w_{s}\right\\}_{s\geq
1}\big{)}\big{\\}}_{t\geq
1}=\big{\\{}Y_{t}\big{(}w_{1:t}\big{)}\big{\\}}_{t\geq 1}$ (5.56)

Then, the set
$\big{\\{}Y_{t}\big{(}w_{1:t}\big{)}:w_{1:t}\in\mathcal{W}^{t}\big{\\}}$
collects all the potential outcomes at time $t$.

######  Assumption 8 (Output).

The output is $\left\\{W_{t},Y_{t}\right\\}_{t\geq
1}=\left\\{W_{t},Y_{t}\big{(}W_{1:t}\big{)}_{t\geq 1}\right\\}$. The
$\left\\{Y_{t}\right\\}_{t\geq 1}$ is called the outcome process.

Furthermore, we assume that the assignmemnt process is sequentially
probabilistic, meaning that any assignment vector may be realized with
positive probability at time $t$ given the history of the observable
stochastic process up to time $t-1$. Let
$\left\\{\mathcal{F}_{t}\right\\}_{t\geq 1}$ denote the natural filtration
generated by the realized $\left\\{w_{t},y_{t}\right\\}_{t\geq 1}$.

######  Assumption 9 (Sequentially probabilistic assignment process).

The assignment process satisfies
$0<\mathbb{P}\big{(}W_{t}=w\big{|}\mathcal{F}_{t-1}\big{)}<1$ with probability
one for all $w\in\mathcal{W}$. Moreover, the probabilities are determined by a
filtered probability space of
$\big{\\{}W_{t},\big{\\{}Y_{t}(w_{1:t}),w_{1:t}\in\mathcal{W}^{t}\big{\\}}\big{\\}}_{t\geq
1}$.

######  Definition 10.

For $t\geq 1$, $h\geq 0$ and any fixed $w,w^{\prime}\in\mathcal{W}$, the
time$-t$, $h-$period ahead:

  * •

$Y_{t+h}(w)-Y_{t+h}(w^{\prime})$ (impulse causal effect).

  * •

$\mathbb{E}\big{[}Y_{t+h}(w)-Y_{t+h}(w^{\prime})\big{|}\mathcal{F}_{t-1}\big{]}$
(impulse causal effect).

$\mathbb{E}\big{[}Y_{t+h}(w)-Y_{t+h}(w^{\prime})\big{]}$ (impulse causal
effect).

######  Remark 21.

Various further applications of the aforementioned frameworks can be
considered starting with the pioneered work of Aigner and Balestra, (1988) who
consider the optimal experimental design for error correction models, the
econometric framework of De Chaisemartin and d’Haultfoeuille, (2020) which
corresponds to the two-way fixed effects estimators with heterogeneous
treatment effects as well as the novel framework of Lewis and Syrgkanis,
(2020) which corresponds to a time series setting based on machine learning
and statistical learning techniques. However, we consider all aforementioned
settings as more advanced topics and beyond the scope of our current
discussion.

##  6\. Advanced Topics

###  6.1. Testing for Unstable Root in Structural ECMs

We follow the framework proposed by Boswijk, (1994). Consider the single-
equation error correction model of a time series $\left\\{y_{t}\right\\}$
conditional upon the $(k\times 1)$ vector time series
$\left\\{z_{t}\right\\}$, for $t\in\left\\{1,...T\right\\}$ as

$\displaystyle\Delta y_{t}=\beta_{0}^{\prime}\Delta
z_{t}+\lambda\big{(}y_{t-1}-\theta^{\prime}\big{)}+\sum_{j=1}^{p-1}\big{(}\gamma_{j}\Delta
y_{t-j}-\beta_{j}^{\prime}\Delta z_{t-j}\big{)}+v_{t},$ (6.1)

where $\left\\{v_{t}\right\\}$ is an innovation process relative to
$\left\\{z_{t},y_{t-j},z_{t-j},j=1,2,...\right\\}$ with positive variance
$\omega^{2}$. Notice that $\theta$ and $\beta_{j}$ are $(k\times 1)$ parameter
vectors such that $j\in\left\\{1,...,p-1\right\\}$, such that $\theta$ defines
the long-run equilibrium relation $y=\theta^{\prime}z$, the deviations from
which lead to a correction of $y_{t}$ by a proportion of $\lambda$, the
adjustment or error correction coefficient.

The conditional model is said to be stable if all roots of the characteristic
equation

$\displaystyle\varphi(\zeta)=(1-\zeta)\left(1-\sum_{j=1}^{p-1}\gamma_{j}\zeta^{j}\right)-\lambda\zeta=0,$
(6.2)

are outside the unit circle. In other words, stability of the model implies
that the disequilibrium error $\left(y_{t}-\theta^{\prime}z_{t}\right)$ is a
stationary process, even though $z_{t}$ and $y_{t}$ are integrated of order
one and hence nonstationary. Specifically, if the model is stable, then
$x_{t}=(y_{t},z_{t}^{\prime})^{\prime}$ is cointegrated with cointegrating
vector $(1,-\theta)^{\prime}$. Thus, the purpose of this exercise is to
develop a class of tests for the null hypothesis that the characteristic
equation has a unit root, so that the model is unstable, against the
alternative hypothesis of stability. Furthermore, the single-equation
conditional model can be seen as a special case if a structural error
correction model. This is a system of Error Correction Equations for a
$(g\times 1)$ vector of time series $\left\\{y_{t}\right\\}$ conditional upon
$\left\\{z_{t}\right\\}$ such that (see, Boswijk, (1994)):

$\displaystyle\boldsymbol{\Gamma}_{0}\Delta\boldsymbol{y}_{t}=\boldsymbol{B}_{0}\Delta\boldsymbol{z}_{t}+\boldsymbol{\Lambda}\big{(}\boldsymbol{\Gamma}\boldsymbol{y}_{t-1}+\boldsymbol{B}\boldsymbol{z}_{t-1}\big{)}+\sum_{j=1}^{p-1}\big{(}\boldsymbol{\Gamma}_{j}\Delta\boldsymbol{y}_{t-j}+\boldsymbol{B}_{j}\Delta\boldsymbol{z}_{t-j}\big{)}+\boldsymbol{v}_{t},$
(6.3)

where $\left\\{\boldsymbol{v}_{t}\right\\}$ is an innovation process with a
positive-definite covariance matrix $\boldsymbol{\Omega}$ such that the above
expression corresponds to a parametrization of a conditional model of $y_{t}$
given $z_{t}$ (if $\boldsymbol{\Gamma}_{0}\neq\boldsymbol{I}_{g})$. Next we
consider the identification of these matrix parameters. In particular, the
above model implies $g$ cointegrating relationships
$\boldsymbol{\Gamma}\boldsymbol{y}+\boldsymbol{B}\boldsymbol{z}=\boldsymbol{0}$,
provided that it is stable, such that the characteristic equation is expressed
as below

$\displaystyle\varphi(\zeta)=\left|(1-\zeta)\left(\boldsymbol{\Gamma}_{0}-\sum_{j=1}^{p-1}\boldsymbol{\Gamma}_{j}\zeta^{j}\right)-\boldsymbol{\Lambda}\boldsymbol{\Gamma}\zeta\right|=0.$
(6.4)

has all roots outside the unit circle.

In other words, unless the parameters are restricted in some way, the
structural model is not identified. Specifically, we identify the long-run
relations by imposing restrictions of the usual form such that

$\displaystyle\boldsymbol{\Gamma}_{ii}=1\ \ \
\boldsymbol{R}_{i}[\boldsymbol{\Gamma}_{i}\ \
\boldsymbol{B}_{i}]=\boldsymbol{0},\ \ \ i\in\left\\{1,...,g\right\\},$ (6.5)

where $\boldsymbol{\Gamma}_{i}$ and $\boldsymbol{B}_{i}$ denote the $i-$th row
of $\boldsymbol{\Gamma}$ and $\boldsymbol{B}$, respectively, and where
$\boldsymbol{R}_{i}$ is a known matrix of approximate order. Thus, the rank
condition for identification of the $i-$th long-run relation

$\displaystyle\mathsf{rank}\big{(}\boldsymbol{R}_{i}\big{[}\boldsymbol{\Gamma}\
\ \boldsymbol{B}\big{]}^{\prime}\big{)}=(g-1)$ (6.6)

Then, the remaining parameters are identified by the normalization
$\boldsymbol{\Gamma}_{0,ii}=1$, and the restriction that
$\boldsymbol{\Lambda}=\mathsf{diag}\big{(}\lambda_{1},...,\lambda_{g}\big{)}$,
that is, where $\boldsymbol{\Lambda}$ is a diagonal matrix. In practice, this
means that only the disequilibrium error of the $i-$th long-run relation
appears in the $i-$th structural error correction equation which means that
the $i-$th equation is (over)-identified if the $i-$th long-run equation is.

######  Remark 22.

The reason for restricting the error correction matrix $\boldsymbol{\Lambda}$
to be diagonal is twofold. Firstly, it allows for an interpretation of these
separate equations as representing economic behaviour of a group of agents,
whose target consists in a particular long-run relationship, such as a money
demand relation or a consumption function. Furthermore, notice that the
possibility that all endogenous variables are affected by each disequilibrium
error is not excluded. However, this is considered to be a property of the
reduced form of the system rather than the structural form. Moreover, imposing
a symmetric matrix
$\boldsymbol{\Lambda}=\mathsf{diag}\big{(}\lambda_{1},...,\lambda_{g}\big{)}$,
facilitates the implementation and interpretation of a test for instability.
Notice that we assume that the number of stable relationships is equal to the
number of cointegrating relationships.

Consequently, under this null hypothesis, there is no error correction in the
$i-$th equation, which suggests that the $i-$th row of the system
$\boldsymbol{\Gamma}y+\boldsymbol{B}\boldsymbol{z}=0$ is not a cointegrating
relationship. Specifically, let $z_{t}$ be generated by the following
expression:

$\displaystyle\Delta\boldsymbol{z}_{t}=\alpha\big{(}\boldsymbol{\Gamma}\boldsymbol{y}_{t-1}+\boldsymbol{B}\boldsymbol{z}_{t-1}\big{)}+\sum_{j=1}^{p-1}\big{(}\boldsymbol{A}_{1j}\Delta\boldsymbol{y}_{t-j}\boldsymbol{A}_{2j}\Delta\boldsymbol{z}_{t-j}\big{)}+\boldsymbol{\varepsilon}_{t},$
(6.7)

In particular, Assumption 1, can be interpreted as a particular type of
exogeneity assumption, because it essentially states that the cointegration
properties of the conditional model carry over to the full VAR system.
Understanding well these concepts are essential especially if one is
interested to apply for advanced identification techniques in SVAR Models such
as the Markov switching regime (see, Lanne and Saikkonen, (2002) and Lanne et
al., (2010)) as well as in Gaussian Mixture Vector Autoregressive Models (see,
Kalliovirta et al., (2016) and Meitz et al., (2023)). In particular,
Kalliovirta et al., (2016) develop a framework for Gaussian mixture vector
autoregression which is a nonlinear vector autoregression and is designed for
analyzing time series that exhibit regime-switching dynamics101010The property
of long memory and regime switching is examined in the study of Diebold and
Inoue, (2001) while a regime switching panel data regression model is proposed
by Cheng et al., (2019). behaviour (see, also Kalliovirta et al., (2015) and
Virolainen, (2020)). Next, we consider another example motivated from the
structural break literature specifically in the context of cointegrated
models.

####  6.1.1. Testing Cointegration Rank when some Cointegrating Directions are
Changing

Consider the following VECM of order $p$ with a structural break such that

$\displaystyle\Delta\boldsymbol{X}_{t}$
$\displaystyle=\left(\boldsymbol{\alpha}_{0}\boldsymbol{\beta}_{0}^{\prime}\boldsymbol{X}_{t-1}+\boldsymbol{\Phi}_{0}\boldsymbol{D}_{t}\right)\boldsymbol{1}\left\\{t\leq
k_{0}\right\\}+\left(\boldsymbol{\alpha}_{1}\boldsymbol{\beta}_{1}^{\prime}\left(\boldsymbol{X}_{t-1}-\boldsymbol{X}_{k_{0}}\right)+\boldsymbol{\Phi}_{1}\boldsymbol{D}_{t}\right)\boldsymbol{1}\left\\{t>k_{0}\right\\}$
$\displaystyle\ \ \
+\sum_{j=1}^{p}\boldsymbol{\Gamma}_{j}\Delta\boldsymbol{X}_{t-j}+\boldsymbol{\varepsilon}_{t},$

Then, the date of the break is characterized by the fraction $\lambda_{0}$ of
the sample size $n$ such that $k_{0}=\lfloor\lambda_{0}\rfloor$, where
$\lambda_{0}\in[\underline{\lambda},\bar{\lambda}]$ and
$0<\underline{\lambda}<\bar{\lambda}<1$. Moreover, the VECM is assumed to
satisfy the stability condition over the two regimes as given by the
Assumption below.

######  Assumption 10.

The roots of the characteristic polynomials of the $\mathsf{VECM}(1)$

$\displaystyle\mathsf{det}\left[(1-z)\boldsymbol{I}_{n}-\alpha_{i}\beta_{i}^{\prime}z-\sum_{j=1}^{p}\Gamma_{j}(1-z)z^{j}\right],\
\ i=0,1,$ (6.8)

satisfy $z=1$ or $|z|>1$.

######  Assumption 11.

Suppose that $\left\\{\varepsilon_{t}\right\\}$ is a vector martingale
difference sequence with respect to $\mathcal{F}_{t-1}$ such that the
variance-covariance matrix
$\boldsymbol{\Sigma}_{\varepsilon}=\mathbb{E}[\varepsilon_{t}\varepsilon_{t}^{\prime}|\mathcal{F}_{t-1}]$,
for $t=1,2,...$.

######  Remark 23.

The Assumption above does not allow for a shift in the innovation covariance
matrix $\Sigma_{\varepsilon}$ across the two regimes. This would split the
model into two separate ones if the short-term coefficient matrices were also
shifting with the break and would thus require a separate analysis over the
two sub-samples. Similarly we do not allow for a different number of
cointegration relationships across the two two regimes. In particular, we
consider the following cases:

  * Case 1.

The break (structural change) does not affect the loading factors. In this
case their estimation requires an identification constraint. In other words,
$\alpha_{0}$ and $\alpha_{1}$ span the same vector space and thus a natural
identification scheme for the two regimes is given by
$\alpha^{\prime}\Sigma_{\varepsilon}\alpha=I_{r}$.

  * Case 2.

The break affects the cointegration vectors and the loading factors. This
implies that the spaces spanned by $\alpha_{0}$ and $\alpha_{1}$ differ across
the two regimes.

######  Theorem 4.

Consider the model defined by the first expression with
$\alpha_{0}=\alpha_{1}=\alpha$ and Assumption 2. Under $H_{0}$, when
$T\to\infty$ it holds that

$\displaystyle\xi_{T}(\lambda_{0})$
$\displaystyle\overset{d}{\to}\frac{1}{\lambda_{0}}\mathsf{trace}\left\\{\left(\int_{0}^{\lambda_{0}}d\boldsymbol{W}_{0}\boldsymbol{G}_{0}^{\prime}\right)\left(\int_{0}^{\lambda_{0}}\boldsymbol{G}_{0}\boldsymbol{G}_{0}^{\prime}du\right)^{-1}\left(\int_{0}^{\lambda_{0}}\boldsymbol{G}_{0}d\boldsymbol{W}_{0}^{\prime}\right)\right\\}$
$\displaystyle+\frac{1}{\left(1-\lambda_{0}\right)}\mathsf{trace}\left\\{\left(\int_{\lambda_{0}}^{1}d\boldsymbol{W}_{0}\boldsymbol{G}_{1}^{\prime}\right)\left(\int_{\lambda_{0}}^{1}\boldsymbol{G}_{1}\boldsymbol{G}_{1}^{\prime}du\right)^{-1}\left(\int_{\lambda_{0}}^{1}\boldsymbol{G}_{1}d\boldsymbol{W}_{0}^{\prime}\right)\right\\}$

###  6.2. Empirical Likelihood Estimation Approach

######  Example 28.

Consider the following constant coefficient autoregressive model with time-
varying variances as below:

$\displaystyle Y_{t}$
$\displaystyle=\beta_{0}+\beta_{1}Y_{t-1}+\beta_{2}Y_{t-2}+...+\beta_{p}Y_{t-p}+u_{t},$
(6.9) $\displaystyle Y_{t}$
$\displaystyle=\boldsymbol{X}^{\top}_{t-1}\boldsymbol{\beta}_{o}+u_{t},\
u_{t}=\sigma_{t}\varepsilon_{t},\ t=1,...,T,$ (6.10)

where the vector of covariates is denoted by
$\boldsymbol{X}_{t}=\big{(}1,Y_{t-1},...,Y_{t-p}\big{)}^{\top}\in\mathbb{R}^{p+1}$
and the true model parameter of interest is denoted by
$\boldsymbol{\beta}_{o}=\big{(}\beta_{0},\beta_{1},...,\beta_{p}\big{)}^{\top}\in\mathbb{R}^{p+1}$,
where the lag order is finite and known.

Based on the aforementioned assumptions the estimation of the unknown
parameter vector $\boldsymbol{\beta}_{o}$ based on the OLS estimator
$\hat{\boldsymbol{\beta}}$ is given by

$\displaystyle\sqrt{T}\left(\boldsymbol{\beta}-\boldsymbol{\beta}_{o}\right)=\left(\frac{1}{T}\sum_{t=1}^{T}\boldsymbol{X}_{t-1}^{\top}\boldsymbol{X}_{t-1}\right)^{-1}\left(\frac{1}{T}\sum_{t=1}^{T}\boldsymbol{X}_{t-1}^{\top}\varepsilon_{t}\right)\overset{d}{\to}\mathcal{N}(\boldsymbol{0},\boldsymbol{\Lambda}),$
(6.11)

where
$\boldsymbol{\Lambda}=\boldsymbol{\Omega}_{1}^{-1}\boldsymbol{\Omega}_{2}\boldsymbol{\Omega}_{1}^{-1}$,
are defined as $(p+1)\times(p+1)$ matrices.

To construct an empirical likelihood function, the estimation equations are
defined as below:

$\displaystyle\boldsymbol{W}_{t}(\boldsymbol{b})=\boldsymbol{X}_{t-1}\cdot\left(Y_{t}-\boldsymbol{X}_{t-1}^{\top}\boldsymbol{b}\right),$
(6.12)

for a generic parameter $\boldsymbol{b}\in\mathbb{R}^{p+1}$. By using the
Lagrange multipliers method, we have that
$\hat{\boldsymbol{\lambda}}=\hat{\boldsymbol{\lambda}}(\boldsymbol{b})\in\mathbb{R}^{p+1}$
is the solution of the following set of equations:

$\displaystyle\frac{1}{T}\sum_{t=1}^{T}\frac{\boldsymbol{W}_{t}(\boldsymbol{b})}{1+\hat{\boldsymbol{\lambda}}^{\prime}\cdot\boldsymbol{W}_{t}(\boldsymbol{b})}=\boldsymbol{0}.$
(6.13)

Then, the corresponding empirical log-likelihood ratio is given by

$\displaystyle\ell(\boldsymbol{b})=2\sum_{t=1}^{T}\mathsf{log}\big{[}1+\hat{\boldsymbol{\lambda}}^{\prime}\cdot\boldsymbol{W}_{t}(\boldsymbol{b})\big{]}$
(6.14)

and it holds that $\ell(\boldsymbol{\beta}_{0})\overset{d}{\to}\chi_{p}^{2}$,
as $T\to\infty$.

####  6.2.1. High Dimensional Generalized Empirical Likelihood Estimation

In this section, we discuss relevant aspects to the empirical likelihood
estimation for high dimensional dependent data, which is applicable to time
series regression models (see, Chang et al., (2015)). Further applications to
high-dimensional time series regression models include Medeiros and Mendes,
(2016) as well as to SVAR models Krampe et al., (2023), Zhang, (2023) and
Adamek et al., (2022). Denote with
$\boldsymbol{\theta}=(\theta_{1},...,\theta_{p})^{\prime}$ be a
$p-$dimensional parameter taking values in a parameter space $\Theta$.
Consider a sequence of $r-$dimensional estimating equation such that

$\displaystyle\mathsf{g}(X_{t},\boldsymbol{\theta})=\big{(}\mathsf{g}_{1}(X_{t},\boldsymbol{\theta}),....,\mathsf{g}_{r}(X_{t},\boldsymbol{\theta})\big{)}$
(6.15)

for some $r\geq p$, Then, the model information regarding the data and the
data parameter is summarized by moment restrictions below:

$\displaystyle\mathbb{E}\big{[}\mathsf{g}(X_{t},\boldsymbol{\theta}_{0})\big{]}=\boldsymbol{0}.$
(6.16)

where $\theta_{0}\in\Theta$ is the true parameter. Furthermore, in order to
preserve the dependence structure among the underlying data, we employ the
blocking technique. Let $M$ and $L$ be two integers denoting the block length
and separation between adjacent blocks, respectively. Then, the total number
of blocks is

$\displaystyle Q=\lfloor\frac{(n-M)}{L}\rfloor+1$ (6.17)

Then, the EL estimator $\boldsymbol{\theta}_{o}$ is
$\widehat{\boldsymbol{\theta}}_{EL}=\mathsf{argmax}_{\theta in\Theta}\
\mathsf{log}\ \mathcal{L}(\boldsymbol{\theta})$. Consequently, the
maximization problem can be carried out more efficiently by solving the
corresponding dual problem, which implies that
$\widehat{\boldsymbol{\theta}}_{EL}$ can be obtained as below:

$\displaystyle\widehat{\boldsymbol{\theta}}_{EL}=\underset{\boldsymbol{\theta}\in\Theta}{\mathsf{arg\
min}}\
\underset{\lambda\in\widehat{\Lambda}_{n}(\boldsymbol{\theta})}{\mathsf{max}}\
\sum_{q=1}^{Q}\mathsf{log}\big{[}1+\boldsymbol{\lambda}^{\top}\phi_{M}(B_{q},\boldsymbol{\theta})\big{]},$
(6.18)
$\displaystyle\widehat{\Lambda}_{n}(\boldsymbol{\theta}):=\left\\{\lambda\in\mathbb{R}^{r}:\lambda^{\top}\cdot\phi_{M}(B_{q},\boldsymbol{\theta})\in\mathcal{N},q=1,...,Q\right\\}$
(6.19)

for any $\boldsymbol{\theta}\in Q$ and $\mathcal{N}$ an open interval
containing zero.

######  Example 29 (Time Series Regression).

Consider a structural model $s-$dimensional time series $Y_{t}$ which involve
unknown parameter $\boldsymbol{\theta}\in\mathbb{R}^{p}$ of interest as well
as time innovations with unknown distributional form. Thus, suppose we have
that

$\displaystyle
h\big{(}Y_{t},...,Y_{t-m};\boldsymbol{\theta}_{0}\big{)}=\boldsymbol{\varepsilon}_{t}\in\mathbb{R}^{r}$
(6.20)

where $m\geq 1$ is some constant. In particular, for conventional vector
autoregressive models such that

$\displaystyle
Y_{t}=\boldsymbol{A}_{1}Y_{t-1}+...+\boldsymbol{A}_{m}Y_{t-m}+\eta_{t},$
(6.21)

Notice that these set of model parameters
$\left\\{\boldsymbol{A}_{1},...,\boldsymbol{A}_{m}\right\\}$ correspond to
coefficient matrices and are usually estimated based on some maximum
likelihood approach (usually the MLS under Gaussianity) and according to the
distributional assumptions of the model. Moreover, suppose that $\eta_{t}$ is
the white noise series such that

$\displaystyle
h\big{(}Y_{t},...,Y_{t-m};\boldsymbol{\theta}_{0}\big{)}=\big{(}Y_{t}-\boldsymbol{A}_{1}Y_{t-1}-...-\boldsymbol{A}_{m}Y_{t-m}\big{)}\otimes\big{(}Y_{t}^{\top},....,Y_{t-m}^{\top}\big{)}^{\top}.$
(6.22)

High dimensional time series analysis, requires to assume that the
dimensionality of $Y_{t}$ is large in relation to sample size, that is,
$s\to\infty$ as $n\to\infty$. Within a high-dimensional environment, the
number of estimating equation and unknown parameters are both $s^{2}m$. On the
other hand, if we replace
$\big{(}Y_{t}^{\top},....,Y_{t-m}^{\top}\big{)}^{\top}$ by
$\big{(}Y_{t}^{\top},....,Y_{t-m-\ell}^{\top}\big{)}^{\top}$ for some fixed
$\ell\geq 1$, then the model will be over-identified. This phenomenon of over-
parametrization in such models is well-known to the literature. Thus, to
implement a consistent estimation approach, the sparsity assumption allows to
employ a penalized estimation methodology.

###  6.3. High Dimensional VARs with Common factors

We follow the framework proposed by Miao et al., (2023) who study high-
dimensional vector autoregressions (VARs) augmented with common factors that
allow for strong cross-sectional dependence. This approach allows to
incorporate in a unified framework a convinient mechanism for accommodating
the interconnectedness and temporal co-variability that are often present in
large dimensional systems.

Consider the $N-$dimensional vector-valued time series
$\left\\{Y_{t}\right\\}=\left\\{\left(y_{1t},...,y_{NT}\right)^{\prime}\right\\}$,
the high-dimensional VAR model of order $p$ with CFs given by

$\displaystyle\boldsymbol{Y}_{t}=\sum_{j=1}^{p}\boldsymbol{A}_{j}^{0}\boldsymbol{Y}_{t-j}+\boldsymbol{\Lambda}^{0}\boldsymbol{f}_{t}^{0}+\boldsymbol{u}_{t},\
\ \ t=1,...,T,$ (6.23)

where $\boldsymbol{A}_{1}^{0},...,\boldsymbol{A}_{p}^{0}$ are the $(N\times
N)$ transition matrices and $\boldsymbol{u}_{t}$ is an $N-$dimensional vector
of unobserved idiosyncratic errors. Moreover, the analytical framework allows
for both the number of cross-sectional units $N$ and the number of time
periods $T$ to pass to infinity. The lag length is also allowed to (slowly)
grow to infinity with $(N,T)$. Estimation then is a natural high-dimensional
problem.

Then, the $N-$dimensional VAR$(p)$ process $\left\\{Y_{t}\right\\}$ can be
rewritten in a companion form as an $Np-$dimensional VAR$(1)$ process with
common factors such that:

$\displaystyle\underbrace{\begin{bmatrix}Y_{t}\\\ Y_{t-1}\\\ \vdots\\\
Y_{t-p-1}\end{bmatrix}}_{\boldsymbol{X}_{t+1}}=\underbrace{\begin{bmatrix}A_{1}^{0}&A_{2}^{0}&\ldots&A_{p-1}^{0}&A_{p}^{0}\\\
I_{N}&\boldsymbol{0}&\ldots&\boldsymbol{0}&\boldsymbol{0}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
\boldsymbol{0}&\boldsymbol{0}&\ldots&\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}}_{\boldsymbol{\Phi}}\underbrace{\begin{bmatrix}Y_{t-1}\\\
Y_{t-2}\\\ \vdots\\\
Y_{t-p}\end{bmatrix}}_{\boldsymbol{X}_{t}}+\underbrace{\begin{bmatrix}\boldsymbol{\Lambda}^{0}\boldsymbol{f}_{t}^{0}\\\
\boldsymbol{0}\\\ \vdots\\\
\boldsymbol{0}\end{bmatrix}}_{\boldsymbol{F}_{t}}+\underbrace{\begin{bmatrix}\boldsymbol{u}_{t}\\\
\boldsymbol{0}\\\ \vdots\\\
\boldsymbol{0}\end{bmatrix}.}_{\boldsymbol{U}_{t}}$ (6.24)

As a result, the reverse characteristic polynomial of $Y_{t}$ can be written
as below:

$\displaystyle\mathcal{A}(z)\equiv I_{N}-\sum_{j=1}^{p}A_{j}^{0}z^{p}.$ (6.25)

In the low-dimensional framework, the process is stationary if
$\mathcal{A}(z)$ has no roots in and on the complex unit circle, or
equivalently the largest modules of the eigenvalues of $\boldsymbol{\Phi}$ is
less than 1. Therefore, to achieve identification, we shall study the Gram or
signal matrix $\boldsymbol{S}_{X}=\boldsymbol{X}^{\prime}\boldsymbol{X}/T$ and
its population counterpart
$\Sigma_{X}=\mathbb{E}\left(\boldsymbol{X}^{\prime}_{t}\boldsymbol{X}_{t}\right)$.
In other words, one can study the deviation bounds for the Gram matrix, under
the Gaussianity assumption and boundedness of the spectral density function.

In order to ensure that the matrix $\Sigma_{X}$ is well-behaved, we write
$\boldsymbol{X}_{t+1}$ as a moving average process of infinite order
MA$(\infty)$ such that

$\displaystyle
X_{t+1}\equiv\sum_{j=0}^{\infty}\boldsymbol{\Phi}^{j}\big{(}\boldsymbol{F}_{t-j}+\boldsymbol{U}_{t-j}\big{)}=\sum_{j=0}^{\infty}\boldsymbol{\Phi}^{j}\boldsymbol{F}_{t-j}+\sum_{j=0}^{\infty}\boldsymbol{\Phi}^{j}\boldsymbol{U}_{t-j}$
(6.26)

Eigenvalue Analysis:

  * •

First, consider
$X_{t+1}^{(f)}=\sum_{j=0}^{\infty}\boldsymbol{\Phi}^{j}\boldsymbol{F}_{t-j}$,
the component due to the common factors. The covariance matrix of
$\boldsymbol{F}_{t}$ is a high-dimensional matrix with rank $R^{0}$ and
explosive non-zero eigenvalues. In other words, even if the largest modules of
the eigenvalues of $\boldsymbol{\Phi}$ is smaller than 1, the variances of the
entries of $X_{t+1}^{(f)}$ are not assumed to be uniformly bounded.

  * •

Consider $y_{it}^{(f)}$, which is the $i-$th entry of $X_{t+1}^{(f)}$. Let
$e_{j,M}$ be the $j-$th column of $I_{M}$. Noting that
$y_{it}^{(f)}=\big{(}e_{1,p}\otimes e_{i,N}\big{)}^{\prime}X_{t+1}^{(f)}$, we
can write $y_{it}^{(f)}$ as the MA$(\infty)$ process given below:

$\displaystyle y_{it}^{(f)}=\sum_{j=0}^{\infty}\big{(}e_{1,p}\otimes
e_{i,N}\big{)}^{\prime}\Phi^{j}\big{(}e_{1,p}\otimes
e_{i,N}\big{)}f_{t-j}^{0}\equiv\sum_{j=0}^{\infty}\alpha_{iN}^{(f)}(j)f_{t-j}^{0},$
(6.27)

in which $f_{t}^{0}$ are allowed to be serially correlated.

######  Assumption 12 (Miao et al., (2023)).

Consider that the following conditions hold:

  * (i).

Let $u_{t}=C^{(u)}\varepsilon_{t}^{(u)}$, where
$\varepsilon_{t}^{(u)}=\big{(}\varepsilon_{1,t}^{(u)},...,\varepsilon_{m,t}^{(u)}\big{)}^{\prime}$
such that $\varepsilon_{i,t}^{(u)}$ are i.i.d random variables across $(i,t)$
with mean zero and variance 1.

  * (ii).

$\left\\{f_{t}^{0}\right\\}$ follows a strictly stationary linear process
given as below:

$\displaystyle
f_{t}^{0}-\mu_{f}=\sum_{j=0}^{\infty}C_{j}^{(f)}\epsilon^{(f)}_{t-j},$ (6.28)

###  6.4. Fast Algorithm for Detection of Breaks in Large VAR models

We follow the framework proposed by Safikhani et al., (2022). Their procedure
is found to be scalable to big high-dimensional time series datasets with a
computational complexity that can achieve $O(\sqrt{n})$, where $n$ is the
length of the time series (sample size), compared to an exhaustive procedure
that requires $O(n)$ steps.

####  6.4.1. Model Formulation

Suppose that there exist $m_{0}$ break points $0<t_{1}<...t_{m_{0}}<T$, with
$t_{0}=0$ and $t_{m_{0}+1}=T$ where $j\in\left\\{1,...,m_{0}+1\right\\}$.
Notice that if there are no break points, then $m_{0}=0$ and so
$(t_{0}=1,t_{1}=T)$ which corresponds to the full sample period.

Thus, we can assume that $m_{0}\geq 1$, which implies that $(t_{0}=1,t_{2}=T)$
and so $j=\left\\{1,2\right\\}$. In other words, for $t_{j-1}\leq t<t_{j}$
(e.g., $t_{0}\leq t<t_{1}$, i.e., $t=1,...,T$) we have the following model:

$\displaystyle
y_{t}=\Phi_{(1,j)}y_{t-1}+...+\Phi_{(q,j)}y_{t-q}+\Sigma_{j}^{1/2}\varepsilon_{t}$
(6.29)

Since in the case that $m_{0}=1$ then $j\in\left\\{1,2\right\\}$ then we have
a two-regime model estimation:

$\displaystyle M_{0}:\boldsymbol{y}_{t}$
$\displaystyle=\boldsymbol{\Phi}_{(1,1)}\boldsymbol{y}_{t-1}+...+\boldsymbol{\Phi}_{(q,1)}\boldsymbol{y}_{t-q}+\boldsymbol{\Sigma}_{1}^{1/2}\boldsymbol{\varepsilon}_{t},\
\ \ t_{0}\leq t<t_{1},\ \ t\in\left\\{0,...,t_{1}\right\\},$ (6.30)
$\displaystyle M_{1}:\boldsymbol{y}_{t}$
$\displaystyle=\boldsymbol{\Phi}_{(1,2)}\boldsymbol{y}_{t-1}+...+\boldsymbol{\Phi}_{(q,2)}\boldsymbol{y}_{t-q}+\boldsymbol{\Sigma}_{2}^{1/2}\boldsymbol{\varepsilon}_{t},\
\ \ t_{1}\leq t<t_{2},\ \ t\in\left\\{t_{1},...,T\right\\},$ (6.31)

There is a single break-point and we fit a different VAR model before the
break, i.e. $M_{0}$ pre-break based on the observations up to the unknown
break-point, and a we fit $M_{1}$ post-break based on the observations from
the unknown break-point and onward until the end of the full sample.

In each segment we estimate the VAR process. Let $\boldsymbol{y}_{t}$ be a
$p-$dimensional vector of observations at time $t$ and
$\boldsymbol{\Phi}_{(\ell,j)}$ be an $(p\times p)$ sparse coefficient matrix
corresponding to the $\ell-$th lag of a VAR process of order $q$ during the
$j-$th stationary segment, and $\varepsilon_{t}$ is a white-noise process with
zero mean and variance matrix $\boldsymbol{\Sigma}_{j}$. Furthermore, in each
segment $[t_{j-1},t_{j})$, all model parameters are assumed to be fixed.
However, the auto-regressive (AR) parameters $\boldsymbol{\Phi}_{(\ell,j)}$
will change values across segments, while the error covariance across all
segments is assumed to be $\boldsymbol{\Sigma}_{j}=\sigma^{2}\boldsymbol{I}$.

Based on the aforementioned setting proposed by Safikhani et al., (2022), the
unknown parameters in each segment are: the number of break-points $m_{0}$,
their locations $t_{j},j=1,...,m_{0}$, the VAR parameters
$\boldsymbol{\Phi}_{(q,j)}$ together with the covariance matrix. Therefore,
according to Safikhani et al., (2022), the statistical problem is then to
detect the break points $t_{j}$, in a computational efficient manner that is
also scalable for very large values of $T$. Additionally, we are also
interested to estimate accurately the VAR parameters
$\boldsymbol{\Phi}_{(\ell,j)}$, under a high-dimensional environment $(p>>T)$.

####  6.4.2. A Block Segmentation Scheme Based Algorithm

The main idea of the BSS is to partition the time points into blocks of size
$b_{n}$ and fix the VAR parameters within each block. To this end, define a
sequence of time points $q=r_{0}<r_{1}<...<r_{k_{n}}=T+1$ which play the role
of the end points for the blocks. In other words, the total number of blocks
is calculated as:

$\displaystyle r_{i+1}-r_{i}=b_{n},\ \ \ \text{for}\
i\in\left\\{0,....,k_{n}-2\right\\},\ k_{n}=\lfloor\frac{n}{b_{n}}\rfloor.$
(6.32)

where $k_{n}$ is the total number of blocks based on the (effective) sample
size and $b_{n}$ is the length size of each block which is kept fixed across
all blocks. Notice that throughout, the dimensions of the vector
$\boldsymbol{y}_{t}$ remains fixed having $p$ elements as well as the
(effective) sample size being $n$.

The theoretical results presented in Safikhani et al., (2022) deal with the
high-dimensional case such that the sparsity levels $d_{k_{j}}$ increase with
the sample size, $T$. Specifically, we define with $p\equiv p(n)$ and
$m_{0}\equiv m_{0}(n)$ and $d_{k_{j}}\equiv d_{k_{j}}(n)$, where $n=T-q+1$. In
other words, $n$ is the effective sample size or the length of the time series
for estimation of model parameters purposes, since the number of lags used for
constructing additional regressors affects the sample size (see, Safikhani et
al., (2022)).

The minimum distance between two consecutive break points is denoted by

$\displaystyle\Delta_{n}=\underset{1\leq j\leq
m_{0}+1}{\mathsf{min}}|t_{j}-t_{j-1}|.$ (6.33)

Let $\boldsymbol{\mathcal{Y}}\in\mathbb{R}^{n\times p}$ and
$\boldsymbol{\mathcal{X}}\in\mathbb{R}^{n\times k_{n}pq}$ and
$\boldsymbol{\theta}\in\mathbb{R}^{k_{n}pq\times p}$ and
$\boldsymbol{E}\in\mathbb{R}^{n\times p}$. In other words, based on this
parametrization, $\theta_{i}\neq 0$ for $i\geq 2$ implies a change in the VAR
coefficients. Therefore, for $j\in\left\\{1,...,m_{0}\right\\}$, the
structural break points $t_{j}$ can be estimated as block-end time point
$r_{i-1}$, where $i\geq 2$ and $\theta_{i}\neq 0$.

The, the model is formulated as
$\boldsymbol{Y}=\boldsymbol{Z}\boldsymbol{\Theta}+\boldsymbol{E}$, where
$\boldsymbol{Y}=\mathsf{vec}(\boldsymbol{\mathcal{Y}})\in\mathbb{R}^{np\times
1}$,
$\boldsymbol{Z}=\boldsymbol{I}_{p}\otimes\boldsymbol{\mathcal{X}}\in\mathbb{R}^{np\times\pi_{b}}$,
where $\pi_{b}=k_{n}p^{2}q$ and
$\boldsymbol{\Theta}=\mathsf{vec}(\boldsymbol{\theta})\in\mathbb{R}^{\pi_{b}\times
1}$. Consequently, the initial estimate of the parameter $\boldsymbol{\Theta}$
is obtained:

$\displaystyle\widehat{\boldsymbol{\Theta}}=\underset{\boldsymbol{\Theta}\in\in\mathbb{R}^{\pi_{b}\times
1}}{\mathsf{arg\ min}}\
\frac{1}{n}\left\lVert\boldsymbol{Y}-\boldsymbol{Z}\boldsymbol{\Theta}\right\rVert_{2}^{2}+\lambda_{1,n}\left\lVert\boldsymbol{\Theta}\right\rVert_{1}+\lambda_{2,n}\sum_{i=1}^{k_{n}}\left\lVert\sum_{j=1}^{i}\theta_{j}\right\rVert_{1}.$
(6.34)

Notice that the above high dimensional problem (convexity is preserved) uses
fused lasso penalty with two $\ell_{1}$ penalties controlling the number of
break points and the sparsity of the VAR model.

Denote the set of indices of blocks with nonzero jumps and corresponding
estimated change points obtained from solving the high-dimensional problem as
below:

$\displaystyle\widehat{I}_{n}$
$\displaystyle=\left\\{\hat{i}_{1},\hat{i}_{2},...,\hat{i}_{\hat{m}}\right\\}=\left\\{i:\left\lVert\widehat{\boldsymbol{\Theta}}_{i}\right\rVert_{F}\neq
0,i=1,...,k_{n}\right\\},$ (6.35) $\displaystyle\mathcal{A}$
$\displaystyle=\left\\{\hat{t}_{1},\hat{t}_{2},...,\hat{t}_{\hat{m}}\right\\}=\left\\{r_{i-1}:i\in\widehat{I}_{n}\right\\}.$
(6.36)

The total number of estimated change points in this step corresponds to the
cardinality of the set $\mathcal{A}_{n}$, where
$\hat{m}=\left|\mathcal{A}_{n}\right|$. Moreover, the block size $b_{n}$ acts
as a tuning parameter that regulates the number of model parameters to be
estimated. In addition to this, the algorithm applies a local screening step.

Since the set $\mathcal{A}_{n}$ of candidate change points tends to
overestimates their numbers, a screening step to eliminate redundant break
points is required. According to Safikhani et al., (2022), the main idea goes
is to estimate the VAR parameters locally on the left and right side of each
selected break point in the first step and compare them to one VAR parameter
estimated from combining the left and right of the selected break point as one
large stationary segment. Now, if the selected break point is close to a true
break point, the sum of squared errors calculated assuming stationarity around
the true break point will be much larger compared to the sum of squared errors
calculated from two separate VAR parameter estimates on the right and left of
the selected break point. Therefore, we can get consistent estimates of the
number of break points by minimizing a localized information criterion (LIC)
comprising of the sum of squared errors and a penalty term on the number of
break points.

Next, we describe the localized screening step in more details. Recall that
$\mathcal{A}=\left\\{\hat{t}_{1},\hat{t}_{2},...,\hat{t}_{\hat{m}}\right\\}$
is the set of candidate break points selected in the first step. Then, for
each subset $A\subset\mathcal{A}_{n}$, we define the following local VAR
parameter estimates for some $\widehat{t}_{i}\in A$ such that

$\displaystyle\widehat{\psi}_{\widehat{t}_{i},1}=\underset{\widehat{t}_{i},1}{\mathsf{arg\
min}}\left\\{\frac{1}{a_{n}}\sum_{t=\widehat{t}_{i}-a_{n}}^{\widehat{t}_{i}-1}\left\lVert
y_{t}-\psi_{\widehat{t}_{i},1}Y_{t-1}\right\rVert_{2}^{2}+\eta_{\widehat{t}_{i},1}\left\lVert\psi_{\widehat{t}_{i},1}\right\rVert_{1}\right\\}$
(6.37)

where $a_{n}$ corresponds to the neighborhood size in which the VAR parameters
are estimated.

####  6.4.3. Consistency of the BSS Estimator

To establish the consistency properties of the BSS-based estimator relevant
regularity conditions are required which we briefly present below.

######  Assumption 13.

For each $j\in\left\\{1,2,...,m_{0}+1\right\\}$, the process

$\displaystyle\boldsymbol{y}_{(j),t}=\boldsymbol{\Phi}_{(1,j)}\boldsymbol{y}_{(j),t-1}+...+\boldsymbol{\Phi}_{(q,j)}\boldsymbol{y}_{(j),t-q}+\boldsymbol{\Sigma}_{j}^{1/2}\boldsymbol{\varepsilon}_{t}$
(6.38)

is a stationary Gaussian time series. Denote the covariance matrices with

$\displaystyle\boldsymbol{\Gamma}_{(j)}(h)=\mathsf{Cov}\left(\boldsymbol{y}_{(j),t},\boldsymbol{y}_{(j),t+h}\right),\
\ \text{for}\ \ t,h\in\mathbb{Z}.$ (6.39)

######  Assumption 14.

The matrices $\boldsymbol{\Phi}_{(1,j)}$ are sparse. Specifically, for all
$k\in\left\\{1,...p\right\\}$ and $j\in\left\\{1,...,m_{0}\right\\}$,
$d_{k_{j}}<<p$, that is, $d_{k_{j}}/p=o(1)$. Moreover, there exists a positive
constant $M_{\Phi}>0$ such that

$\displaystyle\underset{1\leq j\leq
m_{0}+1}{\mathsf{max}}\left\lVert\boldsymbol{\Phi}_{(\cdot,j)}\right\rVert_{\infty}\leq
M_{\Phi}$ (6.40)

######  Remark 24.

Moreover, a relevant study is the paper of Lee et al., (2023) who propose a
change point test for structural vector autoregressive models via independent
component analysis. In practice the test statistic is constructed based on the
fitted residuals and the CUSUM functional.

In summary, there are two streams of literature relevant to structural break
testing around the Vector Autoregressive framework of interest. On the one
hand, Safikhani et al., (2022) propose a fast and scalable algorithm for
detection of structural breaks in large VAR models. The particular modelling
environment corresponds to a piecewise linear Vector Autoregression model.
Thus, testing for structural breaks takes the form of a time-varying VAR model
and therefore in practise the statistical problem of interest is to test for
the presence of time-varying vector autoregressive matrix coefficients across
the blocks. Moreover, since the model is parametrized in a form of a large VAR
model with a corresponding companion matrix, then the focus is on the fast and
accurate break detection algorithm with respect to the dimensionality of the
model. On the other hand, Gouriéroux et al., (2020) propose an econometric
framework for identification and estimation of the structural vector
autoregressive model where the main focus is how the distributional
assumptions on the shocks affect the identification procedure. In practice,
based on the non-Gaussianity assumption of the structural shocks the model is
identifiable and the estimation of model parameters is achieved via the use of
the likelihood function.

######  Example 30.

Suppose we are interested in testing whether there is a change in the
covariance matrix of the stationary multivariate time series
$\left\\{\boldsymbol{Y}_{n1},...,\boldsymbol{Y}_{n1}\right\\}$. Then, this
implies that

$\displaystyle\boldsymbol{Y}_{ni}:=\boldsymbol{Y}_{n1}^{(0)}\boldsymbol{1}\left\\{i\leq
k\right\\}+\boldsymbol{Y}_{n1}^{(1)}\boldsymbol{1}\left\\{i>k\right\\},\ \ \
1\leq i\leq n,$ (6.41)

for some change-point in the interval $1\leq k\leq n$, where
$\boldsymbol{Y}_{n1}^{(0)}$ denotes the stationary time series before and
change-point and $\boldsymbol{Y}_{n1}^{(1)}$ denotes the stationary time
series after the change-point. Therefore, we assume that it induces a change
to the structure of the covariance matrix which implies the following
expressions

$\displaystyle\boldsymbol{\Sigma}_{n}^{(0)}:=\mathsf{Var}\left(Y_{n,1}^{(0)}\right)\
\ \ \text{and}\ \ \
\boldsymbol{\Sigma}_{n}^{(1)}:=\mathsf{Var}\left(Y_{n,k+1}^{(1)}\right)$
(6.42)

Denote with
$\boldsymbol{\Sigma}_{n}([j]):=\mathsf{Var}\left(\boldsymbol{Y}_{nj}\right)$
for some $1\leq j\leq n$. Then, we are interested in testing the change-point
problem at a single location such as

$\displaystyle
H_{0}:\boldsymbol{\Sigma}_{n}([j])\equiv\boldsymbol{\Sigma}_{n}^{(0)}\ \ \
\forall\ j=1,...,n\ \ \ \text{and}\ \ \ H_{1}:\exists\
k\in\left\\{2,...,n\right\\}:\boldsymbol{\Sigma}_{n}([k])\neq\boldsymbol{\Sigma}_{n}^{(0)},$
(6.43)

which tests the null hypothesis of stability of the covariance structure
against the alternative hypothesis of a change. An equivalent formulation of
the statistical testing problem under consideration is given as

$\displaystyle H_{0}:k=n\ \ \ \text{against}\ \ \ H_{1}:k<n.$ (6.44)

The proposed change-point detection test statistics are based are based on the
partial sums of the stationary multivariate time series such that
$\boldsymbol{S}_{nk}=\sum_{i\leq
k}\boldsymbol{Y}_{ni}\boldsymbol{Y}_{ni}^{\prime},\ \ \ k\geq 1$. Under the
assumption that the true change-point $k$ is proportional to the sample size,
$k=\lfloor\lambda n\rfloor$, where $0<r<\lambda<1$, the unknown break-point
can be consistently estimated using the following estimator

$\displaystyle\hat{k}_{n}:=\underset{n_{0}\leq k\leq n}{\mathsf{argmax}}\
\frac{1}{n}\left|\boldsymbol{v}_{n}^{\prime}\left(\boldsymbol{S}_{nk}-\frac{k}{n}\boldsymbol{S}_{nn}\right)\boldsymbol{w}_{n}\right|$
(6.45)

The formulation of the test statistic presented in the above example, implies
that is the (smallest) time index $k\geq n_{0}:=\lfloor nr\rfloor$, for some
small $r>0$, at which the maximum in the definition of the test statistic
$C_{n}$ is attained. On the other hand, in order to robustify these CUSUM-type
test statistics for detecting changes not only in the middle of the sample,
the following statistic is proposed

$\displaystyle C_{n,md}=\frac{T_{n,md}}{\hat{\alpha}_{n}},\ \ \ \text{with}\ \
T_{n,md}=\underset{1\leq j\leq n}{\mathsf{max}}\
\frac{1}{\sqrt{n}}\left|\sum_{1\leq\ell\leq
j}\boldsymbol{v}_{n}^{\prime}\left(\boldsymbol{S}_{n\ell}-\frac{(j-i)}{n}\boldsymbol{S}_{nn}\right)\boldsymbol{w}_{n}\right|$
(6.46)

which considers the maximal deviation from the average of the cumulated sums,
the maximum being taken over all possible subsamples
$\boldsymbol{Y}_{n\ell},i\leq\ell\leq j$, for $1\leq i<j\leq n$. To be more
precise, the particular algorithm implies that each subsample induces a
corresponding CUSUM statistic and then the maximal test statistic is taken
over all these subsamples. Therefore, it can be proved

$\displaystyle T_{n,md}\overset{d}{\to}\ \underset{0<s<t<1}{\mathsf{sup}}\
\left|B^{o}(s)-B^{o}(t)\right|,\ \ \text{as}\ n\to\infty.$ (6.47)

Further studies related to testing for structural breaks in covariance
structures include Kao et al., (2018). In particular,
$\widehat{\boldsymbol{\Sigma}}$ be the sample covariance matrix such that
$\widehat{\boldsymbol{\Sigma}}=\frac{1}{T}\sum_{t=1}^{T}y_{t}y_{t}^{\prime}$.
Then, for a given $\uptau\in[0,1]$, we define a point in time $\lfloor
T\uptau\rfloor$, and we use the subscripts $\uptau$ and $1-\uptau$ to denote
quantities calculated using the subsamples $t=1,...,\lfloor T\uptau\rfloor$
and $t=\lfloor T\uptau\rfloor+1,...,T$, respectively. In particular, we
consider the sequence of partial sample estimators

$\displaystyle\widehat{\boldsymbol{\Sigma}}_{\uptau}=\displaystyle\frac{1}{T\uptau}\sum_{t=1}^{\lfloor
T\uptau\rfloor}y_{t}y_{t}^{\prime}\ \ \ \ \text{and}\ \ \ \
\widehat{\boldsymbol{\Sigma}}_{1-\uptau}=\displaystyle\frac{1}{T(1-\uptau)}\sum_{t=\lfloor
T\uptau\rfloor+1}^{T}y_{t}y_{t}^{\prime}$ (6.48)

We denote with $w_{t}=\mathsf{vec}\big{(}y_{t}y_{t}^{\prime}\big{)}$ and
$\bar{w}_{t}=\mathsf{vec}\big{(}y_{t}y_{t}^{\prime}-\boldsymbol{\Sigma}\big{)}$.

####  6.4.4. Useful Asymptotic theory results

######  Assumption 15.

Let $\mathsf{sup}_{t}\mathbb{E}\left\lVert y_{t}\right\rVert^{2r}<\infty$ for
some $r>2$. Then, we define with

$\displaystyle
V_{\Sigma,T}=\frac{1}{T}\mathbb{E}\left[\left(\sum_{t=1}^{T}\bar{w}_{t}\right)\left(\sum_{t=1}^{T}\bar{w}_{t}\right)^{\prime}\right]$
(6.49)

######  Theorem 5.

Under Assumption 15, as $T\to\infty$

$\displaystyle\frac{1}{\sqrt{T}}\sum_{t=1}^{\lfloor
T\uptau\rfloor}\bar{w}_{t}\to\big{[}V_{\Sigma}\big{]}^{1/2}W_{n^{2}}(\uptau),$
(6.50)

uniformly in $\uptau$. Redefining $\bar{w}_{t}$ in a richer probability space,
under Assumptions 1, there exists a $\delta>0$ such that $\sum_{t=1}^{\lfloor
T\uptau\rfloor}\bar{w}_{t}=\sum_{t=1}^{\lfloor
T\uptau\rfloor}X_{t}+O_{a.s}\left(\lfloor
T\uptau\rfloor^{\frac{1}{2}-\delta}\right)$, uniformly in $\uptau$, where
$X_{t}$ is a zero mean, $i.i.d$ Gaussian sequence with
$\mathbb{E}\left(X_{t}X_{t}^{\prime}\right)=V_{\Sigma}$.

Next, we focus on the estimation of $\boldsymbol{V}_{\Sigma}$. In particular,
if no serial dependence is present, a possible choice is the full sample
estimator

$\displaystyle\widehat{\boldsymbol{V}}_{\Sigma}=\frac{1}{T}\sum_{t=1}^{T}w_{t}w_{t}^{\prime}-\left[\mathsf{vec}\big{(}\widehat{\boldsymbol{\Sigma}}\big{)}\right]\left[\mathsf{vec}\big{(}\widehat{\boldsymbol{\Sigma}}\big{)}\right]^{\prime}$
(6.51)

Alternatively, one could use the sequence of partial sample estimators such
that

$\displaystyle\widehat{\boldsymbol{V}}_{\Sigma}=\frac{1}{T}\sum_{t=1}^{T}w_{t}w_{t}^{\prime}-\bigg{\\{}\uptau\left[\mathsf{vec}\big{(}\widehat{\boldsymbol{\Sigma}}_{\uptau}\big{)}\right]\left[\mathsf{vec}\big{(}\widehat{\boldsymbol{\Sigma}}_{\uptau}\big{)}\right]^{\prime}+(1-\uptau)\left[\mathsf{vec}\big{(}\widehat{\boldsymbol{\Sigma}}_{1-\uptau}\big{)}\right]\left[\mathsf{vec}\big{(}\widehat{\boldsymbol{\Sigma}}_{1-\uptau}\big{)}\right]^{\prime}\bigg{\\}}.$

Furthermore, to accommodate for the case in which
$\Psi_{\ell}\equiv\mathbb{E}\big{(}\bar{w}_{t}\bar{w}_{t-\ell}\big{)}\neq 0$
for some $\ell$, we propose a weighted sum-of-covariance estimator with
bandwidth $m$, such that

$\displaystyle\tilde{\boldsymbol{V}}_{\Sigma}$
$\displaystyle=\widehat{\Psi}_{0}+\sum_{\ell=1}^{m}\left(1-\frac{\ell}{m}\right)\left[\widehat{\Psi}_{\ell}+\widehat{\Psi}_{\ell}^{\prime}\right],$
(6.52) $\displaystyle\widehat{\Psi}_{\ell}$
$\displaystyle=\frac{1}{T}\sum_{t=\ell+1}^{T}\big{[}w_{t}-\mathsf{vec}\left(\widehat{\boldsymbol{\Sigma}}\right)\big{]}\big{[}w_{t-\ell}-\mathsf{vec}\left(\widehat{\boldsymbol{\Sigma}}\right)\big{]}^{\prime}$
(6.53)

Alternatively, we can employ the following matrix

$\displaystyle\tilde{\boldsymbol{V}}_{\Sigma,\uptau}$
$\displaystyle=\left(\widehat{\Psi}_{0,\uptau}+\widehat{\Psi}_{0,1-\uptau}\right)\sum_{\ell=1}^{m}\left(1-\frac{\ell}{m}\right)\left[\left(\widehat{\Psi}_{\ell,\uptau}+\widehat{\Psi}_{\ell,\uptau}^{\prime}\right)+\left(\widehat{\Psi}_{\ell,1-\uptau}+\widehat{\Psi}_{\ell,1-\uptau}^{\prime}\right)\right]$
(6.54) $\displaystyle\widehat{\Psi}_{\ell,\uptau}$
$\displaystyle=\frac{1}{T}\sum_{t=\ell+1}^{\lfloor
T\uptau\rfloor}\big{[}w_{t}-\mathsf{vec}\left(\widehat{\boldsymbol{\Sigma}}_{\uptau}\right)\big{]}\big{[}w_{t-\ell}-\mathsf{vec}\left(\widehat{\boldsymbol{\Sigma}}_{\uptau}\right)\big{]}^{\prime}$
(6.55)

A similar expression holds for $\widehat{\Psi}_{\ell,1-\uptau}$. Furthermore,
denote with

$\displaystyle\Omega_{T}=\frac{1}{T}\mathbb{E}\left\\{\sum_{t=1}^{T}\mathsf{vec}\big{[}\bar{w}_{t}\bar{w}_{t}^{\prime}-\mathbb{E}\left(\bar{w}_{t}\bar{w}_{t}^{\prime}\right)\big{]}\times\mathsf{vec}\big{[}\bar{w}_{t}\bar{w}_{t}^{\prime}-\mathbb{E}\left(\bar{w}_{t}\bar{w}_{t}^{\prime}\right)\big{]}^{\prime}\right\\}$
(6.56)

where $\Omega_{T}$ is positive definite uniformly in $T$, and
$\Omega_{T}\to\Omega$ with $\left\lVert\Omega\right\rVert<\infty$.

######  Theorem 6.

Under the null hypothesis of no structural break in $\Sigma$ then, if
Assumption 15 holds, as $T\to\infty$, there exists a $\delta_{o}>0$ such that

$\displaystyle\underset{1\leq\lfloor T\uptau\rfloor\leq
T}{\mathsf{sup}}\left\lVert\hat{\boldsymbol{V}}_{\Sigma,\uptau}-\boldsymbol{V}_{\Sigma}\right\rVert=o_{p}\left(\frac{1}{T^{\delta_{o}}}\right)$
(6.57)

###  6.5. Sieve Bootstrap for Functional Vector Autoregressions

According to Paparoditis, (2018), bootstrap procedures for Hilbert space-
valued time series proposed so far in the literature, are mainly attempts to
adapt, to the infinite dimensional functional framework, of bootstrap methods
that have been developed for the finite dimensional time series case.
Specifically, Paparoditis, (2018) shows, that under quite general assumptions,
the stochastic process of Fourier coefficients obeys a so-called vector
autoregressive representation and this representation plays a key role in
developing a bootstrap procedure for the functional time series at hand. To
capture the essential driving functional parts of the underlying infinite
dimensional process, the first $m$ functional principal components are used
and the corresponding $m-$dimensional time series of Fourier coefficients is
bootstrapped using a $p$th order vector autoregression fitted to the vector
time series of sample Fourier coefficients. In this way, a $m-$dimensional
pseudo-time series of Fourier coefficients is generated which imitates the
temporal dependence structure of the vector time series of sample Fourier
coefficients. Using the truncated Karhunen-Loeve expansion, these pseudo-
Fourier coefficients are then transformed to functional bootstrap replicates
of the main driving, principal components, of the observed functional time
series.

####  6.5.1. The bootstrap procedure

The basic idea of our procedure is to generate pseudo-replicates
$X_{1}^{*},...,X_{n}^{*}$ of the functional time series at hand by first
bootstrapping the $m-$dimensional time series of Fourier coefficients
$\xi_{t}=(\xi_{1,t},\xi_{2,t},...,\xi_{m,t})$, $t=1,2,...n$, corresponding to
the first $m$ principal components. This $m-$dimensional time series of
Fourier coefficients is bootstrapped using the autoregressive representation
$\xi_{t}$. Then, the generated $m-$dimensional pseudo-time series of Fourier
coefficients is then transformed to functional principal pseudo-components by
means of truncated Karhunen-Loeve expansion $\sum_{j=1}^{n}\xi_{j,t}v_{j}$.
Adding to this, an appropriately resampled functional noise leads to the
functional pseudo-time series $X_{1}^{*},X_{2}^{*},...,X_{n}^{*}$. However,
since the $\xi_{t}^{\prime}$s are not observed, we work with the time series
of estimates scores. This idea is precisely described in the functional sieve
bootstrap algorithm proposed by Paparoditis, (2018):

  * Step 1:

Select a number $m=m(n)$ of functional principal components and an
autoregressive order $p=p(n)$, both finite and depending on $n$

  * Step 2:

Let

$\displaystyle\widehat{\boldsymbol{\xi}}=\left(\widehat{\xi}_{j,t}=\langle
X_{t},\widehat{v}_{j}\rangle,j=1,...,m\right)^{\top},\ \ \ t=1,...,n,$ (6.58)

be the $m-$dimensional time series of estimated Fourier coefficients, where we
denote with $\widehat{v}_{j}$, $j=1,...,m$ the estimated eigenfunctions
corresponding to the estimated eigenvalues
$\widehat{\lambda}_{1}>\widehat{\lambda}_{2}>...>\widehat{\lambda}_{m}$ of the
sample covariance operator

$\displaystyle\widehat{C}_{0}=n^{-1}\sum_{t=1}^{n}\left(X_{t}-\bar{X}_{n}\right)\otimes\left(X_{t}-\bar{X}_{n}\right),\
\ \ \bar{X}_{n}=n^{-1}\sum_{t=1}^{n}X_{t}$ (6.59)

  * Step 3:

Let $\widehat{X}_{t,m}=\sum_{j=1}^{m}\widehat{\xi}_{j,t}\widehat{v}_{j}$ and
define the functional residuals $\widehat{U}_{t,m}=X_{t}-\widehat{X}_{t,m}$
and define the functional residuals
$\widehat{U}_{t,m}=X_{t}-\widehat{X}_{t,m}$, $t=1,...,n$.

  * Step 4:

Fit a $p$th order vector autoregressive process to the $m-$dimensional time
series $\widehat{\xi}_{t}$, $t=1,...,n$, denote by $\widehat{A}_{j,p}(m)$,
$j=1,...,p$, the estimates of the autoregressive matrices and by
$\widehat{e}_{t,p}$ the residuals,

$\displaystyle\widehat{e}_{t,p}=\widehat{\xi}_{t}-\sum_{j=1}^{p}\widehat{A}_{j,p}(m)\widehat{\xi}_{t-j},\
\ \ t=p+1,p+2,...,n.$ (6.60)

We focus on the following Yule-Walker estimators for $\widehat{A}_{j,p}(m)$,
$j=1,...,p$.

  * Step 5:

Generate a $m-$dimensional pseudo time series of scores
$\xi^{*}_{t}=\left(\xi^{*}_{1,t},...,\xi^{*}_{m,t}\right)$, $t=1,2,...,n$,
using

$\displaystyle\xi^{*}_{t}=\sum_{j=1}^{p}\widehat{A}_{j,p}(m)\xi^{*}_{t-j}+e^{*}_{t},$
(6.61)

where $e^{*}_{t}$, $t=1,...,n$ are i.i.d random vectors having as distribution
the empirical distribution of the centered residual vectors
$\widetilde{e}_{t,p}=\widehat{e}_{t,p}-\bar{\widehat{e}}_{n,t}$, for
$t=p+1,p+2,...,n$ and
$\bar{\widehat{e}}=(n-p)^{-1}\sum_{t=p+1}^{n}\widehat{e}_{t,p}$.

  * Step 6:

Generate a pseudo-functional time series $X_{1}^{*},...,X_{n}^{*}$ ,where

$\displaystyle\xi^{*}_{t}=\sum_{j=1}^{m}\xi_{j,t}^{*}\widehat{v}_{j}+U_{t}^{*},\
\ \ t=1,...,n.$ (6.62)

and $U_{1}^{*},...,U_{n}^{*}$ are i.i.d random functions obtained by choosing
with replacement from the set of centered functional residuals
$\widehat{U}_{t,m}-\bar{\widehat{U}}_{t}$ and
$\bar{\widehat{U}}_{t}=n^{-1}\sum_{t=1}^{n}\widehat{U}_{t,m}$, for
$t=1,...,n$.

######  Remark 25.

Notice that $X_{1}^{*},...,X_{n}^{*}$ are functional pseudo-random variables
and the the autoregressive representation of the vector time series of Fourier
coefficients is solely used as a toll to bootstrap the $m$ main functional
principal components of the functional time series. In fact, it is this
autoregressive representation which allows the generation of the pseudo-time
series of Fourier coefficients $\xi^{*}_{1},...,\xi^{*}_{n}$ in Step 4 and
Step 5 in a way that imitates the dependence structure of the sample Fourier
coefficients $\xi_{1},...,\xi_{n}$. These pseudo-Fourier coefficients are
transformed to bootstrapped main principal components by means of the
truncated and estimated Karhunen-Loeve expansion which together with the
additive functional noise $U_{t}^{*}$, lead to the new functional pseudo-
observations $X_{1}^{*},...,X_{n}^{*}$.

Furthermore, an important issue for bootstrap-based inference methods is to
investigate the validity of the functional sieve bootstrap applied in order to
approximate the distribution of some statistic
$T_{n}=T\left(X_{1},...,X_{n}\right)$ of interest, when the bootstrap analogue
$T_{n}^{*}=T\left(X_{1}^{*},...,X_{n}^{*}\right)$ is used. Notice that
establishing validity of a bootstrap procedure for time series heavily depends
on two issues. On the dependence structure of the underlying process which
affects the distribution of the statistic of interest and on the capability of
the bootstrap procedure used to mimic appropriately this dependence structure.

Recall the definition of

$\displaystyle
X_{t}^{*}=\sum_{j=1}^{m}\textbf{1}_{j}^{\top}\xi_{j}^{*}\widehat{v}_{j}+U_{t}^{*}$
(6.63)

and observe that
$\xi^{*}_{t}=\sum_{l=0}^{\infty}\widehat{\Psi}_{l,p}(m)e^{*}_{t-l}$, where
$\widehat{\Psi}_{0,p}(m)=I_{m}$ and the power series

$\displaystyle\widehat{\Psi}_{m,p}(z)=I_{m}+\sum_{l=1}^{\infty}\widehat{\Psi}_{l,p}(m)z^{l}=\left(I_{m}-\sum_{j=1}^{p}\widehat{A}_{j,p}(m)z^{j}\right)^{-1}$
(6.64)

converges for $|z|\leq 1$. Thus, write

$\displaystyle X_{t}^{*}$
$\displaystyle=\sum_{l=0}^{\infty}\sum_{j=1}^{m}\textbf{1}_{j}^{\top}\widehat{\Psi}_{l,p}(m)e^{*}_{t-l}\widehat{v}_{j}+U_{t}^{*}$
(6.65) $\displaystyle X_{t,M}^{*}$
$\displaystyle=\sum_{l=0}^{M-1}\sum_{j=1}^{m}\textbf{1}_{j}^{\top}\widehat{\Psi}_{l,p}(m)e_{t-l}^{*}\widehat{v}_{j}+\sum_{l=M}^{\infty}\sum_{j=1}^{m}\textbf{1}_{j}^{\top}\widehat{\Psi}_{l,p}(m)e_{t-l}^{*}\widehat{v}_{j}+U_{t}^{*}$
(6.66)

where for each $t\in\mathbb{Z},\left\\{e_{s,t}^{*},s\in\mathbb{Z}\right\\}$ is
an independent copy of $\left\\{e_{s}^{*},s\in\mathbb{Z}\right\\}$. Notice
that,

$\displaystyle
X_{M}^{*}-X_{M,M}^{*}=\sum_{l=M}^{\infty}\sum_{j=1}^{m}\textbf{1}_{j}^{\top}\widehat{\Psi}_{l,p}(m)\big{(}e_{M-l}^{*}-e_{M-l,M}^{*}\big{)}\widehat{v}_{j}$
(6.67)

By Minskowski’s inequality, we have that

$\displaystyle\sqrt{\mathbb{E}\left\lVert
X_{M}^{*}-X_{M,M}^{*}\right\rVert^{2}}\leq\sqrt{\mathbb{E}\left\lVert\sum_{l=M}^{\infty}\sum_{j=1}^{m}\textbf{1}_{j}^{\top}\widehat{\Psi}_{l,p}(m)e_{M-l}^{*}\widehat{v}_{j}\right\rVert^{2}}+\sqrt{\mathbb{E}\left\lVert\sum_{l=M}^{\infty}\sum_{j=1}^{m}\textbf{1}_{j}^{\top}\widehat{\Psi}_{l,p}(m)e_{M-l,M}^{*}\widehat{v}_{j}\right\rVert^{2}}.$

Thus, evaluating the first expectation term, we obtain that using $\left\lVert
A\right\rVert_{F}^{2}=tr\left(AA^{\top}\right)$ and the submultiplicative
property of the Frobenius matrix norm, that

$\displaystyle\mathbb{E}\left\lVert\sum_{l=M}^{\infty}\sum_{j=1}^{m}\textbf{1}_{j}^{\top}\widehat{\Psi}_{l,p}(m)e_{M-l,M}^{*}\widehat{v}_{j}\right\rVert^{2}=\sum_{l=M}^{\infty}tr\left(\widehat{\Psi}_{l,p}(m)\Sigma^{*}(m)\widehat{\Psi}_{l,p}(m)\right)\leq\left\lVert\widehat{\Sigma}_{e,p}^{1/2}(m)\right\rVert_{F}^{2}\sum_{l=M}^{\infty}\left\lVert\widehat{\Psi}_{l,p}(m)\right\rVert_{F}^{2},$

where
$\widehat{\Sigma}_{e,p}(m)=\widehat{\Sigma}_{e,p}^{1/2}(m)\widehat{\Sigma}_{e,p}^{1/2}(m)$.
An identical expression appears for the second expectation term on the right-
hand side of 6.5.1.. Applying Minkowski’s inequality again, we get by the
exponential decay of $\left\lVert\widehat{\Psi}_{l,p}(m)\right\rVert_{F}$,
such that the following probability bound holds

$\displaystyle\sum_{M=1}^{\infty}\sqrt{\mathbb{E}\left\lVert
X_{M}^{*}-X_{M,M}^{*}\right\rVert^{2}}\leq
2\left\lVert\widehat{\Sigma}_{e,p}^{1/2}(m)\right\rVert_{F}^{2}\sum_{l=M}^{\infty}\sum_{j=1}^{m}\left\lVert\widehat{\Psi}_{l,p}(m)\right\rVert_{F}=2\left\lVert\widehat{\Sigma}_{e,p}^{1/2}(m)\right\rVert_{F}^{2}\sum_{l=1}^{\infty}l\left\lVert\widehat{\Psi}_{l,p}(m)\right\rVert_{F}=\mathcal{O}_{p}(1).$

##  7\. Conclusion

There are various statistical identification procedures for SVAR models such
as imposing sign-restrictions, short-run restrictions, long-run restrictions,
identification by higher-order moments as well as identification by
conditional heteroscedasticity and changes in volatility (e.g., see, Lanne and
Lütkepohl, 2008a ; Lanne and Lütkepohl, 2008b . More precisely, by exploiting
the heteroscedasticity of the errors $\varepsilon_{t}$, then the
identification of the structural parameters of the system can be achieved. In
particular, Lanne et al., (2010) assume Markov switching and a smooth
transition in the covariance matrix of the error term $\varepsilon_{t}$ of the
model. Specifically, the structural analysis of VARs in this case is based on
the Markov regime switching property to identify shocks, which holds due to
the fact that the reduced form error covariance matrix varies across states.
Furthermore, a different stream of literature considers identification in SVAR
models by assuming that the error terms are non-Gaussian and mutually
independent (e.g., see Lanne et al., (2017) and Gouriéroux et al., (2020)).
For example, Lanne et al., (2010) assume that the errors of the model are
independent over time with a distribution that is a mixture of two Gaussian
distributions with zero means and diagonal covariance matrices, one of which
is an identify matrix and the other one has positive diagonal elements, which
for identifiability have to be distinct. Roughly speaking there are specific
systems where the use of the non-Gaussianity property is a more suitable
identification strategy of structural shocks of the SVAR system which we leave
for discussion in a follow-up paper.

In this note we have reviewed key statistical properties of VAR processes,
cointegrated VAR processes and SVAR processes as well as suitable
identification strategies for the structural parameters of these structural
econometric specifications. Moreover, we have discusses several empirical-
driven applications from the applied macroeconometrics literature and we
studied the proposed identification and estimation methods employed to those
cases. Then, motivated from the literature of dynamic causal effects (see,
Mertens and Ravn, (2013), Stock and Watson, (2018) and Budnik and Rünstler,
(2023)), we have motivated the main statistical and econometric tools for
identification of structural econometric models, commonly used for
counterfactual analysis purposes, and then presented some examples of time
series regression models, both important for understanding suitable techniques
for policy-making evaluation in macroeconometrics. We leave any further
presentation of other relevant topics for further research discussions.
Lastly, some advanced topics were also presented such as break testing for
unstable roots in structural error correction models, estimation techniques
for the optimal lag order, the estimation of a high dimensional VAR model as
well as a fast algorithmic procedure presented in the literature for detecting
structural breaks in large VAR models.

In the Appendix of this set of lecture notes we present an example of the
maximum likelihood estimation for $\alpha-$stable autoregressive processes
(see, Andrews et al., (2009)) as well as the main asymptotic theory results
for inference for the VEC(1) model in Guo and Ling, (2023).

##  8\. Appendix

###  8.1. MLE for $\alpha-$Stable Autoregressive Processes

Following the framework of Andrews et al., (2009), we consider the MLE
estimation for both causal and non-causal autoregressive time series processes
with non-Gaussian $\alpha-$stable noise. In particular, the estimators for the
autoregressive parameters are $n^{-1/\alpha}-$consistent (parametric rate) and
converge in distribution to the maximizer of a random function. Although the
form of this limiting distribution is intractable, the shape of the
distribution can be examined using the bootstrap procedure. Moreover, the
estimators for the parameters of the stable noise distribution have the
traditional $n^{1/2}$ rate of convergence and are asymptotically normal. In
addition, the behavior of the estimators for finite samples is studied via
simulation, and we use the maximum likelihood estimation to fit a noncausal
autoregressive model (see, Andrews et al., (2009) and Andrews and Davis,
(2013)).

Let $\left\\{X_{t}\right\\}$ be the AR process which satisfies the difference
equations

$\displaystyle\phi_{0}(B)X_{t}=Z_{t},$ (8.1)

where the AR polynomial is given by

$\displaystyle\phi_{0}(z)$ $\displaystyle:=1-\phi_{01}z-...-\phi_{0p}z^{p}\neq
0,\ \ \text{for}\ |z|=1,$ (8.2)

Furthermore, because $\phi_{0}(z)\neq 0$ for $|z|=1$, the Laurent series
expansion of $1/\phi_{0}(z)$ is given by

$\displaystyle\frac{1}{\phi_{0}(z)}=\sum_{j=-\infty}^{+\infty}\psi_{j}z^{j},$
(8.3)

and the unique, strictly stationary solution is given by

$\displaystyle X_{t}=\sum_{j=-\infty}^{+\infty}\psi_{j}Z_{t-j},$ (8.4)

  * •

If $\phi_{0}(z)\neq 0$ for $|z|\leq 1$, then $\psi_{j}=0$ for $j<0$ and so
$\left\\{X_{t}\right\\}$ is a causal process, since
$X_{t}=\sum_{j=-\infty}^{+\infty}\psi_{j}Z_{t-j}$, is a function of the past
and present $\left\\{Z_{t}\right\\}$.

  * •

If $\phi_{0}(z)\neq 0$ for $|z|>1$, then
$X_{t}=\sum_{j=0}^{+\infty}\psi_{-j}Z_{t+j}$, and $\left\\{X_{t}\right\\}$ is
called a purely noncausal process.

Moreover, in the purely non-causal process case, the coefficients
$\left\\{\psi_{j}\right\\}$ satisfiy

$\displaystyle\big{(}1-\phi_{01}z-...-\phi_{0p}z^{p}\big{)}\big{(}\psi_{0}+\psi_{-1}z^{-1}+...\big{)}=1,$
(8.5)

which, if $\phi_{0p}\neq 0$, implies that
$\psi_{0}=\psi_{-1}=...=\psi_{1-p}=0$ and $\psi_{1-p}=-\phi_{0p}^{-1}$.

Therefore, to express $\phi_{0}(z)$ as the product of causal and purely
noncausal polynomials, suppose

$\displaystyle\phi_{0}(z)={\color[rgb]{1,0,0}\big{(}1-\theta_{01}z-...-\theta_{0r_{0}}z^{r_{0}}\big{)}}{\color[rgb]{0,0,1}\big{(}1-\theta_{0,r_{0}+1}z-...-\theta_{0,r_{0}+s_{0}}z^{s_{0}}\big{)}},$
(8.6)

where $(r_{0}+s_{0})=p$ and

$\displaystyle\theta_{0}=\begin{cases}{\color[rgb]{1,0,0}\big{(}1-\theta_{01}z-...-\theta_{0r_{0}}z^{r_{0}}\big{)}}=:\theta_{0}^{+}\neq
0&,\text{if}\ |z|\geq 1\\\
{\color[rgb]{0,0,1}\big{(}1-\theta_{0,r_{0}+1}z-...-\theta_{0,r_{0}+s_{0}}z^{s_{0}}\big{)}}=:\theta_{0}^{*}\neq
0&,\text{if}\ |z|\leq 1\end{cases}$ (8.7)

which implies that $\theta_{0}^{+}$ is a causal polynomial and
$\theta_{0}^{*}$ is a purely noncausal polynomial. Therefore, $\phi_{0}(z)$
has a unique representation as the product of causal and noncausal polynomials
if the true order of the AR polynomial $\phi_{0}(z)$ is less than $p$.

Consider the following sequence $\left\\{Z_{t}\right\\}$ such that

$\displaystyle
Z_{t}={\color[rgb]{1,0,0}\big{(}1-\theta_{01}B-...-\theta_{0r_{0}}B^{r_{0}}\big{)}}{\color[rgb]{0,0,1}\big{(}1-\theta_{0,r_{0}+1}B-...-\theta_{0,r_{0}+s_{0}}B^{s_{0}}\big{)}}X_{t}$
(8.8)

Consider the arbitraty autoregressive polynomials as below:

$\displaystyle\theta^{+}(z)$
$\displaystyle={\color[rgb]{1,0,0}\big{(}1-\theta_{1}z-...-\theta_{r}z^{r}\big{)}}$
(8.9) $\displaystyle\theta^{*}(z)$
$\displaystyle={\color[rgb]{0,0,1}\big{(}1-\theta_{r+1}z-...-\theta_{r+s}z^{s}\big{)}}$
(8.10)

Then, we define with

$\displaystyle
Z_{t}(\boldsymbol{\theta},s):={\color[rgb]{1,0,0}\big{(}1-\theta_{1}B-...-\theta_{r}B^{r}\big{)}}{\color[rgb]{0,0,1}\big{(}1-\theta_{r+1}B-...-\theta_{r+s}B^{s}\big{)}}X_{t}$
(8.11)

where $\boldsymbol{\theta}=(\theta_{1},...,\theta_{p})^{\prime}$ and
$\boldsymbol{\theta}=(\theta_{01},...,\theta_{0p})^{\prime}$. Moreover, let
$\boldsymbol{\eta}=(\eta_{1},...,\eta_{p+4})=(\theta_{1},...,\theta_{p},\alpha,\beta,\sigma,\mu)^{\prime}=(\boldsymbol{\theta}^{\prime},\boldsymbol{\tau}^{\prime})^{\prime}$.
In other words, given a realization $\left\\{X_{t}\right\\}_{t=1}^{n}$, then
the log-likelihood of $\boldsymbol{\eta}$ can be approximated by the
conditional log-likelihood such that

$\displaystyle\mathcal{L}(\boldsymbol{\eta},s)=\sum_{t=p+1}^{s}\big{[}\mathsf{ln}f\big{(}Z_{t}(\boldsymbol{\theta},s);\boldsymbol{\tau}\big{)}+\mathsf{ln}|\theta_{p}|\boldsymbol{1}\left\\{s>0\right\\}\big{]},$
(8.12)

where the component
$\left\\{Z_{t}(\boldsymbol{\theta},s)\right\\}_{t=p+1}^{n}$ is computed
separately.

######  Remark 26.

The use of causal and non-causal autoregressive processes can be helpful in
understanding concepts such as identification, granger non-causality and weak
exogeneity while ensuring that these properties are not violated when learning
for dynamic causal effects based on macroeconomic models. Representations for
such univariate processes when modelling explosive bubbles/regimes are studied
by Fries and Zakoian, (2019), Davis and Song, (2020), Cavaliere et al., (2020)
and Hecq and Voisin, (2021) while related representations for multivariate
processes are studied by Lanne and Saikkonen, (2013), Velasco and Lobato,
(2018) and Rygh Swensen, (2022).

###  8.2. Inference for the VEC(1) Model

Consider an $m-$dimensional vector $\boldsymbol{Y}_{t}$ satisfying the
following equation (see, Guo and Ling, (2023))

$\displaystyle\Delta\boldsymbol{Y}_{t}=\boldsymbol{\Pi}_{0}\boldsymbol{Y}_{t-1}+\boldsymbol{u}_{t}\equiv\boldsymbol{\alpha}_{0}\boldsymbol{\beta}_{0}^{\prime}\boldsymbol{Y}_{t-1}+\boldsymbol{u}_{t},$
(8.13)

where
$\boldsymbol{\Pi}_{0}=\boldsymbol{\alpha}_{0}\boldsymbol{\beta}_{0}^{\prime}$
has matrix rank $0\leq r_{0}\leq m$, with $\boldsymbol{\alpha}_{0}$ and
$\boldsymbol{\beta}_{0}$ being $(m\times r_{0})$ full-rank matrices.

Moreover, Johansen, (1995) proposed the following partial sum Granger
representation

$\displaystyle\boldsymbol{Y}_{t}=\boldsymbol{C}\sum_{s=1}^{t}\boldsymbol{u}_{s}+\boldsymbol{\alpha}_{0}\left(\boldsymbol{\beta}_{0}^{\prime}\boldsymbol{\alpha}_{0}\right)^{-1}\boldsymbol{\Xi}(L)\boldsymbol{\beta}_{0}^{\prime}\boldsymbol{u}_{t}+\boldsymbol{C}\boldsymbol{Y}_{0},$
(8.14)

where

$\displaystyle\boldsymbol{C}=\boldsymbol{\alpha}_{0\perp}\left(\boldsymbol{\alpha}_{0\perp}\boldsymbol{\beta}^{\prime}_{0\perp}\right)^{-1}\boldsymbol{\alpha}_{0\perp}^{\prime}\
\ \ \boldsymbol{\Xi}(L)=\sum_{s=0}^{\infty}\boldsymbol{\Xi}_{s}L^{s}.$ (8.15)

Denote with

$\displaystyle\boldsymbol{Q}=\big{[}\boldsymbol{\beta}_{0},\boldsymbol{\alpha}_{0\perp}\big{]}^{\prime}\
\ \
\boldsymbol{Q}^{-1}=\big{[}\boldsymbol{\alpha}_{0}\left(\boldsymbol{\beta}^{\prime}_{0}\boldsymbol{\alpha}_{0}\right)^{-1},\boldsymbol{\beta}^{\prime}_{0\perp}\left(\boldsymbol{\alpha}_{0\perp}\boldsymbol{\beta}^{\prime}_{0\perp}\right)^{-1}\big{]}.$
(8.16)

Thus, we have that

$\displaystyle\boldsymbol{Q}\boldsymbol{\Pi}_{0}=\begin{bmatrix}\boldsymbol{\beta}^{\prime}_{0}\boldsymbol{\alpha}_{0}\boldsymbol{\beta}^{\prime}_{0}\\\
\boldsymbol{0}\end{bmatrix}\ \ \
\boldsymbol{Q}\boldsymbol{\Pi}_{0}\boldsymbol{Q}^{-1}=\begin{bmatrix}\boldsymbol{\beta}^{\prime}_{0}\boldsymbol{\alpha}_{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}$ (8.17)

Denote with

$\displaystyle\boldsymbol{Z}_{t}=\begin{pmatrix}\boldsymbol{\beta}_{0}^{\prime}\boldsymbol{Y}_{t}\\\
\boldsymbol{\alpha}^{\prime}_{0\perp}\boldsymbol{Y}_{t}\end{pmatrix}\equiv\begin{pmatrix}\boldsymbol{Z}_{1,t}\\\
\boldsymbol{Z}_{2,t}\end{pmatrix},\ \ \ \ \
\boldsymbol{W}_{t}=\begin{pmatrix}\boldsymbol{\beta}_{0}^{\prime}\boldsymbol{u}_{t}\\\
\boldsymbol{\alpha}^{\prime}_{0\perp}\boldsymbol{u}_{t}\end{pmatrix}\equiv\begin{pmatrix}\boldsymbol{W}_{1,t}\\\
\boldsymbol{W}_{2,t}\end{pmatrix}$ (8.18)

which implies that the econometric specification can be formulated as below

$\displaystyle\Delta\boldsymbol{Z}_{t}=\boldsymbol{Q}\boldsymbol{\Pi}_{0}\boldsymbol{Q}^{-1}\boldsymbol{Z}_{t-1}+\boldsymbol{W}_{t}.$
(8.19)

Using $\boldsymbol{Z}_{1,t}=\boldsymbol{\beta}_{0}^{\prime}\boldsymbol{Y}_{t}$
and
$\boldsymbol{Z}_{2,t}=\boldsymbol{\alpha}^{\prime}_{0\perp}\boldsymbol{Y}_{t}$
we obtain the following expressions

$\displaystyle{\color[rgb]{0,0,1}\boldsymbol{Z}_{1,t}}$
$\displaystyle\equiv{\color[rgb]{0,0,1}\boldsymbol{\beta}_{0}^{\prime}}\boldsymbol{Y}_{t}={\color[rgb]{0,0,1}\boldsymbol{\beta}_{0}^{\prime}}\boldsymbol{C}\sum_{s=1}^{t}\boldsymbol{u}_{s}+{\color[rgb]{0,0,1}\boldsymbol{\beta}_{0}^{\prime}}\boldsymbol{\alpha}_{0}\left(\boldsymbol{\beta}_{0}^{\prime}\boldsymbol{\alpha}_{0}\right)^{-1}\boldsymbol{\Xi}(L)\boldsymbol{\beta}_{0}^{\prime}\boldsymbol{u}_{t}+{\color[rgb]{0,0,1}\boldsymbol{\beta}_{0}^{\prime}}\boldsymbol{C}\boldsymbol{Y}_{0},$
(8.20) $\displaystyle{\color[rgb]{1,0,0}\boldsymbol{Z}_{2,t}}$
$\displaystyle\equiv{\color[rgb]{1,0,0}\boldsymbol{\alpha}^{\prime}_{0\perp}}\boldsymbol{Y}_{t}={\color[rgb]{1,0,0}\boldsymbol{\alpha}^{\prime}_{0\perp}}\boldsymbol{C}\sum_{s=1}^{t}\boldsymbol{u}_{s}+{\color[rgb]{1,0,0}\boldsymbol{\alpha}^{\prime}_{0\perp}}\boldsymbol{\alpha}_{0}\left(\boldsymbol{\beta}_{0}^{\prime}\boldsymbol{\alpha}_{0}\right)^{-1}\boldsymbol{\Xi}(L)\boldsymbol{\beta}_{0}^{\prime}\boldsymbol{u}_{t}+{\color[rgb]{1,0,0}\boldsymbol{\alpha}^{\prime}_{0\perp}}\boldsymbol{C}\boldsymbol{Y}_{0},$
(8.21)

Therefore, we obtain that

$\displaystyle{\color[rgb]{0,0,1}\boldsymbol{Z}_{1,t}}$
$\displaystyle=\boldsymbol{\Xi}(L)\boldsymbol{\beta}_{0}^{\prime}\boldsymbol{u}_{t}=\sum_{j=0}\left(\sum_{i=0}^{j}\boldsymbol{\Xi}_{i}\boldsymbol{\beta}_{0}^{\prime}\boldsymbol{D}_{j-i}\right)\boldsymbol{\epsilon}_{t-j}\equiv\sum_{j=0}^{\infty}\boldsymbol{C}_{j}\boldsymbol{\varepsilon}_{t-j},$
(8.22) $\displaystyle{\color[rgb]{1,0,0}\boldsymbol{Z}_{2,t}}$
$\displaystyle={\color[rgb]{1,0,0}\boldsymbol{\alpha}^{\prime}_{0\perp}}\boldsymbol{C}\left(\sum_{s=1}^{t}\boldsymbol{u}_{s}+\boldsymbol{Y}_{0}\right)\equiv\big{[}\boldsymbol{0},\boldsymbol{I}_{d}\big{]}\boldsymbol{Q}\left(\sum_{s=1}^{t}\boldsymbol{u}_{s}+\boldsymbol{Y}_{0}\right)\equiv\big{[}\boldsymbol{0},\boldsymbol{I}_{d}\big{]}\sum_{s=1}^{t}\boldsymbol{\gamma}_{s},$
(8.23)

where $d=(m-r_{0})$,
$\boldsymbol{C}_{j}=\sum_{i=0}^{j}\boldsymbol{\Xi}_{i}\boldsymbol{\beta}^{\prime}\boldsymbol{D}_{j-i}=\mathcal{O}_{p}(\rho^{j})$
and

$\displaystyle\boldsymbol{\gamma}_{s}=\sum_{i=0}^{\infty}\boldsymbol{Q}\boldsymbol{D}_{i}\boldsymbol{\varepsilon}_{s-i}\equiv\sum_{i=0}^{\infty}\boldsymbol{\phi}_{i}\boldsymbol{\varepsilon}_{s-i},\
\ \text{with}\
\boldsymbol{\phi}_{i}=\boldsymbol{Q}\boldsymbol{D}_{i}=\mathcal{O}_{p}(\rho^{i})\
\text{for some}\ \rho\in(0,1).$ (8.24)

Therefore, estimating the matrix $\boldsymbol{\Pi}_{0}$ by OLS gives

$\displaystyle\widehat{\boldsymbol{\Pi}}_{ols}:=\underset{\Pi\in\mathbb{R}^{m\times
m}}{\mathsf{argmin}}\
\sum_{t=1}^{n}\left\lVert\Delta\boldsymbol{Y}_{t}-\boldsymbol{\Pi}\boldsymbol{Y}_{t-1}\right\rVert^{2}=\left(\sum_{t=1}^{n}\Delta\boldsymbol{Y}_{t}\boldsymbol{Y}_{t-1}^{\prime}\right)\left(\sum_{t=1}^{n}\boldsymbol{Y}_{t-1}\boldsymbol{Y}_{t-1}^{\prime}\right)^{-1}.$
(8.25)

By Proposition 3.1 in Resnick (1986), the condition for regular variation of
the error terms

$\displaystyle
n\mathbb{P}\left(\frac{\varepsilon_{t}}{a_{n}}\in\cdot\right)\overset{\nu}{\to}\mu(\cdot),$
(8.26)

is equivalent to the following convergence

$\displaystyle\sum_{t=1}^{n}\delta_{\frac{\varepsilon_{t-1}}{a_{n}}}\overset{\nu}{\to}\sum_{i=1}^{\infty}\delta_{P_{i}}\equiv
PRM(\mu),$ (8.27)

as $n\to\infty$, where $PRM(\mu)$ is a Poisson random process with intensity
measure $\mu$ and $\left\\{P_{i}\right\\}$ is a sequence of raondom vectors
such that $\sum_{i=1}^{\infty}\delta_{P_{i}}$ is the point representation of
$PRM(\mu)$. Due to the symmetry of $\varepsilon_{1}$, we can claim that there
exists a stable process $P(r)$ such that

$\displaystyle a_{n}^{-1}\sum_{t=1}^{\lfloor
nr\rfloor}\varepsilon_{t}\overset{D}{\to}P(r),$ (8.28)

as $n\to\infty$, where $\overset{D}{\to}$ mean weak convergence in the
topological space $D[0,1]$ of multivariated functions equipped with the
Skorokhod $J_{1}$ topology. Next, we focus on the limiting properties of the
estimator $\widehat{\boldsymbol{\Pi}}_{ols}$ within the $J_{1}$ topology.

######  Theorem 7.

Let
$\bar{\boldsymbol{\beta}}_{\perp}=\boldsymbol{\beta}_{0\perp}\left(\boldsymbol{\alpha}^{\prime}_{0\perp}\boldsymbol{\beta}_{0\perp}\right)^{-1}$
and
$\bar{\boldsymbol{\beta}}=\boldsymbol{\alpha}_{0}\left(\boldsymbol{\beta}^{\prime}_{0}\boldsymbol{\alpha}_{0}\right)^{-1}$
and $\boldsymbol{\phi}=\boldsymbol{Q}\sum_{i=0}^{\infty}\boldsymbol{D}_{i}$.
Suppose that Assumptions hold and that $\varepsilon_{t}$ has a symmetric
distribution, then it follows that

  * (a).

$\left(\widehat{\boldsymbol{\Pi}}_{ols}-\boldsymbol{\Pi}_{0}\right)\bar{\boldsymbol{\beta}}\to_{d}\boldsymbol{R}_{1}\boldsymbol{\Gamma}_{11}^{-1}$,

  * (b).

${\color[rgb]{1,0,0}n}\left(\widehat{\boldsymbol{\Pi}}_{ols}-\boldsymbol{\Pi}_{0}\right)\bar{\boldsymbol{\beta}}_{\perp}\to_{d}\boldsymbol{R}_{2}^{*}\boldsymbol{\Gamma}_{22}^{-1}-\boldsymbol{R}_{1}^{*}\boldsymbol{\Gamma}_{11}^{-1}\boldsymbol{\Gamma}_{21}\boldsymbol{\Gamma}_{22}^{-1}$,

Denote with $\boldsymbol{R}_{2}$ such that

$\displaystyle\boldsymbol{R}_{2}$
$\displaystyle=\left[\int_{0}^{1}\boldsymbol{P}(r)d\boldsymbol{P}^{\prime}(r)\right]^{\prime}\boldsymbol{\phi}^{\prime}\big{[}\boldsymbol{0},\boldsymbol{I}_{d}\big{]}^{\prime},$
(8.29) $\displaystyle\boldsymbol{\Gamma}_{22}$
$\displaystyle=\big{[}\boldsymbol{0},\boldsymbol{I}_{d}\big{]}\boldsymbol{\phi}\left[\int_{0}^{1}\boldsymbol{P}(r)\boldsymbol{P}^{\prime}(r)dr\right]\boldsymbol{\phi}^{\prime}\big{[}\boldsymbol{0},\boldsymbol{I}_{d}\big{]}^{\prime},$
(8.30) $\displaystyle\boldsymbol{\Gamma}_{11}$
$\displaystyle=\sum_{j=0}^{\infty}\boldsymbol{C}_{j}\boldsymbol{S}_{1}\boldsymbol{C}_{j}^{\prime},$
(8.31) $\displaystyle\boldsymbol{\Gamma}_{21}$
$\displaystyle=\left\\{\boldsymbol{R}_{2}^{\prime}\sum_{i=0}^{\infty}\boldsymbol{C}_{i}^{\prime}+\big{[}\boldsymbol{0},\boldsymbol{I}_{d}\big{]}\boldsymbol{Q}\sum_{i=0}^{\infty}\sum_{j=0}^{i}\boldsymbol{D}_{j}\boldsymbol{S}_{1}\boldsymbol{C}_{i}^{\prime}\right\\}.$
(8.32)

######  Remark 27.

Notice that in order to establish the asymptotic distribution of the OLS
estimator for the VEC(1) model with heavy tailed errors, we shall consider the
weak convergence of the matrix functionals based on the above limit results.
In particular, the OLS related to the short-term parameters is not consistent
and converges to a functional of stable matrix-processes. Moreover, the
inconsistency is because $\left\\{\boldsymbol{u}_{t}\right\\}$ is a series of
dependent random vectors which is a critical difference from the i.i.d white
noises cases as in She et al., (2022).

By defining suitable functionals as above we can establish the joint weak
convergence given below

$\displaystyle\left(\frac{1}{a_{n}^{2}}\sum_{t=1}^{n}\boldsymbol{Y}_{t-i}\boldsymbol{Y}_{t-j}^{\prime},\frac{1}{\tilde{a}_{n}^{2}}\sum_{t=1}^{n}\boldsymbol{\varepsilon}_{t}\boldsymbol{Y}_{t-k}^{\prime}\right)\overset{d}{\to}\left(\sum_{\ell=0}^{\infty}\boldsymbol{\Psi}_{\ell}\boldsymbol{S}_{1}\boldsymbol{\Psi}^{\prime}_{\ell+i-j},\sum_{\ell=0}^{\infty}\boldsymbol{S}_{\ell+k+1}\boldsymbol{\Psi}_{\ell}^{\prime}\right)$
(8.33)

where $1\leq j\leq i\leq p$, $1\leq k\leq p$ and
$\boldsymbol{Y}_{t}=\displaystyle\sum_{\ell=0}^{\infty}\boldsymbol{\Psi}_{\ell}\boldsymbol{\varepsilon}_{t-\ell}$.

Moreover, denoting with
$\boldsymbol{B}_{n}=\left(\boldsymbol{D}_{n}\boldsymbol{Q}\right)^{-1}$ and
$\boldsymbol{H}=\mathsf{diag}\big{\\{}\frac{\tilde{a}_{n}}{a_{n}}\boldsymbol{I}_{r_{0}},n^{-1/2}a_{n}\boldsymbol{I}_{m-r_{0}}\big{\\}}$.
Deduce that

$\displaystyle\boldsymbol{H}^{-1}\boldsymbol{D}_{n}\left(\sum_{t=1}^{n}\boldsymbol{Z}_{t-1}\boldsymbol{Z}_{t-1}^{\prime}\right)\boldsymbol{D}_{n}\boldsymbol{H}^{-1}=\begin{bmatrix}\frac{1}{a_{n}^{2}}\boldsymbol{S}_{11}&\frac{1}{\sqrt{n}}a_{n}^{-2}\boldsymbol{S}_{12n}\\\
\frac{1}{\sqrt{n}}\frac{1}{a_{n}^{2}}\boldsymbol{S}_{11}&\frac{1}{\sqrt{n}}a_{n}^{-2}\boldsymbol{S}_{12n}\end{bmatrix}$
$\displaystyle\overset{d}{\to}\begin{bmatrix}\displaystyle\sum_{\ell=0}^{\infty}\boldsymbol{C}_{\ell}\boldsymbol{S}_{1}\boldsymbol{C}_{\ell}^{\prime}&\boldsymbol{0}\\\
\boldsymbol{0}&\displaystyle\boldsymbol{\phi}\big{[}\int_{0}^{1}\boldsymbol{P}(r)\boldsymbol{P}^{\prime}(r)dr\big{]}\boldsymbol{\phi}\end{bmatrix}$

Therefore, it holds that

$\displaystyle\mathsf{vec}\left(\widetilde{\boldsymbol{\Pi}}_{n,f}-\widehat{\boldsymbol{\Pi}}_{n}\right)^{\prime}\left(\sum_{t=1}^{n}\boldsymbol{Y}_{t-1}\boldsymbol{Y}_{t-1}^{\prime}\otimes\boldsymbol{I}_{m}\right)\mathsf{vec}\left(\widetilde{\boldsymbol{\Pi}}_{n,f}-\widehat{\boldsymbol{\Pi}}_{n}\right)^{\prime}\geq\mu_{2n.\mathsf{min}}\left\lVert(\widehat{\boldsymbol{\Pi}}_{n}-\widetilde{\boldsymbol{\Pi}}_{n,f})\boldsymbol{B}_{n}\boldsymbol{H}\right\rVert^{2},$
(8.34)

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