where $y_{0}=0$ and $\epsilon_{t}\sim_{\textit{i.i.d}}\mathcal{N}(0,1)$.
Define with

$\displaystyle S_{T}=\frac{1}{T}\sum_{t=1}^{T}y_{t-1}\Delta y_{t}\ \ \ \
\text{and}\ \ \ \ H_{T}=\frac{1}{T}\sum_{t=1}^{T}y^{2}_{t-1}$ (4.2)

The large sample behaviour of $\big{(}S_{T},H_{T}\big{)}$ is well understood.
More specifically, under local-to-unity asymptotics with $c=T(\rho-1)$ held
fixed as $T\to\infty$,

$\displaystyle\big{(}S_{T},H_{T}\big{)}\overset{d}{\to}\left(\int_{0}^{1}W_{c}(r)dW_{c}(r),\int_{0}^{1}W_{c}(r)^{2}dr\right),$
(4.3)

where
$W_{c}(r)=\displaystyle\int_{0}^{1}\mathsf{exp}\big{\\{}c(r-s)\big{\\}}dW(s)$
and $W(.)$ is a standard Wiener process.

######  Example 10.

Consider the framework of Phillips, 1987a , such that $\left\\{y_{t}\right\\}$
be a time series generated by

$\displaystyle y_{t}$ $\displaystyle=\alpha y_{t-1}+u_{t},\ \ \ t=1,2,...\ \ \
\text{with}\ \alpha=1$ (4.4)

Consider the sequence of partial sums $\left\\{S_{t}\right\\}$, then we obtain
that

$\displaystyle
X_{T}(r)=\frac{1}{\sqrt{T}\sigma}S_{[Tr]}=\frac{1}{\sqrt{T}\sigma}S_{j-1}$
(4.5)

for $\frac{(j-1)}{T}\leq r\leq\frac{j}{T}$ for $j\in\left\\{1,...,T\right\\}$,
where $[Tr]$ denotes the integral part of Tr. Then, $X_{T}(r)$ lies in
$\mathcal{D}=\mathcal{D}[0,1]$, the space of real valued functions on the
interval $[0,1]$ that are right continuous and have finite left limits. Under
very general conditions the random element $X_{T}(r)$ obeys a central limit
theory on the function space $\mathcal{D}$. In particular, we have that, as
$T\to\infty$,

$\displaystyle X_{T}(r)\Rightarrow W(r).$ (4.6)

which implies a weak convergence on the associated probability measure. In
particular, in this case, the probability measure of $X_{T}(r)$ converges
weakly to the probability measure of the standard Brownian motion $W(r)$. We
focus on the asymptotic behaviour of the sample moments of the process
$\left\\{S_{t}\right\\}$ and the innovations $\left\\{u_{t}\right\\}$. We
express the limit distributions as functions of standard Brownian motion
$W(r)$.

All integrals are understood to be taken over the interval $[0,1]$, while
integrals such as $\int W$, $\int W^{2}$, $\int rW$ are understood to be taken
with respect to Lebesgue measure such that we write $W_{1}=W(1)$. Then, as
$T\to\infty$, we obtain that

$\displaystyle\frac{1}{T\sqrt{T}}$
$\displaystyle\sum_{t=1}^{T}S_{t}\Rightarrow\sigma\int_{0}^{1}Wdr$ (4.7)
$\displaystyle\frac{1}{T^{2}}$
$\displaystyle\sum_{t=1}^{T}S_{t}^{2}\Rightarrow\sigma^{2}\int_{0}^{1}W^{2}dr$
(4.8) $\displaystyle\frac{1}{T\sqrt{T}}$
$\displaystyle\sum_{t=1}^{T}tu_{t}\Rightarrow\sigma^{2}\int_{0}^{1}W^{2}dr=\sigma\left(W_{1}-\int
W\right)$ (4.9)

####  4.1.1. Power Functions for Unit Root Tests

Consider the power functions for unit root tests by considering the sequence
of local alternatives as

$\displaystyle\alpha=e^{c/T}=\left(1+\frac{c}{T}\right)$ (4.10)

where $c=0$ reduces to the null hypothesis, $c>0$ gives local explosive
alternatives and $c<0$ corresponds to local stationary alternatives. Moreover,
the asymptotic theory for the sample moments of near-integrated time series
converge weakly to corresponding functionals of a diffusion process rather
than standard Brownian motion. Specifically, we have that, as $T\to\infty$,

$\displaystyle\frac{1}{T\sqrt{T}}\sum_{t=1}^{T}y_{t}$
$\displaystyle\Rightarrow\sigma\int_{0}^{1}J_{c}$ (4.11)
$\displaystyle\frac{1}{T^{2}}\sum_{t=1}^{T}y^{2}_{t}$
$\displaystyle\Rightarrow\sigma\int_{0}^{1}J^{2}_{c}$ (4.12)
$\displaystyle\frac{1}{T^{5/2}}\sum_{t=1}^{T}ty_{t}$
$\displaystyle\Rightarrow\sigma\int_{0}^{1}rJ_{c}$ (4.13)
$\displaystyle\frac{1}{T}\sum_{t=1}^{T}y_{t-1}u_{t}$
$\displaystyle\Rightarrow\sigma^{2}\int_{0}^{1}J_{c}dW+\lambda,$ (4.14)

where

$\displaystyle J_{c}(r)=\int_{0}^{r}e^{(r-s)c}dW(s)$ (4.15)

is the OU process generated in continuous time by the stochastic difference
equation

$\displaystyle dJ_{c}(r)=cJ_{c}(r)dr+dW(r),\ \ \text{with}\ \ J_{c}(0)=0.$
(4.16)

A particular feature of the local to unity model is that the localizing
parameter is identifiable but it is not consistently estimable.

Consider the following expression which gives the signal to noise ratio

$\displaystyle\frac{\mathsf{Var}(x_{t})}{\mathsf{Var}(u_{t})}\sim\frac{\displaystyle\frac{1}{n}\sum_{t=1}^{n}\left(\frac{y_{t-1}}{n}\right)^{2}}{\displaystyle\sigma^{2}}\overset{p}{\to}0,$
(4.17)

and so the signal from $x_{t}$ is too weak relative to the error variation to
produce a consistent estimator of the localizing coefficient $c$. Although
methods have been developed to utilize the way in which the limit distribution
depends on the localizing coefficient, the failure of a consistent estimation
has been a challenging task when conducting inference in these models. The
dependence of the limit distribution on $c$ also affects resampling procedures
such as the bootstrap, which are known to be inconsistent in models of this
type because exactly of this dependence of functionals to the nuisance
parameter of persistence.

####  4.1.2. Time Series Regression with a Unit Root

The consistent estimates $s_{u}^{2}$ and $s^{2}_{T}$ are used to develop new
tests for unit roots that apply under very general conditions. We define the
statistics

$\displaystyle Z_{\alpha}$
$\displaystyle=T\left(\hat{\alpha}-1\right)-\frac{1}{2}\frac{\displaystyle\left(s^{2}_{T}-s_{u}^{2}\right)}{\displaystyle\left(T^{-2}\sum_{t=1}^{T}y^{2}_{t-1}\right)}$
(4.18) $\displaystyle Z_{t}$
$\displaystyle=\left(\sum_{t=1}^{T}y^{2}_{t-1}\right)^{1/2}\left(\hat{\alpha}-1\right)/s_{T}-\frac{1}{2}\left(s^{2}_{T}-s_{u}^{2}\right)\left[s_{T}\left(T^{-2}\sum_{t=1}^{T}y^{2}_{t-1}\right)^{1/2}\right]^{-1}.$
(4.19)

where $Z_{\alpha}$ is a transformation of the standardized estimator
$T\left(\hat{\alpha}-1\right)$ and $Z_{t}$ is a transformation of the
regression $t$ statistic. Then, the limiting distribution of $Z_{\alpha}$ and
$Z_{t}$ are given by

######  Theorem 10.

If the conditions of Theorem 4.2 are satisfied, then as $T\to\infty$,

$\displaystyle Z_{\alpha}$
$\displaystyle\Rightarrow\frac{\left(W(1)^{2}-1\right)/2}{\int_{0}^{1}W(t)^{2}dt}$
(4.20) $\displaystyle Z_{t}$
$\displaystyle\Rightarrow\frac{\displaystyle\left(W(1)^{2}-1\right)/2}{\displaystyle\left\\{\int_{0}^{1}W(t)^{2}dt\right\\}^{1/2}}$
(4.21)

under the null hypothesis that $\alpha=1$.

######  Remark 10.

The Theorem above demonstrates that the limiting distribution of the two
statistics $Z_{t}$ and $Z_{\alpha}$ are invariant within a very wide class of
weakly dependent and possible heterogeneous distributed innovations
$\left\\{u_{t}\right\\}_{t=1}^{\infty}$. Thus, the limiting distribution of
$Z_{\alpha}$ is identical to that of $T\left(\hat{\alpha}-1\right)$, when
$\sigma_{u}^{2}=\sigma^{2}$.

An important property of $I(1)$ variables is that there can be linear
combinations of these variables that are $I(0)$. If this is so then these
variables are said to be cointegrated. Notice that econometric cointegration
analysis can be used to overcome difficulties associated with stochastic
trends in time series and applied to test whether there exist combinations of
non-stationary series that are themselves stationary.

###  4.2. Regression Asymptotics using Martingale Convergence Methods

Considers weak convergence arguments to stochastic integral approximations. In
particular for the case of a martingale-difference sequence
$\left(\epsilon_{t}^{2}|\mathcal{F}_{t-1}\right)=\sigma^{2}_{\epsilon}$ for
all $t$ and $\mathsf{sup}_{t\in\mathbb{Z}}\mathbb{E}\left(\epsilon_{t}^{p}\
\mathcal{F}_{t-1}\right)=\sigma^{2}_{\epsilon}<\infty$ almost surely for some
$p>2$. Then by Donsker’s theorem for the partial sum process we have that

$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}\epsilon_{t}\to\sigma_{\epsilon}W(r),$ (4.22)

where $\big{\\{}W(s),s\geq 0\big{\\}}$ denotes the standard Brownian motion,
which implies that the bilinear form:

$\displaystyle\frac{1}{n}\sum_{t=2}^{\lfloor
nr\rfloor}\left(\sum_{j=1}^{t-1}\epsilon_{j}\right)\epsilon_{t}\Rightarrow\sigma_{\epsilon}^{2}\int_{0}^{r}W(v)dW(v)$
(4.23)

Notice that the particular approach also has drawbacks. One of the drawbacks
is that the approach is problem specific in certain ways. For instance it
cannot be directly used in the case of statistics such as
$\sum_{t=1}^{n}y_{t-1}u_{t}$, where $y_{t}=\alpha_{n}y_{t-1}+u_{t}$, for
$t=1,...,n$ and $\alpha_{n}\to 1$ as $n\to\infty$, that are central to the
study of local deviations from a unit root in time series regression. Strong
approximations to partial sums of independent random variables, along with an
application of PS device, allow one to obtain invariance principles under
independence and stationarity assumptions with explicit rates of convergence.
For instance, using the Hungarian construction if
$(\epsilon_{t})_{t\in\mathbb{Z}}$ satisfy the conditions then

$\displaystyle\left|\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}\epsilon_{t}-\sigma_{\epsilon}W(r)\right|=o_{a/s}\left(n^{1/p-1/2}\right)$
(4.24)

Notice that some existing results available in the literature on convergence
to stochastic integrals can be applied to obtain convergence results. Denote
with $N_{n,r}=\frac{\epsilon_{0}}{\sqrt{n}}+\sum_{t=1}^{\lfloor
nr\rfloor}\frac{\epsilon_{t}}{\sqrt{n}}$. Then, the following stochastic
integral representation holds for the statistic on the right-hand size

$\displaystyle\frac{1}{n}\sum_{t=2}^{\lfloor
nr\rfloor}\left(\sum_{j=1}^{t-1}\epsilon_{j}\right)\epsilon_{t}\Rightarrow\int_{0}^{r}N_{n,s}dN_{n,s}$
(4.25)

Since $N_{n,r}\to W(r)$, then the latter result implies that the convergence
result to the stochastic integral holds, provided that the sequence of
processes $\left\\{N_{n,t}\right\\}$ satisfies the uniform tightness
condition.

####  4.2.1. Unification of the Limit theory of Autoregression

Consider the following first order autoregression model (e.g., see, Giraitis
and Phillips,  (2006))

$\displaystyle y_{t}=\alpha y_{t-1}+\epsilon_{t},\ \ \ t=1,...,n$ (4.26)

with martingale-difference errors $\epsilon_{t}$. We treat the stationary
$|\alpha|<1$, unit root $\alpha=1$, local to unity, and explosive cases
together in what follows and show how the limit theory for all these cases may
be formulated in a unified manner within the martingale convergence framework.
Consider the stationary and unit root cases. For $r\in(0,1]$, define the
recursive least squares estimator $\hat{\alpha}_{r}=\sum_{t=1}^{\lfloor
nr\rfloor}y_{t-1}y_{t}/\sum_{t=1}^{\lfloor nr\rfloor}y_{t-1}^{2}$,

$\displaystyle\left(\frac{1}{\sigma_{\epsilon}^{2}}\displaystyle\sum_{t=1}^{\lfloor
nr\rfloor}y^{2}_{t-1}\right)^{1/2}\big{(}\hat{\alpha}_{r}-\alpha\big{)}=\frac{\displaystyle\sum_{t=1}^{\lfloor
nr\rfloor}y_{t-1}\epsilon_{t}}{\displaystyle\left(\sum_{t=1}^{\lfloor
nr\rfloor}y^{2}_{t-1}\sigma^{2}_{\epsilon}\right)^{1/2}}=\frac{X_{n}(r)}{\left(\tilde{C}_{n}^{\prime}(r)\right)^{1/2}},$
(4.27)

where $X_{n}(r)$ is the martingale given by

$\displaystyle
X_{n}(r)=\begin{cases}\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}y_{t-1}\epsilon_{t},&\text{for}\ |\alpha|<1\\\ \\\
\displaystyle\frac{1}{n}\sum_{t=1}^{\lfloor
nr\rfloor}y_{t-1}\epsilon_{t},&\text{for}\ \alpha=1\end{cases}$ (4.28)

Then, by Theorem 4.1 it holds that

$\displaystyle X_{n}(r)\to
X(r)=\begin{cases}\sigma_{\alpha}\sigma_{\epsilon}W(r),\ \ \text{for}\ \
|\alpha|<1,\\\ \\\ \displaystyle\sigma_{\epsilon}^{2}\int_{0}^{r}W(r)dW(r),\ \
\text{for}\ \ \alpha=1,\end{cases}$ (4.29)

Then, it can be proved that

$\displaystyle\left(\frac{1}{\sigma_{\epsilon}^{2}}\displaystyle\sum_{t=1}^{\lfloor
nr\rfloor}y^{2}_{t-1}\right)^{1/2}\big{(}\hat{\alpha}_{r}-\alpha\big{)}$
$\displaystyle=\frac{\displaystyle
X_{n}(r)}{\displaystyle\left(\tilde{C}_{n}^{\prime}(r)\right)^{1/2}}\overset{d}{\to}\frac{X(r)}{\left(C(r)\right)^{1/2}}=\begin{cases}\displaystyle\frac{1}{r^{1/2}}W(r),&|\alpha|<1\\\
\\\
\frac{\displaystyle\int_{0}^{r}W(v)dW(v)}{\displaystyle\left(\int_{0}^{r}W(v)^{2}dv\right)^{1/2}},&|\alpha|=1,\end{cases}$

which unifies the limit theory for the stationary and unit root
autoregression.

Defining the error variance estimator $s_{r}^{2}=\frac{1}{\lfloor
nr\rfloor}\sum_{t=1}^{\lfloor
nr\rfloor}\big{(}y_{t}-\hat{\alpha}_{r}y_{t-1}\big{)}^{2}$ and noting that
$s_{r}^{2}\overset{p}{\to}\sigma^{2}_{\epsilon}$ for $r>0$, we have the
corresponding limit theory for the recursive t-statistic such that the
following limit result holds

$\displaystyle
t_{\hat{\alpha}}(r)=\left(\frac{1}{s^{2}_{r}}\sum_{t=1}^{\lfloor
nr\rfloor}y_{t-1}^{2}\right)\big{(}\hat{\alpha}_{r}-\alpha\big{)}=\frac{\displaystyle\sum_{t=1}^{\lfloor
nr\rfloor}y_{t-1}\epsilon_{t}}{\displaystyle\left(\sum_{t=1}^{\lfloor
nr\rfloor}y^{2}_{t-1}\sigma^{2}_{\epsilon}\right)^{1/2}}\frac{\sigma_{\epsilon}}{s_{r}}=\frac{X_{n}(r)}{\left(\tilde{C}^{\prime}_{n}(r)\right)^{1/2}}\frac{\sigma_{\epsilon}}{s_{r}}$
(4.30)

The theory also extends to cases where $\alpha$ lies in the neighborhood of
unity such that $\alpha=\left(1+\frac{c}{n}\right)$. Then, it follows that

$\displaystyle
X_{n}(r)\overset{d}{\to}X(r)=\sigma_{\epsilon}^{2}\int_{0}^{r}J_{c}(v)dW(v),$
(4.31)

where $J_{c}(v)=\displaystyle\int_{0}^{v}e^{c(v-s)}dW(s)$ is a linear
diffusion. Then, we obtain that

$\displaystyle\left(\frac{1}{s^{2}_{r}}\sum_{t=1}^{\lfloor
nr\rfloor}y_{t-1}^{2}\right)^{1/2}\big{(}\hat{\alpha}_{r}-\alpha\big{)}=\frac{\displaystyle
X_{n}(r)}{\displaystyle\left(\tilde{C}_{n}^{\prime}(r)\right)^{1/2}}\overset{d}{\to}\frac{X(r)}{\left(C(r)\right)^{1/2}}\overset{d}{=}\frac{\displaystyle\int_{0}^{1}J_{c}(v)dW(v)}{\displaystyle\left(\int_{0}^{1}J_{c}(v)^{2}dv\right)^{1/2}}.$
(4.32)

Furthermore, we can investigate the effect on the asymptotic theory of the
proposed test statistics when there are moderate deviations from unity of the
form $\alpha=\left(1+\displaystyle\frac{c}{n^{b}}\right)$ for some $b\in(0,1)$
and $c<0$. In particular, under the assumption that the model covariates are
generated as near unit root processes with $b\in(0,1)$ and $c<0$ then the
regressors are considered to be mildly integrated and the functional takes the
following form

$\displaystyle X_{n}(r)=\frac{1}{n^{(1+b)/2}}\sum_{t=1}^{\lfloor
nr\rfloor}y_{t-1}\epsilon_{t},\ \ \ \
\alpha=\left(1+\frac{c}{n^{b}}\right),c<0,\ b\in(0,1),$ (4.33)

######  Theorem 11.

Let $X^{T}=\left\\{X_{t},0\leq t\leq T\right\\}$ be observations of the mean
reversion process with mean reversion function. Then, for any fixed
$0<s_{1}<s_{2}<1$, under the null hypothesis,

$\displaystyle\underset{s\in[s_{1},s_{2}]}{\mathsf{sup}}\Lambda_{T}(s)\overset{\mathcal{D}}{\to}\underset{s\in[s_{1},s_{2}]}{\mathsf{sup}}\frac{\left\lVert
W(s)-sW(1)\right\rVert^{2}}{s(1-s)}$ (4.34)

as $T\to\infty$, where $\left\lVert.\right\rVert$ is the Euclidean norm and
$W$ is a $(p+1)-$dimensional standard Brownian motion.

###  4.3. Time Series Regression

One of the main differences with independent data where the OLS estimator
satisfies certain conditions is that, time-series data commonly violate one of
the classical assumptions. In particular, the independence assumption holds
that the error terms corresponding to different point in time are not
correlated. However, when the error terms are serially correlated
(autocorrelated), the OLS method produces biased estimates of the standard
errors of the regression coefficients. The persistence of a random shock is
what distinguishes an AR error process from an MA error process. An AR(p)
error process can be written as an infinite sum of past random shocks. Each
shock persists indefenitely, although its importance diminishes over time. For
example, in an AR(1) process, the effect persists indefenitely, although it
decreases over time and is effectively zero after some relatively small number
of time periods ( Choudhury et al.,  (1999)).

####  4.3.1. Unit Roots versus Deterministic Trends

Consider the following model

$\displaystyle y_{t}=\mu+\rho y_{t-1}+u_{t}$ (4.35)

Then, assume that the true data generating process is given by

$\displaystyle H_{1}:y_{t}=\mu+\phi t+\rho y_{t-1}+u_{t},\ \ \ \text{with}\ \
|\rho|<1,\phi\neq 0.$ (4.36)

####  4.3.2. The Engle-Granger Approach

Consider a vector time series process $X_{t}\in\mathbb{R}^{k}$ such that each
component is $X_{it}$ is integrated or order one, such that each of the series
$X_{it}$ contains a unit root, but $\Delta X_{it}$ is a zero mean stationary
process. Thus, using the Wold decomposition theorem we can write:

$\displaystyle\Delta
X_{t}=C(L)u_{t}=\left(\sum_{j=0}^{\infty}C_{j}L^{j}\right)u_{t}=\sum_{j=0}^{\infty}c_{j}u_{t-j}$
(4.37)

where $L$ is the lag operator and $u_{t}$ is a $k-$variate white-noise
process.

Working in the context of a bivariate system with at most one cointegrated
vector, Engle and Granger,  (1987) propose estimating the cointegrated vector
$\alpha=\left(1,\alpha_{2}\right)$ by regressing the first component $X_{1t}$
of $X_{t}$ on the second component $X_{2t}$, using OLS, and then testing
whether the OLS residuals $\hat{Z}_{t}$ have a unit root using the augmented
Dickey-Fuller test. Therefore, if the test rejects the unit root hypothesis,
and thus accepts the cointegration hypothesis, one may substitute for
$\alpha^{\prime}X_{t-1}$ the OLS residual $\hat{Z}_{t-1}$ and estimate the
parameter matrices in $A^{*}(L)$ by OLS, assuming that $A^{*}(L)$ is a finite-
order lag polynomial. Notice that the above approach is only applicable if
there is at most one cointegrated vector. Systems with dimension greater than
two, however, may have multiple cointegrated vectors.

####  4.3.3. Testing the hypothesis of no serial correlation

In some cases we are interested to test the hypothesis that the series $Y_{t}$
is serially uncorrelated against the alternative that is serially correlated.
The most appropriate test of this hypothesis is based on the OLS of an AR$(p)$
model. Consider the following model

$\displaystyle
Y_{t}=\alpha_{0}+\alpha_{1}Y_{t-1}+\alpha_{2}Y_{t-2}+...+\alpha_{p}Y_{t-p}+e_{t}$
(4.38)

with $e_{t}$ a MDS. Thus, in this model the series $Y_{t}$ is serially
uncorrelated if the slope coefficients are all zero. For example, when the
cointegrating vector is known and the covariates of the cointegration vector
are strictly exogenous, the optimal test for the null of a unit root in the
known cointegrating vector collapes to the usual unit root test for a single
variable. From a testing perspective, an unknown cointegrating vector is a
nuisance parameter that is identified only under the alternative hypothesis,
so optimal tests against the alternative of cointegration take the form of a
weighted average test where the weights are over the possible cointegrating
vectors.

####  4.3.4. A Cointegrated VAR

A VAR has several equivalent representations that are valuable for
understanding the interactions between exogeneity, cointegration and economic
policy analysis. To start, the levels form of the $s-$th order Gaussian VAR
for $x$ is

$\displaystyle x_{t}=Kq_{t}+\sum_{j=1}^{s}A_{j}x_{t-j}+\varepsilon_{t},\ \ \
\varepsilon_{t}\sim\mathcal{N}\left(0,\Sigma\right).$ (4.39)

where $K$ is an $N\times N_{0}$ matrix of coefficients of the $N_{0}$
deterministic variables $q_{t}$.

Suppose that we have a vector
$Y_{t}=\left[y_{1t},y_{2t},...,y_{nt}\right]^{\prime}$ that does not satisfy
the conditions for stationarity. One way to achieve stationarity might be to
model $\Delta y_{t}$, rather than $y_{t}$ itself. However, differencing can
discard important information about the equilibrium relationships between the
variables. This is because another way to achieve stationarity can be through
linear combinations of the levels of the variables. Thus, if such linear
combinations exist then we have cointegration and the variables are said to be
cointegrated.

Cointegration has some important implications:

  * 1.

Estimates of the cointegrating relationships are super-consistent, they
converge at rate $T$ rather than $\sqrt{T}$. Moreover, the system implies a
set of dynamic long-run equilibria between the variables

  * 2.

Modelling cointegrated variables allows for separate short-run and long-run
dynamic responses.

######  Definition 11.

Suppose that $y_{t}$ is $I(1)$. Then $y_{t}$ is cointegrated if there exists
an $N\times r$ matrix $\beta$, of full column rank and where $0<r<N$, such
that the $r$ linear combinations, $\beta^{\prime}y_{t}=u_{t}$, are $I(0)$.

######  Remark 11.

The dimension $r$ is the cointegration rank and the columns of $\beta$ are the
cointegrating vectors. Testing for cointegrating relations that economic
theory predicts should exist, implies that the null hypothesis of
noncointegration is not rejected. However, under the presence of breaks there
is a need for implementing tests of the null of non-cointegration against
alternatives allowing cointegrating relations subject to breaks. Although, the
dates of breaks as well as their number are unknown a priori (see, Saikkonen
et al.,  (2006) and Shin,  (1994) among others).

Identification, Estimation and Forecasting with Time Series Regressions:

  * 1.

Existence of stationary solution: Usually this refers to finding sufficient
conditions that ensure the existence of a weakly dependent stationary and
ergodic solution $Z_{t}=\left(Y_{t},X_{t}\right)$.

  * 2.

Inference Problem: The inference problem consists of the estimation procedure,
the consistency of the corresponding estimator as well as deriving the
asymptotic distribution of this estimator.

  * 3.

Significance Test of Parameter: Usually we employ a Wald-type test statistic
for linear restrictions on the parameters of the cointegrating regression
model.

  * 4.

Model selection: This step can be done either using a direct model selection
approach, using an information criterion or using an exogenously generated
procedure. The crucial step is to consider conditions that ensure the weak and
strong consistency of the proposed procedure.

######  Example 11.

Consider the following data generating process as below

$\displaystyle\Delta
X_{t}=\alpha\beta^{\prime}X_{t-1}+\sum_{i=1}^{k-1}\Gamma_{i}\Delta
X_{t-i}+\varepsilon_{t},\ \ \ \text{for}\ t=1,...,T,$ (4.40)

where $\left\\{\varepsilon_{t}\right\\}$ is i.i.d with mean zero and full-rank
covariance matrix $\Omega$, and where the initial values $X_{1-k},...,X_{0}$
are fixed. We are interested in the null hypothesis $H_{0}:\beta=\tau$. Thus,
when $\tau$ is a known $(p\times r)$ matrix of full column rank $r$, the
subspace spanned by $\beta$ and $\tau$ are identical.

######  Example 12.

Consider the time series vector $\boldsymbol{Y}_{t}$, which is an
$N-$dimensional random vector generated by the VAR model as below

$\displaystyle\boldsymbol{Y}_{t}=A_{1}\boldsymbol{Y}_{t-1}+...+A_{p}\boldsymbol{Y}_{t-p}+\boldsymbol{u}_{t},\
\ \ t\in\left\\{1,...,T\right\\}.$ (4.41)

Define the $N(p+1)$ vector
$\boldsymbol{X}_{t}=\big{(}\boldsymbol{Y}_{t-p}^{\top},...,\boldsymbol{Y}_{t-1}^{\top},\boldsymbol{Y}_{t}^{\top}\big{)}^{\top}$
and let
$\Sigma_{x}=Var(\boldsymbol{X}_{t})=\mathbb{E}\left[\boldsymbol{X}\boldsymbol{X}^{\top}\right]$
and
$\Gamma_{i}=\mathbb{E}\left[\boldsymbol{Y}_{t}\boldsymbol{Y}_{t-i}^{\top}\right]$
the autocovariance matrix and
$\Sigma_{x}=\begin{pmatrix}\Gamma_{0}&\Gamma_{1}^{\top}&...&\Gamma_{p}^{\top}\end{pmatrix}$.

######  Example 13.

Let $\boldsymbol{x}_{t}$ be an $I(1)$ vector of $n$ components, each with
possibly deterministic trend in mean. Suppose that the system can be written
as a finite-order vector autoregression:

$\displaystyle\boldsymbol{x}_{t}=\boldsymbol{\mu}+\boldsymbol{\pi}_{1}\boldsymbol{x}_{t-1}+\boldsymbol{\pi}_{2}\boldsymbol{x}_{t-2}+...+\boldsymbol{\pi}_{k}\boldsymbol{x}_{t-k}+\boldsymbol{\varepsilon}_{t},\
\ \ t=1,...,T$ (4.42)

Then, the model can be rewritten in error-correction form as below

$\displaystyle\Delta\boldsymbol{x}_{t}=\boldsymbol{\mu}+\boldsymbol{\Gamma}_{1}\Delta\boldsymbol{x}_{t-1}+\boldsymbol{\Gamma}_{2}\Delta\boldsymbol{x}_{t-2}+...+\boldsymbol{\Gamma}_{k-1}\Delta\boldsymbol{x}_{t-k+1}+\boldsymbol{\pi}\boldsymbol{x}_{t-k}+\boldsymbol{\varepsilon}_{t}\equiv\boldsymbol{\mu}+\sum_{i=1}^{k-1}\boldsymbol{\Gamma}_{i}(1-L)L^{i}\boldsymbol{x}_{i}+\boldsymbol{\pi}\boldsymbol{x}_{t-k}+\boldsymbol{\varepsilon}_{t}.$

Therefore, we get the following system equation representation for
$t\in\left\\{1,...,T\right\\}$

$\displaystyle\boldsymbol{\pi}(L)\boldsymbol{x}_{t}$
$\displaystyle=\boldsymbol{\mu}+\boldsymbol{\varepsilon}_{t},\ \
\boldsymbol{\pi}(L)=(1-L)\boldsymbol{I}_{n}-\sum_{i=1}^{k-1}\boldsymbol{\Gamma}_{i}(1-L)L^{i}-\boldsymbol{\pi}L^{k}$
(4.43) $\displaystyle\boldsymbol{\Gamma}_{i}$
$\displaystyle=-\boldsymbol{I}_{n}+\boldsymbol{\pi}_{1}+\boldsymbol{\pi}_{2}+...+\boldsymbol{\pi}_{i},\
\ \ \ i=1,...,k$ (4.44)

######  Example 14.

The asymptotic size and power of the augmented Dickey-Fuller test for a unit
root are studied by Paparoditis and Politis,  (2018). In particular, the
authors show that the limiting distribution of the augmented DF test under the
null hypothesis of a unit root is valid under general set of assumptions on
the dependence structure of innovations. Consider the following specification

$\displaystyle X_{t}=\rho X_{t-1}+\sum_{j=1}^{p}a_{j,p}\Delta X_{t-j}+e_{t,p}$
(4.45)

where $X_{t}$ is a linear, infinite order autoregressive process such that

$\displaystyle X_{t}=X_{t-1}+U_{t},\ \ \
U_{t}=\sum_{j=1}^{\infty}a_{j}U_{t-j}+e_{t}.$ (4.46)

Notice that stationarity and causality of $\left\\{U_{t}\right\\}$ is ensured
by assuming that $\sum_{j=1}^{\infty}|j|^{s}|a_{j}|<\infty$ for some $s\geq 1$
and $\sum_{j=1}^{\infty}a_{j}z^{j}\neq 0$ for all $|z|\leq 1$.

######  Example 15.

Consider a general autoregressive distributed lag ARDL $(p,q)$ model where a
series, $y_{t}$, is a function of a constant term, $\boldsymbol{\alpha}_{0}$,
past values of itself stretching back $p-$periods, contemporaneous and lagged
values of an independent variable, $x_{t}$, of lag order $q$, and i.i.d error
term:

$\displaystyle
y_{t}=\boldsymbol{\alpha}_{0}+\sum_{i=1}^{p}\boldsymbol{\alpha}_{i}y_{t-i}+\sum_{j=0}^{q}\boldsymbol{\beta}_{j}\boldsymbol{x}_{t-j}+\boldsymbol{\epsilon}_{t},$
(4.47)

A commonly used model is the ARDL $(1,1)$ model given by
$y_{t}=\boldsymbol{\alpha}_{0}+\boldsymbol{\alpha}_{1}y_{t-1}+\boldsymbol{\beta}_{0}x_{t}+\boldsymbol{\beta}_{1}x_{t-1}+\boldsymbol{\epsilon}_{t}$.
Then, the contemporaneous effect of $x_{t}$ on $y_{t}$ is given by
$\boldsymbol{\beta}_{0}$. The magnitude of $\boldsymbol{\alpha}_{1}$ informs
us about the memory property of $y_{t}$. Assuming that $0<\alpha_{1}<1$,
larger values indicate that movements in $y_{t}$ take longer to dissipate. The
long-run effect (or long-run multiplier) is the total effect that a change in
$x_{t}$ has on $y_{t}$. The above econometric specification corresponds to a
conditional mean function. One can also consider the corresponding conditional
quantile functional form which implies a cointegrating relation around a
certain quantile level of the distribution function (see, Galvao J et al.,
(2013) and Cho et al.,  (2015)).

###  4.4. Cointegrating Regression

####  4.4.1. Distribution of the OLS Estimate for a Special Case

Following, Hamilton,  (2020) let $y_{1t}$ be a scalar and
$\boldsymbol{y}_{2t}$ be a $(p\times 1)$ vector satisfying

$\displaystyle y_{1t}$
$\displaystyle=\boldsymbol{\alpha}+\boldsymbol{\gamma}^{\top}\boldsymbol{y}_{2t}+u_{1t}$
(4.48) $\displaystyle\boldsymbol{y}_{2t}$
$\displaystyle=\boldsymbol{y}_{2,t-1}+\boldsymbol{u}_{2t}$ (4.49)

In particular, if $y_{1t}$ and $\boldsymbol{y}_{2t}$ are both $I(1)$ but
$u_{1t}$ and $\boldsymbol{u}_{2t}$ are $I(0)$, then $d\equiv(p+1)$, the
$d-$dimensional vector $\left(y_{1t},\boldsymbol{y}_{2,t-1}\right)^{\prime}$
is cointegrated with the cointegrated relation given by expression (4.48).
Therefore, by considering the case that the innovation sequence of the system
follows a Gaussian distribution, we have that the following holds

$\displaystyle\begin{bmatrix}u_{1t}\\\
\boldsymbol{u}_{2t}\end{bmatrix}\overset{\textit{i.i.d}}{\sim}\mathcal{N}\left(\begin{bmatrix}0\\\
\boldsymbol{0}\end{bmatrix},\begin{bmatrix}\sigma_{1}^{2}&\boldsymbol{0}^{\prime}\\\
\boldsymbol{0}&\boldsymbol{\Omega}_{22}\end{bmatrix}\right).$ (4.50)

Notice that although we consider a Gaussian cointegrated system we assume that
the explanatory variables $\boldsymbol{y}_{2,t}$ in (4.48) are independent of
the error term $u_{1t}$ for all $t$. Therefore, conditional on
$\big{(}\boldsymbol{y}_{2,1},\boldsymbol{y}_{2,2},...,\boldsymbol{y}_{2,T}\big{)}$,
the OLS estimates have a Gaussian distribution given by

$\displaystyle\begin{bmatrix}\left(\hat{\boldsymbol{\alpha}}_{T}-\boldsymbol{\alpha}\right)\\\
\left(\hat{\boldsymbol{\gamma}}_{T}-\boldsymbol{\gamma}\right)\end{bmatrix}$
$\displaystyle\big{(}\boldsymbol{y}_{2,1},\boldsymbol{y}_{2,2},...,\boldsymbol{y}_{2,T}\big{)}$
$\displaystyle=\begin{bmatrix}T\ &\
\displaystyle\sum_{t=1}^{T}\boldsymbol{y}_{2,T}^{\prime}\\\
\displaystyle\sum_{t=1}^{T}\boldsymbol{y}_{2,T}\ &\
\displaystyle\sum_{t=1}^{T}\boldsymbol{y}_{2,T}\boldsymbol{y}_{2,T}^{\prime}\end{bmatrix}^{-1}\begin{bmatrix}\displaystyle\sum_{t=1}^{T}u_{1t}\\\
\displaystyle\sum_{t=1}^{T}\boldsymbol{y}_{2,T}u_{1t}\end{bmatrix}\overset{\textit{i.i.d}}{\sim}\mathcal{N}\left(\begin{bmatrix}0\\\
\boldsymbol{0}\end{bmatrix},\begin{bmatrix}T\ &\
\displaystyle\sum_{t=1}^{T}\boldsymbol{y}_{2,T}^{\prime}\\\
\displaystyle\sum_{t=1}^{T}\boldsymbol{y}_{2,T}\ &\
\displaystyle\sum_{t=1}^{T}\boldsymbol{y}_{2,T}\boldsymbol{y}_{2,T}^{\prime}\end{bmatrix}^{-1}\right)$
(4.51)

######  Remark 12.

The above conditional distribution is that is needed to justify the small-
sample application of the usual OLS t and F-tests under the null hypothesis.
Thus, consider a hypothesis test involving $m$ restrictions on
$\boldsymbol{\alpha}$ and $\boldsymbol{\gamma}$ of the following form

$\displaystyle\boldsymbol{R}_{\boldsymbol{\alpha}}\boldsymbol{\alpha}+\boldsymbol{R}_{\boldsymbol{\gamma}}\boldsymbol{\gamma}=\boldsymbol{r},$
(4.52)

where $\boldsymbol{R}_{\boldsymbol{\alpha}}$ and $\boldsymbol{r}$ are known
$(m\times 1)$ vectors and $\boldsymbol{R}_{\boldsymbol{\gamma}}$ is a known
$(m\times d)$ matrix which describes these restrictions. Under the null
hypothesis we have that
$\mathbb{H}_{0}:\boldsymbol{R}\boldsymbol{\beta}=\boldsymbol{r}$ where the
Wald statistic is written as

$\displaystyle
W=\left(\boldsymbol{R}\widehat{\boldsymbol{\beta}}-\boldsymbol{r}\right)^{\top}\left[\boldsymbol{R}\boldsymbol{J}_{\boldsymbol{\beta}\boldsymbol{\beta}}^{-1}(\hat{\boldsymbol{\theta}})\boldsymbol{R}^{\top}\right]^{-1}\left(\boldsymbol{R}\widehat{\boldsymbol{\beta}}-\boldsymbol{r}\right)$
(4.53)

where
$\boldsymbol{\beta}=\big{(}\boldsymbol{\beta}_{1}^{\top},...,\boldsymbol{\beta}_{p}^{\top}\big{)}^{\top}$.

Therefore, the Wald form of the OLS $F-$test of the null hypothesis is given
by

$\displaystyle\big{(}\boldsymbol{R}_{\boldsymbol{\alpha}}\hat{\boldsymbol{\alpha}}+\boldsymbol{R}_{\boldsymbol{\gamma}}\hat{\boldsymbol{\gamma}}-\boldsymbol{r}\big{)}^{\prime}\left\\{s^{2}_{T}\big{[}\boldsymbol{R}_{\boldsymbol{\alpha}}\
\ \boldsymbol{R}_{\boldsymbol{\gamma}}\big{]}\begin{bmatrix}T\ &\
\displaystyle\sum_{t=1}^{T}\boldsymbol{y}_{2,T}^{\prime}\\\
\displaystyle\sum_{t=1}^{T}\boldsymbol{y}_{2,T}\ &\
\displaystyle\sum_{t=1}^{T}\boldsymbol{y}_{2,T}\boldsymbol{y}_{2,T}^{\prime}\end{bmatrix}^{-1}\begin{bmatrix}\boldsymbol{R}_{\boldsymbol{\alpha}}^{\prime}\\\
\\\
\boldsymbol{R}_{\boldsymbol{\gamma}}^{\prime}\end{bmatrix}\right\\}^{-1}\big{(}\boldsymbol{R}_{\boldsymbol{\alpha}}\hat{\boldsymbol{\alpha}}+\boldsymbol{R}_{\boldsymbol{\gamma}}\hat{\boldsymbol{\gamma}}-\boldsymbol{r}\big{)}/m$
(4.54)

where

$\displaystyle
s_{T}^{2}=\frac{1}{T-n}\sum_{t=1}^{T}\big{(}y_{1t}-\hat{\boldsymbol{\alpha}}_{T}-\boldsymbol{\gamma}_{T}^{\prime}\boldsymbol{y}_{2t}\big{)}^{2}.$
(4.55)

In other words, conditional on the vector
$\big{(}\boldsymbol{y}_{2,1},\boldsymbol{y}_{2,2},...,\boldsymbol{y}_{2,T}\big{)}$
it follows that the above expression when we replace with the population
variance, has a $\chi^{2}_{m}$ distribution.

Thus, conditional on the data
$\big{(}\boldsymbol{y}_{2,1},\boldsymbol{y}_{2,2},...,\boldsymbol{y}_{2,T}\big{)}$
the OLS F-test could be viewed as the ratio of a $\chi^{2}_{m}$ variable to
the independent $\chi^{2}_{T-p}$ variable $(T-p)\sigma_{T}^{2}/\sigma_{1}^{2}$
with numerator and denominator each divided by its degree of freedom.
Therefore, the OLS F test has an exact $F(m,T-p)$ distribution. Hence, despite
the $I(1)$ regressors and complications of cointegration, the correct approach
is to is to estimate the model by OLS and use standard $t$ or $F$ statistics
to test any hypotheses about the cointegrating vector.

####  4.4.2. Estimation of cointegrating vectors

We discuss estimation of $\beta$ with the following representation

$\displaystyle y_{t}=\beta^{\prime}x_{t}+e_{t}$ (4.56)

from the observable vector $z_{t}=\left(x_{t}^{\prime},y_{t}\right)^{\prime}$
introduced in the previous section. The setting is or cointegration we assume
that holds requires that the unobservable process $e_{t}$ is $I(d_{e})$,
$d_{e}<d_{y}$, while we assume that $\beta$ is identified for $t=1,...,n$

For a generic column vector or scalar sequence $a_{t}$, $t=1,...,n$ define the
discrete Fourier transform below

$\displaystyle w_{a}(\lambda)=\frac{1}{\left(2\pi
n\right)^{1/2}}\sum_{t=1}^{n}a_{t}e^{it\lambda},$ (4.57)

With also a column vector or scalar sequence $b_{t},t=1,...,n$ possibly
identical to $a_{t}$, define the cross periodogram as below

$\displaystyle
I_{\alpha\beta}=w_{\alpha}\left(\lambda\right)w_{\beta}\left(-\lambda\right).$
(4.58)

Now denote by $\lambda_{j}=2\pi j/n$, for integer $j$, the Fourier
frequencies, and define the averaged cross-periodogram. The case $m=\lfloor
n/2\rfloor$, where $\lfloor.\rfloor$ denotes the integer part, is of
particular interest, as we deduce that

$\displaystyle\widehat{F}_{ab}\left(\lfloor\frac{n}{2}\rfloor\right)=\frac{1}{2}\sum_{t=1}^{n}(a_{t}-\bar{a})(b_{t}-\bar{b})^{\prime},$
(4.59)

the mean-corrected sample covariance, with
$\bar{a}=n^{-1}\sum_{t=1}^{n}a_{t}$. We observe that $\widehat{F}_{ab}(m)$
represent the contributions from frequencies $[1,\lambda_{m}]$ to the sample
covariances above. We estimate $\beta$ by the frequency domain least squares
(FDLS) statistic given by

$\displaystyle\hat{\beta}_{m}=\widehat{F}_{xx}(m)^{-1}\widehat{F}_{xy}(m)$
(4.60)

####  4.4.3. Cointegrating Vector and Equilibrium Points

A typical cointegrating system has the following form

$\displaystyle y_{2t}=Ay_{2t-1}+u_{2t}$ (4.61)

and require the coefficient matrix $A$ to have stable roots, then the new
system is a conventional SEM. Furthermore, it is a triangular system since it
is written in a reduced form. In particular, when $\Sigma$ is not block-
diagonal, the MLE is obtained by using OLS on the augmented regression
equation such that

$\displaystyle y_{1t}=\beta^{\prime}y_{2t}+\gamma^{\prime}\Delta
y_{2t}+u_{1.2t}$ (4.62)

Consider $\widehat{\boldsymbol{\beta}}$ to be the single equation OLS
estimator of $\boldsymbol{\beta}$. Then, it holds that

$\displaystyle
T\left(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right)\Rightarrow\left(\int_{0}^{1}S_{2}S_{2}^{\prime}\right)^{-1}\left(\int_{0}^{1}S_{2}^{\prime}dS_{1}+\sigma_{21}\right).$
(4.63)

Then, consider the fully modified OLS estimator given as below

$\displaystyle\boldsymbol{\beta}^{**}=\big{(}Y_{2}^{\prime}Y_{2}\big{)}^{-1}\big{(}Y_{2}^{\prime}y_{1}-T\widehat{\sigma}_{21}\big{)}$
(4.64)

whose asymptotics are as below

$\displaystyle
T\left(\boldsymbol{\beta}^{**}-\beta\right)\Rightarrow\left(\int_{0}^{1}S_{2}S_{2}^{\prime}\right)^{-1}\left(\int_{0}^{1}S_{2}^{\prime}dS_{1}\right).$
(4.65)

Then, the further modification for the endogeneity of $y_{2t}$ is required to
remove the correlation between the Brownian motion $S_{1}$ and $S_{2}$.

Therefore, this is achieved by constructing

$\displaystyle
y_{1t}^{+}=y_{1t}-\widehat{\sigma}^{\prime}_{21}\widehat{\Sigma}_{22}^{-1}\Delta
y_{2t},$ (4.66)

Then, the fully modified OLS estimator employs both the serial correlation and
endogeneity corrections and is given by the following expression

$\displaystyle\boldsymbol{\beta}^{+}=\big{(}Y_{2}^{\prime}Y_{2}\big{)}^{-1}\big{(}Y_{2}^{\prime}y_{1}^{+}-T\widehat{\delta}^{+}\big{)}$
(4.67)

Therefore, with these corrections the new estimator $\boldsymbol{\beta}^{+}$
has the same asymptotic behaviour as the full system MLE. Observed that it is
a two-step estimator, and relies on the preliminary construction of
$y_{1t}^{+}$ and $\widehat{\delta}^{+}$. Then, fully modified test statistics
are based on $\boldsymbol{\beta}^{+}$ which can be constructed in the usual
way. Therefore, we define the t-ratios as below

$\displaystyle t_{i}^{+}=\left(\beta_{i}^{+}-\beta_{i}\right)/s_{i}^{+}.$
(4.68)

###  4.5. Unit Roots, Cointegration and Structural Breaks

######  Example 16.

Consider that $y_{t}$ is generated by the following model

$\displaystyle y_{t}=\mu_{0}+\mu_{1}t+\alpha y_{t-1}+\beta(L)\Delta
y_{t-1}+\varepsilon_{t},\ \ \ t=1,...,T$ (4.69)

where $\beta(L)$ is the lag polynomial of known order $p$ with the roots of
$1-\beta(L)L$ outside the unit circle. Under the null hypothesis, it holds
that $\alpha=1$ and $\mu_{1}=0$. The above example can be estimated using the
OLS approach without restrictions on $\mu_{0},\mu_{1}$ or $\alpha$, when
$y_{t}$ is regressed on $1,t,y_{t-1},\Delta y_{t-1},...,\Delta y_{t-p}$, the
test statistic for $\alpha=1$, is the standard Dickey-Fuller test for a unit
root against a trend-stationary alternative. We consider the asymptotic
representations for rolling estimators and test statistics, however unlike
rolling coefficient estimators (in expectation) is constant through the
sample.

Thus, the rolling estimator $\bar{\theta}$ is

$\displaystyle\tilde{\theta}(\delta;\delta_{0})=\left(\sum_{t=\lfloor
n(\delta-\delta_{0})\rfloor+1}^{\lfloor
n\delta\rfloor}Z_{t-1}Z_{t-1}^{\prime}\right)^{-1}\left(\sum_{t=\lfloor
n(\delta-\delta_{0})\rfloor+1}^{\lfloor n\delta\rfloor}Z_{t-1}y_{t}\right)$
(4.70)

The estimators and the test statistics are computed using the full $n$
observations for $k\in\left\\{k_{0},k_{0}+1,...,T-k_{0}\right\\}$, where
$k_{0}=\lfloor T\delta_{0}\rfloor$. Then, the stochastic processes constructed
from the sequential estimators and Wald test statistic are as below.

######  Remark 13.

An overview of the use of partial sum processes for break detection is studied
by Katsouris, 2022b (see, also Katsouris, 2023b ). Moreover, Xiao,  (2001)
considers a test statistic for the null hypothesis of stationarity against an
autoregressive unit root alternative. In particular, the author employs
partial-sums of residuals based on the underline econometric specification.
Furthermore, a bootstrap-based approach for detecting multiple breakpoints is
examined by Kejriwal et al.,  (2020).

For $\delta_{0}\leq\delta\leq 1-\delta_{0}$,

$\displaystyle\tilde{\theta}(\delta)$
$\displaystyle=\left(\sum_{t=1}^{n}Z_{t-1}\left(\lfloor
n\delta\rfloor\right)Z_{t-1}\left(\lfloor
n\delta\rfloor\right)^{\prime}\right)^{-1}\left(\sum_{t=1}^{n}Z_{t-1}(\lfloor
n\delta\rfloor)y_{t}\right)$ (4.71)
$\displaystyle\mathcal{Y}_{n}\left(\tilde{\theta}(\delta)-\theta\right)$
$\displaystyle=\Gamma_{n}\left(\delta\right)^{-1}\Psi(\delta),$ (4.72)

and

$\displaystyle\tilde{F}_{n}(\delta)=\big{[}R\tilde{\theta}(\delta)-r\big{]}^{\prime}\left[R\left(\sum_{t=1}^{n}Z_{t-1}\left(\lfloor
n\delta\rfloor\right)Z_{t-1}\left(\lfloor
n\delta\rfloor\right)^{\prime}\right)^{-1}R^{\prime}\right]^{-1}\big{[}R\tilde{\theta}(\delta)-r\big{]}/q\tilde{\sigma}^{2}(\delta)$
(4.73)

where

$\displaystyle\Gamma_{n}(\delta)=\mathcal{Y}_{n}^{-1}\left(\sum_{t=1}^{n}Z_{t-1}\left(\lfloor
n\delta\rfloor\right)Z_{t-1}\left(\lfloor
n\delta\rfloor\right)^{\prime}\right)\mathcal{Y}_{n}^{-1}$ (4.74)

####  4.5.1. Testing for Structural Change in Cointegrated Systems

Following the framework of Seo,  (1998), and by the invariance principle
proposed by Phillips and Durlauf,  (1986), it holds that

$\displaystyle n^{-1/2}\sum_{t=1}^{\lfloor ns\rfloor}u_{t}$
$\displaystyle\Rightarrow W(s)\equiv BM(\Sigma)$ (4.75) $\displaystyle
n^{-1/2}x_{\lfloor ns\rfloor}$ $\displaystyle\Rightarrow C(1)W(s)$ (4.76)

Therefore, we need to show that

$\displaystyle\mathbb{P}\left(\underset{s\in[0,1]}{\text{sup}}n^{-1/2}\left|x_{\lfloor
ns\rfloor}-C(1)\sum_{t=1}^{\lfloor
ns\rfloor}u_{t}\right|>\epsilon\right)\leq\mathbb{P}\left(\underset{s\in[0,1]}{\text{sup}}n^{-1/2}\left|\Phi(L)u_{\lfloor
ns\rfloor}\right|>\epsilon\right)\to 0.$ (4.77)

For instance, if $\left\\{\Phi(L)u_{t}\right\\}$ is uniformly square
integrable, we can then apply the following result.

$\displaystyle n^{-1/2}\sum_{t=1}^{\lfloor
ns\rfloor}w_{t}\Rightarrow\gamma^{\prime}\Phi(1)W(s).$ (4.78)

Therefore, we need to show that

$\displaystyle\mathbb{P}\left(\underset{s\in[0,1]}{\text{sup}}n^{-1/2}\left|w_{t}-\gamma^{\prime}\Phi(1)\sum_{t=1}^{\lfloor
ns\rfloor}u_{t}\right|>\epsilon\right)\leq\mathbb{P}\left(\underset{s\in[0,1]}{\text{sup}}n^{-1/2}\left|\gamma^{\prime}\Phi_{1}(L)u_{t}\right|>\epsilon\right)\to
0$ (4.79)

where $\Phi_{1}(L)=\left(\Phi(L)-\Phi(1)\right)/(1-L)$.

Moreover, a key ingredient for deriving the limiting distribution of the Wald-
type and LR-type statistics is consider the weak convergence of partial-sum
functionals. We define weak convergence on the space $C\left([0,1]\right)$
with respect to the uniform metric. Thus, we define weak convergence of the
projected sequence

$\displaystyle R_{12t}(\lambda)=x_{2t-1}-\left(\sum_{t=1}^{\lfloor\lambda
n\rfloor}x_{2t-1}z_{t-1}^{\prime}\right)\cdot\left(\sum_{t=1}^{\lfloor\lambda
n\rfloor}z_{t-1}z_{t-1}^{\prime}\right)^{-1}\cdot z_{t-1}$ (4.80)

Related asymptotic moments for the particular framework are as below:

$\displaystyle\frac{1}{n}\sum_{t=1}^{\lfloor\lambda
n\rfloor}R_{12}(\lambda)u_{t}^{\prime}\Rightarrow\int_{0}^{\lambda}W_{2}(\lambda)dW^{\prime}(s),$
(4.81) $\displaystyle\frac{1}{n^{2}}\sum_{t=1}^{\lfloor\lambda
n\rfloor}R_{12t}(\lambda)R_{12t}(\lambda)^{\prime}\Rightarrow\int_{0}^{\lambda}W_{2}(\lambda)W_{2}^{\prime}(s)ds.$
(4.82)

Therefore, under the null hypothesis of no structural change, the model can be
estimated by existing methods. Then, it holds that

$\displaystyle F(\lambda)^{b}$ $\displaystyle=F(\lambda)-V(\lambda)V(1)F(1),$
$\displaystyle F(\lambda)^{b}$
$\displaystyle=\int_{0}^{\lambda}B_{2}(s,\lambda)B_{2}(s,\lambda)^{\prime}ds,$
$\displaystyle B_{2}(s,\lambda)$
$\displaystyle=B_{2}(s)-\frac{1}{\lambda}B_{2}(s)ds.$

Notice that the expression $\boldsymbol{B}(s)=BM(\boldsymbol{\Omega})$
represents a Brownian motion with long-run variance $\boldsymbol{\Omega}$. Due
to uniformly integrability, the following limit result holds

$\displaystyle\frac{1}{n}\sum_{t=1}^{\lfloor\lambda
n\rfloor}x_{t}w_{t}^{\prime}=\frac{1}{n}\sum_{t=1}^{\lfloor\lambda
n\rfloor}\big{(}x_{t-1}+\Delta
x_{t}\big{)}w_{t}^{\prime}\Rightarrow\int_{0}^{\lambda}C(1)WdW^{\prime}\Phi(1)^{\prime}\gamma+\lambda\Lambda_{1}$
(4.83)

By Lemma 9 of Seo,  (1998), it holds that

$\displaystyle\frac{1}{n}\sum_{t=1}^{\lfloor\lambda
n\rfloor}R_{2t}(\lambda)R_{2t}(\lambda)^{\prime}$
$\displaystyle=\left(\frac{1}{n}\sum_{t=1}^{\lfloor\lambda
n\rfloor}w_{t-1}w_{t-1}^{\prime}\right)-\left(\frac{1}{n}\sum_{t=1}^{\lfloor\lambda
n\rfloor}w_{t-1}z_{t-1}^{\prime}\right)=\left(\frac{1}{n}\sum_{t=1}^{\lfloor\lambda
n\rfloor}z_{t-1}z_{t-1}^{\prime}\right)^{-1}\left(\frac{1}{n}\sum_{t=1}^{\lfloor\lambda
n\rfloor}z_{t-1}w_{t-1}^{\prime}\right)$
$\displaystyle\overset{p}{\to}\lambda\mathbb{E}\big{(}w_{0}w_{0}^{\prime}\big{)}-\lambda\mathbb{E}\big{(}w_{0}z_{0}^{\prime}\big{)}\mathbb{E}\big{(}z_{0}z_{0}^{\prime}\big{)}^{-1}\big{(}z_{0}w_{0}^{\prime}\big{)}\equiv\lambda\boldsymbol{Q}$

uniformly in $\uptau\in\mathcal{T}$.

Thus, to derive the limit distribution under the null hypothesis,
$\frac{1}{n}\sum_{t=1}^{\lfloor\lambda n\rfloor}z_{t}z_{t}^{\prime}\lambda Q$,
uniformly in $\lambda\in[0,1]$, for Q some positive definite matrix and for
all $t$ and $n^{-1/2}\sum_{t=1}^{\lfloor\lambda n\rfloor}z_{t}u_{t}\Rightarrow
sQ^{-1/2}W_{q}(s)$, where $W_{q}(s)$ is a $q-$vector of independent Wiener
processes. This imposes a form of moment homogeneity, that is, a homogeneous
distribution throughout the sample. The particular assumption holds only under
the null of no structural break and is a necessary condition for deriving the
limiting distributions of the test statistics.

######  Remark 14.

According to Seo,  (1998), even though $F(\lambda)$ is distributed as mixed
normal with covariance matrix, it follows that $F(\lambda)^{b}$ is not a
Brownian bridge, as defined in the stationary case, although it is still tied
down. Hence, the asymptotic distributions here are different from those found
by Andrews,  (1993). Specifically, because the distribution of
$LM_{1}^{\beta}$ is a $\chi^{2}$ for a known $\lambda$, these tests for
structural change of the cointegrating vector are standard only if we know the
change point. Therefore, the distribution of the tests depends only on the
number of parameters and the admissible range of the break point.

####  4.5.2. Trend Stationarity and Structural Break Testing

Consider the following model for the scalar random variable $x_{t}$ such that

$\displaystyle
x_{t}=\beta_{1}+\beta_{2}t+\beta_{3}DT_{t}\left(\tau^{*}\right)+e_{t},\ \ \
t=1,...,T$ (4.84)

The shocks, $e_{t}$, are assumed to follow a zero mean fractionally integrated
process of order $d$, denoted by $e_{t}\in I(d)$. Furthermore, the
deterministic trend break term, $DT_{t}\left(\uptau^{*}\right)$, is defined
for a generic $\uptau$ such that
$DT_{t}\left(\uptau\right):=\left(t-\lfloor\uptau
T\rfloor\right)\boldsymbol{1}\left\\{t\geq\lfloor\uptau T\rfloor\right\\}$.
Therefore, where a true trend break occurs, which implies that we are under
the alternative hypothesis and so $\beta_{3}\neq 0$, we assume that the true
break fraction is such that
$\uptau^{*}\in[\uptau_{U},\uptau_{L}]=:\Lambda\subset[0,1]$, where the
quantiles $\uptau_{L}$ and $\uptau_{U}$ are trimming parameters. The
formulation of the model is obtained using the vectors such that
$z_{t}(\tau):=\left(1,t,DT_{t}(\uptau)\right)^{\prime}$ and
$\beta:=\left(\beta_{1},\beta_{2},\beta_{3}\right)^{\prime}$. Therefore, we
can obtain the OLS estimate of $\beta$ as below

$\displaystyle\widehat{\beta}(\uptau):=\left(\sum_{t=j}^{T}z_{t}(\uptau)z_{t}(\uptau)^{\prime}\right)^{-1}\left(\sum_{t=j}^{T}z_{t}(\uptau)y_{t}\right),$
(4.85)

Moreover, we define the corresponding de-trended residuals as below

$\displaystyle\widehat{u}_{t}(\uptau):=y_{t}-z_{t}(\uptau)^{\prime}\widehat{\beta}(\uptau)$
(4.86)

where the particular sequence is obtained for $t=1,...,T$.

Therefore, under the null hypothesis we have that the estimated $\eta_{t}$ are
obtained by taking the corresponding fractional differences of these OLS de-
trended residuals. In other words, if the true break fraction $\uptau^{*}$,
was known then one would simply evaluate the Likelihood ratio for break
testing, denoted with $LM(\uptau)$ at the location $\uptau=\uptau^{*}$, and
thus the resulting test statistic is denoted by $LM(\uptau^{*})$. However,
most of the empirical applications in the literature do not have a priori
information regarding the exact location of the break-point location within
the full sample. Our focus, is on the case where $\uptau^{*}$ is unknown and
so, our proposed test will be based on evaluating $LM(\uptau)$ at
$\hat{\uptau}$, which is obtained as the minimum RSS estimate

$\displaystyle\hat{\uptau}:=\underset{\uptau\in\Lambda}{\mathsf{argmin}}\sum_{t=1}^{T}\big{(}\widehat{u}_{t}(\uptau)\big{)}^{2},$
(4.87)

whose exact form is determined according the exact value of $d_{0}$ being
tested under the null hypothesis.

Specifically if $d_{0}$ lies in the region $(-0.5,0.5)$ then we estimate
$\uptau^{*}$ using the levels of the data and testing the null hypothesis that
the long memory parameter in the levels data is $d_{0}$, whereas if $d_{0}$
lies in the range $(0.5,1.5)$ we instead estimate $\uptau^{*}$ using the first
differences of the data and test the null hypothesis that the long memory
parameter in the first difference data is $(d_{0}-1)$ (see, Arai and Kurozumi,
(2007)).

Therefore, in order to be able to determine the limiting distribution of the
test statistic, we will need to determine the large sample behaviour of LM
evaluated at the estimated break-point location, denoted by
$LM(\hat{\uptau})$, which is done by comparing it to the infeasible LM
statistic. In other words, the large sample behaviour of $LM(\hat{\uptau})$
clearly depends on the large sample properties of the estimates $\hat{\uptau}$
and $\widehat{\beta}(\widehat{\uptau})$ evaluated at
$\uptau=\widehat{\uptau}$. Now assume that the underline data generating
process has a break under the alternative hypothesis, which implies that we
can define the Pitman drift such that $H_{c}:\theta:=\theta_{T}=c/\sqrt{T}$.

Then, for $d_{0}\in\left(-0.5,0.5\right)$, define a generic element $\alpha$
and the diagonal matrix such that

$\displaystyle
K_{T}(\alpha):=\mathsf{diag}\big{\\{}T^{\frac{3}{2}-\alpha},T^{\frac{3}{2}-\alpha}\big{\\}}.$
(4.88)

######  Remark 15.

Therefore, a consequence of Theorem 1 is that, $LM(\hat{\uptau})-LM=o_{p}(1)$
irrespective of whether $\beta_{3}\neq 0$ or $\beta_{3}=0$. Conseqenently,
regardless of the value of $\beta_{3}$, the LM converges in distribution to a
$\chi^{2}-$random variate with centrality parameter

$\displaystyle
LM(\widehat{\uptau})\overset{d}{\to}\chi_{1}^{2}(c^{2}\omega^{2})$ (4.89)

which implies that the test statistic retains asymptotic optimality. Moreover,
since $LM(\widehat{\uptau})\overset{d}{\to}\chi^{2}_{1}$ under $H_{0}$,
standard critical values can still be easily obtained.

######  Remark 16.

Theorem 1 is based on establishing that the difference between the LM-type
statistics based on $\widehat{\varepsilon}_{t}$ and
$\widehat{\varepsilon}_{t}(\widehat{\tau})$ is asymptotically negligible. A
key part of the derivation of the theorem is proving that
$\widehat{A}-\widehat{A}(\widehat{\tau})=o_{p}(T^{-1/2})$ and since the
difference between the LM-type test statistic based on
$\widehat{\varepsilon}_{t}$ and $\widehat{\varepsilon}_{t}(\widehat{\uptau})$
crucially depends on the term
$\Delta_{+}^{d_{0}}\big{(}DT_{t}(\widehat{\uptau})-DT_{t}(\uptau^{*})\big{)}$,
on showing that

$\displaystyle\sum_{t=1}^{T}\left(\sum_{j=1}^{t-1}j^{-1}\Delta_{+}^{d_{0}}\big{(}DT_{t}(\widehat{\uptau})-DT_{t}(\uptau^{*})\big{)}\right)\widehat{\varepsilon}_{t}=o_{p}(T^{1/2})$
(4.90)

Under the alternative hypothesis $H_{c}$, by setting $r=0$ and
$\alpha=\delta_{0}$, the FCLT holds such that

$\displaystyle T^{-\left(\frac{1}{2}+\delta_{0}\right)}\sum_{t=1}^{\lfloor\tau
T\rfloor}u_{t}\Rightarrow\sigma_{\infty}W(\uptau;\delta_{0}).$ (4.91)

######  Remark 17.

More typically in applied work, it is natural that the break-point $\uptau$ is
thought to be unknown. Therefore, in this case the testing procedure is
nonstandard because a nuisance parameter $\uptau$ appears only under the
alternative hypothesis. Therefore, testing problems in which the nuisance
parameter appears only under the alternative hypothesis require the
development of tests with specific optimality properties.

####  4.5.3. Cointegration Testing under Structural Breaks

Further to the two previous examples, in this section we consider
cointegration testing under structural breaks (see, Campos et al.,  (1996),
Johansen et al.,  (2000) and Kasparis and Phillips,  (2012)). In order to test
for structural breaks in cointegration models a residual-based test can be
derived from single equation models. However, the system equation approach
appears limited when a structural break in cointegrating vectors is
incorporated in the conventional specification form due to the presence of
parameter instability (see, Saikkonen et al.,  (2006)). The literature
proposes two different estimation and testing approaches, the first approach
consists of the test for the null of cointegration while the second approach
considers the test for the null of no cointegration (see, Wagner,  (2022)).
Specifically, the authors consider the Lagrange Multiplier (LM) test and
derive its limiting distribution which is free of nuisance parameter
dependencies except for the number of $I(1)$ regressors (degrees of freedom)
and the location of the structural break.

To derive the asymptotic theory of the test statistic we derive an invariance
principle and a corresponding functional central limit theorem to the partial-
sum process $n^{-1/2}\sum_{t=1}^{\lfloor nr\rfloor}v_{t}$, which weakly
converges to a two-dimensional standard Brownian motion
$B(r)=\left(B_{1}(r),B_{2}(r)\right)^{\prime}$. In particular, the test
proposed by Shin,  (1994) is constructed based on the regression residual of
$y_{1t}$ on 1 and $y_{2t}$, denoted by $\hat{v}_{1t}$.

Thus, the test statistic is given by

$\displaystyle V_{n}=n^{-2}\frac{1}{\hat{\sigma}^{2}}\sum_{t=1}^{n}S_{t}^{2},\
\ \ \ S_{t}=\sum_{j=1}^{t}\hat{v}_{1j}\ \ \ \text{and}\ \ \
\hat{\sigma}^{2}=n^{-1}\sum_{t=1}^{n}\hat{v}^{2}_{1t}.$ (4.92)

We show that $V_{n}$ diverges to infinity but $V_{n}/n$ weakly converges to a
distribution that is positive almost surely. Denote with
$x_{t}=\left(1,y_{2t}\right)^{\prime}$ and $e_{t}=v_{1t}+\mu_{2}d_{2t}$ such
that $y_{1t}=\left(\mu_{1},\beta\right)x_{t}+e_{t}$. The residual
$\hat{v}_{1t}$ is

$\displaystyle\hat{v}_{1t}=e_{t}-\left(\sum_{t=1}^{n}e_{t}x_{t}^{\prime}\right)\left(\sum_{t=1}^{n}x_{t}x_{t}^{\prime}\right)^{-1}x_{t}.$
(4.93)

The following weak convergence results hold

$\displaystyle\boldsymbol{D}_{n}^{-1}\left(\sum_{t=1}^{n}\boldsymbol{x}_{t}\boldsymbol{x}_{t}^{\prime}\right)$
$\displaystyle\Rightarrow\int_{0}^{1}\boldsymbol{X}(s)\boldsymbol{X}(s)^{\prime}ds$
(4.94) $\displaystyle
n^{-1/2}\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{n}\boldsymbol{x}_{t}e_{t}$
$\displaystyle\Rightarrow\mu_{2}\int_{0}^{1}\boldsymbol{X}(s)d_{2}(s)ds$
(4.95)

Furthermore, it holds that

$\displaystyle n^{-1}\sum_{t=1}^{\lfloor
nr\rfloor}e_{t}\Rightarrow\mu_{2}\int_{0}^{r}d_{2}(s)ds,\ \ \
n^{-1/2}\boldsymbol{D}_{n}^{-1}\sum_{t=1}^{\lfloor
nr\rfloor}\boldsymbol{x}_{t}\Rightarrow\int_{0}^{r}\boldsymbol{X}(s)ds.$
(4.96)

where the normalization matrix is given by
$\boldsymbol{D}_{n}=\mathsf{diag}\left(n^{1/2},n\right)$,
$\boldsymbol{X}(s)=\left[1,B_{2}(s)\right]^{\prime}$.

Moreover, $d_{2}(s)$ is a step-function such that, $d_{2}(s)=1$ if $s>0.5$ and
$0$ otherwise. Then, it holds that

$\displaystyle
n^{-3}\sum_{t=1}^{n}S_{t}^{2}\Rightarrow\mu_{2}^{2}\int_{0}^{1}\left\\{\int_{0}^{r}\left[d_{2}(s)-\left(\int_{0}^{1}\boldsymbol{X}(u)d_{2}(u)du\right)^{\prime}\times\left(\int_{0}^{1}\boldsymbol{X}(u)\boldsymbol{X}(u)^{\prime}du\right)^{-1}\boldsymbol{X}(s)\right]ds\right\\}^{2}dr.$

and the variance under the null hypothesis is estimated to be

$\displaystyle\hat{\sigma}^{2}\Rightarrow
1+\mu_{2}^{2}\left\\{\int_{0}^{1}d_{2}(s)ds-\left(\int_{0}^{1}\boldsymbol{X}(s)d_{2}ds\right)^{-1}\times\left(\int_{0}^{1}X(s)X(s)^{\prime}ds\right)^{-1}\int_{0}^{1}X(s)d_{2}(s)ds\right\\}.$
(4.97)

####  4.5.4. Testing for a Structural Break in Cointegrating Regression

Consider the following predictive regression model

$\displaystyle y_{t}$
$\displaystyle=\beta_{0}+\boldsymbol{\beta}^{\prime}_{1}\boldsymbol{x}_{t}+\epsilon_{t},$
(4.98) $\displaystyle\boldsymbol{x}_{t}$
$\displaystyle=\boldsymbol{R}_{T}\boldsymbol{x}_{t-1}+\boldsymbol{\eta}_{t}$
(4.99)

where $y_{t}$ is a scalar vector and $\boldsymbol{x}_{t}\in\mathbb{R}^{p\times
1}$ is $p-$dimensional regressor vector. Furthermore, consider the case in
which $\boldsymbol{\Phi}_{T}=\boldsymbol{I}_{p}$. Define the vector
$\boldsymbol{u}_{t}=\big{(}\epsilon_{t},\boldsymbol{\eta}_{t}^{\prime}\big{)}^{\prime}$
and the following long-run covariance matrices

$\displaystyle\boldsymbol{\Omega}_{T}:=\underset{T\to\infty}{\mathsf{lim}}\frac{1}{T}\sum_{t=1}^{T}\sum_{j=1}^{T}\mathbb{E}\big{[}\boldsymbol{u}_{j}\boldsymbol{u}_{t}^{\prime}\big{]}=\begin{pmatrix}\displaystyle\underset{T\to\infty}{\mathsf{lim}}\frac{1}{T}\sum_{t=1}^{T}\sum_{j=1}^{T}\mathbb{E}\big{[}\epsilon_{j}^{\prime}\epsilon_{t}\big{]}\
&\
\displaystyle\underset{T\to\infty}{\mathsf{lim}}\frac{1}{T}\sum_{t=1}^{T}\sum_{j=1}^{T}\mathbb{E}\big{[}\epsilon_{j}^{\prime}\boldsymbol{\eta}_{t}\big{]}\\\
\\\
\displaystyle\underset{T\to\infty}{\mathsf{lim}}\frac{1}{T}\sum_{t=1}^{T}\sum_{j=1}^{T}\mathbb{E}\big{[}\boldsymbol{\eta}_{j}^{\prime}\epsilon_{t}\big{]}\
&\
\displaystyle\underset{T\to\infty}{\mathsf{lim}}\frac{1}{T}\sum_{t=1}^{T}\sum_{j=1}^{T}\mathbb{E}\big{[}\boldsymbol{\eta}_{j}^{\prime}\boldsymbol{\eta}_{t}\big{]}\end{pmatrix}$

such that

$\displaystyle\boldsymbol{\Omega}_{T}\overset{d}{\to}\boldsymbol{\Omega}:=\begin{pmatrix}\boldsymbol{\Omega}_{\epsilon\epsilon}\
&\ \boldsymbol{\Omega}_{\epsilon\eta}\\\ \\\
\boldsymbol{\Omega}_{\eta\epsilon}\ &\
\boldsymbol{\Omega}_{\eta\eta}\end{pmatrix}$ (4.100)

Similarly, we define

$\displaystyle\boldsymbol{\Lambda}_{T}:=\underset{T\to\infty}{\mathsf{lim}}\frac{1}{T}\sum_{t=1}^{T}\sum_{j=1}^{t}\mathbb{E}\big{[}\boldsymbol{u}_{j}\boldsymbol{u}_{t}^{\prime}\big{]}=\begin{pmatrix}\displaystyle\underset{T\to\infty}{\mathsf{lim}}\frac{1}{T}\sum_{t=1}^{T}\sum_{j=1}^{t}\mathbb{E}\big{[}\epsilon_{j}^{\prime}\epsilon_{t}\big{]}\
&\
\displaystyle\underset{T\to\infty}{\mathsf{lim}}\frac{1}{T}\sum_{t=1}^{T}\sum_{j=1}^{t}\mathbb{E}\big{[}\epsilon_{j}^{\prime}\boldsymbol{\eta}_{t}\big{]}\\\
\\\
\displaystyle\underset{T\to\infty}{\mathsf{lim}}\frac{1}{T}\sum_{t=1}^{T}\sum_{j=1}^{t}\mathbb{E}\big{[}\boldsymbol{\eta}_{j}^{\prime}\epsilon_{t}\big{]}\
&\
\displaystyle\underset{T\to\infty}{\mathsf{lim}}\frac{1}{T}\sum_{t=1}^{T}\sum_{j=1}^{t}\mathbb{E}\big{[}\boldsymbol{\eta}_{j}^{\prime}\boldsymbol{\eta}_{t}\big{]}\end{pmatrix}$

$\displaystyle\boldsymbol{\Lambda}_{T}\overset{d}{\to}\boldsymbol{\Lambda}:=\begin{pmatrix}\boldsymbol{\Lambda}_{\epsilon\epsilon}\
&\ \boldsymbol{\Lambda}_{\epsilon\eta}\\\ \\\
\boldsymbol{\Lambda}_{\eta\epsilon}\ &\
\boldsymbol{\Lambda}_{\eta\eta}\end{pmatrix}$ (4.101)

Furthermore, we define with
$\boldsymbol{\Omega}_{\epsilon\eta}=\boldsymbol{\Omega}_{\epsilon\epsilon}-\boldsymbol{\Omega}_{\epsilon\eta}\boldsymbol{\Omega}_{\eta\eta}^{-1}\boldsymbol{\Omega}_{\eta\epsilon}$
and
$\boldsymbol{\Lambda}_{\eta\epsilon}^{+}=\boldsymbol{\Lambda}_{\eta\epsilon}-\boldsymbol{\Omega}_{\eta\eta}\boldsymbol{\Lambda}_{\eta\eta}^{-1}\boldsymbol{\Lambda}_{\eta\epsilon}$.
Then, we estimate the cointegrating regression model via OLS and obtain the
residuals
$\hat{\epsilon}=y_{t}-\boldsymbol{\beta}^{\prime}\boldsymbol{z}_{t}$, where
$\boldsymbol{z}_{t}=\big{(}1,\boldsymbol{x}_{t}^{\prime}\big{)}^{\prime}$ and
$\boldsymbol{\beta}=\big{(}\beta_{0},\boldsymbol{\beta}_{1}^{\prime}\big{)}$.
Moreover, we define the following vector of residuals

$\displaystyle\hat{\boldsymbol{u}}_{t}=\big{(}\hat{\epsilon}_{t},\left(\Delta
z_{t}-\Delta\bar{z}\right)^{\prime}\big{)}^{\prime},$ (4.102)

Notice that $\Delta\bar{z}$ represents the sample mean of $\Delta z$.
Furthermore, we denote with $\hat{u}_{t}$ the estimated vector of residuals as
well as with $\hat{\boldsymbol{\Omega}}$ and $\hat{\boldsymbol{\Lambda}}$ the
corresponding estimators of the long-run covariance matrices. We also define
the transformed dependent variable as

$\displaystyle
y_{t}^{+}:=y_{t}-\hat{\boldsymbol{\Omega}}_{\epsilon\eta}\hat{\boldsymbol{\Omega}}_{\eta\eta}^{-1}\big{(}\Delta
z_{t}-\Delta\bar{z}\big{)}.$ (4.103)

Therefore, we obtain the fully modified (FM) estimator as below

$\displaystyle\hat{\boldsymbol{\beta}}^{+}$
$\displaystyle=\left(\sum_{t=1}^{T}\boldsymbol{z}_{t}\boldsymbol{z}_{t}^{\prime}\right)^{-1}\left(\sum_{t=1}^{T}y_{t}^{+}\boldsymbol{z}_{t}^{\prime}-\big{[}0\
\ \hat{\boldsymbol{\Lambda}}^{+\prime}_{\eta\epsilon}\big{]}\right)$ (4.104)
$\displaystyle\hat{\boldsymbol{\beta}}^{+}$
$\displaystyle=\left(\begin{bmatrix}\displaystyle\sum_{t=1}^{T}1\ \ &\ \
\displaystyle\sum_{t=1}^{T}\boldsymbol{x}_{t}\\\ \\\
\displaystyle\sum_{t=1}^{T}\boldsymbol{x}_{t}^{\prime}\ \ &\ \
\displaystyle\sum_{t=1}^{T}\boldsymbol{x}_{t}^{\prime}\boldsymbol{x}_{t}\end{bmatrix}^{-1}\right)\left(\sum_{t=1}^{T}y_{t}^{+}\boldsymbol{z}_{t}^{\prime}-\big{[}0\
\ \hat{\boldsymbol{\Lambda}}^{+\prime}_{\eta\epsilon}\big{]}\right)$ (4.105)

The associated residual vector is given by
$\hat{\epsilon}^{+}=y_{t}^{+}-\boldsymbol{\beta}^{+}\boldsymbol{x}_{t}$.
Furthermore, we define the following

$\displaystyle\hat{s}_{t}=\left[\boldsymbol{x}_{t}\hat{\epsilon}^{+}-\begin{bmatrix}0\\\
\hat{\boldsymbol{\Lambda}}^{+\prime}_{\eta\epsilon}\end{bmatrix}\right].$
(4.106)

Define the following test statistic

$\displaystyle\boldsymbol{F}_{k}=\boldsymbol{S}_{Tk}^{\prime}\left[\hat{\Omega}_{\epsilon.\eta}\boldsymbol{V}_{Tk}\right]^{-1}\boldsymbol{S}_{Tk}$
(4.107) $\displaystyle\boldsymbol{S}_{Tk}=\sum_{t=1}^{k}\hat{s}_{t},\ \ \
\boldsymbol{V}_{Tk}=\big{(}\boldsymbol{M}_{Tk}-\boldsymbol{M}_{Tk}\boldsymbol{M}_{TT}^{-1}\boldsymbol{M}_{Tk}\big{)},\
\ \
\boldsymbol{M}_{Tk}=\sum_{t=1}^{k}\boldsymbol{x}_{t}\boldsymbol{x}_{t}^{\prime}$
(4.108)

The F-test is a test for a structural break at a known point which is
asymptotically distributed under the null hypothesis as a
$\chi^{2}-$distributed random variable with $p$ degree of freedoms. However,
when the break-point is unknown, then the test is
$F_{\mathsf{sup}}=\mathsf{sup}_{k\in\mathcal{B}}F_{k}$, where
$\mathcal{B}\in(0,1)$. Moreover, a framework for structural break detection in
predictive regression models is proposed by Katsouris, 2023b ; Katsouris,
2023d .

####  4.5.5. Unit root and Cointegrating limit theory

######  Example 17.

Consider the following data generating process

$\displaystyle y_{t}$ $\displaystyle=\mu+x_{t}^{\prime}\beta+u_{t}$ (4.109)
$\displaystyle x_{t}$ $\displaystyle=x_{t-1}+v_{t}.$ (4.110)

Stacking the error process defines $\eta_{t}=\left[u_{t},\
v_{t}^{\prime}\right]^{\prime}$. Furthermore, it is assumed that $\eta_{t}$ is
a vector of $I(0)$ processes in which case $x_{t}$ is a non-cointegrating
vector of $I(1)$ processes and there exists a cointegrating relationship among
$[y_{t},x_{t}^{\prime}]^{\prime}$ with cointegrating vector
$[1,-\beta^{\prime}]^{\prime}$. To review existing theory and to obtain the
key theoretical results in the paper, assumptions about $\eta_{t}$ are
required. It is sufficient to assume that $\eta_{t}$ satisfies a functional
central limit theorem (FCLT) of the form given below

$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}\eta_{t}\Rightarrow B(r)\Rightarrow\Omega^{1/2}W(r),\ \ \
r\in[0,1],$ (4.111)

Define the partial sum process such that

$\displaystyle\widehat{S}_{t}=\sum_{j=1}^{t}\widehat{\eta}_{t}$ (4.112)

We start by establishing functional central theorems for

$\displaystyle\frac{1}{T}\sum_{t=1}^{\lfloor nr\rfloor}\widehat{u}_{t}$
$\displaystyle=\frac{1}{\sqrt{T}}\sum_{t=1}^{\lfloor
nr\rfloor}u_{t}-\frac{\lfloor
nr\rfloor}{n}n^{1/2}\left(\widehat{\mu}-\mu\right)-\frac{1}{n\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}x_{t}^{\prime}n\left(\widehat{\beta}-\beta\right)$
$\displaystyle\Rightarrow\int_{0}^{r}dB_{u}(s)-r\int_{0}^{r}B_{v}^{*}(s)^{\prime}ds\Theta.$

Using the definition of
$\widehat{\eta}=\left[\widehat{u}_{t},v_{t}^{\prime}\right]^{\prime}$ and
stacking now leads to the following asymptotic theory result

$\displaystyle\frac{1}{\sqrt{n}}\widehat{S}_{\lfloor
nr\rfloor}=\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}\widehat{\eta}_{t}\Rightarrow\begin{bmatrix}\displaystyle\int_{0}^{r}dB_{u}(s)-\int_{0}^{r}B_{v}^{*}(s)^{\prime}ds\Theta\\\
\displaystyle B_{v}(r)\end{bmatrix}$ (4.113)

Under the stated assumption it holds that

$\displaystyle\frac{1}{T}\sum_{t=2}^{T}S_{t-1}^{\eta}\eta_{t}^{\prime}\Rightarrow\int
B(r)dB(r)+\Lambda.$ (4.114)

######  Proposition 6.

As $b\to 0$, the fixed-b limiting distribution of $\widehat{\theta}_{b}^{+}$
converges in probability to the traditional limit distribution.

######  Remark 18.

Notice that these results show that the performance of the FM-OLS estimator
relies critically on the consistency approximation of the long-run variance
estimators being accurate and that moving around the bandwidth and kernel
impacts the sampling behaviour of the FM-OLS estimator. However, it is well-
known that non-parametric kernel long run variance estimators suffer from bias
and and sampling variability which as a result can affect the the accuracy of
the traditional approximation. Further details on fixed-b asymptotics are
discussed in Vogelsang and Wagner,  (2013) and Vogelsang and Wagner,  (2014).

######  Example 18.

Consider the cointegrating system as below

$\displaystyle\boldsymbol{y}_{t}$
$\displaystyle=\boldsymbol{A}x_{t}+\boldsymbol{u}_{t},$ (4.115)
$\displaystyle\boldsymbol{x}_{t}$
$\displaystyle=\boldsymbol{x}_{t-1}+\boldsymbol{v}_{t}$ (4.116)

where $\boldsymbol{y}_{t}$ is an $m-$dimensional vector and
$\boldsymbol{x}_{t}$ is $p-$dimensional such that
$\boldsymbol{\eta}_{t}=\left(\boldsymbol{u}_{t}^{\prime},\boldsymbol{v}_{t}^{\prime}\right)^{\prime}$
is an $(m+K)-$dimensional vector of innovations and $\boldsymbol{A}$ is an
$(m\times K)$ matrix of cointegrating coefficients.

The mixture process of the limit theory is given by

$\displaystyle\mathsf{vec}\left\\{n\left(\hat{\boldsymbol{A}}^{+}-\boldsymbol{A}\right)\right\\}\Rightarrow\mathcal{MN}\left(\boldsymbol{0},\left(\int_{0}^{1}\boldsymbol{B}_{\uptau}^{+}\boldsymbol{B}_{\uptau}^{+\prime}\right)^{-1}\otimes\boldsymbol{\Omega}_{yy.x}\right),$
(4.117)

where $\hat{A}^{+}$ is the FM regression estimator with a conditional long-run
covariance matrix of $u_{t}$ given $v_{t}$ expressed as below

$\displaystyle\boldsymbol{\Omega}_{yy.x}:=\boldsymbol{\Omega}_{yy}-\boldsymbol{\Omega}_{yx}\boldsymbol{\Omega}_{xx}^{-1}\boldsymbol{\Omega}_{xy}$
(4.118)

Then, the FM regression estimator has the explicit form given by

$\displaystyle\hat{\boldsymbol{A}}^{+}=\left(\hat{\boldsymbol{Y}}^{+\prime}\boldsymbol{X}-n\hat{\boldsymbol{\Delta}}^{+}_{yx}\right)\times\left(\boldsymbol{X}^{\prime}\boldsymbol{X}\right)^{-1}$
(4.119)

where

$\displaystyle\boldsymbol{X}=\big{[}x_{1}^{\prime},...,x_{n}^{\prime}\big{]},\
\ \
\hat{\boldsymbol{Y}}^{+}=\big{[}\hat{y}_{1}^{+\prime},...,\hat{y}_{n}^{+\prime}\big{]}^{\prime}\in\mathbb{R}^{n\times
m}$ (4.120)

such that

$\displaystyle\hat{\boldsymbol{y}}_{t}^{+}=\boldsymbol{y}_{t}-\hat{\boldsymbol{\Omega}}_{yx}\hat{\boldsymbol{\Omega}}_{xx}^{-1}\Delta\boldsymbol{x}_{t},\
\ \ \text{and}\ \ \
\boldsymbol{\Delta}_{yx}^{+}=\boldsymbol{\Delta}_{yx}-\boldsymbol{\Omega}_{yx}\boldsymbol{\Omega}_{xx}^{-1}\boldsymbol{\Delta}_{xx}$
(4.121)

where $\hat{\boldsymbol{\Omega}}_{yx}\hat{\boldsymbol{\Omega}}_{xx}^{-1}$ and
$\hat{\boldsymbol{\Delta}}_{yx}^{+}$ are consistent estimates of
$\boldsymbol{\Omega}_{yx}\hat{\boldsymbol{\Omega}}_{xx}^{-1}$ and
$\boldsymbol{\Delta}_{yx}^{+}$ respectively.

These matrices can be constructed in the familiar fashion using semiparametric
lag kernel methods with residuals from a preliminary cointegrating least
squares regression. Notice that setting with
$u_{y.xt}=u_{yt}-\Omega_{yx}\Omega_{xx}^{-1}\Delta x_{t}$ and
$U_{y.x}=\big{[}u_{y.x1}^{\prime},...,u_{y.xn}^{\prime}\big{]}^{\prime}$ as
the corresponding data matrix, we have that

$\displaystyle\hat{\boldsymbol{y}}_{t}^{+}=\boldsymbol{y}_{t}-\hat{\boldsymbol{\Omega}}_{yx}\hat{\boldsymbol{\Omega}}_{xx}^{-1}\Delta\boldsymbol{x}_{t}=\boldsymbol{A}\boldsymbol{x}_{t}+\boldsymbol{u}_{0.xt}$
(4.122)

In terms of asymptotic theory we have that

$\displaystyle\xi_{y.xn}^{+}(s):=\xi_{yn}(s)-\boldsymbol{\Omega}_{yx}\boldsymbol{\Omega}_{xx}^{-1}\xi_{xn}(s)\Rightarrow\boldsymbol{B}_{y.x}(s)\equiv
BM\big{(}\boldsymbol{\Omega}_{y.xx}\big{)}$ (4.123)

Notice that the unit root limit theory given by expressions (17) and (18) of
Magdalinos and Phillips,  (2009) involves the demeaned process $B^{\mu}(s)$
although there is no intercept in the regression. In particular, the demeaning
effects arises because as shown in expression (16), in the direction of the
initial condition, the time series is dominated by a component that behaves
like a constant. Therefore, the sample moment matrix is no longer
asymptotically singular such that

$\displaystyle\frac{1}{n^{2}}\sum_{t=1}^{n}\boldsymbol{x}_{t-1}\boldsymbol{x}_{t-1}^{\prime}\Rightarrow\int_{0}^{1}J_{c}^{*}(r)J_{c}^{*\prime}(r)$
(4.124)

Then, from the first-order vector autoregression of $y_{t}$ on $y_{t-1}$ we
obtain the regression coefficient matrix

$\displaystyle\hat{A}=\left(\sum_{t=1}^{T}y_{t}y_{t-1}^{\prime}\right)\left(\sum_{t=1}^{T}y_{t-1}y_{t-1}^{\prime}\right)^{-1}.$
(4.125)

The asymptotic behaviour of $\hat{A}$ is described by a corresponding
functional Brownian motion. To be more precise consider standardized
deviations of $\hat{A}$ about $I_{n}$ such that

$\displaystyle
T\left(\hat{A}-I_{p}\right)=\left(\frac{1}{n}\sum_{t=1}^{T}u_{t}y_{t-1}^{\prime}\right)\left(\frac{1}{n^{2}}\sum_{t=1}^{T}y_{t-1}y_{t-1}^{\prime}\right)^{-1}.$
(4.126)

Therefore, to obtain the asymptotic behaviour of the above statistic we write
the sample second moment
$\left(\frac{1}{T^{2}}\sum_{t=1}^{n}y_{t-1}y_{t-1}^{\prime}\right)$ as a
quadratic functional of the random element $X_{n}(r)$, at least up to a term
of $o_{p}(1)$.

$\displaystyle\frac{1}{n^{2}}\sum_{t=1}^{n}y_{t-1}y_{t-1}^{\prime}=\int_{0}^{1}X_{n}(r)X_{n}(r)^{\prime}dr+o_{p}(1)$
(4.127)

Then, by an application of the continuous mapping theorem we can establish
that

$\displaystyle\frac{1}{n^{2}}\sum_{t=1}^{n}y_{t-1}y_{t-1}^{\prime}$
$\displaystyle\Rightarrow\int_{0}^{1}B(r)B(r)^{\prime}dr,\ \ \ \text{as}\ \
T\to\infty.$ (4.128)
$\displaystyle\frac{1}{n}\sum_{t=1}^{n}u_{t}y_{t-1}^{\prime}$
$\displaystyle\Rightarrow\int_{0}^{1}dB(r)B(r)^{\prime}$ (4.129)

####  4.5.6. FM transformation in Cointegrating Regression

A challenging issue in the cointegrating regression literature is that when
$u_{t}$ is uncorrelated with $v_{t}$ and hence uncorrelated with $x_{t}$, it
follows that (i) $\lambda_{uv}=\boldsymbol{0}$, $\Delta_{uv}=\boldsymbol{0}$,
and (ii) $B_{u}(r)$ is independent of $B_{v}(r)$. Furthermore, because of the
independence between the Brownian motions $B_{u}(r)$ and $B_{v}(r)$ in this
case, one can condition on $B_{v}(r)$ to show that the limiting distribution
of $T\left(\widehat{\beta}-\beta\right)$ is a zero mean Gaussian mixture.
Therefore, one can also show that the $t$ and Wald statistics for testing the
hypotheses about $\beta$ have the usual $\mathcal{N}(0,1)$ and chi-square
limits assuming serial correlation in $u_{t}$ is handled using consistent
robust standard errors. On the other hand, when regressors are endogenous, the
limiting distribution of $T\left(\widehat{\beta}-\beta\right)$ is obviously
more complicated because of the correlation between $B_{u}(r)$ and $B_{v}(r)$
and the presence of nuisance parameters in the vector $\Delta_{vu}$.

Thus, we can no longer condition on $B_{v}(r)$ to obtain an asymptotic normal
result and $\Delta_{vu}$ introduces an asymptotic bias. In particular,
inference is difficult in this situation because nuisance parameters cannot be
removed by simple scaling methods. Therefore, the FM-OLS estimator of Phillips
and Hansen,  (1990) (see, also Phillips and Ouliaris,  (1990)) is employed to
asymptotically remove $\Delta_{vu}$ and to deal with the correlation between
$B_{u}(r)$ and $B_{v}(r)$. In practise FM-OLS estimation requires the choice
of bandwidth and kernel. While bandwidth and kernel play no role
asymptotically when considering the consistency results for $\widehat{\Omega}$
and $\widehat{\Delta}$, in finite samples they affect the sampling
distributions of $\widehat{\theta}^{+}$ and thus of $t$ and Wald statistics
based on the FM-OLS estimator of $\widehat{\theta}^{+}$. In other words, to
obtain an approximation for $\widehat{\theta}^{+}$ that reflects the choice of
bandwidth and kernel, the natural asymptotic theory to use is the fixed-b
theory. However, fixed-b theory has primarily been developed for models with
stationary regressors, which means that some additional work is required to
obtain analogous results for cointegrating regressions. A major difference is
that the first component of $\widehat{\eta}_{t}$, that is, $\widehat{u}_{t}$,
is the residual from a cointegrating regression, which leads to dependence of
the corresponding limit partial sum process on the number of integrated
regressors and the deterministic components.

Specifically, the FM-OLS estimator for the cointegrating predictive regression
model is defined as below

$\displaystyle
A_{FM}=\left(\sum_{t=1}^{T}\boldsymbol{y}_{t}^{+}\boldsymbol{x}_{t}^{\prime}-T\hat{\kappa}\hat{\boldsymbol{\Gamma}}\right)\left(\sum_{t=1}^{T}\boldsymbol{x}_{t}\boldsymbol{x}_{t}^{\prime}\right)^{-1}$
(4.130)

where
$y_{t}^{+}=y_{t}-\hat{\boldsymbol{\Omega}}_{12}\hat{\boldsymbol{\Omega}}_{22}^{-1}D\boldsymbol{x}_{t}$.
Then, the asymptotic distribution of the FM-OLS estimator

$\displaystyle
T\big{(}\dot{A}_{FM}-A\big{)}\Rightarrow\left(\int_{0}^{1}dB_{1.2}(r)B_{2}(r)^{\prime}\right)\left(\int_{0}^{1}B_{2}(r)B_{2}(r)^{\prime}dr\right)^{-1},$
(4.131)

which is the same as the asymptotic distribution of the CCR estimator. Now, we
may use the residuals

$\displaystyle\dot{u}_{t}^{+}=\boldsymbol{y}_{t}^{+}-\dot{A}_{FM}\boldsymbol{x}_{t}=u_{t}-\hat{\boldsymbol{\Omega}}_{12}\hat{\boldsymbol{\Omega}}_{22}^{-1}\Delta\boldsymbol{x}_{t}-\big{(}\dot{A}_{FM}-A\big{)}\boldsymbol{x}_{t}$
(4.132)

The FM-OLS is optimal in the sense that is has the same limiting distribution
as Gaussian maximum likelihood. Then estimators can estimated with long-run
covariance matrices.

The OLS residuals can be written as below

$\displaystyle\hat{u}_{t}=y_{t}-\hat{A}\boldsymbol{x}_{t}=u_{t}-\left(\hat{A}-A\right)\boldsymbol{x}_{t}$
(4.133)

Consider the partial sum processes $S^{u}_{t}=\sum_{i=1}^{t}u_{i}$ and
$S^{x}_{t}=\sum_{i=1}^{t}x_{i}$, which implies that

$\displaystyle\frac{1}{T^{2}}\sum_{t=1}^{T}\hat{S}_{t}\hat{S}_{t}^{\prime}\Rightarrow\int_{0}^{1}\bigg{[}\boldsymbol{B}_{1}(r)-\alpha\boldsymbol{B}_{2}(r)\bigg{]}\bigg{[}\boldsymbol{B}_{1}(r)-\alpha\boldsymbol{B}_{2}(r)\bigg{]}^{\prime}dr,$
(4.134)

where

$\displaystyle\alpha=\bigg{\\{}\int_{0}^{1}\boldsymbol{B}_{2}(r)d\boldsymbol{B}_{1}(r)^{\prime}+\boldsymbol{\Gamma}_{21}\bigg{\\}}\left\\{\int_{0}^{1}\boldsymbol{B}_{2}(r)\boldsymbol{B}_{2}(r)^{\prime}\right\\}$
(4.135)

In particular, these results show that eliminating nuisance parameters is not
easy unless $\boldsymbol{x}_{t}$ is strictly exogenous. Thus, the endogeneity
aspect of the model implies that there is a dependence between the
nonstationary regressor $X_{t}$ and the stationary error $u_{t}$.

####  4.5.7. FM estimation with nearly integrated regressors

In particular, the autoregressive root of the regressor which is parametrized
with the local-to-unity specification allows to capture the near unit root
behaviour of many predictor variables and is less restrictive than the pure
unit root specification. Moreover, the OLS estimator of $\beta$ does not have
an asymptotically mixed normal distribution due to the correlation between
$u_{t}$ and $v_{t}$. From the paper of hjalmarsson2007fully the fully modified
OLS estimator is given by

$\displaystyle\hat{\beta}^{+}=\left(\sum_{t=1}^{n}\underline{\boldsymbol{y}}^{+}_{t}\underline{\boldsymbol{x}}_{t-1}n-\widehat{\boldsymbol{\Lambda}}_{12}\right)\left(\sum_{t=1}^{n}\underline{\boldsymbol{x}}_{t-1}\underline{\boldsymbol{x}}_{t-1}^{\prime}\right)^{-1}.$
(4.136)

Define with $\omega_{11.2}=\omega_{11}-\omega_{21}^{2}\omega_{22}^{-1}$ and
$B_{1.2}=B_{1}-\omega_{21}\omega_{22}^{-1}B_{2}\equiv BM(\omega_{11.2})$.
Then, as $T\to\infty$, the following limiting distribution holds

$\displaystyle
T\left(\hat{\beta}^{+}-\beta\right)\Rightarrow\left(\int_{0}^{1}dB_{1.2}\underline{J}_{c}\right)\left(\int_{0}^{1}\underline{J}_{c}^{2}\right)^{-1}\equiv\mathcal{MN}\left(0,\omega_{11.2}\left(\int_{0}^{1}\underline{J}_{c}^{2}\right)^{-1}\right).$
(4.137)

Furthermore, under the assumption that both $u_{t}$ and $v_{t}$ are martingale
difference sequences it can be shown that the OLS estimation of the augmented
regression

$\displaystyle y_{t}=\alpha+\beta
x_{t-1}+\gamma\Delta_{c}\boldsymbol{x}_{t}+u_{t.v},$ (4.138)

yields an estimator of $\beta$ with an asymptotic distribution identical to
that of $\hat{\beta}^{+}$.

###  4.6. Dynamic Seemingly Unrelated Cointegrating Regression

We follow the framework proposed by Mark et al.,  (2005). More specifically,
we consider a fixed number of $N$ cointegrating regressions each with $T$
observations.

######  Assumption 6.

Each equation of the SUR system $i\in\left\\{1,...,N\right\\}$ has the
triangular representation:

$\displaystyle y_{it}$
$\displaystyle=\boldsymbol{x}_{it}^{\prime}\boldsymbol{\beta}_{i}+u_{it}$
(4.139) $\displaystyle\Delta\boldsymbol{x}_{it}$
$\displaystyle=\boldsymbol{e}_{it}$ (4.140)

where $\boldsymbol{x}_{it}$ and $\boldsymbol{e}_{it}$ are $(k\times 1)$
dimensional vectors.

Moreover, we denote with
$\boldsymbol{u}_{t}=\left(u_{1t},...,u_{NT}\right)^{\prime}$ and
$\boldsymbol{e}_{t}=\left(e^{\prime}_{1t},...,e^{\prime}_{Nt}\right)$ and with
$\boldsymbol{w}_{t}=\left(\boldsymbol{u}_{t}^{\prime},\boldsymbol{e}_{t}^{\prime}\right)^{\prime}$
an $N(k+1)$dimensional vector with the orthonormal Wold moving average
representation such that
$\boldsymbol{w}_{t}=\boldsymbol{\Psi}(\boldsymbol{L})\boldsymbol{\epsilon}_{t}$.
Notice that $\boldsymbol{\epsilon}_{t}$ is a martingale difference sequence,
such that $\mathbb{E}\left[\boldsymbol{\epsilon}_{t}\right]=\boldsymbol{0}$
and
$\mathbb{E}\big{[}\boldsymbol{\epsilon}_{t}\boldsymbol{\epsilon}_{t}^{\prime}\big{]}=\boldsymbol{I}_{k}$
and finite fourth moments. Therefore, the endogeneity problem shows up as
correlation between the $i-$th equilibrium error $u_{it}$ and potentially an
infinite number of leads and lags of the first differenced regressors from all
of the equations of the system such as
$\Delta\boldsymbol{x}_{jt}=\boldsymbol{e}_{jt}$ for
$(i,j)\in\left\\{1,...,N\right\\}$.

Consider the following formulation of the model

$\displaystyle
y_{it}=\boldsymbol{x}_{it}^{\prime}\boldsymbol{\beta}_{i}+\boldsymbol{z}_{pt}^{\prime}\boldsymbol{\delta}_{pi}+u_{it}$
(4.141)

Let $\boldsymbol{y}_{t}=\left(y_{1t},...,y_{Nt}\right)^{\prime}$,
$\boldsymbol{u}_{t}=\left(u_{1t},...,u_{Nt}\right)^{\prime}$ and
$\boldsymbol{\beta}=\left(\boldsymbol{\beta}_{1}^{\prime},...,\boldsymbol{\beta}_{N}^{\prime}\right)^{\prime}$,
$\boldsymbol{\delta}=\left(\boldsymbol{\delta}_{1}^{\prime},...,\boldsymbol{\delta}_{N}^{\prime}\right)^{\prime}$.
Moreover we have the following matrices

$\displaystyle\boldsymbol{Z}_{pt}=\left(\mathbf{I}_{N}\otimes\boldsymbol{z}_{pt}\right),\
\
\boldsymbol{X}_{t}=\text{diag}\left(\boldsymbol{x}_{1t},...,\boldsymbol{x}_{Nt}\right)\
\ \text{and}\ \
\boldsymbol{W}_{t}=\left(\boldsymbol{X}_{t}^{\prime},\boldsymbol{Z}_{pt}^{\prime}\right)^{\prime}.$
(4.142)

Then, the equations can be stacked together in a system as below

$\displaystyle\boldsymbol{y}_{t}=\left(\boldsymbol{\beta}^{\prime},\boldsymbol{\delta}_{p}^{\prime}\right)\boldsymbol{W}_{t}+\boldsymbol{u}_{t}.$
(4.143)

Moreover, denote the long-run covariance matrix of $\boldsymbol{u}_{t}$ by
$\boldsymbol{\Omega}_{uu}$. Then, an expression for the DSUR estimator with
known $\boldsymbol{\Omega}_{uu}$ is given by

$\displaystyle\begin{bmatrix}\hat{\boldsymbol{\beta}}_{DSUR}\\\ \\\
\hat{\boldsymbol{\delta}}_{DSUR}\end{bmatrix}=\left(\sum_{t=p+1}^{T-p}\boldsymbol{W}_{t}\boldsymbol{\Omega}_{uu}^{-1}\boldsymbol{W}_{t}^{\prime}\right)^{-1}\left(\sum_{t=p+1}^{T-p}\boldsymbol{W}_{t}\boldsymbol{\Omega}_{uu}^{-1}\boldsymbol{y}_{t}\right).$
(4.144)

This, it can be shown that $\hat{\boldsymbol{\beta}}_{DSUR}$ is asymptotically
mixed normal.

Hypothesis testing for the linear restrictions
$\boldsymbol{R}\boldsymbol{\beta}=\boldsymbol{r}$, where $\boldsymbol{R}$ is a
$q\times Nk$ matrix of constants and $\boldsymbol{r}$ is a $q-$dimensional
vector of constants. In practise, the testing hypothesis can be also written
as below

$\displaystyle\mathbb{H}_{0}:\beta_{1}=...=\beta_{N}$ (4.145)

Notice that the above formulation of the null hypothesis provides a way for
conveniently formulating a test of homogeneity restrictions on the
cointegrating vectors.

Then, the DSUR Wald test statistic is expressed as below

$\displaystyle\mathcal{W}_{dsur}=\left(\boldsymbol{R}\hat{\boldsymbol{\beta}}_{dsur}-\boldsymbol{r}\right)^{\prime}\big{[}\boldsymbol{R}\hat{\boldsymbol{V}}_{dsur}\boldsymbol{R}^{\prime}\big{]}^{-1}\left(\boldsymbol{R}\hat{\boldsymbol{\beta}}_{dsur}-\boldsymbol{r}\right)$
(4.146)

where

$\displaystyle\hat{\boldsymbol{V}}_{dsur}=\sum_{t=p+1}^{T-p}\boldsymbol{X}_{t}\boldsymbol{\Omega}_{uu}^{-1}\boldsymbol{X}_{t}^{\prime}$
(4.147)

The Wald statistic $\mathcal{W}_{DSUR}$ is asymptotically distributed as a
chi-square variate with $q$ degrees of freedom under the null hypothesis.
Moreover, in applications we replace $\boldsymbol{\Omega}_{uu}$ with a
consistent estimator,
$\hat{\boldsymbol{\Omega}}_{uu}\overset{p}{\to}\boldsymbol{\Omega}_{uu}$.
Estimation of the long-run covariance matrix is discussed below. Such an
estimator might be called a "feasible" DSUR estimator.

##### Two Step DSUR

The first step purges endogeneity by least squares and the second step
estimates $\boldsymbol{\beta}$ by running SUR on the least squares residuals
obtained from the first-step regression. Under standard regularity conditions,
the two-step DSUR estimator is asymptotically equivalent to the DSUR estimator
$\hat{\boldsymbol{\beta}}_{DSUR}$ discussed above. Stacking the equations
together as
$\hat{\boldsymbol{y}}_{t}=\hat{\boldsymbol{X}}_{t}^{\prime}\boldsymbol{\beta}+\hat{\boldsymbol{u}}_{t}$
and running SUR gives the two-step DSUR estimator gives,

$\displaystyle\hat{\boldsymbol{\beta}}_{2sdsur}=\left[\sum_{t=p+1}^{T-p}\hat{\boldsymbol{X}}_{t}\boldsymbol{\Omega}_{uu}^{-1}\hat{\boldsymbol{X}}_{t}^{\prime}\right]^{-1}\left[\sum_{t=p+1}^{T-p}\hat{\boldsymbol{X}}_{t}\boldsymbol{\Omega}_{uu}^{-1}\hat{\boldsymbol{y}}_{t}\right]$
(4.148)

##### Restricted DSUR

Next, we consider the estimation of the cointegration vector under the
homogeneity restrictions
$\boldsymbol{\beta}_{1}=...=\boldsymbol{\beta}_{N}=\boldsymbol{\beta}$. As in
two-step DSUR, endogeneity can first be purged by regressing $y_{it}$ and each
element of $\boldsymbol{x}_{it}$ on $\boldsymbol{z}_{pt}$. Let $\hat{y}_{it}$
and $\hat{x}_{it}$ denote the resulting regression errors. Therefore, the
estimation problem becomes one of estimating $\boldsymbol{\beta}$, in the
system of equations
$\hat{y}_{it}=\hat{\boldsymbol{x}}_{it}^{\prime}\boldsymbol{\beta}+\hat{u}_{it}$.
Stacking the system equations together:

$\displaystyle\hat{\boldsymbol{y}}_{t}=\hat{\boldsymbol{x}}_{t}^{\prime}\boldsymbol{\beta}+\hat{\boldsymbol{u}}_{t}$
(4.149)

##### Asymptotic Properties

Let $\boldsymbol{W}(r)$ be a vector standard Brownian motion for $0\leq r\leq
1$.

######  Proposition 7.

Let $T^{*}=(T-2p)$. Under the conditions of Assumptions above,

  1. (i)

$T^{*}\left(\hat{\boldsymbol{\beta}}_{dsur}-\boldsymbol{\beta}_{dsur}\right)$
and
$\sqrt{T^{*}}\left(\hat{\boldsymbol{\delta}}_{p,dsur}-\boldsymbol{\delta}_{p,dsur}\right)$
are asymptotically independent.

  2. (ii)

If
$\boldsymbol{B}_{e}=\text{diag}\left(\boldsymbol{B}_{e_{1}},...,\boldsymbol{B}_{e_{N}}\right)$,
$\hat{\boldsymbol{V}}_{dsur}=\sum_{t=p+1}^{T-p}\boldsymbol{X}_{t}\boldsymbol{\Omega}_{uu}^{-1}\boldsymbol{X}_{t}^{\prime}$
and $\boldsymbol{R}$ is a $q\times Nk$ matrix of constants such that
$\boldsymbol{R}\boldsymbol{\beta}=\boldsymbol{r}$, then as $T^{*}\to\infty$,

$\displaystyle
T^{*}\left(\hat{\boldsymbol{\beta}}_{dsur}-\boldsymbol{\beta}\right)\Rightarrow\left(\int\boldsymbol{B}_{e}\boldsymbol{\Omega}_{uu}^{-1}\boldsymbol{B}_{e}^{\prime}\right)^{-1}\left(\int\boldsymbol{B}_{e}\boldsymbol{\Omega}_{uu}^{-1}d\boldsymbol{B}_{u}\right)$
(4.150)

and

$\displaystyle\left(\boldsymbol{R}\hat{\boldsymbol{\beta}}_{dsur}-\boldsymbol{r}\right)^{\prime}\big{[}\boldsymbol{R}\hat{\boldsymbol{V}}_{dsur}\boldsymbol{R}^{\prime}\big{]}^{-1}\left(\boldsymbol{R}\hat{\boldsymbol{\beta}}_{dsur}-\boldsymbol{r}\right)\overset{D}{\to}\chi^{2}_{q}.$
(4.151)

Notice that a functional central limit theory applies such that

$\displaystyle\displaystyle\frac{1}{\sqrt{T^{*}}}\sum_{t=p+1}^{\lfloor(T-p)r\rfloor}\boldsymbol{w}_{t}\overset{D}{\to}\left(\boldsymbol{B}_{u}^{\prime},\boldsymbol{B}_{v}^{\prime}\right)^{\prime}$
(4.152)

with the long-run covariance matrix
$\boldsymbol{\Omega}=\mathsf{diag}\big{(}\boldsymbol{\Omega}_{uu},\boldsymbol{\Omega}_{ee}\big{)}$.

Due to the block diagonality of $\boldsymbol{\Omega}$, it can be easily
verified that $\boldsymbol{B}_{u}$ and $\boldsymbol{B}_{e}$ are independent as
discussed in the frameworks of Mark et al.,  (2005) and Saikkonen,  (1993).
Similarly Phillips and Moon,  (1999) develop the asymptotic theory for
regression models with nonstationary panel data for which the time series
component is an integrated process and where both $T$ and $n$ are large.
Recently, Chen et al.,  (2023) consider a framework for robust estimation of a
Seemingly Unrelated Regression System for VAR models with explosive roots
based on the approach of Magdalinos and Phillips,  (2009) and Magdalinos,
(2020).

######  Remark 19.

Notice that the possibility of interpreting cointegration vectors as economic
long-run relations is the main reason why the vector autoregressive model has
become widely used in the empirical analysis of economic data. Specifically,
the statistical concept of cointegration in the $I(1)$ model, involving linear
combination of levels of the variables, corresponds to the economic concept of
a long-run static steady-state relation. Similarly, the statistical concept of
multicointegration or polynomial cointegration, involving linear combinations
of both levels and differences, corresponds to the economic concept of a long-
run dynamic steady-state relation.

###  4.7. Predictive Regression Models

####  4.7.1. Regression with Integrated Regressors

Various examples related to integrated processes and the limit theory in
regression with integrated predictors can be found in the book of Banerjee et
al.,  (1993). Among others, related literature includes the papers of Cavanagh
et al.,  (1995), Stock,  (1991), Jansson and Moreira,  (2006). Below we
present an example (see, corresponding chapter from Banerjee et al.,  (1993)),
to shed light on the related asympotics.

######  Example 19.

(Cointegrating Regression)

Consider the following bivariate system of co-integrated variables
$\\{y_{t}\\}_{t=1}^{\infty}$ and $\\{x_{t}\\}_{t=1}^{\infty}$.

$\displaystyle y_{t}$ $\displaystyle=\beta x_{t}+u_{t}$ (4.153)
$\displaystyle\Delta x_{t}$ $\displaystyle=\epsilon_{t}$ (4.154)

with $u_{t}\sim N(0,\sigma^{2}_{u})$, $\epsilon_{t}\sim
N(0,\sigma^{2}_{\epsilon})$ and
$\mathbb{E}(u_{t}\epsilon_{s})=\sigma_{u\epsilon}\ \forall\ t\neq s$.

The OLS estimator of $\beta$ is given by

$\displaystyle\hat{\beta}=\left(\sum_{t=1}^{T}x_{t}^{2}\right)^{-1}\left(\sum_{t=1}^{T}y_{t}x_{t}\right)$
(4.155)

Thus,

$\displaystyle
T\left(\hat{\beta}-\beta\right)=\left(T^{-2}\sum_{t=1}^{T}x_{t}^{2}\right)^{-1}\left(T^{-1}\sum_{t=1}^{T}x_{t}u_{t}\right)$
(4.156)

Note that we have a regression with an integrated regressor, since

$\displaystyle x_{t}=x_{t-1}+\epsilon_{t}$ (4.157)

Therefore, it follows that

$\displaystyle\left(\frac{1}{T^{2}}\sum_{t=1}^{T}x_{t}^{2}\right)\Rightarrow\sigma^{2}_{\epsilon}\int_{0}^{1}W_{\epsilon}(r)^{2}dr.$
(4.158)

In order to derive the limiting distribution of the model parameter in the
case of integrated regressors, we shall first derive the limiting distribution
of $\left(T^{-1}\sum_{t=1}^{T}x_{t}u_{t}\right)$.

To do this, we condition $u_{t}$ on $\epsilon_{t}$ as given below

$\displaystyle u_{t}=\phi\epsilon_{t}+v_{t},\ \
\phi=\frac{\sigma_{u\epsilon}}{\sigma^{2}_{\epsilon}}\ \ \text{and}\ \
\sigma^{2}_{v}=\sigma^{2}_{u}-\frac{\sigma^{2}_{u\epsilon}}{\sigma^{2}_{\epsilon}}$
(4.159)

Define $W_{\epsilon}(r)$ and $W_{v}(r)$ to be two independent Wiener processes
on $\mathcal{C}[0,1]$. Therefore,

$\displaystyle\left(T^{-1}\sum_{t=1}^{T}x_{t}u_{t}\right)$
$\displaystyle=T^{-1}\sum_{t=1}^{T}x_{t}\left(\phi\epsilon_{t}+v_{t}\right)$
(4.160)
$\displaystyle=\phi\left(T^{-1}\sum_{t=1}^{T}x_{t}\epsilon_{t}\right)+\left(T^{-1}\sum_{t=1}^{T}x_{t}v_{t}\right)$
(4.161)

Substituting $x_{t}=x_{t-1}+\epsilon_{t}$ into the above gives

$\displaystyle\left(T^{-1}\sum_{t=1}^{T}x_{t}u_{t}\right)$
$\displaystyle=\phi\left(T^{-1}\sum_{t=1}^{T}(x_{t-1}+\epsilon_{t})\epsilon_{t}\right)+\left(T^{-1}\sum_{t=1}^{T}(x_{t-1}+\epsilon_{t})v_{t}\right)$
(4.162)
$\displaystyle=\phi\left(T^{-1}\sum_{t=1}^{T}x_{t-1}\epsilon_{t}\right)+\phi\left(T^{-1}\sum_{t=1}^{T}\epsilon^{2}_{t}\right)$
(4.163) $\displaystyle\ \ \
+\left(T^{-1}\sum_{t=1}^{T}x_{t-1}v_{t}\right)+\left(T^{-1}\sum_{t=1}^{T}\epsilon_{t}v_{t}\right)$
(4.164)

We have the following asymptotic results

$\displaystyle T^{-1}\sum_{t=1}^{T}\epsilon^{2}_{t}$
$\displaystyle\overset{p}{\to}\sigma^{2}_{\epsilon}$ (4.165) $\displaystyle
T^{-1}\sum_{t=1}^{T}\epsilon_{t}v_{t}$ $\displaystyle\overset{p}{\to}0$
(4.166) $\displaystyle T^{-1}\sum_{t=1}^{T}x_{t-1}v_{t}$
$\displaystyle\Rightarrow\sigma_{\epsilon}\sigma_{v}\int_{0}^{1}W_{\epsilon}(r)dW_{v}(r)\equiv\int_{0}^{1}B_{\epsilon}(r)dB_{v}(r)$
(4.167) $\displaystyle T^{-1}\sum_{t=1}^{T}x_{t-1}\epsilon_{t}$
$\displaystyle\Rightarrow\frac{\sigma^{2}_{\epsilon}}{2}\bigg{[}W^{2}_{\epsilon}(1)-1\bigg{]}\
\ \text{(see, \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{phillips1987time}{\@@citephrase{(}}{\@@citephrase{)}}})}$
(4.168)

Therefore, we have that

$\displaystyle
T^{-1}\sum_{t=1}^{T}x_{t}u_{t}\Rightarrow\left\\{\phi\left(\frac{\sigma^{2}_{\epsilon}}{2}\bigg{[}W^{2}_{\epsilon}(1)-1\bigg{]}\right)+\phi\sigma^{2}_{\epsilon}+\sigma_{\epsilon}\sigma_{v}\int_{0}^{1}W_{\epsilon}(r)dW_{v}(r)\right\\}$
(4.169)

Furthermore, Phillips and Park,  (1988) proved the following result

$\displaystyle\int_{0}^{1}W_{\epsilon}(r)dW_{v}(r)\Rightarrow
N\left(0,\int_{0}^{1}W_{\epsilon}(r)dr\right)$ (4.170)

Therefore, under the null hypothesis $H_{0}:\beta=0$,

$\displaystyle
T\hat{\beta}\Rightarrow\left\\{\phi\frac{\sigma^{2}_{\epsilon}}{2}\bigg{[}W^{2}_{\epsilon}(1)+1\bigg{]}+\sigma_{\epsilon}\sigma_{v}\bigg{[}\int_{0}^{1}W_{\epsilon}(r)dW_{v}(r)\bigg{]}\right\\}\left(\sigma^{2}_{\epsilon}\int_{0}^{1}W_{\epsilon}(r)^{2}dr\right)^{-1}.$
(4.171)

Thus, the $t-$statistic, denoted as $\mathcal{T}_{\beta=0}$ for testing the
null hypothesis, $H_{0}:\beta=0$,

$\displaystyle\mathcal{T}_{\beta=0}=\frac{\hat{\beta}}{\hat{\sigma}^{2}_{u}\left(\displaystyle\sum_{t=1}^{T}x_{t}^{2}\right)^{-\frac{1}{2}}}=T\frac{\hat{\beta}}{\hat{\sigma}^{2}_{u}\left(\displaystyle
T^{-2}\sum_{t=1}^{T}x_{t}^{2}\right)^{-\frac{1}{2}}}$ (4.172)

has the following limiting distribution

$\displaystyle\mathcal{T}_{\beta=0}$
$\displaystyle\Rightarrow\left\\{\phi\frac{\sigma^{2}_{\epsilon}}{2}\bigg{[}W^{2}_{\epsilon}(1)+1\bigg{]}+\sigma_{\epsilon}\sigma_{v}\bigg{[}\int_{0}^{1}W_{\epsilon}(r)dW_{v}(r)\bigg{]}\right\\}\left(\sigma^{2}_{\epsilon}\int_{0}^{1}W_{\epsilon}(r)^{2}dr\right)^{-\frac{1}{2}}\times\frac{1}{\sigma_{u}^{2}}$
(4.173)
$\displaystyle\equiv\frac{\phi}{2}\frac{\sigma_{\epsilon}}{\sigma_{u}}\bigg{[}W^{2}_{\epsilon}(1)+1\bigg{]}\left(\int_{0}^{1}W_{\epsilon}(r)^{2}dr\right)^{-\frac{1}{2}}+\frac{\sigma_{v}}{\sigma_{u}}N(0,1)$
(4.174)

The above limiting distribution indicates that the $t-$ratio of $\hat{\beta}$
does not follow a standard normal distribution unless $\phi=0$, which in that
cases implies that $x_{t}$ is exogenous for the estimation of $\beta$. In
particular, when $\phi\neq 0$ then the first term of the above limiting
distribution gives rise to second-order or endogeneity bias, which although
asymptotically negligible in estimating $\beta$ due to super consistency, can
be important in finite samples.

######  Example 20.

We follow Cavanagh et al.,  (1995). Consider the following recursive system

$\displaystyle y_{t}$ $\displaystyle=\mu_{y}+\gamma x_{t-1}+\epsilon_{2t}$
(4.175) $\displaystyle x_{t}$ $\displaystyle=\mu_{x}+v_{t}$ (4.176)

where the sequence $v_{t}$ is generated as

$\displaystyle\big{(}1-\alpha L\big{)}b(L)v_{t}=\epsilon_{1t}$ (4.177)

where $b(L)=\sum_{j=0}^{k}b_{j}L^{j}$, $b_{0}=1$, and
$\epsilon_{t}=(\epsilon_{1t},\epsilon_{2t})^{\prime}$.

######  Assumption 7.

The vector $\epsilon_{t}=(\epsilon_{1t},\epsilon_{2t})^{\prime}$ is a
martingale difference sequence such that

  1. (i)

$\mathbb{E}\left(\epsilon_{t}\epsilon_{t}^{\prime}|\epsilon_{t-1},\epsilon_{t-2},...\right)=\Sigma>0.$

  2. (ii)

$\mathbb{E}\left(\epsilon_{it}^{4}<\infty\right)$, $i=1,2$.

Let $\delta=\text{corr}(\epsilon_{1t},\epsilon_{2t})$ and assume that
$Ev_{0}^{2}<\infty$. The roots of $b(L)$ are assumed to be fixed and less than
1 in absolute value.

  * •

If $|\alpha|<1$ and $\alpha$ is fixed, then $x_{t}$ is integrated of order 0,
that is, $I(0)$.

  * •

If $\alpha=1$ and $\rho$ is fixed, then $x_{t}$ is integrated of order 1, that
is, $I(1)$.

Therefore, we consider $\alpha$ to be the largest autoregressive root of the
univariate representation of $x_{t}$. Thus, we can jointly write expressions
$x_{t}=\mu_{x}+v_{t}$ and $\big{(}1-\alpha L\big{)}b(L)v_{t}=\epsilon_{1t}$ in
augmented Dickey-Fuller, (ADF) form as below

$\displaystyle\Delta x_{t}=\tilde{\mu}_{x}+\beta x_{t-1}+\alpha(L)\Delta
x_{t-1}+\epsilon_{1t}$ (4.178)

where

$\displaystyle\tilde{\mu}_{x}=(1-\alpha)b(1)\mu_{x},\ \ \beta=(\alpha-1)b(1),\
\ \ a_{i}=-\sum_{j=i+1}^{k}\tilde{\alpha}_{j}$ (4.179)

with $\tilde{\alpha}(L)=L^{-1}\big{[}1-\big{(}1-\alpha L\big{)}b(L)\big{]}$.
We consider the problem of testing the null hypothesis that
$\gamma=\gamma_{0}$ or, equivalently, constructing confidence intervals for
$\gamma$. Thus, for this problem, the root $\alpha$ is a nuisance parameter.
Let $B=(B_{1},B_{2})$ be a two-dimensional Brownian motion with covariance
matrix $\tilde{\Sigma}$ such that

$\displaystyle\tilde{\Sigma}=\begin{bmatrix}1&\delta\\\ \delta&1\\\
\end{bmatrix}$ (4.180)

and let $J_{c}$ be the diffusion process222Notice that the Ornstein–Uhlenbeck
stochastic process is a time-homogeneous Ito diffusion process. defined by

$\displaystyle J_{c}(r)=cJ_{c}(r)ds+dB_{1}(s),$ (4.181)

where $J_{c}(0)=0$. Let $J_{c}^{\mu}(s)=J_{c}(s)-\int_{0}^{1}J_{c}(r)dr$.
Under the local-to-unity model $\alpha=(1+c/T)$ then,

$\displaystyle\left\\{\sigma_{11}^{-1/2}T^{-1/2}\sum_{t=1}^{\lfloor
T\pi\rfloor}\epsilon_{1t},\ \sigma_{22}^{-1/2}T^{-1/2}\sum_{t=1}^{\lfloor
T\pi\rfloor}\epsilon_{2t},\ \omega^{-1}T^{-1/2}x^{\mu}_{\lfloor
T\pi\rfloor}\right\\}\Rightarrow\bigg{\\{}B_{1}(\pi),B_{2}(\pi),J_{c}^{\mu}(\pi)\bigg{\\}}$
(4.182)

jointly, where $\omega^{2}=\sigma_{11}/b(1)^{2}$, $\displaystyle
x_{t}^{\mu}=x_{t}-\frac{1}{T-1}\sum_{t=2}^{T}x_{t-1}$. Therefore, it follows
that $t_{\beta}$ and $t_{\gamma}$ have the following joint limiting
representation:

$\displaystyle\big{(}t_{\beta},t_{\gamma}\big{)}\Rightarrow\bigg{\\{}\tau_{1c}+c\Theta_{c},\tau_{2c}\bigg{\\}}\equiv\bigg{\\{}\tau_{1c}+c\theta_{c},\delta\tau_{1c}+(1-\delta^{2})^{1/2}\mathcal{Z}\bigg{\\}}$
(4.183)

where

$\displaystyle\tau_{1c}$
$\displaystyle=\left[\int\bigg{(}J_{c}^{\mu}(r)\bigg{)}^{2}\right]^{-1/2}\times\left\\{\int
J_{c}^{\mu}(r)dB_{1}\right\\}$ (4.184) $\displaystyle\tau_{2c}$
$\displaystyle=\left[\int\bigg{(}J_{c}^{\mu}(r)\bigg{)}^{2}\right]^{-1/2}\times\left\\{\int
J_{c}^{\mu}(r)dB_{2}\right\\}$ (4.185)

Furthermore,

$\displaystyle\Theta_{c}=\left[\int\bigg{(}J_{c}^{\mu}(r)\bigg{)}^{2}\right]^{1/2}$
(4.186)

and $\mathcal{Z}$ is a standard normal random variable distributed
independently of $\big{(}B_{1},J_{c}\big{)}$. The final expression in (4.183)
is obtained by writing

$\displaystyle B_{2}=\delta B_{1}+(1-\delta^{2})^{1/2}\widetilde{B}_{2}$
(4.187)

where $\widetilde{B}_{2}$ is a standard Brownian motion distributed
independently of $B_{1}$. The limiting distribution of $t_{\gamma}$ depends on
both $c$ and $\delta$. However, $\delta$ is consistently estimated by the
sample correlation between $\hat{\epsilon}_{1t}$ and $\hat{\epsilon}_{2t}$,
therefore we treat $\delta$ as known for the purposes of the asymptotic
theory. Moreover, a joint test of $c$ and $\gamma$ can be performed using an
appropriate Wald statistic.

We define the following vector

$\displaystyle\phi_{T}(\gamma_{0},c_{0})=\bigg{[}T\hat{\beta}-c_{0}\hat{b}(1),T(\hat{\gamma}-\gamma_{0})\bigg{]}^{\prime}$
(4.188)

where $\hat{b}(1)=1-\sum_{j=1}^{k}\hat{a}_{j-1}$ with
$\left\\{\hat{a}_{j}\right\\}$ are the estimators of $\left\\{a_{j}\right\\}$
from the OLS estimation of the ADF parametrization. Let $\hat{\Sigma}$ be the
$2\times 2$ matrix with typical element

$\displaystyle\hat{\sigma}_{ij}=\displaystyle\frac{1}{T-1}\sum_{t=1}^{T}e_{it}e_{jt}$
(4.189)

where $e_{1t}$ and $e_{2t}$ are the residuals of the corresponding equations.

Consider the test statistic

$\displaystyle
W\left(\gamma_{0},c_{0}\right)=\frac{1}{2}\phi_{T}(\gamma_{0},c_{0})^{\prime}\left(\hat{\Sigma}\
T^{-2}\sum_{t=2}^{T}\left(x^{\mu}_{t-1}\right)^{2}\right)\phi_{T}(\gamma_{0},c_{0}).$
(4.190)

Furthermore, extensions of the calculations in Stock,  (1991) show that, under
the null hypothesis $\mathbb{H}_{0}:\gamma=\gamma_{0}$ and $\beta=0$
(jointly), that is, $(\gamma,c)=(\gamma_{0},c_{0})$

$\displaystyle
W\left(\gamma_{0},c_{0}\right)\Rightarrow\frac{1}{2}\left(\tau^{2}_{1c_{0}}+\mathcal{Z}^{2}\right).$
(4.191)

######  Remark 20.

The key difficulty for tests of the hypothesis $\gamma=\gamma_{0}$ using
either $t_{\gamma}$ or $W\left(\gamma_{0},c_{0}\right)$ is that the limiting
distributions of these statistics333The exception is if $\delta=0$, in which
case $t_{\gamma}$ has a standard normal distribution for all values of $c$, as
well as for $\alpha$ fixed, that is, $|\alpha|<1$. depend on the local-to-
unity parameter $c$. Although $\alpha$ is constently estimable, $c$ is not, so
the asymptotic inference in this case cannot in general rely on simply
substituting a suitable estimator $\hat{c}$ for $c$ when selecting critical
values for tests of $\gamma$.

####  4.7.2. Optimal Inference in Predictive Regressions with Persistent
Regressors

Following Jansson and Moreira,  (2006) consider the case where
$\left\\{(y_{t},x_{t}^{\prime}\right\\}$ is generated by the predictive
regression model with the local to unity specification for the autocorrelation
matrix.

Let $\hat{\Omega}$ be a consistent estimator of

$\displaystyle\Omega=\begin{bmatrix}\omega_{yy}&\omega_{yx}\\\
\omega_{xy}&\omega_{xx}\end{bmatrix}=\underset{T\to\infty}{\text{lim}}T^{-1}\sum_{t=1}^{T}\sum_{s=1}^{T}\mathbb{E}\left[\begin{pmatrix}\epsilon_{yt}\\\
\Psi(L)\epsilon_{xt}\end{pmatrix}\begin{pmatrix}\epsilon_{ys}\\\
\Psi(L)\epsilon_{xs}\end{pmatrix}^{\prime}\right],$ (4.192)

which is the long-run variance of
$\left(\epsilon_{yt},\Psi(L)\epsilon_{xt}\right)^{\prime}$.

######  Example 21.

Consider the general I(1) vector process examined by Phillips and Durlauf,
(1986)

$\displaystyle x_{t}=x_{t-1}+v_{t},\ \ \text{where}\ \ x_{0}=0,$ (4.193)

and $v_{t}$ is a weakly stationary stochastic process with unconditional
variance $\mathbb{E}\left(v_{t}v_{t}^{\prime}\right)=G$ and long run
covariance matrix $\Omega=G+\Lambda+\Lambda^{\prime}$. Notice that the matrix
$\Lambda$ has the following representation

$\displaystyle\Lambda=\sum_{j=1}^{\infty}\mathbb{E}\left(v_{0}v_{j}^{\prime}\right)$
(4.194)

By considering the case of non-IID errors $x_{T}/\sqrt{T}$ converges to the
vector Brownian motion BM$(\Omega)$:

$\displaystyle T^{-1/2}\sum_{t=1}^{[Tr]}v_{t}\Rightarrow B(r).$ (4.195)

Then, we obtain that

$\displaystyle T^{-1}\sum_{t=1}^{T}x_{t-1}v_{t}^{\prime}$
$\displaystyle\Rightarrow\int_{0}^{1}B(r)dB(r)^{\prime}+\Lambda,$ (4.196)
$\displaystyle T^{-1}\sum_{t=1}^{T}x_{t}v_{t}^{\prime}$
$\displaystyle\Rightarrow\int_{0}^{1}B(r)dB(r)^{\prime}+G+\Lambda.$ (4.197)

######  Example 22.

(Cointegration in Systems of Equations) Consider the following DGP

$\displaystyle y_{t}$ $\displaystyle=\beta z_{t}+v_{t}$ (4.198) $\displaystyle
v_{t}$ $\displaystyle=\rho v_{t-1}+\epsilon_{1t}$ (4.199) $\displaystyle
z_{t}$ $\displaystyle=z_{t-1}+\epsilon_{2t}$ (4.200)

and

$\displaystyle\begin{pmatrix}\epsilon_{1t}\\\
\epsilon_{2t}\end{pmatrix}\sim\mathcal{N}\left(\begin{pmatrix}0\\\
0\end{pmatrix},\begin{pmatrix}\sigma_{1}^{2}&\theta\sigma_{1}\sigma_{2}\\\
\theta\sigma_{1}\sigma_{2}&\sigma_{2}^{2}\end{pmatrix}\right)$ (4.201)

Moreover, the system can be represented using the usual ECM form

$\displaystyle\begin{bmatrix}\Delta y_{t}\\\ \Delta
z_{t}\end{bmatrix}=\begin{bmatrix}\rho-1&\beta(1-\rho)\\\
0&0\end{bmatrix}\begin{bmatrix}y_{t-1}\\\
z_{t-1}\end{bmatrix}+\begin{bmatrix}u_{1t}\\\ u_{2t}\end{bmatrix}$ (4.202)

where $u_{1t}=\beta\epsilon_{2t}+\epsilon_{1t},u_{2t}=\epsilon_{2t}$, and
$\mathbb{E}\left(uu^{\top}\right)=\Lambda$, with

$\displaystyle\Lambda=\begin{bmatrix}\beta^{2}\sigma_{2}^{2}+\sigma_{1}^{2}+2\beta\theta\sigma_{1}\sigma_{2}&\beta\sigma_{2}^{2}+\theta\sigma_{1}\sigma_{2}\\\
\beta^{2}\sigma_{2}^{2}+\theta\sigma_{1}\sigma_{2}&\sigma_{2}^{2}\end{bmatrix}$
(4.203)

Therefore, within our setting we aim to compare the efficiency of the
likelihood ratio test and the Wald type statistics as structural break
detectors for the predictive regression model with persistent regressors at an
unknown break-point. In order to study the asymptotic behaviour of the
individual components we consider limit theorems involving invariance
principles for partial sums within the moderate deviations framework developed
by Phillips and Magdalinos,  (2007).

Notice that the stochastic process $x_{\lfloor ns\rfloor,n}:=x_{\lfloor
ns\rfloor}/d_{n}$ on the Skorohod space $\mathcal{D}[0,1]$ converges weakly to
a Gaussian process $G(s)$. Furthermore, on a suitably expanded probability
space there exists a process $\left(x_{t,n}^{0},1\leq t\leq
n\right)=_{d}\left(x_{t,n},1\leq t\leq n\right)$ such that

$\displaystyle\underset{0\leq s\leq 1}{\text{sup}}\left|x_{\lfloor
ns\rfloor,n}^{0}-G(s)\right|=o_{p}(1)$

####  4.7.3. Moderate Deviations from a Unit Root in Autoregressive Processes

######  Example 23.

Consider the autoregressive time series model (see, Phillips and Magdalinos,
(2005))

$\displaystyle x_{t}=\rho_{n}x_{t-1}+\epsilon_{t},\ \
t\in\left\\{1,...,n\right\\}$ (4.204)

such that the autocorrelation coefficient is defined as

$\displaystyle\rho_{n}=\left(1+\frac{c}{n^{\gamma}}\right),\ \ \gamma\in(0,1)$
(4.205)

Consider the component random elements $x_{[n^{\gamma}\bullet]}$ of the
Skorohod space. Furthermore, denote with

$\displaystyle
W_{n^{\gamma}}(\bullet):=\frac{1}{n^{\gamma/2}}\sum_{j=1}^{[n^{\gamma}\bullet]}\epsilon_{j}$
(4.206)

It is possible to approximate the partial sum process on
$\mathcal{D}[0,+\infty)$ for $x_{[n^{\gamma}\bullet]}$ by the Stieltjes
integral

$\displaystyle
U_{n^{\gamma}}(\bullet):=\int_{0}^{\bullet}e^{c(\bullet-r)}dW_{n^{\gamma}}(r)=\frac{1}{n^{\gamma/2}}\sum_{i=1}^{[n^{\gamma}\bullet]}e^{\frac{c}{n^{\gamma}}\left(n^{\gamma}\bullet-j\right)}\epsilon_{j}.$
(4.207)

Therefore for each $\gamma\in(0,1)$ and $c<0$, we have that

$\displaystyle\underset{t\in[0,n^{1-\gamma}]}{\text{sup}}\left|\frac{1}{n^{\gamma/2}}x_{[n^{\gamma}t]}-U_{n^{\gamma}}(t)\right|=o_{p}(1)\
\ \ \text{as}\ \ \ n\to\infty.$ (4.208)

######  Remark 21.

Therefore, by considering the above invariance law we can operate in the
familiar framework of Phillips, 1987b where $U_{n^{\gamma}}(t)$, and hence the
time series $x_{n}$ with appropriate normalization, converges to the linear
diffusion $\int_{0}^{t}e^{c(t-s)}dW(s)$, where $W$ is Brownian motion with
variance $\sigma^{2}$. Theorem 4.6 in the paper of PM provides a bridge
between stationary and local to unity autoregressions with weakly dependent
innovation errors. Furthermore, when the innovation error sequence is a linear
process, then the least squares estimator has been found to satisfy a Gaussian
limit theory with an asymptotic bias.

Notice that with the seminal paper of Phillips and Magdalinos,  (2007), the
authors introduce limit theorems and invariance principles for moderate
deviations from unit root. The particular framework is employed in time series
models such as autoregressive and predictive regression models with predictors
assumed to be generated with the LUR specification. A strong approximation
over the interval $[0,n^{1-\alpha}]$ for the partial sum process of i.i.d
errors can be constructed via an expanded probability space with a Brownian
motion $W(.)$ with variance $\sigma^{2}$ for which

$\displaystyle\underset{t\in[0,n^{1-\alpha}]}{\text{sup}}\left|W_{n^{\alpha}}-W(t)\right|=o_{a.s}\left(\frac{1}{n^{\frac{\alpha}{2}-\frac{1}{\nu}}}\right)$
(4.209)

Therefore, we have that one of the main results of the paper which provide a
uniform approximation of $n^{-\alpha/2}y_{\lfloor n^{\alpha}.\rfloor}$ by
$V_{n^{\alpha}}$ on $[0,n^{1-\alpha}]$. For each $\alpha\in(0,1)$ and $c<0$ we
have that

$\displaystyle\underset{t\in[0,n^{1-\alpha}]}{\text{sup}}\
\left|\frac{1}{n^{\alpha/2}}y_{\lfloor
n^{\alpha}t\rfloor}-V_{n^{\alpha}}(t)\right|=o_{p}(1),\ \ \text{as}\
n\to\infty$ (4.210)

The importance of the above result lies in the fact that an embedding of the
random element $V_{n^{\alpha}}(t)$ to the linear diffusion
$J_{c}(t):=\displaystyle\int_{0}^{t}e^{c(t-r)}dB(r)$ is possible. Using
integration by parts it can be shown that,

$\displaystyle\underset{t\in[0,n^{1-\alpha}]}{\text{sup}}\
\left|V_{n^{\alpha}}(t)-J_{c}(t)\right|\leq
2\underset{t\in[0,n^{1-\alpha}]}{\text{sup}}\
\left|B_{n^{\alpha}}(t)-B(t)\right|.$ (4.211)

Moreover, by Lemma 3.1 we obtain the following expression

$\displaystyle\underset{t\in[0,n^{1-\alpha}]}{\text{sup}}\
\left|V_{n^{\alpha}}(t)-J_{c}(t)\right|=\mathcal{O}\left(\frac{1}{n^{\frac{\alpha}{2}-\frac{1}{\nu}}}\right),\
\ \text{as}\ \ n\to\infty$ (4.212)

Therefore, we obtain that

$\displaystyle\underset{t\in[0,n^{1-\alpha}]}{\text{sup}}\
\left|\frac{1}{n^{\alpha/2}}y_{\lfloor
n^{\alpha}t\rfloor}-J_{c}(t)\right|=\mathcal{O}\left(\frac{1}{n^{\frac{\alpha}{2}-\frac{1}{\nu}}}\right),\
\ \text{as}\ \ n\to\infty.$ (4.213)

We note that the limit theory is established through a combination of a
functional law to a diffusion and a central limit law to a Gaussian random
variable. A more immediate consequence is the limit law of the random element
$y_{\lfloor n^{\alpha}.\rfloor}$ on the original probability space. For all
$j=0,...,\lfloor n^{1-\alpha}\rfloor-1$ and $p\in[0,1]$ we obtain
$\frac{1}{n^{\alpha/2}}y_{\lfloor n^{\alpha}j\rfloor+\lfloor
n^{\alpha}p\rfloor}$.

######  Lemma 9.

(Lemma 5.1 in Phillips and Magdalinos,  (2005)) For each $\alpha\in(0,1)$ and
$c>0$, we have that

  1. (a)

$\displaystyle\underset{t\in[0,n^{1-\alpha}]}{\text{sup}}\left|\int_{0}^{t}\rho_{n}^{-n^{\alpha}s}dB_{n^{\alpha}}(s)-\int_{0}^{t}e^{-cs}dB(s)\right|=o_{p}\left(\frac{1}{n^{\frac{\alpha}{2}-\frac{1}{\nu}}}\right)$
(4.214)

  2. (b)

$\displaystyle\underset{t\in[0,n^{1-\alpha}]}{\text{sup}}\left|\int_{0}^{t}\rho_{n}^{-\left(\lfloor
n^{\alpha}t\rfloor-\lfloor
n^{\alpha}s\rfloor\right)}dB_{n^{\alpha}}(s)-J_{-c}(t)\right|=o_{p}\left(\frac{1}{n^{\frac{\alpha}{2}-\frac{1}{\nu}}}\right)$
(4.215)

as $n\to\infty$, on the same probability space.

We analyze each of the two terms separately. The term containing the block
components can be written:

$\displaystyle U_{1n}$
$\displaystyle=\rho_{n}^{-2\kappa_{n}}\sum_{j=0}^{\lfloor
n^{1-\alpha}\rfloor-1}\frac{1}{n^{2\alpha}}\sum_{k=1}^{\lfloor
n^{\alpha}\rfloor}y^{2}_{\lfloor n^{\alpha}j\rfloor+k}$
$\displaystyle=\rho_{n}^{-2\kappa_{n}}\int_{0}^{\lfloor
n^{1-\alpha}\rfloor}\left(\int_{0}^{r}\rho_{n}^{\lfloor
n^{\alpha}r\rfloor-n^{\alpha}s}dB_{n^{\alpha}}(s)\right)^{2}dr+o_{p}(1).$

Taking the inner integral along $[0,r]=[0,\lfloor n^{1-\alpha}\rfloor]$ we
have, up to $o_{p}(1)$ that

$\displaystyle U_{1n}$ $\displaystyle=\left(\int_{0}^{\lfloor
n^{1-\alpha}\rfloor}\rho_{n}^{-n^{\alpha}s}dB_{n^{\alpha}}(s)\right)^{2}\rho_{n}^{-2\kappa_{n}}\int_{0}^{\lfloor
n^{1-\alpha}\rfloor}\rho_{n}^{2\lfloor n^{\alpha}r\rfloor}dr+R_{n},$ (4.216)

where the remainder term $R_{n}$ is shown in the Appendix to be $o_{p}(1)$.

The second integral on the right side can be evaluated directly to obtain

$\displaystyle\int_{0}^{\lfloor n^{1-\alpha}\rfloor}\rho_{n}^{2\lfloor
n^{\alpha}r\rfloor}dr=\frac{\rho_{n}^{2\kappa_{n}}}{2c}\left[1+O(1)\right],\ \
\text{as}\ \ n\to\infty.$ (4.217)

Furthermore, we obtain that

$\displaystyle U_{1n}=\frac{1}{2c}\left(\int_{0}^{\lfloor
n^{1-\alpha}\rfloor}e^{-cs}dB(s)\right)^{2}+o_{p}\left(\frac{1}{n^{\frac{\alpha}{2}-\frac{1}{\nu}}}\right)$
$\displaystyle=\frac{1}{2c}\left(\int_{0}^{\infty}e^{-cs}dB(s)\right)^{2}+o_{p}\left(\frac{1}{n^{\frac{\alpha}{2}-\frac{1}{\nu}}}\right)$

Then,
$\Psi_{n}^{\prime}\to_{\text{a.s}}\Psi=\sum_{j=1}^{\infty}\rho^{-j}u_{j}^{\prime}$,
and it follows by the Shorokhod representation theorem that
$\Psi_{n}\to_{d}\Psi$. Then, by joint weak convergence of $\Psi_{n}$ and
$Z_{n}$ it follows that
$\left(Z_{n},\Psi_{n}\right)\Rightarrow\left(Z,\Psi\right)$ as $n\to\infty$,
with $Z=_{d}\Psi$. The limiting random variables $\Psi$ and $Z$ can be shown
to be independent by modifying Anderson’s argument adjusted for weakly
dependent errors. The idea is that, as $n\to\infty$, $Z_{n}$ can be
approximated by the first $\lfloor L_{n}\rfloor$ elements of the sum
$\sum_{j=1}^{n}\rho^{-j}u_{j}$ whereas $\Psi_{n}$ can be approximated by the
last $\lfloor L_{n}\rfloor$ elements of the sum
$\sum_{j=1}^{n}\rho^{-(n-j)-1}u_{j}$, where
$\left(L_{n}\right)_{n\in\mathbb{N}}$ is a sequence increasing to $\infty$
with $L_{n}\leq n/3$ for each $n$.

Accordingly, we define

$\displaystyle Z_{n}^{*}:=\sum_{j=1}^{\lfloor L_{n}\rfloor}\rho^{-j}u_{j}\ \
\text{and}\ \ \Psi_{n}^{*}:=\sum_{j=n-\lfloor
L_{n}\rfloor+1}^{n}\rho^{-(n-j)}u_{j}=\sum_{k=1}^{\lfloor
L_{n}\rfloor-1}\rho^{-k}u_{n-k+1}.$ (4.218)

We may further approximate

$\displaystyle\Psi_{n}^{*}=\sum_{k=1}^{\lfloor
L_{n}\rfloor-1}\rho^{-k}\sum_{s=0}^{\infty}c_{s}\epsilon_{n-k+1-s}$
$\displaystyle=\sum_{k=1}^{\lfloor L_{n}\rfloor-1}\rho^{-k}\sum_{s=0}^{\lfloor
L_{n}\rfloor}c_{s}\epsilon_{n-k+1-s}+\sum_{k=1}^{\lfloor
L_{n}\rfloor-1}\rho^{-k}\sum_{s=\lfloor
L_{n}\rfloor+1}^{\infty}c_{s}\epsilon_{n-k+1-s}$
$\displaystyle=\Psi_{n}^{**}+\sum_{k=1}^{\lfloor
L_{n}\rfloor-1}\rho^{-k}\sum_{s=\lfloor
L_{n}\rfloor+1}^{\infty}c_{s}\epsilon_{n-k+1-s},$

We have that, $\Psi_{n}-\Psi_{n}^{*}=\sum_{k=\lfloor
L_{n}\rfloor}^{n}\rho^{-k}u_{n-k+1}$, and so

$\displaystyle\mathbb{E}\left|\Psi_{n}-\Psi_{n}^{*}\right|\leq\mathbb{E}\left|u_{1}\right|\sum_{k=\lfloor
L_{n}\rfloor+1}^{n}|\rho|^{-k}=\mathcal{O}\left(|\rho|\right)^{-L_{n}}$
(4.219)

Similar to the sample variance, the asymptotic behaviour of the sample
covariance is partly determined by elements of the time series $y_{t-1}u_{t}$
that do not belong to the block components

$\displaystyle\bigg{\\{}y_{\lfloor n^{\alpha}j\rfloor+k-1}u_{\lfloor
n^{\alpha}j\rfloor+k}:j=0,...,\lfloor n^{1-\alpha}\rfloor-1,k=1,...,\lfloor
n^{\alpha}\rfloor\bigg{\\}}$ (4.220)

Obtaining limits for the block components and the remaining time series
separately in a method similar to that used for the sample variance will work.
For the limit as $\alpha\to 1$, we have that $n^{1-\alpha}\to 1$, and so
$\lfloor n^{1-\alpha}\rfloor=1$ for $\alpha=1$, in which case $j=0$ in the
aforementioned blocking scheme. Therefore, the invariance principle proposed
by Phillips, 1987a $n^{-1/2}y_{\lfloor np\rfloor}\Rightarrow J_{c}(p)$ on
$\mathcal{D}\left([0,1]\right)$ which yield the usual local to unity limit
result

$\displaystyle
n\left(\hat{\rho}-\rho\right)\Rightarrow\frac{\displaystyle\int_{0}^{1}J_{c}(r)dB(r)}{\displaystyle\int_{0}^{1}J_{c}(r)^{2}dr}.$
(4.221)

In other words continuity in the limit theory cannot be achieved at the
boundary with the conventional local to unity asympotics, at least without
using the blocking construction.

##### Supplementary Results: Main Proofs

######  Proposition 8.

For each $x\in[0,M],M>0$, possibly depending on $n$, and real- valued,
measurable function $f$ on $[0,\infty)$

$\displaystyle\frac{1}{n^{\alpha/2}}\sum_{i=1}^{\lfloor
n^{\alpha}x\rfloor}f\left(\frac{i}{n^{\alpha}}\right)u_{i}=\int_{0}^{x}f(r)dB_{n^{\alpha}}(r).$
(4.222)

An immediate consequence of the above Proposition is the following identity.
For each $x\in[0,n^{1-\alpha}]$ and $m\in\mathbb{N}$

$\displaystyle\frac{1}{n^{\alpha/2}}\sum_{i=1}^{\lfloor
n^{\alpha}x\rfloor}f\left(\frac{i}{n^{\alpha}}\right)u_{i+m}=\int_{0}^{x}f(r)dB_{n^{\alpha}}\left(r+\frac{m}{n^{\alpha}}\right).$
(4.223)

######  Proposition 9.

For each $\alpha\in(0,1)$ we have that

$\displaystyle\underset{0\leq t\leq
n}{\text{max}}\left|\frac{\tilde{\epsilon}_{t}}{n^{\alpha/2}}\right|=o_{p}(1),\
\ \text{as}\ \ n\to\infty.$ (4.224)

###### Proof.

The arguments follows Phillips (1999). Summability conditions of
$\sum_{j=1}^{\infty}j|c_{j}|$ ensures that
$\tilde{\epsilon}_{t}=\sum_{j=1}^{\infty}\tilde{c}_{j}\epsilon_{t-j}$
converges absolutely almost surely. Therefore, Fatou’s lemma and the Minkowski
inequality give the following

$\displaystyle\mathbb{E}|\tilde{\epsilon}_{t}|^{\nu}$
$\displaystyle\leq\underset{N\to\infty}{\text{ lim inf}}\
\mathbb{E}\left|\sum_{j=0}^{N}\tilde{c}_{j}\epsilon_{t-j}\right|\leq\underset{N\to\infty}{\text{
lim
inf}}\left[\sum_{j=0}^{N}\big{(}\mathbb{E}\left|\tilde{c}_{j}\epsilon_{t-j}\right|^{\nu}\big{)}^{\frac{1}{\nu}}\right]^{\nu}$
$\displaystyle=\mathbb{E}|\epsilon_{0}|^{\nu}\underset{N\to\infty}{\text{ lim
inf}}\left(\sum_{j=0}^{N}\left|\tilde{c}_{j}\right|\right)^{\nu}=\mathbb{E}|\epsilon_{0}|^{\nu}C_{2}^{\nu}$

where $C_{2}=\sum_{j=0}^{\infty}\left|\tilde{c}_{j}\right|<\infty$.

Thus, for any $\delta>0$ the Markov inequality gives

$\displaystyle\mathbb{P}\left(\underset{0\leq t\leq
n}{\text{max}}|\tilde{\epsilon}_{t}|>\delta
n^{\alpha/2}\right)\leq\sum_{t=0}^{n}\mathbb{P}\left(|\tilde{\epsilon}_{t}|>\delta
n^{\alpha/2}\right)\leq\sum_{t=0}^{n}\frac{\mathbb{E}|\tilde{\epsilon}_{t}|^{\nu}}{\delta^{\nu}n^{\nu\alpha/2}}$
$\displaystyle\leq\frac{\mathbb{E}|\tilde{\epsilon}_{0}|^{\nu}C_{2}^{\nu}}{\delta^{\nu}}\frac{n+1}{n^{\nu\alpha/2}}=o(1).$

if and only if $\frac{\nu\alpha}{2}>1$. ∎

######  Proposition 10.

The following two results hold

  1. (a)

Let $y_{nt}^{*}:=\sum_{i=1}^{n}$. Then, for each $\alpha\in(0,\frac{1}{2}]$

$\displaystyle\frac{1}{n^{\frac{1+3\alpha}{2}}}\sum_{t=1}^{n}y_{t-1}\widetilde{\epsilon}_{t}=\frac{1}{n^{\frac{1+3\alpha}{2}}}\sum_{t=1}^{n}y^{*}_{nt-1}\widetilde{\epsilon}_{t}+o_{p}(1),\
\ \text{as}\ n\to\infty.$ (4.225)

  2. (b)

Let
$\gamma_{m}(h)=\mathbb{E}\left[\tilde{\epsilon}_{t}u_{t-h}\right]=\sigma^{2}\sum_{j=0}^{\infty}c_{j}\tilde{c}_{j+h}$
for $h\geq 0$ and $m_{n}=\sum_{i=1}^{\infty}\rho_{n}^{i-1}\gamma_{m}(i)$. Then
it holds that,

$\displaystyle\underset{n\to\infty}{\text{lim}}m_{n}=\sum_{i=1}^{\infty}\gamma_{m}(i).$
(4.226)

######  Proposition 11.

For part (a), we can write the following expression

$\displaystyle\frac{1}{n^{\frac{1+3\alpha}{2}}}\sum_{t=1}^{n}\left(y^{*}_{nt-1}-y_{t-1}\right)\tilde{\epsilon}_{t}$
$\displaystyle=\frac{1}{n^{\frac{1+3\alpha}{2}}}\sum_{t=1}^{n}\left[\left(\sum_{t=1}^{n}\rho_{n}^{i}u_{t-i-1}-y_{0}\rho_{n}^{t}\right)\right]\tilde{\epsilon}_{t}$
$\displaystyle=\frac{1}{n^{\frac{1+3\alpha}{2}}}\sum_{t=1}^{n}\sum_{t=1}^{n}\rho_{n}^{i}u_{t-i-1}\tilde{\epsilon}_{t}+o_{p}\left(\frac{1}{n^{\frac{1+3\alpha}{2}}}\right),$

Since, by Proposition A3 and the fact that
$\sum_{t=1}^{n}\left|\rho_{n}\right|^{t}=\mathcal{O}\left(n^{\alpha}\right)$,

$\displaystyle\left|\frac{1}{n^{\frac{1+3\alpha}{2}}}\sum_{t=1}^{n}y_{0}\rho_{n}^{t}\tilde{\epsilon}_{t}\right|\leq\left|\frac{y_{0}}{n^{\alpha/2}}\right|\underset{1\leq
t\leq
n}{\text{max}}\left|\frac{\tilde{\epsilon}_{t}}{n^{\alpha/2}}\right|\frac{1}{n^{\frac{1+\alpha}{2}}}\sum_{t=1}^{n}\left|\rho_{n}\right|^{t}=o_{p}\left(\frac{1}{n^{\frac{1+3\alpha}{2}}}\right).$
(4.227)

##### Proof of Lemma 4.1

Consider the stochastic process

$\displaystyle\xi_{t}:=\sum_{j=0}^{\infty}\sum_{j=0}^{\infty}c_{j}\tilde{c}_{j}\epsilon_{t-j}\epsilon_{t-i}$
(4.228)

is a stationary process with autocovariance function given by

$\displaystyle\gamma_{\xi}(h)=\sigma^{4}\sum_{j=h}^{\infty}\sum_{i=j+1}^{\infty}c_{j-h}\tilde{c}_{i-h}\tilde{c}_{i},\
\ h\in\mathbb{Z}.$ (4.229)

Then by Theorem of Inrangimov and Linnik (1971)

$\displaystyle\mathbb{E}\left[\left(\sum_{t=1}^{n}\xi_{t}\right)^{2}\right]\leq
n\sum_{h=-\infty}^{\infty}\left|\gamma_{\xi}(h)\right|,$ (4.230)

Therefore,

$\displaystyle\mathbb{E}\left[\left(\frac{1}{n^{\frac{1+\alpha}{2}}}\sum_{t=1}^{n}\xi_{t}\right)^{2}\right]\leq\frac{1}{n^{\alpha}}\sum_{h=-\infty}^{\infty}\left|\gamma_{\xi}(h)\right|<\frac{2}{n^{\alpha}}\sum_{h=0}^{\infty}\left|\gamma_{\xi}(h)\right|=\mathcal{O}\left(\frac{1}{n^{\alpha}}\right)$
(4.231)

provided that $\sum_{h=0}^{\infty}\left|\gamma_{\xi}(h)\right|<\infty$.

To show summability of the covariance function $\gamma_{\xi}$, write

$\displaystyle\sum_{h=0}^{\infty}\left|\gamma_{\xi}(h)\right|$
$\displaystyle\leq\sigma^{4}\sum_{h=0}^{\infty}\sum_{j=h}^{\infty}\left|c_{j-h}c_{j}\right|\left|\sum_{i=j+1}^{\infty}\tilde{c}_{i-h}\tilde{c}_{i}\right|$
$\displaystyle\leq\sigma^{4}\sum_{h=0}^{\infty}\sum_{j=h}^{\infty}\left|c_{j-h}c_{j}\right|\left(\sum_{i=j+1}^{\infty}\tilde{c}_{i-h}^{2}\right)^{1/2}\left(\sum_{i=j+1}^{\infty}\tilde{c}_{i}^{2}\right)^{1/2}$
$\displaystyle\leq\sigma^{4}\left(\sum_{i=0}^{\infty}\tilde{c}_{i}^{2}\right)\sum_{h=0}^{\infty}\sum_{j=h}^{\infty}\left|c_{j-h}\right|\left|c_{j}\right|$
$\displaystyle=C_{4}\sum_{i=0}^{\infty}\left|c_{j}\right|\sum_{j=h}^{\infty}\left|c_{h}\right|\leq\sigma^{4}C_{4}C_{1}^{2}<\infty.$

Once the fitted stationary process is obtained, the variance $\Sigma$ of
$\left\\{w_{t}\right\\}$ can be consistently estimated as

$\displaystyle\hat{\Sigma}=\frac{1}{n}\sum_{t=1}^{n}\hat{w}_{t}\hat{w}_{t}.$
(4.232)

##### Summary

Although it is not shown explicitly, the idea of stationary transformations
can be used also for the essentially singular model to construct the
regression which yields asympotically invariant, with respect to nuisance
parameters, tests.

######  Lemma 10.

Suppose that $X_{ni}=\left[1+\frac{z_{n}}{n}\right]X_{n,i-1}+v_{ni}$, for
$i=1,...,n,X_{n0}=0$, where $z_{n}\to z$ and
$v_{ni}=\sum_{j=0}^{\infty}c_{j}(a_{n})e_{i-j}$ with $\alpha_{n}\to\alpha$ and
with the $c_{j}$’s satisfying
$\sum_{j=0}^{\infty}\left|c_{j}(\alpha)\right|<\infty$. Then, the stochastic
process $n^{-1/2}X_{\lfloor nt\rfloor}$ converges in law in the Skorokhod
space to $\int_{0}^{t}\text{exp}\left[z(t-s)\right]dW(t)$, where $W(t)$ is the
two-dimensional Brownian motion with variance given by

$\displaystyle\left(\sum_{r=0}^{\infty}c_{r}(\alpha)\right)\left(\sum_{r=0}^{\infty}c_{r}(\alpha)\right)^{\prime}$
(4.233)

is finite and non-singular.

The use of moderate deviation principles are useful in understanding the
various asymptotic theory results that apply across the parameter space of the
autoregressive coefficient of the nonstationary autoregressive model. In
particular, for the case of quantile autoregressions and quantile predictive
regression models a relevant framework is proposed by Katsouris, 2022a .

####  4.7.4. Multivariate Predictive Regression Model

Consider the multivariate predictive regression model with lag-augmentation

$\displaystyle\boldsymbol{y}_{t}$
$\displaystyle=\boldsymbol{A}\boldsymbol{y}_{t-1}+\boldsymbol{u}_{t},\ \ \ \ \
\ \ $ (4.234) $\displaystyle\boldsymbol{x}_{t}$
$\displaystyle=\mathbf{R}_{n}\boldsymbol{x}_{t-1}+\boldsymbol{v}_{t},\ \ \ \ \
\ \ \text{with}\ \ \
\mathbf{R}_{n}=\left(\boldsymbol{I}_{p}-\frac{\boldsymbol{C}_{p}}{n^{\alpha}}\right)$
(4.235)

where
$\left(\boldsymbol{y}_{t},\boldsymbol{x}_{t}\right)\in\mathbb{R}^{p\times 1}$
are $p-$dimensional vectors for $t\in\left\\{1,...,n\right\\}$ and
$\left\\{\boldsymbol{A}\right\\}\in\mathbb{R}^{p\times p}$ are $(p\times p)$
matrices of coefficients and
$\boldsymbol{C}=\mathsf{diag}\left\\{c_{1},...,c_{p}\right\\}$ is a $(p\times
p)$ diagonal matrix with the coefficients of persistence. The degree of
persistence of the regressors is determined by the value of the unknown
persistence coefficients $c_{i}$’s which are assumed to be positive constants.
Moreover, the common exponent rate for the persistence coefficients take
values, $\alpha=1$, which covers the local-unit-root (LUR) regressors and
$\alpha\in(0,1)$ the case of mildly integrated (MI) regressors as defined by
Magdalinos and Phillips,  (2009) and Kostakis et al.,  (2015). When $\alpha=0$
we assume the presence of stationary regressors, for a suitably restricted
coefficient matrix $\left(\boldsymbol{I}_{p}-\boldsymbol{C}_{p}\right)$. A
more recent application is given by Yang et al.,  (2020).

######  Assumption 8.

Let
$\boldsymbol{e}_{t}=\left(\boldsymbol{u}^{\prime}_{t},\boldsymbol{\epsilon}_{t}^{\prime}\right)^{\prime}$
be a $2p-$dimensional vector. The innovation sequence $\boldsymbol{e}_{t}$ is
a conditionally homoscedastic martingale difference sequence (m.d.s) such that
the following conditions hold:

  1. A1.

$\mathbb{E}\left(\boldsymbol{e}_{t}|\mathcal{F}_{t-1}\right)=\boldsymbol{0}$,
where
$\mathcal{F}_{t}=\sigma\left(\boldsymbol{e}_{t},\boldsymbol{e}_{t-1},...\right)$
is an increasing sequence of $\sigma-$fields.

  2. A2.

$\mathbb{E}\left(\boldsymbol{e}_{t}\boldsymbol{e}_{t}^{\prime}|\mathcal{F}_{t-1}\right)=\boldsymbol{\Sigma}_{ee}$,
where $\boldsymbol{\Sigma}_{ee}\in\mathbb{R}^{p\times p}$ is a positive-
definite covariance matrix, which has the following form:

$\displaystyle\boldsymbol{\Sigma}_{ee}=\begin{bmatrix}\boldsymbol{\Sigma}_{uu}&\boldsymbol{\Sigma}_{uv}\\\
\boldsymbol{\Sigma}_{vu}&\boldsymbol{\Sigma}_{vv}\end{bmatrix}>0.$

such that
$\boldsymbol{\Sigma}_{uu},\boldsymbol{\Sigma}_{uv},\boldsymbol{\Sigma}_{vv}\in\mathbb{R}^{p\times
p}$ where $p$ is the number of regressors.

  3. A3.

The innovation to $x_{t}$ is a linear process with the representation below

$\displaystyle\boldsymbol{v}_{t}:=\boldsymbol{\Psi}(\boldsymbol{L})\boldsymbol{\epsilon}_{t}\equiv\displaystyle\sum_{j=0}^{\infty}\boldsymbol{\Psi}_{j}\boldsymbol{\epsilon}_{t-j},\
\ \
\boldsymbol{\epsilon}_{t}\sim^{i.i.d}\left(\boldsymbol{0},\boldsymbol{\Sigma}_{ee}\right)$

where $\left\\{\boldsymbol{\Psi}\right\\}_{j=0}^{\infty}$ is a sequence of
absolute summable constant matrices such that
$\sum_{j=0}^{\infty}\boldsymbol{\Psi}_{j}$ has full rank and
$\boldsymbol{\Psi}_{0}=\boldsymbol{I}_{p}$ with $\boldsymbol{\Psi}(1)\neq 0$,
allowing for the presence of serial correlation in the innovations of the
predictive regression model. Suppose that(A1)-(A3) hold, the following FCLT
applies

$\displaystyle\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}\boldsymbol{w}_{t}=\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}\begin{bmatrix}\boldsymbol{u}_{t}\\\
\boldsymbol{v}_{t}\end{bmatrix}\Rightarrow\
\text{BM}\left(\boldsymbol{\Sigma}_{w}\right)=\begin{bmatrix}\boldsymbol{B}_{u}(r)\\\
\boldsymbol{B}_{v}(r)\end{bmatrix}\equiv\boldsymbol{\Sigma}_{w}^{1/2}\boldsymbol{W}(r),$
(4.236)

where
$\boldsymbol{w}_{t}=\left(\boldsymbol{u}_{t},\boldsymbol{v}_{t}\right)^{\prime}$
and $\left\\{\boldsymbol{W}(r):0\leq r\leq 1\right\\}$ is the standard
(vector) BM, in $\mathcal{D}\left([0,1]\right)$.

Moreover, it holds that $B(r)=\omega W(r)$ for some $r\in[0,1]$ and
$\omega:=\sum_{j=-\infty}^{\infty}\gamma_{u}(j)=C(1)^{2}\sigma^{2}$, the one-
sided covariance of $v_{t}$. Moreover, by definition it holds that
$W_{c}(r)=\int_{0}^{r}e^{c(r-s)}dW(s)$. Notice that, expression (4.236)
provides an invariance principle for the partial sum process of
$\boldsymbol{w}_{t}$, which implies that the partial sum of
$\boldsymbol{w}_{t}$, weakly converges to the stochastic quantity
$\boldsymbol{\Sigma}_{w}^{1/2}\boldsymbol{W}(r)$, that is, a $p-$dimensional
Brownian process with covariance matrix $\boldsymbol{\Sigma}_{w}$. Within our
context we employ the local-unit-root limit theory as proposed by the seminal
paper of Phillips, 1987a , such that
$\boldsymbol{x}_{[nr]}/\sqrt{n}\Rightarrow\boldsymbol{J}_{c}(r)$, where
$\boldsymbol{J}_{c}(r)$ is a $p-$dimensional Gaussian process defined as below

$\displaystyle\displaystyle\boldsymbol{J}_{c}(r)=\int_{0}^{r}e^{(r-s)\boldsymbol{C}_{p}}d\boldsymbol{W}(s),\
\ \ \ r\in(0,1).$ (4.237)

that satisfies the Black-Scholes differential equation
$d\boldsymbol{J}_{c}(r)\equiv\boldsymbol{C}_{p}\boldsymbol{K}_{c}(r)+d\boldsymbol{B}_{v}(r)$,
with $\boldsymbol{K}_{c}(r)=0$, implying also that
$\boldsymbol{K}_{c}(r)\equiv\sigma_{v}\boldsymbol{J}_{c}(r)$, where
$\displaystyle\boldsymbol{J}_{c}(r)=\int_{0}^{r}e^{(r-s)\boldsymbol{C}_{p}}d\boldsymbol{W}(s)$.
Moreover, since in this paper we consider time series nonstationarity in the
form of regressors exhibiting mildly integradeness, then we also employ the
asymptotic results proposed by Phillips and Magdalinos,  (2007) since the
invariance principle $x_{[nr]}$ requires appropriate normalization. Notice
that $\boldsymbol{K}_{c}(r)$ represents the Ornstein-Uhlenbeck, (OU) process,
which encompasses the unit root case such that
$\boldsymbol{J}_{c}(r)\equiv\boldsymbol{B}_{v}(r)$, for
$\boldsymbol{C}_{p}=0$. Moreover, the assumption of a local-unit-root
specification for the predictors of the model such that
$\boldsymbol{x}_{t}=\left(\boldsymbol{I}_{p}-\frac{\boldsymbol{C}_{p}}{n^{\alpha}}\right)$
$\boldsymbol{x}_{t-1}+\boldsymbol{v}_{t}$ and more specifically by allowing
the autocorrelation coefficient to be of the form
$\boldsymbol{\Phi}_{n}=\left(\boldsymbol{I}_{p}-\frac{\boldsymbol{C}_{p}}{n^{\alpha}}\right)$,
permits to consider the $\boldsymbol{K}_{c}(r)$ Gaussian process given by
(4.237), as a stochastic approximation.

Proof. Expanding the expression $x_{t}z_{t}^{\prime}$, vectorizing and summing
over $t\in(1,...,n)$

$\displaystyle
x_{t}z_{t}^{\prime}=R_{n}x_{t-1}z_{t-1}^{\prime}R_{n}^{\prime}+R_{n}x_{t-1}u_{xt}^{\prime}+u_{xt}z_{t-1}^{\prime}R_{n}^{\prime}+u_{xt}u_{xt}^{\prime}$

We have that

$\displaystyle\mathcal{L}=\left[I_{K^{2}}-R_{zn}\otimes
R_{n}\right]\frac{1}{n}\sum_{t=1}^{n}\text{vec}\left(x_{t-1}z_{t-1}^{\prime}\right)$
$\displaystyle=\left[I_{K}+o_{p}(1)\right]\text{vec}\left[\frac{1}{n}\sum_{t=1}^{n}x_{t-1}u_{xt}^{\prime}+\frac{1}{n}\sum_{t=1}^{n}u_{xt}z_{t-1}^{\prime}+\frac{1}{n}\sum_{t=1}^{n}u_{xt}u_{xt}^{\prime}\right]$
$\displaystyle=\text{vec}\left[\frac{1}{n}\sum_{t=1}^{n}x_{t-1}u_{xt}^{\prime}+\Lambda^{\prime}_{xx}+\mathbb{E}\left(u_{x1}u_{x1}^{\prime}\right)\right]+o_{p}(1)$

Therefore, for $\delta<\gamma$, which holds since $\gamma=1$ and
$\delta\in(0,1)$, then we obtain

$\displaystyle\left[I_{K^{2}}-R_{zn}\otimes
R_{n}\right]=-\frac{1}{n^{\delta}}\left(C_{z}\otimes
I_{K}\right)\left[I_{K}+O_{p}\left(\frac{1}{n^{\gamma-\delta}}\right)\right]$
(4.238)

Note for example, that

$\displaystyle R_{zn}\otimes
R_{n}=\left(I_{K}+\frac{C_{z}}{n^{\delta}}\right)\otimes\left(I_{K}+\frac{C}{n^{\gamma}}\right)=I_{K^{2}}+\frac{C\otimes
I_{K}}{n^{\gamma}}+\frac{C_{z}\otimes I_{K}}{n^{\delta}}+\frac{C_{z}\otimes
C}{n^{\gamma+\delta}}$

Thus,

$\displaystyle-\frac{1}{n^{1+\delta}}\left(C_{z}\otimes
I_{K}\right)\sum_{t=1}^{n}\text{vec}\left(x_{t-1}z_{t-1}\right)$
$\displaystyle\Rightarrow\left(\int_{0}^{1}J_{C}dB^{\prime}_{x}+\Omega_{xx}\right)$
(4.239)
$\displaystyle\frac{1}{n^{1+\delta}}\sum_{t=1}^{n}x_{t-1}z_{t-1}^{\prime}$
$\displaystyle\Rightarrow-\left(\int_{0}^{1}J_{C}dB^{\prime}_{x}+\Omega_{xx}\right)C_{z}^{-1}$
(4.240)