Part (L2). Since, $J_{c}(r)=\int_{0}^{r}e^{-c(r-s)}dB_{v}(s)$ and using
expression (14) in PM for the univariate predictive regression with single
regressor we have that

$\displaystyle\frac{\sum_{t=1}^{n}z_{t-1}^{2}}{n^{1+\delta}}\overset{p}{\to}V_{c_{z}}:=\int_{0}^{\infty}\sigma_{v}^{2}e^{-2rc_{z}}dr=\frac{\sigma^{2}_{v}}{2c_{z}}$
(4.241)

The corresponding result for the the univariate predictive regression with
multiple regressors is as below

$\displaystyle\frac{1}{T^{1+\delta}}\sum_{t=1}^{T}z_{t-1}z_{t-1}^{\prime}\overset{p}{\to}V_{zz}:=\int_{0}^{\infty}e^{rC_{z}}\Omega_{xx}e^{rC_{z}}dr$
(4.242)

Recall that by definition the functional
$\underline{J}_{c}(r)=\int_{0}^{r}e^{(r-s)C}d\underline{B}(s)$, is considered
to be a vector diffusion process and satisfies the stochastic differential
equation system

$\displaystyle
d\underline{J}_{c}(r)=C\underline{J}_{c}(r)dr+d\underline{B}(r),\ \
\underline{J}_{c}(0)=0$ (4.243)

Therefore, using the initial condition as well we can also write

$\displaystyle\underline{J}_{c}(r)=\underline{B}(r)+C\int_{0}^{r}e^{(r-s)C}\underline{B}(s)ds$
(4.244)

Since $\underline{J}_{c}(r)$ is a Gaussian process and for fixed $r$ then it
can be easily proved that the finite dimensional distribution (see e.g.,
Phillips,  (1988))

$\displaystyle\underline{J}_{c}(r)\Rightarrow N(0,Q)\ \ \text{where}\
Q=\int_{0}^{r}e^{(r-s)C}\Omega e^{(r-s)C^{\prime}}ds$ (4.245)

where $\Omega$ the $p\times p$ covariance matrix of $\underline{B}(r)$ the
$p-$dimensional Brownian motion.

Part (L3). The weakly convergence result to a mixed Gaussian distribution
below

$\displaystyle\frac{1}{T^{\frac{1+\delta}{2}}}\sum_{t=1}^{T}\left(z_{t-1}\otimes
u_{t}\right)\Rightarrow N\left(0,V_{zz}\otimes\Sigma_{uu}\right)$ (4.246)

shows that the limit distribution of
$T^{-(1+\delta)/2}\sum_{t=1}^{T}\left(z_{t-1}\otimes u_{t}\right)$ is Gaussian
with mean zero and covariance matrix equal to the probability limit of
$T^{-(1+\delta)/2}\sum_{t=1}^{T}\left(z_{t-1}\otimes u_{t}\right)$, which is
equal to $V_{zz}\otimes\Sigma_{uu}$, where
$V_{zz}:=\int_{0}^{\infty}e^{rC_{z}}\Omega_{vv}e^{rC_{z}}dr$. Similarly the
limit distribution of $T^{-(1+\delta)/2}\sum_{t=1}^{T}\left(x_{t-1}\otimes
u_{t}\right)\Rightarrow N\left(0,V_{xx}\otimes\Sigma_{uu}\right)$, where
$V_{xx}:=\int_{0}^{\infty}e^{rC}\Omega_{vv}e^{rC}dr$, is proved in Lemma 3.3
of Magdalinos and Phillips,  (2009).

We prove the weakly convergence result given above for the univariate
predictive regression (applied to both the case of single versus multiple
regressors). See also case 2 of Lemma B3 in Appendix of KMS, where the result
for the multivariate predictive regression is proved. We consider the partial
sum process $S_{t}=\sum_{j=1}^{t}z_{j}$ where $z_{j}$ is the IVX instrument.
Therefore, we obtain $\Delta S_{t-1}\equiv S_{t-1}-S_{t-2}=z_{t-1}$, i.e.,

$\displaystyle\frac{1}{T^{1+\delta}}\sum_{t=1}^{T}x_{t-1}z_{t-1}$
$\displaystyle=\frac{1}{T^{1+\delta}}\sum_{t=1}^{T}x_{t-1}\Delta
S_{t-1}=\frac{1}{T^{1+\delta}}\sum_{t=1}^{T}x_{t-1}\left(S_{t-1}-S_{t-2}\right)$
$\displaystyle=\frac{1}{T^{1+\delta}}\sum_{t=1}^{T}S_{t-1}x_{t-1}-\frac{1}{T^{1+\delta}}\sum_{t=1}^{T}S_{t-2}x_{t-1}$
$\displaystyle=\frac{1}{T^{1+\delta}}\left\\{\left[\sum_{t=1}^{T}S_{t-1}x_{t-1}-\sum_{t=1}^{T}S_{t-2}x_{t-2}\right]-\sum_{t=1}^{T}S_{t-2}\Delta
x_{t-1}\right\\}$
$\displaystyle=\frac{1}{T^{1+\delta}}\left(S_{T-1}x_{T-1}-S_{0}x_{0}\right)-\frac{1}{T^{1+\delta}}\sum_{t=1}^{T}S_{t-2}\Delta
x_{t-1}$

The first term of the above expression is considered to be asymptotically
negligible, i.e., $o_{p}(1)$. Thus, we consider the asymptotic behaviour of
the second term by expanding the expression further. We have that
$\displaystyle\Delta x_{t}=\frac{C}{T}x_{t-1}+v_{t}$, implying

$\displaystyle\frac{1}{T^{1+\delta}}\sum_{t=1}^{T}S_{t-2}\Delta
x_{t-1}=\frac{1}{T^{1+\delta}}\sum_{t=1}^{T}S_{t-2}v_{t-1}+\frac{C}{T^{2+\delta}}\sum_{t=1}^{T}S_{t-2}x_{t-2}$
(4.247)

Therefore, the above gives the following

$\displaystyle\frac{1}{T^{1+\delta}}\sum_{t=1}^{T}x_{t-1}z_{t-1}$
$\displaystyle=-\frac{1}{T^{1+\delta}}\left\\{\sum_{t=1}^{T}S_{t-2}v_{t-1}+\frac{C}{T}\sum_{t=1}^{T}S_{t-2}x_{t-2}\right\\}=-\frac{1}{T^{1+\delta}}\left\\{\sum_{t=1}^{T}S_{t-2}\left(v_{t-1}+\frac{C}{T}x_{t-2}\right)\right\\}$
$\displaystyle=-\frac{1}{T^{1+\delta}}\left\\{\sum_{t=1}^{T}\sum_{j=1}^{t-2}z_{j}\left(v_{t-1}+\frac{C}{T}x_{t-2}\right)\right\\}$

We prove the limiting behaviour of the above terms using the convergence of
the martingale expression (16) in PM and the fact that $x_{t}$ is a local to
unity process.

Then, the following weakly convergence of (4.248) and (4.249) follows

$\displaystyle\frac{1}{T^{1+\delta}}\sum_{t=1}^{T}\sum_{j=1}^{t-2}z_{j}v_{t-1}$
$\displaystyle\Rightarrow\int_{0}^{1}\underline{J}_{c}(r)dB_{v}C_{z}^{-1}$
(4.248)
$\displaystyle\frac{C}{n^{2+\delta}}\sum_{t=1}^{n}\sum_{j=1}^{t-2}z_{j}x_{t-2}$
$\displaystyle\Rightarrow\Omega_{vv}C_{z}^{-1}$ (4.249)
$\displaystyle\frac{1}{T^{1+\delta}}\sum_{t=1}^{T}x_{t-1}z_{t-1}$
$\displaystyle\Rightarrow-\left(\int_{0}^{1}\underline{J}_{c}(r)dB_{v}+\Omega_{vv}\right)C^{-1}_{z}$
(4.250)

####  4.7.5. Predictive Regression with Multiple Regressors

Consider the predictive regression model such as

$\displaystyle y_{t}$
$\displaystyle=\beta_{0}+\boldsymbol{\beta}^{\prime}_{1}\boldsymbol{x}_{t-1}+u_{t},$
(4.251) $\displaystyle\boldsymbol{x}_{t}$
$\displaystyle=\boldsymbol{R}_{T}\boldsymbol{x}_{t-1}+\boldsymbol{v}_{t}$
(4.252)

where $y_{t}$ is a scalar vector and $\boldsymbol{x}_{t-1}$ is a
$k-$dimensional vector of regressors. Denote with
$\boldsymbol{\eta}_{t}=\big{[}u_{t},\boldsymbol{v}_{t}^{\prime}\big{]}^{\prime}$.

Then, the predictive regression above includes a model intercept and
furthermore we do not demean the random variables in order to remove the model
intercept. In addition, we assume that
$\boldsymbol{R}_{T}\neq\boldsymbol{I}_{T}$ which would correspond to the
integrated regressor case.

Fully Modified OLS Under the aforementioned conditions the functional law

$\displaystyle\displaystyle T^{-1/2}\sum_{t=1}^{\lfloor
Tr\rfloor}\overset{d}{\to}\boldsymbol{B}(r)\equiv BM(\Omega)$

holds for partial sums of $\boldsymbol{\eta}_{t}$. Define the partition
$\boldsymbol{B}=\big{(}B_{u},\boldsymbol{B}_{v}^{\prime}\big{)}^{\prime}$.

######  Proposition 12.

Under Assumptions it holds that

$\displaystyle
T\left(\widehat{A}^{+}-A\right)\overset{d}{\to}\left(\int_{0}^{1}d\boldsymbol{B}_{u.v}\boldsymbol{B}^{\prime}_{v}\right)\left(\int_{0}^{1}\boldsymbol{B}_{v}\boldsymbol{B}^{\prime}_{v}\right)$
(4.253)

When $\Omega_{uu.v}$ has full rank, the rate of convergence of the FM-OLS
estimator is determined by the rates of weak convergence of the sample
covariances and the rate of nonparametric estimation of $\Omega$ and
$\Gamma^{+}$ does not apply any role. In particular, the OLS estimator
$\widehat{A}^{+}=Y^{\prime}X\left(X^{\prime}X\right)^{-1}$ and employs
corrections for endogeneity in the regressor $x_{t}$, leading to the
transformed dependent variable

$\displaystyle\widehat{y}_{t}^{+}=y_{t}-\widehat{\Omega}_{0x}\widehat{\Omega}_{xx}^{-1}\big{(}x_{t}-x_{t-1}\big{)}$
(4.254)

and a bias correction term involving

$\displaystyle\widehat{\Delta}_{0x}^{+}=\widehat{\Delta}_{0x}-\widehat{\Omega}_{0x}\widehat{\Omega}_{xx}^{-1}\widehat{\Delta}_{xx}$
(4.255)

which is constructed in the usual way using consistent nonparametric
estimators of submatrices of the long run and one sided long run quantities
$\Omega$ and $\Gamma^{+}$. In particular, the FM-OLS estimators removes
asymptotic bias and increases efficiency by correcting both the long run
serial correlation in $u_{t}$ and endogeneity in $x_{t}$ causes by the long
run correlation between $u_{0t}$ and $u_{xt}$.

######  Example 24 (Mildly Explosive Autoregression).

Consider the martingale difference property of
$\left\\{\hat{\epsilon}_{n,t}-\mathbb{E}_{\mathcal{F}_{t-1}}\left[\epsilon_{1t}\right]:t\geq
1\right\\}$ which yields that

$\displaystyle\left\lVert\sum_{t=1}^{k_{n}}a_{n,t}\big{(}\epsilon_{1t}-\mathbb{E}_{\mathcal{F}_{t-1}}\left[\epsilon_{1t}\right]\big{)}^{2}\right\rVert\leq\Delta_{n}^{2}\sum_{t=1}^{k_{n}}a_{n,t}^{2}\to
0.$ (4.256)

by some choice of $(\Delta_{n})_{n\in\mathbb{N}}$. Furthermore, the fact that
$(y_{t})$ is $\mathcal{F}_{t}-$adapted implies that, for any fixed integer
$M>0$,

$\displaystyle\sum_{t=1}^{k_{n}}a_{n,t}y_{t}=\sum_{m=0}^{M-1}\sum_{t=1}^{k_{n}}a_{n,t}\big{(}\mathbb{E}_{\mathcal{F}_{t-m}}\left[y_{t}\right]-\mathbb{E}_{\mathcal{F}_{t-m-1}}\left[y_{t}\right]\big{)}+\sum_{t=1}^{k_{n}}a_{n,t}\mathbb{E}_{\mathcal{F}_{t-M}}[y_{t}]$
(4.257)

Thus, for each $m$, it holds that

$\displaystyle\epsilon_{t}^{(m)}:=\mathbb{E}_{\mathcal{F}_{t-m}}[y_{t}]-\mathbb{E}_{\mathcal{F}_{t-m-1}}[y_{t}]$
(4.258)

is a $\mathcal{F}_{t-m}-$martingale difference process that inherits the
uniform integrability property from $y_{t}$, and so, $\epsilon_{t}^{(m)}$
satisfies the required conditions. Then, applying the triangle inequality we
obtain the following

$\displaystyle\left\lVert\sum_{t=1}^{n}a_{n,t}y_{t}\right\rVert_{1}$
$\displaystyle\leq\sum_{m=0}^{M-1}\left\lVert\sum_{t=1}^{k_{n}}a_{n,t}\epsilon_{t}^{(m)}\right\rVert_{1}+\sum_{t=1}^{k_{n}}\left|a_{n,t}\right|\left\lVert\mathbb{E}_{\mathcal{F}_{t-M}}[y_{t}]\right\rVert_{1}$
(4.259) $\displaystyle\leq M\times\underset{0\leq
m<M}{\mathsf{max}}\left\lVert\sum_{t=1}^{k_{n}}a_{n,t}\epsilon_{t}^{(m)}\right\rVert_{1}+\zeta\underset{n\in\mathbb{N}}{\mathsf{sup}}\sum_{t=1}^{k_{n}}\left|a_{n,t}\right|\psi_{M}.$
(4.260)

Let $\delta>0$ be arbitray. Since
$C:=\zeta\mathsf{sup}_{n\in\mathbb{N}}\sum_{t=1}^{k_{n}}\left|a_{n,t}\right|<\infty$
and $\psi_{M}\to\infty$, there exists $M_{0}(\delta)\in\mathbb{N}$ such that
$\psi_{M}\leq\delta/(2C)$.

######  Example 25.

Consider the case in which all regressors, $X_{t}$, is a $p\times 1$ vector of
$I(1)$ variables such that

$\displaystyle X_{t}=X_{t-1}+\eta_{t},\ \ 1\leq t\leq n$ (4.261)

where $X_{0}=\mathcal{O}_{p}(1)$ and $\underset{1\leq t\leq n}{\mathsf{max}}\
\mathbb{E}\left(\left\lVert\eta_{t}\right\rVert^{q}\right)\leq C<\infty$ for
some $q>8$.

Furthermore, $\left\\{u_{t}\right\\}_{t=1}^{n}$ is independent of
$\left\\{(X_{t},Z_{t})\right\\}_{t=1}^{n}$ and
$\left\\{\big{(}\left(u_{t},\eta_{t}\right),\mathcal{F}_{n,t-1}\big{)}\right\\}_{2\leq
t\leq n}$ and forms a martingale difference sequence with
$\sigma_{v}^{2}=\mathbb{E}\left(v_{t}^{2}\right)<\infty$, such that

$\displaystyle\underset{2\leq t\leq
2}{\mathsf{sup}}\left|\mathbb{E}\left(v_{t}^{2}|\mathcal{F}_{n,t-1}\right)-\sigma_{v}^{2}\right|\to
0$ (4.262)

Denote with

$\displaystyle B_{n,\eta}(r)\equiv\frac{1}{\sqrt{n}}\sum_{t=1}^{\lfloor
nr\rfloor}\eta_{t}.$ (4.263)

There exists a vector Brownian motion $B_{\eta}$ such that
$B_{n,\eta}(r)\Rightarrow B_{\eta}(r)$, on $D[0,1]^{d}$ as $n\to\infty$, where
$D[0,1]^{d}$ is the space of cadlag functions on $[0,1]^{d}$ equipped with
Skorohod topology, and $B_{\eta}$ is a $d-$dimensional multivariate Brownian
motion with a finite positive definite covariance matrix $\Sigma_{\eta}$ such
that

$\displaystyle\Sigma_{\eta}=\underset{n\to\infty}{\mathsf{lim}}\text{Var}\left(n^{-1/2}\sum_{t=1}^{n}\eta_{t}\right)$
(4.264)

The martingale difference condition is required to employ the generalized
martingale central limit theorem when we derive the limiting distribution of
our proposed test statistic under the null hypothesis. Furthermore, taking the
weak convergence result in (4.263) as an assumption is commonly done in the
econometrics literature. The conditions for the multivariate functional
central limit theorem for partial sums of weakly dependent random vectors can
be found Wooldridge and White,  (1988). Thus, by applying the continuous
mapping theorem it follows that

$\displaystyle\underset{0\leq r\leq
1}{\mathsf{sup}}\left|B_{n,\eta}\right|\overset{d}{\to}\underset{0\leq r\leq
1}{\mathsf{sup}}\left|B_{\eta}\right|$ (4.265)

Since it is bounded in probability, that is, $\underset{0\leq r\leq
1}{\mathsf{sup}}\left|B_{\eta}(r)\right|=\mathcal{O}_{p}(1)$, hence we have
that

$\displaystyle\underset{1\leq t\leq
n}{\mathsf{max}}\left|X_{t}\right|=\mathcal{O}_{p}\left(\sqrt{n}\right),$
(4.266)

####  4.7.6. Testing for Structural Break in Predictive Regression

In this section, we present the framework of Georgiev et al.,  (2018). The
approach of the particular paper is to consider between two important test
statistics which are commonly used in detecting structural change, that is,
the sup Wald test of Andrews,  (1993) and the Cramer-von-Mises type statistics
of Nyblom,  (1989). On the one hand the sup Wald test proposed by Andrews,
(1993) is motivated by alternatives where the parameters display a small
number of breaks at deterministic points in the sample, while under the
alternative hypothesis of the Cramer-von-Mises test statistic the coefficients
of the model are random and slowly evolve through time. Using a fixed
regressor wild bootstrap procedure their framework allows for both conditional
and unconditional heteroscedasticity in the data.

Consider the time-varying predictive regression model

$\displaystyle y_{t}=\alpha_{t}+\beta_{t}x_{t-1}+\epsilon_{yt},\ \ t=1,...,T.$
(4.267)

where $\epsilon_{yt}$ is a mean zero innovation process and $x_{t}$ is an
observed process, generated by

$\displaystyle x_{t}$ $\displaystyle=\mu+s_{xt},\ \
s_{xt}=\rho_{x}s_{xt-1}+\epsilon_{xt}$ (4.268)

where $\rho_{x}:=\left(1-\frac{c_{x}}{T}\right)$, with $c_{x}\leq 0$ which
allows to consider two types of persistence properties for the regressors of
the model, that is, (i) strongly persistence unit root, and (ii) local-to-unit
root.

######  Remark 22.

Note that (4.267) provides a time-varying representation of the predictive
regression model, which allows both the model intercept and the slope
coefficients to vary over time. In particular, the parameters are expressed as

$\displaystyle\alpha_{t}:=\alpha+as_{\alpha t},\ \ \
\beta_{t}:=\beta+bs_{\beta t}$ (4.269)

For instance, in the context of a time-invariant model, that is,
$\alpha_{t}=\alpha$ and $\beta_{t}=\beta$. Therefore, with near-unit root
predictors then the framework of Cavanagh et al (1995) applies. Moreover, when
$\beta$ is fixed, then the predictive regression model can be interpreted as a
co-integrating regression because $y_{t}$ is a near-unit root process. The
main focus of the proposed framework is on testing the null hypotheses that
the intercept and the slope parameters are constant over time against the
alternative that they vary over time through the sequences of associated time-
varying coefficients, $s_{\alpha t}$ and $s_{\beta t}$. In particular, this
can be done by testing the restrictions that $a=0$ and $b=0$.

##### Stochastic Coefficient Variation

The assumption of time-varying parameters in the predictive regression model
implies that the time-varying components of the model parameters, that is,
$s_{\alpha t}$ and $s_{\beta t}$ follow (near-) unit root processes expressed
as below

$\displaystyle\begin{bmatrix}s_{\alpha t}\\\ s_{\beta
t}\end{bmatrix}=\begin{bmatrix}\rho_{\alpha}&0\\\ 0&\rho_{\beta}\\\
\end{bmatrix}\begin{bmatrix}s_{\alpha t-1}\\\ s_{\beta
t-1}\end{bmatrix}+\begin{bmatrix}\epsilon_{\alpha t}\\\ \epsilon_{\beta
t}\end{bmatrix}$ (4.270)

where the autocorrelation coefficient is expressed as below

$\displaystyle\rho_{\alpha}:=\left(1-\frac{c_{\alpha}}{T}\right),\ \
\rho_{\beta}:=\left(1-\frac{c_{\beta}}{T}\right),\ \ \text{with}\ \
c_{\alpha}\geq 0\ \ \text{and}\ \ c_{\beta}\geq 0.$ (4.271)

which are unit-root or local-to-unit root autoregressive processes.

The non-stochastic coefficient variation mechanism includes the testing
framework proposed by Andrews,  (1993). Under the assumption of one-time
breakpoint, then $s_{\alpha t}$ and $s_{\beta t}$ are modelled as below

$\displaystyle s_{\alpha t}=s_{\beta
t}=D_{t}\left(\lfloor\pi_{0}T\rfloor\right)$ (4.272)

where we define
$D_{t}\left(\lfloor\pi_{0}T\rfloor\right):=\mathbf{1}\left(t\leq\lfloor\pi
T\rfloor\right)$ with $\lfloor\pi T\rfloor$ denoting a generic shift point
with associated break fraction $\pi$. We take the true shift fraction
$\pi_{0}\in\Pi$, where $\Pi=[\pi_{1},\pi_{2}]$ with $0<\pi_{1}<\pi_{2}<1$.

Consider the testing hypothesis given by

$\displaystyle H_{0}:a\neq 0\ \ \text{versus}\ \ \ b\neq 0.$ (4.273)

Moreover, consider the fitted regression model

$\displaystyle y_{t}=\hat{\alpha}+\hat{\beta}x_{t-1}+\hat{\beta}_{0}\Delta
x_{t}+\hat{e}_{t},\ \ \ t=1,...,T$ (4.274)

Let $\hat{\sigma}^{2}:=T^{-1}\sum_{t=1}^{T}\hat{e}_{t}^{2}$, with
$\hat{e}_{t}^{2}$ the OLS residual from the fitted regression given by
(4.274). Moreover, we denote with $\underline{X}=[1\ x_{t-1}]^{\prime}$.

Then, the LM statistic is constructed as

$\displaystyle
LM:=\frac{1}{T\hat{\sigma}^{2}}\sum_{j=1}^{T}\left(\sum_{t=1}^{j}\underline{X}_{t}\hat{e}_{t}\right)\left(\sum_{t=1}^{T}\underline{X}_{t}\underline{X}_{t}^{\prime}\right)^{-1}\left(\sum_{t=1}^{j}\underline{X}_{t}\hat{e}_{t}\right)$
(4.275)

Note that the corresponding single parameter LM statistics are given by

$\displaystyle LM_{1}$
$\displaystyle:=\frac{1}{T^{2}\hat{\sigma}^{2}}\sum_{j=1}^{T}\left(\sum_{t=1}^{j}\hat{e}_{t}\right)^{2}$
(4.276) $\displaystyle LM_{2}$
$\displaystyle:=\frac{1}{\left(T^{2}\hat{\sigma}^{2}\sum_{t=1}^{T}x_{t-1}^{2}\right)}\sum_{j=1}^{T}\left(\sum_{t=1}^{j}x_{t-1}\hat{e}_{t}\right)^{2}$
(4.277)

where LM1 corresponds to the test statistic relating the the intercept alone
and LM2 to the slope coefficient alone.

######  Theorem 12.

Consider the model in (1)-(3) and let Assumption 2 hold. Under the null
hypothesis and under the same local alternatives , the following converge
jointly as $T\to\infty$, in the sense of weak convergence of random measures
on $\mathbb{R}:$

$\displaystyle
LM_{x}\big{|}x\to_{w}\int_{0}^{1}\mathbf{J}^{\prime}(r)\left[\mathbf{V}(1)\right]^{-1}\mathbf{J}^{\prime}(r)dr\bigg{|}B_{1}$
(4.278)

and

$\displaystyle
LM^{*}_{x}\big{|}x,y\to_{w}\int_{0}^{1}\mathbf{J}_{0}^{\prime}(r)\left[\mathbf{V}(1)\right]^{-1}\mathbf{J}_{0}^{\prime}(r)dr\bigg{|}B_{1}$
(4.279)

where $\mathbf{J}_{0}(r):=\int_{0}^{r}\mathbf{A}(s)dY(s),r\in[0,1]$.

Similarly, in the sense of weak convergence of random measures on
$\mathbb{R}$, the following convergence jointly as $T\to\infty:$

$\displaystyle\text{supF}_{x}\big{|}x\to_{w}\text{sup}_{r\in\Lambda}\bigg{\\{}\mathbf{J}^{\prime}(r)\bigg{[}\mathbf{V}(r)-\mathbf{V}(r)\mathbf{V}(1)^{-1}\mathbf{V}(r)\bigg{]}^{-1}\mathbf{J}(r)\bigg{\\}}\bigg{|}B_{1}$
(4.280)

and

$\displaystyle\text{supF}^{*}_{x}\big{|}x,y\to_{w}\text{sup}_{r\in\Lambda}\bigg{\\{}\mathbf{J}_{0}^{\prime}(r)\bigg{[}\mathbf{V}(r)-\mathbf{V}(r)\mathbf{V}(1)^{-1}\mathbf{V}(r)\bigg{]}^{-1}\mathbf{J}_{0}(r)\bigg{\\}}\bigg{|}B_{1}:=\mathcal{J}_{0}\big{|}B_{1}.$
(4.281)

##### Supplementary Examples

In this section we provide additional examples related to testing for
parameter instability in predictive regression models. Further derivations can
be found in the studies of Katsouris, 2023b ; Katsouris, 2023d ; Katsouris,
2023e . First, consider the case in which the predictive regression model
includes a model intercept but is assumed to be stable before conducting any
statistical inference on the existence of a structural break on the slopes of
the model.

######  Example 26.

Consider the following predictive regression model

$\displaystyle y_{t}$
$\displaystyle=\alpha+\boldsymbol{\beta}_{1}^{\prime}x_{t-1}I\\{t\leq
k\\}+\boldsymbol{\beta}_{2}^{\prime}x_{t-1}I\\{t>k\\}+u_{t}$ (4.282)

where $y_{t}$ is a scalar (i.e., univariate predictant) and $x_{t}$ is
$p-$dimensional vector of predictors (i.e., multiple predictors). Moreover,
$k=\lfloor T\pi\rfloor$ denotes the unknown break-point location in the sample
such that $\pi\in(0,1)$. The predictors of the predictive regression model are
generated as a LUR process given by

$\displaystyle
x_{t}=\left(\mathbf{I}_{p}-\frac{\mathbf{C}}{T^{\gamma_{x}}}\right)x_{t-1}+v_{t}$
(4.283)

where $C>0$, i.e., $c_{i}>0$ for all $i\in\left\\{1,...,p\right\\}$ and
$\gamma_{x}\in(0,1)$ (MI). We are interested to test:

$\displaystyle\mathbb{H}_{0}:\boldsymbol{\beta}_{1}=\boldsymbol{\beta}_{2}\ \
\ \text{against}\ \ \
\mathbb{H}_{1}:\boldsymbol{\beta}_{1}\neq\boldsymbol{\beta}_{2},$ (4.284)
$\displaystyle\mathcal{W}_{T}^{OLS}(\pi)$
$\displaystyle=\frac{1}{\hat{\sigma}_{u}^{2}}\left(\hat{\boldsymbol{\beta}}_{1}-\hat{\boldsymbol{\beta}}_{2}\right)^{\prime}\left[\boldsymbol{\mathcal{R}}\left(\boldsymbol{X}^{*\prime}\boldsymbol{X}^{*}\right)^{-1}\boldsymbol{\mathcal{R}}^{\prime}\right]^{-1}\left(\hat{\boldsymbol{\beta}}_{1}-\hat{\boldsymbol{\beta}}_{2}\right)$
(4.285)

Therefore, when a model intercept is included in the model and is assumed to
remain stable, then we shall consider the demeaned versions of the variables
and reformulated the regression specification. The model can be written as:

$\displaystyle y_{t}^{*}=\beta_{1}x_{t-1}^{*}I\\{t\leq
k\\}+\beta_{2}x_{t-1}^{*}I\\{t>k\\}+u_{t}^{*}\ \ \ \ \ \ \ $ (4.286)

such that the above predictive regression model considers the demeaned
variables as below:

$\displaystyle y_{t}^{*}=y_{t}-\frac{1}{T}\sum_{t=1}^{T}y_{t},\
x_{1t-1}^{*}=x_{1t-1}-\frac{1}{T}\sum_{t=1}^{T}x_{1t-1},\
x_{2t-1}^{*}=x_{2t-1}-\frac{1}{T}\sum_{t=1}^{T}x_{2t-1},\
u_{t}^{*}=u_{t}-\frac{1}{T}\sum_{t=1}^{T}u_{t}.$

where $x_{1t-1}:=x_{t-1}I\\{t\leq k\\}$ and $x_{2t-1}:=x_{t-1}I\\{t>k\\}$. In
matrix notation we use $X_{1}^{*}=\left(X_{1}-\bar{X}_{1}\right)$ and
$X_{2}^{*}=\left(X_{2}-\bar{X}_{2}\right)$ where
$\bar{X}_{1}=\frac{1}{T}\sum_{t=1}^{T}x_{1t-1}$ and
$\bar{X}_{2}=\frac{1}{T}\sum_{t=1}^{T}x_{2t-1}$. Also, $u_{t}^{*}$ denotes the
average-corrected error sequence of the predictive regression model given by
(4.316).

######  Remark 23.

Notice that while the regressors $\boldsymbol{X}_{1}$ and $\boldsymbol{X}_{2}$
are orthogonal regressors, the corresponding regressors
$\boldsymbol{X}_{1}^{*}$ and $\boldsymbol{X}_{2}^{*}$ which occur after
considering the demeaned versions of their counterparts are not orthogonal.
For this reason one has to be careful when constructing the estimators for the
parameters $\hat{\boldsymbol{\beta}}_{1}$ and $\hat{\boldsymbol{\beta}}_{2}$
with respect to the regressors $\boldsymbol{X}_{1}^{*}$ and
$\boldsymbol{X}_{2}^{*}$, as well as the corresponding covariance matrix.

More specifically, while asymptotically the conventional OLS estimators can be
used without further modifications in finite-samples the standard formula will
not hold. Similarly the covariance matrix can be simplified only
asymptotically, which implies having off-diagonal terms converging in
probability to zero, however this result does not hold in finite samples. In
practise, we have the following specification i.e., a predictive regression
model with multiple regressors written as below

$\displaystyle y_{t}$
$\displaystyle=\alpha+\beta_{1}^{\prime}\boldsymbol{x}_{1t-1}+\boldsymbol{x}_{2t-1}\beta_{2}^{\prime}+u_{t}$
$\displaystyle\bar{y}_{T}$
$\displaystyle=\alpha+\beta_{1}^{\prime}\bar{\boldsymbol{x}}_{1T}+\beta_{2}^{\prime}\bar{\boldsymbol{x}}_{2T}+\bar{u}_{T}$
$\displaystyle\left(y_{t}-\bar{y}_{T}\right)$
$\displaystyle=\beta_{1}^{\prime}\left(\boldsymbol{x}_{1t-1}-\bar{\boldsymbol{x}}_{1T}\right)+\beta_{2}^{\prime}\left(\boldsymbol{x}_{2t-1}-\bar{\boldsymbol{x}}_{2T}\right)+\left(u_{t}-\bar{u}_{T}\right)$

And by simplifying the notation we write

$\displaystyle
y_{t}^{*}=\beta_{1}x_{1t-1}^{*}+\beta_{2}x_{2t-1}^{*}+u_{t}^{*}\ \ \ \ \ \ \ $

where $x_{1t-1}:=x_{t-1}I\\{t\leq k\\}$ and $x_{2t-1}:=x_{t-1}I\\{t>k\\}$.

######  Remark 24.

Specifically, continuing the discussion from Remark 1, the expressions
$\hat{\boldsymbol{\beta}}_{1}=\left(\boldsymbol{X}_{1}^{*\prime}\boldsymbol{X}_{1}^{\prime}\right)\boldsymbol{X}_{1}^{*\prime}u^{*}$
and
$\hat{\boldsymbol{\beta}}_{2}=\left(\boldsymbol{X}_{2}^{*\prime}\boldsymbol{X}_{2}^{\prime}\right)\boldsymbol{X}_{2}^{*\prime}u^{*}$
will not hold in finite samples due to the fact that
$\boldsymbol{X}_{1}^{*\prime}\boldsymbol{X}_{1}^{\prime}\neq 0$ in finite
samples. Moreover since we operate within the predictive regression model
where regressors are generated as LUR processes, we have to note that these
results hold for the mildly integrated case. For instance, in the case of near
stationary regressors these effects are not contributing even after demeaning,
which just implies that are still valid asymptotically for the near stationary
case. On the other hand, for the case of LUR regressors, i.e., $\gamma_{x}=1$,
then the demeaning effect is contributing to finite samples and therefore we
need to check the terms for the parameter estimators and the corresponding
covariance matrix.

Therefore, we have the following expression

$\displaystyle
T^{\frac{1+\gamma_{x}}{2}}\left(\hat{\boldsymbol{\beta}}_{1}-\boldsymbol{\beta}^{0}\right)$
$\displaystyle=\left(\frac{1}{T^{1+\gamma_{x}}}\sum_{t=1}^{T}x^{*}_{1t-1}x_{1t-1}^{*\prime}\right)^{-1}\left(\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x^{*}_{1t-1}u^{*}_{t}\right)+o_{p}(1).$
(4.287)

Consider the first term of (4.287)

$\displaystyle\sum_{t=1}^{T}x^{*}_{1t-1}x_{1t-1}^{*\prime}=\sum_{t=1}^{T}\left(x_{1t-1}-\frac{1}{T}\sum_{t=1}^{T}x_{1t-1}\right)\left(x_{1t-1}-\frac{1}{T}\sum_{t=1}^{T}x_{1t-1}\right)^{\prime}=\sum_{t=1}^{T}x_{1t-1}x_{1t-1}^{\prime}-T\bar{x}_{1t-1}\bar{x}_{1t-1}^{\prime}$

Therefore, we have that

$\displaystyle\frac{1}{T^{1+\gamma_{x}}}\sum_{t=1}^{T}x^{*}_{1t-1}x_{1t-1}^{*\prime}$
$\displaystyle=\frac{1}{T^{1+\gamma_{x}}}\sum_{t=1}^{T}x_{1t-1}x_{1t-1}^{\prime}-\frac{T}{T^{1+\gamma_{x}}}\left(\frac{1}{T}\sum_{t=1}^{T}x_{1t-1}\right)\left(\frac{1}{T}\sum_{t=1}^{T}x_{1t-1}\right)^{\prime}$
$\displaystyle=\frac{1}{T^{1+\gamma_{x}}}\sum_{t=1}^{T}x_{1t-1}x_{1t-1}^{\prime}-\left(\frac{\sum_{t=1}^{T}x_{1t-1}}{T^{\frac{\gamma_{x}}{2}+1}}\right)\left(\frac{\sum_{t=1}^{T}x_{1t-1}}{T^{\frac{\gamma_{x}}{2}+1}}\right)^{\prime}$

Since as $T\to\infty$,
$\frac{\sum_{t=1}^{T}x_{1t-1}}{T^{\frac{\gamma_{x}}{2}+1}}\to_{p}0$ for
$\gamma_{x}\in(0,1)$, then we obtain that

$\displaystyle\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{1+\gamma_{x}}}\sum_{t=1}^{T}x^{*}_{1t-1}x_{1t-1}^{*\prime}\right\\}$
$\displaystyle\equiv\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{1+\gamma_{x}}}\sum_{t=1}^{T}x_{1t-1}x_{1t-1}^{\prime}\right\\}=\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{1+\gamma_{x}}}\sum_{t=1}^{T}x_{t-1}x_{t-1}^{\prime}I\left\\{t\leq
k\right\\}\right\\}$
$\displaystyle=\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{1+\gamma_{x}}}\sum_{t=1}^{\lfloor
T\pi\rfloor}x_{t-1}x_{t-1}^{\prime}\right\\}\Rightarrow\pi\mathbf{V}_{c}$

Consider the second term of (4.287)

$\displaystyle\sum_{t=1}^{T}x^{*}_{1t-1}u^{*}_{t}=\sum_{t=1}^{T}\left(x_{1t-1}-\frac{1}{T}\sum_{t=1}^{T}x_{1t-1}\right)\left(u_{t}-\frac{1}{T}\sum_{t=1}^{T}u_{t}\right)=\sum_{t=1}^{T}x_{1t-1}u_{t}-\bar{u}_{t}\sum_{t=1}^{T}x_{1t-1}-\bar{x}_{t-1}\sum_{t=1}^{T}\left(u_{t}-\bar{u}_{t}\right)$

Since the last term above is zero, we have that

$\displaystyle\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x^{*}_{1t-1}u^{*}_{t}=\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x_{1t-1}u_{t}-\bar{u}_{T}\frac{\sum_{t=1}^{T}x_{1t-1}}{T^{\frac{1+\gamma_{x}}{2}}}$
(4.288)

In other words,

$\displaystyle\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x^{*}_{1t-1}u^{*}_{t}=\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x_{1t-1}u_{t}+o_{p}(1).$

###### Proof.

Notice that $\bar{u}_{T}=\frac{1}{T}\sum_{t=1}^{T}u_{t}$, where
$u_{t}\sim_{\textit{i.i.d}}(0,\sigma^{2})$ which implies that
$\bar{u}_{T}=\mathcal{O}_{p}\left(\frac{1}{\sqrt{T}}\right)$. Another way to
think about it is that since
$T^{-1}\sum_{t=1}^{T}u_{t}\overset{p}{\to}\mathbb{E}\left(u_{t}\right)$ then
$T^{-3/2}\sum_{t=1}^{T}u_{t}=o_{p}(1)$. Notice that for the LUR process,
$x_{t}=\rho x_{t-1}+u_{t}$, we have that $u_{t}$ is a martingale difference
sequence with respect to
$\mathcal{F}_{t}=\sigma\left(u_{t},u_{t-1},...\right)$ satisfying
$\mathbb{E}\left(u_{t}^{2}|\mathcal{F}_{t}\right)=\sigma^{2}>0$ for all $t$,
and $\left(u_{t}^{2}\right)_{t\in\mathbb{Z}}$ is a uniformly integrable
sequence. Therefore, by expanding the autoregressive model, where
$\rho_{T}=\left(1-\frac{c}{T}^{\gamma_{x}}\right)$, for some $c>0$ and
$\gamma_{x}\in(0,1)$, we obtain the following expression:

$\displaystyle\left(1-\rho_{T}\right)\sum_{t=1}^{T}x_{t-1}$
$\displaystyle=\sum_{t=1}^{T}u_{t}-\left(x_{T}-x_{0}\right)$

Notice that $x_{T}$ dominates (faster asymptotic convergence) the term
$\sum_{t=1}^{T}x_{t-1}$ in the mildly integrated case, i.e.,
$\gamma_{x}\in(0,1)$ since the quantity $x_{T}$ is
$\mathcal{O}_{p}\left(T^{-\frac{\gamma_{x}}{2}}\right)$ while the quantity
$\sum_{t=1}^{T}x_{t-1}$ is $o_{p}(1)$. We can see the first result by
considering the order of convergence of the term
$\mathbb{E}\left[x_{T}\right]=\sum_{j=1}^{t}\rho_{T}^{t-j}x_{t-j}$ which is
$\mathcal{O}_{p}\left(T^{\frac{\gamma_{x}}{2}}\right)$. Moreover, it also
holds that $\sum_{t=1}^{T}u_{t}=o_{p}(1)$. Thus, the order of convergence is

$\displaystyle\bar{u}_{T}\frac{\sum_{t=1}^{T}x_{1t-1}}{T^{\frac{1+\gamma_{x}}{2}}}\overset{p}{\to}\mathcal{O}_{p}(1)\frac{1}{\sqrt{T}}\frac{\sum_{t=1}^{T}x_{1t-1}}{T^{\frac{\gamma_{x}}{2}}}=\mathcal{O}_{p}(1)o_{p}(1)=o_{p}(1)$
(4.289)

∎

Recall that if for some sequence $M_{n}$ it holds that
$m_{n}=\mathcal{O}(\frac{1}{n})$ then the sequence $nM_{n}=\mathcal{O}(1)$ is
a bounded sequence. Since as $T\to\infty$, $\bar{u}_{t}\to 0$ while
$T^{-\frac{1+\gamma_{x}}{2}}\sum_{t=1}^{T}x_{1t-1}$ is bounded, then we have
that

$\displaystyle\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x^{*}_{1t-1}u^{*}_{t}\right\\}$
$\displaystyle\equiv\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x_{1t-1}u_{t}\right\\}=\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x_{t-1}u_{t}I\left\\{t\leq
k\right\\}\right\\}$
$\displaystyle=\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{\lfloor
T\pi\rfloor}x_{t-1}u_{t}\right\\}\Rightarrow\mathcal{N}\bigg{(}0,\pi\sigma_{u}^{2}\otimes\mathbf{V}_{c}\bigg{)}$

Therefore, we obtain the convergence result below:

$\displaystyle
T^{\frac{1+\gamma_{x}}{2}}\left(\hat{\boldsymbol{\beta}}_{1}-\boldsymbol{\beta}^{0}\right)\Rightarrow\bigg{(}\pi\mathbf{V}_{c}\bigg{)}^{-1}\mathcal{N}\bigg{(}0,\pi\sigma_{u}^{2}\otimes\mathbf{V}_{c}\bigg{)}=\frac{1}{\pi}\mathbf{V}_{c}^{-1}\mathcal{N}\bigg{(}0,\pi\sigma_{u}^{2}\otimes\mathbf{V}_{c}\bigg{)}$
(4.290)

We also have the following expression

$\displaystyle
T^{\frac{1+\gamma_{x}}{2}}\left(\hat{\boldsymbol{\beta}}_{2}-\boldsymbol{\beta}^{0}\right)$
$\displaystyle=\left(\frac{1}{T^{1+\gamma_{x}}}\sum_{t=1}^{T}x^{*}_{2t-1}x_{2t-1}^{*\prime}\right)^{-1}\left(\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x^{*}_{2t-1}u^{*}_{t}\right)+o_{p}(1).$
(4.291)

Consider the second term of (4.291)

$\displaystyle\sum_{t=1}^{T}x^{*}_{2t-1}u^{*}_{t}=\sum_{t=1}^{T}\left(x_{2t-1}-\frac{1}{T}\sum_{t=1}^{T}x_{2t-1}\right)\left(u_{t}-\frac{1}{T}\sum_{t=1}^{T}u_{t}\right)=\sum_{t=1}^{T}x_{2t-1}u_{t}-\bar{u}_{t}\sum_{t=1}^{T}x_{2t-1}-\bar{x}_{t-1}\sum_{t=1}^{T}\left(u_{t}-\bar{u}_{t}\right)$

Since the last term above is zero, we have that

$\displaystyle\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x^{*}_{2t-1}u^{*}_{t}=\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x_{2t-1}u_{t}-\bar{u}_{T}\frac{\sum_{t=1}^{T}x_{2t-1}}{T^{\frac{1+\gamma_{x}}{2}}}$
(4.292)

In other words,

$\displaystyle\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x^{*}_{2t-1}u^{*}_{t}=\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x_{2t-1}u_{t}+o_{p}(1).$

and the proof of the above holds due to similar arguments as in the case of
$\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x^{*}_{1t-1}u^{*}_{t}$.
Using similar arguments we can show that

$\displaystyle\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{1+\gamma_{x}}}\sum_{t=1}^{T}x^{*}_{2t-1}x_{2t-1}^{*\prime}\right\\}$
$\displaystyle\equiv\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{1+\gamma_{x}}}\sum_{t=1}^{T}x_{2t-1}x_{2t-1}^{\prime}\right\\}=\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{1+\gamma_{x}}}\sum_{t=1}^{T}x_{t-1}x_{t-1}^{\prime}I\left\\{t>k\right\\}\right\\}$
$\displaystyle=\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{1+\gamma_{x}}}\sum_{t=\lfloor
T\pi\rfloor+1}^{T}x_{t-1}x_{t-1}^{\prime}\right\\}\Rightarrow(1-\pi)\mathbf{V}_{c}$
(4.293)

and

$\displaystyle\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x^{*}_{2t-1}u^{*}_{t}\right\\}$
$\displaystyle\equiv\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x_{2t-1}u_{t}\right\\}=\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=1}^{T}x_{t-1}u_{t}I\left\\{t>k\right\\}\right\\}$
$\displaystyle=\underset{T\to\infty}{\text{plim}}\left\\{\frac{1}{T^{\frac{1+\gamma_{x}}{2}}}\sum_{t=\lfloor
T\pi\rfloor+1}^{T}x_{t-1}u_{t}\right\\}\Rightarrow\mathcal{N}\bigg{(}0,(1-\pi)\sigma_{u}^{2}\otimes\mathbf{V}_{c}\bigg{)}$

Therefore, we obtain the convergence result below:

$\displaystyle
T^{\frac{1+\gamma_{x}}{2}}\left(\hat{\boldsymbol{\beta}}_{2}-\boldsymbol{\beta}^{0}\right)\Rightarrow\bigg{(}(1-\pi)\mathbf{V}_{c}\bigg{)}^{-1}\mathcal{N}\bigg{(}0,(1-\pi)\sigma_{u}^{2}\otimes\mathbf{V}_{c}\bigg{)}\equiv\frac{1}{1-\pi}\mathbf{V}_{c}^{-1}\mathcal{N}\bigg{(}0,(1-\pi)\sigma_{u}^{2}\otimes\mathbf{V}_{c}\bigg{)}$
(4.294)

Now, using expressions (4.290) and (4.294) we obtain the following simplified
expression

$\displaystyle
T^{\frac{1+\gamma_{x}}{2}}\left(\hat{\boldsymbol{\beta}}_{1}-\hat{\boldsymbol{\beta}}_{2}\right)$
$\displaystyle\Rightarrow\left\\{\frac{1}{\pi}\mathbf{V}_{c}^{-1}\mathcal{N}\bigg{(}0,\pi\sigma_{u}^{2}\otimes\mathbf{V}_{c}\bigg{)}-\frac{1}{1-\pi}\mathbf{V}_{c}^{-1}\mathcal{N}\bigg{(}0,(1-\pi)\sigma_{u}^{2}\otimes\mathbf{V}_{c}\bigg{)}\right\\}$
$\displaystyle=\left\\{\frac{\mathbf{V}_{c}^{-1}\mathcal{N}\bigg{(}0,\pi\sigma_{u}^{2}\otimes\mathbf{V}_{c}\bigg{)}-\mathbf{V}_{c}^{-1}\mathcal{N}\bigg{(}0,\sigma_{u}^{2}\otimes\mathbf{V}_{c}\bigg{)}}{\pi(1-\pi)}\right\\}$
$\displaystyle=\left\\{\frac{\mathbf{V}_{c}^{-1/2}\mathbf{V}_{c}^{-1/2}\mathcal{N}\bigg{(}0,\pi\sigma_{u}^{2}\otimes\mathbf{V}_{c}\bigg{)}-\mathbf{V}_{c}^{-1/2}\mathbf{V}_{c}^{-1/2}\mathcal{N}\bigg{(}0,\sigma_{u}^{2}\otimes\mathbf{V}_{c}\bigg{)}}{\pi(1-\pi)}\right\\}$
$\displaystyle=\left\\{\frac{\mathbf{V}_{c}^{-1/2}}{\pi(1-\pi)}\left[\mathbf{V}_{c}^{-1/2}\mathcal{N}\bigg{(}0,\pi\sigma_{u}^{2}\otimes\mathbf{V}_{c}\bigg{)}-\mathbf{V}_{c}^{-1/2}\mathcal{N}\bigg{(}0,\sigma_{u}^{2}\otimes\mathbf{V}_{c}\bigg{)}\right]\right\\}$
$\displaystyle=\left\\{\frac{\mathbf{V}_{c}^{-1/2}}{\pi(1-\pi)}\left[\mathcal{N}\bigg{(}0,\pi\sigma_{u}^{2}\otimes\mathbf{I}_{p}\bigg{)}-\mathcal{N}\bigg{(}0,\sigma_{u}^{2}\otimes\mathbf{I}_{p}\bigg{)}\right]\right\\}$
(4.295)

where $\gamma_{x}\in(0,1)$ and $\pi$ the unknown break fraction. We construct
the OLS Wald statistic for testing the null hypothesis
$H_{0}:\boldsymbol{\mathcal{R}}\boldsymbol{\beta}=0$, which is equivalent to
the null of no parameter instability, such that
$\boldsymbol{\mathcal{R}}=\left[\mathbf{I}_{p}\ -\mathbf{I}_{p}\right]$ and
$\boldsymbol{\beta}=\left(\boldsymbol{\beta}_{1},\boldsymbol{\beta}_{2}\right)$,
where $\boldsymbol{\beta}$ is the model parameter and
$\hat{\boldsymbol{\beta}}=\left(\boldsymbol{\beta}_{1},\boldsymbol{\beta}_{2}\right)$
the OLS estimator.

$\displaystyle\mathcal{W}_{T}(\pi)$
$\displaystyle=\frac{1}{\hat{\sigma}_{u}^{2}}\left[T^{\frac{1+\gamma_{x}}{2}}\left(\hat{\boldsymbol{\beta}}_{1}-\hat{\boldsymbol{\beta}}_{2}\right)\right]^{\prime}\left[\boldsymbol{\mathcal{R}}\left(\frac{\boldsymbol{X}^{*\prime}\boldsymbol{X}^{*}}{T^{1+\gamma_{x}}}\right)^{-1}\boldsymbol{\mathcal{R}}^{\prime}\right]^{-1}\left[T^{\frac{1+\gamma_{x}}{2}}\left(\hat{\boldsymbol{\beta}}_{1}-\hat{\boldsymbol{\beta}}_{2}\right)\right]$
(4.296)

where

$\displaystyle\left[\boldsymbol{\mathcal{R}}\left(\frac{\boldsymbol{X}^{*\prime}\boldsymbol{X}^{*}}{T^{1+\gamma_{x}}}\right)^{-1}\boldsymbol{\mathcal{R}}^{\prime}\right]\Rightarrow\left[\bigg{(}\pi\mathbf{V}_{c}\bigg{)}^{-1}+\bigg{(}(1-\pi)\mathbf{V}_{c}\bigg{)}^{-1}\right]=\left[\frac{1}{\pi}\mathbf{V}_{c}+\frac{1}{1-\pi}\mathbf{V}_{c}\right]=\frac{\mathbf{V}_{c}^{-1}}{\pi(1-\pi)}$

Therefore, based on expressions (26), (4.296) and (26) the simplified
asymptotic result for the OLS-Wald statistic is given by

$\displaystyle\mathcal{W}_{T}(\pi)$
$\displaystyle\Rightarrow\frac{1}{\sigma_{u}^{2}}\left\\{\frac{\mathbf{V}_{c}^{-1/2}}{\pi(1-\pi)}\left[\mathcal{N}\bigg{(}0,\pi\sigma_{u}^{2}\otimes\mathbf{I}_{p}\bigg{)}-\mathcal{N}\bigg{(}0,\sigma_{u}^{2}\otimes\mathbf{I}_{p}\bigg{)}\right]\right\\}^{\prime}\left[\frac{\mathbf{V}_{c}^{-1}}{\pi(1-\pi)}\right]^{-1}$
$\displaystyle\ \ \ \ \
\times\left\\{\frac{\mathbf{V}_{c}^{-1/2}}{\pi(1-\pi)}\left[\mathcal{N}\bigg{(}0,\pi\sigma_{u}^{2}\otimes\mathbf{I}_{p}\bigg{)}-\mathcal{N}\bigg{(}0,\sigma_{u}^{2}\otimes\mathbf{I}_{p}\bigg{)}\right]\right\\}$
$\displaystyle=\frac{1}{\sigma_{u}^{2}}\frac{1}{\pi(1-\pi)}\left[\mathcal{N}\bigg{(}0,\pi\sigma_{u}^{2}\otimes\mathbf{I}_{p}\bigg{)}-\mathcal{N}\bigg{(}0,\sigma_{u}^{2}\otimes\mathbf{I}_{p}\bigg{)}\right]^{\prime}\left(\mathbf{V}_{c}^{-1/2}\right)^{\prime}\mathbf{V}_{c}\mathbf{V}_{c}^{-1/2}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\times\left[\mathcal{N}\bigg{(}0,\pi\sigma_{u}^{2}\otimes\mathbf{I}_{p}\bigg{)}-\mathcal{N}\bigg{(}0,\sigma_{u}^{2}\otimes\mathbf{I}_{p}\bigg{)}\right]$
(4.297)

Notice that since

$\displaystyle\left(\mathbf{V}_{c}^{-1/2}\right)^{\prime}\mathbf{V}_{c}\mathbf{V}_{c}^{-1/2}=\left(\mathbf{V}_{c}^{-1/2}\right)^{\prime}\mathbf{V}_{c}^{1/2}\left(\mathbf{V}_{c}^{1/2}\right)^{\prime}\mathbf{V}_{c}^{-1/2}=\mathbf{I}_{p}$
(4.298)

Then the limit result for the Wald OLS statistic simplifies to the following
expression

$\displaystyle\mathcal{W}_{T}(\pi)$
$\displaystyle\Rightarrow\frac{1}{\sigma_{u}^{2}}\frac{1}{\pi(1-\pi)}\left[\mathcal{N}\bigg{(}0,\pi\sigma_{u}^{2}\otimes\mathbf{I}_{p}\bigg{)}-\mathcal{N}\bigg{(}0,\sigma_{u}^{2}\otimes\mathbf{I}_{p}\bigg{)}\right]^{\prime}\left[\mathcal{N}\bigg{(}0,\pi\sigma_{u}^{2}\otimes\mathbf{I}_{p}\bigg{)}-\mathcal{N}\bigg{(}0,\sigma_{u}^{2}\otimes\mathbf{I}_{p}\bigg{)}\right]$
$\displaystyle\Rightarrow\frac{\bigg{[}\mathbf{W}_{p}(\pi)-\pi\mathbf{W}_{p}(1)\bigg{]}^{\prime}\bigg{[}\mathbf{W}_{p}(\pi)-\pi\mathbf{W}_{p}(1)\bigg{]}}{\pi(1-\pi)}:=\frac{\boldsymbol{\mathcal{BB}}_{p}(\pi)^{\prime}\boldsymbol{\mathcal{BB}}_{p}(\pi)}{\pi(1-\pi)}$

where $\mathbf{W}_{p}(\pi)$ is a Brownian motion with covariance matrix
$\sigma_{u}^{2}$. In summary, we have proved that the limiting distribution of
the OLS-Wald statistic for testing the null hypothesis of parameter constancy
for the parameters of the predictive regression model when: (i) a model
intercept is included but is assumed to remain stable through out the sample
and (ii) the regressors are modelled as mildly integrated regressors,it weakly
convergence to a normalized Brownian Bridge, i.e., the standard NBB limit
result of Andrews is preserved. Therefore, similarly the corresponding
limiting distribution of the sup OLS-Wald statistic is the sup of a NBB
process. In other words, it follows that

$\displaystyle\widetilde{\mathcal{W}}^{OLS}(\pi)\Rightarrow\underset{\pi\in[\pi_{1},\pi_{2}]}{\text{sup}}\frac{\boldsymbol{\mathcal{BB}}_{p}(\pi)^{\prime}\boldsymbol{\mathcal{BB}}_{p}(\pi)}{\pi(1-\pi)},$
(4.299)

where $\boldsymbol{\mathcal{BB}}_{p}(.)$ is a $p-$dimensional standard
Brownian bridge, such that
$\boldsymbol{\mathcal{BB}}_{p}(.):=\left[\mathbf{W}_{p}(.)-\pi\mathbf{W}_{p}(1)\right]$.

Notice that in the case of a fixed break-point, say $\pi\equiv\pi_{0}$, then
since
$\mathbf{W}_{p}(\pi_{0})-\pi_{0}\mathbf{W}_{p}(1)\equiv\mathcal{N}\big{(}0,\pi_{0}(1-\pi_{0})\big{)}$,
then it holds that
$\displaystyle\frac{\boldsymbol{\mathcal{BB}}_{p}(\pi_{0})^{\prime}\boldsymbol{\mathcal{BB}}_{p}(\pi_{0})}{\pi_{0}(1-\pi_{0})}\equiv\sum_{j=1}^{p}\mathcal{N}(0,1)^{2}=\chi^{2}(p)$
which shows that the OLS-Wald is free of any nuisance parameters.

####  4.7.7. Main Aspects on Predictive Regressions

A research question of interest is whether future values of one series
$\left\\{Y_{t}\right\\}$ can be predicted from lagged values of another series
$\left\\{X_{t}\right\\}$, where the two series obey the following model:

$\displaystyle Y_{t}$ $\displaystyle=\alpha+\beta X_{t-1}+u_{t},$ (4.300)
$\displaystyle X_{t}$ $\displaystyle=\mu+\rho X_{t-1}+v_{t},$ (4.301)

It is well known that the OLS estimate of $\beta$ is biased when the errors
$\left\\{u_{t},v_{t}\right\\}$ are correlated, with the amount of bias
increasing as $\rho$ get closer to the unit boundary. There is currently no
known theoretical justification that inference (e.g., a t-test) based on the
bias-corrected OLS estimate will have improved size properties relative to the
test based on the uncorrected OLS estimate. In practise, there are several
studies in the literature that provide examples where even the use of the
exact bias correction does not result in accurate finite-sample inference
because the resulting t-statistic can still be very far from normal.
Therefore, from an inferential point of view, issues such as the bias of point
estimates may be irrelevant in small samples, and thus it is also crucial to
examine the likelihood. At first glance, it may seem surprising that a
likelihood ratio test (LRT) may provide well-behaved hypothesis tests in
situations when the t-statistic does not because the to test statistics are
closely related (see, Phillips and Chen,  (2014)).

In this section, we consider the estimation of the predictive regression model
from the statistical point of view, motivating the use of the restricted
likelihood, which is free of the nuisance intercept parameter and hence able
to imitate the likelihood of the no-intercept univariate model with its
attendant small curvature. A curvature-related approach to tackle the
predictive regression problem for the model was also taken by Jansson and
Moreira,  (2006). Due to the presence of model intercepts $(\mu,\eta)$, we are
motivated to seek a likelihood that does not involve the location parameters
and yet possesses small curvature properties similar to those of the model
with known location parameters. Thus, the restricted likelihood has exactly
these properties. In practise, the idea of restricted likelihood was
originally proposed by KS (1970) precisely as a means of eliminating the
effect of nuisance location parameters when estimating the parameters of the
model covariance structure in a linear model.

We focus our attention on the use of the restricted likelihood for carrying
inference on $\beta$ in the bivariate predictive regression model. Notice that
in the particular framework, the restricted likelihood is the exact likelihood
of the vector
$\big{\\{}(Y_{t}-Y_{t-1})_{t=2}^{n},(X_{t}-X_{t-1})_{t=2}^{n}\big{\\}}$. In
practise, the restricted maximum likelihood (REML) estimates are found to be
asymptotically efficient. A standard method for testing the two-sided
hypothesis, $H_{0}:\beta=0$ versus $H_{A}:\beta\neq 0$ is the LRT, which
compares the log-likelihood evaluated at the unrestricted estimates of the
parameters to the log-likelihood evaluated at the parameter estimates obtained
under the restriction that the null hypothesis $H_{0}:\beta=0$ is true. Then,
it is easy to generalize the REML likelihood in two directions that are both
of practical interest. One generalization is to the case where the predictor
series is a multivariate AR(1) process. The other generalization is to the
case where the univariate predictor follows a higher order AR process.

In other words, the inclusion of an intercept in the model causes the
likelihood ratio to lose this property, thus pointing to the intercept as the
source of the problem. Therefore, this motivates the use of the restricted
likelihood, which is free of the nuisance intercept parameter and hence able
to imitate the likelihood of the no-intercept univariate model with its
attendant small curvature. Thus, we are indeed able to obtain theoretical
results that demonstrate the the LRT based on the restricted likelihood (RLRT)
has good finite-sample performance for both estimation and inference in this
context.

##### Multivariate Regression

Assume that the data
$\big{(}Y_{1},...,Y_{n},\boldsymbol{X}_{0},...,\boldsymbol{X}_{n}^{\prime}\big{)}$
follow

$\displaystyle Y_{t}$
$\displaystyle=\mu+\boldsymbol{\beta}^{\prime}\boldsymbol{X}_{t-1}+u_{t},$
(4.302) $\displaystyle\boldsymbol{X}_{t}$
$\displaystyle=\boldsymbol{\eta}+\boldsymbol{R}\boldsymbol{X}_{t-1}+\boldsymbol{v}_{t},$
(4.303)

where $u_{t}=\boldsymbol{\phi}^{\prime}\boldsymbol{v}_{t}+e_{t}$, such that
$\big{(}e_{t},\boldsymbol{v}_{t}^{\prime}\big{)}^{\prime}\sim\mathcal{N}\big{(}\boldsymbol{0},\mathsf{diag}\left(\sigma_{e}^{2},\boldsymbol{\Sigma}_{v}\right)\big{)}$
is an i.i.d series and $\boldsymbol{R}$ is a $(k\times k)$ matrix with all
eigenvalues less than unity in absolute value. Define with
$\boldsymbol{\Sigma}_{v}\equiv\mathsf{Var}(\boldsymbol{v}_{t})$ and
$\boldsymbol{\Sigma}_{\boldsymbol{X}}\equiv\mathsf{Var}\left(\boldsymbol{X}_{t}\right)$
as

$\displaystyle\mathsf{vec}\left(\boldsymbol{\Sigma}_{\boldsymbol{X}}\right)=\big{(}I_{K^{2}}-\boldsymbol{R}\otimes\boldsymbol{R}\big{)}^{-1}\mathsf{vec}\left(\boldsymbol{\Sigma}_{v}\right)$
(4.304)

and define with

$\displaystyle\hat{\boldsymbol{K}}=\left[\boldsymbol{\Sigma}_{\boldsymbol{X}}^{-1}+n(\boldsymbol{I}-\boldsymbol{R})^{\prime}\boldsymbol{\Sigma}_{v}^{-1}(\boldsymbol{I}-\boldsymbol{R})\right]^{-1}\left[\boldsymbol{\Sigma}_{\boldsymbol{X}}^{-1}\boldsymbol{X}_{0}+(\boldsymbol{I}-\boldsymbol{R})^{\prime}\boldsymbol{\Sigma}_{v}^{-1}(\boldsymbol{I}-\boldsymbol{R})\sum_{t=1}^{n}\boldsymbol{X}_{t}\right]$
(4.305)

Then, the REML log-likelihood up to an additive constant for the model is
given by

$\displaystyle L_{M}$
$\displaystyle=-\left(\frac{n-1}{2}\right)\mathsf{log}\sigma_{e}^{2}-\frac{1}{2\sigma_{e}^{2}}S\left(\boldsymbol{\phi},\boldsymbol{\beta},\boldsymbol{R}\right)-\frac{1}{2}\mathsf{log}|\boldsymbol{\Sigma}_{\boldsymbol{X}}|-\frac{n}{2}\mathsf{log}|\boldsymbol{\Sigma}_{v}|$
$\displaystyle\ \ \
-\frac{1}{2}\mathsf{log}\left|\boldsymbol{\Sigma}_{\boldsymbol{X}}+n(\boldsymbol{I}-\boldsymbol{R})^{\prime}\boldsymbol{\Sigma}_{v}^{-1}(\boldsymbol{I}-\boldsymbol{R})\right|$
$\displaystyle\ \ \
-\frac{1}{2}\left\\{\left(\boldsymbol{X}_{0}-\hat{\boldsymbol{K}}\right)^{\prime}\boldsymbol{\Sigma}_{\boldsymbol{X}}^{-1}\left(\boldsymbol{X}_{0}-\hat{\boldsymbol{K}}\right)+\sum_{t=1}^{n}\left[\boldsymbol{X}_{t}-\hat{\boldsymbol{K}}-\boldsymbol{R}\left(\boldsymbol{X}_{t-1}-\hat{\boldsymbol{K}}\right)\right]^{\prime}\boldsymbol{\Sigma}_{v}^{-1}\left[\boldsymbol{X}_{t}-\hat{\boldsymbol{K}}-\boldsymbol{R}\left(\boldsymbol{X}_{t-1}-\hat{\boldsymbol{K}}\right)\right]\right\\}$

where

$\displaystyle
S\left(\boldsymbol{\phi},\boldsymbol{\beta},\boldsymbol{R}\right)=\sum_{t=1}^{n}\bigg{[}Y_{t}^{\mu}-\boldsymbol{\phi}^{\prime}\boldsymbol{X}_{t}^{\mu}-\big{(}\boldsymbol{\beta}^{\prime}-\boldsymbol{\phi}^{\prime}\boldsymbol{R}\big{)}\boldsymbol{X}_{t-1}^{\mu}\bigg{]}^{2}$
(4.306)

and

$\displaystyle\boldsymbol{X}_{t}^{\mu}=\boldsymbol{X}_{t}-\frac{1}{n}\sum_{t=1}^{n}\boldsymbol{X}_{t}\
\ \ \ \text{and}\ \ \ \
\boldsymbol{X}_{t-1}^{\mu}=\boldsymbol{X}_{t-1}-\frac{1}{n}\sum_{t=1}^{n}\boldsymbol{X}_{t-1}$
(4.307)

######  Remark 25.

Notice that in the case where $\boldsymbol{R}$ is assumed to be diagonal
matrix, the predictive regression model is no longer a seemingly unrelated
regression (SUR) system and hence OLS will no longer be efficient. However,
REML will clearly retain efficiency, no matter what the form of
$\boldsymbol{R}$ is, thus giving it an advantage in terms of both asymptotic
efficiency and power over any OLS-based procedure.

Furthermore, since the dimension of the parameter space is very large in the
vector case, it is not feasible to obtain a result such as Theorem 3 in the
most general case. However, in the case where $\boldsymbol{R}$ is a diagonal
matrix and where
$\big{(}\sigma_{e}^{2},\boldsymbol{\phi},\boldsymbol{\Sigma}_{v}\big{)}$ are
assumed known with $\boldsymbol{\Sigma}_{v}$ diagonal, we are able to obtain
the following result on the finite-sample behaviour of the RLRT for testing
$\mathbb{H}_{0}:\boldsymbol{\beta}=\boldsymbol{0}$.

##### Higher Order Autoregressive Regressors

Let the observed data $\big{(}Y_{1},...,Y_{n},X_{-p+2},...,X_{n}\big{)}$
follow

$\displaystyle Y_{t}$ $\displaystyle=\mu+\beta X_{t-1}+u_{t},$ (4.308)
$\displaystyle X_{t}$
$\displaystyle=\eta+\alpha_{1}X_{t-1}+....+\alpha_{p}X_{t-p}+v_{t}$ (4.309)

where $u_{t}=\phi v_{t}+e_{t}$ and
$\big{(}e_{t},v_{t}\big{)}\sim\mathcal{N}\big{(}\boldsymbol{0},\mathsf{diag}\left(\sigma_{e}^{2},\sigma_{v}^{2}\right)\big{)}$
are an i.i.d series. Moreover, assume that all the roots of the polynomial
$z^{p}-\sum_{s=1}^{p}z^{p-s}\alpha_{s}$ lie within the unit circle. Define
with

$\displaystyle\boldsymbol{Y}^{\mu}=\boldsymbol{Y}-\boldsymbol{1}\bar{Y},\ \ \
\
\boldsymbol{X}^{\mu}=\bigg{[}\boldsymbol{X}_{1}-\boldsymbol{1}\bar{X}_{1},...,\boldsymbol{X}_{-p+1}-\boldsymbol{1}\bar{X}_{-p+1}\bigg{]},$
(4.310)

where $X_{i}=\frac{1}{n}\boldsymbol{1}\boldsymbol{X}_{i}$.

######  Exercise 1.

Consider the predictive regression model formulated as below

$\displaystyle y_{t}$
$\displaystyle=\alpha\left(\frac{t}{n}\right)+\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{\beta}\left(\frac{t}{n}\right)+u_{t}$
(4.311) $\displaystyle\boldsymbol{x}_{t}$
$\displaystyle=\left(\boldsymbol{I}_{p}-\frac{\boldsymbol{C}_{p}}{n^{\gamma}}\right)\boldsymbol{x}_{t-1}+\boldsymbol{v}_{t}$
(4.312)

  * •

Under the null hypothesis we have a predictive regression model with fixed
parameter vector which is expresses as
$y_{t}=\alpha+\boldsymbol{x}_{t-1}^{\prime}\boldsymbol{\beta}+u_{t}$.

  * •

Under the alternative hypothesis, $\mathcal{H}_{1}$,
$\alpha_{t}=\alpha\left(\frac{t}{n}\right)$ and
$\beta_{t}=\boldsymbol{\beta}\left(\frac{t}{n}\right)$ are changing over time.

Therefore, the OLS estimator might not be suitable since there exists no
parameter vector $\boldsymbol{\theta}$ such that
$\mathbb{E}\left(y_{t}|\boldsymbol{X}_{t-1}\right)=\boldsymbol{X}_{t-1}^{\prime}\boldsymbol{\theta}$
almost surely under the alternative hypothesis, where
$\boldsymbol{X}_{t-1}=(1,\boldsymbol{x}_{t-1}^{\prime})^{\prime}$ such that
$\boldsymbol{\theta}=\left(\alpha,\boldsymbol{\beta}^{\prime}\right)^{\prime}$.
What about a corresponding IVX estimator? Can it consistently estimate the
time-varying parameter $\theta_{t}$ given the assumptions of the predictive
regression modelling framework?

Provide a suitable asymptotic theory analysis, clearly indicating assumptions
and theorems employed for the literature that justify the use of a parametric
or nonparametric estimation approach with desirable statistical properties.
You can also provide a small Monte Carlo simulation study.

######  Exercise 2.

Consider the following predictive regression model

$\displaystyle y_{t}$
$\displaystyle=\beta_{1}x_{1,t-1}+\beta_{2}x_{2,t-1}+\beta_{3}x_{3,t-1}+\epsilon_{t}$
(4.313) $\displaystyle x_{t}$
$\displaystyle=\begin{pmatrix}1-\frac{c_{1}}{n}&0&0\\\
0&1-\frac{c_{2}}{n}&0\\\ 0&0&1-\frac{c_{3}}{n}\end{pmatrix}x_{t-1}+u_{t}$
(4.314) $\displaystyle u_{t}$ $\displaystyle=\begin{pmatrix}0.28&0&0\\\
0&0.32&0\\\ 0&0&-0.14\end{pmatrix}u_{t-1}+v_{t}$ (4.315)

  * •

Write an R Script that simulates the vector
$\mathsf{data}:=[y_{t},\mathbf{x}_{t}]$ based on the above DGP.

  * •

Hence, using suitable values for the sample size $n$ and for the nuisance
parameters of persistence $(c_{1},c_{2},c_{3})$ such that regressors are
either stationary or mildly explosive, demonstrates whether or not the
predictive accuracy of the model is improved in the case when all regressors
are stationary in comparison to the case of mixed integration order.

######  Exercise 3.

Consider the following predictive regression models

$\displaystyle y_{t}$
$\displaystyle=\boldsymbol{x}_{1t-1}^{\prime}\boldsymbol{\delta}_{1}+u_{t}$
(4.316) $\displaystyle y_{t}$
$\displaystyle=\boldsymbol{x}_{1t-1}^{\prime}\boldsymbol{\beta}_{1}+\boldsymbol{x}_{2t-1}^{\prime}\boldsymbol{\beta}_{2}+v_{t}$
(4.317)

where the resulting out of sample forecast errors are obtained as

$\displaystyle\hat{e}_{1,t}=y_{t}-\boldsymbol{x}_{1t-1}^{\prime}\hat{\boldsymbol{\delta}}_{1t}\
\ \ \ \text{and}\ \ \ \
\hat{e}_{2,t}=y_{t}-\boldsymbol{x}_{t-1}^{\prime}\hat{\boldsymbol{\beta}}_{t}$
(4.318)

Denote with
$\boldsymbol{x}_{t-1}=\left(\boldsymbol{x}_{1t-1}^{\prime},\boldsymbol{x}_{2t-1}^{\prime}\right)$
and
$\boldsymbol{\beta}=\left(\boldsymbol{\beta}_{1}^{\prime},\boldsymbol{\beta}_{2}^{\prime}\right)$
and set $p=p_{1}+p_{2}$. The nested environment is commonly employed for
predictive accuracy testing (see Hansen and Timmermann,  (2015)). Therefore,
one step ahead forecasts of $y_{t+1}$ based on the econometric specifications
given by expressions (4.316) and (4.317) are generated recursively as
$\hat{y}_{1,t|t-1}=\boldsymbol{x}_{1t-1}^{\prime}\hat{\boldsymbol{\delta}}_{1t}$
and
$\hat{y}_{2,t|t-1}=\boldsymbol{x}_{t-1}^{\prime}\hat{\boldsymbol{\beta}}_{t}$,
for $t\in\left\\{k_{0},...,T-1\right\\}$,

$\displaystyle\hat{\boldsymbol{\delta}}_{1t}$
$\displaystyle=\left(\sum_{j=1}^{t}\boldsymbol{x}_{1j-1}\boldsymbol{x}_{j-1}^{\prime}\right)^{-1}\left(\sum_{j=1}^{t}\boldsymbol{x}_{1j-1}^{\prime}y_{j}\right)$
(4.319) $\displaystyle\hat{\boldsymbol{\beta}}_{t}$
$\displaystyle=\left(\sum_{j=1}^{t}\boldsymbol{x}_{j-1}\boldsymbol{x}_{j-1}^{\prime}\right)^{-1}\left(\sum_{j=1}^{t}\boldsymbol{x}_{j-1}^{\prime}y_{j}\right)$
(4.320)

Moreover, based on the following test statistic provide suitable asymptotic
theory analysis.

$\displaystyle
T_{n}=\frac{1}{\hat{\sigma}^{2}_{\epsilon}}\left\\{\sum_{t=n_{\kappa}+1}^{n}\big{(}y_{t}-\tilde{y}_{t|t-1}\big{)}^{2}-\big{(}y_{t}-\hat{y}_{t|t-1}\big{)}^{2}\right\\}$
(4.321)

######  Exercise 4.

Consider the structural change monitoring environment of Chu et al.,  (1996)

$\displaystyle y_{t}=x_{t}^{\prime}\beta+\epsilon_{t},\ t=1,...,n$ (4.322)

Under the null the parameter vector is constant $(\beta_{t}=\beta\ \forall t)$
against the alternative of temporary shifts. The alternative of a single
structural change occurs at an unknown change point $k>n$.

$\displaystyle
y_{t}=\begin{cases}x_{t}^{\prime}\beta_{1}+\epsilon_{t},&\text{if }\
t=1,...,k\\\ x_{t}^{\prime}\beta_{2}+\epsilon_{t},&\text{if}\
t=k+1,...,n\end{cases}$ (4.323)

Treating $k\geq n$ then the two model estimates ex-ante and ex-post the
structural change are given

$\displaystyle\hat{\beta}_{1,T}(k)$
$\displaystyle=\left(\sum_{t=1}^{k}x_{t}x_{t}^{\prime}\right)^{-1}\left(\sum_{t=1}^{k}x_{t}y_{t}\right)$
(4.324) $\displaystyle\hat{\beta}_{2,T}(k)$
$\displaystyle=\left(\sum_{t=k+1}^{T}x_{t}x_{t}^{\prime}\right)^{-1}\left(\sum_{t=k+1}^{T}x_{t}y_{t}\right)$
(4.325)

Moreover, consider the construction of a test statistic based on the
difference between the model estimates in-sample and the the full-sample given
by

$\displaystyle\underset{n\leq k\leq T}{\text{max}}\
\frac{k}{\hat{\sigma}_{T}\sqrt{T}}\left\lVert
Q^{1/2}_{t}\left(\hat{\beta}_{1,T}(k)-\hat{\beta}_{T}\right)\right\rVert$
(4.326)

where $\hat{\sigma}_{T}$ is an unbiased estimator of $\sigma_{2}$ and
$Q_{T}=\frac{1}{T}\sum_{i=1}^{T}x_{i}x_{i}^{\prime}$. Then, the model
parameter that depends on the window length $h$ is given by

$\displaystyle\tilde{\beta}\big{(}k,[Th]\big{)}=\left(\sum_{t=k+1}^{k+[Th]}x_{t}x_{t}^{\prime}\right)^{-1}\left(\sum_{t=k+1}^{k+[Th]}x_{t}y_{t}\right)$
(4.327)

Therefore, assuming that the structural change occurs in the monitoring period
with length exactly the size of the moving window, $\lfloor Th\rfloor$, then
the proposed class of ME tests against the alternative of structural change
has the following form

$\displaystyle\text{ME}_{T,[Th]}=\underset{n\leq k\leq T}{\text{max}}\
\frac{[Th]}{\hat{\sigma}_{T}\sqrt{T}}\left\lVert
Q^{1/2}_{t}\left(\hat{\beta}_{1,T}(k)-\hat{\beta}_{T}\right)\right\rVert$
(4.328)

  * (a)

Consider suitable modifications of the asymptotic theory that corresponds to
the above test in order to accommodate the IVX estimator of the predictive
regression model below:

$\displaystyle y_{t}$ $\displaystyle=\beta x_{t-1}+u_{0t}$ (4.329)
$\displaystyle x_{t}$ $\displaystyle=\rho x_{t-1}+u_{xt},\ \ \
\rho=\left(1+\frac{c}{n^{\alpha}}\right)$ (4.330)

where the IVX-instrumentation is obtained with

$\displaystyle\tilde{z_{t}}=\sum_{j=1}^{t}\rho_{nz}^{t-j}\Delta x_{j},\ \ \
\rho_{nz}=1+\frac{c_{z}}{n^{\beta}},\ \text{where }\ \beta\in(0,1),c_{z}<0$
(4.331)

and the IVX estimator has the following form:

$\displaystyle\hat{\beta}_{ivx}=\frac{\displaystyle\sum_{t=1}^{n}\tilde{z}_{t-1}y_{t}}{\displaystyle\sum_{t=1}^{n}\tilde{z}_{t-1}x_{t-1}}=\beta+\frac{\displaystyle\sum_{t=1}^{n}\tilde{z}_{t-1}u_{0t}}{\displaystyle\sum_{t=1}^{n}\tilde{z}_{t-1}x_{t-1}}$
(4.332)

Does the limiting distribution of $n^{\frac{1+\beta}{2}}\left(\hat{\beta}_{ME-
ivx}-\beta\right)\implies\tilde{\Psi}$ is mixed Gaussian? Clearly indicate all
necessary assumptions and related limit results.

  * (b)

Consider the following alternative hypothesis which implies the presence of
heterogeneous persistence and temporary parameter instability in the
predictive regression system

$\displaystyle y_{t}=\begin{cases}\beta_{1}x_{1,t-1}^{\prime}+u_{0t},\
x_{1,t}=\rho_{1}x_{1,t-1}+u_{xt}&\ t=1,...,k\\\
\beta_{2}x_{2,t-1}^{\prime}+u_{0t},\ x_{2,t}=\rho_{2}x_{2,t-1}+u_{xt}&\
t=k+1,...,k+l\\\ \beta_{1}x_{1,t-1}^{\prime}+u_{0t},\
x_{1,t}=\rho_{1}x_{1,t-1}+u_{xt}&\ t=k+l+1,...,T\end{cases}$ (4.333)

where $\rho_{1}=1+\frac{c_{1}}{n^{\alpha}}$,
$\rho_{2}=1+\frac{c_{2}}{n^{\alpha}}$ with $c_{1}<0$ and $c_{2}>0$ and
$\alpha\in(0,1)$ in both cases. Since, we have a moving window in practise the
model estimation and IVX instrumentation occurs in each rolling window.
Moreover, you may assume that a different instrumentation is constructed for
each regime such that
$\tilde{z}_{1,t}=z_{1,t}+\frac{c_{1}}{n^{\beta_{1}}}\psi^{(1)}_{nt}$ and
$\tilde{z}_{2,t}=z_{2,t}+\frac{c_{2}}{n^{\beta_{2}}}\psi^{(2)}_{nt}$.

Using the ME-IVX test function which corresponds to a self-normalized test
statistic representing a difference of the IVX estimators between the in-
sample against the estimator within the monitoring period with a corresponding
fixed window of size $h$:

$\displaystyle
M_{n}(r|h)=\frac{[nh]}{\hat{\sigma}\sqrt{n}}.\Omega^{1/2}_{(n)}.\bigg{[}\hat{\beta}_{t,{([nr]-[nh],[nh])}}^{\text{IVX}}-\hat{\beta}_{t,(n)}^{\text{IVX}}\bigg{]}$
(4.334)

for some $h$, $0<h\leq 1$ the moving data window as a percentage of the
historical period, obtain suitable asymptotic theory analysis that
demonstrates the statistical properties of the test. Relevant studies to check
include: Busetti and Taylor,  (2004) and Horváth et al.,  (2020).

##  5\. Resampling for Time Series Regressions

Seminal studies for aspects related to implementation of resampling methods in
time series analysis include Politis and Romano,  (1994), Paparoditis and
Politis,  (2001, 2003) among others.

###  5.1. Preliminary Theory on Bootstrap Methodologies

Let $\left\\{X_{n}\right\\}_{n\in\mathbb{Z}}$ be a stationary sequence of
random variables with common continuous distribution function
$F(t)=\mathbb{P}\left(X_{0}\leq t\right)$ on a probability space. Assume
$0\leq X_{0}\leq 1$. The empirical process is defined as

$\displaystyle F_{n}(t)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\mathbf{1}\left\\{X_{i}\leq
t\right\\}$ (5.1) $\displaystyle B_{n}(t)$
$\displaystyle=\sqrt{n}\big{(}F_{n}(t)-F(t)\big{)},$ (5.2)

where $F_{n}(t)$ is the empirical distribution of $\left\\{X_{n}\right\\}$. We
have that $B_{n}(t)\to_{d}B(t)$ in the spaces $D[0,1]$ endowed with Skorohod
topology where $B$ is a Gaussian process specified by
$\mathbb{E}\left(B(t)\right)=0$ for every $t$, and for every $t$ and $s$,

$\displaystyle
cov\big{(}B(s),B(t)\big{)}=\sum_{h=-\infty}^{h=+\infty}cov\big{(}\mathbf{1}\left\\{X_{0}\leq
s\right\\},\mathbf{1}\left\\{X_{k}\leq t\right\\}\big{)}.$ (5.3)

In addition,

$\displaystyle\underset{n\to\infty}{\text{lim}}\
cov\big{(}B_{n}(s),B_{n}(s)\big{)}$ (5.4)

The block-based bootstrap estimators of the mean and empirical process are
defined as below. Let $k$ and $l$ be two integers such that $n=kl$. Let
$T_{n1},T_{n2},...,T_{nk}$ be i.i.d random variables each having uniform
distribution on $\left\\{1,...,n\right\\}$. Define the triangular array
$\left\\{X_{ni},1\leq i\leq n+l\right\\}$by $X_{ni}=X_{i}$ for $1\leq i\leq n$
and $X_{ni}=X_{i-n}$ for $1\leq i\leq n+l$. In other words, we extend our
sample of size $n$ by another $l$ observations, namely, $X_{1},...,X_{l}$.
Then, the bootstrapped estimator of the mean is defined as below

$\displaystyle\bar{X}_{n}^{*}=\frac{1}{k}\sum_{i=1}^{k}\frac{1}{l}\sum_{j=T_{ni}}^{T_{ni}+l-1}X_{j},$
(5.5)

and the bootstrapped estimator of the empirical process is given by

$\displaystyle
F_{n}(t)^{*}=\frac{1}{k}\sum_{i=1}^{k}\frac{1}{l}\sum_{j=T_{ni}}^{T_{ni}+l-1}\mathbf{1}\left\\{X_{j}\leq
t\right\\}.$ (5.6)

Then, the boostrapped empirical process is defined as

$\displaystyle B_{n}(t)^{*}=n^{1/2}\big{[}F_{n}(t)^{*}-F_{n}(t)\big{]}.$ (5.7)

We denote with $\mathbb{E}^{*}$, $var^{*}$ the moments under the conditional
probability measure $\mathcal{P}^{*}$ induced by the resampling mechanism,
that is, $\mathcal{P}^{*}$, is the conditional probability given
$\left(X_{1},...,X_{n}\right)$. In this paper, we focus on the weak
convergence of $B_{n}^{*}(t)$ to $B(t)$ in the Skorohod topology on $D[0,1]$,
almost surely.

##### Simulating the bootstrap distribution

We follow the steps below to obtain the approximation to the bootstrap
distribution $G_{n}\left(x,\widehat{F}_{n}\right)$. We consider that
$b=1,...,B$ are the bootstrap replications.

  1. Step 1.

Draw a random sample from the cdf $\widehat{F}_{n}$. The realized sample
$x_{1}^{*b},....,x_{n}^{*b}$ is your bootstrap sample.

  2. Step 2.

Calculate the bootstrap version of $Q_{n}(x,F)$, which is $Q_{n}(x^{*b},F)$.
We denote this as short-hand by $Q_{n}^{*b}$.

  3. Step 3.

After repeating steps 1 and 2 B times, then we collect all the estimated
bootstrapped quantities $Q_{n}^{*1},...,Q_{n}^{*B}$ to calculate a related
test statistic as function of $G_{n}\left(x,\widehat{F}_{n}\right)$.

Now, how the sample in Step 1 of Algorithm 1 is drawn, depends on the form of
the estimator $\widehat{F}_{n}$. Two main bootstrap methodologies are commonly
used:

  * 1.

Parametric bootstrap: $F$ is estimated by
$F\left(x|\widehat{\theta}_{n}\right)$. In this case, the bootstrap sample,
$X_{1}^{*},...,X_{n}^{*}$ is drawn from the cdf
$F\left(x|\widehat{\theta}_{n}\right)$, where $\widehat{\theta}_{n}$ is an
estimate of $\theta$ based on the original sample $x_{1},...,x_{n}$.

  * 2.

Nonparametric (iid) bootstrap: $F$ is estimated by EDF $\widehat{F}_{n}^{E}$.
In this case, the bootstrap sample, $X_{1}^{*},...,X_{n}^{*}$ is drawn from
the cdf $\widehat{F}_{n}^{E}$, which implies that $X_{1}^{*},...,X_{n}^{*}$ is
drawn with replacement from the original sample $x_{1},...,x_{n}$.

####  5.1.1. Bootstrap in regression models

Consider the following regression model for $i=1,...,n$

$\displaystyle Y_{i}=\alpha+\beta X_{i}+\epsilon_{i}$ (5.8)

##### Pair bootstrap

The pairs bootstrap is a direct extension of the iid bootstrap and is based on
the bivariate EDF $\widehat{F}_{n}(x,y)$ for the sample
$(X_{1},Y_{1}),...,(X_{n},Y_{n})$ are a random sample with bivariate cdf F.
The pairs bootstrap builds the bootstrap sample
$(X_{1}^{*},Y_{1}^{*}),...,(X_{n}^{*},Y_{n}^{*})$ by drawing pairs with
replacement from the sample $(X_{1},Y_{1}),...,(X_{n},Y_{n})$. It is crucial
that $(X_{i},Y_{i})$ are kept together as a pair, otherwise if
$\textbf{Y}^{*}$ and $\textbf{X}^{*}$ were drawn separately the bootstrap
would assume there is no relation between the two.

###  5.2. On Bootstrap Asymptotics

The bootstrap is a method for estimating the distribution of an estimator or
test statistic by resampling one’s data or a model estimated from the data.
The methods that are available for implementing the bootstrap and the
improvements in accuracy that it achieves relative to first-order asymptotic
approximations depend on whether the data are an i.i.d random sample or a time
series. If the data are iid, the bootstrap can be implemented by sampling the
data randomly with replacement or by sampling a parametric model of the
distribution of the data. The distribution of a statistic is estimated by its
empirical distribution under sampling from the data or parametric model (see,
Härdle et al.,  (2003)). Therefore, the situation is more complicated when the
data are a time series because bootstrap sampling must be carried out in a way
that suitably captures the dependence structure of the DGP.

####  5.2.1. Why the bootstrap provides Asymptotic Refinements

The term asymptotic refinements refers to approximations to distribution
functions, coverage probabilities that are more accurate than those of first-
order asymptotic distribution theory (see, Beran,  (1988)). We focus on the
distribution function of the asymptotically $N(0,1)$ statistic
$T_{n}=n^{1/2}\left(\theta_{n}-\theta\right)/s_{n}$.

Let $\widehat{\mathcal{P}}$ denote the probability measure induced by
bootstrap sampling, and let $\widehat{T}_{n}$ denote a bootstrap analogue of
$T_{n}$. If the data are i.i.d, then it suffices to let
$\widehat{\mathcal{P}}$ be the empirical distribution of the data. Bootstrap
samples are drawn by sampling the data $\left\\{X_{i}:i=1,...,n\right\\}$
randomly with replacement. If $\left\\{\widehat{X}_{i}:i=1,...,n\right\\}$ is
such a sample, then

$\displaystyle\widehat{T}_{n}=n^{1/2}\left(\widehat{\theta}_{n}-\theta_{n}\right)/\widehat{s}_{n},$
(5.9)

where $\hat{\theta}_{n}=\theta(\hat{m}_{n})$ and
$\hat{m}_{n}=n^{-1}\sum_{i=1}^{n}\widehat{X}_{i}$, and $\widehat{s}_{n}$ is
obtained by replacing the $\left\\{X_{i}\right\\}$ with
$\left\\{\hat{X}_{i}\right\\}$ in the formula for $s_{n}^{2}$.

####  5.2.2. The Block Bootstrap

The block bootstrap is the best-known method for implementing the bootstrap
with time-series data. It consists of dividing the data into blocks of
observations and sampling the blocks randomly with replacement. The blocks may
be non-overlapping. Define $Y_{i}=\left\\{X_{i},....,X_{i-q}\right\\}$. With
non-overlapping blocks of length $\ell$, block 1 is observations
$\left\\{Y_{j}:j=1,...,\ell\right\\}$, block 2 is observations
$\left\\{Y_{j+1}:j=1,...,\ell\right\\}$, and so forth. The bootstrap sample is
obtained by sampling blocks randomly with replacement and laying them end-to-
end in the order sampled. The procedure of sampling blocks of $Y_{i}$’s
instead of $X_{i}$’s is called the blocks-of-blocks bootstrap (see, Andrews,
(2004)). The bootstrap estimators of distribution functions are smaller with
overlapping blocks that with non-overlapping ones. The rates of convergence of
the error made with overlapping and non-overlapping blocks are the same, and
theoretical arguments will be based on non-overlapping blocks with the
understanding that the results also apply to overlapping blocks (see, Politis
and Romano,  (1994)).

Moreover, the errors made by the stationary bootstrap are larger than those of
the bootstrap with non-stochastic block lengths and either overlapping or non-
overlapping blocks. Therefore, the stationary bootstrap is unattractive
relative to the bootstrap with non-stochastic block lengths. Regardless of
whether the blocks are overlapping or non-overlapping, the block length must
increase with increasing sample size $n$ to make bootstrap estimators of
moments and distribution functions consistent. The block length must also
increase with increasing $n$ to enable the block bootstrap to achieve
asymptotically correct coverage probabilities for confidence intervals and
rejection probabilities for tests. Thus, when the objective is to estimate a
moment of distribution function, the asymptotically optimal block length may
be defined as the one that minimizes the asymptotic mean-square error of the
block bootstrap estimator.

####  5.2.3. Studentization

We address the problem of Studentizing
$n^{1/2}\hat{\Delta}_{n}=n^{1/2}\left[\theta\left(\hat{m}_{n}\right)-\theta\left(\hat{E}\hat{m}_{n}\right)\right]$.
The source of blocking distorts the dependence structure of the DGP. To
illustrate the essential issues with a minimum of complexity, assume that the
blocks are non-overlapping, $\theta$ is the identity function, and
$\left\\{X_{i}\right\\}$ is a sequence of uncorrelated (though not necessarity
independent) scalar random variables. For example, many economic time series
are martingale difference sequences, and therefore, serially uncorrelated.

Let $V$ denote the variance operator relative to the process that generates
$\left\\{X_{i}\right\\}$. Then,

$\displaystyle n^{1/2}\Delta_{n}=n^{1/2}\left(m_{n}-\mu\right),\ \ \text{and}\
\ n^{1/2}\widehat{\Delta}_{n}=n^{1/2}\left(\hat{m}_{n}-\mu\right)$ (5.10)

with
$V\left(n^{1/2}\Delta_{n}\right)=\boldsymbol{E}\left(X_{1}-\mu\right)^{2}$.
The natural choice of $s_{n}^{2}$ is the sample variance,
$s_{n}^{2}=n^{-1}\sum_{i=1}^{n}\left(X_{i}-\mu\right)^{2}$, in which case
$s_{n}^{2}-\text{var}\left(n^{1/2}\Delta_{n}\right)=\mathcal{O}_{p}\left(\Delta_{n}\right)=\mathcal{O}_{p}\left(n^{-1/2}\right)$.
Let $\ell$ and $B$ denote the block length and number of blocks respectively,
and assume that $B\ell=n$. Let $\hat{V}$ denote the variance operator relative
to the block bootstrap DGP. An obvious bootstrap analog of $s_{n}^{2}$ is
$\hat{s}_{n}^{2}=n^{-1}\sum_{i=1}^{n}\left(\hat{X}_{i}-\hat{m}_{n}\right)^{2}$,
which leads to the Studentized statistic, $\widetilde{T}_{n}\equiv
n^{1/2}\hat{\Delta}_{n}/\hat{s}_{n}$. However, we have that
$\widehat{V}\left(n^{1/2}\hat{\Delta}_{n}\right)=\tilde{s}_{n}^{2}$, where

$\displaystyle\tilde{s}_{n}^{2}=n^{-1}\sum_{b=0}^{B}\sum_{i=1}^{\ell}\sum_{j=1}^{\ell}\left(X_{b\ell}-m_{n}\right)\left(X_{b\ell+j}-m_{n}\right)$
(5.11)

Moreover,
$\hat{s}_{n}^{2}-\tilde{s}_{n}^{2}=\mathcal{O}_{p}\left(\left(\ell/n\right)^{1/2}\right)$
almost surely. The consequences of this relatively large error in the
estimator of the variance of $n^{1/2}\hat{\Delta}_{n}$ can be seen by carrying
out Edgeworth expansions of

$\displaystyle\mathbb{P}\left(n^{1/2}\Delta_{n}/s_{n}\leq z\right)\ \ \
\hat{\mathbb{P}}\left(n^{1/2}\hat{\Delta}_{n}/\hat{s}_{n}\leq z\right)$ (5.12)

This problem can be mitigated by Studentizing $n^{1/2}\hat{\Delta}_{n}$ with
$\tilde{s}_{n}$ or the estimator

$\displaystyle
n^{-1}\sum_{b=0}^{B-1}\sum_{i=1}^{\ell}\sum_{j=1}^{\ell}\left(X_{b\ell+i}-\hat{m}_{n}\right)\left(X_{b\ell+j}-\hat{m}_{n}\right)$
(5.13)

####  5.2.4. The Sieve Bootstrap for Linear Processes

A substantial improvement over the performance of the block bootstrap is
possible if the DGP is known to be a linear process. That is, the DGP has the
form

$\displaystyle
X_{i}-\mu=\sum_{j=1}^{\infty}\alpha_{j}\left(X_{i-j}-\mu\right)+U_{i},$ (5.14)

where $\mu=\mathbb{E}\left(X_{i}\right)$ for all $i$, $\left\\{U_{i}\right\\}$
of i.i.d random variables, and $\left\\{X_{i}\right\\}$ may be a scalar or a
vector process. Assume that $\sum_{j=1}^{\infty}\alpha_{j}^{2}<\infty$ and all
of the roots of the power series $1-\sum_{j=1}^{\infty}\alpha_{j}z^{j}$ are
outside of the unit circle. For instance, Paparoditis,  (1996) proposed
approximating (5.14) by an AR$(p)$ model in which $p=p(n)$ increases with
increasing sample size.

Let $\left\\{a_{nj}:j=1,...,p\right\\}$ denote least squares or Yule-Walker
estimates of the coefficients of the approximating process, and let
$\left\\{U_{nj}\right\\}$ denote the centered residuals. The sieve bootstrap
consists of generating bootstrap samples according to the process

$\displaystyle\hat{X}_{i}-m=\sum_{j=1}^{p}a_{nj}\left(\hat{X}_{i-j}-m\right)+\hat{U}_{j},$
(5.15)

where $m=n^{-1}\sum_{i=1}^{n}X_{i}$ and the $\hat{U}_{j}$ are sampled randomly
with replacement from the $U_{nj}$.

Furthermore, Paparoditis,  (2018) considers the Sieve bootstrap for functional
time series. In particular, bootstrap procedures for Hilbert space-valued time
series proposed so far in the literature, are mainly attempts to adapt, to the
infinite dimensional functional framework, of bootstrap methods that have been
developed for the finite dimensional time series case.

####  5.2.5. Discussion on Bootstrap Asymptotics

A different not so popular approach when deriving asymptotics for
bootstrapped-based estimators and test statistics is the use of the Stein’s
approximation. In terms of the implementation of Stein’s method via
econometric models, Bandi and Phillips,  (2007) consider a parametric
estimation method that matches parametric estimates of the drift and diffusion
functions to their functional counterparts. More precisely, the authors
consider a corresponding estimate of the distance between the process and the
limiting Wiener process, with respect to a certain metric. On the other hand,
it is argued that the Stein’s method proves weak convergence type results
through approximations to expectations without direct use of characteristic
functions, allowing it to be used in complex problems with dependence.
Consequently, in the particular paper the authors show that consistency can be
proved for any type of bootstrap with exchangeable weights using Stein’s
method. Furthermore, Barbour,  (1990) employs Stein’s method to derive rates
of convergence to a Wiener process limits. These limit results demonstrate
that the Stein operator can be utilized to find the solution of an equilibrium
distribution based on a collection of OU processes.

###  5.3. Bootstrapping in Time Series Regressions

The bootstrap proposed by Efron (1979) has proven to be a powerful
nonparametric tool for approximating the sampling distribution and variance of
a statistic of interest. In particular, the bootstrap estimates the asymptotic
distribution of $\tau_{n}$, that is,

$\displaystyle G_{n}^{*}(x):=\mathbb{P}^{*}(\tau_{n}\leq
x)\to_{p}G_{\infty}(x)$ (5.16)

######  Example 27.

Consider the following triangular array (see, Parker et al.,  (2006)).
Previous results on the residual-based block bootstrap are established by
Paparoditis and Politis,  (2003))

$\displaystyle x_{t,n}=\rho_{n}x_{t-1,n}+u_{t},\ \ \
\rho_{n}=\left(1+\frac{c}{n}\right),c<0$ (5.17)

Denote with $y_{t}=\displaystyle\sum_{j=1}^{t}\hat{u}_{j}$. Furthermore,
recall that

$\displaystyle S_{n}(r)=S_{n}\left(\frac{t-1}{n}\right),\ \ \ \text{where}\ \
\frac{t-1}{n}\leq r\leq\frac{t}{n}.$ (5.18)

Then, it follows from Theorem 1 and the continuous mapping theorem:

$\displaystyle\frac{1}{n^{2}}\sum_{t=2}^{n}y^{2}_{t-1}$
$\displaystyle=\frac{1}{n^{2}}\sum_{t=2}^{n}\left(\sum_{j=1}^{t}\hat{u}_{j}\right)^{2}=\frac{1}{n}\sigma_{n}^{2}\sum_{t=2}^{n}S^{2}_{n}\left(\frac{t-1}{n}\right)dr$
$\displaystyle=\sigma_{n}^{2}\sum_{t=2}^{n}\int_{(t-1)/n}^{t/n}S^{2}_{n}(r)dr=\sigma_{n}^{2}\int_{0}^{1}S^{2}_{n}(r)dr\overset{d}{\to}\int_{0}^{1}W^{2}(r)dr.$

Notice that to obtain the limit results as shown in the example above, we
consider the behaviour of the continuous standardized partial sum process
$\big{\\{}S_{n}(r),0\leq r\leq 1\big{\\}}$ given by

$\displaystyle S_{n}(r)$
$\displaystyle=\frac{1}{\hat{\sigma}_{n}^{2}}\frac{1}{\sqrt{n}}\sum_{j=1}^{\lfloor
nr\rfloor}\hat{u}_{j},$ (5.19) $\displaystyle\hat{\sigma}_{n}^{2}$
$\displaystyle=\text{Var}\left(\frac{1}{\sqrt{n}}\sum_{j=1}^{\lfloor
nr\rfloor}\hat{u}_{j}\right).$ (5.20)

To establish asymptotic theory results one considers the limit behaviour of
the following expression

$\displaystyle\begin{bmatrix}n^{-1/2}\hat{\beta}\\\ \\\
n\left(\hat{\rho}-1\right)\end{bmatrix}=\begin{bmatrix}1&\displaystyle\frac{1}{n^{3/2}}\sum_{t=2}^{n}x_{t-1}\\\
\\\
\displaystyle\frac{1}{n^{3/2}}\sum_{t=2}^{n}x_{t-1}&\displaystyle\frac{1}{n^{2}}\sum_{t=2}^{n}x^{2}_{t-1}\end{bmatrix}$
(5.21)

######  Example 28 ($I(1)$ Bootstrap Samples).

Estimate the regression

$\displaystyle\Delta y_{t}=\sum_{j=1}^{\ell_{\tau}}\pi_{j}\Delta
y_{t-j}+e_{t}^{*},\ \ t=\ell_{T}+2,....,T$ (5.22)

Consider the following wild bootstrap (see, Cavaliere et al.,  (2015) and
Cavaliere and Georgiev,  (2020))

$\displaystyle\tilde{\tau}_{n}:=a_{n}^{-1}\sum_{t=1}^{n}\epsilon_{t}^{*}$
(5.23)

where $\epsilon_{t}^{*}=\hat{\epsilon}_{t}w_{t}^{*}$, $w_{t}^{*}\sim(0,1)$. It
can be shown that

$\displaystyle P^{*}\left(\tilde{\tau}_{n}^{*}\leq
x\right)=\mathbb{P}\left(\frac{1}{a_{n}}\sum_{t=1}^{n}\hat{\epsilon}_{t}w_{t}^{*}\leq
x\bigg{|}\left\\{\epsilon_{t}\right\\}\right)\overset{w}{\to}\mathbb{P}\left(\sum_{t=1}^{\infty}\delta_{t}Z_{t}\leq
x\bigg{|}Z\right)$ (5.24)

The bootstrap mimics a particular conditional distribution of the original
statistic. In classic bootstrap inference, asymptotic bootstrap validity is
understood and established as convergence in probability (or almost surely) of
the CDF of $\tau$, say $F$. This convergence, along with continuity of $F$
implies that sup${}_{x\in\mathbb{R}}\left|F_{n}^{*}(x)-F(x)\right|\to 0$, in
probability (or almost surely). In other words, it is often the case that
bootstrap validity can be addressed through the lens of a conditioning
argument. In this regard, we consider related asymptotic results which concern
the probability that, for a sequence of random elements $X_{n}$, it holds that
the bootstrap $p-$value is uniformly distributed in large sample conditionally
on $X_{n}$

$\displaystyle\mathbb{P}\left(p_{n}^{*}\leq q|X_{n}\right)\overset{p}{\to}q,\
\ \ \forall\ q\in(0,1).$ (5.25)

This property that we call "bootstrap validity conditional on $X_{n}$",
implies unconditional validity. Thus, conditional bootstrap validity given
$X_{n}$ implies that the bootstrap replicates asymptotically the property of
conditional tests and confidence intervals to have, conditionally on $X_{n}$,
constant null rejection probability and coverage probability respectively.
When dealing with random limit distributions, the usual convergence concept
employed to establish bootstrap validity, that is, weak convergence in
probability, can only be used in some special cases. Instead, we use the
probabilistic concept of weak convergence of random measures. As $F$ is sample
path-continuous then,
$\left(\tau_{n},F_{n}^{*}\right)\overset{p}{\to}\left(\tau,F\right)$ on the
Skorokhod-representation space, whereas
$\left(\tau_{n},F_{n}^{*}\right)\overset{w}{\to}\left(\tau,F\right)$ on a
general probabilistic space. In other applications, the idea of similar proof
would be to choose $X_{n}$ through $D_{n}-$measurable random elements such
that $\tau_{n}^{*}$ depends on the data essentially through $X_{n}$ (see,
Cavaliere and Georgiev,  (2020)). Further details on the bootstrap for
nonstationary autoregressions can be found in Katsouris, 2023a (see, also
Reichold and Jentsch,  (2022)).

Overall, the following bootstrap consistency result holds

$\displaystyle\underset{x\in\mathbb{R}}{\text{sup}}\big{|}\mathbb{P}^{*}\left(\tau_{n}^{*}\leq
x\right)-\mathbb{P}\left(\tau_{n}^{*}\leq x\right)\big{|}\to_{p}0$ (5.26)

Therefore, the bootstrap p-value satisfies
$p_{n}^{*}:=\mathbb{P}^{*}\left(\tau_{n}^{*}\leq
x\right)|_{x=\tau_{n}}=G_{n}^{*}(\tau_{n})\to_{w}U[0,1]$.

######  Example 29.

Consider the first order autoregression with a unit root given by

$\displaystyle y_{t}=\rho y_{t-1}+\epsilon_{t},\ \ \rho=1\ \ \epsilon_{t}\sim\
i.i.d\left(0,\sigma^{2}\right)$ (5.27)

Let $J_{c}(r)$ represent the OU process with mean reversion parameter $c$. We
define with

$\displaystyle
H_{c}(x):=\mathbb{P}\left(\frac{\displaystyle\int_{0}^{1}J_{c}dJ_{c}}{\displaystyle\int_{0}^{1}J_{c}^{2}}\leq
x\right)$ (5.28)

Let
$\tau_{n}:=n\left(\hat{\rho}-1\right)\to\tau_{\infty}:=\displaystyle\int_{0}^{1}J_{0}dJ_{0}\bigg{/}\displaystyle\int_{0}^{1}J_{0}^{2}du$,
that is, $\mathbb{P}\left(\tau_{n}\leq x\right)\to H_{0}(x)$.

######  Example 30.

Consider the location model with non-stationary stochastic volatility

$\displaystyle y_{t}=\mu+\sigma_{t}\epsilon_{t}$ (5.29)

Then, the assumption of non-stationary stochastic volatility implies that,
$\sigma_{n}(r)$ on $\mathcal{D}[0,1]$ is the approximate of
$\left\\{\sigma_{t}\right\\}$, that is, $\sigma_{n}(.):=\sigma_{[nr]}\
\text{with}\ r\in[0,1]$. It can be proved that the standardized estimator of
the mean is mixed normal, as $n\to\infty$, that is,

$\displaystyle\tau_{n}:=\sqrt{n}\left(\hat{\mu}_{n}-\mu\right)=n^{-1/2}\sum_{t=1}^{n}\sigma_{t}\epsilon_{t}\to\tau_{\infty}=\int_{0}^{1}\sigma(u)dB(u)\equiv\mathcal{MN}\left(0,V\right)$
(5.30)

with $V=\displaystyle\int_{0}^{1}\sigma^{2}(u)du$.

######  Example 31.

Consider the Gaussian Wild Bootstrap, given by the following form

$\displaystyle y_{t}^{*}:=\epsilon_{t}\eta_{t}^{*},\ \
\eta^{*}_{t}\sim^{\text{i.i.d}}N(0,1)$ (5.31)

and the bootstrap statistic given by

$\displaystyle\tau_{n}^{*}:=n^{-1/2}\sum_{t=1}^{n}y_{t}^{*}=n^{-1/2}\sum_{t=1}^{n}\epsilon_{t}\eta_{t}^{*}$
(5.32)

Conditional on the original data, with
$\hat{V}_{n}:=n^{-1}\sum_{t=1}^{n}\epsilon_{t}$,
$\tau_{n}^{*}|\left\\{y_{t}\right\\}\sim N\left(0,\hat{V}_{n}\right)$, or
equivalently, denoting with $\Phi$ the CDF for $N(0,1)$ we have that

$\displaystyle\mathbb{P}^{*}\left(\tau_{n}^{*}\leq
x\right)=\mathbb{P}^{*}\left(N(0,\hat{V}_{n})\leq
x\right)=\Phi\left(\hat{V}_{n}^{-1/2}x\right),x\in\mathbb{R}.$ (5.33)

Therefore, since $\hat{V}_{n}\to V:=\displaystyle\int\sigma^{2}(u)du$, we have
a random limit given by

$\displaystyle\mathbb{P}^{*}\big{(}\tau_{n}^{*}\leq
x\big{)}\to\Phi\left(V_{n}^{-1/2}x\right)=\mathbb{P}\big{(}N(0,V)\leq
x|V\big{)},\ x\in\mathbb{R}\ \text{which implies that}\ \tau_{n}^{*}\to
N(0,V)|V.$ (5.34)

##  6\. Open Problems

Many open problems remain in the econometrics and statistics literature. We
give emphasis to the aspects related to the development of an econometric
environmnet that incorporates network dependence and time series
nonstationarity. Below we present related literature and ask some relevant
research questions.

###  6.1. Literature on Measures of Connectedness

Financial connectedness is a crucial concept in understanding the structure of
financial systems as well as to monitor financial contagion, spillover effects
and the amplification of systemic risk. In particular, the rise of systemic
risk is considered to be due to shocks transmitted via system-wide
connectedness. Moreover, such type of connectedness can be traced in the long
memory properties of time series within a network and therefore appropriate
modelling methodologies require to capture the evolution of spillover effects
and the network’s spectral response within a infinite time-horizon (e.g.,
Schennach,  (2013)). On the other hand, if we are only interested on capturing
financial contagion then for example, Glasserman and Young,  (2015) provide a
framework which depends less on the network topology and link formation.
However, our approach aims to include such information and additionally
provide a more robust representative of the interactions within financial
networks by incorporating features of time series.

Three important lines of literature provide an in depth examination of this
concept. Firstly, Diebold and Yılmaz,  (2014), introduce a robust framework to
study network connectedness via the use of forecast error variance
decomposition (FEVD) of an estimated VAR process, as a transformation method
that summarizes connectedness between a set of institutions. The significance
of their methodology is the provision of directional connectedness measures
which are analogous to bilateral imports and exports for each of a set of $N$
countries. Consider, an $N$ dimensional covariance-stationary data generating
process (DGP) which describes the evolution of the economic network under
examination within a VAR(p) topology, i.e.,
$\boldsymbol{Y_{t}}=(y_{1,t},...,y_{N,t})$ for $t=1,...,T$. Then, by the Wold
decomposition theorem gives
$\boldsymbol{\Phi(L)}\boldsymbol{Y}_{t}=\boldsymbol{\epsilon_{t}}$, where
$\boldsymbol{\Phi(L)}=\sum_{h}\boldsymbol{\Phi}_{h}L^{h}$ and assuming that
$|\Phi(z)|$ lie outside the unit-circle, the VAR process has the MA($\infty$)
representation,
$\boldsymbol{Y_{t}}=\boldsymbol{\Theta(L)}\boldsymbol{\epsilon_{t}}$, where
$\boldsymbol{\Theta(L)}$ is an $(NxN)$ matrix of infinite lag polynomials.
Note that many of the aspects of connectedness are described with this
topology. Also, $\boldsymbol{\epsilon}_{t}$ is considered to be a white-noise
generated by the covariance matrix $\boldsymbol{\Sigma}$. Therefore, the
H-step generalized variance decomposition matrix DgH = $[d_{ij}^{gH}]$ is
given by 444See also, Lanne and Nyberg,  (2016)

$\displaystyle
d_{ij}=\frac{\sigma_{ij}^{-1}\sum_{h=0}^{H-1}(e^{\prime}_{i}\boldsymbol{\Theta}_{h}\boldsymbol{\Sigma}e_{j})^{2}}{\sum_{h=0}^{H-1}(e^{\prime}_{i}\boldsymbol{\Theta}_{h}\boldsymbol{\Sigma}\boldsymbol{\Theta}^{{}^{\prime}}_{h}e_{i})}$
(6.1)

where $e_{j}$ are selection matrices e.g.,
$e_{j}=(0,...1,...0)^{{}^{\prime}}$, $\boldsymbol{\Theta}_{h}$ is the
coefficient matrix evaluated at the $h$-lagged shock vector and $\sigma_{ij}$
is the $j$th diagonal element of $\boldsymbol{\Sigma}$.

The particular methodology is build on the previous work of Diebold and
Yilmaz,  (2012), who introduce generalized spillover measurements, separating
directional spillovers (i.e. how much of the shocks to the volatility
spillovers across major asset classes) and net pairwise spillovers (i.e., how
much each market contributes to the volatility in other markets). The H-step
variance decomposition gives the fraction of variable $i$’s H-step forecast
error variance due to shocks in variable $j$ and practically is associated
with connectedness measures since it provides information regarding the
connectedness among the components of the series (e.g., as a network adjacency
matrix).

Secondly, Billio et al.,  (2012), examine the applications of two econometric
methodologies as measures of connectedness which are designed to capture
changes in correlation and causality among financial institutions, that is,
(i) principal components (PCA) and (ii) Linear Granger causality. Therefore,
using the PCA methodology, the authors observe that during periods when a
subset of principal components has an explanatory power greater than a
threshold $H$ of the total volatility in the system this is indicative of
increased interconnectedness between financial institutions. Furthermore, if
we let $\lambda_{j}$ to represent the $j$th eigenvalue of the diagonal matrix
of loadings $\Lambda$, then the critical threshold of increased
interconnectedness is defined to be

$\displaystyle\frac{\sum_{j=1}^{n}\lambda_{j}}{\sum_{j=1}^{N}\lambda_{j}}\equiv
h_{n}\geq H$ (6.2)

Moreover, there is a mapping between the contribution and exposure of
institution $j$ to the risk of the system, which is given by

$\displaystyle\text{PCAS}_{j,n}=\frac{1}{2}\frac{\sigma^{2}_{j}}{\sigma^{2}_{S}}\frac{\partial\sigma^{2}_{S}}{\partial\sigma^{2}_{S}}\biggr{\rvert}_{h_{n}\geq
H}=\sum_{k=1}^{n}\frac{\sigma^{2}_{j}}{\sigma^{2}_{S}}L^{2}_{jk}\lambda_{k}\biggr{\rvert}_{h_{n}\geq
H}$ (6.3)

The above measure gives a degree of connectedness, thus in order to also
measure the directionality of connectedness the authors use the linear Granger
causality. Linear Granger causality as first introduced by Granger,  (1969,
1980) has been further tested in various frameworks (e.g., VAR(p) models) and
for different moments (e.g., first or higher order). For example, let two time
series be $\\{Y_{i,t}\\}_{t=1}^{T}$ and $\\{Y_{j,t}\\}_{t=1}^{T}$ with the
corresponding information set available at time $t-1$ be
$I_{t-1}\equiv(I_{i,t-1},I_{j,t-1})$. Then we say that if there are enough
statistical evidence to support the hypothesis

$\displaystyle H_{1}:F(Y_{i,t}|I_{i,t-1})\neq F(Y_{i,t}|I_{t-1}),\text{for}\
\forall\ F(.|I_{t-1})\in\mathbb{R}$ (6.4)

then there is additional information in the set of $Y_{j,t-1}$ which helps
predict the future values of $Y_{i,t-1}$, that is, time series
$\\{Y_{j,t}\\}_{t=1}^{T}$ is said to "Granger-cause"
$\\{Y_{i,t}\\}_{t=1}^{T}$. Based on this idea, the authors introduce various
measures of directionality connectedness such as degree of Granger Causality,
number of connections, sector-conditional connections, closeness and
centrality. The authors also present an extension of this framework to the
case of non-linear Granger Causality which captures volatility-based
interconnectedness.

###  6.2. Further Research

The network and multivariate time series literature have seen plethora of
independent applications in recent years, but a unified framework bridging the
gap remains an open research question. In this course we have studies various
estimation and inference methods from the time series econometrics literature
as well as useful limit theorems relevant to network econometrics. Therefore,
the study statistical estimation methodologies under network dependence and
nonstationarity can indeed be fruitful. A crucial component in this stream of
literature is the identification of network structures via shock propagation.
Specifically, one can consider a single network with directed links. Although
most of the current literature considers some dependence structure for time
series data when considering nonstationary processes, some form of weak
dependence provides enough information for identification purposes. This
motivates the study of cointegration dynamics under network dependence.

More specifically, Schennach,  (2018) consider a diffusion process to identify
shocks across the nodes of a network. Furthermore, recently Bykhovskaya,
(2022) consider an estimation and prediction framework where the model
specifies the temporal evolution of a weighted network that combines classical
autoregression with non-negativity, a positive probability of vanishing and
peer effect interactions between weights assigned to edges in the process.
Although nonstationarity in time series can be interpreted as time-varying
model parameters, we shall focus on cointegrating and unit root dynamics when
the underline data structure is defined across nodes of a single network
(graph). In other words, we are interested for conditions of network
stationarity against explosiveness of the underline network evolution process.

Consider distance as a measure of dependence, so-called $m-$dependence case. A
sequence of random variables $\left\\{X_{i},i\geq 1\right\\}$ is a sequence of
$m-$dependent random variables if the two sets of random variables
$\left\\{X_{1},X_{2},...,X_{i}\right\\}$ is a sequence of $m-$dependent random
variables if the two sets of random variables
$\left\\{X_{1},...,X_{i}\right\\}$ and
$\left\\{X_{j},X_{j+1},...,X_{n}\right\\}$ are independent whenever $j-i>m$.
Therefore, interesting research questions we are interested to examine is how
$m-$dependence expressed in terms of the distance of nodes in a graph can
affect the stability of the system as approximated by the persistence
properties of regressors as well as the degree of network dependence.

##  Appendix A Elements of Stochastic Processes

###  A1. Probability Theory

######  Proposition 13 (Martingale Difference CLT).

In the notation of the previous theorem, let $\left\\{X_{in}:i\leq n\right\\}$

$\displaystyle X_{in}=\frac{X_{i}-\mathbb{E}(X_{i})}{\sqrt{n}}$ (A.1)

and suppose that for each $n$,
$\left\\{\left(X_{in},\mathcal{A}_{in}\right):i\leq n\right\\}$ is a
martingale difference sequence satisfying the Linderberg condition, such that

$\displaystyle
W_{n}=\sum_{i=1}^{n}\mathbb{E}\left[X_{in}^{2}\boldsymbol{1}\left\\{|X_{in}|\geq\frac{1}{r}\right\\}\big{|}\mathcal{F}_{i-1,n}\right]$
(A.2)

The following statements hold true:

  * (i)

If
$\displaystyle\sum_{i=1}^{n}\mathbb{E}\left[X_{in}^{2}|\mathcal{F}_{i-1,n}\right]\to\sigma^{2}$,
then $\displaystyle
z_{n}=\sum_{i=1}^{n}X_{in}^{2}\overset{d}{\to}\zeta\sim\mathcal{N}\left(0,\sigma^{2}\right)$

  * (ii)

If $\sum_{i=1}^{n}X_{in}^{2}\overset{p}{\to}\sigma^{2}$, then
$z_{n}\to\zeta\sim\mathcal{N}(0,\sigma^{2})$.

The stochastic process
$X:\mathbb{R}_{+}\times\Omega\mapsto\mathcal{B}\left(\mathbb{R}_{+}\right)\otimes\mathcal{E}$
is measurable when this mapping is measurable, that is, for all
$A\in\mathcal{B}\left(\mathbb{R}\right)$,

$\displaystyle\left\\{(t,\omega):X(t,\omega)\in
A\right\\}\in\mathcal{B}\left(\mathbb{R}\right),$ (A.3)

Therefore, as a consequence of this measurability and Fubini’s theorem,
$X(.,\omega):\mathbb{R}_{+}\mapsto\mathbb{R}$ is almost surely measurable,
while for measurable functions $h:\mathbb{R}\mapsto\mathbb{R}$,

$\displaystyle
Y(\omega)\equiv\int_{\mathbb{R}_{+}}h\left(X(t,\omega)\right)dt$ (A.4)

is a random variable provided this integral exists. A stochastic process on
$\mathbb{R}_{+}$, is necessarily measurable, if for example, the trajectories
are either almost surely continuous or almost surely monotonic and right-
continuous. We are interested to examine the evolution of a stochastic
process, that is, we observe $\left\\{X(s,\omega):0<s\leq t\right\\}$ for some
(unknown) $\omega$ and finite time interval $(0,t]$. It is then natural to
consider the $\sigma-$algebra,
$\mathcal{F}_{t}^{(X)}\equiv\sigma\left\\{X(s,\omega):0<s\leq t\right\\}$,
generated by all possible such evolutions.

Clearly, $\mathcal{F}_{s}^{(X)}\subset\mathcal{F}_{t}^{(X)}$, for
$0<s<t<\infty$. Generally speaking, an expanding family
$\boldsymbol{\mathcal{F}}=\left\\{\mathcal{F}_{t}:0\leq t<\infty\right\\}$ of
$\sigma-$algebras of $\mathcal{E}$ is called a filtration or history. Suppose
$X$ is measurable and $\mathcal{F}_{t}-$adapted. A stronger condition to
impose on $X$ is that of progressive measurability with respect to
$\mathcal{F}$, meaning that for every $t\in\mathbb{R}_{+}$ and any
$A\in\mathcal{B}\left(\mathbb{R}\right)$,

$\displaystyle\left\\{(s,\omega):)<s\leq t,X(s,\omega)\in
A\right\\}\in\mathcal{B}\left((0,t]\right)\times\mathcal{F}_{t}.$ (A.5)

Therefore, given any measurable $\mathcal{F}-$adapted $\mathbb{R}-$valued
process $X$, we can find an $\mathcal{F}-$progressively measurable process
$Y$, that is modification of X, in the sense of being defined on
$(\Omega,\mathcal{E},\mathbb{P})$ and satisfying

$\displaystyle\mathbb{P}\left\\{\omega:X(t,\omega)=Y(t,\omega)\right\\}=1$
(A.6)

Therefore, the sets of the form $[s,t]\times U,0\leq
s<t,U\in\mathcal{F}_{t},t\geq 0$, generate a $\sigma-$algebra on
$\mathbb{R}_{+}\times\Omega$, which may be called the
$\boldsymbol{\mathcal{F}}-$progressively measurable may be rephrased as the
requirement that $X(t,\omega)$ be measurable with respect to the
$\boldsymbol{\mathcal{F}}-$progressive $\sigma-$algebra. A more restrictive
condition to impose on $X$ is that it be
$\boldsymbol{\mathcal{F}}-$predictable. Then, call the sub-$\sigma-$algebra of
$\mathcal{B}\left(\mathbb{R}_{+}\right)\otimes\mathcal{E}$ generated by the
product sets of the form $(s,t]\times U$, where $U\in\mathcal{F}_{s},t\geq s$,
and $0\leq s<\infty$, the predictable $\sigma-$algebra, denoted by
$\Psi^{\mathcal{F}}$. This terminology is chosen because it reflects what can
be predicted at some future time $t$ given the evolution of the process- as
revealed by sets $U\in\mathcal{F}_{s}$-up to the present time $s$. Then $X$ is
$\mathcal{F}-$predictable when it is $\Psi^{\mathcal{F}}-$measurable, that is,
for any $A\in\mathcal{B}\left(\mathbb{R}\right)$,

$\displaystyle\left\\{(t,\omega):X(t,\omega)\in
A\right\\}\in\Psi^{\mathcal{F}}.$ (A.7)

Moreover, the $\mathcal{F}-$predictable process is left-continuous and the
left-continuous history
$\mathcal{F}_{(-)}\equiv\left\\{\mathcal{F}_{-}\right\\}$ associated with
$\mathcal{F}$ appears, here $\mathcal{F}_{0-}=\mathcal{F}_{0}$ and
$\mathcal{F}_{t-}=\text{lim
sup}_{s<t}\mathcal{F}_{s}\vee_{s<t}\mathcal{F}_{s}$. Notice that if
$X(t,\omega)$ is $\mathcal{F}_{t-}-$measurable, its value at $t$ is in fact
determined by information at times prior to $t$.

######  Definition 12.

Given a history $\mathcal{F}$, a nonnegative r.v $T:\Omega\mapsto[0,\infty]$
is an $\mathcal{F}-$stopping time if,

$\displaystyle\left\\{\omega:T(\omega)\leq t\right\\}\in\mathcal{F}_{t},\ \ \
0\leq t<\infty.$ (A.8)

If $S,T$ are stopping times, then so are $S\vee T$ and $S\wedge T$. Indeed,
given a family $\left\\{T_{n}:n=1,2,...\right\\}$ of stopping times,
$\text{sup}_{n\leq 1}T_{n}$ is an $\mathcal{F}-$stopping time, while
$\text{inf}_{n\geq 1}T_{n}$ is an $\mathcal{F}_{(+)}-$stopping time.

######  Lemma 11.

Let $X$ be an $\mathcal{F}-$adapted monotonically increasing right-continuous
process, and let $Y$ be an $\mathcal{F}_{0}-$measurable r.v. Then
$T(\omega)\equiv\text{inf}\left\\{t:X(t,\omega)\geq Y(\omega)\right\\}$ is an
$\mathcal{F}-$stopping time, possibly extended, while if $X$ is
$\mathcal{F}-$predictable, then $T$ is an (extended)
$\mathcal{F}_{(-)}-$stopping time.

As an important corollary to this result, observe that if $X$ is
$\mathcal{F}-$progressive and almost surely integrable on finite intervals,
then

$\displaystyle Y(t,\omega)=\int_{0}^{t}X(s,\omega)ds$ (A.9)

is $\mathcal{F}-$progressive, $Y(T)$ is a r.v if $T<\infty$ and $Y(t\wedge T)$
is again $\mathcal{F}-$progressive.

##### Martingales

######  Definition 13.

Let $(\Omega,\mathcal{E},\mathbb{P})$ be a probability space, $\mathcal{F}$ a
history on $(\Omega,\mathcal{E})$ and $X(.)\equiv\left\\{X(t):0\leq
t<\infty\right\\}$ a real-valued process adapted to $\mathcal{F}$ and such
that $\mathbb{E}\left(|X(t)|\right)<\infty$ for $0\leq t<\infty$. Then, $X$ is
an $\mathcal{F}-$martingale if $0\leq s<t<\infty$,

$\displaystyle\mathbb{E}\left[X(t)|\mathcal{F}_{s}\right]=X(s)\ \text{almost
surely},$ (A.10)

and an $\mathcal{F}-$submartingale if

$\displaystyle\mathbb{E}\left[X(t)|\mathcal{F}_{s}\right]\geq X(s)\
\text{almost surely},$ (A.11)

and an $\mathcal{F}-$supmartingale if the reverse inequality holds.

######  Theorem 13.

(Doob-Meyer). Let $\mathcal{F}-$ be a history and $X(.)$ a bounded
$\mathcal{F}-$submartingale with right-continuous trajectories. Then, there
exists a unique (up to equivalence) uniformly integrable
$\mathcal{F}-$martingale $Y(.)$ and a unique $\mathcal{F}-$predictable
cumulative process $A(.)$ such that

$\displaystyle X(t)=Y(t)+A(t).$ (A.12)

Therefore, for nondecreasing process $A(.)$ with right-continuous
trajectories, it can be shown that $\mathcal{F}-$predictability is equivalent
to the property that for every bounded $\mathcal{F}-$martingale $Z(.)$ and
positive $u$,

$\displaystyle\mathbb{E}\left[\int_{0}^{u}Z(t)A(dt)\right]=\mathbb{E}\left[\int_{0}^{u}Z(t-)A(dt)\right].$
(A.13)

Since for any $\mathcal{F}-$adapted cumulative process $\xi$ and any
$\mathcal{F}-$martingale $Z$,
$\mathbb{E}\left[Z(u)\int_{0}^{u}\xi(dt)\right]=\mathbb{E}\left[Z(t)\int_{0}^{u}\xi(dt)\right]$,
the property above is then equivalent to

$\displaystyle\mathbb{E}\left[Z(u)A(u)\right]=\mathbb{E}\left[\int_{0}^{u}Z(t-)A(dt)\right].$
(A.14)

A cumulative process with this property is referred to in many texts as a
natural increasing process. The theorem can then be reshaped as: every bounded
submartingale has a unique decomposition into the sum of a uniformly
integrable martingale and a natural increasing function. Therefore, there is a
relation between natural increasing and predictable processes.

##### Monotone Convergence Theorem

######  Theorem 14.

Let $\left(X_{n}\right)_{n\geq 1}$ be a monotonically sequence, that is,
$X_{n}\leq X_{n+1}$ almost surely. Assume that
$\mathbb{E}\left[|X_{n}|\right]<\infty$ for every $n\geq 1$. Then
$X_{n}(\omega)\to X(\omega)$ for all $\omega$ and some limiting random
variable $X$ (that is possibly degenerate), and

$\displaystyle\mathbb{E}\left[X_{n}\right]\to\mathbb{E}\left[X\right].$ (A.15)

##### The Supercritical Regime

The main result in this section, is a law of large numbers for the size of the
maximal connected component. Below, we write
$\zeta_{\lambda}=1-\eta_{\lambda}$, so the following theorem shows that there
exists a giant component:

######  Theorem 15.

(LLN for giant components) Fix $\lambda>1$. Then, for every
$\nu\in(\frac{1}{2},1)$, there exists $\delta=\delta(\nu,\lambda)>0$ such that

$\displaystyle\mathbb{P}_{\lambda}\bigg{(}|\mathcal{E}|-\zeta_{\lambda}\eta\geq\eta^{\nu}\bigg{)}=\mathcal{O}\left(\eta^{-\delta}\right).$
(A.16)

The Theorem says that: a vertex has a large connected component with
probability $\zeta_{\lambda}$. Therefore, at least in expectation, there are
roughly $\zeta_{\lambda}\eta$ vertices with large connected components.

###### Proof.

(page 130) Here, we give an overview of the proof. We rely on an analysis of
the number of vertices in connected components of size at least $k$,

$\displaystyle Z_{\geq
k}=\sum_{v\in[n]}\boldsymbol{1}\left\\{|\mathcal{E}(v)|\geq k\right\\}.$
(A.17)

The proof contains 4 main steps. In the first step, for $k_{n}=K\text{log}n$
and $K$ sufficiently large, we compute

$\displaystyle\mathbb{E}_{\lambda}\left[Z_{\geq
k}\right]=n\mathbb{P}_{\lambda}\big{(}|\mathcal{E}(v)|\geq k\big{)}$ (A.18)

We evaluate $\mathbb{P}_{\lambda}\big{(}|\mathcal{E}(v)|\geq k\big{)}$ using
the bound of the theorem above. Moreover, the Proposition below

$\displaystyle\mathbb{P}_{\lambda}\big{(}|\mathcal{E}(v)|\geq
k_{n}\big{)}=\zeta_{\lambda}(1+o(1)).$ (A.19)

In the second step, we use a variance estimate on $Z_{\geq k}$ in Proposition
4.10, implying that $\forall\ \nu\in(\frac{1}{2},1)$,

$\displaystyle|Z_{\geq k}-\mathbb{E}\left[Z_{\geq k}\right]|\leq n^{\nu}$
(A.20)

In the third step, we show that for $k=k_{n}=K\text{log}n$ for some $K>0$
sufficiently large and assuming that there is no connected component of size
in between $k_{n}$ and $\alpha n$, for any $\alpha<\zeta_{\lambda}$. This is
done by a first moment argument: the expected number of vertices in such
connected components is equal to $\mathbb{E}_{\lambda}\left[Z_{\geq
k}-Z_{\geq\alpha n}\right]$, and we use the bound in Proposition and another
Proposition, which states that, for any $\alpha<\zeta_{\lambda}$, there exists
$J=J(\alpha)>0$ such that, for all $n$ sufficiently large,

$\displaystyle\mathbb{P}_{\lambda}\big{(}k_{n}\leq|\mathcal{E}(v)|<\alpha
n\big{)}\leq e^{-k_{n}J}.$ (A.21)

In the forth step, we prove that for $2\alpha>\zeta_{\lambda}$, and when there
are no clusters of size in between $k_{n}$ and $\alpha_{n}$, then it holds
that $Z_{\geq k_{n}}=|\mathcal{E}_{\text{max}}|$. ∎

###  A2. Elements of Large Deviations Principles

Following the framework of Gao and Zhao,  (2011), let
$C\left(\bar{B}(\theta_{0},\eta)\right)$ denote the space of continuous
$\mathbb{R}^{d}-$valued functions on $\bar{B}(\theta_{0},\eta)$ and define
$\left\lVert
f\right\rVert=\text{sup}_{\theta\in\bar{B}(\theta_{0},\eta)}|f(\theta)|$ for
$f\in C\left(\bar{B}(\theta_{0},\eta)\right)$. Let $\Psi_{0}(\theta)$ and
$\Psi_{0n}$ be the restrictions of $\Psi$ and $\Psi_{n}$ on
$\bar{B}(\theta_{0},\eta)$. Moreover, we have that $\left\\{a(n),n\geq
1\right\\}$ satisfies $a(n)\to\infty$ and $\frac{a(n)}{\sqrt{n}}\to 0$ and
$\left\\{\psi(X_{i},\theta),i\geq 1\right\\}$ satisfies

$\displaystyle\frac{\sqrt{n}}{a(n)}\underset{\theta\in\bar{B}(\theta_{0},\eta)}{\text{sup}}\left|\Psi_{n}(\theta)-\Psi(\theta)\right|\to
0$ (A.22)

and

$\displaystyle\underset{n\to\infty}{\text{lim sup}}\frac{1}{a^{2}(n)}\
\text{log}\left(n\mathbb{P}\left(\underset{\theta\in\bar{B}(\theta_{0},\eta)}{\text{sup}}\left|\psi(X,\theta)\right|\geq\sqrt{n}a(n)\right)\right)=-\infty.$
(A.23)

Let $Y$ be a random variable taking its values in a Banach space and
$\mathbb{E}(Y)=0$. If there exists a sequence of increasing nonnegative
functions $\left\\{H_{k},k\geq 1\right\\}$ on $(0,+\infty)$ satisfying

$\displaystyle\underset{u\to\infty}{\text{lim}}u^{-2}H_{k}(u)=+\infty,\ \
\underset{k\to\infty}{\text{lim}}\underset{n\to\infty}{\text{lim}}\frac{1}{a^{2}(n)}\text{log}\frac{H_{k}(\sqrt{n}a(n))}{n}=+\infty,$
(A.24)

and

$\displaystyle\mathbb{E}\left(H_{k}(\left\lVert Y\right\rVert)\right)<\infty,\
\ \text{for any}\ \ k\geq 1,$ (A.25)

then

$\displaystyle\underset{n\to\infty}{\text{lim
sup}}\frac{1}{a^{2}(n)}\text{log}\bigg{(}n\mathbb{P}\big{(}\left\lVert
Y\right\rVert\geq\sqrt{n}a(n)\big{)}\bigg{)}=-\infty.$ (A.26)

Let $\left(\xi_{k},\mathcal{F}_{k}\right)$ be a martingale difference
sequence. The paper concerns the tail behvaiour of the quadratic form
$S_{n}=\sum_{k=1}^{n}\sum_{j=1}^{k-1}\beta_{n}^{k-j}\xi_{k}\xi_{j}$, where
$\beta_{n}\to 1$ as $n\to\infty$. Then, the main conclusions about
$\mathbb{P}\left(\frac{1}{n}S_{n}>x_{n}\right)$, where $x_{n}\to\infty$ as
$n\to\infty$, are obtained using the tail behaviour of a martingale with
values in a certain Hilbert space.

We set with

$\displaystyle
S_{n}=\sum_{k=1}^{n}\sum_{j=1}^{k-1}\beta_{n}^{k-j}\xi_{k}\xi_{j},$ (A.27)

where $\left\\{\beta_{n}\right\\}$ is a sequence of real numbers such that
$\beta_{n}\to 1$ as $n\to\infty$.

In particular, this paper considers the tail behaviour of the quadratic form
$S_{n}$, which strongly depends on a near the unity boundary behaviour of
$\beta_{n}$. We assume that the sequence $\left\\{\xi_{k}\right\\}$ satisfies

$\displaystyle
n^{-1}\sum_{i=1}^{[nt]}\mathbb{E}\left[\xi^{2}\big{|}\mathcal{F}_{i-1}\right]\to
t,\ \ \text{as}\ \ n\to\infty.$ (A.28)

for each $t\in[0,1]$ and

$\displaystyle
n^{-1}\sum_{i=1}^{n}\mathbb{E}\left[\xi^{2}_{i}\chi\left\\{|\xi_{i}|>\epsilon\sqrt{n}\right\\}\big{|}\mathcal{F}_{i-1}\right]\to
0,\ \ \text{as}\ n\to\infty$ (A.29)

for each $\epsilon>0$. Then, we have that

$\displaystyle n^{-1}S_{n}\to_{D}\int_{0}^{1}Y(t)dW(t),\ \ \text{as}\ \
n\to\infty.$ (A.30)

if $n\left(n-\beta_{n}^{2}\right)\to\gamma\geq 0$, as $n\to\infty$. We have
that $W(t),t\in[0,1]$ is a standard Wiener process where $Y(t)$ is an OU
process defined by the Ito stochastic differential equation

$\displaystyle dY(t)=-\frac{\gamma}{2}Y(t)dt+dW(t).$ (A.31)

Moreover, if condition (A.28) is satisfied and
$\mathbb{E}\left[\xi_{k}^{2}|\mathcal{F}_{k-1}\right]=1$ almost surely for
every $k=1,2,...$, then from the martingale central limit theorem it follows
that

$\displaystyle\left(1-\beta_{n}^{2}\right)^{1/2}n^{-1/2}S_{n}\to\mathcal{N}(0,1),\
\text{as}\ n\to\infty.$ (A.32)

We shall investigate the tail behaviour of the quadratic form $S_{n}$ assuming
the limit theorems above hold. Particularly, it follows that

$\displaystyle\frac{\displaystyle\mathbb{P}\left(n^{-1}S_{n}>x\right)}{\displaystyle\mathbb{P}\left(\int_{0}^{1}Y(t)dW(t)>x\right)}\to
1,\ \text{as}\ n\to\infty,$ (A.33)

uniformly with respect to $x$ in the interval $0\leq x\leq O(1)$.

The aim of this section is the extension of the above convergence result to
the uniform convergence over a wider range $0\leq x\leq x_{n}$, where
$x_{n}\to\infty$. Usually, these results refer to as limit theorems for large
deviation probabilities. In particular, this topic is well investigated for
sums of i.i.d random variables, for sums of i.i.d random elements taking
values in certain Banach spaces, for martingales and semimartingales. Limit
theorems for large deviation probabilities of quadratic form, where
$\beta_{n}\to 1$, are motivated by investigations of nearly nonstationary
autoregressive processes.

######  Theorem 16.

Let $S_{n}$ be defined as $\eqref{def}$, where $1-\beta^{2}\geq 0$,
$n(1-\beta_{n}^{2})\to\gamma\geq 0$, as $n\to\infty$ and let
$\left\\{\mathcal{F}_{k}\right\\}-$mds and $\left\\{\xi_{k}\right\\}$ satisfy
the following conditions:

  1. (i)

there exists a finite constant $M>0$ such that $|\xi_{k}|\leq M$ almost surely
for every $k\geq 1$,

  2. (ii)

$\mathbb{E}\left[\xi_{k}^{2}|\mathcal{F}_{k-1}\right]=1$ almost surely for
every $k\geq 1$.

Then the following holds

$\displaystyle\underset{0\leq x\leq
x_{n}}{\text{sup}}\left|\frac{\displaystyle\mathbb{P}\left(n^{-1}S_{n}>x\right)}{\displaystyle\mathbb{P}\left(\int_{0}^{1}Y(t)dW(t)>x\right)}-1\right|$
$\displaystyle\to 0,\text{as}\ \ n\to\infty$ (A.34) $\displaystyle
x_{n}.\text{max}\bigg{\\{}n^{-1/3}:\left|\gamma-n\left(1-\beta_{n}^{2}\right)\right|^{1/3}\bigg{\\}}$
$\displaystyle\to\ \text{as}\ n\to\infty.$ (A.35)

where for each $n$, $\left\\{\hat{\xi}_{nk},k=1,2,...\right\\}$ is a
martingale difference sequence in the space $L_{2}\left([0,1],\mu\right)$
where $\mu(dt)=\delta_{1}(dt)+\gamma dt$, and $\delta_{\alpha}$ is the
probability measure concentrated at the point $\alpha$. The second fact is a
limit theorem for probabilities of large deviations for martingales taking
values in Hilbert spaces.

##  Appendix B Elements of Real Analysis

###  B1. Convergence and Cauchy Sequences

######  Example 32.

Consider the space $C^{0}\left([0,1]\right)$ with the usual metric

$\displaystyle d(f,g)=\underset{0\leq t\leq
1}{\text{max}}\left|f(t)-g(t)\right|.$ (B.1)

A sequence $(f_{n})_{n\in\mathbb{N}}$ converges to $f_{0}$ in
$C^{0}\left([0,1]\right)$ if, and only if,

$\displaystyle\forall\ \epsilon>0\exists N\in\mathbb{N}\forall n\leq N\
\forall t\in[0,1]:\left|f_{n}(t)-f_{0}(t)\right|<\epsilon.$ (B.2)

###  B2. Polynomial and Power Series Operators

We have seen that if the output function $y_{t}$ and the input function
$x_{t}$ are connected by a linear time invariant filter, then we have the
relation below

$\displaystyle y_{t}=\sum_{k=0}^{\infty}\phi_{k}x_{t-k},\ \ \
\phi=\left\\{\phi_{k}:k=0,1,2,...\right\\}$ (B.3)

Lets suppose that we introduce the lag operator $L$, defined by
$L^{k}x_{t}=x_{t-k},\ \ \ k=0,1,2,...$. Therefore, the model can be rewritten
in the following form

$\displaystyle
y_{t}=\left(\sum_{k=0}^{\infty}\phi_{k}x_{t-k}\right)x_{t}=\Phi(L)x_{t}$ (B.4)

######  Example 33.

Consider the polynomial operator given by

$\displaystyle a(L)=\sum_{j=0}^{k}a_{j}L^{j}$ (B.5)

which implies that if $y_{t}=a(L)x_{t}$ this gives us

$\displaystyle y_{t}=\sum_{j=0}^{k}a_{j}x_{t-j}\equiv
a_{0}x_{t}+a_{1}x_{t-1}+\alpha_{2}x_{t-2}+...+a_{k}x_{t-k},$ (B.6)

######  Example 34.

The limiting form becomes as below

$\displaystyle
x_{t}=\sum_{j=0}^{\infty}\lambda^{j}\varepsilon_{t-j}=\frac{1}{1-\lambda
L}\varepsilon_{t}$ (B.7)

######  Example 35.

Consider the following time series representation

$\displaystyle x_{t}=\sum_{j=0}^{k-1}\psi_{j}u_{t-j},\ \ \ t=1,2,...$ (B.8)

where the error terms $u_{j}$, have the following properties

$\displaystyle\mathbb{E}\left(u_{t}\right)=0\ \ \ \ \text{and}\ \ \
\mathbb{E}\left(u_{t}u_{t}^{\prime}\right)=\delta_{tt^{\prime}}\sigma^{2}\ \ \
\forall\ \ k$ (B.9)

Moreover, we can assume that $u_{t}\sim\mathcal{N}\left(0,\sigma^{2}\right)$
for all $t$, then indeed the process represents a strictly stationary process.
Then, its covariance kernel is given by the following expression

$\displaystyle
K(s,t):=\mathsf{Cov}(X_{s},X_{t})\equiv\mathbb{E}\left[X_{s},X_{t}\right]$
$\displaystyle=\mathbb{E}\left[\sum_{i=0}^{k-1}\sum_{j=0}^{k-1}\psi_{i}\psi_{j}u_{t-i}u_{t-j}\right]$
$\displaystyle=\sum_{i=0}^{k-1}\sum_{j=0}^{k-1}\psi_{i}\psi_{j}\delta_{i-i,t-j}\sigma^{2}=\sigma^{2}\sum_{i=0}^{k-|s-t|-1}\psi_{i}\psi_{j}.$

###  B3. Dynamic Multipliers Models

######  Example 36.

Consider the general structural model given by

$\displaystyle y_{1t}$
$\displaystyle=\beta_{21}y_{2t}+\gamma_{11}y_{1t-1}+\gamma_{011}^{\star}x_{1t}+\gamma_{121}^{\star}x_{2t-1}+u_{1t}$
(B.10) $\displaystyle y_{2t}$
$\displaystyle=\beta_{12}y_{1t}+\gamma_{12}y_{1t-1}+\gamma_{22}y_{2t-1}+\gamma^{\star}_{022}x_{2t}+\gamma_{112}^{\star}x_{1t-1}+u_{2t}$
(B.11)

Therefore, the system can be written in the following form

$\displaystyle\begin{bmatrix}I-\gamma_{11}L&-\beta_{21}I\\\
-\beta_{21}I-\gamma_{12}L&I-\gamma_{22}L\end{bmatrix}\begin{pmatrix}y_{1t}\\\
y_{2t}\end{pmatrix}=\begin{bmatrix}\gamma_{011}^{\star}I&\gamma^{\star}_{121}L\\\
\gamma^{\star}_{122}L&\gamma^{\star}_{022}I\end{bmatrix}\begin{pmatrix}x_{1t}\\\
x_{2t}\end{pmatrix}+\begin{pmatrix}u_{1t}\\\ u_{2t}\end{pmatrix}$ (B.12)

######  Remark 26.

If we change the value of the $j-$th exogenous variable from $\bar{w}_{j}$ to
$\bar{w}_{j}+1$ and maintain the new level forever, all other exogenous
variables remaining fixed, then the ultimate impact of this on the expectation
of the $i-$th jointly dependent variable is given by the $(i,j)-$th element of
$[C(1)]^{-1}C^{\star}(1)$. Therefore, we may call the $(i,j)$ element of the
latter the dynamic multiplier of the $j-$th exogenous variable relative to the
$i-$th endogenous variable. Hence, $[C(1)]^{-1}C^{\star}(1)$, is said to be
the matrix of dynamic multipliers.

######  Example 37.

Further examples can be found in Hatanaka,  (1996). Suppose that $\\{\Delta
x_{t}\\}$ is a linear process with zero mean, that is,

$\displaystyle\Delta x_{t}=b_{0}\epsilon_{t}+b_{1}\epsilon_{t-1}+...\ \ \ \ \
\ b_{0}=1,$ (B.13)

where $\\{\epsilon_{t}\\}$ is i.i.d., with $E(\epsilon_{t})=0$ and
$E(\epsilon^{2}_{t})=\sigma^{2}_{\epsilon}$. Suppose that $\\{x_{t}\\}$ is a
stationary linear process such that
$x_{t}=c_{0}\epsilon_{t}+c_{1}\epsilon_{t-1}+...$. Then,

$\displaystyle\Delta
x_{t}=c_{0}\epsilon_{t}+(c_{1}-c_{0})\epsilon_{t-1}+(c_{2}-c_{1})\epsilon_{t-2}+...$
(B.14)

which implies that $b_{0}=c_{0}$, $b_{1}=c_{1}-c_{0}$, which it follows that
$\sum_{j=0}^{\infty}b_{j}=0$. Thus, the long-run component of $\Delta x_{t}$
vanishes. In fact, being a stationary linear process implies that no part of
$x_{t}$ has a permanent impact upon its future. We can use a lag operator such
that $\Delta x_{t}=b(L)\epsilon_{t}$, where $b(L)\equiv
b_{0}+b_{1}L+b_{2}L^{2}+...$. Thus, it also follows that (i.e., by
substitution of L with 1), $b(1)=\sum_{j=1}^{\infty}b_{j}$. Therefore, the
long-run component of $\Delta x_{t}$ is written as $b(1)\epsilon_{t}$.
Furthermore, it follows from

$\displaystyle
b(L)-b(1)=-(1-L)\left[b_{1}+b_{2}(1+L)+b_{3}(1+L+L^{2})+...\right]$ (B.15)

Then, we get a useful identity

$\displaystyle b(L)-b(1)=(1-L)b^{*}(L),$ (B.16)

where $b^{*}(L)=b_{0}^{*}+b_{1}^{*}L+b_{2}^{*}L+...$,
$b_{0}^{*}=-\sum_{j=1}^{\infty}b_{j}$, $b_{1}^{*}=-\sum_{j=2}^{\infty}b_{j}$
,….

Thus, using the identity (B.16), then (B.13) becomes

$\displaystyle\Delta x_{t}\equiv
b(L)\epsilon_{t}=b(1)+(1-L)b^{*}(L)\epsilon_{t}.$ (B.17)

which gives that

$\displaystyle x_{t}=x_{0}+\sum_{s=1}^{t}\Delta
x_{s}=b(1)\sum_{s=1}^{t}\epsilon_{s}+b^{*}(L)\epsilon_{t}+x_{0}-b^{*}(L)\epsilon_{0}.$
(B.18)

In terms of the econometric interpretation of the above asymptotic results the
mixed Gaussian assumption of the sample moments indicates the stochastic
nature of the covariance which depends on the persistence properties of the
regressors included in the model. However, the covariance matrix is free of
nuisance parameters which makes inference based on the cointegrated predictive
regression system, for example in the form of linear restriction testing
feasible. Below, we present an example to demonstrate the definition of the
mixed normal distribution (Example 13.1.2 derived from Hatanaka,  (1996)).

######  Example 38.

Consider the cointegrated regression with an uncorrelated error

$\displaystyle y_{t}=\beta x_{t}+\epsilon_{2t}$ (B.19)

where $x_{t}\equiv v_{1t}=\sum_{j=1}^{t}\epsilon_{1t}$ and the cointegrating
vector $(-\beta,1)$ to correspond to the bivariate process $(x_{t},y_{t})$.
Set $\left(W_{1}(r),W_{2}(r)\right)^{\prime}$ with the covariance matrix
$\text{diag}\left\\{\sigma_{11},\sigma_{22}\right\\}$ and setting
$\sigma_{12}=0$ then we have the following asymptotic result

$\displaystyle n\left(\hat{\beta}-\beta\right)\overset{d}{\to}\left(\int
W_{1}(r)dW_{2}(r)\right)\left(\int W_{1}(r)^{2}dr\right)^{-1}\equiv\xi$ (B.20)

Since $W_{1}(r)$ and $W_{2}(r)$ are independent standard Wiener processes then
further simplification follow directly from the property of independent
Gaussian processes. Define

$\displaystyle g\equiv\left(\int W_{1}(r)^{2}dr\right)^{-1}$ (B.21)

Then, it can be proved that $\xi|g\sim N(0,\sigma_{22}g)$ with a conditional
p.d.f given by

$\displaystyle\left(2\pi\sigma_{22}g\right)^{-1/2}\text{exp}\left\\{-\frac{1}{2\sigma_{22}g}\xi^{2}\right\\}f(g)dg$
(B.22)

where $f(g)$ is the marginal p.d.f. of g and the marginal p.d.f of $\xi$ is
given by

$\displaystyle\int_{0}^{\infty}\left(2\pi\sigma_{22}g\right)^{-1/2}\text{exp}\left\\{-\frac{1}{2\sigma_{22}g}\xi^{2}\right\\}f(g)dg$
(B.23)

The above distribution is exactly the mixed Gaussian and represents a mixture
of zero mean normal distributions with different variances.

##  Appendix C Elements of Weak Convergence

###  C1. Joint Weak Convergence in $J_{1}$ topology

In this section we explain in more details the implications of having a random
limit distribution in the related theory for bootstrapping. In other words,
the theoretical result which we aim to prove involves inference based on
stochastic limit bootstrap measures555Notice that our approach in this paper
is different from examining the conditional versus unconditional validity of
the bootstrap for predictive regression models. . To begin with, we assume
that the standard conditional weak convergence result applies,
$\underset{x\in\mathbb{R}}{\text{sup}}\big{|}F_{n}^{*}(x)-F(x)\big{|}\to_{p}0$.
Furthermore, denote with $\mathbb{E}^{*}$ the expectation under the
probability measure induced by the standard bootstrap.

######  Theorem 17.

Suppose that $\left\\{X_{n,j},\mathcal{F}_{n,j}\right\\}$ is a martingale
difference array. Let $\left\\{\mathcal{J}_{n}(r),r\in[0,1]\right\\}$ be a
sequence of adapted time scales and
$\left\\{\mathcal{J}(r),r\in[0,1]\right\\}$ a continuous, nonrandom function.
If it holds,

$\displaystyle\forall\epsilon>0,\
\sum_{j=1}^{\mathcal{J}(1)}\mathbb{E}\left(X_{n,j}^{2}\boldsymbol{1}\left\\{|X_{n,j}|>\epsilon\right\\}\big{|}\mathcal{F}_{n,j-1}\right)$
$\displaystyle\to_{p}0,\ \ \text{as}\ \ n\to\infty$ (C.1)
$\displaystyle\sum_{j=1}^{\mathcal{J}_{n}(r)}\mathbb{E}\left(X^{2}_{n,j}\big{|}\mathcal{F}_{n,j-1}\right)$
$\displaystyle\to_{p}\mathcal{J}(r),\ \ \text{as}\ \ n\to\infty,r\in[0,1],$
(C.2)

Then

$\displaystyle\sum_{j=1}^{\mathcal{J}_{n}(r)}X_{n,j}\to_{d}W\left(\mathcal{J}(r)\right),\
\ \text{as}\ \ n\to\infty,\ \ \text{in}\ \ \mathcal{D}[0,1].$ (C.3)

A sequence $\left\\{P,P_{n}\right\\}$ of probability measures on the metric
space $(S,d)$ converges weakly, when

$\displaystyle\int\phi(x)dP_{n}(x)\to\int\phi(x)dP(x),\ \ n\to\infty,$ (C.4)

holds true for all $\phi\in C_{b}(S,\mathbb{R})$.

######  Theorem 18.

(Continuous Mapping Theorem) Let $\left\\{X,X_{n}\right\\}$ be a sequence of
random elements taking values in some metric space $(S,d)$ equipped with the
associated Borel $\sigma-$field. Assume that

$\displaystyle X_{n}\Rightarrow X,\ \ \text{as}\ \ n\to\infty,$ (C.5)

If $\phi:S\to S^{\prime}$, is a mapping into another metric space $S^{\prime}$
with metric $d^{\prime}$ that is almost surely, continuous on
$X(\Omega)\subset S$, then

$\displaystyle\phi(X_{n})\overset{d}{\to}\phi(X),\ \ \text{as}\ \ n\to\infty,$
(C.6)

######  Theorem 19.

(Joint Weak Convergence) Let $\left\\{X,X_{n}\right\\}$ and
$\left\\{Y,Y_{n}\right\\}$ be two sequences taking values in $(S_{1},d_{1})$,
respectively, $(S_{2},d_{2})$, such that some conditions hold.
Then,$(X_{n},Y_{n})\Rightarrow(X,Y)$, as $n\to\infty$, provided at least one
of the following conditions is satisfied

  1. (i)

$Y=c\in S_{2}$ is a constant, that is, non-random.

  2. (ii)

$X_{n}$ and $Y_{n}$ are independent for all $n$ as well as $X$ and $Y$ are
independent.

Recall that we are given a metric space $(S,d)$ such that as
$\mathcal{D}\left([0,1],\mathbb{R}\right)$ equipped with the Borel
$\sigma-$field and the associated set $\mathcal{P}(S)$ of probability
measures. Then, the space $\mathcal{P}(S)$ can be metrized by the Prohorov
metric $\pi$ and the convergence with respect to the Prohorov metric is the
weak convergence,

$\displaystyle P_{n}\Rightarrow P\ \ \text{if and only if}\
\phi\left(P_{n},P\right)\to 0.$ (C.7)

In a metric space convergence can be characterized by a sequence criterion: A
sequence converges if and only if any subsequence contains a further
convergent subsequence. Therefore, the weak convergence of a sequence
$\left\\{P_{n},P\right\\}$ of probability measures can be characterized in the
following way:

$\displaystyle P_{n}\Rightarrow P\iff\pi(P_{n},P)\to 0,\ \ \text{as}\
n\to\infty$ (C.8)

if and only if any subsequence $\left\\{P_{n_{k}}:k\geq 1\right\\}$ contains a
further subsequence $\left\\{P_{n_{k^{\prime}}}:k\geq 1\right\\}$ such that
$P_{n_{k^{\prime}}}\Rightarrow P\iff\pi(P_{n_{k^{\prime}}},P)\to 0$, as
$k\to\infty$, which is equivalent to $P_{n_{k^{\prime}}}\Rightarrow P$, as
$k\to\infty$.

Further, in any metric space a subset $A$ is relatively compact, that is, has
a compact closure, if every subsequence $\left\\{P_{n}\right\\}\subset A$ has
a subsequence $\left\\{P_{n_{k^{\prime}}}:k\geq 1\right\\}$ with
$P_{n_{k^{\prime}}}\to P^{\prime}$ as $k\to\infty$, where the limit
$P^{\prime}$ is in the closure of $A$ is relatively compact. Applied to out
setting this means: A subset $A\subset\mathcal{P}(\mathcal{S})$ of probability
measures has compact closure $\bar{A}$ if and only if every sequence
$\left\\{P_{n}\right\\}\subset A$ has a subsequence
$\left\\{P_{n_{k}}\right\\}$ with converges weakly to some
$P^{\prime}\in\bar{A}$, such that, $P_{n_{k}}\Rightarrow P^{\prime}$ as
$k\to\infty$.

Here the limit $P^{\prime}$ may depend on the subsequence. Therefore, the weak
convergence $P_{n}\Rightarrow P$ as $n\to\infty$. First, one shows that
$\left\\{P_{n}\right\\}$ is relatively compact and then one verifies that all
possible limits are equal to $P$. A theorem due to Prohorov allows us to
relate the compact sets of $\mathcal{P}(S)$ to the compact sets of $S$. This
is achieved by the concept of tightness. A subset $A\subset\mathcal{P}(S)$ of
probability measures in $\mathcal{P}(S)$ is called tight if for all
$\epsilon>0$ there exists a compact subset $K_{\epsilon}\subset S$ such that

$\displaystyle P\left(K_{\epsilon}\right)>1-\epsilon,\ \ \text{for all}\ \
P\in A.$ (C.9)

####  C1.1. Functional Central Limit Theorems

######  Theorem 20.

(Donsker i.i.d case) Let $\xi_{1},\xi_{2}...$ be a sequence of i.i.d random
variables with $\mathbb{E}\left(\xi_{1}\right)=0$ and
$\sigma^{2}=\mathbb{E}\left(\xi_{1}^{2}\right)<\infty$. Then,

$\displaystyle\frac{1}{\sqrt{T}}\sum_{t=1}^{\lfloor
Ts\rfloor}\xi_{t}\Rightarrow\sigma B(s),\ \ \text{as}\ T\to\infty$ (C.10)

where $B$ denotes the standard Brownian motion and $\Rightarrow$ signifies
weak convergence in the Skorohod space
$\mathcal{D}\left([0,1],\mathbb{R}\right)$.

######  Theorem 21.

Suppose $\xi_{1},\xi_{2},...$ satisfies a weak invariance principle. Then,

$\displaystyle S_{T}(u)=\frac{1}{\sqrt{T}}S\left(\lfloor
Tu\rfloor\right)\Rightarrow B(u),\ \ \text{as}\ T\to\infty.$ (C.11)

###### Proof.

We have that $\left\\{B(u):u\geq 0\right\\}$ is equal in distribution to
$\left\\{\frac{1}{\sqrt{T}}B(Tu):u\geq 0\right\\}$ for each $T$. Therefore,

$\displaystyle\underset{u\in[0,1]}{\mathsf{sup}}\left|S_{T}(u)-B(u)\right|\overset{d}{=}\underset{u\in[0,1]}{\mathsf{sup}}\left|\frac{1}{\sqrt{T}}\sum_{t=1}^{\lfloor
Tu\rfloor}\xi_{t}-\frac{1}{\sqrt{T}}B(Tu)\right|.$ (C.12)

Therefore, we can conclude that on a new probability space,

$\displaystyle\underset{u\in[0,1]}{\mathsf{sup}}\left|S_{T}(u)-B(u)\right|$
$\displaystyle\overset{d}{=}\underset{u\in[0,1]}{\mathsf{sup}}\frac{1}{\sqrt{T}}\left|\sum_{t=1}^{\lfloor
Tu\rfloor}\xi_{t}-B(Tu)\right|$ $\displaystyle=\frac{1}{\sqrt{T}}\
\underset{n\leq T}{\mathsf{max}}\ \left|S(n)-B(n)\right|\overset{p}{\to}0$

as $T\to\infty$, which implies that $S_{T}\Rightarrow B$ as $T\to\infty$, for
the original processes. ∎

###  C2. Uniform Inference in the $J_{1}$ topology

The FCLT allows us to obtain weak convergence of functionals to the $J_{1}$
topology which is used to derive the asymptotic distribution of estimators,
test statistics, which are considered to be point-estimates. One might also be
interested to obtain interval estimates such as the case when constructing
confidence intervals. More specifically, uniform inference provides the
theoretical framework to construct confidence intervals for the sum of
coefficients in autoregressive models. In particular the paper of Mikusheva,
(2007) provides a good overview on the subject. Within the proposed framework,
the author clarify the difference between uniform and pointwise asymptotic
approximations, and show that a pointwise convergence of coverage
probabilities for all values of the parameter does not guarantee the validity
of the confidence set (see, also Phillips,  (2014)).

######  Example 39.

Assume that we have an AR(1) process with an intercept

$\displaystyle y_{t}=c+x_{t},\ \ x_{t}=\rho x_{t-1}+\epsilon_{t},\ \
t=\\{1,...,T\\},\ \ \ x_{0}=0.$ (C.13)

Consider that the autoregressive coefficient $\rho$ takes only values on the
parameter space $\Theta\in(-1,1)$. The uniform inference framework provides a
way for constructing a confidence set for the parameter $\rho$.

######  Definition 14.

A subset $C(Y)$ of the parameter space $\Theta$ is said to be a confidence set
at a confidence level

$\displaystyle(1-\alpha)\ \ \text{if}\ \ \ \
\underset{\rho\in\Theta}{\text{inf}}\ \mathbb{P}\big{[}\rho\in C(Y)\big{]}\geq
1-\alpha.$ (C.14)

######  Definition 15.

A subset $C(Y)$ of the parameter space $\Theta$ is said to be an asymptotic
confidence set at a confidence level $1-\alpha$ (or is said to have a uniform
asymptotic convergence probability $1-\alpha)$ if

$\displaystyle\underset{T\to\infty}{\text{lim inf}}\
\underset{\rho\in\Theta}{\text{inf}}\ \mathbb{P}\big{[}\rho\in C(Y)\big{]}\geq
1-\alpha$ (C.15)

Therefore, the requirement of uniform convergence is much stronger than a
requirement of pointwise convergence of coverage probabilities

$\displaystyle\underset{T\to\infty}{\text{lim}}\ \mathbb{P}\big{[}\rho\in
C(Y)\big{]}\geq 1-\alpha\ \ \ \text{for every}\ \rho\in\Theta.$ (C.16)

Specifically, the uniform convergence implies that for every value of the
parameter space and for any given accuracy, we can find a large enough sample
size to provide the required accuracy at this value. More importantly, the
condition given by (C.16) does not necessarily ensures that there is a sample
size that provides the required accuracy for all values of the parameter
space. Therefore, a solution to the problem of poor coverage probabilities in
certain regions of the parameter space, is to use the methodology of inverting
tests.

####  C2.1. Testing Methodology

Let $A(\rho_{o})$ be an acceptance region of an asymptotic level-$\alpha$ test
for testing the null hypothesis, $\mathbb{H}_{0}:\rho=\rho_{0}$. A set is
constructed as a set of parameter values for which the corresponding simple
hypothesis is accepted, such that $C(Y)=\\{\rho:Y\in A(\rho)\\}$. Let the
testing procedure for a test of the hypothesis, $\mathbb{H}_{0}:\rho=\rho_{0}$
be based on a test statistic $\phi(Y,T,\rho_{0})$, and critical values
$c_{1}(T,\rho_{0})$ and $c_{2}(T,\rho_{0})$. Then, a set $C(Y)$ is defined as
below

$\displaystyle
C(Y)=\bigg{\\{}\rho\in\Theta:c_{1}(T,\rho_{0})\leq\phi(Y,T,\rho_{0})\leq
c_{2}(T,\rho_{0})\bigg{\\}}.$ (C.17)

####  C2.2. Class of Test Statistics

Let $y_{t}^{\mu}$ be the demeaned process that corresponds to $y_{t}$, that
is, $y_{t}^{\mu}=y_{t}-\frac{1}{T}\sum_{t=1}^{T}y_{t-1}$. We consider a class
of test statistics based on a pair of statistics

$\displaystyle\big{(}S(T,\rho),R(T,\rho)\big{)}=\left(\frac{1}{\sqrt{g(T,\rho)}}\sum_{t=1}^{T}y_{t-1}^{\mu}\big{(}y_{t}-\rho
y_{t-1}\big{)},\