On Time-Varying VAR Models:
Estimation, Testing and Impulse Response Analysis ††Corresponding author: Jiti
Gao, Department of Econometrics and Business Statistics, Monash University,
Caulfield East, Victoria 3145, Australia. Email<EMAIL_ADDRESS>The
authors of this paper would like to thank George Athanasopoulos, Rainer
Dahlhaus, David Frazier, Oliver Linton, Gael Martin, Peter CB Phillips and Wei
Biao Wu for their constructive comments on earlier versions of this paper. Gao
acknowledges financial support from the Australian Research Council Discovery
Grants Program under Grant Numbers: DP170104421 and DP200102769. Peng also
acknowledges the Australian Research Council Discovery Grants Program for its
financial support under Grant Number DP210100476.

Yayi Yan, Jiti Gao and Bin Peng

Monash University, Melbourne, Australia

Vector autoregressive (VAR) models are widely used in practical studies, e.g.,
forecasting, modelling policy transmission mechanism, and measuring connection
of economic agents. To better capture the dynamics, this paper introduces a
new class of time-varying VAR models in which the coefficients and covariance
matrix of the error innovations are allowed to change smoothly over time.
Accordingly, we establish a set of theories, including the impulse responses
analyses subject to both of the short-run timing and the long-run
restrictions, an information criterion to select the optimal lag, and a Wald-
type test to determine the constant coefficients. Simulation studies are
conducted to evaluate the theoretical findings. Finally, we demonstrate the
empirical relevance and usefulness of the proposed methods through an
application to the transmission mechanism of U.S. monetary policy.

Keywords: Multivariate Dynamic Time Series; Time-Varying Impulse Response;
Testing for Parameter Stability

JEL Classification: C14, C32, E52

##  1 Introduction

Vector autoregressive (VAR) models as well as their extensions are among some
of the most popular frameworks for modelling dynamic interactions among
multiple variables. These models arise mainly as a response to the
“incredible” identification conditions embedded in the large-scale
macroeconomic models Sims (1980). VAR modelling begins with minimal
restrictions on the multivariate dynamic models. Gradually armed with
identification information, VAR models and their statistical tool-kits like
impulse response functions become powerful tools for conducting policy
analysis. We refer interested readers to Stock and Watson (2001) for a
comprehensive review. Despite the popularity, linear VAR models can always be
rejected by data in empirical studies (Tsay, 1998). For example, Stock and
Watson (2016) point out, “changes associated with the Great Moderation go
beyond reduction in variances to include changes in dynamics and reduction in
predictability”.

To go beyond linear VAR models, various parametric time-varying VAR models
have been proposed (e.g., Tsay, 1998, Sims and Zha, 2006, and the references
therein) in order to allow for certain changes in economic relationship.
However, model misspecification and parameter instability may undermine the
performance of the proposed parametric models. As pointed out by Hansen
(2001), it may seem unlikely that structural change could be immediate and
might seem more reasonable to allow structural change to take a period of time
to take effect. Practically, it may be more reasonable to allow smooth
structural changes over a period of time rather than in an abrupt manner. To
model the dynamic transit, an important strand of the VAR literature assumes
that the coefficients of VAR models evolve in a random way (e.g., Primiceri,
2005; Petrova, 2019), and the estimation procedure relies on extensive Markov
Chain Monte Carlo (MCMC) draws plus the use of a variety of filters, such as
Kalman, or related filters. However, by doing so, the asymptotic properties of
the estimated model coefficients as well as the corresponding impulse
responses are still unclear (Giraitis et al., 2014).

Meanwhile, there is a separate literature about using nonparametric methods to
estimate deterministically unknown time–varying parameters for autoregressive
models. Up to this point, it is worth bringing up the terminology “local
stationarity”, as a bulk of the literature is carried out using the local
stationarity technique, which at least dates back to the seminal work by
Dahlhaus (1996). Recent developments focus on univariate autoregressive models
(Dahlhaus and Rao, 2006; Zhang and Wu, 2012; Richter and Dahlhaus, 2019; Sun
et al., 2021). To the best of our knowledge, it has had very little success to
extend the local stationarity technique to multivariate settings, such as VAR
models with deterministic time–varying parameters. In some specific cases
where different locally stationary univariate time series may be approximated
by their stationary versions on the same segments, the local stationarity
technique may be applicable to such specific multivariate settings. However,
univariate time series of general multivariate time series may have quite
different patterns and behaviours, such as the three univariate time series
plotted in Figure 1 of Section 5.

To address the aforementioned issues, this paper therefore proposes a class of
deterministic time-varying VAR models where both VAR coefficients and
covariance matrix of the model’s error innovations are allowed to change
smoothly over time. We develop a time-varying vector moving average infinity
(VMA$(\infty)$) representation for a class of VAR models before we are able to
establish uniform consistency results and a joint central limit theory for
kernel-based estimators of both VAR coefficients and covariance matrix, which
facilitate the inference on time-varying structural impulse responses. This
type of impulse responses is of major importance in typical VAR applications
Inoue and Kilian (2013, 2020); Paul (2020). In addition, inferences on
structural impulse responses subject to both short-run timing and long-run
restrictions are considered, so that the proposed model can better capture the
simultaneous relations among multiple variables of interest over time. Such
modelling strategy is especially useful for analyzing multivariate time series
over a long horizon, since it offers a comprehensive treatment on tracking
interests which are affected by frequently updated policies, environment,
system, etc. In an economy system consisting of inflation, unemployment and
interest rates, we discuss time-varying impacts of the interest rate change,
which helps stabilize fluctuations in inflation and unemployment in the long-
run. Under the proposed framework, it is achieved by investigating the
corresponding time-varying impulse response functions.

We now comment on the literature closely related to the parameter stability
testing problem we are considering in this paper. Detecting and estimating
parametric components in univariate time-varying autoregressive models has
been studied by Zhang and Wu (2012). Recently, Truquet (2017) considers
parameter stability testing for univariate time-varying autoregressive
conditional heteroscedasticity. More recently, Lütkepohl and Schlaak (2021)
discuss testing for time–varying impact effects through a change–point
mechanism. We develop a simple test for checking whether some of the
time–varying coefficients (if not all) reduce to constant coefficients
involved in the VAR models. It can be used to test whether the policy
transmission mechanism is changing over time, which is of great importance in
the macroeconomic literature (e.g., Primiceri, 2005; Paul, 2020). To give
another example, consider our empirical study on the ongoing debate about the
high inflation during 1970-1980, i.e., whether the high inflation is due to
bad monetary policy (Primiceri, 2005; Sims and Zha, 2006). Mathematically, the
discrepancy comes down to specification testing on the coefficient matrices of
the VAR models by the proposed test.

In summary, our contributions are in three-fold. First, we propose a class of
time-varying VAR($p$) models, and develop a time-varying VMA$(\infty)$
representation for the VAR($p$) models before we establish an estimation
theory for time-varying structural impulse responses, which are often of great
interest to describe how the economy reacts over time to structural shocks.
The proposed estimation is conducted subject to both short-run timing and
long-run restrictions, which have attracted considerable attention in the
literature (Kilian and Lütkepohl, 2017). Second, we develop a Wald–type test
statistic for detecting time-invariant parameters in time-varying VAR models
before we show that the proposed test statistic is asymptotically normally
distributed under both the null hypothesis and a sequence of local
alternatives. Third, in an empirical study, we investigate the changing
dynamics of three key U.S. macroeconomic variables (inflation, unemployment,
and interest rate), and uncover a fall in the volatilities of exogenous
shocks. We observe that there exists a substantial time-variation in the
policy transmission mechanism, and the “price puzzle” is limited to periods of
bad monetary policy.

The organization of this paper is as follows. Section 2 considers a class of
time-varying VAR models, and establish asymptotic properties for the estimated
time-varying impulse response functions. Section 3 describes the proposed test
and establishes the corresponding asymptotic theory. Section 4 discusses some
implementation issues and presents comprehensive simulation studies. Section 5
provides a case study to demonstrate the empirical relevance of the proposed
models and estimation theory. Section 6 concludes. The proofs of the main
results are given in Appendix A. Preliminary lemmas and their proofs are given
in Appendix B.

Before proceeding further, it is convenient to introduce some notations:
$\|\cdot\|$ denotes the Euclidean norm of a vector or the Frobenius norm of a
matrix; $\otimes$ denotes the Kronecker product; $\bm{I}_{a}$ stands for an
$a\times a$ identity matrix; $\bm{0}_{a\times b}$ stands for an $a\times b$
matrix of zeros, and we write $\bm{0}_{a}$ for short when $a=b$; for a
function $g(w)$, let $g^{(j)}(w)$ be the $j^{th}$ derivative of $g(w)$, where
$j\geq 0$ and $g^{(0)}(w)\equiv g(w)$; $K_{h}(\cdot)=K(\cdot/h)/h$, where
$K(\cdot)$ and $h$ stand for a nonparametric kernel function and a bandwidth
respectively; let $\tilde{c}_{k}=\int_{-1}^{1}u^{k}K(u)du$ and
$\tilde{v}_{k}=\int_{-1}^{1}u^{k}K^{2}(u)du$ for integer $k\geq 0$;
$\mathrm{vec}(\cdot)$ stacks the elements of an $m\times n$ matrix as an
$mn\times 1$ vector; for any $a\times a$ square matrix $\bm{A}$,
$\mathrm{vech}\left(\bm{A}\right)$ denotes the $\frac{1}{2}a(a+1)\times 1$
vector obtained from $\mathrm{vec}\left(\bm{A}\right)$ by eliminating all
supra–diagonal elements of $\bm{A}$; $\mathrm{tr}\left(\bm{A}\right)$ denotes
the trace of $\bm{A}$. Finally, $\to_{P}$ and $\to_{D}$ denote convergence in
probability and convergence in distribution, respectively.

##  2 The Time-Varying VAR($p$) Model

Suppose that we observe
$\\{\bm{x}_{-p+1},\ldots,\bm{x}_{0},\bm{x}_{1},\ldots,\bm{x}_{T}\\}$ from the
following data generating process:

$\bm{x}_{t}=\bm{a}(\tau_{t})+\sum_{j=1}^{p}\bm{A}_{j}(\tau_{t})\bm{x}_{t-j}+\bm{\eta}_{t}\
\ \mbox{with}\ \ \bm{\eta}_{t}=\bm{\omega}(\tau_{t})\bm{\epsilon}_{t},$ (2.1)

where $\tau_{t}=t/T$, $\bm{a}(\cdot)$ and $\bm{A}_{j}(\cdot)$ are respectively
a $d\times 1$ vector and $d\times d$ matrices of unknown functional
coefficients, and $\bm{\omega}(\tau)$ is a $d\times d$ matrix of unknown
functions capturing the heteroskedasticity of the error components. Allowing
$\bm{\omega}(\cdot)$ to vary over time is important theoretically and
practically, because a constant covariance matrix implies that the shock to
the $i^{th}$ variable of $\bm{x}_{t}$ has a time-invariant effect on the
$j^{th}$ variable of $\bm{x}_{t}$, restricting simultaneous interactions among
multiple variables to be time-invariant (Primiceri, 2005). For the time being,
we assume $p$ is known, and shall come back to the choice of $p$ in Section
2.3.

The following conditions are necessary for our development.

######  Assumption 1.

  1. 1.

The roots of
$\bm{I}_{d}-\bm{A}_{1}(\tau)L-\cdots-\bm{A}_{p}(\tau)L^{p}=\bm{0}_{d}$ all lie
outside the unit circle uniformly in $\tau\in[0,1]$.

  2. 2.

Each element of
$\bm{A}(\tau)=\left[\bm{a}(\tau),\bm{A}_{1}(\tau),\ldots,\bm{A}_{p}(\tau)\right]$
is second order continuously differentiable on $[0,1]$ and
$\bm{A}(\tau)=\bm{A}(0)$ for $\tau<0$.

  3. 3.

Each element of $\bm{\omega}(\tau)$ is second order continuously
differentiable on $[0,1]$. Moreover,
$\bm{\Omega}(\tau)=\bm{\omega}(\tau)\bm{\omega}(\tau)^{\top}$ is positive
definite uniformly in $\tau\in[0,1]$ and $\bm{\omega}(\tau)=\bm{\omega}(0)$
for $\tau<0$.

######  Assumption 2.

$\\{\bm{\epsilon}_{t}\\}_{t=-\infty}^{\infty}$ is a martingale difference
sequence (m.d.s.) adapted to the filtration
$\left\\{\mathcal{F}_{t}\right\\}$, where
$\mathcal{F}_{t}=\sigma\left(\bm{\epsilon}_{t},\bm{\epsilon}_{t-1},\ldots\right)$
is the $\sigma$-field generated by
$\left(\bm{\epsilon}_{t},\bm{\epsilon}_{t-1},\ldots\right)$,
$E[\bm{\epsilon}_{t}\bm{\epsilon}_{t}^{\top}|\mathcal{F}_{t-1}]=\bm{I}_{d}$
almost surely (a.s.), and
$\max_{t}E\left\|\bm{\epsilon}_{t}\right\|^{\delta}<\infty$ for some
$\delta>4$.

Assumption 1.1 ensures the eigenvalues of the companion matrix
$\bm{\Phi}(\tau)$ of (2.1) below all lie inside the unit circle uniformly over
$\tau\in[0,1]$. As a consequence, (2.1) cannot include any unit-root or
explosive components. Similar treatments have also been adopted to investigate
univariate locally stationary models in the literature (e.g., Assumption T3 of
Zhang and Wu, 2012). Assumptions 1.2 and 1.3 allow the underlying data
generating process to evolve over time in a smooth manner. The conditions
$\bm{A}(\tau)=\bm{A}(0)$ and $\bm{\omega}(\tau)=\bm{\omega}(0)$ for $\tau<0$
gives

$\displaystyle\bm{x}_{t}=\bm{a}(0)+\sum_{j=1}^{p}\bm{A}_{j}(0)\bm{x}_{t-j}+\bm{\omega}(0)\bm{\epsilon}_{t},$
(2.2)

which basically assumes that $\bm{x}_{t}$ behaves like a parametric VAR$(p)$
model for $t\leq 0$. A similar condition can be found in Vogt (2012) for a
nonparametric time-varying time series model.

Assumption 2 imposes some conditions on the innovation error terms, and are
standard in the VAR literature (cf., Lütkepohl, 2005).

With these conditions, the following proposition says (2.1) admits a time-
varying vector moving average infinity (VMA$(\infty)$) representation, which
sheds a light on how to recover the time-varying structural impulse responses.

######  Proposition 2.1.

Under Assumptions 1 and 2, there exists a time-varying VMA$(\infty)$ process
of the form:

$\widetilde{\bm{x}}_{t}=\bm{\mu}(\tau_{t})+\bm{B}_{0}(\tau_{t})\bm{\epsilon}_{t}+\bm{B}_{1}(\tau_{t})\bm{\epsilon}_{t-1}+\bm{B}_{2}(\tau_{t})\bm{\epsilon}_{t-2}+\cdots$

such that $\bm{x}_{t}$ of (2.1) satisfies $\max_{t\geq
1}\\{E\left\|\bm{x}_{t}-\widetilde{\bm{x}}_{t}\right\|^{\delta}\\}^{1/\delta}=O(T^{-1})$,
where

$\displaystyle\bm{\mu}(\tau)=\bm{a}(\tau)+\sum_{j=1}^{\infty}\bm{\Psi}_{j}(\tau)\bm{a}(\tau),\quad\bm{\Psi}_{j}(\tau)=\bm{J}\bm{\Phi}^{j}(\tau)\bm{J}^{\top}\text{
for }j\geq 1,$ $\displaystyle\bm{J}=\left[\bm{I}_{d},\bm{0}_{d\times
d(p-1)}\right],\quad\bm{B}_{0}(\tau)=\bm{\omega}(\tau),\quad\bm{B}_{j}(\tau)=\bm{\Psi}_{j}(\tau)\bm{\omega}(\tau),$
$\displaystyle\bm{\Phi}(\tau)=\left[\begin{matrix}\bm{A}_{1}(\tau)&\cdots&\bm{A}_{p-1}(\tau)&\bm{A}_{p}(\tau)\\\
\bm{I}_{d}&\cdots&\bm{0}_{d}&\bm{0}_{d}\\\ \vdots&\ddots&\vdots&\vdots\\\
\bm{0}_{d}&\cdots&\bm{I}_{d}&\bm{0}_{d}\\\ \end{matrix}\right].$ (2.3)

From Proposition 2.1, we can see that the $d\times 1$ vector of the
orthogonalized impulse response function of a unit shock at time $t$ to the
$j^{th}$ equation on $\bm{x}_{t+n}$ is given by
$\bm{B}_{n}(\tau_{t+n})\bm{e}_{j}$, where $\bm{e}_{j}$ is a $d\times 1$
selection vector with unity as its $j^{th}$ element and zeros elsewhere.
Hence, the impulse responses produced by our model are deterministic functions
of rescaled time, so that the TV-VAR model captures potential drifts in the
transmission mechanism and produces impulse responses which are not history-
and shock-dependent.

###  2.1 Estimation

To estimate $\\{\bm{B}_{j}(\tau):j\geq 0\\}$, we need a joint central limit
theorem for the estimators of the coefficients and the innovation covariance
matrix. That said, we consider the estimation of $\bm{A}(\cdot)$ and
$\bm{\Omega}(\cdot)$ using the local linear kernel method111The local linear
kernel method allows us to address the so-called boundary effects of the
kernel estimation when establishing the first result of Theorem 2.1 below.
Alternatively, one may consider some boundary adjustment approaches as
mentioned in Hong and Li (2005) and Connor et al. (2012).. Intuitively, when
$\tau_{t}$ is in a small neighbourhood of $\tau$, we can write (2.1) as

$\displaystyle\bm{x}_{t}=\bm{A}(\tau_{t})\bm{z}_{t-1}+\bm{\eta}_{t}\approx\left[\bm{A}(\tau),h\bm{A}^{(1)}(\tau)\right]\bm{z}_{t-1}^{*}+\bm{\eta}_{t},$
(2.4)

where
$\bm{z}_{t-1}=\left[1,\bm{x}_{t-1}^{\top},\ldots,\bm{x}_{t-p}^{\top}\right]^{\top}$
and
$\bm{z}^{*}_{t-1}=\left[\bm{z}_{t-1}^{\top},\frac{\tau_{t}-\tau}{h}\bm{z}_{t-1}^{\top}\right]^{\top}$.
The local linear estimators222It is worth pointing out that the estimation of
the covariance matrix using the local linear kernel method (such as the second
estimator of (2.1)) is a non-trivial problem, and even has its own literature.
We refer interested readers to Zhang and Wu (2012) for more details. of
$\bm{A}(\tau)$ and $\bm{\Omega}(\tau)$ are then respectively given by

$\displaystyle\mathrm{vec}[\bm{\widehat{A}}(\tau)]$ $\displaystyle=$
$\displaystyle[\bm{I}_{d^{2}p+d},\bm{0}_{d^{2}p+d}]\cdot\left(\sum_{t=1}^{T}\bm{Z}_{t-1}^{*}\bm{Z}_{t-1}^{*,\top}K_{h}(\tau_{t}-\tau)\right)^{-1}\sum_{t=1}^{T}\bm{Z}_{t-1}^{*}\bm{x}_{t}K_{h}(\tau_{t}-\tau),$
$\displaystyle\bm{\widehat{\Omega}}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{T}\sum_{t=1}^{T}\bm{\widehat{\eta}}_{t}\bm{\widehat{\eta}}_{t}^{\top}\omega_{t}(\tau),$
(2.5)

where $\bm{Z}_{t}^{*}=\bm{z}_{t}^{*}\otimes\bm{I}_{d}$,
$\bm{\widehat{\eta}}_{t}=\bm{x}_{t}-\bm{\widehat{A}}(\tau_{t})\bm{z}_{t-1}$,
$\omega_{t}(\tau)=K_{h}(\tau_{t}-\tau)\frac{P_{h,2}(\tau)-\frac{\tau_{t}-\tau}{h}P_{h,1}(\tau)}{P_{h,0}(\tau)P_{h,2}(\tau)-P_{h,1}^{2}(\tau)}$
is the local linear weight, and
$P_{h,k}(\tau)=\frac{1}{T}\sum_{t=1}^{T}\left(\frac{\tau_{t}-\tau}{h}\right)^{k}K_{h}(\tau_{t}-\tau)$
for $k=0,1,2$.

We require the following conditions to hold for the kernel function and the
bandwidth.

######  Assumption 3.

Let $K(\cdot)$ be a symmetric and positive kernel function defined on $[-1,1]$
with $\int_{-1}^{1}K(u)\mathrm{d}u=1$. Moreover, $K(\cdot)$ is Lipschitz
continuous on $[-1,1]$. As $(T,h)\to(\infty,0)$, $Th\to\infty$.

With Assumption 3 in hand, we summarize the first theorem of this paper below.

######  Theorem 2.1.

Let Assumptions 1-3 hold. Suppose that $\max_{t\geq
1}E[\|\bm{\epsilon}_{t}\|^{4}|\mathcal{F}_{t-1}]<\infty$ a.s., and
$\frac{T^{1-\frac{4}{\delta}}h}{\log T}\to\infty$. Then

  1. 1.

$\sup_{\tau\in[0,1]}\|\bm{\widehat{A}}(\tau)-\bm{A}(\tau)\|=O_{P}\left(h^{2}+(\frac{\log
T}{Th})^{1/2}\right)$.

In addition, suppose that conditional on $\mathcal{F}_{t-1}$, the third and
fourth moments of $\bm{\epsilon}_{t}$ are identical to the corresponding
unconditional moments a.s., and $Th^{5}\to\alpha\in[0,\infty)$. Then the
following two results also hold for $\forall\tau\in(0,1)$:

  1. 2.

$\sqrt{Th}\widehat{\bm{V}}^{-1/2}(\tau)\left[\begin{matrix}\mathrm{vec}\left(\bm{\widehat{A}}(\tau)-\bm{A}(\tau)-\frac{1}{2}h^{2}\tilde{c}_{2}\bm{A}^{(2)}(\tau)\right)\\\
\mathrm{vech}\left(\bm{\widehat{\Omega}}(\tau)-\bm{\Omega}(\tau)-\frac{1}{2}h^{2}\tilde{c}_{2}\bm{\Omega}^{(2)}(\tau)\right)\end{matrix}\right]\to_{D}N\left(\bm{0},\bm{I}\right),$

where $\bm{V}(\tau)$ and $\widehat{\bm{V}}(\tau)$ are defined in (A.3) and
(A.5) for the sake of presentation.

The first result of Theorem 2.1 establishes a uniform convergence rate for
$\bm{\widehat{A}}(\tau)$, which further allows us to establish a joint
asymptotic distribution in the second result. If $\delta>5$, the usual optimal
bandwidth $h_{opt}=O\left(T^{-1/5}\right)$ satisfies the condition
$\frac{T^{1-\frac{4}{\delta}}h}{\log T}\to\infty$.

###  2.2 On Impulse Responses

Having established the joint CLT in Theorem 2.1, we are now ready to study the
impulse responses. As
$\bm{\Omega}(\cdot)=\bm{\omega}(\cdot)\bm{\omega}^{\top}(\cdot)$, we cannot
infer the elements of $\bm{\omega}(\cdot)$ unless certain identification
restrictions are imposed. In the following, we consider two types of
identification conditions: (i) the short-run timing restrictions, and (ii) the
long-run restrictions. The economic interpretations of the two types of
identification conditions can be found in Kilian and Lütkepohl (2017), and we
do not repeat them here for the sake of space.

Under the short-run timing restrictions, $\bm{\omega}(\cdot)$ is a lower-
triangular matrix. Thus, $\widehat{\bm{\omega}}(\tau)$ is chosen as the lower
triangular matrix from the Cholesky decomposition of
$\bm{\widehat{\Omega}}(\tau)$, i.e.,
$\bm{\widehat{\Omega}}(\tau)=\bm{\widehat{\omega}}(\tau)\bm{\widehat{\omega}}^{\top}(\tau)$.
Alternatively, one can impose the conditions on the long-run impacts of the
shocks (i.e., $\bm{B}(\tau)$ defined below). Specifically, define

$\displaystyle\bm{B}(\tau):=\sum_{j=0}^{\infty}\bm{B}_{j}(\tau)=\left(\bm{I}_{d}-\sum_{i=1}^{p}\bm{A}_{i}(\tau)\right)^{-1}\bm{\omega}(\tau),$
$\displaystyle\bm{\Psi}(\tau):=\sum_{j=0}^{\infty}\bm{\Psi}_{j}(\tau)=\left(\bm{I}_{d}-\sum_{i=1}^{p}\bm{A}_{i}(\tau)\right)^{-1},$
(2.6)

where the last equalities of both lines follow in an obvious matter. Thus, the
elements of $\bm{B}(\tau)$ may be recovered from
$\bm{B}(\tau)\bm{B}^{\top}(\tau)=\bm{\Psi}(\tau)\bm{\Omega}(\tau)\bm{\Psi}^{\top}(\tau)$.
It is then convenient to assume that $\bm{B}(\tau)$ is a lower-triangular
matrix, so $\widehat{\bm{B}}(\tau)$ is chosen as the lower triangular matrix
from the Cholesky decomposition of
$\widehat{\bm{\Psi}}(\tau)\widehat{\bm{\Omega}}(\tau)\widehat{\bm{\Psi}}^{\top}(\tau)$,
where $\widehat{\bm{\Psi}}(\tau)$ is defined in the same way as
$\bm{\Psi}(\tau)$ in (2.1) but replacing $\bm{A}_{i}(\tau)$ with
$\widehat{\bm{A}}_{i}(\tau)$. Under the long-run restrictions,
$\widehat{\bm{\omega}}(\tau)=\widehat{\bm{\Psi}}^{-1}(\tau)\widehat{\bm{B}}(\tau)$.

Either way, the estimator of the impulse response function $\bm{B}_{j}(\tau)$
for each given $j\geq 0$ is given by

$\displaystyle\bm{\widehat{B}}_{j}(\tau)=\bm{\widehat{\Psi}}_{j}(\tau)\bm{\widehat{\omega}}(\tau),$
(2.7)

where
$\bm{\widehat{\Psi}}_{j}(\tau)=\bm{J}\bm{\widehat{\Phi}}^{j}(\tau)\bm{J}^{\top}$,
$\bm{J}$ is defined in Proposition 2.1, and $\bm{\widehat{\Phi}}^{j}(\tau)$ is
defined under (A.6).

We summarize the asymptotic properties of the estimation of the impulse
responses in the following theorem.

######  Theorem 2.2.

Under the conditions of Theorem 2.1. For any fixed integer $j\geq 0$

$\displaystyle\sqrt{Th}\left(\mathrm{vec}\left(\bm{\widehat{B}}_{j}(\tau)-\bm{B}_{j}(\tau)\right)-\frac{1}{2}h^{2}\tilde{c}_{2}\bm{B}_{j}^{(2)}(\tau)\right)\to_{D}N\left(0,\bm{\Sigma}_{\bm{B}_{j}}(\tau)\right),$

where

$\displaystyle\bm{B}_{j}^{(2)}(\tau)$ $\displaystyle=$
$\displaystyle\bm{C}_{j,1}(\tau)\mathrm{vec}\left(\bm{A}^{(2)}(\tau)\right)+\bm{C}_{j,2}(\tau)\mathrm{vech}\left(\bm{\Omega}^{(2)}(\tau)\right),$
$\displaystyle\bm{\Sigma}_{\bm{B}_{j}}(\tau)$ $\displaystyle=$
$\displaystyle\left[\bm{C}_{j,1}(\tau),\bm{C}_{j,2}(\tau)\right]\bm{V}(\tau)\left[\bm{C}_{j,1}(\tau),\bm{C}_{j,2}(\tau)\right]^{\top}.$

Specifically, we have the following expressions.

  1. 1.

Under the short-run timing restrictions, we have

$\displaystyle\bm{C}_{0,1}(\tau)$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\bm{C}_{j,1}(\tau)$ $\displaystyle=$
$\displaystyle\left(\bm{\omega}^{\top}(\tau)\otimes\bm{I}_{d}\right)\left(\sum_{m=0}^{j-1}\bm{J}(\bm{\Phi}^{\top}(\tau))^{j-1-m}\otimes\bm{\Psi}_{m}(\tau)\right)\left[\bm{0}_{d^{2}p\times
d},\bm{I}_{d^{2}p}\right],\ j\geq 1,$ $\displaystyle\bm{C}_{j,2}(\tau)$
$\displaystyle=$
$\displaystyle\left(\bm{I}_{d}\otimes\bm{\Psi}_{j}(\tau)\right)\bm{L}_{d}^{\top}\left(\bm{L}_{d}\bm{N}_{1}(\tau)\bm{L}_{d}^{\top}\right)^{-1},\
j\geq 0,$

in which
$\bm{N}_{1}(\tau)=(\bm{I}_{d^{2}}+\bm{K}_{d,d})(\bm{\omega}(\tau)\otimes\bm{I}_{d})$,
the elimination matrix $\bm{L}_{d}$ satisfies that
$\mathrm{vech}(\bm{F})=\bm{L}_{d}\mathrm{vec}(\bm{F})$ for any $d\times d$
matrix $\bm{F}$, and the commutation matrix $\bm{K}_{m,n}$ satisfies
$\bm{K}_{m,n}\mathrm{vec}(\bm{G})=\mathrm{vec}(\bm{G}^{\top})$ for any
$m\times n$ matrix $\bm{G}$.

  2. 2.

Under the long-run restrictions, we have

$\displaystyle\bm{C}_{0,1}(\tau)$ $\displaystyle=$
$\displaystyle(\bm{I}_{d}\otimes\bm{\Psi}_{j}(\tau))\left(\bm{N}_{1}^{\top}(\tau)\bm{N}_{1}(\tau)+\bm{N}_{2}^{\top}(\tau)\bm{N}_{2}(\tau)\right)^{-1}\bm{N}_{2}^{\top}(\tau)\bm{D}_{2}(\tau),$
$\displaystyle\bm{C}_{j,1}(\tau)$ $\displaystyle=$
$\displaystyle\left(\bm{\omega}^{\top}(\tau)\otimes\bm{I}_{d}\right)\left(\sum_{m=0}^{j-1}\bm{J}(\bm{\Phi}^{\top}(\tau))^{j-1-m}\otimes\bm{\Psi}_{m}(\tau)\right)\left[\bm{0}_{d^{2}p\times
d},\bm{I}_{d^{2}p}\right]+(\bm{I}_{d}\otimes\bm{\Psi}_{j}(\tau))$
$\displaystyle\times\left(\bm{N}_{1}^{\top}(\tau)\bm{N}_{1}(\tau)+\bm{N}_{2}^{\top}(\tau)\bm{N}_{2}(\tau)\right)^{-1}\bm{N}_{2}^{\top}(\tau)\bm{D}_{2}(\tau)\left[\bm{0}_{d^{2}p\times
d},\bm{I}_{d^{2}p}\right],\ j\geq 1,$ $\displaystyle\bm{C}_{j,2}(\tau)$
$\displaystyle=$
$\displaystyle(\bm{I}_{d}\otimes\bm{\Psi}_{j}(\tau))\left(\bm{N}_{1}^{\top}(\tau)\bm{N}_{1}(\tau)+\bm{N}_{2}^{\top}(\tau)\bm{N}_{2}(\tau)\right)^{-1}\bm{N}_{1}^{\top}(\tau)\bm{D}_{1},\
j\geq 0,$

in which
$\bm{N}_{2}(\tau)=\bm{Q}\left(\bm{I}_{d}\otimes\bm{A}_{\tau}^{-1}(1)\right)$,
$\bm{D}_{2}(\tau)=\bm{Q}\left(\bm{B}^{\top}(\tau)\otimes\bm{A}_{\tau}^{-1}(1)\right)\triangledown_{\bm{\alpha}(\tau)}\bm{A}_{\tau}(1)$
with $\bm{A}_{\tau}(1)=\bm{I}_{d}-\sum_{i=1}^{p}\bm{A}_{i}(\tau)$ and
$\triangledown_{\bm{\alpha}(\tau)}\bm{A}_{\tau}(1)=-\left[\bm{I}_{d^{2}},...,\bm{I}_{d^{2}}\right]$
$(d^{2}\times d^{2}p)$, the duplication matrix $\bm{D}_{1}$ satisfies
$\mathrm{vec}\left(\bm{\Omega}(\tau)\right)=\bm{D}_{1}\mathrm{vech}\left(\bm{\Omega}(\tau)\right)$,
and $\bm{Q}$ is a $d(d-1)/2\times d^{2}$ selection matrix of $0$ and $1$ such
that $\bm{Q}\mathrm{vec}\left(\bm{B}(\tau)\right)=0$.

For ease of presentation, we provide the definitions of the respective
estimators of $\bm{V}(\tau)$ and $\bm{\Phi}(\tau)$ (i.e.,
$\widehat{\bm{V}}(\tau)$ and $\widehat{\bm{\Phi}}(\tau)$) in Appendix A.1. It
is easy to see that $\widehat{\bm{\Phi}}(\tau)\to_{P}\bm{\Phi}(\tau)$,
$\widehat{\bm{\omega}}(\tau)\to_{P}\bm{\omega}(\tau)$, and
$\widehat{\bm{V}}(\tau)\to_{P}\bm{V}(\tau)$ by Theorem 2.1. As a result,
$\widehat{\bm{\Sigma}}_{\bm{B}_{j}}(\tau)\to_{P}\bm{\Sigma}_{\bm{B}_{j}}(\tau)$,
where $\widehat{\bm{\Sigma}}_{\bm{B}_{j}}(\tau)$ has a form identical to
$\bm{\Sigma}_{\bm{B}_{j}}(\tau)$ but replacing $\bm{\Phi}(\tau)$,
$\bm{\omega}(\tau)$ and $\bm{V}(\tau)$ with their estimators, respectively.

###  2.3 Selection of the Optimal Lag

We next consider the choice of the optimal number of lags, i.e., the
estimation of $p$. Specifically, we consider the minimization of the next
information criterion.

$\displaystyle\widehat{\mathsf{p}}=\operatorname*{\arg\\!\min}_{1\leq\mathsf{p}\leq\mathsf{P}}(\log\left\\{\text{RSS}(\mathsf{p})\right\\}+\mathsf{p}\cdot\chi_{T})\equiv\operatorname*{\arg\\!\min}_{1\leq\mathsf{p}\leq\mathsf{P}}\text{IC}(\mathsf{p}),$
(2.8)

where
$\text{RSS}(\mathsf{p})=\frac{1}{T}\sum_{t=1}^{T}\widehat{\bm{\eta}}_{\mathsf{p},t}^{\top}\widehat{\bm{\eta}}_{\mathsf{p},t}$,
$\chi_{T}$ is the penalty term, $\widehat{\bm{\eta}}_{\mathsf{p},t}$ is the
value of $\widehat{\bm{\eta}}_{t}$ by letting the lag be $\mathsf{p}$, and
$\mathsf{P}$ is a sufficiently large fixed positive integer. The next theorem
shows the validity of (2.8).

######  Theorem 2.3.

Let Assumptions 1-3 hold. Suppose $\frac{T^{1-\frac{4}{\delta}}h}{\log
T}\to\infty$, $\max_{t\geq
1}E(\|\bm{\epsilon}_{t}\|^{4}|\mathcal{F}_{t-1})<\infty$ a.s., $\chi_{T}\to
0$, and $c_{T}^{-2}\chi_{T}\to\infty$, where $c_{T}=h^{2}+\left(\frac{\log
T}{Th}\right)^{1/2}$. Then $\Pr\left(\widehat{\mathsf{p}}=p\right)\to 1$.

In view of the conditions on $\chi_{T}$, a natural choice is

$\displaystyle\chi_{T}=\max\left\\{h^{4},\frac{\log
T}{Th}\right\\}\cdot\log(\log(Th)).$

To this end, we have established the asymptotic properties of the proposed
estimators. In the following, we discuss some model specification issues when
building (2.1).

##  3 Testing for Parameter Stability

To close our investigation, we consider testing whether some (if not all)
components of the coefficient matrices are time-invariant, which as explained
in the introduction may help settle the discrepancy on the ongoing debate
about the high inflation in the U.S. during 1970-1980.

For model (2.1), we are specifically interested in testing

$\mathbb{H}_{0}:\bm{C}\bm{\beta}(\cdot)=\bm{c}\text{ for some unknown
$c\in\mathbb{R}^{s}$},$ (3.1)

where $\bm{\beta}(\tau):=\mathrm{vec}\left(\bm{A}(\tau)\right)$ and $\bm{C}$
is a selection matrix. Practically, the choice of $\bm{C}$ should be
theory/application driven, and $\bm{c}$ needs to be estimated. For example, in
the context of monetary policy analysis Primiceri (2005), one can let
$\bm{C}=\left[\bm{0}_{d^{2}p\times d},\bm{I}_{d^{2}p}\right]$ to test whether
the policy transmission mechanism is varying over time.

The test statistic is constructed based on the weighted integrated squared
errors:

$\widehat{Q}_{\bm{C},\bm{H}}=\int_{0}^{1}\left\\{\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right\\}^{\top}\bm{H}(\tau)\left\\{\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right\\}\mathrm{d}\tau,$
(3.2)

where $\widehat{\bm{\beta}}(\cdot):=\text{vec}[\widehat{\bm{A}}(\cdot)]$
should be obvious, and $\widehat{\bm{c}}$ is the estimator of $\bm{c}$ to be
constructed below. In (3.2), $\bm{H}(\cdot)$ is an $s\times s$ positive
definite weighting matrix, and is typically set as the precision matrix
associated with $\widehat{\bm{\beta}}(\cdot)$.

Let $\bm{C}$ be a pre-specified matrix and
$\bm{c}=\int_{0}^{1}\bm{C}\bm{\beta}(\tau)\mathrm{d}\tau$. Then a natural
estimator of $\bm{c}$ is given by

$\widehat{\bm{c}}=\int_{0}^{1}\bm{C}\widehat{\bm{\beta}}(\tau)\mathrm{d}\tau,$
(3.3)

of which the asymptotic properties are summarized in the following lemma.

######  Lemma 3.1.

Let Assumptions 1-3 hold. Suppose further that $\max_{t\geq
1}E[\|\bm{\epsilon}_{t}\|^{4}|\mathcal{F}_{t-1}]<\infty$ a.s.,
$\frac{T^{1-\frac{4}{\delta}}h}{\log T}\to\infty$, each element of
$\bm{A}(\cdot)$ has finite third-order derivative, $Th^{2}/(\log
T)^{2}\to\infty$, and $Th^{6}\to 0$. As $T\to\infty$, we have

$\sqrt{T}\left(\widehat{\bm{c}}-\bm{c}-\frac{1}{2}h^{2}\tilde{c}_{2}\int_{0}^{1}\bm{C}\bm{\beta}^{(2)}(\tau)\mathrm{d}\tau\right)\to_{D}N\left(\bm{0},\int_{0}^{1}\bm{C}\bm{V}_{\bm{\beta}}(\tau)\bm{C}^{\top}\mathrm{d}\tau\right),$

where
$\bm{V}_{\bm{\beta}}(\tau):=\bm{\Sigma}^{-1}(\tau)\otimes\bm{\Omega}(\tau)$
and $\bm{\Sigma}(\tau)$ is defined in (A.1).

Lemma 3.1 indicates that under the null, $\widehat{\bm{c}}$ will achieve a
parametric rate (i.e., $\sqrt{T}$). This is important for the finite sample
study, since more reliable information can be retrieved from the estimators.

Having established Lemma 3.1, we move on to investigate the test statistic
(3.2), and examine the local alternatives. After some tedious development, one
actually can show that

$\displaystyle\widehat{Q}_{\bm{C},\bm{H}}=\frac{1}{Th}\cdot\frac{\widetilde{v}_{0}}{T}\sum_{t=1}^{T}\bm{\eta}_{t}^{\top}\bm{Z}_{t-1}^{\top}\bm{H}_{0}(\tau_{t})\bm{Z}_{t-1}\bm{\eta}_{t}+O_{P}\left(\frac{1}{T\sqrt{h}}\right)$
(3.4)

where $\bm{Z}_{t}=\bm{z}_{t}\otimes\bm{I}_{d}$,
$\bm{H}_{0}(\tau)=\bm{\Sigma}_{\bm{Z}}^{-1}(\tau)\bm{C}^{\top}\bm{H}(\tau)\bm{C}\bm{\Sigma}_{\bm{Z}}^{-1}(\tau)$,
$\bm{\Sigma}_{\bm{Z}}(\tau)=\bm{\Sigma}(\tau)\otimes\bm{I}_{d}$. Clearly,

$\displaystyle\frac{\widetilde{v}_{0}}{T}\sum_{t=1}^{T}\bm{\eta}_{t}^{\top}\bm{Z}_{t-1}^{\top}\bm{H}_{0}(\tau_{t})\bm{Z}_{t-1}\bm{\eta}_{t}$
(3.5)

converges to a fixed value in probability, so the first term on the right hand
side of (3.4) is the leading one. Moreover,
$\frac{\widetilde{v}_{0}}{T}\sum_{t=1}^{T}\bm{\eta}_{t}^{\top}\bm{Z}_{t-1}^{\top}\bm{H}_{0}(\tau_{t})\bm{Z}_{t-1}\bm{\eta}_{t}$
can be considered as a weighted average of the second moment of
$\bm{\eta}_{t}$, it thus does not yield any distribution for us to conduct the
hypothesis testing. In this regard, it can be considered a bias term which
should be removed. The term which really yields an asymptotic distribution is
in fact the second part on the right hand side of (3.4). Thus, we present the
asymptotic distribution of the test statistic as follows.

######  Theorem 3.1.

Let the conditions of Lemma 3.1 hold. Under the null hypothesis, if
$Th^{5.5}\to 0$ and
$E[\|\bm{\epsilon}_{t}\|^{\delta}|\mathcal{F}_{t-1}]<\infty$ a.s., we have

$\displaystyle
T\sqrt{h}\left(\widehat{Q}_{\bm{C},\widehat{\bm{H}}}-\frac{1}{Th}s\widetilde{v}_{0}\right)\to_{D}N(0,4sC_{B}),$

where
$\widehat{\bm{H}}(\tau)=(\bm{C}\widehat{\bm{V}}_{\bm{\beta}}(\tau)\bm{C}^{\top})^{-1}$,
$\widehat{\bm{V}}_{\bm{\beta}}(\tau)=\widehat{\bm{\Sigma}}^{-1}(\tau)\otimes\widehat{\bm{\Omega}}(\tau)$,
$C_{B}=\int_{0}^{2}\left(\int_{-1}^{1-v}K(u)K(u+v)\mathrm{d}u\right)^{2}\mathrm{d}v$,
and $\widehat{\bm{\Sigma}}(\tau)$ is defined in (A.5).

Theorem 3.1 states that the test statistic converges to a normal distribution.
The term $s\widetilde{v}_{0}$ is the limit associated with (3.5). Due to the
use of $\widehat{\bm{H}}(\cdot)$, the second moment of $\bm{\eta}_{t}$
disappears from the asymptotic distribution automatically.

Here, we would like to emphasize that the proposed test is in fact a one-sided
test, which is reflected in the investigation for the local alternative below.
An intuitive explanation is that due to the quadratic form of (3.2), any
departure from the true value will eventually yield a squared term when
analysing the asymptotic power. Therefore, the null of (3.1) will be rejected
at the level $\alpha$ if

$\displaystyle\widehat{Q}_{\bm{C},\widehat{\bm{H}}}^{*}=\frac{T\sqrt{h}\left(\widehat{Q}_{\bm{C},\widehat{\bm{H}}}-\frac{1}{Th}s\widetilde{v}_{0}\right)}{\sqrt{4sC_{B}}}>q_{1-\alpha},$
(3.6)

where $q_{1-\alpha}$ stands for the $(1-\alpha)^{th}$ quantile of the standard
normal distribution.

In what follows, we consider a sequence of local alternatives of the form:

$\displaystyle\mathbb{H}_{1}:\bm{C}\bm{\beta}(\tau)=\bm{c}+d_{T}\bm{f}(\tau),$
(3.7)

where $\bm{f}(\tau)$ is a twice continuously differentiable vector of
functions, and $d_{T}\to 0$. The term $d_{T}\bm{f}(\tau)$ characterizes the
departure of the time-varying coefficient $\bm{C}\bm{\beta}(\tau)$ from the
constant $\bm{c}$. Following the development of Theorem 3.1, it is
straightforward to obtain the following corollary.

######  Corollary 3.1.

Let the conditions of Theorem 3.1 hold. Under the $\mathbb{H}_{1}$ of (3.7),
if $d_{T}=T^{-1/2}h^{-1/4}$, then

$\displaystyle
T\sqrt{h}\left(\widehat{Q}_{\bm{C},\widehat{\bm{H}}}-\frac{1}{Th}s\widetilde{v}_{0}\right)\to_{D}N\left(\delta_{1},4sC_{B}\right),$

where
$\delta_{1}=\int_{0}^{1}\bm{f}(\tau)^{\top}(\bm{C}\bm{V}_{\bm{\beta}}(\tau)\bm{C}^{\top})^{-1}\bm{f}(\tau)\mathrm{d}\tau$.

Moreover, we have
$\Pr\left(\widehat{Q}_{\bm{C},\widehat{\bm{H}}}^{*}>q_{1-\alpha}\right)\to\Phi\left(q_{\alpha}+\frac{\delta_{1}}{2\sqrt{sC_{B}}}\right)$.

Corollary 3.1 shows that the test has a non-trivial power against
$\mathbb{H}_{1}$ when $d_{T}=T^{-1/2}h^{-1/4}$. If
$T^{-1/2}h^{-1/4}=o(d_{T})$, the power of the test converges to 1, i.e.,

$\displaystyle\Pr(\widehat{Q}_{\bm{C},\widehat{\bm{H}}}^{*}>q_{1-\alpha})\to
1.$

Before we conclude this section, we would like to add some comments on the
assumptions imposed and the main results established in relation to the
relevant literature. The construction of the proposed test is similar to those
discussed in Zhang and Wu (2012) and then Truquet (2017). Because we have
developed and then employed Proposition 2.1 for the time–varying VMA($\infty$)
representation, the assumptions, such as requiring $Th^{5.5}\to 0$, are less
restrictive than those assumed in the relevant literature, see, for example,
$Th^{3.5}\to 0$ by Truquet (2017). As a consequence, the main techniques
employed in our proofs may be of general interest and applicability in dealing
with similar problems.

In what follows, we will examine the finite sample performance of the
asymptotic properties for the proposed estimators and test statistic by
simulation studies.

##  4 Simulation

In this section, we first provide some details of the numerical implementation
in Section 4.1, and then respectively examine the estimation and hypothesis
testing in Sections 4.2 and 4.3.

###  4.1 Numerical Implementation

Throughout the numerical studies, Epanechnikov kernel
$K(u)=0.75(1-u^{2})I(|u|\leq 1)$ is adopted.

When selecting the optimal lag by (2.8), the bandwidth $\widehat{h}_{cv}$ is
always chosen by minimizing the following cross-validation criterion function
for each $\mathsf{p}$.

$\widehat{h}_{cv}=\arg\min_{h}\sum_{t=1}^{T}\|\bm{x}_{t}-\widehat{\bm{a}}_{-t}(\tau_{t})-\sum_{j=1}^{\mathsf{p}}\widehat{\bm{A}}_{j,-t}(\tau_{t})\bm{x}_{t-j}\|^{2},$
(4.1)

where $\widehat{\bm{a}}_{-t}(\cdot)$, and $\widehat{\bm{A}}_{j,-t}(\cdot)$ are
obtained using (2.1) but leaving the $t^{th}$ observation out. Once
$\widehat{\mathsf{p}}$ and $\widehat{h}_{cv}$ are obtained, the estimation
procedure is relatively straightforward. It is pointed out that it might be
possible to extend the Bayesian bandwidth selection method proposed for the
conventional time–varying regression model by Cheng et al. (2019) to the
time–varying VAR setting under study for an optimal choice of the bandwidth
parameter.

We then comment on the testing procedure. To improve the finite sample
performance of the test, we propose a simulation-assisted testing procedure. A
similar procedure has also been adopted by Zhang and Wu (2012) and Truquet
(2017) to for the same purpose in the context of univariate time-varying
models. Alternatively, one may consider the moving block bootstrap as in
Jentsch and Lunsford (2021). For simplicity, we adopt the former as follows.

Algorithm — a simulation-assisted testing procedure

  * Step 1: Use the sample $\\{\bm{x}_{t}\\}$ to estimate the unrestricted model and the restricted model, and then compute $\widehat{Q}_{\bm{C},\widehat{\bm{H}}}$ based on (3.2).

Step 2: Generate i.i.d. standard multivariate normal random vectors
$\\{\bm{x}_{t}^{*}\\}$.

Step 3: Compute the bootstrap statistic
$\widetilde{Q}_{\bm{C},\widehat{\bm{H}}}^{b}$ in the same way as
$\widehat{Q}_{\bm{C},\widehat{\bm{H}}}$, with $\\{\bm{x}_{t}^{*}\\}$ replacing
the original sample $\\{\bm{x}_{t}\\}$.

Step 4: Repeat Steps 2-3 $B$ times to obtain $B$ bootstrap test statistics
$\\{\widetilde{Q}_{\bm{C},\widehat{\bm{H}}}^{b}\\}_{b=1}^{B}$, as well as its
empirical quantile $\widehat{q}_{1-\alpha}$. We reject the null hypothesis
(3.1) at level $\alpha$ if
$\widehat{Q}_{\bm{C},\widehat{\bm{H}}}>\widehat{q}_{1-\alpha}$.

###  4.2 Examining the Model Estimation

We now examine the finite sample performance of the theoretical findings. The
data generating process (DGP) is as follows.

$\bm{x}_{t}=\bm{a}(\tau_{t})+\bm{A}_{1}(\tau_{t})\bm{x}_{t-1}+\bm{A}_{2}(\tau_{t})\bm{x}_{t-2}+\bm{\eta}_{t}\
\ \mbox{with}\ \ \bm{\eta}_{t}=\bm{\omega}(\tau_{t})\bm{\epsilon}_{t}\ \
\text{for}\ \ t=1,\ldots,T,$ (4.2)

where $\bm{\epsilon}_{t}$’s are i.i.d. draws from $N(\bm{0}_{2\times
1},\bm{I}_{2})$,

$\displaystyle\bm{a}(\tau)$ $\displaystyle=$
$\displaystyle\left[0.5\sin(2\pi\tau),0.5\cos(2\pi\tau)\right]^{\top},$
$\displaystyle\bm{A}_{1}(\tau)$ $\displaystyle=$
$\displaystyle\left[\begin{matrix}0.8\exp{(-0.5+\tau)}&0.8(\tau-0.5)^{3}\\\
0.8(\tau-0.5)^{3}&0.8+0.3\sin(\pi\tau)\end{matrix}\right],$
$\displaystyle\bm{A}_{2}(\tau)$ $\displaystyle=$
$\displaystyle\left[\begin{matrix}-0.2\exp{(-0.5+\tau)}&0.8(\tau-0.5)^{2}\\\
0.8(\tau-0.5)^{2}&-0.4+0.3\cos(\pi\tau)\end{matrix}\right],$
$\displaystyle\bm{\omega}(\tau)$ $\displaystyle=$
$\displaystyle\left[\begin{matrix}1.5+0.2\exp{(0.5-\tau)}&0\\\
0.1\exp{(0.5-\tau)}&1.5+0.5(\tau-0.5)^{2}\end{matrix}\right].$

Let the sample size be $T\in\\{200,400,800\\}$, and conduct 1000 replications
for each choice of $T$.

First, we check whether the coefficient matrices $\bm{A}_{1}(\tau)$ and
$\bm{A}_{2}(\tau)$ satisfy Assumption 1.1. Thus, for each generated dataset,
we compute the largest eigenvalue of the true companion matrix
$\bm{\Phi}(\tau)$ in $\tau\in[0,1]$, which varies from 0.54 to 0.88 over
replications indicating the validity of Assumption 1.1.

Next, we report the percentages of $\widehat{\mathsf{p}}<2$,
$\widehat{\mathsf{p}}=2$, and $\widehat{\mathsf{p}}>2$ respectively based on
1000 replications. Table 1 shows that the information criterion (2.8) performs
reasonably well, as the percentages associated with $\widehat{\mathsf{p}}=2$
are sufficiently close to 1 even for $T=200$.

Table 1: The percentages of $\widehat{\mathsf{p}}<2$, $\widehat{\mathsf{p}}=2$, and $\widehat{\mathsf{p}}>2$ $T$ | $\widehat{\mathsf{p}}<2$ | $\widehat{\mathsf{p}}=2$ | $\widehat{\mathsf{p}}>2$
---|---|---|---
200 | 0.004 | 0.976 | 0.020
400 | 0.004 | 0.986 | 0.010
800 | 0.000 | 1.000 | 0.000

Finally, we evaluate the estimates of $\bm{A}(\tau)$, $\bm{\Omega}(\tau)$, as
well as the estimates of the impulse responses (say, $\bm{B}_{1}(\tau)$ and
$\bm{B}_{5}(\tau)$ without loss of generality) based on the short-run timing
restrictions. For each parameter of interest, we calculate the root mean
square error (RMSE) as follows:

$\left\\{\frac{1}{1000T}\sum_{n=1}^{1000}\sum_{t=1}^{T}\|\widehat{\bm{\theta}}^{(n)}(\tau_{t})-\bm{\theta}(\tau_{t})\|^{2}\right\\}^{1/2},$

where
$\bm{\theta}(\cdot)\in\left\\{\bm{A}(\cdot),\bm{\Omega}(\cdot),\bm{B}_{1}(\tau),\bm{B}_{5}(\tau)\right\\}$,
and $\widehat{\bm{\theta}}^{(n)}(\tau)$ is the estimate of $\bm{\theta}(\tau)$
for the $n^{th}$ replication. Of interest, we also report the finite sample
coverage probabilities of the confidence intervals. The nominal coverage is
95%. Given $\bm{\theta}(\cdot)$, for each generated dataset, the coverage
probability is first calculated for each functional component of
$\bm{\theta}(\cdot)$ over the grid points $\\{\tau_{t},t=1,\ldots,T\\}$, and
then we further take an average across the elements of $\bm{\theta}(\cdot)$.
After 1000 replications, we present the averaged value of these coverage
probabilities in Table 2.

As shown in Table 2, the RMSEs decrease as the sample size increases. The
finite sample coverage probabilities are smaller than their nominal level when
$T=200$, but are fairly close to 95% as $T=800$.

Table 2: The RMSEs and the empirical coverage probabilities (in parentheses) $T$ | $\bm{A}(\tau)$ | $\bm{\Omega}(\tau)$ | $\bm{B}_{1}(\tau)$ | $\bm{B}_{5}(\tau)$
---|---|---|---|---
200 | 0.54 (0.89) | 0.83 (0.87) | 0.46 (0.87) | 0.31 (0.89)
400 | 0.40 (0.91) | 0.71 (0.91) | 0.30 (0.91) | 0.34 (0.89)
800 | 0.29 (0.92) | 0.62 (0.93) | 0.29 (0.92) | 0.30 (0.90)

###  4.3 Examining the Parameter Stability Testing

To evaluate the size and local power of the proposed test statistic, we
consider the following DGP:

$\bm{x}_{t}=\bm{A}_{1}(\tau_{t})\bm{x}_{t-1}+\bm{A}_{2}(\tau_{t})\bm{x}_{t-2}+\bm{\eta}_{t},$
(4.3)

where $\bm{A}_{2}(\cdot)$ and $\bm{\eta}_{t}$ are generated in the same way as
in Section 4.2, and

$\bm{A}_{1}(\tau)=\left[\begin{matrix}0.4&-0.1\\\
-0.1&0.4\end{matrix}\right]+b\times
d_{T}\times\left[\begin{matrix}2\exp(\tau-1)-1&\exp(\tau-1)-1\\\
\exp(\tau-1)-1&2\exp(\tau-1)-1\end{matrix}\right]$

in which $d_{T}=T^{-1/2}h^{-1/4}$ and $b$ is set to be $0$, $2$ or $4$ in
order to investigate the size and local power of the proposed test. We use the
proposed testing procedure to test whether the coefficient $\bm{A}_{1}(\cdot)$
is time-varying.

Again, we let $T\in\\{200,400,800\\}$ and conduct $1000$ replications for each
choice of $T$. We use the simulation-assisted testing procedure to get the
empirical critical value $\widehat{q}_{1-\alpha}$ after $1000$ bootstrap
replications. We consider a sequence of bandwidths to check the robustness of
the proposed test with respect to the bandwidth choice:

$h=\alpha_{1}T^{-1/5},\quad\alpha_{1}=0.2,0.4,\ldots,1.8.$ (4.4)

Table 3 reports the rejection rates at the $5\%$ and $10\%$ nominal levels. A
few facts emerge from the table. First, our test has reasonable sizes using
empirical critical values obtained by the bootstrap procedure. Second, the
size behaviour of our test is not sensitive to the choices of bandwidths. As
discussed in Gao and Gijbels (2008), the estimation-based optimal bandwidths
may also be optimal for testing purposes, so for simplicity one can use the
rule-of-thumb in practice. Third, the local power of our test increases
rapidly as $b$ increases.

Table 3: Size and power evaluation |  |  | $5\%$ |  | $10\%$
---|---|---|---|---|---
| Bandwidth | T | 200 | 400 | 800 |  | 200 | 400 | 800
$b=0$ (size) | $0.2T^{-1/5}$ |  | 0.046 | 0.060 | 0.058 |  | 0.081 | 0.113 | 0.117
$0.4T^{-1/5}$ |  | 0.044 | 0.050 | 0.066 |  | 0.095 | 0.093 | 0.104
$0.6T^{-1/5}$ |  | 0.057 | 0.070 | 0.046 |  | 0.110 | 0.121 | 0.102
$0.8T^{-1/5}$ |  | 0.056 | 0.066 | 0.050 |  | 0.113 | 0.131 | 0.105
$1.0T^{-1/5}$ |  | 0.033 | 0.060 | 0.047 |  | 0.066 | 0.104 | 0.100
$1.2T^{-1/5}$ |  | 0.055 | 0.042 | 0.042 |  | 0.115 | 0.093 | 0.086
$1.4T^{-1/5}$ |  | 0.039 | 0.045 | 0.043 |  | 0.085 | 0.103 | 0.078
$1.6T^{-1/5}$ |  | 0.065 | 0.047 | 0.038 |  | 0.114 | 0.120 | 0.088
$1.8T^{-1/5}$ |  | 0.041 | 0.066 | 0.046 |  | 0.094 | 0.117 | 0.088
$b=2$ (local power) | $0.2T^{-1/5}$ |  | 0.095 | 0.103 | 0.143 |  | 0.177 | 0.189 | 0.242
$0.4T^{-1/5}$ |  | 0.145 | 0.155 | 0.162 |  | 0.204 | 0.247 | 0.236
$0.6T^{-1/5}$ |  | 0.148 | 0.129 | 0.170 |  | 0.208 | 0.255 | 0.239
$0.8T^{-1/5}$ |  | 0.124 | 0.168 | 0.168 |  | 0.233 | 0.258 | 0.240
$1.0T^{-1/5}$ |  | 0.144 | 0.176 | 0.145 |  | 0.230 | 0.250 | 0.225
$1.2T^{-1/5}$ |  | 0.123 | 0.189 | 0.158 |  | 0.196 | 0.274 | 0.246
$1.4T^{-1/5}$ |  | 0.160 | 0.173 | 0.183 |  | 0.246 | 0.275 | 0.279
$1.6T^{-1/5}$ |  | 0.142 | 0.139 | 0.189 |  | 0.208 | 0.218 | 0.292
$1.8T^{-1/5}$ |  | 0.172 | 0.170 | 0.194 |  | 0.269 | 0.268 | 0.287
$b=4$ (local power) | $0.2T^{-1/5}$ |  | 0.477 | 0.448 | 0.655 |  | 0.635 | 0.604 | 0.783
$0.4T^{-1/5}$ |  | 0.340 | 0.561 | 0.648 |  | 0.513 | 0.705 | 0.758
$0.6T^{-1/5}$ |  | 0.428 | 0.582 | 0.708 |  | 0.583 | 0.709 | 0.803
$0.8T^{-1/5}$ |  | 0.506 | 0.579 | 0.640 |  | 0.615 | 0.701 | 0.773
$1.0T^{-1/5}$ |  | 0.501 | 0.579 | 0.585 |  | 0.598 | 0.699 | 0.695
$1.2T^{-1/5}$ |  | 0.473 | 0.527 | 0.620 |  | 0.592 | 0.657 | 0.728
$1.4T^{-1/5}$ |  | 0.482 | 0.532 | 0.553 |  | 0.614 | 0.648 | 0.655
$1.6T^{-1/5}$ |  | 0.519 | 0.559 | 0.583 |  | 0.617 | 0.664 | 0.665
$1.8T^{-1/5}$ |  | 0.460 | 0.523 | 0.613 |  | 0.584 | 0.653 | 0.720

##  5 Empirical Study

In this section, we discuss the transmission mechanism of the monetary policy.
As well documented, inflation is higher and more volatile in the U.S. during
1970-1980, but substantially decreases in the subsequent period, which is
often referred to as the Great Moderation (Primiceri, 2005). Two explanations
(i.e., bad policy and bad luck) have been debated repeatedly in the
literature. The first explanation focuses on the changes in the transmission
mechanism (e.g., Cogley and Sargent, 2005), while the second regards it as a
consequence of changes in the size of exogenous shocks (e.g., Sims and Zha,
2006). In what follows, we revisit the arguments associated with the Great
Moderation using our approach. The estimation is conducted in exactly the same
way as in Section 4, so we no longer repeat the details.

First, we estimate the time-varying VAR$(p)$ model using three commonly
adopted macroeconomic variables of the literature (Primiceri, 2005; Cogley et
al., 2010), which are the inflation rate (measured by 100 times the year-over-
year log change in the GDP deflator), the unemployment rate representing the
non-policy block, and the interest rate (measured by the average value for the
Federal funds rates over the quarter) representing the monetary policy block.
Following Primiceri (2005), the short-run timing restrictions are employed to
identify the monetary policy shocks. The interest rate is included as the
third variable and being treated as the monetary policy instrument. The
identification requirement is that the monetary policy actions affect the
inflation and the unemployment with at least one period of lag (i.e., short-
run timing restrictions). The data are quarterly observations measured at an
annual rate from 1954:Q3 to 2015:Q4, which are collected from the Federal
Reserve Bank of St. Louis economic database. Figure 1 plots the three
variables.

Figure 1: Plots of the inflation (left), the unemployment rate (middle) and
the interest rate (right)

For the time-varying VAR$(p)$ model, the optimal lag is
$\widehat{\mathsf{p}}=3$ by our approach, while it is often assumed to be
known with the values varying from $2$ to $4$ in the literature. Thus, the
following analyses focus on the time-varying VAR$(3)$ model (referred to as
TV-VAR$(3)$ hereafter). We then conduct a robustness check to see whether the
innovation process $\bm{\epsilon}_{t}$ exhibits serial correlation. We use the
Breusch-Godfrey LM test (cf., Breusch, 1978; Godfrey, 1978) for testing the
first-order autocorrelation of $\bm{\epsilon}_{t}$’s. The null hypothesis is
$H_{0}:E\left(\bm{\epsilon}_{t}\bm{\epsilon}_{t-1}^{\top}\right)=0$. Based on
the estimates
$\widehat{\bm{\epsilon}}_{t}=\widehat{\bm{\omega}}^{-1}(\tau_{t})\widehat{\bm{\eta}}_{t}$,
the $p$-value is 0.81 suggesting that the TV-VAR$(3)$ model fits the data
quite well.

Before investigating the time-varying effects of monetary policy on real
economy, it is reasonable to check whether the VAR coefficients (i.e., the
policy transmission mechanism) are time-varying. To this end, we employ the
proposed test statistic to examine the constancy of model coefficients, and
summarize the results in Table 4. Specifically, we first set $\bm{C}$ to be an
identity matrix for selecting between parametric and time-varying VAR models.
From Table 4 (the row labelled “Constancy”), we conclude that the VAR
coefficients are not constant, implying that there exists significant time-
variations in policy transmission mechanism. We then apply the testing
procedures to examine whether the selected variables (i.e., intercept,
$\bm{x}_{t-1}$, $\bm{x}_{t-2}$, $\bm{x}_{t-3}$) have time-varying
contributions. By Table 4, at the 5% significance level we conclude that all
$\bm{a}(\cdot)$, $\bm{A}_{1}(\cdot)$, $\bm{A}_{2}(\cdot)$ and
$\bm{A}_{3}(\cdot)$ are time-varying by examining them
individually333Certainly, one may examine each element of these matrices.
However, it will lead to a quite lengthy presentation. In order not to deviate
from our main goal, we no longer conduct more testing along this line.,
suggesting that the TV-VAR(3) is more appropriate than a constant VAR model.

Table 4: Summary statistics of the test | test statistic | p-value
---|---|---
Constancy | 341.27 | 0.00
$\bm{a}(\cdot)$ | 50.15 | 0.00
$\bm{A}_{1}(\cdot)$ | 63.02 | 0.00
$\bm{A}_{2}(\cdot)$ | 23.27 | 0.00
$\bm{A}_{3}(\cdot)$ | 17.23 | 0.01

We now consider the changes in the size of exogenous shocks. Figure 2 plots
the estimated residuals $\widehat{\bm{\eta}}_{t}$. As can be seen, the
magnitudes of estimated residuals change over time. For example, the
magnitudes of the estimated residuals in the interest rate component increase
from 1970 to 1980, but decrease quickly after 1980. The evidence favours the
TV-VAR$(3)$ model which accounts for heteroscedasticity. More precisely,
Figure 3 plots the time-varying standard deviation of the estimated residuals
$\widehat{\bm{\eta}}_{t}$ as well as the associated 95% confidence intervals.
We can see that a decline in unconditional volatilities of exogenous shocks
after 1980 in every subplot of Figure 3. Thus, our results support the
explanation of “bad luck” (e.g., Primiceri, 2005; Sims and Zha, 2006).

Figure 2: Estimation residuals in the inflation equation (left), the
unemployment equation (middle) and the interest rate equation (right).

We then discuss how inflation and unemployment respond to monetary policy
shocks. Figure 4 plots the time-varying cumulative impulse responses444The
cumulative impulse responses are constructed in exactly the same way as in
Primiceri (2005), and the standard errors are computed based on Theorem 2.2.
of inflation and unemployment to a monetary shock subject to the short-run
timing restrictions. Clearly, these responses vary over time, indicating a
substantial time-variation in the policy transmission mechanism. The result
differs from Primiceri (2005) that has found no evidence of time variation in
the response of the economy to monetary policy shocks using a Bayesian
approach. Interestingly, our results show that the responses of inflation
exhibit a price puzzle during 1970-1980, which suggests that inflation
increases in response to a monetary tightening although standard macroeconomic
theory predicts the opposite. Therefore, our results also support the
explanation of “bad policy” (e.g., Cogley and Sargent, 2005).

Figure 3: The standard deviation of estimation residuals
$\widehat{\bm{\eta}}_{t}$ in the inflation equation (left), the unemployment
equation (middle) and the interest rate equation (right) as well as the
associated 95% confidence intervals. The red line denotes the estimated
standard deviation using the VAR(3) model with constant parameters.

Figure 4: Time-varying cumulative impulse responses to the monetary policy
shocks. Left front axis left: Quarters (Horizon). Right front axis left: Time.

Additionally, Figure 5 plots the impulse responses of inflation to a monetary
policy shock in 1978:Q1 and 1994:Q1. The left panel in Figure 5 exhibits the
price puzzle, while the right panel shows a response that corresponds to the
mainstream prior: the price level declines soon after a tightening. This
result shows that the price puzzle is limited to periods of bad monetary
policy, and is consistent with a stream of literature (e.g., Elbourne and de
Haan, 2006; Rusnák et al., 2013), which suggests that the price puzzle emerges
when using data for different monetary policy regimes and VARs estimated
within a period of single monetary policy rarely encounter the price puzzle.
Hence, the existence of time-varying policy transmission mechanism favours the
TV-VAR(3) model.

Figure 5: Impulse responses of inflation to the monetary policy shocks in
1978:Q1 and 1994:Q1 with one-standard error confidence intervals.

##  6 Conclusions and Discussion

In this paper, we have proposed a new class of time-varying VAR models where
the VAR coefficients and covariance matrix of the error innovations are
allowed to evolve over time. Accordingly, we have established a set of
asymptotic results, including the impulse responses analyses subject to both
short-run timing and long-run restrictions, an information criterion to select
the optimal lag, and a Wald–type test to determine the constant coefficients.
Simulation studies are conducted to evaluate the theoretical findings.
Finally, we demonstrate the empirical relevance and usefulness of the proposed
methods through an application to the transmission mechanism of U.S. monetary
policy. We have revealed that there exists a substantial time-variation in the
policy transmission mechanism, and the “price puzzle” is limited to periods of
bad monetary policy.

There are several directions for possible extensions. The first one is about
how to consistently estimate the $d$-dimensional components of the VAR$(p)$
process for the case where the dimensionality, $d$, and the number of lags,
$p$, may diverge along with the sample size, $T$. The second one is to allow
for some time-varying structure in cointegrated dynamic models. We wish to
leave such issues for future study.

## Appendix A

For the sake of presentation, we first provide some notations and mathematical
symbols in Appendix A.1. The proofs of main results are provided in Appendix
A.2. In what follows, $M$ and $O(1)$ always stand for constants, and may be
different at each appearance.

##  A.1 Notations and Mathematical Symbols

For ease of notation, we define three matrices $\bm{\Sigma}(\tau)$,
$\bm{V}(\tau)$ and $\bm{\Phi}(\tau)$ with their estimators respectively. For
$\forall\tau\in(0,1)$, let

$\displaystyle\bm{\Sigma}(\tau)$ $\displaystyle=$
$\displaystyle\left[\begin{matrix}1&\bm{\mu}^{\top}(\tau)&\cdots&\bm{\mu}^{\top}(\tau)\\\
\bm{\mu}(\tau)&\bm{\Sigma}_{0}(\tau)&\cdots&\bm{\Sigma}_{p-1}^{\top}(\tau)\\\
\vdots&\vdots&\ddots&\vdots\\\
\bm{\mu}(\tau)&\bm{\Sigma}_{p-1}(\tau)&\cdots&\bm{\Sigma}_{0}(\tau)\end{matrix}\right],$
(A.1)

in which $\bm{\mu}(\tau)$ and $\bm{B}_{j}(\tau)$ are defined in Proposition
2.1 and
$\bm{\Sigma}_{m}(\tau)=\bm{\mu}(\tau)\bm{\mu}(\tau)^{\top}+\sum_{j=0}^{\infty}\bm{B}_{j}(\tau)\bm{B}_{j+m}^{\top}(\tau)$
for $m=0,\ldots,p-1$. We define the estimator of $\bm{\Sigma}(\tau)$ as

$\displaystyle\widehat{\bm{\Sigma}}(\tau)=\left(\frac{1}{T}\sum_{t=1}^{T}K_{h}(\tau_{t}-\tau)\right)^{-1}\frac{1}{T}\sum_{t=1}^{T}\bm{z}_{t-1}\bm{z}_{t-1}^{\top}K_{h}(\tau_{t}-\tau),$
(A.2)

where $\bm{z}_{t}$ is defined in (2.4).

Next, we let

$\displaystyle\bm{V}(\tau)=\left[\begin{matrix}\bm{V}_{1,1}(\tau)&\bm{V}_{2,1}^{\top}(\tau)\\\
\bm{V}_{2,1}(\tau)&\bm{V}_{2,2}(\tau)\end{matrix}\right],$ (A.3)

where
$\bm{V}_{1,1}(\tau)=\tilde{v}_{0}\bm{\Sigma}^{-1}(\tau)\otimes\bm{\Omega}(\tau)$,

$\displaystyle\bm{V}_{2,1}(\tau)$ $\displaystyle=$
$\displaystyle\lim_{T\rightarrow\infty}\frac{h}{T}\sum_{t=1}^{T}E\left(\mathrm{vech}(\bm{\eta}_{t}\bm{\eta}_{t}^{\top})\bm{\eta}_{t}^{\top}\bm{Z}_{t-1}^{\top}\right)K_{h}(\tau_{t}-\tau)^{2}\cdot(\bm{\Sigma}^{-1}(\tau)\otimes\bm{I}_{d}),$
$\displaystyle\bm{V}_{2,2}(\tau)$ $\displaystyle=$
$\displaystyle\lim_{T\rightarrow\infty}\frac{h}{T}\sum_{t=1}^{T}E\left(\mathrm{vech}(\bm{\eta}_{t}\bm{\eta}_{t}^{\top})\mathrm{vech}(\bm{\eta}_{t}\bm{\eta}_{t}^{\top})^{\top}\right)K_{h}(\tau_{t}-\tau)^{2}$
(A.4)
$\displaystyle-\tilde{v}_{0}\mathrm{vech}\left(\bm{\Omega}(\tau)\right)\mathrm{vech}\left(\bm{\Omega}(\tau)\right)^{\top}.$

The estimator of $\bm{V}(\tau)$ is then defined as follows:

$\displaystyle\widehat{\bm{V}}(\tau)$ $\displaystyle=$
$\displaystyle\left[\begin{matrix}\widehat{\bm{V}}_{1,1}(\tau)&\widehat{\bm{V}}_{2,1}^{\top}(\tau)\\\
\widehat{\bm{V}}_{2,1}(\tau)&\widehat{\bm{V}}_{2,2}(\tau)\end{matrix}\right],$
(A.5)

where $\bm{\widehat{V}}_{1,1}(\tau)$, $\bm{\widehat{V}}_{2,1}(\tau)$ and
$\bm{\widehat{V}}_{2,2}(\tau)$ have the forms identical to their counterparts
of (A.1), but we replace $\bm{\Sigma}(\tau)$, $\bm{\eta}_{t}$ and
$\bm{\Omega}(\tau)$ with their estimators presented in (A.2) and (2.1).

Finally, recall the following definition:

$\displaystyle\bm{\Phi}(\tau)=\left[\begin{matrix}\bm{A}_{1}(\tau)&\cdots&\bm{A}_{p-1}(\tau)&\bm{A}_{p}(\tau)\\\
\bm{I}_{d}&\cdots&\bm{0}_{d}&\bm{0}_{d}\\\ \vdots&\ddots&\vdots&\vdots\\\
\bm{0}_{d}&\cdots&\bm{I}_{d}&\bm{0}_{d}\\\ \end{matrix}\right].$ (A.6)

Replacing $\bm{A}_{j}(\tau)$’s of (A.6) with their estimators obtained from
(2.1) yields an estimator, $\widehat{\bm{\Phi}}(\tau)$, for $\bm{\Phi}(\tau)$.

##  A.2 Proofs of the Main Results

###### Proof of Proposition 2.1.

Consider the VMA representation of $\bm{x}_{t}$:
$\bm{x}_{t}=\bm{\mu}_{t}+\bm{B}_{0,t}\bm{\epsilon}_{t}+\bm{B}_{1,t}\bm{\epsilon}_{t-1}+\bm{B}_{2,t}\bm{\epsilon}_{t-2}+\cdots$,
where $\bm{B}_{0,t}=\bm{\omega}(\tau_{t})$,
$\bm{B}_{j,t}=\bm{\Psi}_{j,t}\bm{\omega}(\tau_{t-j})$,
$\bm{\Psi}_{j,t}=\bm{J}\prod_{m=0}^{j-1}\bm{\Phi}(\tau_{t-m})\bm{J}^{\top}$
for $j\geq 1$,
$\bm{\mu}_{t}=\bm{a}(\tau_{t})+\sum_{j=1}^{\infty}\bm{\Psi}_{j,t}\bm{a}(\tau_{t-j})$
and $\tau_{t-j}=\frac{t-j}{T}I(t\geq j)$.

First, we investigate the validity of the VMA representations of $\bm{x}_{t}$
and $\widetilde{\bm{x}}_{t}$. Let $\rho_{A}$ denote the largest eigenvalue of
$\bm{\Phi}(\tau)$ uniformly over $\tau\in[0,1]$. Then, $\rho_{A}<1$ by
Assumption 4.1. Similar to the proof of Proposition 2.4 in Dahlhaus and
Polonik (2009), we have $\max_{t\geq
1}\left\|\prod_{m=0}^{j-1}\bm{\Phi}(\tau_{t-m})\right\|\leq M\rho_{A}^{j}$. It
follows that $\left\|{\rm
E}(\bm{x_{t}})\right\|\leq\sum_{j=0}^{\infty}\left\|\bm{\Psi}_{j,t}\right\|\cdot\left\|\bm{a}(\tau_{t-j})\right\|\leq
M\sum_{j=0}^{\infty}\rho_{A}^{j}<\infty$ and

$\displaystyle\left\|{\rm Var}(\bm{x_{t}})\right\|$ $\displaystyle=$
$\displaystyle\left\|\sum_{j=0}^{\infty}\bm{B}_{j,t}\bm{B}_{j,t}^{\top}\right\|\leq\sum_{j=0}^{\infty}\left\|\bm{B}_{j,t}\right\|^{2}\leq
M\sum_{j=0}^{\infty}\rho_{A}^{2j}<\infty.$

Similarly, we have $\left\|{\rm
E}\left(\widetilde{\bm{x}}_{t}\right)\right\|<\infty$ and $\left\|{\rm
Var}\left(\widetilde{\bm{x}}_{t}\right)\right\|<\infty$.

Then, we need to verify that $\max_{t\geq
1}\\{E\left\|\bm{x}_{t}-\widetilde{\bm{x}}_{t}\right\|^{\delta}\\}^{1/\delta}=O(T^{-1})$.
For any conformable matrices $\\{\bm{A}_{i}\\}$ and $\\{\bm{B}_{i}\\}$, since
$\prod_{i=1}^{r}\bm{A}_{i}-\prod_{i=1}^{r}\bm{B}_{i}=\sum_{j=1}^{r}\left(\prod_{k=1}^{j-1}\bm{A}_{k}\right)\left(\bm{A}_{j}-\bm{B}_{j}\right)\left(\prod_{k=j+1}^{r}\bm{B}_{k}\right)$,
we then obtain

$\displaystyle\left\|\bm{B}_{j,t}-\bm{B}_{j}(\tau_{t})\right\|=\left\|\bm{J}\prod_{m=0}^{j-1}\bm{\Phi}(\tau_{t-m})\bm{J}^{\top}\bm{\omega}(\tau_{t-j})-\bm{J}\bm{\Phi}^{j}(\tau_{t})\bm{J}^{\top}\bm{\omega}(\tau_{t})\right\|$
$\displaystyle=\left\|\left(\bm{J}\prod_{m=0}^{j-1}\bm{\Phi}(\tau_{t-m})\bm{J}^{\top}-\bm{J}\bm{\Phi}^{j}(\tau_{t})\bm{J}^{\top}\right)\bm{\omega}(\tau_{t})+\bm{J}\prod_{m=0}^{j-1}\bm{\Phi}(\tau_{t-m})\bm{J}^{\top}\left(\bm{\omega}(\tau_{t-j})-\bm{\omega}(\tau_{t})\right)\right\|$
$\displaystyle\leq
M\sum_{i=1}^{j-1}\left\|\bm{\Phi}^{i}(\tau_{t})(\bm{\Phi}(\tau_{t-i})-\bm{\Phi}(\tau_{t}))\prod_{m=i+1}^{j-1}\bm{\Phi}(\tau_{t-m})\right\|+M\rho_{A}^{j}\frac{j}{T}\leq
M\sum_{i=1}^{j-1}\frac{i}{T}\rho_{A}^{j-1}+M\rho_{A}^{j}\frac{j}{T}=O(T^{-1}),$

which implies for the same $\delta>4$ as in Assumption 2,

$\displaystyle\\{E\left\|\bm{x}_{t}-\widetilde{\bm{x}}_{t}\right\|^{\delta}\\}^{1/\delta}$
$\displaystyle\leq$
$\displaystyle\sum_{j=1}^{\infty}\left\|\bm{\Psi}_{j,t}\bm{a}(\tau_{t-j})-\bm{\Psi}_{j}(\tau_{t})\bm{a}(\tau_{t})\right\|+\sum_{j=1}^{\infty}\left\|\bm{B}_{j,t}-\bm{B}_{j}(\tau_{t})\right\|\cdot\\{E\left\|\bm{\epsilon}_{t}\right\|^{\delta}\\}^{1/\delta}$
$\displaystyle\leq$ $\displaystyle
M\sum_{j=1}^{\infty}\left(\sum_{i=1}^{j-1}\frac{i}{T}\rho_{A}^{j-1}+\rho_{A}^{j}\frac{j}{T}\right)=O\left(T^{-1}\right).$

The proof is now completed. ∎

###### Proof of Theorem 2.1.

(1). For notation simplicity, let
$\bm{S}_{T,k}(\tau)=\frac{1}{T}\sum_{t=1}^{T}\bm{z}_{t-1}\bm{z}_{t-1}^{\top}\left(\frac{\tau_{t}-\tau}{h}\right)^{k}K_{h}(\tau_{t}-\tau)$,

$\bm{S}_{T}(\tau)=\left(\begin{matrix}\bm{S}_{T,0}(\tau)&\bm{S}_{T,1}(\tau)\\\
\bm{S}_{T,1}(\tau)&\bm{S}_{T,2}(\tau)\end{matrix}\right),$

and
$\bm{M}(\tau_{t})=\bm{A}(\tau_{t})-\bm{A}(\tau)-\bm{A}^{(1)}(\tau)(\tau_{t}-\tau)-\frac{1}{2}\bm{A}^{(2)}(\tau)(\tau_{t}-\tau)^{2}$.

We now begin our investigation. Since

$\displaystyle\bm{x}_{t}$ $\displaystyle=$
$\displaystyle\left(\bm{A}(\tau)+\bm{A}^{(1)}(\tau)(\tau_{t}-\tau)+\frac{1}{2}\bm{A}^{(2)}(\tau)(\tau_{t}-\tau)^{2}+\bm{M}(\tau_{t})\right)\bm{z}_{t-1}+\bm{\eta}_{t}$
$\displaystyle=$
$\displaystyle\bm{Z}_{t-1}^{*,\top}\left[\begin{matrix}\mathrm{vec}\left(\bm{A}(\tau)\right)\\\
h\mathrm{vec}\left(\bm{A}^{(1)}(\tau)\right)\end{matrix}\right]+\frac{1}{2}h^{2}\left(\frac{\tau_{t}-\tau}{h}\right)^{2}\left(\bm{z}_{t-1}^{\top}\otimes\bm{I}_{d}\right)\mathrm{vec}\left(\bm{A}^{(2)}(\tau)\right)$
$\displaystyle+\left(\bm{z}_{t-1}^{\top}\otimes\bm{I}_{d}\right)\mathrm{vec}\left(\bm{M}(\tau_{t})\right)+\bm{\eta}_{t},$

we write

$\displaystyle\mathrm{vec}(\bm{\widehat{A}}(\tau)-\bm{A}(\tau))$
$\displaystyle=$
$\displaystyle[\bm{I}_{d^{2}p+d},\bm{0}_{d^{2}p+d}]\cdot\left(\frac{1}{T}\sum_{t=1}^{T}\bm{Z}_{t-1}^{*}\bm{Z}_{t-1}^{*,\top}K_{h}(\tau_{t}-\tau)\right)^{-1}\left(\frac{1}{T}\sum_{t=1}^{T}\bm{Z}_{t-1}^{*}\bm{x}_{t}K_{h}(\tau_{t}-\tau)\right)-\mathrm{vec}\left(\bm{A}(\tau)\right)$
$\displaystyle=$
$\displaystyle[\bm{I}_{d^{2}p+d},\bm{0}_{d^{2}p+d}]\left(\bm{S}_{T}^{-1}(\tau)\left(\begin{matrix}\bm{S}_{T,2}(\tau)\\\
\bm{S}_{T,3}(\tau)\end{matrix}\right)\otimes\bm{I}_{d}\right)\left\\{\frac{1}{2}h^{2}\mathrm{vec}\left(\bm{A}^{(2)}(\tau)\right)\right\\}$
$\displaystyle+[\bm{I}_{d^{2}p+d},\bm{0}_{d^{2}p+d}]\left(\bm{S}_{T}^{-1}(\tau)\otimes\bm{I}_{d}\right)\left(\frac{1}{T}\sum_{t=1}^{T}(\bm{z}_{t-1}^{*}\bm{z}_{t-1}^{\top}\otimes\bm{I}_{d})\mathrm{vec}\left(\bm{M}(\tau_{t})\right)K_{h}(\tau_{t}-\tau)\right)$
$\displaystyle+[\bm{I}_{d^{2}p+d},\bm{0}_{d^{2}p+d}]\left(\bm{S}_{T}^{-1}(\tau)\otimes\bm{I}_{d}\right)\left(\frac{1}{T}\sum_{t=1}^{T}(\bm{z}_{t-1}^{*}\otimes\bm{I}_{d})\bm{\eta}_{t}K_{h}(\tau_{t}-\tau)\right)$
$\displaystyle:=$ $\displaystyle I_{T,1}+I_{T,2}+I_{T,3}.$

By standard arguments for the local linear kernel estimator and the uniform
convergence results in Lemmas B.5.2-3, we have
$\left\|I_{T,1}+I_{T,2}\right\|=O(h^{2})+O_{P}(h^{2}\sqrt{\log T/(Th)})$
uniformly over $\tau\in[0,1]$. By Lemma B.6.1, we have
$I_{T,3}=O_{P}\left((\frac{\log T}{Th})^{\frac{1}{2}}\right)$ uniformly over
$\tau\in[0,1]$. Therefore, the first result follows.

(2). We begin our investigation on the asymptotic normality by writing that
for $\forall\tau\in(0,1)$,

$\displaystyle\widehat{\bm{\Omega}}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{Th}\sum_{t=1}^{T}\bm{\widehat{\eta}}_{t}\bm{\widehat{\eta}}_{t}^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)+O_{P}\left(\frac{1}{Th}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{Th}\sum_{t=1}^{T}\left(\bm{\eta}_{t}+\bm{\widehat{\eta}}_{t}-\bm{\eta}_{t}\right)\left(\bm{\eta}_{t}+\bm{\widehat{\eta}}_{t}-\bm{\eta}_{t}\right)^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)+O_{P}\left(\frac{1}{Th}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{Th}\sum_{t=1}^{T}\bm{\eta}_{t}\bm{\eta}_{t}^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)+\frac{1}{Th}\sum_{t=1}^{T}(\bm{\widehat{\eta}}_{t}-\bm{\eta}_{t})(\bm{\widehat{\eta}}_{t}-\bm{\eta}_{t})^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle+\frac{1}{Th}\sum_{t=1}^{T}\bm{\eta}_{t}(\bm{\widehat{\eta}}_{t}-\bm{\eta}_{t})^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)+\frac{1}{Th}\sum_{t=1}^{T}(\bm{\widehat{\eta}}_{t}-\bm{\eta}_{t})\bm{\eta}_{t}^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)+O_{P}\left(\frac{1}{Th}\right)$
$\displaystyle:=$
$\displaystyle\bm{I}_{T,4}+\bm{I}_{T,5}+\bm{I}_{T,6}+\bm{I}_{T,7}+O_{P}\left(\frac{1}{Th}\right).$

Let $\rho_{T}=h^{2}+\sqrt{\frac{\log T}{Th}}$. By what we have just proved for
Theorem 2.1(i), for $\forall\tau\in[0,1]$ we have

$\displaystyle\left\|\frac{1}{Th}\sum_{t=1}^{T}(\bm{\widehat{\eta}}_{t}-\bm{\eta}_{t})(\bm{\widehat{\eta}}_{t}-\bm{\eta}_{t})^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)\right\|$
$\displaystyle\leq$
$\displaystyle\sup_{\tau_{t}\in[0,1]}\|\bm{\widehat{A}}(\tau_{t})-\bm{A}(\tau_{t})\|^{2}\cdot\frac{1}{Th}\sum_{t=1}^{T}\left\|\bm{z}_{t-1}\right\|^{2}K\left(\frac{\tau_{t}-\tau}{h}\right)=O_{P}(\rho_{T}^{2}).$

By Lemma B.6.2, $\bm{I}_{T,6}$ and $\bm{I}_{T,7}$ are both
$o_{P}((Th)^{-1/2})$. Hence,

$\displaystyle\sqrt{Th}\left(\frac{1}{Th}\sum_{t=1}^{T}\bm{\widehat{\eta}}_{t}\bm{\widehat{\eta}}_{t}^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)-\frac{1}{Th}\sum_{t=1}^{T}\bm{\eta}_{t}\bm{\eta}_{t}^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)-o_{P}(h^{4})\right)=o_{P}(1).$

Combined with the convergence results of the sample covariance matrix stated
in Lemma B.5, the above development yields that

$\displaystyle\sqrt{Th}\left[\begin{matrix}\mathrm{vec}\left(\bm{\widehat{A}}(\tau)-\bm{A}(\tau)-\frac{1}{2}h^{2}\tilde{c}_{2}\bm{A}^{(2)}(\tau)\right)+o_{P}(h^{2})\\\
\mathrm{vech}\left(\bm{\widehat{\Omega}}(\tau)-\bm{\Omega}(\tau)-\frac{1}{2}h^{2}\tilde{c}_{2}\bm{\Omega}^{(2)}(\tau)\right)+o_{P}(h^{2})\end{matrix}\right]$
$\displaystyle=$
$\displaystyle\left[\begin{matrix}\left(\bm{\Sigma}^{-1}(\tau)\otimes\bm{I}_{d}\right)\left(\frac{1}{\sqrt{Th}}\sum_{t=1}^{T}\bm{Z}_{t-1}\bm{\eta}_{t}K\left(\frac{\tau_{t}-\tau}{h}\right)\right)\\\
\frac{1}{\sqrt{Th}}\sum_{t=1}^{T}\mathrm{vech}\left(\bm{\eta}_{t}\bm{\eta}_{t}^{\top}-\bm{\Omega}(\tau_{t})\right)K\left(\frac{\tau_{t}-\tau}{h}\right)\end{matrix}\right]+o_{P}(1):=\bm{I}_{T,8}+o_{P}(1).$

Below, we focus on $\bm{I}_{T,8}$. First, we show
$\text{Var}(\bm{I}_{T,8})\to\bm{V}(\tau)$. Let
$\text{Var}(\bm{I}_{T,8})=\left[\begin{matrix}\widetilde{\bm{V}}_{1,1}(\tau)&\widetilde{\bm{V}}_{2,1}^{\top}(\tau)\\\
\widetilde{\bm{V}}_{2,1}(\tau)&\widetilde{\bm{V}}_{2,2}(\tau)\end{matrix}\right]$,
where the definition of each block should be obvious. Moreover, simple algebra
shows that $\widetilde{\bm{V}}_{i,j}(\tau)\to\bm{V}_{i,j}(\tau)$ for
$i,j\in\\{1,2\\}$.

By construction and Assumption 2, $\bm{I}_{T,8}$ is a summation of m.d.s., we
thus use Lemma B.1 and Cramér-Wold device to prove its asymptotic normality.
It suffices to show that
$\bm{d}^{\top}\bm{I}_{T,8}\to_{D}N\left(\bm{0},\bm{d}^{\top}\bm{V}(\tau)\bm{d}\right)$
for any conformable unit vector $\bm{d}$. Let

$\bm{Z}_{T,t}(\tau)=\frac{1}{\sqrt{Th}}\bm{d}^{\top}\left[\begin{matrix}\left(\bm{\Sigma}^{-1}(\tau)\otimes\bm{I}_{d}\right)\left(\bm{Z}_{t-1}\bm{\eta}_{t}K\left(\frac{\tau_{t}-\tau}{h}\right)\right)\\\
\mathrm{vech}\left(\bm{\eta}_{t}\bm{\eta}_{t}^{\top}-\bm{\Omega}(\tau_{t})\right)K\left(\frac{\tau_{t}-\tau}{h}\right)\end{matrix}\right].$

By the law of large numbers for martingale differences, we have

$\sum_{t=1}^{T}\bm{Z}_{T,t}^{2}(\tau)-\sum_{t=1}^{T}E(\bm{Z}_{T,t}^{2}(\tau)|\mathcal{F}_{t-1})\to_{P}0.$

Since conditional on $\mathcal{F}_{t-1}$ the third and fourth moments of
$\bm{\epsilon}_{t}$ are identical to the corresponding unconditional moments
a.s., by Lemma B.5.1 we can prove that
$\sum_{t=1}^{T}E(\bm{Z}_{T,t}^{2}(\tau)|\mathcal{F}_{t-1})\to_{P}\bm{d}^{\top}\bm{V}(\tau)\bm{d}$.

Furthermore, for any $\nu>0$ and $\tau\in(0,1)$, by both Holder’s and Markov’s
inequalities, we have

$\displaystyle\sum_{t=1}^{T}E\left(\left(\bm{Z}_{T,t}(\tau)\right)^{2}I\left(|\bm{Z}_{T,t}(\tau)|>\nu\right)\right)$
$\displaystyle\leq$
$\displaystyle\sum_{t=1}^{T}\left[E|\bm{Z}_{T,t}(\tau)|^{\delta/2}\right]^{4/\delta}\left[\frac{E|\bm{Z}_{T,t}(\tau)|^{\delta/2}|}{\nu^{\delta/2}}\right]^{(\delta-4)/\delta}$
$\displaystyle=$ $\displaystyle O((Th)^{(\delta-4)/4})=o(1).$

Thus, the CLT follows.

Finally, we consider $\widehat{\bm{V}}(\cdot)$. By Lemma B.5 and the above
proof, we have $\widehat{\bm{V}}_{1,1}(\tau)\to_{P}\bm{V}_{1,1}(\tau)$. By the
uniform convergence results of $\widehat{\bm{A}}(\tau)$, we can replace
$\widehat{\bm{\eta}}_{t}$ with $\bm{\eta}_{t}$ in the following derivations.
Therefore, we have

$\displaystyle\widehat{\bm{V}}_{2,1}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{Th}\sum_{t=1}^{T}\mathrm{vech}\left(\bm{\eta}_{t}\bm{\eta}_{t}^{\top}\right)\bm{\eta}_{t}^{\top}\bm{Z}_{t-1}^{\top}K^{2}\left(\frac{\tau_{t}-\tau}{h}\right)\left(\bm{\Sigma}^{-1}(\tau)\otimes\bm{I}_{d}\right)+o_{P}(1)\to_{P}\bm{V}_{2,1}(\tau),$

and

$\displaystyle\widehat{\bm{V}}_{2,2}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{Th}\sum_{t=1}^{T}\mathrm{vech}(\bm{\eta}_{t}\bm{\eta}_{t}^{\top})\mathrm{vech}(\bm{\eta}_{t}\bm{\eta}_{t}^{\top})^{\top}K^{2}\left(\frac{\tau_{t}-\tau}{h}\right)-\widetilde{v}_{0}\mathrm{vech}\left(\bm{\Omega}(\tau)\right)\mathrm{vech}\left(\bm{\Omega}(\tau)\right)^{\top}+o_{P}(1)$
$\displaystyle\to_{P}$ $\displaystyle\bm{V}_{2,2}(\tau).$

The proof is now completed. ∎

###### Proof of Theorem 2.2.

Let $\bm{A}(\bm{\theta})$ be a real, differentiable, $m\times n$ matrix
function of real $p\times 1$ vector $\bm{\theta}$. Define
$\triangledown_{\bm{\theta}}\bm{A}=\frac{\partial\mathrm{vec}(\bm{A})}{\partial\bm{\theta}^{\top}}$,
and thus
$\mathrm{vec}(\mathrm{d}\bm{A})=\triangledown_{\bm{\theta}}\bm{A}\mathrm{d}\bm{\theta}$.

Let
$\bm{\alpha}(\tau)=\mathrm{vec}\left(\bm{A}_{1}(\tau),...,\bm{A}_{p}(\tau)\right)$,
$\bm{\sigma}(\tau)=\mathrm{vech}\left(\bm{\Omega}(\tau)\right)$ and
$\bm{\phi}(\tau)=[\bm{\alpha}^{\top}(\tau),\bm{\sigma}^{\top}(\tau)]^{\top}$.
Given the joint distribution of $\mathrm{vec}(\bm{\widehat{A}}(\tau))$ and
$\mathrm{vech}(\bm{\widehat{\Omega}}(\tau))$ in Theorem 2.1, Theorem 2.2 can
be obtained by the Delta method. By the first-order approximation of
$\mathrm{vec}\left(\bm{\widehat{B}}_{j}(\tau)\right)$ around
$\mathrm{vec}\left(\bm{B}_{j}(\tau)\right)$, we have

$\sqrt{Th}\mathrm{vec}\left(\bm{\widehat{B}}_{j}(\tau)-\bm{B}_{j}(\tau)\right)\simeq\triangledown_{\bm{\phi}(\tau)}\bm{B}_{j}(\tau)\sqrt{Th}\left(\widehat{\bm{\phi}}(\tau)-\bm{\phi}(\tau)\right)$

and thus

$\displaystyle\sqrt{Th}\left(\mathrm{vec}\left(\bm{\widehat{B}}_{j}(\tau)-\bm{B}_{j}(\tau)\right)-\frac{1}{2}h^{2}\tilde{c}_{2}\bm{B}_{j}^{(2)}(\tau)+o_{P}(h^{2})\right)\to_{D}N\left(0,\bm{\Sigma}_{\bm{B}_{j}}(\tau)\right),$

where $\bm{B}_{j}^{(2)}(\tau)$ and $\bm{\Sigma}_{\bm{B}_{j}}(\tau)$ have been
defined in the body of the theorem.

To complete the proof, we first derive an analytic form for the derivative
$\triangledown_{\bm{\phi}(\tau)}\bm{B}_{j}(\tau)$ under each of the
identification restrictions.

We have two sets of restrictions: (a) $d(d+1)/2$ restrictions implied by
$\bm{\Omega}(\tau)=\bm{\omega}(\tau)\bm{\omega}^{\top}(\tau)$ and (b)
additional $d(d-1)/2$ structural restrictions based on short-run or long-run
restrictions.

Consider type (a) restrictions. We begin by considering
$\mathrm{d}\bm{\Omega}(\tau)=\mathrm{d}\bm{\omega}(\tau)\cdot\bm{\omega}^{\top}(\tau)+\bm{\omega}(\tau)\cdot\mathrm{d}\bm{\omega}^{\top}(\tau)$.
Let $\bm{B}$ and $\bm{C}$ be $n\times q$ and $q\times r$ matrices,
respectively. By
$\mathrm{vec}(\bm{A}\bm{B}\bm{C})=\bm{C}^{\top}\otimes\bm{A}\mathrm{vec}(\bm{B})$,
$\mathrm{vec}(\bm{A}^{\top})=\bm{K}_{m,n}\mathrm{vec}(\bm{A})$ and
$\bm{K}_{m,q}(\bm{A}\otimes\bm{C})=(\bm{C}\otimes\bm{A})\bm{K}_{n,r}$, we have
$\bm{N}_{1}(\tau)\mathrm{vec}(\mathrm{d}\bm{\omega}(\tau))=\mathrm{vec}(\mathrm{d}\bm{\Omega}(\tau))$,
where
$\bm{N}_{1}(\tau)=(\bm{I}_{d^{2}}+\bm{K}_{d,d})(\bm{\omega}(\tau)\otimes\bm{I}_{d})$.
Let $\bm{D}_{1}$ be the duplication matrix such that
$\mathrm{vec}[\bm{\Omega}(\tau)]=\bm{D}_{1}\mathrm{vech}[\bm{\Omega}(\tau)]$,
which follows that
$\bm{N}_{1}(\tau)\mathrm{vec}(\mathrm{d}\bm{\omega}(\tau))=\bm{D}_{1}\mathrm{d}\bm{\sigma}(\tau)$
and

$\bm{N}_{1}(\tau)\triangledown_{\bm{\sigma}(\tau)}\bm{\omega}(\tau)=\bm{D}_{1}.$
(A.1)

We then illustrate how to combine equation (A.1) with gradient equations from
type (b) restrictions in order to compute
$\triangledown_{\bm{\phi}(\tau)}\bm{\omega}(\tau)$.

In the case of short-run timing restrictions, because types (a) and (b)
restrictions do not involve $\bm{\alpha}$,
$\triangledown_{\bm{\phi}(\tau)}\bm{\omega}(\tau)$ has the form
$[\bm{0},\triangledown_{\bm{\sigma}(\tau)}\bm{\omega}(\tau)]$. Let
$\bm{L}_{d}$ be the elimination matrix defined by
$\mathrm{vech}[\bm{\omega}(\tau)]=\bm{L}_{d}\mathrm{vec}[\bm{\omega}(\tau)]$.
Because $\bm{\omega}(\tau)$ is lower triangular subject to short-run
restrictions, $\bm{L}^{\top}$ is a duplication matrix such that
$\mathrm{vec}[\bm{\omega}(\tau)]=\bm{L}_{d}^{\top}\mathrm{vech}[\bm{\omega}(\tau)]$.
Write

$\displaystyle\bm{N}_{1}(\tau)\mathrm{vec}(\mathrm{d}\bm{\omega}(\tau))$
$\displaystyle=$ $\displaystyle\bm{D}_{1}\mathrm{d}\bm{\sigma}(\tau),$
$\displaystyle\bm{L}_{d}\bm{N}_{1}(\tau)\bm{L}_{d}^{\top}\mathrm{vech}(\mathrm{d}\bm{\omega}(\tau))$
$\displaystyle=$
$\displaystyle\bm{L}_{d}\bm{D}_{1}\mathrm{d}\bm{\sigma}(\tau)=\mathrm{d}\bm{\sigma}(\tau),$
$\displaystyle\mathrm{vech}(\mathrm{d}\bm{\omega}(\tau))$ $\displaystyle=$
$\displaystyle\left(\bm{L}_{d}\bm{N}_{1}(\tau)\bm{L}_{d}^{\top}\right)^{-1}\mathrm{d}\bm{\sigma}(\tau),$
$\displaystyle\mathrm{vec}(\mathrm{d}\bm{\omega}(\tau))$ $\displaystyle=$
$\displaystyle\bm{L}_{d}^{\top}\left(\bm{L}_{d}\bm{N}_{1}(\tau)\bm{L}_{d}^{\top}\right)^{-1}\mathrm{d}\bm{\sigma}(\tau).$

Hence,
$\triangledown_{\bm{\sigma}(\tau)}\bm{\omega}(\tau)=\bm{L}_{d}^{\top}\left(\bm{L}_{d}\bm{N}_{1}(\tau)\bm{L}_{d}^{\top}\right)^{-1}$.
Recall that
$\triangledown_{\bm{\phi}(\tau)}\bm{B}_{j}(\tau)=[\triangledown_{\bm{\alpha}(\tau)}\bm{B}_{j}(\tau),\triangledown_{\bm{\sigma}(\tau)}\bm{B}_{j}(\tau)]$.
For $\triangledown_{\bm{\alpha}(\tau)}\bm{B}_{j}(\tau)$,

$\displaystyle\triangledown_{\bm{\alpha}(\tau)}\bm{B}_{j}(\tau)$
$\displaystyle=$
$\displaystyle\frac{\partial\mathrm{vec}\left[\bm{\Psi}_{j}(\tau)\bm{\omega}(\tau)\right]}{\partial\bm{\alpha}^{\top}(\tau)}=(\bm{\omega}^{\top}(\tau)\otimes\bm{I}_{d})\frac{\partial\mathrm{vec}\left[\bm{\Psi}_{j}(\tau)\right]}{\partial\bm{\alpha}^{\top}(\tau)}$
$\displaystyle=$
$\displaystyle(\bm{\omega}^{\top}(\tau)\otimes\bm{I}_{d})\frac{\partial\mathrm{vec}\left[\bm{J}\bm{\Phi}^{j}(\tau)\bm{J}^{\top}\right]}{\partial\bm{\alpha}^{\top}(\tau)}$
$\displaystyle=$
$\displaystyle(\bm{\omega}^{\top}(\tau)\otimes\bm{I}_{d})\left(\sum_{m=0}^{j-1}\bm{J}(\bm{\Phi}^{\top}(\tau))^{j-1-m}\otimes\bm{\Psi}_{m}(\tau)\right).$

For $\triangledown_{\bm{\sigma}(\tau)}\bm{B}_{j}(\tau)$,

$\displaystyle\triangledown_{\bm{\sigma}(\tau)}\bm{B}_{j}(\tau)$
$\displaystyle=$
$\displaystyle\frac{\partial\mathrm{vec}\left[\bm{\Psi}_{j}(\tau)\bm{\omega}(\tau)\right]}{\partial\bm{\sigma}^{\top}(\tau)}=(\bm{I}_{d}\otimes\bm{\Psi}_{j}(\tau))\frac{\partial\mathrm{vec}\left[\bm{\omega}(\tau)\right]}{\partial\bm{\sigma}^{\top}(\tau)}$
$\displaystyle=$
$\displaystyle(\bm{I}_{d}\otimes\bm{\Psi}_{j}(\tau))\bm{L}_{d}^{\top}\left(\bm{L}_{d}\bm{N}_{1}(\tau)\bm{L}_{d}^{\top}\right)^{-1}.$

In the case of long-run restrictions, type (b) restrictions involve
$\bm{\alpha}(\tau)$, so that
$\triangledown_{\bm{\phi}(\tau)}\bm{\omega}(\tau)$ has the form
$[\triangledown_{\bm{\alpha}(\tau)}\bm{\omega}(\tau),\triangledown_{\bm{\sigma}(\tau)}\bm{\omega}(\tau)]$.
First, equation (A.1) must be extended in the form
$\bm{N}_{1}\triangledown_{\bm{\phi}(\tau)}\bm{\omega}(\tau)=[\bm{0},\bm{D}_{1}]$.
Second, long-run restrictions can be expressed as
$\bm{Q}\mathrm{vec}\left[\bm{A}_{\tau}^{-1}(1)\bm{\omega}\right]=0$, where
$\bm{Q}$ is a $d(d-1)/2\times d^{2}$ matrix of $0$ and $1$, and
$\bm{A}_{\tau}(1)=\bm{I}_{d}-\sum_{i=1}^{p}\bm{A}_{i}(\tau)$. By
$\mathrm{d}\bm{A}^{-1}=-\bm{A}^{-1}\cdot\mathrm{d}\bm{A}\cdot\bm{A}^{-1}$, we
have

$\displaystyle\bm{Q}\mathrm{vec}\left[\bm{A}_{\tau}^{-1}(1)\bm{\omega}\right]=0$
$\displaystyle\bm{Q}\mathrm{vec}\left[\mathrm{d}(\bm{A}_{\tau}^{-1}(1))\bm{\omega}+\bm{A}_{\tau}^{-1}(1)\mathrm{d}\bm{\omega}\right]=0$
$\displaystyle\bm{Q}\mathrm{vec}\left[-\bm{A}_{\tau}^{-1}(1)\mathrm{d}(\bm{A}_{\tau}(1))\bm{A}_{\tau}^{-1}(1)\bm{\omega}+\bm{A}_{\tau}^{-1}(1)\mathrm{d}\bm{\omega}\right]=0$
$\displaystyle\bm{Q}\left[\bm{I}_{d}\otimes\bm{A}_{\tau}^{-1}(1)\right]\mathrm{vec}\left[\mathrm{d}\bm{\omega}\right]=\bm{Q}[\bm{B}^{\top}(\tau)\otimes\bm{A}_{\tau}^{-1}(1)]\mathrm{vec}[\mathrm{d}\bm{A}_{\tau}(1)]$
$\displaystyle\bm{N}_{2}(\tau)\triangledown_{\bm{\phi}(\tau)}\bm{\omega}(\tau)=[\bm{D}_{2}(\tau),\bm{0}],$

where
$\bm{N}_{2}(\tau)=\bm{Q}\left[\bm{I}_{d}\otimes\bm{A}_{\tau}^{-1}(1)\right]$
and
$\bm{D}_{2}(\tau)=\bm{Q}[\bm{B}^{\top}(\tau)\otimes\bm{A}_{\tau}^{-1}(1)]\triangledown_{\bm{\alpha}(\tau)}\bm{A}_{\tau}(1)$
with
$\triangledown_{\bm{\alpha}(\tau)}\bm{A}_{\tau}(1)=-[\bm{I}_{d^{2}},...,\bm{I}_{d^{2}}]$
$(d^{2}\times d^{2}p)$. Hence,

$\displaystyle\triangledown_{\bm{\phi}(\tau)}\bm{\omega}(\tau)$
$\displaystyle=$
$\displaystyle[\triangledown_{\bm{\alpha}(\tau)}\bm{\omega}(\tau),\triangledown_{\bm{\sigma}(\tau)}\bm{\omega}(\tau)]$
$\displaystyle=$
$\displaystyle\left[\left(\bm{N}_{1}^{\top}(\tau),\bm{N}_{2}^{\top}(\tau)\right)\left(\begin{matrix}\bm{N}_{1}(\tau)\\\
\bm{N}_{2}(\tau)\end{matrix}\right)\right]^{-1}\left[\bm{N}_{1}^{\top}(\tau),\bm{N}_{2}^{\top}(\tau)\right]\left[\begin{matrix}\bm{0}&\bm{D}_{1}\\\
\bm{D}_{2}(\tau)&\bm{0}\end{matrix}\right]$ $\displaystyle=$
$\displaystyle\left(\bm{N}_{1}^{\top}(\tau)\bm{N}_{1}(\tau)+\bm{N}_{2}^{\top}(\tau)\bm{N}_{2}(\tau)\right)^{-1}\left[\bm{N}_{2}^{\top}(\tau)\bm{D}_{2}(\tau),\bm{N}_{1}^{\top}(\tau)\bm{D}_{1}\right].$

For $\triangledown_{\bm{\alpha}(\tau)}\bm{B}_{j}(\tau)$,

$\displaystyle\triangledown_{\bm{\alpha}(\tau)}\bm{B}_{j}(\tau)$
$\displaystyle=$
$\displaystyle\frac{\partial\mathrm{vec}\left[\bm{\Psi}_{j}(\tau)\bm{\omega}(\tau)\right]}{\partial\bm{\alpha}^{\top}(\tau)}$
$\displaystyle=$
$\displaystyle(\bm{\omega}^{\top}(\tau)\otimes\bm{I}_{d})\frac{\partial\mathrm{vec}\left[\bm{\Psi}_{j}(\tau)\right]}{\partial\bm{\alpha}^{\top}(\tau)}+(\bm{I}_{d}\otimes\bm{\Psi}_{j}(\tau))\frac{\partial\mathrm{vec}\left[\bm{\omega}(\tau)\right]}{\partial\bm{\alpha}^{\top}(\tau)}$
$\displaystyle=$
$\displaystyle(\bm{\omega}^{\top}(\tau)\otimes\bm{I}_{d})\left(\sum_{m=0}^{j-1}\bm{J}(\bm{\Phi}^{\top}(\tau))^{j-1-m}\otimes\bm{\Psi}_{m}(\tau)\right)$
$\displaystyle+(\bm{I}_{d}\otimes\bm{\Psi}_{j}(\tau))\left(\bm{N}_{1}^{\top}(\tau)\bm{N}_{1}(\tau)+\bm{N}_{2}^{\top}(\tau)\bm{N}_{2}(\tau)\right)^{-1}\bm{N}_{2}^{\top}(\tau)\bm{D}_{2}(\tau).$

For $\triangledown_{\bm{\sigma}(\tau)}\bm{B}_{j}(\tau)$,

$\displaystyle\triangledown_{\bm{\sigma}(\tau)}\bm{B}_{j}(\tau)$
$\displaystyle=$
$\displaystyle\frac{\partial\mathrm{vec}\left[\bm{\Psi}_{j}(\tau)\bm{\omega}(\tau)\right]}{\partial\bm{\sigma}^{\top}(\tau)}=(\bm{I}_{d}\otimes\bm{\Psi}_{j}(\tau))\frac{\partial\mathrm{vec}\left[\bm{\omega}(\tau)\right]}{\partial\bm{\sigma}^{\top}(\tau)}$
$\displaystyle=$
$\displaystyle(\bm{I}_{d}\otimes\bm{\Psi}_{j}(\tau))\bm{L}_{d}^{\top}\left(\bm{N}_{1}^{\top}(\tau)\bm{N}_{1}(\tau)+\bm{N}_{2}^{\top}(\tau)\bm{N}_{2}(\tau)\right)^{-1}\bm{N}_{1}^{\top}(\tau)\bm{D}_{1}.$

The proof is now completed. ∎

###### Proof of Theorem 2.3.

We need to prove that
$\lim_{T\to\infty}\Pr\left(\text{IC}(\mathsf{p})<\text{IC}(p)\right)=0$ for
all $\mathsf{p}\neq p$ and $\mathsf{p}\leq\mathsf{P}$.

Note that

$\text{IC}(\mathsf{p})-\text{IC}(p)=\log[\text{RSS}(\mathsf{p})/\text{RSS}(p)]+(\mathsf{p}-p)\chi_{T}.$

(i) For $\mathsf{p}<p$, Lemma B.7 implies that
$\text{RSS}(\mathsf{p})/\text{RSS}(p)>1+\nu$ for some $\nu>0$ with large
probability for all large $T$. Thus,
$\log[\text{RSS}(\mathsf{p})/\text{RSS}(p)]\geq\nu/2$ for large $T$. Because
$\chi_{T}\to 0$, we have
$\text{IC}(\mathsf{p})-\text{IC}(p)\geq\nu/2-(p-\mathsf{p})\chi_{T}\geq\nu/3$
for large $T$ with large probability. Thus
$\Pr\left(\text{IC}(\mathsf{p})<\text{IC}(p)\right)\to 0$ for $\mathsf{p}<p$.

(ii) We then consider $\mathsf{p}>p$. Lemma B.7 implies that
$\log[\text{RSS}(\mathsf{p})/\text{RSS}(p)]=1+O_{P}(\rho_{T}^{2})$ with
$\rho_{T}=h^{2}+\sqrt{\frac{\log(T)}{Th}}$. Hence,
$\log[\text{RSS}(\mathsf{p})/\text{RSS}(p)]=O_{P}(\rho_{T}^{2})$. Because
$(\mathsf{p}-p)\chi_{T}\geq\chi_{T}$, which converges to zero at a slower rate
than $\rho_{T}^{2}$, it follows that

$\Pr\left(\text{IC}(\mathsf{p})<\text{IC}(p)\right)\leq\Pr\left(O_{P}(\rho_{T}^{2})+\chi_{T}<0\right)\to
0.$

The proof is now completed. ∎

###### Proof of Lemma 3.1.

By part(1) of Theorem 2.1, we have
$\sup_{\tau\in[0,1]}\|\widehat{\bm{\beta}}(\tau)-\bm{\beta}(\tau)\|=O_{P}(h^{2}+\sqrt{\log
T/(Th)})$. Hence, by $Th^{6}\to 0$, we have

$\sqrt{T}\left(\widehat{\bm{c}}-\bm{c}\right)=\sqrt{T}\int_{h}^{1-h}\bm{C}\left(\widehat{\bm{\beta}}(\tau)-\bm{\beta}(\tau)\right)\mathrm{d}\tau+o_{P}(1).$

In addition, by the proof of part(1) of Theorem 2.1, the uniform convergence
results of Lemmas B.5-B.6 and the condition that $\bm{\beta}(\tau)$ has third-
order derivative, we have

$\displaystyle\widehat{\bm{\beta}}(\tau)-\bm{\beta}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{2}h^{2}\widetilde{c}_{2}\bm{\beta}^{(2)}(\tau)+\bm{\Sigma}_{\bm{Z}}^{-1}(\tau)\left(\frac{1}{Th}\sum_{t=1}^{T}\bm{Z}_{t-1}\bm{\eta}_{t}K\left(\frac{\tau_{t}-\tau}{h}\right)\right)$
$\displaystyle+O_{P}(h^{2}\sqrt{\log T/(Th)})+O_{P}(h^{3})+O_{P}(\log T/(Th))$

uniformly over $\tau\in[h,1-h]$.

As $Th^{6}\to 0$ and $Th^{2}/(\log T)^{2}\to\infty$, we have

$\displaystyle\sqrt{T}\left(\widehat{\bm{c}}-\bm{c}-\frac{1}{2}h^{2}\widetilde{c}_{2}\int_{0}^{1}\bm{C}\bm{\beta}^{(2)}(\tau)\mathrm{d}\tau\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\bm{C}\left(\int_{-1}^{1}\bm{\Sigma}_{\bm{Z}}^{-1}(\tau)K\left(\frac{\tau_{t}-\tau}{h}\right)\mathrm{d}\tau\right)\bm{Z}_{t-1}\bm{\eta_{t}}+o_{P}(1)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\bm{\Sigma}_{\bm{Z}}^{-1}(\tau_{t})\bm{Z}_{t-1}\bm{\eta_{t}}+o_{P}(1)$
$\displaystyle\to_{D}$ $\displaystyle
N\left(\bm{0},\int_{0}^{1}\bm{C}\left(\bm{\Sigma}^{-1}(\tau)\otimes\bm{\Omega}(\tau)\right)\bm{C}^{\top}\mathrm{d}\tau\right)$

by Lemma B.1 under the conditions of this lemma.

The proof is now completed. ∎

###### Proof of Theorem 3.1.

First, we introduce some additional notation to facilitate the development.
Let
$A_{Q}=\widetilde{v}_{0}\cdot\mathrm{tr}\left\\{\int_{0}^{1}\bm{\Sigma}_{Q}(\tau)\mathrm{d}\tau\right\\}$
and
$B_{Q}=4C_{B}\cdot\mathrm{tr}\left\\{\int_{0}^{1}\bm{\Sigma}_{Q}(\tau)^{2}\mathrm{d}\tau\right\\}$,
where
$\bm{\Sigma}_{Q}(\tau)=\bm{H}(\tau)^{1/2}\bm{C}\bm{V}_{\bm{\beta}}(\tau)\bm{C}^{\top}\bm{H}(\tau)^{1/2}$.
Recall $\rho_{T}=h^{2}+\sqrt{\frac{\log T}{Th}}$. By Theorem 2.1.1, we have

$\displaystyle\sup_{\tau\in[0,1]}\left\|\widehat{\bm{\beta}}(\tau)-\bm{\beta}(\tau)\right\|=O_{P}\left(\rho_{T}\right).$
(A.2)

Then we can conclude that as $Th^{11/2}=o(1)$,

$\displaystyle\int_{\mathcal{B}_{T}}\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{c}\right]^{\top}\bm{H}(\tau)\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{c}\right]\mathrm{d}\tau=O_{P}\left(\log
T/T+h^{5}\right)=o_{P}\left(T^{-1}h^{-1/2}\right),$ (A.3)

where $\mathcal{B}_{T}=[0,h]\cup[1-h,1]$.

In addition, by Lemma B.5 and Lemma B.6.1, we have

$\displaystyle\sup_{[h,1-h]}\left\|\bm{S}_{T}(\tau)-\bm{\Sigma}_{\bm{Z}}(\tau)\otimes\bm{\Lambda}_{1}\right\|=O_{P}\left(h^{2}+\sqrt{\frac{\log
T}{Th}}\right),$
$\displaystyle\sup_{[0,1]}\left\|\bm{R}_{T}(\tau)\right\|=O_{P}\left(\sqrt{\frac{\log
T}{Th}}\right),$ (A.4)

where $\bm{\Sigma}_{\bm{Z}}(\tau)=\bm{\Sigma}(\tau)\otimes\bm{I}_{d}$,
$\bm{\Lambda}_{1}=\mathrm{diag}(\widetilde{c}_{0},\widetilde{c}_{2})$ and
$\bm{R}_{T}(\tau)=\frac{1}{T}\sum_{t=1}^{T}\bm{Z}_{t-1}\bm{\eta}_{t}K_{h}(\tau_{t}-\tau)$.
Hence, under the null hypothesis, we have

$\displaystyle\sup_{[h,1-h]}\left\|\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{c}-\bm{C}\bm{\Sigma}_{\bm{Z}}^{-1}(\tau)\bm{R}_{T}(\tau)\right\|=O_{P}(\rho_{T}^{2}).$
(A.5)

By (A.2) and $Th^{11/2}\to 0$, we have

$\displaystyle\int_{0}^{1}\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{c}\right]^{\top}\bm{H}(\tau)\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{c}\right]\mathrm{d}\tau$
$\displaystyle=$
$\displaystyle\int_{h}^{1-h}\bm{R}_{T}^{\top}(\tau)\bm{H}_{0}(\tau)\bm{R}_{T}(\tau)\mathrm{d}\tau+O_{P}\left(\log
T/T+h^{5}\right)+O_{P}\left(\rho_{T}^{2}\sqrt{\frac{\log T}{Th}}\right)$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\bm{R}_{T}^{\top}(\tau)\bm{H}_{0}(\tau)\bm{R}_{T}(\tau)\mathrm{d}\tau+o_{P}(T^{-1}h^{-1/2}),$

where
$\bm{H}_{0}(\tau)=\bm{\Sigma}_{\bm{Z}}^{-1}(\tau)\bm{C}^{\top}\bm{H}(\tau)\bm{C}\bm{\Sigma}_{\bm{Z}}^{-1}(\tau)$.

Consider
$\int_{0}^{1}\bm{R}_{T}^{\top}(\tau)\bm{H}_{0}(\tau)\bm{R}_{T}(\tau)\mathrm{d}\tau$,
and write

$\displaystyle\int_{0}^{1}\bm{R}_{T}^{\top}(\tau)\bm{H}_{0}(\tau)\bm{R}_{T}(\tau)\mathrm{d}\tau=\frac{1}{T^{2}h^{2}}\sum_{t=1}^{T}\bm{\eta}_{t}^{\top}\bm{Z}_{t-1}^{\top}\left\\{\int_{0}^{1}\bm{H}_{0}(\tau)K^{2}\left(\frac{\tau-\tau_{t}}{h}\right)\mathrm{d}\tau\right\\}\bm{Z}_{t-1}\bm{\eta}_{t}$
$\displaystyle+\frac{1}{T^{2}h^{2}}\sum_{t=1}^{T}\sum_{s=1,\neq
t}^{T}\bm{\eta}_{t}^{\top}\bm{Z}_{t-1}^{\top}\left\\{\int_{0}^{1}\bm{H}_{0}(\tau)K\left(\frac{\tau-\tau_{t}}{h}\right)K\left(\frac{\tau-\tau_{s}}{h}\right)\mathrm{d}\tau\right\\}\bm{Z}_{s-1}\bm{\eta}_{s}$
$\displaystyle:=$ $\displaystyle I_{T,1}+I_{T,2},$

where the definitions of $I_{T,1}$ and $I_{T,2}$ should be obvious.

For $I_{T,1}$, simple algebra shows that

$\displaystyle\bm{\eta}_{t}^{\top}\bm{Z}_{t-1}^{\top}\bm{\Sigma}_{\bm{Z}}^{-1}(\tau)\bm{C}^{\top}\bm{H}(\tau)\bm{C}\bm{\Sigma}_{\bm{Z}}^{-1}(\tau)\bm{Z}_{t-1}\bm{\eta}_{t}$
$\displaystyle=$
$\displaystyle\mathrm{tr}\left\\{\left[\left(\bm{\Sigma}^{-1}(\tau)\bm{z}_{t-1}\bm{z}_{t-1}^{\top}\bm{\Sigma}^{-1}(\tau)\right)\otimes\bm{\eta}_{t}\bm{\eta}_{t}^{\top}\right]\cdot\bm{C}^{\top}\bm{H}(\tau)\bm{C}\right\\}.$

Then we have

$\displaystyle I_{T,1}$ $\displaystyle=$
$\displaystyle\widetilde{v}_{0}\frac{1}{T^{2}h}\sum_{t=1}^{T}\bm{\eta}_{t}^{\top}\bm{Z}_{t-1}^{\top}\left[\bm{H}_{0}(\tau_{t})+O(h)\right]\bm{Z}_{t-1}\bm{\eta}_{t}$
$\displaystyle=$
$\displaystyle\widetilde{v}_{0}\frac{1}{T^{2}h}\sum_{t=1}^{T}\mathrm{tr}\left\\{\left[\left(\bm{\Sigma}^{-1}(\tau_{t})\bm{z}_{t-1}\bm{z}_{t-1}^{\top}\bm{\Sigma}^{-1}(\tau_{t})\right)\otimes\bm{\eta}_{t}\bm{\eta}_{t}^{\top}\right]\cdot\bm{C}^{\top}\bm{H}(\tau_{t})\bm{C}\right\\}+O_{P}(T^{-1})$
$\displaystyle=$
$\displaystyle\widetilde{v}_{0}\frac{1}{T^{2}h}\sum_{t=1}^{T}\mathrm{tr}\left\\{\left[\bm{\Sigma}^{-1}(\tau_{t})\otimes\bm{\Omega}(\tau_{t})\right]\cdot\bm{C}^{\top}\bm{H}(\tau_{t})\bm{C}\right\\}+O_{P}(T^{-1}+T^{-3/2}h^{-1})$
$\displaystyle\to_{P}$ $\displaystyle(Th)^{-1}A_{Q}.$

Consider $I_{T,2}$. Let
$w_{s,t}=\frac{1}{T\sqrt{h}}\int_{-1}^{1}K\left(u\right)K\left(u+\frac{t-s}{Th}\right)\mathrm{d}u$.
Since

$\int_{0}^{1}\bm{H}_{0}(\tau)K\left(\frac{\tau-\tau_{t}}{h}\right)K\left(\frac{\tau-\tau_{s}}{h}\right)\mathrm{d}\tau=h\int_{-1}^{1}\bm{H}_{0}(\tau_{t}+uh)K\left(u\right)K\left(u+\frac{t-s}{Th}\right)du,$

we have

$T\sqrt{h}I_{T,2}=2\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{\eta}_{t}^{\top}\bm{Z}_{t-1}^{\top}\bm{H}_{0}(\tau_{t})\bm{Z}_{s-1}\bm{\eta}_{s}w_{s,t}(1+o(1))=2\widetilde{U}+o_{P}(1),$

where the definition of $\widetilde{U}$ is obvious. By Lemma B.12, we have

$\widetilde{U}\to_{D}N\left(0,\sigma_{\widetilde{U}}^{2}\right),$

where
$\sigma_{\widetilde{U}}^{2}=\lim_{T\to\infty}\sum_{t=2}^{T}\mathrm{tr}\left\\{E\left(\bm{H}_{0}^{\top}(\tau_{t})\bm{Z}_{t-1}\bm{\eta}_{t}\bm{\eta}_{t}^{\top}\bm{Z}_{t-1}^{\top}\bm{H}_{0}(\tau_{t})\right)E\left(\sum_{s=1}^{t-1}\bm{Z}_{s-1}\bm{\eta}_{s}\bm{\eta}_{s}^{\top}\bm{Z}_{s-1}^{\top}\right)w_{s,t}^{2}\right\\}$.

We then show that
$\sigma_{\widetilde{U}}^{2}=C_{B}\mathrm{tr}\left\\{\int_{0}^{1}\bm{\Sigma}_{Q}(\tau)^{2}\mathrm{d}\tau\right\\}$.
Let $\bm{V}_{1}(\tau)=\bm{H}_{0}(\tau)\bm{V}_{2}(\tau)\bm{H}_{0}(\tau)$ and
$\bm{V}_{2}(\tau)=\bm{\Sigma}(\tau)\otimes\bm{\Omega}(\tau)$. Write

$\displaystyle\sum_{t=2}^{T}E\left(\bm{H}_{0}^{\top}(\tau_{t})\bm{Z}_{t-1}\bm{\eta}_{t}\bm{\eta}_{t}^{\top}\bm{Z}_{t-1}^{\top}\bm{H}_{0}(\tau_{t})\right)E\left(\sum_{s=1}^{t-1}\bm{Z}_{s-1}\bm{\eta}_{s}\bm{\eta}_{s}^{\top}\bm{Z}_{s-1}^{\top}\right)w_{s,t}^{2}$
$\displaystyle=$
$\displaystyle\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{V}_{1}(\tau_{t})\bm{V}_{2}(\tau_{s})w_{s,t}^{2}=\frac{1}{T^{2}h}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{V}_{1}(\tau_{t})\bm{V}_{2}(\tau_{s})\left[\int_{-1}^{1}K\left(u\right)K\left(u+\frac{t-s}{Th}\right)\mathrm{d}u\right]^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{T^{2}h}\sum_{s=1}^{T-1}\sum_{j=1}^{T-s}\bm{V}_{1}(\tau_{s}+j/T)\bm{V}_{2}(\tau_{s})\left[\int_{-1}^{1}K\left(u\right)K\left(u+\frac{t-s}{Th}\right)\mathrm{d}u\right]^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{T^{2}h}\sum_{s=1}^{T-1}\sum_{j=1}^{T-s}\bm{V}_{1}(\tau_{s}+j/T)\bm{V}_{2}(\tau_{s})\left[\int_{-1}^{1}K\left(u\right)K\left(u+\frac{j}{Th}\right)\mathrm{d}u\right]^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{T^{2}h}\sum_{s=1}^{T-1}\sum_{j=1}^{T-s}\bm{V}_{1}(\tau_{s})\bm{V}_{2}(\tau_{s})\left[\int_{-1}^{1}K\left(u\right)K\left(u+\frac{j}{Th}\right)\mathrm{d}u\right]^{2}$
$\displaystyle+\frac{1}{T^{2}h}\sum_{s=1}^{T-1}\sum_{j=1}^{T-s}O(j/T)\bm{V}_{2}(\tau_{s})\left[\int_{-1}^{1}K\left(u\right)K\left(u+\frac{j}{Th}\right)\mathrm{d}u\right]^{2}:=I_{T,3}+I_{T,4},$

where the definitions of $I_{T,3}$ and $I_{T,4}$ are obvious.

It is easy to verify $\mathrm{tr}\left\\{I_{T,3}\right\\}\to
C_{B}\mathrm{tr}\left\\{\int_{0}^{1}\bm{\Sigma}_{Q}(\tau)^{2}\mathrm{d}\tau\right\\}$.
For $I_{T,4}$, we have

$\displaystyle\left\|I_{T,4}\right\|$ $\displaystyle\leq$ $\displaystyle
M\frac{1}{Th}\sum_{j=1}^{T}j/T\left[\int_{-1}^{1}K\left(u\right)K\left(u+\frac{j}{Th}\right)\mathrm{d}u\right]^{2}$
$\displaystyle\simeq$ $\displaystyle
M\int_{0}^{2}vh\left[\int_{-1}^{1}K\left(u\right)K\left(u+v\right)\mathrm{d}u\right]^{2}\mathrm{d}v=o(1).$

Combining the above results, we have proved

$\displaystyle
T\sqrt{h}\left[\int_{0}^{1}\bm{R}_{T}^{\top}(\tau)\bm{H}_{0}(\tau)\bm{R}_{T}(\tau)\mathrm{d}\tau-(Th)^{-1}A_{Q}\right]\to_{D}N\left(0,B_{Q}\right).$

Note that

$\displaystyle\int_{0}^{1}\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right]^{\top}\bm{H}(\tau)\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right]\mathrm{d}\tau-\int_{0}^{1}\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{c}\right]^{\top}\bm{H}(\tau)\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{c}\right]\mathrm{d}\tau$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\left(\widehat{\bm{c}}-\bm{c}\right)^{\top}\bm{H}(\tau)\left(\widehat{\bm{c}}-\bm{c}\right)\mathrm{d}\tau-2\int_{0}^{1}\left(\widehat{\bm{c}}-\bm{c}\right)^{\top}\bm{H}(\tau)\left(\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{c}\right)\mathrm{d}\tau$
$\displaystyle:=$ $\displaystyle I_{T,5}-2I_{T,6},$

where the definitions of $I_{T,5}$ and $I_{T,6}$ are obvious.

Since $\widehat{\bm{c}}=\bm{c}+O_{P}\left(T^{-1/2}\right)$, we have
$I_{T,5}=O_{P}(T^{-1})$. For $I_{T,6}$, by (A.2) and (A.5), we have

$\displaystyle I_{T,6}$ $\displaystyle=$
$\displaystyle\left(\widehat{\bm{c}}-\bm{c}\right)^{\top}\int_{h}^{1-h}\bm{H}(\tau)\bm{C}\bm{\Sigma}_{\bm{Z}}^{-1}(\tau)\bm{R}_{T}(\tau)\mathrm{d}\tau+o_{P}\left(T^{-1}h^{-1/2}\right)$
$\displaystyle=$ $\displaystyle
O_{P}(T^{-1})+o_{P}\left(T^{-1}h^{-1/2}\right)=o_{P}\left(T^{-1}h^{-1/2}\right)$

provided that

$\displaystyle\int_{h}^{1-h}\bm{H}(\tau)\bm{C}\bm{\Sigma}_{\bm{Z}}^{-1}(\tau)\bm{R}_{T}(\tau)\mathrm{d}\tau$
$\displaystyle=$
$\displaystyle\frac{1}{T}\sum_{t=1}^{T}\int_{-1}^{1}\bm{H}(\tau_{t}+uh)\bm{C}\bm{\Sigma}_{\bm{Z}}^{-1}(\tau_{t}+uh)K(u)\mathrm{d}u\bm{Z}_{t-1}\bm{\eta}_{t}=O_{P}(T^{-1/2}).$

We thenconclude that
$T\sqrt{h}\left[\widehat{Q}_{\bm{C},\bm{H}}-(Th)^{-1}A_{Q}\right]\to_{D}N\left(0,B_{Q}\right)$.

Observe that

$\displaystyle\int_{0}^{1}\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right]^{\top}\widehat{\bm{H}}(\tau)\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right]\mathrm{d}\tau-\int_{0}^{1}\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{c}\right]^{\top}\bm{H}(\tau)\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{c}\right]\mathrm{d}\tau$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right]^{\top}\widehat{\bm{H}}(\tau)\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right]\mathrm{d}\tau-\int_{0}^{1}\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right]^{\top}\bm{H}(\tau)\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right]\mathrm{d}\tau$
$\displaystyle+\int_{0}^{1}\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right]^{\top}\bm{H}(\tau)\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right]\mathrm{d}\tau-\int_{0}^{1}\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{c}\right]^{\top}\bm{H}(\tau)\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{c}\right]\mathrm{d}\tau.$

Then we just need to focus on

$\int_{0}^{1}\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right]^{\top}\widehat{\bm{H}}(\tau)\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right]\mathrm{d}\tau-\int_{0}^{1}\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right]^{\top}\bm{H}(\tau)\left[\bm{C}\widehat{\bm{\beta}}(\tau)-\widehat{\bm{c}}\right]\mathrm{d}\tau:=I_{T,7}$

Hence, it suffices to show $T\sqrt{h}I_{T,7}=o_{P}(1)$. Using Lemma B.6.3, it
is easy to know that

$\displaystyle|I_{T,7}|$ $\displaystyle\leq$
$\displaystyle\sup_{\tau\in[0,1]}\left\|\widehat{\bm{H}}(\tau)-\bm{H}(\tau)\right\|\times\widehat{Q}_{\bm{C},\bm{I}_{s}}$
$\displaystyle=$ $\displaystyle O_{P}\left(h+\sqrt{\frac{\log
T}{Th}}\right)O_{P}\left((Th)^{-1}+1/(T\sqrt{h})\right)=o_{P}(1/(T\sqrt{h})).$

The proof is now completed. ∎

###### Proof of Corollary 3.1.

Under the local alternative (3.7), we have
$\bm{C}\bm{\beta}(\tau)=\bm{c}+d_{T}\bm{f}(\tau)$ and thus

$\displaystyle\widehat{Q}_{\bm{C},\bm{H}}-\int_{0}^{1}\bm{R}_{T}^{\top}(\tau)\bm{H}_{0}(\tau)\bm{R}_{T}(\tau)\mathrm{d}\tau$
$\displaystyle=$ $\displaystyle
d_{T}^{2}\int_{0}^{1}\bm{f}(\tau)^{\top}\bm{H}(\tau)\bm{f}(\tau)\mathrm{d}\tau+2d_{T}\int_{0}^{1}\bm{f}(\tau)^{\top}\bm{H}(\tau)\left(\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{C}\bm{\beta}(\tau)\right)\mathrm{d}\tau$
$\displaystyle+\left[\int_{0}^{1}\left(\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{C}\bm{\beta}(\tau)\right)^{\top}\bm{H}(\tau)\left(\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{C}\bm{\beta}(\tau)\right)\mathrm{d}\tau-\int_{0}^{1}\bm{R}_{T}^{\top}(\tau)\bm{H}_{0}(\tau)\bm{R}_{T}(\tau)\mathrm{d}\tau\right]$
$\displaystyle=$ $\displaystyle
d_{T}^{2}\int_{0}^{1}\bm{f}(\tau)^{\top}\bm{H}(\tau)\bm{f}(\tau)\mathrm{d}\tau+I_{T,1}+I_{T,2}.$

Since
$\bm{C}\widehat{\bm{\beta}}(\tau)-\bm{C}\bm{\beta}(\tau)=O_{P}\left(d_{T}\rho_{T}+\sqrt{\frac{\log
T}{Th}}\rho_{T}\right)+\bm{C}\bm{\Sigma}_{\bm{Z}}^{-1}(\tau)\bm{R}_{T}(\tau)$
uniformly over $\tau\in[h,1-h]$ and

$\int_{0}^{1}\bm{f}(\tau)^{\top}\bm{H}(\tau)\bm{C}\bm{\Sigma}_{\bm{Z}}^{-1}(\tau)\bm{R}_{T}(\tau)\mathrm{d}\tau=O_{P}(T^{-1/2}),$

we have $I_{T,1}=O_{P}\left(d_{T}(d_{T}\rho_{T}+\sqrt{\frac{\log
T}{Th}}\rho_{T}+T^{-1/2})\right)=o_{P}(T^{-1}h^{-1/2})$.

For $I_{T,2}$, since
$\sup_{\tau\in[0,1]}\left\|\bm{R}_{T}(\tau)\right\|=O_{P}\left(\sqrt{\frac{\log
T}{Th}}\right)$, we have

$I_{T,2}=O_{P}\left(d_{T}^{2}\rho_{T}^{2}+d_{T}\rho_{T}\sqrt{\frac{\log
T}{Th}}\right)=o_{P}(T^{-1}h^{-1/2}).$

As
$T\sqrt{h}\left(\int_{0}^{1}\bm{R}_{T}^{\top}(\tau)\bm{H}_{0}(\tau)\bm{R}_{T}(\tau)-(Th)^{-1}A_{Q}\right)\to
N(0,B_{Q})$, we have

$T\sqrt{h}\left(\widehat{Q}_{\bm{C},\bm{H}}-(Th)^{-1}A_{Q}\right)\to
N(\delta_{1},B_{Q}).$

In addition, similar to the proofs of Theorem 3.1, we have
$T\sqrt{h}\left(\widehat{Q}_{\bm{C},\bm{H}}-\widehat{Q}_{\bm{C},\widehat{\bm{H}}}\right)=o_{P}(1)$.
The proof is now completed. ∎

## Appendix B

In this appendix, we present the results omitted from the main text of this
paper. Specifically, Appendix B.1 includes the preliminary lemmas. Appendix
B.2 provides the proofs of the preliminary lemmas.

##  B.1 Preliminary Lemmas

######  Lemma B.1.

Suppose $\\{Z_{t},\mathcal{F}_{t}\\}$ is a martingale difference sequence,
$S_{T}=\sum_{t=1}^{T}Z_{t}$, $U_{T}=\sum_{t=1}^{T}Z_{t}^{2}$ and
$s_{T}^{2}=E(U_{T}^{2})=E(S_{T}^{2})$. If $s_{T}^{-2}U_{T}^{2}\to_{P}1$ and
$\sum_{t=1}^{T}E[Z_{T,t}^{2}I(|Z_{T,t}|>\nu)]\to 0$ for any $\nu>0$ with
$Z_{T,t}=s_{T}^{-1}Z_{t}$, then as $T\to\infty$,
$s_{T}^{-1}S_{T}\to_{D}N(0,1).$

Lemma B.1 can be found in Hall and Heyde (1980).

######  Lemma B.2.

Let $\\{Z_{t},\mathcal{F}_{t}\\}$ be a martingale difference sequence. Suppose
that $\left|Z_{t}\right|\leq M$ for a constant $M$, $t=1,\ldots,T$. Let
$V_{T}=\sum_{t=1}^{T}\text{Var}\left(Z_{t}|\mathcal{F}_{t-1}\right)\leq V$ for
some $V>0$. Then for any given $\nu>0$,

$\Pr\left(\left|\sum_{t=1}^{T}Z_{t}\right|>\nu\right)\leq\exp\left\\{-\frac{\nu^{2}}{2(V+M\nu)}\right\\}.$

Lemma B.2 is the Proposition 2.1 of Freedman (1975).

######  Assumption A.1.

$\max_{t}\|\bm{\mu}_{t}\|<\infty$,
$\max_{t}\sum_{j=1}^{\infty}j\|\bm{B}_{j,t}\|<\infty$,
$\limsup_{T\to\infty}\sum_{t=1}^{T-1}\|\bm{\mu}_{t+1}-\bm{\mu}_{t}\|<\infty$
and
$\limsup_{T\to\infty}\sum_{t=1}^{T-1}\sum_{j=1}^{\infty}j\|\bm{B}_{j,t+1}-\bm{B}_{j,t}\|<\infty$.
Define the stochastic process of the form
$\bm{h}_{t}=\bm{\mu}_{t}+\sum_{j=0}^{\infty}\bm{B}_{j,t}\bm{\epsilon}_{t}$ for
$t=1,...,T$.

######  Lemma B.3.

Let Assumptions 2 and A.1 hold and $\max_{t\geq
1}E\left(\left\|\bm{\epsilon}_{t}\right\|^{\delta}|\mathcal{F}_{t-1}\right)<\infty$
a.s.. In addition, let $\left\\{\bm{W}_{T,t}(\cdot)\right\\}_{t=1}^{T}$ be a
sequence of $q\times d$ matrices of deterministic functions, in which $q$ is
fixed, each functional component is Lipschitz continuous and defined on a
compact set $[a,b]$. Moreover, suppose that

  1. 1.

$\sup_{\tau\in[a,b]}\sum_{t=1}^{T}\|\bm{W}_{T,t}(\tau)\|=O(1)$;

  2. 2.

$\sup_{\tau\in[a,b]}\sum_{t=1}^{T-1}\|\bm{W}_{T,t+1}(\tau)-\bm{W}_{T,t}(\tau)\|=O(d_{T})$,
where $d_{T}=\sup_{\tau\in[a,b],t\geq 1}\|\bm{W}_{T,t}(\tau)\|$.

Then as $T\to\infty$,

  1. 1.

$\sup_{\tau\in[a,b]}\|\sum_{t=1}^{T}\bm{W}_{T,t}(\tau)\left(\bm{h}_{t}-E(\bm{h}_{t})\right)\|=O_{P}(\sqrt{d_{T}\log
T})$ provided $T^{\frac{2}{\delta}}d_{T}\log T\rightarrow 0$;

  2. 2.

$\sup_{\tau\in[a,b]}\|\sum_{t=1}^{T}\bm{W}_{T,t}(\tau)\left(\bm{h}_{t}\bm{h}_{t+p}^{\top}-E(\bm{h}_{t}\bm{h}_{t+p}^{\top})\right)\|=O_{P}(\sqrt{d_{T}\log
T})$ for any fixed integer $p\geq 0$ provided $T^{\frac{4}{\delta}}d_{T}\log
T\rightarrow 0$, where $\delta$ is the same as in Assumption 2.

Lemma B.3 is the Theorem 2.1 in Yan et al. (2020).

######  Lemma B.4.

Let Assumptions 2 and A.1 hold. In addition, let
$\left\\{\bm{W}_{T,t}(\cdot)\right\\}_{t=1}^{T}$ be a sequence of $q\times d$
matrices of deterministic functions, in which $q$ is fixed, each functional
component is Lipschitz continuous and defined on a compact set $[a,b]$.
Moreover, suppose that

  1. 1.

$\sup_{\tau\in[a,b]}\sum_{t=1}^{T}\|\bm{W}_{T,t}(\tau)\|=O(1)$;

  2. 2.

$\sup_{\tau\in[a,b]}\sum_{t=1}^{T-1}\|\bm{W}_{T,t+1}(\tau)-\bm{W}_{T,t}(\tau)\|=O(d_{T})$,
where $d_{T}=\sup_{\tau\in[a,b],t\geq 1}\|\bm{W}_{T,t}(\tau)\|$.

Then as $T\to\infty$, for any $\tau\in[a,b]$

  1. 1.

$\|\sum_{t=1}^{T}\bm{W}_{T,t}(\tau)\left(\bm{h}_{t}-E(\bm{h}_{t})\right)\|=O_{P}(\sqrt{d_{T}})$;

  2. 2.

$\|\sum_{t=1}^{T}\bm{W}_{T,t}(\tau)\left(\bm{h}_{t}\bm{h}_{t+p}^{\top}-E(\bm{h}_{t}\bm{h}_{t+p}^{\top})\right)\|=O_{P}(\sqrt{d_{T}})$
for any fixed integer $p\geq 0$.

The proof of Lemma B.4 is similar with that of Theorem 2.1 in Yan et al.
(2020).

######  Lemma B.5.

Suppose Assumptions 1-3 hold. Let $\bm{W}(\cdot)$ be a twice-differentiable
functional matrix in $R^{m\times d}$. As $T\to\infty$,

  1. 1.

for $\tau\in(0,1)$,

$\displaystyle\frac{1}{T}\sum_{t=1}^{T}\bm{W}(\tau_{t})\bm{x}_{t}\left(\frac{\tau_{t}-\tau}{h}\right)^{k}K_{h}(\tau_{t}-\tau)-\widetilde{c}_{k}\bm{W}(\tau)\bm{\mu}(\tau)=O_{P}(h^{2}+1/(\sqrt{Th})),$
$\displaystyle\frac{1}{T}\sum_{t=1}^{T}\bm{W}(\tau_{t})\bm{x}_{t}\bm{x}_{t+p}^{\top}\left(\frac{\tau_{t}-\tau}{h}\right)^{k}K_{h}(\tau_{t}-\tau)-\widetilde{c}_{k}\bm{W}(\tau)\bm{\Sigma}_{p}(\tau)=O_{P}(h^{2}+1/(\sqrt{Th})),$

where
$\bm{\Sigma}_{p}(\tau)=\bm{\mu}(\tau)\bm{\mu}^{\top}(\tau)+\sum_{j=0}^{\infty}\bm{B}_{j}(\tau)\bm{B}_{j+p}^{\top}(\tau)$
for fixed integers $k$ and $p\geq 0$;

  2. 2.

given $\frac{T^{1-\frac{2}{\delta}}h}{\log T}\rightarrow\infty$ and
$\rho_{T}=h^{2}+\sqrt{\frac{\log(T)}{Th}}$,

$\sup_{\tau\in[h,1-h]}\left\|\frac{1}{T}\sum_{t=1}^{T}\bm{W}(\tau_{t})\bm{x}_{t}\left(\frac{\tau_{t}-\tau}{h}\right)^{k}K_{h}(\tau_{t}-\tau)-\widetilde{c}_{k}\bm{W}(\tau)\bm{\mu}(\tau)\right\|=O_{P}\left(\rho_{T}^{2}\right);$

  3. 3.

given $\frac{T^{1-\frac{4}{\delta}}h}{\log T}\rightarrow\infty$ and
$\max_{t\geq 1}E[\|\bm{\epsilon}_{t}\|^{4}|\mathcal{F}_{t-1}]<\infty$ a.s.,

$\sup_{\tau\in[h,1-h]}\left\|\frac{1}{T}\sum_{t=1}^{T}\bm{W}(\tau_{t})\bm{x}_{t}\bm{x}_{t+p}^{\top}\left(\frac{\tau_{t}-\tau}{h}\right)^{k}K_{h}(\tau_{t}-\tau)-\widetilde{c}_{k}\bm{W}(\tau)\bm{\Sigma}_{p}(\tau)\right\|=O_{P}\left(\rho_{T}^{2}\right).$

  4. 4.

$\frac{1}{T}\sum_{t=1}^{T}\bm{W}(\tau_{t})(\bm{x}_{t}-E(\bm{x}_{t}))=O_{P}(1/\sqrt{T})$
and
$\frac{1}{T}\sum_{t=1}^{T}\bm{W}(\tau_{t})(\bm{x}_{t}\bm{x}_{t+p}^{\top}-E(\bm{x}_{t}\bm{x}_{t+p}^{\top}))=O_{P}(1/\sqrt{T})$.

######  Lemma B.6.

Let Assumptions 1-3 hold. Suppose $\frac{T^{1-\frac{4}{\delta}}h}{\log
T}\rightarrow\infty$ and $\max_{t\geq
1}E[\left\|\bm{\epsilon}_{t}\right\|^{4}|\mathcal{F}_{t-1}]<\infty$ a.s. As
$T\to\infty$,

  1. 1.

$\sup_{\tau\in[0,1]}\left\|\frac{1}{Th}\sum_{t=1}^{T}\bm{Z}_{t-1}\bm{\eta}_{t}K\left(\frac{\tau_{t}-\tau}{h}\right)\right\|=O_{P}\left(\left(\frac{\log
T}{Th}\right)^{\frac{1}{2}}\right)$;

  2. 2.

$\frac{1}{\sqrt{Th}}\sum_{t=1}^{T}\bm{\eta}_{t}\left(\bm{\eta}_{t}-\bm{\widehat{\eta}}_{t}\right)^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)=o_{P}(1)$
for $\forall\tau\in[0,1]$;

  3. 3.

$\sup_{\tau\in[h,1-h]}\left\|\widehat{\bm{V}}_{\bm{\beta}}(\tau)-\bm{V}_{\bm{\beta}}(\tau)\right\|=O_{P}(h^{2}+\left(\frac{\log
T}{Th}\right)^{\frac{1}{2}})$.

######  Lemma B.7.

Let Assumptions 1-3 hold. Suppose $\frac{T^{1-\frac{4}{\delta}}h}{\log
T}\rightarrow\infty$ and $\max_{t\geq
1}E[\left\|\bm{\epsilon}_{t}\right\|^{4}|\mathcal{F}_{t-1}]<\infty$ a.s. As
$T\to\infty$,

  1. 1.

if $\mathsf{p}\geq p$, then
$\text{RSS}(\mathsf{p})=\frac{1}{T}\sum_{t=1}^{T}E\left(\bm{\eta}_{t}^{\top}\bm{\eta}_{t}\right)+O_{P}\left(\rho_{T}^{2}\right)$
with $\rho_{T}=h^{2}+\sqrt{\frac{\log(T)}{Th}}$;

  2. 2.

if $\mathsf{p}<p$, then
$\text{RSS}(\mathsf{p})=\frac{1}{T}\sum_{t=1}^{T}E\left(\bm{\eta}_{t}^{\top}\bm{\eta}_{t}\right)+c+o_{P}\left(1\right)$
with some constant $c>0$.

Define
$w_{s,t}=\frac{1}{T\sqrt{h}}\int_{-1}^{1}K\left(u\right)K\left(u+\frac{t-s}{Th}\right)\mathrm{d}u$.
Let $a_{t}=\sum_{s=1}^{t-1}w_{s,t}^{2}$, $b_{s}=\sum_{t=s+1}^{T}w_{s,t}^{2}$
and $\sigma^{2}=\sum_{t=2}^{T}a_{t}$.

######  Lemma B.8.

Suppose Assumption 3 hold. Then, we have

  1. 1.

$\sigma^{2}\to\int_{0}^{2}\left[\int_{-1}^{1-v}K\left(u\right)K\left(u+v\right)\mathrm{d}u\right]^{2}\mathrm{d}v$;

  2. 2.

$\max_{2\leq t\leq T}a_{t}=O(1/T)$;

  3. 3.

for any fixed $J\in\mathbb{N}$, $\sum_{s=1}^{T-J}w_{s,s+J}^{2}=O(1/(Th))$;

  4. 4.

$T\sum_{s=1}^{T-1}b_{s}^{2}=O(1)$;

  5. 5.

$\sum_{k=1}^{T-1}\sum_{t=1}^{k-1}\left[\sum_{j=k+1}^{T}w_{k,j}w_{t,j}\right]^{2}=O(1/T)$;

For a random vector $\bm{z}$, we write $\bm{z}\in\mathcal{L}^{q}$, $q>0$, if
${\left|\kern-0.96873pt\left|\kern-0.96873pt\left|\bm{z}\right|\kern-0.96873pt\right|\kern-0.96873pt\right|}_{q}=\left[E\left(\left\|\bm{z}\right\|^{q}\right)\right]^{1/q}<\infty$,
and we denote
${\left|\kern-0.96873pt\left|\kern-0.96873pt\left|\cdot\right|\kern-0.96873pt\right|\kern-0.96873pt\right|}={\left|\kern-0.96873pt\left|\kern-0.96873pt\left|\cdot\right|\kern-0.96873pt\right|\kern-0.96873pt\right|}_{2}$.

######  Lemma B.9.

Let $\bm{\xi}_{1}$,…,$\bm{\xi}_{T}$ be a $d$-dimensional martingale difference
for which $\bm{\xi}_{t}\in\mathcal{L}^{p}$, $p>1$. Let $p^{*}=\min(2,p)$. Then

${\left|\kern-0.96873pt\left|\kern-0.96873pt\left|\sum_{t=1}^{T}\bm{\xi}_{t}\right|\kern-0.96873pt\right|\kern-0.96873pt\right|}_{p}^{p^{*}}\leq
M\sum_{t=1}^{T}{\left|\kern-0.96873pt\left|\kern-0.96873pt\left|\bm{\xi}_{t}\right|\kern-0.96873pt\right|\kern-0.96873pt\right|}_{p}^{p^{*}}.$

Let $\bm{h}_{t-1}^{*}:=\sum_{s=1}^{t-1}w_{s,t}\bm{y}_{s}$, where
$\bm{y}_{t}\in\mathcal{L}^{\delta}$, $t=1,2,...,T$ are martingale differences
subject to the filtration $\mathcal{F}_{t}$ and $\delta>4$.

######  Lemma B.10.

Suppose Assumption 3 hold. Assume $\bm{w}_{t}^{*}\in\mathcal{L}^{\delta/2}$
for some $\delta>4$ is $\mathcal{F}_{t}$-measurable. Then, as $T\to\infty$,

$E\left|\sum_{t=2}^{T}\mathrm{tr}\left[\left(\bm{w}_{t}^{*}-E(\bm{w}_{t}^{*})\right)\bm{h}_{t-1}^{*}\bm{h}_{t-1}^{*,\top}\right]\right|\to
0.$

######  Lemma B.11.

Suppose Assumptions 1-3 hold and $\bm{y}_{t}=\bm{Z}_{t-1}\bm{\eta}_{t}$. Then,
as $T\to\infty$,

$\sum_{t=2}^{T}\mathrm{tr}\left[\bm{H}_{t}\left(\bm{h}_{t-1}^{*}\bm{h}_{t-1}^{*,\top}-E(\bm{h}_{t-1}^{*}\bm{h}_{t-1}^{*,\top})\right)\right]\to_{P}0,$

where $\bm{H}_{t}$ is a $d^{2}\times d^{2}$ deterministic weighting function
satisfying $\left\|\bm{H}_{t}\right\|<\infty$.

Let $Q_{T}=\sum_{t=2}^{T}\bm{y}_{t}^{\top}\bm{H}_{t}\bm{h}_{t-1}^{*}$, where
$\bm{H}_{t}$ is a $d^{2}\times d^{2}$ functional weighting matrix.

######  Lemma B.12.

Suppose Assumptions 1-3 hold and $\bm{y}_{t}=\bm{Z}_{t-1}\bm{\eta}_{t}$. Then
as $T\to\infty$,

$Q_{T}\to_{D}N\left(0,\sigma_{Q}^{2}\right).$

where
$\sigma_{Q}^{2}=\lim_{T\to\infty}\sum_{t=2}^{T}\mathrm{tr}\left[E\left(\bm{H}_{t}^{\top}\bm{y}_{t}\bm{y}_{t}^{\top}\bm{H}_{t}\right)E\left(\bm{h}_{t-1}^{*}\bm{h}_{t-1}^{*,\top}\right)\right]$.

##  B.2 Proofs of the Preliminary Lemmas

###### Proof of Lemma B.5.

(1). First, for any fixed $\tau\in(0,1)$, let
$\bm{W}_{T,t}(\tau)=\frac{1}{T}\bm{W}(\tau_{t})\left(\frac{\tau_{t}-\tau}{h}\right)^{k}K_{h}\left(\tau_{t}-\tau\right)$.
It is straightforward to show that
$\sup_{\tau\in[0,1]}\sum_{t=1}^{T}\|\bm{W}_{T,t}(\tau)\|=O(1)$,
$\sup_{\tau\in[0,1],t\geq 1}\|\bm{W}_{T,t}(\tau)\|=O(1/(Th))$ and

$\sup_{\tau\in[0,1]}\sum_{t=1}^{T-1}\|\bm{W}_{T,t+1}(\tau)-\bm{W}_{T,t}(\tau)\|=O(1/(Th)).$

Second, by triangle inequality, Cauchy-Schwarz inequality and Proposition 2.1,

$\displaystyle\sum_{t=1}^{T}\left\|\bm{x}_{t}\bm{x}_{t}^{\top}-\widetilde{\bm{x}}_{t}\widetilde{\bm{x}}_{t}^{\top}\right\|\leq\sum_{t=1}^{T}\left(\left\|\bm{x}_{t}\right\|+\left\|\widetilde{\bm{x}}_{t}\right\|\right)\left\|\bm{x}_{t}-\widetilde{\bm{x}}_{t}\right\|$
$\displaystyle\leq$
$\displaystyle\left(\sum_{t=1}^{T}\left(\left\|\bm{x}_{t}\right\|+\left\|\widetilde{\bm{x}}_{t}\right\|\right)^{2}\right)^{1/2}\left(\sum_{t=1}^{T}\left\|\bm{x}_{t}-\widetilde{\bm{x}}_{t}\right\|^{2}\right)^{1/2}=O_{P}(\sqrt{T})\cdot
O_{P}(1/\sqrt{T})=O_{P}(1).$

In addition, similar to the proof of Proposition 2.1, we have

$\sum_{j=0}^{\infty}j\left\|\bm{B}_{j}(\tau)\right\|=\sum_{j=0}^{\infty}j\left\|\bm{\Psi}_{j}(\tau)\bm{\omega}(\tau)\right\|\leq
M\sum_{j=0}^{\infty}j\rho_{A}^{j}<\infty,$

and

$\displaystyle\sum_{t=1}^{T-1}\sum_{j=0}^{\infty}j\left\|\bm{B}_{j}(\tau_{t+1})-\bm{B}_{j}(\tau_{t})\right\|$
$\displaystyle\leq$
$\displaystyle\sup_{\tau\in[0,1]}\sum_{j=1}^{\infty}j\|\bm{\Psi}_{j}(\tau)\|\sum_{t=1}^{T}\left\|\bm{\omega}(\tau_{t+1})-\bm{\omega}(\tau_{t})\right\|+M\sum_{t=1}^{T-1}\sum_{j=0}^{\infty}j\|\bm{\Psi}_{j}(\tau_{t+1})-\bm{\Psi}_{j}(\tau_{t})\|$
$\displaystyle\leq$ $\displaystyle
M\sum_{j=0}^{\infty}(j^{2}\rho_{A}^{j-1}+j\rho_{A}^{j})<\infty.$

Therefore, Lemma B.4 are still valid for the TV-VAR($p$) process. The proof of
part (1) is completed.

(2)-(4). The proofs of part (2)-(4) can be done in a similar way to that of
part (1). ∎

###### Proof of Lemma B.6.

(1). Let $\\{S_{l}\\}$ be a finite number of sub-intervals covering the
interval $[0,1]$, which are centered at $s_{l}$ with the length
$\delta_{T}=o(h^{2})$. Denote the number of these intervals by $N_{T}$ then
$N_{T}=O(\delta_{T}^{-1})$. Hence,

$\displaystyle\sup_{\tau\in[0,1]}\left\|\frac{1}{T}\sum_{t=1}^{T}\bm{Z}_{t-1}\bm{\eta}_{t}K_{h}(\tau_{t}-\tau)\right\|$
$\displaystyle\leq$ $\displaystyle\max_{1\leq l\leq
N_{T}}\left\|\frac{1}{T}\sum_{t=1}^{T}\bm{Z}_{t-1}\bm{\eta}_{t}K_{h}(\tau_{t}-s_{l})\right\|$
$\displaystyle+\max_{1\leq l\leq N_{T}}\sup_{\tau\in
S_{l}}\left\|\frac{1}{T}\sum_{t=1}^{T}\bm{Z}_{t-1}\bm{\eta}_{t}\left(K_{h}(\tau_{t}-\tau)-K_{h}(\tau_{t}-s_{l})\right)\right\|$
$\displaystyle:=$ $\displaystyle I_{T,1}+I_{T,2}.$

By the continuity of kernel function $K(\cdot)$ and taking
$\delta_{T}=O(\gamma_{T}h^{2})$ with $\gamma_{T}=\left(\frac{\log
T}{Th}\right)^{\frac{1}{2}}$, then we have

$E|I_{T,2}|\leq
M\frac{\delta_{T}}{h^{2}}E\|\bm{Z}_{t-1}\bm{\eta}_{t}\|=O(\gamma_{T}).$

We then apply the truncation method again. Define
$\bm{u}_{t}=\bm{Z}_{t-1}\bm{\eta}_{t}$,
$\bm{u}_{t}^{\prime}=\bm{u}_{t}I(\|\bm{u}_{t}\|\leq T^{\frac{2}{\delta}})$ and
$\bm{u}_{t}^{\prime\prime}=\bm{u}_{t}-\bm{u}_{t}^{\prime}$. Then we have

$\displaystyle I_{T,1}$ $\displaystyle=$ $\displaystyle\max_{1\leq l\leq
N_{T}}\left\|\frac{1}{T}\sum_{t=1}^{T}\bm{u}_{t}^{\prime}+\bm{u}_{t}^{\prime\prime}-E(\bm{u}_{t}^{\prime}+\bm{u}_{t}^{\prime\prime}|\mathcal{F}_{t-1})K_{h}(\tau_{t}-s_{l})\right\|$
$\displaystyle\leq$ $\displaystyle\max_{1\leq l\leq
N_{T}}\left\|\frac{1}{T}\sum_{t=1}^{T}\left(\bm{u}_{t}^{\prime}-E(\bm{u}_{t}^{\prime}|\mathcal{F}_{t-1})\right)K_{h}(\tau_{t}-s_{l})\right\|+\max_{1\leq
l\leq
N_{T}}\left\|\frac{1}{T}\sum_{t=1}^{T}\bm{u}_{t}^{\prime\prime}K_{h}(\tau_{t}-s_{l})\right\|$
$\displaystyle+\max_{1\leq l\leq
N_{T}}\left\|\frac{1}{T}\sum_{t=1}^{T}E(\bm{u}_{t}^{\prime\prime}|\mathcal{F}_{t-1})K_{h}(\tau_{t}-s_{l})\right\|$
$\displaystyle:=$ $\displaystyle I_{T,11}+I_{T,12}+I_{T,13}.$

Now consider $I_{T,12}$. Let $d_{T}=\max_{1\leq t\leq T,1\leq l\leq
N_{T}}K_{h}(\tau_{t}-s_{l})/T$. By Holder’s inequality and Chebyshev
inequality, we have

$\displaystyle E|I_{T,12}|$ $\displaystyle\leq$ $\displaystyle
d_{T}\sum_{t=1}^{T}E\left\|\bm{\eta}_{t}^{\prime\prime}\right\|\leq
d_{T}\sum_{t=1}^{T}E\left(\left\|\bm{Z}_{t-1}\bm{\eta}_{t}\right\|^{\delta/2}\right)^{\frac{2}{\delta}}\left(\frac{E\left(\left\|\bm{Z}_{t-1}\bm{\eta}_{t}\right\|^{\delta/2}\right)}{T}\right)^{\frac{\delta-2}{\delta}}$
$\displaystyle=$ $\displaystyle
O(T^{\frac{2}{\delta}}d_{T})=o\left(\sqrt{\frac{\log T}{Th}}\right).$

Similarly, $I_{T,13}=O_{P}(T^{\frac{2}{\delta}}d_{T})$.

For any fixed $1\leq l\leq N_{T}$, let
$\bm{Y}_{t}:=\frac{1}{T}(\bm{u}_{t}^{\prime}-E(\bm{u}_{t}^{\prime}|\mathcal{F}_{t-1}))K_{h}(\tau_{t}-s_{l})$,
then we have $E\left(\bm{Y}_{t}|\mathcal{F}_{t-1}\right)=0$ and
$\left\|\bm{Y}_{t}\right\|\leq 2T^{2/\delta}d_{T}$ with $d_{T}=\max_{1\leq
t\leq T}K_{h}(\tau_{t}-s_{l})/T$. Also, by Lemma B.5, we have

$\sup_{0\leq\tau\leq
1}\frac{1}{T}\sum_{t=1}^{T}E\left(\left\|\bm{Z}_{t-1}\bm{\eta}_{t}\right\|^{2}|\mathcal{F}_{t-1}\right)K_{h}(\tau_{t}-\tau)=O_{P}(1)$

which follows that

$\max_{1\leq l\leq
N_{T}}\left\|\sum_{t=1}^{T}E(\bm{Y}_{t}\bm{Y}_{t}^{\top}|\mathcal{F}_{t-1})\right\|\leq
4\max_{1\leq l\leq
N_{T}}\sum_{t=1}^{T}E\left(\left\|\bm{u}_{t}\right\|^{2}|\mathcal{F}_{t-1}\right)\frac{K_{h}^{2}(\tau_{t}-s_{l})}{T^{2}}=O_{P}\left(d_{T}\right).$

Therefore, we have $\max_{1\leq l\leq
N_{T}}\left\|\sum_{t=1}^{T}E(\bm{Y}_{t}\bm{Y}_{t}^{\top}|\mathcal{F}_{t-1})\right\|\leq\frac{M}{Th}$
in probability. By Lemma B.2, we have

$\begin{split}\Pr\left(I_{T,11}>\sqrt{8M}\gamma_{T}\right)&\leq\Pr\left(I_{T,11}>\sqrt{8M}\gamma_{T},\max_{1\leq
l\leq
N_{T}}\left\|\sum_{t=1}^{T}E(\bm{Y}_{t}\bm{Y}_{t}^{\top}|\mathcal{F}_{t-1})\right\|\leq\frac{M}{Th}\right)\\\
&+\Pr\left(\max_{1\leq l\leq
N_{T}}\left\|\sum_{t=1}^{T}E(\bm{Y}_{t}\bm{Y}_{t}^{\top}|\mathcal{F}_{t-1})\right\|>\frac{M}{Th}\right)\\\
&\leq
N_{T}\exp\left(-\frac{8M\gamma_{T}^{2}}{2(\frac{M}{Th}+\gamma_{T}2T^{\frac{2}{\delta}}d_{T})}\right)+o(1)\\\
&\leq N_{T}\exp\left(-4\log(T)\right)+o(1)=o(1),\end{split}$

if $\frac{T^{1-\frac{4}{\delta}}h}{\log T}\rightarrow\infty$, which completes
the proof of part (1).

(2). Note that

$\displaystyle\frac{1}{\sqrt{Th}}\sum_{t=1}^{T}\bm{\eta}_{t}\left(\bm{\eta}_{t}-\bm{\widehat{\eta}}_{t}\right)^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)=\frac{1}{\sqrt{Th}}\sum_{t=1}^{T}\bm{\eta}_{t}\bm{z}_{t-1}^{\top}\left(\bm{\widehat{A}}(\tau_{t})-\bm{A}(\tau_{t})\right)^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{Th}}\sum_{t=1}^{T}\bm{\eta}_{t}[\bm{z}_{t-1}^{\top},\bm{0}_{(d^{2}p+d)\times
1}^{\top}]\frac{1}{2}h^{2}\bm{S}_{T}^{-1}(\tau_{t})\left(\begin{matrix}\bm{S}_{T,2}(\tau_{t})\\\
\bm{S}_{T,3}(\tau_{t})\end{matrix}\right)\bm{A}^{(2),\top}(\tau_{t})K\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle+\frac{1}{\sqrt{Th}}\sum_{t=1}^{T}\bm{\eta}_{t}[\bm{z}_{t-1}^{\top},\bm{0}_{(d^{2}p+d)\times
1}^{\top}]\bm{S}_{T}^{-1}(\tau_{t})\left(\frac{1}{Th}\sum_{s=1}^{T}\left(\bm{z}_{s-1}^{*}\bm{z}_{s-1}^{\top}\otimes\bm{I}_{d}\right)\bm{M}^{\top}(\tau_{s})K\left(\frac{\tau_{s}-\tau_{t}}{h}\right)\right)K\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle+\frac{1}{\sqrt{Th}}\sum_{t=1}^{T}\bm{\eta}_{t}[\bm{z}_{t-1}^{\top},\bm{0}_{(d^{2}p+d)\times
1}^{\top}]\bm{S}_{T}^{-1}(\tau_{t})\left(\frac{1}{Th}\sum_{s=1}^{T}\bm{Z}_{s-1}\bm{\eta}_{s}^{\top}K\left(\frac{\tau_{s}-\tau_{t}}{h}\right)\right)K\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle:=$ $\displaystyle J_{T,1}+J_{T,2}+J_{T,3}.$

For $J_{T,1}$ to $J_{T,2}$, using Lemmas B.5.2-3, we can replace the sample
covariance matrix with its converged and deterministic value with rate
$O_{P}\left(\sqrt{\frac{\log T}{Th}}\right)$ and hence it’s easy to show that
$J_{T,1}$ to $J_{T,2}$ are $o_{P}(1)$.

For $J_{T,3}$, for notational simplicity, we ignore
$\bm{S}_{T}^{-1}(\tau_{t})$ and hence,

$\begin{split}J_{T,3}=\
&\frac{1}{(Th)^{3/2}}\sum_{t=1}^{T}\bm{\eta}_{t}\bm{z}_{t-1}^{\top}\bm{z}_{t-1}\bm{\eta}_{t}^{\top}K(0)K\left(\frac{\tau_{t}-\tau}{h}\right)\\\
&+\frac{1}{(Th)^{3/2}}\sum_{i=1}^{T-1}\sum_{t=1}^{T-i}\bm{\eta}_{t}\bm{z}_{t-1}^{\top}\bm{z}_{t+i-1}\bm{\eta}_{t+i}^{\top}K\left(\frac{i}{Th}\right)K\left(\frac{\tau_{t}-\tau}{h}\right)\\\
&+\frac{1}{(Th)^{3/2}}\sum_{i=1}^{T-1}\sum_{t=1}^{T-i}\bm{\eta}_{t+i}\bm{z}_{t+i-1}^{\top}\bm{z}_{t-1}\bm{\eta}_{t}^{\top}K\left(\frac{i}{Th}\right)K\left(\frac{\tau_{t}-\tau}{h}\right)\\\
:=\ &J_{T,31}+J_{T,32}+J_{T,33}.\\\ \end{split}$

It’s easy to see $J_{T,31}=O_{P}\left((Th)^{-1/2}\right)$. For $J_{T,32}$,

$\begin{split}J_{T,32}=\
&\frac{1}{(Th)^{3/2}}\sum_{i=1}^{T-1}\sum_{t=1}^{T-i}\bm{\eta}_{t}E\left(\bm{z}_{t-1}^{\top}\bm{z}_{t+i-1}\right)\bm{\eta}_{t+i}^{\top}K\left(\frac{i}{Th}\right)K\left(\frac{\tau_{t}-\tau}{h}\right)\\\
&+\frac{1}{(Th)^{3/2}}\sum_{i=1}^{T-1}\sum_{t=1}^{T-i}\bm{\eta}_{t}\left(\bm{z}_{t-1}^{\top}\bm{z}_{t+i-1}-E\left(\bm{z}_{t-1}^{\top}\bm{z}_{t+i-1}\right)\right)\bm{\eta}_{t+i}^{\top}K\left(\frac{i}{Th}\right)K\left(\frac{\tau_{t}-\tau}{h}\right)\\\
:=\ &J_{T,321}+J_{T,322}.\end{split}$

For $J_{T,321}$,

$\begin{split}E\left\|J_{T,321}\right\|^{2}\leq\
&\frac{1}{(Th)^{3}}\sum_{i=1}^{T-1}\sum_{t=1}^{T-i}\left\\{E\left(\bm{z}_{t-1}^{\top}\bm{z}_{t+i-1}\right)E\left(\bm{\eta}_{t}^{\top}\bm{\eta}_{t}\right)E\left(\bm{\eta}_{t+i}^{\top}\bm{\eta}_{t+i}\right)\right\\}^{2}K^{2}\left(\frac{i}{Th}\right)K^{2}\left(\frac{\tau_{t}-\tau}{h}\right)\\\
=\ &O\left(\frac{1}{Th}\right),\end{split}$

which then yields that $J_{T,321}=O_{P}\left((Th)^{-1/2}\right)$. For
$J_{T,322}$,

$J_{T,322}=\frac{1}{(Th)^{3/2}}\sum_{i=1}^{T-1}\sum_{t=1}^{T-i}\bm{\eta}_{t}\sum_{m=1}^{p}\left(\bm{x}_{t-m}^{\top}\bm{x}_{t+i-m}-E\left(\bm{x}_{t-m}^{\top}\bm{x}_{t+i-m}\right)\right)\bm{\eta}_{t+i}^{\top}K\left(\frac{i}{Th}\right)K\left(\frac{\tau_{t}-\tau}{h}\right).$

For notational simplicity, let $p=1$ and thus

$\displaystyle J_{T,322}$ $\displaystyle=$
$\displaystyle\frac{1}{(Th)^{3/2}}\sum_{i=1}^{T-1}\sum_{t=1}^{T-i}\bm{\eta}_{t}\left(\bm{\mu}_{t-1}^{\top}\sum_{j=0}^{\infty}\bm{\Psi}_{j,t+i-1}\bm{\eta}_{t+i-1-j}\right)\bm{\eta}_{t+i}^{\top}K\left(\frac{i}{Th}\right)K\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle+\frac{1}{(Th)^{3/2}}\sum_{i=1}^{T-1}\sum_{t=1}^{T-i}\bm{\eta}_{t}\left(\sum_{j=0}^{\infty}\bm{\eta}_{t-1-j}^{\top}\bm{\Psi}_{j,t-1}^{\top}\bm{\mu}_{t+i-1}\right)\bm{\eta}_{t+i}^{\top}K\left(\frac{i}{Th}\right)K\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle+\frac{1}{(Th)^{3/2}}\sum_{i=1}^{T-1}\sum_{t=1}^{T-i}\bm{\eta}_{t}\left(\sum_{j=0}^{\infty}\left(\bm{\eta}_{t-1-j}^{\top}\otimes\bm{\eta}_{t-1-j}^{\top}-E\left(\bm{\eta}_{t-1-j}^{\top}\otimes\bm{\eta}_{t-1-j}^{\top}\right)\right)\right.$
$\displaystyle\cdot\left.\mathrm{vec}\left(\bm{\Psi}_{j,t-1}^{\top}\bm{\Psi}_{j+i,t+i-1}\right)\right)\bm{\eta}_{t+i}^{\top}K\left(\frac{i}{Th}\right)K\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle+\frac{1}{(Th)^{3/2}}\sum_{i=1}^{T-1}\sum_{t=1}^{T-i}\bm{\eta}_{t}\left(\sum_{j=0}^{\infty}\sum_{m=0,\neq
j+i}^{\infty}\left(\bm{\eta}_{t+i-1-m}^{\top}\otimes\bm{\eta}_{t-1-j}^{\top}\right)\mathrm{vec}\left(\bm{\Psi}_{j,t-1}^{\top}\bm{\Psi}_{m,t+i-1}\right)\right)$
$\displaystyle\cdot\bm{\eta}_{t+i}^{\top}K\left(\frac{i}{Th}\right)K\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle:=$ $\displaystyle J_{T,3221}+J_{T,3222}+J_{T,3223}+J_{T,3224}.$

For $J_{T,3221}$,

$\displaystyle E\left\|J_{T,3221}\right\|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{(Th)^{3}}\sum_{i=1}^{T-1}\sum_{t=1}^{T-i}E\left\|\bm{\eta}_{t}\left(\bm{\mu}_{t-1}^{\top}\sum_{j=0}^{\infty}\bm{\Psi}_{j,t+i-1}\bm{\eta}_{t+i-1-j}\right)\right\|^{2}E\left\|\bm{\eta}_{t+i}\right\|^{2}K^{2}\left(\frac{i}{Th}\right)K^{2}\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle=$ $\displaystyle O\left(\frac{1}{Th}\right).$

Similarly, $J_{T,3222}$ and $J_{T,3223}$ are $O_{P}\left((Th)^{-1/2}\right)$.
For $J_{T,3224}$,

$\displaystyle J_{T,3224}$ $\displaystyle=$
$\displaystyle\frac{1}{(Th)^{3/2}}\sum_{i=1}^{T-1}\sum_{t=1}^{T-i}\bm{\eta}_{t}\left(\sum_{j=0}^{\infty}\left(\bm{\eta}_{t}^{\top}\otimes\bm{\eta}_{t-1-j}^{\top}\right)\mathrm{vec}\left(\bm{\Psi}_{j,t-1}^{\top}\bm{\Psi}_{i-1,t+i-1}\right)\right)\bm{\eta}_{t+i}^{\top}K\left(\frac{i}{Th}\right)K\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle+\frac{1}{(Th)^{3/2}}\sum_{i=1}^{T-1}\sum_{t=1}^{T-i}\bm{\eta}_{t}\left(\sum_{j=0}^{\infty}\sum_{m=0,\neq
j+i,\neq
i-1}^{\infty}\left(\bm{\eta}_{t+i-1-m}^{\top}\otimes\bm{\eta}_{t-1-j}^{\top}\right)\right.$
$\displaystyle\cdot\left.\mathrm{vec}\left(\bm{\Psi}_{j,t-1}^{\top}\bm{\Psi}_{m,t+i-1}\right)\right)\bm{\eta}_{t+i}^{\top}K\left(\frac{i}{Th}\right)K\left(\frac{\tau_{t}-\tau}{h}\right):=J_{T,32241}+J_{T,32242}.$

Similar to the proof of $J_{T,3221}$, we can show that
$J_{T,32242}=O_{P}\left((Th)^{-1/2}\right)$.

Let
$\bm{w}_{t,i}=\sum_{j=0}^{\infty}\left(\bm{\eta}_{t}\bm{\eta}_{t}^{\top}\otimes\bm{\eta}_{t-1-j}^{\top}\right)\mathrm{vec}\left(\bm{\Psi}_{j,t-1}^{\top}\bm{\Psi}_{i-1,t+i-1}^{\top}\right)$.
For $J_{T,32241}$,

$\displaystyle
E\left\|J_{T,32241}\right\|^{2}=\frac{1}{(Th)^{3}}\sum_{i_{1}=1}^{T-1}\sum_{i_{2}=1}^{T-1}\sum_{t_{1}=1}^{T-i_{1}}E\left\|\bm{\eta}_{t_{1}+i_{1}}^{\top}\bm{\eta}_{t_{1}+i_{1}}\right\|E\left\|\bm{w}_{t_{1}+i_{1}-i_{2},i_{2}}^{\top}\bm{w}_{t_{1},i_{1}}\right\|$
$\displaystyle\cdot
K\left(\frac{i_{1}}{Th}\right)K\left(\frac{i_{2}}{Th}\right)K\left(\frac{\tau_{t_{1}}-\tau}{h}\right)K\left(\frac{\tau_{t_{1}+i_{1}-i_{2}}-\tau}{h}\right)$
$\displaystyle\leq\frac{M}{(Th)^{3}}\sum_{t_{1}=1}^{T}\left(\max_{t}\sum_{i=1}^{T-1}\left\|\bm{\Psi}_{i,t}\right\|\right)^{2}\left(\max_{t}\sum_{j=0}^{\infty}\left\|\bm{\Psi}_{j,t}\right\|\right)^{2}K\left(\frac{\tau_{t_{1}}-\tau}{h}\right)=O\left((Th)^{-2}\right).$

Hence, $J_{T,32}=O_{P}\left((Th)^{-1/2}\right)$. Similar to $J_{T,32}$,
$J_{T,33}=O_{P}\left((Th)^{-1/2}\right)$. The proof is now completed.

(3). By Lemma B.5, we have

$\displaystyle\sup_{\tau\in[h,1-h]}\left\|\widehat{\bm{\Sigma}}(\tau)-\bm{\Sigma}(\tau)\right\|=O_{P}\left(h^{2}+\left(\frac{\log
T}{Th}\right)^{1/2}\right).$

Then we just need to focus on the rate associated with
$\widehat{\bm{\Omega}}(\tau)$. For notational simplicity, we ignore the
$\frac{1}{T}\sum_{t=1}^{T}K_{h}(\tau_{t}-\tau)$, because

$\displaystyle\frac{1}{T}\sum_{t=1}^{T}K_{h}(\tau_{t}-\tau)=1+O((Th)^{-1})$

uniformly over $\tau\in[h,1-h]$.

Write

$\displaystyle\widehat{\bm{\Omega}}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{Th}\sum_{t=1}^{T}\bm{\widehat{\eta}}_{t}\bm{\widehat{\eta}}_{t}^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{Th}\sum_{t=1}^{T}\left(\bm{\eta}_{t}+\bm{\widehat{\eta}}_{t}-\bm{\eta}_{t}\right)\left(\bm{\eta}_{t}+\bm{\widehat{\eta}}_{t}-\bm{\eta}_{t}\right)^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{Th}\sum_{t=1}^{T}\bm{\eta}_{t}\bm{\eta}_{t}^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)+\frac{1}{Th}\sum_{t=1}^{T}(\bm{\widehat{\eta}}_{t}-\bm{\eta}_{t})(\bm{\widehat{\eta}}_{t}-\bm{\eta}_{t})^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle+\frac{1}{Th}\sum_{t=1}^{T}\bm{\eta}_{t}(\bm{\widehat{\eta}}_{t}-\bm{\eta}_{t})^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)+\frac{1}{Th}\sum_{t=1}^{T}(\bm{\widehat{\eta}}_{t}-\bm{\eta}_{t})\bm{\eta}_{t}^{\top}K\left(\frac{\tau_{t}-\tau}{h}\right)$
$\displaystyle:=$ $\displaystyle I_{T,1}+I_{T,2}+I_{T,3}+I_{T,4}.$

Consider $I_{T,1}$. Similar to the proof of part (1), we have

$\displaystyle\sup_{\tau\in[0,1]}\left\|\frac{1}{T}\sum_{t=1}^{T}\left[\bm{\eta}_{t}\bm{\eta}_{t}^{\top}-E\left(\bm{\eta}_{t}\bm{\eta}_{t}^{\top}\right)\right]K_{h}(\tau_{t}-\tau)\right\|=O_{P}\left(\sqrt{\frac{\log
T}{Th}}\right).$

Next, consider $I_{T,2}$. By Lemma B.5, we have

$\displaystyle\sup_{\tau\in[0,1]}\frac{1}{T}\sum_{t=1}^{T}\left\|\bm{Z}_{t-1}\right\|^{2}K_{h}(\tau_{t}-\tau)$
$\displaystyle\leq$
$\displaystyle\sup_{\tau\in[0,1]}\left|\frac{1}{T}\sum_{t=1}^{T}\mathrm{tr}\left[\bm{Z}_{t-1}^{\top}\bm{Z}_{t-1}-E(\bm{Z}_{t-1}^{\top}\bm{Z}_{t-1})\right]K_{h}(\tau_{t}-\tau)\right|$
$\displaystyle+\sup_{\tau\in[0,1]}\left|\frac{1}{T}\sum_{t=1}^{T}\mathrm{tr}\left[E(\bm{Z}_{t-1}^{\top}\bm{Z}_{t-1})\right]K_{h}(\tau_{t}-\tau)\right|$
$\displaystyle=$ $\displaystyle o_{P}(1)+O(1)=O_{P}(1).$

Hence, by the first result of Theorem 2.1

$\displaystyle\sup_{\tau\in[0,1]}\left\|I_{T,2}\right\|$ $\displaystyle\leq$
$\displaystyle\sup_{\tau\in[0,1]}\|\widehat{\bm{\beta}}(\tau)-\bm{\beta}(\tau)\|^{2}\cdot\sup_{\tau\in[0,1]}\frac{1}{T}\sum_{t=1}^{T}\left\|\bm{Z}_{t-1}\right\|^{2}K_{h}(\tau_{t}-\tau)=o_{P}\left(h^{2}+\sqrt{\frac{\log
T}{Th}}\right).$

Similarly, for $I_{T,3}$ and $I_{T,4}$, we have

$\displaystyle\sup_{\tau\in[0,1]}\left\|I_{T,3}\right\|$ $\displaystyle\leq$
$\displaystyle\sup_{\tau\in[0,1]}\left\|\widehat{\bm{\beta}}(\tau)-\bm{\beta}(\tau)\right\|\cdot\sup_{\tau\in[0,1]}\frac{1}{T}\sum_{t=1}^{T}\left\|\bm{Z}_{t-1}\bm{\eta}_{t}\right\|K_{h}(\tau_{t}-\tau)$
$\displaystyle\leq$
$\displaystyle\sup_{\tau\in[0,1]}\left\|\widehat{\bm{\beta}}(\tau)-\bm{\beta}(\tau)\right\|\cdot\left\\{\sup_{\tau\in[0,1]}\frac{1}{T}\sum_{t=1}^{T}\left\|\bm{Z}_{t-1}\right\|^{2}K_{h}(\tau_{t}-\tau)\right\\}^{1/2}$
$\displaystyle\cdot\left\\{\sup_{\tau\in[0,1]}\frac{1}{T}\sum_{t=1}^{T}\left\|\bm{\eta}_{t}\right\|^{2}K_{h}(\tau_{t}-\tau)\right\\}^{1/2}$
$\displaystyle=$ $\displaystyle O_{P}\left(h^{2}+\sqrt{\frac{\log
T}{Th}}\right).$

The proof is now completed. ∎

Define
$\bm{\Lambda}_{\mathsf{p}}(\tau)=[\bm{a}(\tau),\bm{A}_{\mathsf{p},1}(\tau),...,\bm{A}_{\mathsf{p},\mathsf{p}}(\tau)]$,
where $\bm{A}_{\mathsf{p},j}(\tau)=\bm{A}_{j}(\tau)$ for $1\leq j\leq p$ and
$\bm{A}_{\mathsf{p},j}(\tau)=0$ for $j>p$. Let
$\bm{z}_{\mathsf{p},t-1}=[1,\bm{x}_{t-1}^{\top},...,\bm{x}_{t-\mathsf{p}}^{\top}]^{\top}$,
$\bm{z}_{\mathsf{p},t-1}^{*}=\left[\bm{z}_{\mathsf{p},t-1}^{\top},\frac{\tau_{t}-\tau}{h}\bm{z}_{\mathsf{p},t-1}^{\top}\right]^{\top}$,
$\bm{Z}_{\mathsf{p},t}^{*}=\bm{z}_{\mathsf{p},t}^{*}\otimes\bm{I}_{d}$,
$\bm{M}_{\mathsf{p}}(\tau_{t})=\bm{\Lambda}_{\mathsf{p}}(\tau_{t})-\bm{\Lambda}_{\mathsf{p}}(\tau)-\bm{\Lambda}_{\mathsf{p}}^{(1)}(\tau)(\tau_{t}-\tau)-\frac{1}{2}h^{2}\bm{\Lambda}_{\mathsf{p}}^{(2)}(\tau)(\tau_{t}-\tau)^{2}$,
$\bm{\Lambda}_{\overline{\mathsf{p}}}(\tau)=[\bm{A}_{\mathsf{p},\mathsf{p}+1}(\tau),...,\bm{A}_{\mathsf{p},\mathsf{P}}(\tau)]$
and
$\bm{z}_{\overline{\mathsf{p}},t-1}=[\bm{x}_{t-\mathsf{p}-1}^{\top},...,\bm{x}_{t-\mathsf{P}}^{\top}]^{\top}$.

###### Proof of Lemma B.7.

(1). Since $\mathsf{p}\geq p$, we have