Therefore, we conclude item 3 by taking $\tilde{C}_{\sigma}=E[u_{i}^{2}]/2$.

For item 4, we note that item 2 implies

$P(|n^{-1}\sum_{i=1}^{n}\tilde{u}_{i}^{4k}-E[u_{i}^{4k}]|>1)\leq
Cn^{-1-\epsilon}~{}.$

Therefore, we conclude item 4 by taking $M=E[u_{i}^{4k}]+1$.

###  D.2 Uniform Consistency

Additional Notation: Define
$\xi_{n,i}(\rho_{n},h_{n})=y_{n,i+h}-\beta(\rho_{n},h_{n})y_{n,i}$ and recall
$\beta(\rho,h)\equiv\rho^{h}$. Algebra shows

$\xi_{n,i}(\rho_{n},h_{n})=\sum_{\ell=1}^{h_{n}}\rho_{n}^{h_{n}-\ell}u_{n,i+\ell}~{}~{}.$
(A.20)

###### Proof of Proposition B.1.

We use the general subsequence approach of Andrews et al. (2020) to show that
the uniform result in the proposition holds. We prove that for any sequence
$\\{\rho_{n}:n\geq 1\\}$ such that $|\rho_{n}|\leq 1+M/n$ and any sequence
$\\{\sigma_{n}^{2}:n\geq 1\\}\subset[\underline{\sigma},\>\overline{\sigma}]$,
there exists subsequences $\\{\rho_{n_{k}}:k\geq 1\\}$ and
$\\{\sigma_{n_{k}}^{2}:k\geq 1\\}$ such that

$\lim_{K\to\infty}~{}\lim_{k\to\infty}~{}P\left(~{}\frac{g(\rho_{n_{k}},n_{k})^{-2}\sum_{i=1}^{n_{k}}y_{n_{k},i-1}^{2}}{n_{k}~{}}\geq
1/K~{}\right)=1~{}.$ (A.21)

We consider two cases to prove (A.21). The first case is
$n_{k}(1-|\rho_{n_{k}}|)\to\infty$ for some subsequence $\\{n_{k}:k\geq 1\\}$,
which considers the subsequence of $\rho_{n}$ that stay on the stationary
region or go to the boundary at slower rates. The second case is
$n_{k}(1-|\rho_{n_{k}}|)\to c\in[-M,+\infty)$ for some subsequence
$\\{n_{k}:k\geq 1\\}$, which considers the subsequence of $\rho_{n}$ that goes
to the boundary (local-unit-model) or are on it (unit-root model). For both
cases, we assume $\sigma_{n_{k}}^{2}\to\sigma_{0}^{2}$ since any sequence
$\\{\sigma_{n}^{2}:n\geq 1\\}\subset[\underline{\sigma},\>\overline{\sigma}]$
has always a convergent subsequence. To avoid complicated sub-index notation,
we present the algebra derivation using the original sequence.

Case 1: Suppose $n(1-|\rho_{n}|)\to\infty$ and
$\sigma_{n}^{2}\to\sigma_{0}^{2}$ as $n\to\infty$. This condition implies that
there exists $N_{0}$ such that $|\rho_{n}|\leq 1$ for all $n\geq N_{0}$,
otherwise there is a subsequence $|\rho_{n_{k}}|$ in $(1,1+M/{n_{k}}]$ but
this cannot occur since $n_{k}(1-|\rho_{n_{k}}|)\in[-M,0]$. As a result, we
have
$g(\rho_{n},n)^{2}=\sum_{\ell=0}^{n-1}~{}\rho_{n}^{2\ell}\leq(1-\rho_{n}^{2})^{-1}$
that implies

$P\left(~{}\frac{g(\rho_{n},n)^{-2}\sum_{i=1}^{n}y_{n,i-1}^{2}}{n~{}}\geq
1/K~{}\right)\geq
P\left(~{}\frac{(1-\rho_{n}^{2})}{n}\sum_{i=1}^{n}y_{n,i-1}^{2}~{}\geq
1/K~{}\right)~{}.$

Therefore, to verify (A.21) is sufficient to prove that

$\lim_{K\to\infty}~{}\lim_{n\to\infty}~{}P\left(~{}\frac{(1-\rho_{n}^{2})}{n}\sum_{i=1}^{n}y_{n,i-1}^{2}~{}\geq
1/K~{}\right)=1~{},$

which follows if we prove

$\frac{(1-\rho_{n}^{2})\sum_{i=1}^{n}y_{n,i-1}^{2}}{n}\overset{p}{\to}\sigma_{0}^{2}~{}.$
(A.22)

To prove (A.22), we first prove that the expected value of the left-hand side
on (A.22) converges to $\sigma_{0}^{2}$. We then show that the variance of the
left-hand side on (A.22) goes to zero.

Step 1: Using Assumption B.1 and
$y_{n,i-1}=\sum_{\ell=1}^{i-1}\rho_{n}^{i-1-\ell}u_{n,\ell}$, we derive the
following:

$\displaystyle
E\left[\frac{(1-\rho_{n}^{2})}{n}\sum_{i=1}^{n}y_{n,i-1}^{2}\right]$
$\displaystyle=\frac{(1-\rho_{n}^{2})}{n}\sum_{i=1}^{n}\sum_{\ell=1}^{n-1}E[u_{n,\ell}^{2}]~{}\rho_{n}^{2(i-1-\ell)}~{}I\\{1\leq\ell\leq
i-1\\}$
$\displaystyle=\frac{(1-\rho_{n}^{2})}{n}\sum_{\ell=1}^{n-1}\sigma_{n}^{2}\sum_{i=\ell+1}^{n}\rho_{n}^{2(i-1-\ell)}$
$\displaystyle=\frac{\sigma_{n}^{2}(n-1)}{n}-\frac{\sigma_{n}^{2}}{n}\sum_{\ell=1}^{n-1}\rho_{n}^{2(n-\ell)}$

We conclude the right-hand side of the previous display converges to
$\sigma_{0}^{2}$ since
$n^{-1}\sum_{\ell=1}^{n}\rho_{n}^{2(n-\ell)}\leq\\{n(1-|\rho_{n}|)(1+|\rho_{n}|)\\}^{-1}$,
$n(1-|\rho_{n}|)\to\infty$, and $\sigma_{n}^{2}\to\sigma_{0}^{2}$.

Step 2: We use $E[y_{n,i-1}^{2}]=g(\rho_{n},i-1)^{2}\sigma_{n}^{2}$ to derive
the following decomposition

$\displaystyle\sum_{i=1}^{n}y_{n,i-1}^{2}-E[y_{n,i-1}^{2}]$
$\displaystyle=\sum_{i=1}^{n}\sum_{\ell=1}^{i-1}\left(u_{n,\ell}^{2}-\sigma_{n}^{2}\right)\rho_{n}^{2(i-1-\ell)}+2\sum_{i=1}^{n}\sum_{\ell_{1}=1}^{i-1}u_{n,\ell_{1}}\sum_{\ell_{2}=1}^{\ell_{1}-1}u_{n,\ell_{2}}\rho_{n}^{i-1-\ell_{1}}\rho_{n}^{i-1-\ell_{2}}~{}$
$\displaystyle=\sum_{\ell=1}^{n-1}\left(u_{n,\ell}^{2}-\sigma_{n}^{2}\right)b_{n,\ell}+2\sum_{\ell=1}^{n-1}u_{n,\ell}~{}d_{n,\ell}~{},$

where $b_{n,\ell}=\sum_{i=1+\ell}^{n}\rho_{n}^{2(i-1-\ell)}$ and
$d_{n,\ell}=b_{n,\ell}\sum_{\ell_{2}=1}^{\ell-1}u_{n,\ell_{2}}\rho_{n}^{\ell-\ell_{2}}$.
Note $d_{n,\ell}$ is measurable with respect to the $\sigma$-algebra defined
by $\\{u_{n,k}:1\leq k\leq\ell-1\\}$.

The decomposition above, Loeve’s inequality (see Theorem 9.28 in Davidson
(1994)), and Assumption B.1 imply that

$\displaystyle\text{Var}\left(\frac{(1-\rho_{n}^{2})\sum_{i=1}^{n}y_{n,i-1}^{2}}{n}\right)\leq\frac{2(1-\rho_{n}^{2})^{2}}{n^{2}}\left(\sum_{\ell=1}^{n-1}E\left[\left(u_{n,\ell}^{2}-\sigma_{n}^{2}\right)^{2}\right]b_{n,\ell}^{2}+4\sum_{\ell=1}^{n-1}E\left[u_{n,\ell}^{2}\right]E\left[d_{n,\ell}^{2}\right]\right)~{}.$

Since $b_{n,\ell}^{2}\leq(1-\rho_{n}^{2})^{-2}$ and
$E[d_{n,\ell}^{2}]=b_{n,\ell}^{2}\sigma_{n}^{2}\sum_{\ell_{2}=1}^{\ell-1}\rho_{n}^{2(\ell-\ell_{2})}\leq\sigma_{n}^{2}(1-\rho_{n}^{2})^{-3}$
for all $\ell=1,\dots,n-1$, and $E[(u_{n,\ell}^{2}-\sigma_{n}^{2})^{2}]\leq
E[u_{n,\ell}^{4}]\leq K_{4}$ by Assumption B.1, the right-hand side of the
previous display is lower than

$\frac{2(n-1)K_{4}}{n^{2}}+\frac{8(n-1)\sigma_{n}^{4}}{n^{2}(1-\rho_{n}^{2})}~{},$

which goes to 0 since $\sigma_{n}^{4}=E[u_{n,\ell}^{2}]^{2}\leq
E[u_{n,\ell}^{4}]\leq K_{4}$ and
$n(1-\rho_{n}^{2})=n(1-|\rho_{n}|)(1+|\rho_{n}|)\to\infty$ as $n\to\infty$.
This proves that the variance of the left-hand side on (A.22) goes to zero,
which prove (A.22) due to step 1.

Case 2: Suppose $n(1-|\rho_{n}|)\to c\in[-M,+\infty)$ and
$\sigma_{n}^{2}\to\sigma_{0}^{2}$ as $n\to\infty$. We first observe that
$g(\rho_{n},n)^{2}\leq n\exp(2M)$ due to $|\rho_{n}|\leq 1+M/n$ and the
following derivation:

$\displaystyle g(\rho_{n},n)^{2}$
$\displaystyle=\sum_{\ell=0}^{n-1}\rho_{n}^{2\ell}\leq
n(1+M/n)^{2n}=n\exp\left(2n\log\left(1+M/n\right)\right)\leq
n\exp\left(2M\right)~{},$

where the last inequality uses that $\log(1+x)\leq x$ for all $x>-1$. The
previous observation implies

$P\left(~{}\frac{g(\rho_{n},n)^{-2}\sum_{i=1}^{n}y_{n,i-1}^{2}}{n~{}}\geq
1/K~{}\right)\geq
P\left(~{}\frac{\exp(-2M)}{n^{2}}\sum_{i=1}^{n}y_{n,i-1}^{2}~{}\geq
1/K~{}\right)~{},$

where $\exp(-2M)$ is constant that does not change as $n\to\infty$ and
$K\to\infty$. Therefore, to verify (A.21) is sufficient to prove that

$\lim_{K\to\infty}~{}\lim_{n\to\infty}~{}P\left(~{}\frac{1}{n^{2}}\sum_{i=1}^{n}y_{n,i-1}^{2}~{}\geq
1/K~{}\right)=1~{},$

which follows if we prove

$\frac{1}{n^{2}}\sum_{i=1}^{n}y_{n,i-1}^{2}\overset{d}{\to}\sigma_{0}^{2}\int_{0}^{1}J_{-c}(r)^{2}dr~{},$
(A.23)

where $J_{-c}(r)=\int_{0}^{r}e^{-(r-s)c}dW(s)$ and $W(s)$ is a standard
Brownian motion.

To prove (A.23), we rely on the results and techniques presented in Phillips
(1987). Specifically, we adapt his Lemma 1 part (c) for the sequence of models
and the drifting parameter that we consider in this paper. We proceed in two
steps. First, we construct a triangular array $\\{\tilde{y}_{n,i}:1\leq i\leq
n;n\geq 1\\}$ that verify (A.23). Then, we prove that the constructed sequence
approximates the original triangular array $\\{y_{n,i}:1\leq i\leq n\\}$.

Step 1: Define $\tilde{u}_{n,i}=u_{n,i}I\\{\rho_{n}\geq
0\\}+(-1)^{i}u_{n,i}I\\{\rho_{n}<0\\}$ for all $i=1,\dots,n$. Note that the
sequence $\\{\tilde{u}_{n,i}:1\leq i\leq n\\}$ defines a martingale difference
array with the same variance $E[\tilde{u}_{n,i}^{2}]=\sigma_{n}^{2}$ and
satisfies that
$E[\tilde{u}_{n,i}^{2}]\in[\underline{\sigma},\overline{\sigma}]$, and
$E[\tilde{u}_{n,i}^{4}]\leq K_{4}$. Using this notation, we construct the
following triangular array:

$\tilde{y}_{n,i}=e^{-c/n}\tilde{y}_{n,i-1}+\tilde{u}_{n,i}~{},\quad\tilde{y}_{n,0}=0~{},$

where $c=\lim_{n\to\infty}n(1-|\rho_{n}|)$. Denote the sequence of partial
sums by $S_{n,j}=\sum_{i=1}^{j}u_{n,i}~{}$ for any $j=1,\dots,n$ and
$S_{n,0}=0$. Let us define the following random process

$X_{n}(r)=\frac{1}{\sqrt{n}}\frac{1}{\sigma_{n}}S_{n,[nr]}=\frac{1}{\sqrt{n}}\frac{1}{\sigma_{0}}S_{n,j-1}~{}\text{
if }(j-1)/n\leq r<j/n~{},$

and $X_{n}(1)=\frac{1}{\sqrt{n}}\frac{1}{\sigma_{n}}S_{n,n}$. By a functional
central limit theorem for martingale difference arrays (see Theorem 27.14 in
Davidson (1994)), we claim that $\\{X_{n}(r):r\in[0,1]\\}$ converges to the
standard Brownian motion process $\\{W(r):r\in[0,1]\\}$. To use this result,
we prove

$\displaystyle\text{(a)}\quad\sum_{i=1}^{n}\frac{\tilde{u}_{i,n}^{2}}{n\sigma_{n}^{2}}\overset{p}{\to}1~{},$
$\displaystyle\text{(b)}\quad\max_{i\leq
n}\left|\frac{\tilde{u}_{i,n}}{\sqrt{n}\sigma_{n}}\right|\overset{p}{\to}0~{},$
$\displaystyle\text{(c)}\quad\lim_{n}\sum_{i=1}^{[nr]}\frac{E[\tilde{u}_{i,n}^{2}]}{n\sigma_{n}^{2}}=r~{}.$

We can verify condition (a) using $\tilde{u}_{i,n}^{2}={u}_{i,n}^{2}$,
Chebyshev’s inequality and Assumption B.1:

$\displaystyle
P\left(\left|\sum_{i=1}^{n}\frac{\tilde{u}_{i,n}^{2}}{n\sigma_{n}^{2}}-1\right|>\epsilon\right)\leq\frac{1}{\epsilon^{2}}\frac{\sum_{i=1}^{n}E[{u}_{i,n}^{4}]}{n^{2}}\leq\frac{K_{4}}{\epsilon^{2}n}\to
0\quad\text{as }n\to\infty~{},$

for any $\epsilon>0$. To verify condition (b) holds is sufficient to show that

$nE\left[\frac{\tilde{u}_{i,n}^{2}}{n\sigma_{n}^{2}}I\left\\{\left|\frac{\tilde{u}_{i,n}}{\sqrt{n}\sigma_{n}}\right|>c\right\\}\right]\to
0~{},$

for any $c>0$, where $I\\{\cdot\\}$ is the indicator function. If the previous
display holds, then condition (b) follows by theorem 23.16 in Davidson (1994).
To verify the previous condition, note that

$\displaystyle
nE\left[\frac{\tilde{u}_{i,n}^{2}}{n\sigma_{n}^{2}}I\left\\{\left|\frac{\tilde{u}_{i,n}}{\sqrt{n}\sigma_{n}}\right|>c\right\\}\right]$
$\displaystyle\leq
nE\left[\frac{\tilde{u}_{i,n}^{4}}{n^{2}\sigma_{n}^{4}c^{2}}I\left\\{\left|\frac{\tilde{u}_{i,n}}{\sqrt{n}\sigma_{n}}\right|>c\right\\}\right]$
$\displaystyle\leq\frac{1}{n\sigma_{n}^{4}c^{2}}E[\tilde{u}_{i,n}^{4}]$
$\displaystyle\leq\frac{K_{4}}{n\underline{\sigma}^{2}c^{2}}~{},$

were the last inequality uses $\tilde{u}_{i,n}^{4}={u}_{i,n}^{4}$ and
Assumption B.1. Finally, condition (c) holds since
$E[\tilde{u}_{i,n}^{2}]=\sigma_{n}^{2}$.

Using the functional central limit theorem, the continuous mapping theorem,
and $\sigma_{n}\to\sigma_{0}$, we can repeat the arguments presented in the
proof of Lemma 1 in Phillips (1987) to conclude that

$\frac{1}{n^{2}}\sum_{i=1}^{n}\tilde{y}_{n,i-1}^{2}\overset{d}{\to}\sigma_{0}^{2}\int_{0}^{1}J_{-c}(r)^{2}dr~{}.$

Step 2: Define $a_{n}=|\rho_{n}|e^{c/n}$. Using that
$y_{n,i}=\sum_{\ell=1}^{i}\rho_{n}^{i-\ell}u_{n,i}$ and
$\tilde{y}_{n,i}=\sum_{\ell=1}^{i}e^{-c(i-\ell)/n}\tilde{u}_{n,i}$, we obtain
that

$\tilde{y}_{n,i}=\begin{cases}~{}y_{n,i}-R_{n,i},&\text{if $\rho_{n}\geq
0$,}\\\ ~{}(-1)^{i}y_{n,i}-R_{n,i},&\text{if $\rho_{n}<0$,}\end{cases}$

where
$R_{n,i}=\sum_{\ell=1}^{i}\left(a_{n}^{i-\ell}-1\right)e^{-c(i-\ell)/n}\tilde{u}_{n,\ell}~{}.$
Therefore, we conclude that
$y_{n,i}^{2}=\tilde{y}_{n,i}^{2}+2\tilde{y}_{n,i}R_{n,i}+R_{n,i}^{2}$. This
implies that

$\left|\frac{1}{n^{2}}\sum_{i=1}^{n}y_{n,i-1}^{2}-\frac{1}{n^{2}}\sum_{i=1}^{n}\tilde{y}_{n,i-1}^{2}\right|\leq\left|\frac{2}{n^{2}}\sum_{i=1}^{n}\tilde{y}_{n,i-1}R_{n,i}\right|+\left|\frac{1}{n^{2}}\sum_{i=1}^{n}R_{n,i}^{2}\right|~{}.$

By Cauchy-Schwartz’s inequality, the right-hand side of the previous
expression is lower or equal than

$2\left(\frac{1}{n^{2}}\sum_{i=1}^{n}\tilde{y}_{n,i-1}^{2}\right)^{1/2}\left(\frac{1}{n^{2}}\sum_{i=1}^{n}R_{n,i}^{2}\right)^{1/2}+\left|\frac{1}{n^{2}}\sum_{i=1}^{n}R_{n,i}^{2}\right|~{}.$
(A.24)

By the result at the end of Step 1, we have
$n^{-2}\sum_{i=1}^{n}\tilde{y}_{n,i-1}^{2}$ is $O_{p}\left(1\right)$.
Therefore, we claim that

$n^{-2}\sum_{i=1}^{n}R_{n,i}^{2}\overset{p}{\to}0~{},$

which is sufficient to conclude that (A.24) converges to zero in probability.

To verify the claim, we first observe that
$a_{n}^{j}e^{-(i-\ell)c/n}=|\rho_{n}|^{j}e^{-(i-\ell-j)c/n}\leq|\rho_{n}|^{j}$
for all $j=0,\dots,i-\ell-1$, which implies that

$\displaystyle|R_{n,i}|$
$\displaystyle=\left|\sum_{\ell=1}^{i}(a_{n}-1)(1+a_{n}+...+a_{n}^{i-\ell-1})e^{-(i-\ell)c/n}\tilde{u}_{n,\ell}\right|$
$\displaystyle\leq|a_{n}-1|\sum_{\ell=1}^{i}(1+|\rho_{n}|+...+|\rho_{n}|^{i-\ell-1})|\tilde{u}_{n,\ell}|~{}.$

Using the previous inequality and
$|\rho_{n}|^{j}\leq(1+M/n)^{j}\leq(1+M/n)^{n}\leq e^{M}$, we obtain

$|R_{n,i}|\leq e^{M}|a_{n}-1|\sum_{\ell=1}^{i}(i-\ell)|\tilde{u}_{n,\ell}|\leq
e^{M}|n(a_{n}-1)|\sum_{\ell=1}^{n}|u_{n,\ell}|~{},$

for all $i=1,\dots,n$, where we used that $|\tilde{u}_{n,\ell}|=|u_{n,\ell}|$
in the last inequality. Then, we derive

$n^{-2}\sum_{i=1}^{n}R_{n,i}^{2}\leq
e^{2M}|n(a_{n}-1)|^{2}\left(n^{-1}\sum_{\ell=1}^{n}|u_{n,\ell}|\right)^{2}~{}.$

By Markovs’s inequality and Assumption B.1, we obtain that
$n^{-1}\sum_{\ell=1}^{n}|u_{n,\ell}|$ is $O_{p}\left(1\right)$. Analyzing
$a_{n}-1=e^{c/n}\left(|\rho_{n}|-e^{-c/n}\right)$, we can conclude that
$n(a_{n}-1)=o\left(1\right)$, which implies that the right-hand side of the
previous display converges to zero in probability. As a result, we conclude
that (A.24) converges to zero in probability, which implies that

$\frac{1}{n^{2}}\sum_{i=1}^{n}y_{n,i-1}^{2}-\frac{1}{n^{2}}\sum_{i=1}^{n}\tilde{y}_{n,i-1}^{2}=o_{p}\left(1\right)~{},$

and by the result at the end of Step 1, we conclude (A.23).

###### Proof of Theorem B.1.

The derivations presented on page 1811 in Montiel Olea and Plagborg-Møller
(2021) implies

$R_{n}(h)=\frac{\sum_{i=1}^{n-h}\xi_{n,i}(h)\hat{u}_{n,i}(h)}{\left(\sum_{i=1}^{n-h}\hat{\xi}_{n,i}(h)^{2}\hat{u}_{n,i}(h)^{2}\right)^{1/2}}~{},$

which is equal to

$\left(\frac{\sum_{i=1}^{n-h}\xi_{n,i}(h)u_{n,i}}{(n-h)^{1/2}V(\rho,h)^{1/2}}+\frac{\sum_{i=1}^{n-h}\xi_{n,i}(\rho,h)(\hat{u}_{n,i}(h)-u_{n,i})}{(n-h)^{1/2}V(\rho,h)^{1/2}}\right)\times\left(\frac{(n-h)V(\rho,h)}{\sum_{i=1}^{n-h}\hat{\xi}_{n,i}(h)^{2}\hat{u}_{n,i}(h)^{2}}\right)^{1/2}~{},$

where $V(\rho,h)=E[\xi_{n,i}(h)^{2}u_{n,i}^{2}]$. We then follow their
approach and prove that under Assumption B.1: for any sequences
$\\{\rho_{n}:n\geq 1\\}\subset[-1,1]$, $\\{\sigma_{n}^{2}:n\geq
1\\}\subset[\underline{\sigma},\>\overline{\sigma}]$, and $\\{h_{n}:n\geq
1\\}$ satisfying $h_{n}=o(n)$ and $h_{n}\leq n$, we have

$\displaystyle(i)$
$\displaystyle\quad\frac{\sum_{i=1}^{n-h_{n}}\xi_{n,i}(h_{n})u_{n,i}}{(n-h_{n})^{1/2}V(\rho_{n},h_{n})^{1/2}}\overset{d}{\to}N(0,1)~{},$
$\displaystyle(ii)$
$\displaystyle\quad\frac{\sum_{i=1}^{n-h}\xi_{n,i}(\rho,h)(\hat{u}_{n,i}(h)-u_{n,i})}{(n-h)^{1/2}V(\rho,h)^{1/2}}\overset{p}{\to}0~{},$
$\displaystyle(iii)$
$\displaystyle\quad\frac{\sum_{i=1}^{n-h}\hat{\xi}_{n,i}(h)^{2}\hat{u}_{n,i}(h)^{2}}{(n-h)V(\rho,h)}\overset{p}{\to}1~{}$

Finally, Lemmas D.4, D.5, and D.7 imply (i), (ii), and (iii), respectively.

######  Lemma D.1.

Suppose Assumption B.1 holds. Then, for any $(\rho_{n},\sigma_{n},h_{n})$ such
that $|\rho_{n}|\leq 1+M/n$,
$\sigma_{n}^{2}\in[\underline{\sigma},\>\overline{\sigma}]$, and $h_{n}\leq
n$, we have

$E\left[\xi_{n,i}(\rho_{n},h_{n})^{4}\right]\leq
4g(\rho_{n},h_{n})^{4}K_{4}~{},$

where $g(\rho,k)=\left(\sum_{\ell=0}^{k-1}~{}\rho^{2\ell}\right)^{1/2}$ and
$\xi_{n,i}(\rho_{n},h_{n})$ is in (A.20).

###### Proof.

Using the definition of $\xi_{n,i}(\rho_{n},h_{n})$ in (A.20), we obtain

$\displaystyle E\left[\xi_{n,i}(\rho_{n},h_{n})^{4}\right]$
$\displaystyle=E\left[\left(\sum_{\ell=1}^{h_{n}}\rho_{n}^{h_{n}-\ell}u_{n,i+\ell}\right)^{4}\right]$
$\displaystyle=\sum_{\ell_{1}=1}^{h_{n}}\rho^{4(h_{n}-\ell_{2})}E\left[u_{n,i+\ell_{1}}^{4}\right]+3\sum_{\ell_{1}=1}^{h_{n}}\sum_{\ell_{2}=1}^{\ell_{1}-1}\rho^{2(h_{n}-\ell_{1})}\rho^{2(h_{n}-\ell_{2})}E\left[u_{n,i+\ell_{1}}^{2}u_{n,i+\ell_{1}}^{2}\right]~{}$
$\displaystyle\leq
4\sum_{\ell_{1}=1}^{h_{n}}\sum_{\ell_{2}=1}^{h_{n}}\rho^{2(h_{n}-\ell_{1})}\rho^{2(h_{n}-\ell_{2})}K_{4}~{}=4g(\rho_{n},h_{n})^{4}K_{4}~{},$

where the second equality follows by noting that terms in the sum
$u_{n,i_{1}}^{3}u_{n,i_{2}}$, $u_{n,i_{1}}^{2}u_{n,i_{2}}u_{n,i_{3}}$, and
$u_{n,i_{1}}u_{n,i_{2}}u_{n,i_{3}}u_{n,i_{4}}$ have zero mean whenever the
four sub-index $i_{1}$,$i_{2}$, $i_{3}$, $i_{4}$ are different due to
Assumption B.1; therefore, only the terms with even exponent appears. The
inequality above follows by Assumption B.1 since
$E\left[u_{n,i+\ell_{1}}^{2}u_{n,i+\ell_{1}}^{2}\right]=\sigma_{n}^{4}\leq
E[u_{n,i}^{4}]\leq K_{4}$ for $\ell_{1}\neq\ell_{2}$. Finally, the last
equality follows by the definition of $g(\rho_{n},h_{n})$.

######  Lemma D.2.

Suppose Assumption B.1 holds. Then, for any $(\rho_{n},\sigma_{n},h_{n})$ such
that $|\rho_{n}|\leq 1+M/n$,
$\sigma_{n}^{2}\in[\underline{\sigma},\>\overline{\sigma}]$, and $h_{n}\leq
n$, we have

$\displaystyle
E\left[\left(\frac{\sum_{i=1}^{n-h_{n}}\xi_{n,i}(\rho_{n},h_{n})y_{n,i-1}}{(n-h_{n})g(\rho_{n},n-h_{n})g(\rho_{n},h_{n})\sigma_{n}^{2}}\right)^{2}\right]\leq\frac{n}{n-h_{n}}\times\frac{h_{n}}{n-h_{n}}\times\frac{\sqrt{4K_{4}}}{\underline{\sigma}}~{}.$

where $g(\rho,k)=\left(\sum_{\ell=0}^{k-1}~{}\rho^{2\ell}\right)^{1/2}$ and
$\xi_{n,i}(\rho_{n},h_{n})$ is in (A.20).

###### Proof.

Using the definition of $\xi_{n,i}(\rho_{n},h_{n})$ in (A.20), we write

$\displaystyle\sum_{i=1}^{n-h_{n}}\xi_{n,i}(\rho_{n},h_{n})y_{n,i-1}$
$\displaystyle=\sum_{i=1}^{n-h_{n}}\sum_{\ell=1}^{h_{n}}\rho_{n}^{h_{n}-\ell}u_{n,i+\ell}~{}y_{n,i-1}$
$\displaystyle=\sum_{i=1}^{n-h_{n}}\sum_{j=i+1}^{i+h}\rho_{n}^{i+h-j}u_{n,j}~{}y_{n,i-1}$
$\displaystyle=\sum_{j=1}^{n}u_{n,j}b_{n,j}~{},$

where

$b_{n,j}=\sum_{i=j-h_{n}}^{j-1}\rho_{n}^{i+h_{n}-j}~{}y_{n,i-1}I\\{1\leq i\leq
n-h\\}~{}.$

Note that $b_{n,j}$ is measurable with respect to the $\sigma$-algebra defined
by $\\{u_{n,k}:1\leq k\leq j-2\\}$. Using Assumption B.1, we obtain

$E\left[\left(\sum_{j=1}^{n}u_{n,j}b_{n,j}\right)^{2}\right]=\sum_{j=1}^{n}E[u_{n,j}^{2}b_{n,j}^{2}]=\sigma_{n}^{2}\sum_{j=1}^{n}E[b_{n,j}^{2}]~{}.$

Therefore, the derivation above implies

$E\left[\left(\sum_{i=1}^{n-h_{n}}\xi_{n,i}(\rho_{n},h_{n})y_{n,i-1}\right)^{2}\right]=\sigma_{n}^{2}\sum_{j=1}^{n}E[b_{n,j}^{2}]~{}.$

We claim that

$E[b_{n,j}^{2}]\leq
h_{n}g(\rho_{n},h_{n})^{2}g(\rho_{n},n-h_{n})^{2}\sqrt{4K_{4}}~{},$ (A.25)

for any $j=1,\dots,n$. The previous claim and Assumption B.1 imply

$E\left[\left(\frac{\sum_{i=1}^{n-h_{n}}\xi_{n,i}(\rho_{n},h_{n})y_{n,i-1}}{(n-h_{n})g(\rho_{n},n-h_{n})g(\rho_{n},h_{n})\sigma_{n}^{2}}\right)^{2}\right]\leq\frac{nh_{n}}{(n-h_{n})^{2}\sigma_{n}^{2}}\times\sqrt{4K_{4}}\leq\frac{nh_{n}}{(n-h_{n})^{2}\underline{\sigma}}\times\sqrt{4K_{4}}~{}.$

To verify (A.25), we consider three cases. The first case is $j\leq h_{n}$, in
which we derive that

$\displaystyle E[b_{j}^{2}]$
$\displaystyle=E[(\sum_{i=1}^{j-1}\rho^{i+h-j}y_{n,i-1})^{2}]$
$\displaystyle\leq(j-1)E[\sum_{i=1}^{j-1}\rho^{2(i+h-j)}y_{n,i-1}^{2}]$
$\displaystyle\leq
h_{n}\left(\sum_{i=1}^{j-1}\rho^{2(i+h-j)}\right)g(\rho_{n},n-h_{n})^{2}\sqrt{4K_{4}}~{}$
$\displaystyle\leq
h_{n}g(\rho_{n},h_{n})^{2}g(\rho_{n},n-h_{n})^{2}\sqrt{4K_{4}}~{},$

where we use Loeve’s inequality (see Theorem 9.28 in Davidson (1994)) in the
first inequality above. In the second inequality, we use $E[y_{n,i-1}^{2}]\leq
E[y_{n,i-1}^{4}]^{1/2}$, $y_{n,i-1}=\xi_{n,0}(\rho_{n},i-1)$, Lemma D.1, and
$g(\rho_{n},i-1)^{2}\leq g(\rho_{n},n-h_{n})^{2}$ for all $i=1,\dots,n-h_{n}$.
Note that we also use $j\leq h_{n}$, which also implies the last inequality
above. The second case is $h_{n}+1\leq j\leq n-h_{n}+1$. We follow the same
approach as before and derive

$\displaystyle E[b_{j}^{2}]$
$\displaystyle=E[(\sum_{i=j-h_{n}}^{j-1}\rho_{n}^{i+h_{n}-j}y_{n,i-1})^{2}]$
$\displaystyle\leq
h_{n}E[\sum_{i=j-h_{n}}^{j-1}\rho^{2(i+h_{n}-j)}y_{n,i-1}^{2}]$
$\displaystyle\leq
h_{n}g(\rho_{n},h_{n})^{2}g(\rho_{n},n-h_{n})^{2}\sqrt{4K_{4}}~{},$

where the inequalities are obtained using the same tools described for the
first case. In the final case, we have $j\geq n-h_{n}+2$. As before, we obtain

$\displaystyle E[b_{j}^{2}]$
$\displaystyle=E[(\sum_{i=j-h_{n}}^{n-h_{n}}\rho^{i+h-j}y_{n,i-1})^{2}]$
$\displaystyle\leq
h_{n}g(\rho_{n},h_{n})^{2}g(\rho_{n},n-h_{n})^{2}\sqrt{4K_{4}}~{}.$

The same arguments presented for the first case can be used to derive the
previous inequality; in this case, we use $n-j+1\leq h_{n}$. The derivation
above verifies (A.25).

######  Lemma D.3.

Suppose Assumption B.1 holds. Then, for any $(\rho_{n},\sigma_{n},h_{n})$ such
that $|\rho_{n}|\leq 1+M/n$,
$\sigma_{n}^{2}\in[\underline{\sigma},\>\overline{\sigma}]$, and $h_{n}\leq
n$, we have

$E\left[\left(\frac{\sum_{i=1}^{n-h_{n}}u_{n,i}y_{n,i-1}}{(n-h_{n})^{1/2}g(\rho_{n},n-h_{n})}\right)^{2}\right]\leq
2K_{4}~{},$

where $g(\rho,k)=\left(\sum_{\ell=0}^{k-1}~{}\rho^{2\ell}\right)^{1/2}$.

###### Proof.

Using Assumption B.1 we obtain

$\displaystyle
E\left[\left(\frac{\sum_{i=1}^{n-h_{n}}u_{n,i}y_{n,i-1}}{(n-h_{n})^{1/2}g(\rho_{n},n-h_{n})}\right)^{2}\right]$
$\displaystyle=\frac{\sum_{i=1}^{n-h_{n}}E[u_{n,i}^{2}]E[y_{n,i-1}^{2}]}{(n-h_{n})g(\rho_{n},n-h_{n})^{2}}$
$\displaystyle\leq\frac{\sigma_{n}^{2}}{n-h_{n}}\sum_{i=1}^{n-h_{n}}\left(\frac{E[y_{n,i-1}^{4}]}{g(\rho_{n},n-h_{n})^{4}}\right)^{1/2}~{}.$

Note that $y_{n,i-1}=\xi_{n,0}(\rho_{n},i-1)$ and $g(\rho_{n},i-1)^{2}\leq
g(\rho_{n},n-h_{n})^{2}$ for all $i=1,\dots,n-h_{n}$. By Lemma D.1, we
conclude that the right-hand side of the expression above is lower or equal
than

$\sigma_{n}^{2}\left(4K_{4}\right)^{1/2}~{},$

which is lower than $\sqrt{K_{4}}\sqrt{4K_{4}}$ by Assumption B.1. This
completes the proof.

######  Lemma D.4.

Suppose Assumptions B.1 hold. Then, for any sequences $\\{\rho_{n}:n\geq 1\\}$
such that $|\rho_{n}|\leq 1+M/n$, $\\{\sigma_{n}^{2}:n\geq
1\\}\subset[\underline{\sigma},\>\overline{\sigma}]$, and $\\{h_{n}:n\geq
1\\}$ satisfying $h_{n}=o(n)$ and $h_{n}\leq n$, we have

$\frac{\sum_{i=1}^{n-h_{n}}\xi_{n,i}(\rho_{n},h_{n})u_{n,i}}{(n-h_{n})^{1/2}g(\rho_{n},h_{n})\sigma_{n}^{2}}\overset{d}{\to}N(0,1)~{},$
(A.26)

where $\xi_{n,i}(\rho_{n},h_{n})$ is in (A.20) and
$g(\rho,h)^{2}=\sum_{\ell=1}^{h}\rho^{2h}$.

###### Proof.

We adapt the proof of Lemma A1 in Montiel Olea and Plagborg-Møller (2021). We
start by writing the term on the left-hand side term in (A.26) as follows

$\sum_{i=1}^{n-h_{n}}\chi_{n,i}~{},$

where

$\chi_{n,i}=\frac{\xi_{n,n-h_{n}+1-i}(\rho_{n},h_{n})u_{n,n-h_{n}+1-i}}{(n-h_{n})^{1/2}g(\rho_{n},h_{n})\sigma_{n}^{2}}~{},$

for $i=1,\dots,n-h_{n}$. Define the $\sigma$-algebra
$\mathcal{F}_{n,i}=\sigma\left(u_{n-h_{n}+j-i}:j\geq 1\right)$. Note that for
any $i=1,\dots,n-h_{n}$, $\chi_{n,i}$ is measurable with respect to
$\mathcal{F}_{n,i}$. Therefore, the sequence $\\{\chi_{n,i}:1\leq i\leq
n-h_{n}\\}$ is adapted to the filtration $\\{\mathcal{F}_{n,i}:1\leq i\leq
n-h_{n}\\}$. Moreover, $\xi_{n,n-h_{n}+1-i}(\rho_{n},h_{n})$ is measurable
with respect to $\mathcal{F}_{n,i-1}$ since it is a function of
$\\{u_{n,n-h_{n}+j-(i-1)}:1\leq j\leq h_{n}\\}$. This implies that
$E[\chi_{n,i}|\mathcal{F}_{n,i-1}]=0$ since

$E[\chi_{n,i}|\mathcal{F}_{n,i-1}]=\frac{\xi_{n,n-h_{n}+1-i}(\rho_{n},h_{n})}{(n-h_{n})^{1/2}g(\rho_{n},h_{n})\sigma_{n}^{2}}E[u_{n,n-h_{n}+1-i}|\mathcal{F}_{n,i-1}]$

and by Assumption B.1 we conclude
$E[u_{n,n-h_{n}+1-i}|\mathcal{F}_{n,i-1}]=E[u_{n,n-h_{n}+1-i}]=0$.

The derivation presented above proves that the sequence $\\{\chi_{n,i}:1\leq
i\leq n-h_{n}\\}$ is a martingale difference array with respect to the
filtration $\\{\mathcal{F}_{n,i}:1\leq i\leq n-h_{n}\\}$. The result in (A.26)
then follows by a martingale central limit theorem (see Theorem 24.3 in
Davidson (1994)), which requires verifying that

$\displaystyle(i)\quad\sum_{i=1}^{n-h_{n}}E[\chi_{n,i}^{2}]=1~{},$
$\displaystyle(ii)\quad\sum_{i=1}^{n-h_{n}}\chi_{n,i}^{2}\overset{p}{\to}1~{},$
$\displaystyle(iii)\quad\max_{1\leq i\leq
n-h_{n}}|\chi_{n,i}|\overset{p}{\to}0~{}.$

The condition $(i)$ follows by using that
$E[\xi_{n,i}(\rho_{n},h_{n})^{2}u_{n,i}^{2}]=g(\rho_{n},h_{n})^{2}\sigma_{n}^{4}$.
To prove the condition $(ii)$ is sufficient to show

$\text{Var}\left(\sum_{i=1}^{n-h_{n}}\chi_{n,i}^{2}\right)\to 0~{}.$ (A.27)

To prove (A.27), we first recall that

$\sum_{i=1}^{n-h_{n}}\chi_{n,i}^{2}=\frac{1}{(n-h_{n})g(\rho_{n},h_{n})^{2}\sigma_{n}^{4}}\times\sum_{i=1}^{n-h_{n}}\xi_{n,i}(\rho_{n},h_{n})^{2}u_{n,i}^{2}~{},$

where the second term of the right-hand side of the previous display can be
decomposed into the sum of its expected value and another three zero mean
terms:

$(n-h_{n})g(\rho_{n},h_{n})^{2}\sigma_{n}^{4}+\sum_{j=1}^{n}(u_{n,j}^{2}-\sigma_{n}^{2})b_{n,j}+\sum_{j=1}^{n}u_{n,j}d_{n,j}+\sum_{j=1}^{n}(u_{n,j}^{2}-\sigma_{n}^{2})g(\rho_{n},h_{n})^{2}\sigma_{n}^{2}~{},$

where

$\displaystyle b_{n,j}$
$\displaystyle=\sum_{i=j-h_{n}}^{j-1}\rho_{n}^{2(h_{n}+i-j)}u_{n,i}^{2}I\\{1\leq
i\leq n-h_{n}\\}~{},$ $\displaystyle d_{n,j}$
$\displaystyle=\sum_{i=j-h_{n}}^{j-1}\sum_{\ell_{2}=1}^{j-i-1}u_{n,i+\ell_{2}}\rho_{n}^{2h-j+i-\ell_{2}}u_{n,i}^{2}I\\{1\leq
i\leq n-h_{n}\\}~{}.$

Note that $b_{n,j}$ and $d_{n,j}$ are measurable with respect to the
$\sigma-$algebra $\sigma(u_{n,k}:1\leq k\leq j-1\\}$. By Assumption B.1 and
Loeve’s inequality (Theorem 9.28 in Davidson (1994)), we conclude

$\displaystyle E[b_{n,j}^{2}]$ $\displaystyle\leq
h_{n}\sum_{i=j-h_{n}}^{j-1}E[\rho_{n}^{4(h_{n}+i-j)}u_{n,i}^{4}]$
$\displaystyle\leq h_{n}g(\rho_{n},h_{n})^{4}K_{4}$

and

$\displaystyle E[d_{n,j}^{2}]$ $\displaystyle\leq
h_{n}E\left[\left(\sum_{\ell_{2}=1}^{j-i-1}u_{n,i+\ell_{2}}\rho_{n}^{2h-j+i-\ell_{2}}u_{n,i}^{2}\right)^{2}\right]$
$\displaystyle\leq h_{n}g(\rho_{n},h_{n})^{4}\sigma_{n}^{2}K_{4}$

for all $j=1,\dots,n$.

We use the decomposition presented above, Assumption B.1, and Loeve’s
inequality (see Theorem 9.28 in Davidson (1994)) imply that the left-hand side
of (A.27) is lower or equal than

$\frac{3\sum_{j=1}^{n}E[(u_{n,j}^{2}-\sigma_{n}^{2})^{2}b_{n,j}^{2}]+3\sum_{j=1}^{n}E[u_{n,j}^{2}d_{n,j}^{2}]+3g(\rho_{n},h_{n})^{4}\sigma_{n}^{4}\sum_{j=1}^{n}E[(u_{n,j}^{2}-\sigma_{n}^{2})^{2}]}{(n-h_{n})^{2}g(\rho_{n},h_{n})^{4}\sigma_{n}^{8}}~{}.$

By Assumption B.1 and the upper bounds that we found for $E[b_{n,j}^{2}]$ and
$E[d_{n,j}^{2}]$, the previous expression is lower or equal than

$\frac{3nh_{n}K_{4}^{2}}{(n-h_{n})^{2}\underline{\sigma}^{4}}+\frac{3nh_{n}K_{4}}{(n-h_{n})^{2}\underline{\sigma}^{2}}+\frac{3nK_{4}}{{(n-h_{n})^{2}\underline{\sigma}^{2}}}~{}.$

The previous expression converges to zero since $h_{n}/n\to 0$ as
$n\to\infty$. This implies that (A.27) holds.

Finally, to verify that condition (iii) holds is sufficient to show

$(n-h_{n})E\left[\chi_{n,i}^{2}I\\{|\chi_{n,i}^{2}|>c\\}\right]\to 0~{},$
(A.28)

for any $c>0$, where $I\\{\cdot\\}$ is the indicator function. If the
condition in (A.35) holds, then condition (iii) follows by Theorem 23.16 in
Davidson (1994). To verify (A.28), note that

$\displaystyle(n-h_{n})E\left[\chi_{n,i}^{2}I\\{|\chi_{n,i}^{2}|>c\\}\right]$
$\displaystyle\leq~{}(n-h_{n})E\left[\frac{\chi_{n,i}^{4}}{c^{2}}I\\{|\chi_{n,i}^{2}|>c\\}\right]$
$\displaystyle\leq~{}(n-h_{n})\frac{E\left[\chi_{n,i}^{4}\right]}{c^{2}}$
$\displaystyle=\frac{E\left[\xi_{n,n-h_{n}+1-i}(\rho_{n},h_{n})^{4}\right]E\left[u_{n,n-h_{n}+1-i}^{4}\right]}{(n-h_{n})\sigma_{n}^{8}g(\rho_{n},h_{n})^{4}c^{2}}$
$\displaystyle\leq\frac{4K_{4}E\left[u_{n,n-h_{n}+1-i}^{4}\right]}{(n-h_{n})\sigma_{n}^{8}c^{2}}~{},$

where the equality above uses Assumption B.1, and the last inequality follows
by Lemma D.1. By Assumption B.1 we obtain

$(n-h_{n})E\left[\chi_{n,i}^{2}I\\{|\chi_{n,i}^{2}|>c\\}\right]\leq\frac{4K_{4}^{2}}{(n-h_{n})\underline{\sigma}^{4}c^{2}}~{},$

which is sufficient to conclude (A.28).

######  Lemma D.5.

Suppose Assumption B.1 holds. Then, for any sequences $\\{\rho_{n}:n\geq 1\\}$
such that $|\rho_{n}|\leq 1+M/n$, $\\{\sigma_{n}^{2}:n\geq
1\\}\subset[\underline{\sigma},\>\overline{\sigma}]$, and $\\{h_{n}:n\geq
1\\}$ satisfying $h_{n}=o(n)$ and $h_{n}\leq n$, we have

$\frac{\sum_{i=1}^{n-h}\xi_{n,i}(\rho_{n},h_{n})(\hat{u}_{n,i}(h_{n})-u_{n,i})}{(n-h_{n})^{1/2}g(\rho_{n},h_{n})\sigma_{n}^{2}}\overset{p}{\to}0~{},$
(A.29)

where $\hat{u}_{n,i}(h_{n})=y_{n,i}-\hat{\rho}_{n}(h_{n})y_{n,i-1}$,
$\hat{\rho}_{n}(h_{n})$ is defined in (A.7), and
$g(\rho,h)^{2}=\sum_{\ell=1}^{h}\rho^{2h}$.

###### Proof.

We use $\hat{u}_{n,i}(h_{n})=y_{n,i}-\hat{\rho}_{n}(h_{n})y_{n,i-1}$ and the
definition of $\hat{\rho}_{n}(h_{n})$ to rewrite the left hand side in (A.29)
equal to

$\left(\frac{\sum_{i=1}^{n-h}\xi_{n,i}(\rho,h)y_{n,i-1}}{(n-h_{n})g(\rho_{n},n-h_{n})g(\rho_{n},h_{n})\sigma_{n}^{2}}\right)\times\left(\frac{\sum_{i=1}^{n-h_{n}}u_{n,i}y_{n,i-1}}{(n-h_{n})^{1/2}g(\rho_{n},n-h_{n})}\right)\times
C_{n}~{},$ (A.30)

where

$C_{n}\equiv~{}\left(\frac{g(\rho_{n},n-h_{n})^{-2}\sum_{i=1}^{n-h_{n}}y_{n,i-1}^{2}}{n-h_{n}~{}}\right)^{-1}~{}.$

By Lemma D.2, we claim that

$E\left[\left(\frac{\sum_{i=1}^{n-h}\xi_{n,i}(\rho,h)y_{n,i-1}}{(n-h_{n})g(\rho_{n},n-h_{n})g(\rho_{n},h_{n})\sigma_{n}^{2}}\right)^{2}\right]\to
0~{},$ (A.31)

which implies that the first term in (A.30) converges to zero in probability
since we have $E[\xi_{n,i}(\rho,h)y_{n,i-1}]=0$ and (A.31) is equivalent to
say that the variance goes to zero.

By Lemma D.3, we have

$E\left[\left(\frac{\sum_{i=1}^{n-h_{n}}u_{n,i}y_{n,i-1}}{(n-h_{n})^{1/2}g(\rho_{n},n-h_{n})}\right)^{2}\right]\leq
2K_{4}~{}.$ (A.32)

which implies that the second term in (A.30) is bounded in probability.

Finally, by Proposition B.1 we claim that $C_{n}$ is bounded in probability.
Therefore, the expression in (A.30) converges to zero in probability since the
first term is $o_{p}(1)$ while the other two terms are $O_{p}(1)$. This
concludes the proof of the lemma.

######  Lemma D.6.

Suppose Assumption B.1 holds. Then, for any sequences $\\{\rho_{n}:n\geq 1\\}$
such that $|\rho_{n}|\leq 1+M/n$, $\\{\sigma_{n}^{2}:n\geq
1\\}\subset[\underline{\sigma},\>\overline{\sigma}]$, and $\\{h_{n}:n\geq
1\\}$ satisfying $h_{n}=o(n)$ and $h_{n}\leq n$, we have

$\displaystyle(i)$
$\displaystyle\quad\frac{\hat{\beta}_{n}(h_{n})-\beta(\rho_{n},h_{n})}{g(\rho_{n},h_{n})}\overset{p}{\to}0~{},$
$\displaystyle(ii)$
$\displaystyle\quad\frac{g(\rho_{n},n-h_{n})\left(\hat{\eta}(\rho_{n},h_{n})-\eta(\rho_{n},h_{n})\right)}{g(\rho_{n},h_{n})}\overset{p}{\to}0~{},$
$\displaystyle(iii)$ $\displaystyle\quad(n-h_{n})^{1/2}\times
g(\rho_{n},n-h_{n})\times\left(\hat{\rho}_{n}(h_{n})-\rho_{n}\right)=O_{p}(1)~{},$

where
$\hat{\eta}(\rho_{n},h_{n})=\rho_{n}\hat{\beta}_{n}(h_{n})+\hat{\gamma}_{n}(h_{n})$,
$\eta(\rho_{n},h_{n})=\rho_{n}\beta(\rho_{n},h_{n})=\rho_{n}^{h_{n}+1}$,
$\hat{\beta}_{n}(h_{n})$ and $\hat{\gamma}_{n}(h_{n})$ are defined in (A.6),
$\hat{\rho}_{n}(h_{n})$ is in (A.7), and
$g(\rho,h)^{2}=\sum_{\ell=1}^{h}\rho^{2h}$.

###### Proof.

This proof follows Lemma A3 in Montiel Olea and Plagborg-Møller (2021). We
claim that $\hat{\beta}_{n}(h_{n})$ and $\hat{\eta}_{n}(\rho_{n},h_{n})$ are
the coefficients in the linear regression between $y_{n,i+h}$ and
$(u_{i},y_{i-1})$, since $(u_{i},y_{i-1})$ is a linear transformation of
$(y_{i},y_{i-1})$.

Using that
$y_{n,i+h_{n}}=\beta(\rho_{n},h_{n})u_{n,i}+\eta(\rho_{n},h_{n})y_{n,i-1}+\xi_{n,i}(\rho_{n},h_{n})$,
where $\xi_{n,i}(\rho_{n},h_{n})$ is in (A.20), and the usual ordinary linear
regression formulas, we obtain

$\frac{1}{g(\rho_{n},h_{n})}\begin{pmatrix}\hat{\beta}_{n}(h_{n})-\beta(\rho_{n},h_{n})\\\
g(\rho_{n},n-h_{n})(\hat{\eta}_{n}(h_{n})-\hat{\eta}(\rho_{n},h_{n}))\end{pmatrix}=\hat{M}_{n}^{-1}\begin{pmatrix}\hat{m}_{n,1}\\\
\hat{m}_{n,2}\end{pmatrix}~{},$ (A.33)

where

$\displaystyle\hat{M}_{n}=\large\begin{pmatrix}\dfrac{\sum_{i=1}^{n-h_{n}}u_{n,i}^{2}}{n-h_{n}}&\dfrac{\sum_{i=1}^{n-h_{n}}u_{n,i}y_{n,i-1}}{g(\rho_{n},n-h_{n})(n-h_{n})}\\\
\dfrac{\sum_{i=1}^{n-h_{n}}u_{n,i}y_{n,i-1}}{g(\rho_{n},n-h_{n})(n-h_{n})}&\dfrac{g(\rho_{n},n-h_{n})^{-2}\sum_{i=1}^{n-h_{n}}y_{n,i-1}^{2}}{(n-h_{n})}\end{pmatrix}$

and

$\displaystyle\hat{m}_{n,1}$
$\displaystyle=\frac{\sum_{i=1}^{n-h_{n}}\xi_{n,i}(\rho_{n},h_{n})u_{n,i}}{(n-h_{n})g(\rho_{n},h_{n})}$
$\displaystyle\hat{m}_{n,2}$
$\displaystyle=\frac{\sum_{i=1}^{n-h_{n}}\xi_{n,i}(\rho_{n},h_{n})y_{n,i-1}}{(n-h_{n})g(\rho_{n},h_{n})g(\rho_{n},n-h_{n})}~{}.$

We claim that (a) $\hat{M}_{n}^{-1}$ is $O_{p}\left(1\right)$, (b)
$\hat{m}_{n,1}$ is $o_{p}\left(1\right)$, and (c) $\hat{m}_{n,2}$ is
$o_{p}\left(1\right)$. These claims imply that the right-hand side in (A.33)
converges to zero in probability, which implies part (i) and (ii) of the
lemma.

To prove (a), we proceed by contradiction. Suppose (a) is false: there is a
subsequence of $\\{(\rho_{n_{k}},\sigma_{n_{k}},h_{n_{k}}:k\geq 1\\}$ such
that $\hat{M}_{n_{k}}^{-1}$ is not $O_{p}\left(1\right)$. To avoid complex
notation in the sub-index, we assume that the original sequence is such that
$\hat{M}_{n}^{-1}$ is not $O_{p}\left(1\right)$. We will prove that
$\hat{M}_{n}$ converges in distribution to $M_{0}$ along some subsequence,
where $M_{0}$ is an invertible matrix. By continuous mapping theorem, this
will imply that along the subsequence $\hat{M}_{n}^{-1}$ is
$O_{p}\left(1\right)$, which is a contradiction to assume that (a) is false.

Now, let us find the desirable subsequence. By Lemma D.3, the off-diagonal
terms of $\hat{M}_{n}$ converge to zero in probability. Without loss of
generality, we can assume that $\sigma_{n}\to\sigma_{0}$. By Chebyshev’s
inequality and Assumption B.1, we conclude that the first diagonal term of
$\hat{M}_{n}$ converges to $\sigma_{0}^{2}$ in probability. By Proposition
B.1, the last diagonal term of $\hat{M}_{n}$ is larger than $1/K$ with high-
probability as $n\to\infty$. Therefore, $\hat{M}_{n}$ converge to a positive
diagonal matrix along some subsequence.

To prove (b), we use Lemma D.4 to conclude that
$\hat{m}_{n,1}=(n-h_{n})^{-1}O_{p}\left(1\right)=o_{p}\left(1\right)$. To
prove (c), we use to lemma D.2 and $h_{n}/n\to 0$ as $n\to\infty$ to conclude
that $E[\hat{m}_{n,2}^{2}]\to 0$; therefore, $\hat{m}_{n,2}\in
o_{p}\left(1\right)$.

Finally, using the definition of $\hat{\rho}_{n}(h_{n})$ we obtain that

$(n-h_{n})^{1/2}\times
g(\rho_{n},n-h_{n})\times\left(\hat{\rho}_{n}(h_{n})-\rho_{n}\right)$

is equal to

$\left(\frac{\sum_{i=1}^{n-h_{n}}u_{n,i}y_{n,i-1}}{(n-h_{n})^{1/2}g(\rho_{n},n-h_{n})}\right)\times\left(\frac{g(\rho_{n},n-h_{n})^{-2}\sum_{i=1}^{n-h_{n}}y_{n,i-1}^{2}}{n-h_{n}}\right)^{-1}~{}.$

By Lemma D.3, the first term in the expression above is $O_{p}\left(1\right)$.
By Proposition B.1, the second term in the expression above is
$O_{p}\left(1\right)$. This proves part (iii) of the lemma.

######  Lemma D.7.

Suppose Assumption B.1 holds. Then, for any sequences $\\{\rho_{n}:n\geq 1\\}$
such that $|\rho_{n}|\leq 1+M/n$, $\\{\sigma_{n}^{2}:n\geq
1\\}\subset[\underline{\sigma},\>\overline{\sigma}]$, and $\\{h_{n}:n\geq
1\\}$ satisfying $h_{n}=o(n)$ and $h_{n}\leq n$, we have

$\frac{\sum_{i=1}^{n-h_{n}}\hat{\xi}_{n,i}(h_{n})^{2}\hat{u}_{n,i}(h_{n})^{2}}{(n-h_{n})g(\rho_{n},h_{n})^{2}\sigma_{n}^{4}}\overset{p}{\to}1~{},$

where
$\hat{\xi}_{n,i}(h_{n})=y_{n,i+h_{n}}-\hat{\beta}_{n}(h_{n})y_{n,i}-\hat{\gamma}_{n}(h_{n})y_{n,i-1}$,
$\hat{u}_{n,i}(h_{n})=y_{n,i}-\hat{\rho}_{n}(h_{n})y_{n,i-1}$,
$\hat{\beta}_{n}(h_{n})$ and $\hat{\gamma}_{n}(h_{n})$ are defined in (A.6),
$\hat{\rho}_{n}(h_{n})$ is defined in (A.7), and
$g(\rho,h)^{2}=\sum_{\ell=1}^{h}\rho^{2h}.$

###### Proof.

We adapt the proof of Lemma A2 in Montiel Olea and Plagborg-Møller (2021)
presented in Appendix E2. They claim that is sufficient to prove

$\frac{\sum_{i=1}^{n-h_{n}}\hat{\xi}_{n,i}(h_{n})^{2}\hat{u}_{n,i}(h_{n})^{2}}{(n-h_{n})g(\rho_{n},h_{n})^{2}\sigma_{n}^{4}}-\frac{\sum_{i=1}^{n-h_{n}}{\xi}_{n,i}(\rho_{n},h_{n})^{2}{u}_{n,i}^{2}}{(n-h_{n})g(\rho_{n},h_{n})^{2}\sigma_{n}^{4}}\overset{p}{\to}0~{},$
(A.34)

since they then can conclude using their Lemma A6, which implies

$\frac{\sum_{i=1}^{n-h_{n}}{\xi}_{n,i}(\rho_{n},h_{n})^{2}{u}_{n,i}^{2}}{(n-h_{n})g(\rho_{n},h_{n})^{2}\sigma_{n}^{4}}\overset{p}{\to}1~{}.$

We avoid using their Lemma A6 since its proof requires that the shocks have a
finite 8th moment. Instead, we observe that (A.27) presented in the proof of
Lemma D.4 implies the previous claim.

To verify (A.34), Montiel Olea and Plagborg-Møller prove that is sufficient to
show that

$\frac{\sum_{i=1}^{n-h_{n}}\left[\hat{\xi}_{n,i}(h_{n})^{2}\hat{u}_{n,i}(h_{n})^{2}-{\xi}_{n,i}(\rho_{n},h_{n})^{2}{u}_{n,i}^{2}\right]^{2}}{(n-h_{n})g(\rho_{n},h_{n})^{2}\sigma_{n}^{4}}$
(A.35)

converges in probability to zero. To prove that, they derive the following
upper bound for (A.35):

$3[(\hat{R}_{1})^{1/2}\times(\hat{R}_{2})^{1/2}]+3[(\hat{R}_{3})^{1/2}\times(\hat{R}_{4})^{1/2}]+3[(\hat{R}_{1})^{1/2}\times(\hat{R}_{3})^{1/2}]~{},$

where

$\displaystyle\hat{R}_{1}$
$\displaystyle=\frac{\sum_{i=1}^{n-h_{n}}\left[\xi_{n,i}(\rho_{n},h_{n})-\hat{\xi}_{n,i}(h_{n})\right]^{4}}{(n-h_{n})g(\rho_{n},h_{n})^{4}\sigma_{n}^{8}}$
$\displaystyle\hat{R}_{2}$
$\displaystyle=\frac{\sum_{i=1}^{n-h_{n}}u_{n,i}^{4}}{n-h_{n}}$
$\displaystyle\hat{R}_{3}$
$\displaystyle=\frac{\sum_{i=1}^{n-h_{n}}\left[\hat{u}_{n,i}(h_{n})-u_{n,i}\right]^{4}}{n-h_{n}}$
$\displaystyle\hat{R}_{4}$
$\displaystyle=\frac{\sum_{i=1}^{n-h_{n}}\xi_{n,i}(\rho_{n},h_{n})^{4}}{(n-h_{n})g(\rho_{n},h_{n})^{4}\sigma_{n}^{8}}~{}.$

In what follows, we use Assumption B.1 to prove that (i) $\hat{R}_{1}$ and
$\hat{R}_{3}$ are $o_{p}\left(1\right)$ and (ii) $\hat{R}_{2}$ and
$\hat{R}_{4}$ are $O_{p}\left(1\right)$, which are sufficient to conclude that
(A.35) converges to zero in probability.

To verify $\hat{R}_{1}$ is $o_{p}\left(1\right)$, let us first observe that

$\xi_{n,i}(\rho_{n},h_{n})-\hat{\xi}_{n,i}(h_{n})=[\hat{\beta}_{n}(h_{n})-\beta(\rho_{n},h_{n})]u_{n,i}+[\hat{\eta}_{n}(\rho_{n},h_{n})-\eta(\rho_{n},h_{n})]y_{n,i-1}~{},$

where
$\hat{\eta}(\rho_{n},h_{n})=\rho_{n}\hat{\beta}_{n}(h_{n})+\hat{\gamma}_{n}(h_{n})$
and $\eta(\rho_{n},h_{n})=\rho_{n}\beta(\rho_{n},h_{n})$. Then, using Loeve’s
inequality (see Theorem 9.28 in Davidson (1994)), we obtain

$\displaystyle\hat{R}_{1}$ $\displaystyle\leq
8\left(\frac{[\hat{\beta}_{n}(h_{n})-\beta(\rho_{n},h_{n})]}{g(\rho_{n},h_{n})}\right)^{4}\left(\frac{\sum_{i=1}^{n-h_{n}}u_{n,i}^{4}}{(n-h_{n})\sigma_{n}^{8}}\right)$
$\displaystyle~{}+8\left(\frac{g(\rho_{n},n-h_{n})[\hat{\eta}_{n}(\rho_{n},h_{n})-\eta(\rho_{n},h_{n})]}{g(\rho_{n},h_{n})}\right)^{4}\left(\frac{\sum_{i=1}^{n-h_{n}}y_{n,i-1}^{4}}{(n-h_{n})g(\rho_{n},n-h_{n})^{4}\sigma_{n}^{8}}\right)~{}.$

Note that the first term on the right-hand side in the previous expression
goes to zero in probability due to part (i) in Lemma D.6, Markov’s inequality,
and Assumption B.1. The second term on the right-hand side in the previous
expression goes to zero in probability due to part (ii) in Lemma D.6, Markov’s
inequality, and using that

$E[y_{n,i-1}^{4}]=E[\xi_{n,0}(\rho_{n},i-1)^{4}]\leq
g(\rho_{n},n-h_{n})^{4}4K_{4}~{},$ (A.36)

where the inequality holds due to Lemma D.1 and $g(\rho_{n},i-1)^{4}\leq
g(\rho_{n},n-h_{n})^{4}$ for all $i\leq n-h_{n}$. This completes the proof of
$\hat{R}_{1}$ is $o_{p}\left(1\right)$.

To prove that $\hat{R}_{3}$ is $o_{p}\left(1\right)$, note that we can write:

$\hat{R}_{1}=(g(\rho_{n},h_{n})[\hat{\rho}_{n}(h_{n})-\rho])^{4}\frac{\sum_{i=1}^{n-h_{n}}y_{i-1}^{4}}{(n-h_{n})g(\rho_{n},h_{n})^{4}}$

since $\hat{u}_{n,i}(h_{n})-u_{n,i}=(\hat{\rho}_{n}(h_{n})-\rho)y_{i-1}~{}.$
Note that the right-hand side in the previous expression goes to zero in
probability due to part (iii) in Lemma D.6, Markov’s inequality, and using
(A.36). This completes the proof of $\hat{R}_{3}$ is $o_{p}\left(1\right)$.
Finally, Markov’s inequality and Assumption B.1 implies that $\hat{R}_{2}$ is
$O_{p}\left(1\right)$. While Markov’s inequality and Lemma D.1 implies
$\hat{R}_{4}$ is $O_{p}\left(1\right)$.

###  D.3 Asymptotic Refinements

###### Proof of Theorem B.2.

The proof of this theorem is based on two claims. Our first claim is that
$\tilde{J}_{n}(x,h,P,\rho)$ has a valid Edgeworth expansion up to an error of
size $o\left(n^{-3/2}\right)$, that is

$\sup_{x\in\textbf{R}}\left|\tilde{J}_{n}(x,h,P,\rho)-\left(\Phi(x)+\sum_{j=1}^{3}n^{-j/2}q_{j}(x,h,P,\rho)\phi(x)\right)\right|=o\left(n^{-3/2}\right)~{},$
(A.37)

where $q_{j}(x,h,P,\rho)$ are polynomials on $x$ with coefficients that are
function of moments of $P$ (up-to order $3(j+2)$) and $\rho$. This claim
implies the result of Theorem B.2 since
$\sup_{x\in\textbf{R}}|q_{3}(x,h,P,\rho)\phi(x)|$ is bounded because
$q_{3}(x,h,P,\rho)$ is a polynomial on $x$ and $\phi(x)$ is the pdf of the
standard normal distribution.

Our second claim is that $(n-h)^{-1/2}\sum_{i=1}^{n-h}X_{i}$ has a valid
Edgeworth expansion, that is

$\sup_{A}\left|P\left(\frac{1}{(n-h)^{1/2}}\sum_{i=1}^{n-h}X_{i}\in
A\right)-\int_{A}d\Psi_{n,5}(z)\right|=o\left(n^{-3/2}\right)~{},$

where the supremum is taken over sets $A\subset\textbf{R}^{8}$ that satisfy
condition (1.6) in Bhattacharya and Ghosh (1978) and

$d\Psi_{n,5}(z)=\left[1+\sum_{j=1}^{3}n^{-j/2}\tilde{Q}_{j}(-D,P,\rho)\right]\phi_{\Sigma}(z)~{}.$
(A.38)

Here we use $D=(D_{1},\dots,D_{8})$ to denote the differential operator, the
function $\tilde{Q}_{j}(z,P,\rho)$ is a polynomial on $z\in\textbf{R}^{8}$
with the coefficients depending continuously on the moments of $P$ (up-to
order $3(j+2)$) and $\rho$, and $\phi_{\Sigma}(z)$ is the pdf of a
multivariate normal distribution with zero mean and non-singular variance
$\Sigma$. This claim and the proof of Theorem 2 in Bhattacharya and Ghosh
(1978) implies our first claim. The continuous dependent of the coefficient in
$\tilde{Q}_{j}(z,P,\rho)$ implies the continuous dependency of the
coefficients in $q_{j}(x,h,P,\rho)$.

The proof of our second claim follows from Theorem 1.2 in Götze and Hipp
(1994) that guarantee that we can use results from Götze and Hipp (1983) such
as their Theorem 2.8 and Remark 2.12 that guarantees that the polynomials
$\tilde{Q}_{j}(z,P,\rho)$ does not depend on $n$. We can use Theorem 1.2 in
Götze and Hipp (1994) since the sequence
$\\{X_{i}=F(z_{i-1},z_{i},z_{i+h}):1\leq i\leq n-h\\}$ in (A.9) is defined by
the function $F$ and we have assumed that shocks the $u_{i}$ have a positive
continuous density in part (ii) of Assumption B.2. The polynomials
$\tilde{Q}_{j}(z,P,\rho)$ are computed by a formal expansion and using the
Fourier-Stieltjes transformation, which guarantees that the coefficients of
the polynomials are continuous functions of the cumulants of the vector
$(n-h)^{-1/2}\sum_{i=1}^{n-h}X_{i}$. Since the sequence $\\{X_{i}:1\leq i\leq
n-h\\}$ is strictly stationary, $X_{i}$ is a polynomial of
$(z_{i-1},z_{i},z_{i+h})$ as in (A.8), and $z_{i}$ is strictly stationary, it
follows that the cumulants of $(n-h)^{-1/2}\sum_{i=1}^{n-h}X_{i}$ can be
computed as continuous functions of the moments of $z_{i}$, $\rho$,
$\sigma^{2}$ and $V$. Since the moments of $z_{i}$ can be computed using the
moments of $u_{i}$ and $\rho$, we conclude the coefficients in
$\tilde{Q}_{j}(z,P,\rho)$ are continuous functions of moments of $P$ (up-to
order $3(j+2)$) and $\rho$.

###### Proof of Theorem B.3.

The proof has three steps. We first consider a sequence of event $E_{n}$ such
that $P(E_{n}^{c})\leq Cn^{-1-\epsilon}$. We then show that, conditional on
the event $E_{n}$, the (scaled) bootstrap sample average

$\frac{1}{(n-h)^{1/2}}\sum_{i=1}^{n-h}X_{b,i}^{*}$

has an Edgeworth expansion. We conclude relying on the arguments presented in
Lemma 2.1 in Bhattacharya and Ghosh (1978) to conclude that, conditional on
the event $E_{n}$, there exists an Edgeworth expansion for
$\tilde{J}_{n}(x,h,\hat{P}_{n},\hat{\rho}_{n})$.

Step 1: Define the events $E_{n,1}=\\{|\hat{\rho}_{n}|\leq 1-a/2\\}$,
$E_{n,2}=\\{n^{-1}\sum_{i=1}^{n}\tilde{u}_{i}^{2}\geq\tilde{C}_{\sigma}\\}$,
and $E_{n,3}=\\{n^{-1}\sum_{i=1}^{n}\tilde{u}_{i}^{4k}<M\\}$, where
$\tilde{C}_{\sigma}$ and $M$ are as in Lemma B.5. Define $E_{n}=E_{n,1}\cap
E_{n,2}\cap E_{n,3}$. By Lemma B.5 and Assumption 4.1, it follows that
$P(E_{n}^{c})\leq C_{2}n^{-1-\epsilon}$ for some constant $C_{2}$ that depends
on the moments of $u_{i}$. Since $k>8$, it follows that, conditional on
$E_{n}$, the empirical distribution $\hat{P}_{n}$ verifies part (i) and (iii)
of Assumption B.2. It is important to mention that Götze and Hipp (1994) use
part (ii) of Assumption B.2 to guarantee the dependent-data version of the
Cramer condition that appears in Götze and Hipp (1983); see Lemma 2.3 in Götze
and Hipp (1994).

Step 2: Condition (iii) in Lemma 2.3 in Götze and Hipp (1994) holds for the
the bootstrap sequence $X_{b,i}^{*}$ since it holds for the original sequence
$X_{i}$, otherwise the function $F$ verifies equation (8) in Götze and Hipp
(1994). Therefore, the dependent-data version of the Cramer condition holds
for the bootstrap sequence $X_{b,i}^{*}$. The results in Götze and Hipp (1994)
implied that Edgeworth expansion exists for the sample average.

Step 3: The arguments presented in Lemma 2.1 in Bhattacharya and Ghosh (1978)
can be applied. It is important to mention that the approximation error in
equation (2.13) in Bhattacharya and Ghosh (1978) only depends on the
coefficients of the polynomial $T$ (that we consider) and the maximum
eigenvalues of $\Sigma$ (asymptotic covariate matrix); therefore the error
term is locally-uniform in the coefficients of $T$ and $\Sigma$, i.e., the
result of Lemma 2.1 applies for a sequence of $T_{n}$ where the coefficients
of the polynomial are close to the original $T$ and matrix $\hat{\Sigma}_{n}$
close to $\Sigma$.

The properties of the polynomial $q_{j}$ follow by the same arguments
presented in the proof of Theorem B.2. The main difference is that the
coefficients of the polynomials $q_{j}$ are based on the moments of the
distribution $\hat{P}_{n}$ (up to order 12) and $\hat{\rho}_{n}$.

######  Lemma D.8.

Let $\\{W_{n,j}:1\leq j\leq r\\}$ be a sequence of random variables. Suppose
that there exist constants $c_{j}$ and $C$ such that

$\displaystyle P(n^{1/2}|W_{n,j}|>c_{j}\delta)\leq C\delta^{-k}~{},$

for $j=1,..,r$ and some $k\in\mathbf{N}$. Then, for any $r\geq 2$ and
$\delta<n^{1/2}$, we have

  1. 1.

$P(n^{1/2}|\sum_{j=1}^{r}W_{n,j}|>(\sum_{j=1}^{r}c_{j})\delta)\leq
rC\delta^{-k}~{}.$

  2. 2.

$P(n^{r/2}|\prod_{j=1}^{r}W_{n,j}|>(\prod_{j=1}^{r}c_{j})\delta^{r})\leq
rC\delta^{-k}~{}.$

  3. 3.

$P(n^{1/2}|W_{n,1}+\prod_{j=2}^{r}W_{n,j}|>(c_{1}+\prod_{j=2}^{r}c_{j})\delta)\leq
2C\delta^{-k}~{}.$

  4. 4.

If $c_{1}n^{-1/2}\delta<1$. Then, $P(|W_{n,1}|>1-b)\leq C\delta^{-k}~{}$ for
any $b\in(0,1-c_{1}n^{-1/2}\delta)$.

  5. 5.

If $c_{1}n^{-1/2}\delta<1$. Then, for any $b\in(0,1-c_{1}n^{-1/2}\delta)$, we
have

$P\left(n^{3/2}|(1+W_{n,1})^{-1/2}-(1-\frac{1}{2}W_{n,1}+\frac{3}{8}W_{n,1}^{2})|>\frac{5}{16}b^{-7/2}c_{1}^{3}\delta^{3}\right)\leq
4C\delta^{-k}~{}$

and

$P\left(n^{2/2}|(1+W_{n,1})^{-1}-(1-W_{n,1})|>\frac{2}{1+(1-b)^{3}}\delta^{2}\right)\leq
3C\delta^{-k}~{}.$

###### Proof.

Bonferroni’s inequality implies items 1 and 2. To prove item 3, we use
Bonferroni’s inequality, item 2, and $\delta<n^{1/2}$. To prove item 4, we use
that $\\{|W_{n,1}|>1-b\\}\subseteq\\{n^{1/2}|W_{n,1}|>c_{1}\delta\\}$ for any
$b\in(0,1-c_{1}n^{-1/2}\delta)$. To prove the first part of item 5, we use
Bonferroni’s inequality to conclude that the left-hand side in item 5 is lower
or equal to the sum of $P(|W_{n,1}|>1-b)$ and

$P\left(n^{3/2}|(1+W_{n,1})^{-1/2}-(1-\frac{1}{2}W_{n,1}+\frac{3}{8}W_{n,1}^{2})|>\frac{5}{16}b^{-7/2}c_{1}^{3}\delta^{3},|W_{n,1}|\leq
1-b\right)~{}.$

Item 4 implies that the former term is bounded by $C\delta^{-k}$, while the
latter term is lower or equal to

$P\left(n^{3/2}\frac{5}{16}b^{-7/2}|W_{n,1}|^{3}>\frac{5}{16}b^{-7/2}c_{1}^{3}\delta^{3},|W_{n,1}|\leq
1-b\right)~{},$

where the left-hand side term inside the previous probability used the Taylor
Polynomial error and $|W_{n,1}|\leq 1-b$. By item 2, the above probability is
lower or equal to $3C\delta^{-k}$. Finally, adding the upper and lower bounds
concludes the first part of item 5. Similarly for the second part of item 5.

######  Lemma D.9.

Let $\\{Z_{i}:1\leq i\leq n\\}$ be a martingale difference sequence. Then, for
any $k\geq 2$, we have

$E\left[\left|n^{-1/2}\sum_{i=1}^{n}Z_{i}\right|^{k}\right]\leq
d_{k}\beta_{n,k}~{},$

where $\beta_{n,k}=n^{-1}\sum_{i=1}^{n}E[|Z_{i}|^{k}]$ and
$d_{k}=(8(k-1)\max\\{1,2^{k-3}\\})^{k}$.

###### Proof.

See Dharmadhikari et al. (1968), where this lemma is the main theorem.

######  Lemma D.10.

Suppose Assumption 4.1 holds. For any fixed $h,k\in\mathbf{N}$ and
$a\in(0,1)$. Define

$f_{n}=\frac{\sum_{i=1}^{n-h}\xi_{i}(\rho,h)\hat{u}_{i}(h)}{n-h}$

where $\xi_{i}(\rho,h)=\sum_{\ell=1}^{h}\rho^{h-\ell}u_{i+\ell}$,
$\hat{u}_{i}(h)=y_{i}-\hat{\rho}_{n}(h)y_{i-1}$, and $\hat{\rho}_{n}(h)$ is as
in (6). Then, for any $\rho\in[-1+a,1-a]$, there exists a constant
$C=C(h,k,a,C_{\sigma})$ such that we can write

$f_{n}=f_{1,n}+f_{2,n}+f_{3,n}+f_{4,n}~{},$

where $P\left((n-h)^{j/2}|f_{j,n}|\geq\delta^{j}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)$ for any
$\delta<(n-h)^{1/2}$ and $j\in\\{1,2,3,4\\}$.

###### Proof.

In what follows we use $\xi_{i}=\xi_{i}(\rho,h)$. Using the definition of
$\hat{\rho}_{n}(h)$, we obtain

$f_{n}=\frac{\sum_{i=1}^{n-h}\xi_{i}u_{i}}{n-h}-\left(\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}\right)\left(\frac{\sum_{i=1}^{n-h}\xi_{i}y_{i-1}}{n-h}\right)\left(\frac{\sum_{i=1}^{n-h}y_{i-1}^{2}}{n-h}\right)^{-1}~{}.$

Let us define the components of $f_{n}$ as follows:

$\displaystyle f_{1,n}$
$\displaystyle=\left(\frac{\sum_{i=1}^{n-h}\xi_{i}u_{i}}{n-h}\right)$
$\displaystyle f_{2,n}$
$\displaystyle=-\frac{(1-\rho^{2})}{\sigma^{2}}\left(\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}\right)\left(\frac{\sum_{i=1}^{n-h}\xi_{i}y_{i-1}}{n-h}\right)$
$\displaystyle f_{3,n}$
$\displaystyle=\frac{(1-\rho^{2})}{\sigma^{4}}\left(\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}\right)\left(\frac{\sum_{i=1}^{n-h}\xi_{i}y_{i-1}}{n-h}\right)\left(2\rho\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}+\frac{\sum_{i=1}^{n-h}u_{i}^{2}-\sigma^{2}}{n-h}\right)$
$\displaystyle f_{4,n}$
$\displaystyle=f_{n}-\left(f_{1,n}+f_{2,n}+f_{3,n}\right)~{}.$

Note that by construction $f_{n}=\sum_{j=1}^{4}f_{j,n}$. Lemma D.14 guarantees
that each sample average in $f_{1,n}$, $f_{2,n}$, and $f_{3,n}$ verify the
conditions to use Lemma D.8, which imply that
$P\left((n-h)^{j/2}|f_{j,n}|\geq\delta^{j}\right)\leq
C\delta^{-k}E[u_{i}^{2k}]$ for $j=1,2,3$, where the constant $C$ includes
$C_{\sigma}$ and $a$.

To prove $P\left((n-h)^{4/2}|f_{j,n}|\geq\delta^{4}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)$, we
proceed in two steps.

Step 1: Define

$W_{n}=\frac{\sum_{i=1}^{n-h}(1-\rho^{2})\sigma^{-2}y_{i-1}^{2}}{n-h}-1~{}.$

We can use (1) and algebra to derive the following identity

$W_{n}=-\frac{\sigma^{-2}y_{n-h}^{2}}{n-h}+2\rho\sigma^{-2}\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}+\sigma^{-2}\frac{\sum_{i=1}^{n-h}u_{i}^{2}-\sigma^{2}}{n-h}~{}.$

This implies that

$f_{3,n}=\frac{(1-\rho^{2})}{\sigma^{2}}\left(\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}\right)\left(\frac{\sum_{i=1}^{n-h}\xi_{i}y_{i-1}}{n-h}\right)\left(W_{n}+\frac{\sigma^{-2}y_{n-h}^{2}}{n-h}\right)~{}.$

Therefore, $f_{4,n}=f_{n}-f_{1,n}-f_{2,n}-f_{3,n}$ is equal to

$-\frac{(1-\rho^{2})}{\sigma^{2}}\left(\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}\right)\left(\frac{\sum_{i=1}^{n-h}\xi_{i}y_{i-1}}{n-h}\right)\left(\left(1+W_{n}\right)^{-1}-(1-W_{n})+\frac{\sigma^{-2}y_{n-h}^{2}}{n-h}\right)~{}.$

Step 2: Due to Lemma D.8, it is sufficient to show that

$P\left((n-h)^{2/2}|(1+W_{n})^{-1}-(1-W_{n})|>\delta^{2}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)~{}$ (A.39)

and

$P\left((n-h)^{2/2}\left|\frac{\sigma^{-2}y_{n-h}^{2}}{n-h}\right|>\delta^{2}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)~{}.$
(A.40)

Note that (A.39) follows by part 5 in Lemma D.8 since
$P(n^{1/2}|W_{n}|>\delta)\leq C\delta^{-k}E[u_{i}^{2k}]$ due to Lemma D.14 and
$\delta<n^{1/2}$. Finally, (A.40) follows by Markov’s inequality and Lemma
D.13. As we mentioned before, the constant $C$ includes the constants $C$’s
that appear in Lemmas D.13 and D.14 that only depends on $a$, $h$, $k$, and
$C_{\sigma}$.

######  Lemma D.11.

Suppose Assumption 4.1 holds. For any fixed $h,k\in\mathbf{N}$ and
$a\in(0,1)$. Define

$g_{n}=\frac{\sum_{i=1}^{n-h}\hat{\xi}_{i}(h)^{2}\hat{u}_{i}(h)^{2}}{V(n-h)}-1$

where $V=\sigma^{2}\sum_{\ell=1}^{h}\rho^{2(h-\ell)}$, $\hat{\xi}_{i}(h)$ is
as in (4), $\hat{u}_{i}(h)=y_{i}-\hat{\rho}_{n}(h)y_{i-1}$, and
$\hat{\rho}_{n}(h)$ is as in (6). Then, for any $\rho\in[-1+a,1-a]$, there
exists a constant $C=C(h,k,a,C_{\sigma})$ such that we can write

$g_{n}=g_{1,n}+g_{2,n}+g_{3,n}~{},$

where $P\left((n-h)^{j/2}|g_{j,n}|\geq\delta^{j}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)$ for any
$\delta<(n-h)^{1/2}$ and $j\in\\{1,2,3\\}$.

###### Proof.

In what follows we use
$\xi_{i}=\xi_{i}(\rho,h)=\sum_{\ell=1}^{h}\rho^{h-\ell}u_{i+\ell}$. As we did
for the case of $f_{n}$ in Lemma D.10, we utilize the linear regression
formulas to define the components of $g_{n}$ as functions of the sample
average of functions of $\xi_{i}$, $u_{i}$, and $y_{i-1}$:

$\displaystyle Vg_{1,n}$
$\displaystyle=\frac{\sum_{i=1}^{n-h}(\xi_{i}^{2}u_{i}^{2}-V)}{n-h}$
$\displaystyle Vg_{2,n}$
$\displaystyle=\frac{\psi_{4}^{4}}{\sigma^{4}}\left(\frac{\sum_{i=1}^{n-h}\xi_{i}u_{i}}{n-h}\right)^{2}-\frac{2}{\sigma^{2}}\left(\frac{\sum_{i=1}^{n-h}\xi_{i}u_{i}}{n-h}\right)\left(\frac{\sum_{i=1}^{n-h}\xi_{i}u_{i}^{3}}{n-h}\right)$
$\displaystyle+(1-\rho^{2})g(\rho,h)^{2}\left(\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}\right)^{2}-\frac{2(1-\rho^{2})}{\sigma^{2}}\left(\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}\right)\left(\frac{\sum_{i=1}^{n-h}\xi_{i}^{2}u_{i}y_{i-1}}{n-h}\right)$
$\displaystyle+(1-\rho^{2})\left(\frac{\sum_{i=1}^{n-h}\xi_{i}y_{i-1}}{n-h}\right)^{2}-\frac{2(1-\rho^{2})}{\sigma^{2}}\left(\frac{\sum_{i=1}^{n-h}\xi_{i}y_{i-1}}{n-h}\right)\left(\frac{\sum_{i=1}^{n-h}\xi_{i}u_{i}^{2}y_{i-1}}{n-h}\right)$
$\displaystyle Vg_{3,n}$
$\displaystyle=Vg_{n}-V\left(g_{1,n}+g_{2,n}\right)~{},$

where $\psi_{4}^{4}=E[u_{i}^{4}]$ and
$g(\rho,h)^{2}=\sum_{\ell=1}^{h}\rho^{2(h-\ell)}$. By construction
$g_{n}=\sum_{j=1}^{3}g_{j,n}$. Lemma D.14 guarantees that each sample average
in $g_{1,n}$ and $g_{2,n}$ verify the conditions to use Lemma D.8, which imply
$P\left((n-h)^{j/2}|g_{j,n}|\geq\delta^{j}\right)\leq
C\delta^{-k}\left(E[u_{i}^{4k}]\right)$ for $\delta<(n-h)^{1/2}$ and
$j\in\\{1,2\\}$, , where the constant $C$ includes $C_{\sigma}$ and $a$. In
what follows we prove $P\left((n-h)^{3/2}|g_{3,n}|\geq\delta^{3}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)$.

First, we write $Vg_{3,n}=R_{g,1}+R_{g,2}$, where $R_{g,1}$ and $R_{g,2}$ are
specified below. We will prove that
$P((n-h)^{3/2}|V^{-1}R_{g,j}|>\delta^{3})\leq C\delta^{-k}E[u_{i}^{4k}]$ for
$j=1,2$. To compute $g_{3,n}$ we use equation (4) and the following
derivations

$\displaystyle\hat{\xi}_{i}(h)$
$\displaystyle=y_{i+h}-\hat{\beta}_{n}(h)y_{i}-\hat{\gamma}_{n}(h)y_{i-1}$
$\displaystyle=y_{i+h}-\hat{\beta}_{n}(h)u_{i}-\hat{\eta}_{n}(\rho,h)y_{i-1}~{},$
$\displaystyle=\xi_{i}-(\hat{\beta}_{n}(h)-\beta(\rho,h))u_{i}-\hat{\eta}_{n}(\rho,h)y_{i-1}~{},$

where $\hat{\eta}_{n}(\rho,h)=\rho\hat{\beta}_{n}(h)+\hat{\gamma}_{n}(h)$. We
also use that
$\hat{u}_{i}(h)=y_{i}-\hat{\rho}_{n}(h)y_{i-1}=u_{i}-(\hat{\rho}_{n}(h)-\rho)y_{i-1}$.
In what follows, we use $\hat{\beta}=\hat{\beta}_{n}(h)$,
$\hat{\rho}=\hat{\rho}_{n}(h)$ and $\hat{\eta}=\hat{\eta}_{n}(\rho,h)$, and
$\sum=\sum_{i=1}^{n-h}$ to simplified the heavy notation. We obtain $R_{g,2}$
is equal to

$\displaystyle
2\left[(\eta-\hat{\eta})+\frac{(1-\rho^{2})}{\sigma^{2}}\left(\frac{\sum\xi_{i}y_{i-1}}{n-h}\right)\right]\left(\frac{\sum\xi_{i}u_{i}^{2}y_{i-1}}{n-h}\right)+\frac{\sigma^{4}}{1-\rho^{2}}\left\\{(\eta-\hat{\eta})^{2}-\frac{(1-\rho^{2})^{2}}{\sigma^{4}}\left(\frac{\sum\xi_{i}y_{i-1}}{n-h}\right)^{2}\right\\}$
$\displaystyle+2\left[(\rho-\hat{\rho})+\frac{(1-\rho^{2})}{\sigma^{2}}\left(\frac{\sum
u_{i}y_{i-1}}{n-h}\right)\right]\left(\frac{\sum\xi_{i}^{2}u_{i}y_{i-1}}{n-h}\right)+\psi_{4}^{4}\left\\{(\beta-\hat{\beta})^{2}-\frac{1}{\sigma^{4}}\left(\frac{\sum\xi_{i}u_{i}}{n}\right)^{2}\right\\}$
$\displaystyle+2\left[(\beta-\hat{\beta})+\frac{1}{\sigma^{2}}\left(\frac{\sum\xi_{i}u_{i}}{n-h}\right)\right]\left(\frac{\sum\xi_{i}u_{i}^{3}}{n-h}\right)+\frac{\sigma^{4}g_{2}^{2}}{1-\rho^{2}}\left\\{(\rho-\hat{\rho})^{2}-\frac{(1-\rho^{2})^{2}}{\sigma^{4}}\left(\frac{\sum
u_{i}y_{i-1}}{n-h}\right)^{2}\right\\}~{},$

and we obtain that $R_{g,1}$ is equal to

$\displaystyle(\beta-\hat{\beta})^{2}\left(\frac{\sum
u_{i}^{4}}{n-h}-\psi_{4}^{4}\right)+(\rho-\hat{\rho})^{2}\left(\frac{\sum\xi_{i}^{2}y_{i-1}^{2}}{n-h}-\frac{\sigma^{4}g(\rho,h)^{2}}{1-\rho^{2}}\right)+(\eta-\hat{\eta})^{2}\left(\frac{\sum
u_{i}^{2}y_{i-1}^{2}}{n-h}-\frac{\sigma^{4}}{1-\rho^{2}}\right)$
$\displaystyle 2(\beta-\hat{\beta})^{2}(\rho-\hat{\rho})\left(\frac{\sum
u_{i}^{3}y_{i-1}}{n-h}\right)+2(\eta-\hat{\eta})^{2}(\rho-\hat{\rho})\left(\frac{\sum
u_{i}y_{i-1}^{3}}{n-h}\right)+2(\beta-\hat{\beta})(\rho-\hat{\rho})^{2}\left(\frac{\sum\xi_{i}u_{i}y_{i-1}^{2}}{n-h}\right)$
$\displaystyle+4(\beta-\hat{\beta})(\rho-\hat{\rho})\left(\frac{\sum\xi_{i}u_{i}^{2}y_{i-1}}{n-h}\right)+2(\eta-\hat{\eta})(\rho-\hat{\rho})^{2}\left(\frac{\sum\xi_{i}y_{i-1}^{3}}{n-h}\right)+4(\eta-\hat{\eta})(\rho-\hat{\rho})\left(\frac{\sum\xi_{i}u_{i}y_{i-1}^{2}}{n-h}\right)$
$\displaystyle+2(\beta-\hat{\beta})(\eta-\hat{\eta})\left(\frac{\sum
u_{i}^{3}y_{i-1}}{n-h}\right)+2(\beta-\hat{\beta})(\eta-\hat{\eta})(\rho-\hat{\rho})^{2}\left(\frac{\sum
u_{i}y_{i-1}^{3}}{n-h}\right)$
$\displaystyle+(\beta-\hat{\beta})^{2}(\rho-\hat{\rho})^{2}\left(\frac{\sum
u_{i}^{2}y_{i-1}^{2}}{n-h}-\frac{\sigma^{4}}{1-\rho^{2}}\right)+4(\beta-\hat{\beta})(\eta-\hat{\eta})(\rho-\hat{\rho})\left(\frac{\sum
u_{i}^{2}y_{i-1}^{2}}{n-h}-\frac{\sigma^{4}}{1-\rho^{2}}\right)$
$\displaystyle+(\eta-\hat{\eta})^{2}(\rho-\hat{\rho})^{2}\left(\frac{\sum
y_{i-1}^{4}}{n-h}-\frac{E[u_{i}^{4}]}{1-\rho^{4}}-6\frac{\rho^{2}\sigma^{4}}{(1-\rho^{2})(1-\rho^{4})}\right)+(\eta-\hat{\eta})^{2}(\rho-\hat{\rho})^{2}\frac{E[u_{i}^{4}]}{1-\rho^{4}}$
$\displaystyle+\frac{\sigma^{4}}{1-\rho^{2}}(\beta-\hat{\beta})^{2}(\rho-\hat{\rho})^{2}+4(\beta-\hat{\beta})(\eta-\hat{\eta})(\rho-\hat{\rho})\frac{\sigma^{4}}{1-\rho^{2}}+(\eta-\hat{\eta})^{2}(\rho-\hat{\rho})^{2}\frac{6\rho^{2}\sigma^{4}}{(1-\rho^{2})(1-\rho^{4})}$

Note that $P((n-h)^{3/2}|V^{-1}R_{g,1}|>\delta^{3})\leq
C\delta^{-k}E[u_{i}^{4k}]$ follows by Lemmas D.8, D.14, and D.15, since each
term between parenthesis in the definition of $R_{g,1}$ appears in Lemma D.14
or in items 5-8 of Lemma D.15. Similarly,
$P((n-h)^{3/2}|V^{-1}R_{g,2}|>\delta^{3})\leq C\delta^{-k}E[u_{i}^{4k}]$
follows by Lemmas D.8, D.14, and D.15, since each term between parenthesis in
the definition of $R_{g,2}$ appears in Lemma D.14, the terms in brackets
appears in items 1-4 of Lemma D.15, and the terms between curly brackets can
be written as the product of terms like parenthesis and brackets terms.

######  Lemma D.12.

Suppose Assumption 4.1 holds. For any fixed $h\in\mathbf{N}$ and $a\in(0,1)$.
Then, for any $\rho\in[-1+a,1-a]$ and $\epsilon\in(0,1/2)$, there exist a
constant $C=C(a,h,k)>0$, where $k\geq 8(1+\epsilon)/(1-2\epsilon)$, and a
real-valued function
$T(\cdot~{};\sigma^{2},\psi_{4}^{4},\rho):\mathbf{R}^{8}\to\mathbf{R},$ such
that

  1. 1.

$T(\mathbf{0};\sigma^{2},\psi_{4}^{4},\rho)=0$,

  2. 2.

$T(x;\sigma^{2},\psi_{4}^{4},\rho)$ is a polynomial of degree 3 in
$x\in\mathbf{R}^{8}$ with coefficients depending continuously differentiable
on $\sigma^{2}$, $\psi_{4}^{4}$, and $\rho$,

  3. 3.

$P_{\rho}\left(\left|R_{n}(h)-\tilde{R}_{n}(h)\right|>n^{-1-\epsilon}\right)\leq
Cn^{-1-\epsilon}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right),$

where $\sigma^{2}=E_{P}[u_{1}^{2}]$, $\psi_{4}^{4}=E_{P}[u_{1}^{4}]$,

$\tilde{R}_{n}(h)\equiv(n-h)^{1/2}T\left(\frac{1}{n-h}\sum_{i=1}^{n-h}X_{i};\sigma^{2},\psi_{4}^{4},\rho\right)$

and the sequence $\\{X_{i}:1\leq i\leq n-h\\}$ is defined in (A.9).
Furthermore, the asymptotic variance of $\tilde{R}_{n}(h)$ equals one.

###### Proof.

The proof has two main parts. We first use Lemmas D.10 and D.11 to approximate
$R_{n}(h)$ using functions based on $\xi_{i}$, $u_{i}$, and $y_{i-1}$. We then
replace $y_{i-1}$ by $z_{i-1}$. We specifically define the polynomial $T$.

Part 1: The derivations presented on page 1811 in Montiel Olea and Plagborg-
Møller (2021) implies

$R_{n}(h)=\frac{\sum_{i=1}^{n-h}\xi_{i}(h)\hat{u}_{i}(h)}{\left(\sum_{i=1}^{n-h}\hat{\xi}_{i}(h)^{2}\hat{u}_{i}(h)^{2}\right)^{1/2}}~{},$

where $\xi_{i}(\rho,h)=\sum_{\ell=1}^{h}\rho^{h-\ell}u_{i+\ell}$,
$\hat{u}_{i}(h)=y_{i}-\hat{\rho}_{n}(h)y_{i-1}$,
$\hat{\xi}_{i}(h)=y_{i+h}-\left(\hat{\beta}_{n}(h)y_{i}+\hat{\gamma}_{n}(h)y_{i-1}\right)$,
and the coefficients $(\hat{\beta}_{n}(h),\hat{\gamma}_{n}(h))$ is as in (4)
and $\hat{\rho}_{n}(h)$ is defined in (6). Define

$f_{n}=\frac{\sum_{i=1}^{n-h}\xi_{i}(\rho,h)\hat{u}_{i}(h)}{n-h}$

and

$g_{n}=\frac{\sum_{i=1}^{n-h}\hat{\xi}_{i}(h)^{2}\hat{u}_{i}(h)^{2}}{V(n-h)}-1~{},$

where
$V=E[\xi_{i}(\rho,h)^{2}u_{i}^{2}]=\sigma^{4}\sum_{\ell=1}^{h}\rho^{2(h-\ell)}$.
It follows that

$R_{n}(h)=(n-h)^{1/2}V^{-1/2}f_{n}\left(1+g_{n}\right)^{-1/2}~{}.$

Lemmas D.8, D.10 and D.11 imply

$P((n-h)^{1/2}|V^{-1/2}f_{n}|>\delta)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)$

and

$P((n-h)^{1/2}|g_{n}|>\delta)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)~{}.$

Step 1: Define
$\tilde{R}_{g,n}=(n-h)^{1/2}f_{n}V^{-1/2}\left(1-\frac{1}{2}g_{n}+\frac{3}{8}g_{n}^{2}\right)$.
Due to Lemma D.8, we have

$P\left((n-h)^{3/2}\left|R_{n}(h)-\tilde{R}_{g,n}\right|>\delta^{4}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)~{}$

since

$P\left(n^{3/2}\left|\left(1+g_{n}\right)^{-1/2}-\left(1-\frac{1}{2}g_{n}+\frac{3}{8}g_{n}^{2}\right)\right|>\delta^{3}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)~{}.$

Step 2: Define
$\tilde{R}_{f,n}=(n-h)^{1/2}V^{-1/2}\left(f_{1,n}+f_{2,n}+f_{3,n}\right)\left(1-\frac{1}{2}g_{n}+\frac{3}{8}g_{n}^{2}\right)$,
where $f_{n}=\sum_{j=1}^{4}f_{j,n}$ as in Lemma D.10. We conclude

$\displaystyle
P\left((n-h)^{3/2}\left|\tilde{R}_{g,n}-\tilde{R}_{f,n}\right|>\delta^{4}\right)$
$\displaystyle=P\left((n-h)^{4/2}\left|V^{-1/2}f_{4,n}\left(1-\frac{1}{2}g_{n}+\frac{3}{8}g_{n}^{2}\right)\right|\geq\delta^{4}\right)$
$\displaystyle\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)~{},$

where the last inequality follows by Lemmas D.8 and D.10.

Step 3: Define
$\tilde{R}_{fg,n}=n^{1/2}V^{-1/2}\left(f_{1,n}+f_{2,n}+f_{3,n}-\frac{1}{2}f_{1,n}g_{n}-\frac{1}{2}f_{2,n}g_{n}+\frac{3}{8}f_{1,n}g_{n}^{2}\right)$.
Lemmas D.8, D.10, and D.11 imply

$P\left((n-h)^{3/2}\left|\tilde{R}_{f,n}-\tilde{R}_{fg,n}\right|>\delta^{4}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)~{}.$

Step 4: Define

$\tilde{R}_{y,n}(h)=n^{1/2}V^{-1/2}\left(f_{1,n}+f_{2,n}+f_{3,n}-\frac{1}{2}f_{1,n}g_{1,n}-\frac{1}{2}f_{1,n}g_{2,n}-\frac{1}{2}f_{2,n}g_{1,n}+\frac{3}{8}f_{1,n}g_{1,n}^{2}\right)~{},$
(A.41)

where $g_{n}=\sum_{j=1}^{3}g_{j,n}$ as in Lemma D.11. We use Lemmas D.8, D.10,
and D.11 to conclude

$P\left((n-h)^{3/2}\left|\tilde{R}_{fg,n}-\tilde{R}_{y,n}(h)\right|>\delta^{4}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)~{}.$

Step 5: We conclude by Bonferroni’s inequality that

$P\left((n-h)^{3/2}\left|R_{n}(h)-\tilde{R}_{y,n}(h)\right|>\delta^{4}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)~{}.$

Part 2: We consider $V^{1/2}T(x;\sigma^{2},\psi_{4}^{4},\rho)$ is equal to the
following polynomial

$\left(f_{1}(x)+f_{2}(x)+f_{3}(x)-\frac{1}{2}\left(f_{1}(x)g_{1}(x)+f_{1}(x)g_{2}(x)+f_{2}(x)g_{1}(x)\right)+\frac{3}{8}f_{1}(x)g_{1}(x)^{2}\right)~{},$
(A.42)

where $f_{1}(x)=x_{1}$, $f_{2}(x)=-\sigma^{-2}(1-\rho^{2})x_{5}x_{6}$,
$f_{3}(x)=\sigma^{-4}(1-\rho^{2})x_{5}x_{6}(2\rho x_{5}+x_{2})$,
$g_{1}(x)=V^{-1}x_{3}$, and

$\displaystyle
g_{2}(x)=V^{-1}\left(\frac{\psi_{4}^{4}}{\sigma^{4}}x_{1}^{2}-\frac{2}{\sigma^{2}}x_{1}x_{4}+(1-\rho^{2})\left(g(\rho,h)^{2}x_{5}^{2}-\frac{2}{\sigma^{2}}x_{5}x_{7}+x_{6}^{2}-\frac{2}{\sigma^{2}}x_{6}x_{8}\right)\right)$

Note that
$\tilde{R}_{y,n}(h)=V^{1/2}T\left((n-h)^{-1}\sum_{i=1}^{n-h}\tilde{X}_{i};\sigma^{2},\psi_{4}^{4},\rho\right)$,
where

$\tilde{X}_{i}=(\xi_{i}u_{i},u_{i}^{2}-\sigma^{2},(\xi_{i}u_{i})^{2}-V,\xi_{i}u_{i}^{3},u_{i}y_{i-1},\xi_{i}y_{i-1},\xi_{i}^{2}u_{i}y_{i-1},\xi_{i}u_{i}^{2}y_{i-1})~{}.$

Since $z_{i}=y_{i}+\rho^{i}z_{0}$, it follows that

$P\left((n-h)^{1/2}\left|\frac{\sum_{i=1}^{n-h}f_{i}y_{i-1}}{n-h}-\frac{\sum_{i=1}^{n-h}f_{i}z_{i-1}}{n-h}\right|>\delta\right)\leq
C\delta^{-k}\left(E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)$

for $f_{i}=u_{i},\xi_{i},\xi_{i}^{2}u_{i},\xi_{i}u_{i}^{2}$. Then, Lemma D.8
and step 5 in part 1 implies

$P\left((n-h)^{3/2}\left|R_{n}(h)-\tilde{R}_{n}(h)\right|>\delta^{4}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]+E[u_{i}^{4k}]\right)~{},$

where
$\tilde{R}_{n}(h)=V^{1/2}T\left((n-h)^{-1}\sum_{i=1}^{n-h}{X}_{i};\sigma^{2},\psi_{4}^{4},\rho\right)$
and the sequence $\\{X_{i}:1\leq i\leq n-h\\}$ is defined in (A.20). As we
mentioned before, the constant $C$ includes the constants $C$’s that appear in
Lemmas D.10, D.11, D.13 and D.14 that only depends on $a$, $h$, $k$, and
$C_{\sigma}$. Finally, we take $\delta=n^{(1/2-\epsilon)/4}$

######  Lemma D.13.

Suppose Assumption 4.1 holds. For fixed $a\in(0,1)$ and $k\geq 1$. Then, for
any $|\rho|\leq 1-a$, there exists a constant $C=C(a,k)>0$ such that

$E[|y_{n}|^{k}]\leq C\>E[|u_{n}|^{k}]~{},~{}\forall n\geq 1~{}.$

and $P(n^{-1/2}|y_{n}^{s}|>\delta)\leq
C\>\delta^{-k}E[|u_{n}|^{sk}]~{},~{}\forall n\geq 1~{}.$

###### Proof.

The proof goes by induction. For $k=1$, we have
$|y_{n}|\leq|\rho||y_{n-1}|+|u_{n}|$, which implies that

$E[|y_{n}|]\leq E[|u_{n}|]\left(1+|\rho|+...+|\rho|^{n-1}\right)\leq
E[|u_{n}|]~{}a^{-1}~{}.$

Therefore, the constant $C=a^{-1}$. We can also derive
$y_{n}^{2}=\rho^{2}y_{n-1}^{2}+2\rho y_{n-1}u_{n}+u_{n}^{2}$, which implies
$E[y_{n}^{2}]\leq\rho^{2}E[y_{n-1}^{2}]+2|\rho|E[|y_{n-1}u_{n}|]+E[u_{n}^{2}]$.
Using that $E[|y_{n-1}u_{n}|]=E[|y_{n-1}|]E[|u_{n}|]\leq a^{-1}E[u_{n}^{2}]$,
we conclude
$E[y_{n}^{2}]\leq\rho^{2}E[y_{n-1}^{2}]+(2|\rho|a^{-1}+1)E[u_{n}^{2}]$, which
implies
$E[y_{n}^{2}]\leq(2|\rho|a^{-1}+1)E[u_{n}^{2}]\left(1+\rho^{2}+...+\rho^{2(n-1)}\right)\leq
C_{1}(a)E[u_{n}^{2}]$, where $C_{1}(a)=(2(1-a)/a+1)/a$. In this case, the
constant $C=C_{1}(a)$.

Now, let us use $C_{1}(a)$ to construct $C_{2}(a)$ and so on. Suppose we
already compute $C_{k}(a)$. Now, let us compute $C_{k+1}(a)$. We have

$y_{n}^{2k+1}=\rho^{2k+1}y_{n-1}^{2k+1}+\sum_{\ell=1}^{2k}C_{\ell}^{2k+1}\rho^{2k+1-\ell}y_{n-1}^{2k+1-\ell}u_{n}^{\ell}+u_{n}^{2k+1}$

By triangular inequality and similar arguments as before, we obtain

$E[y_{n}^{2k+1}]\leq|\rho|^{2k+1}E[|y_{n-1}|^{2k+1}]+\sum_{\ell=1}^{2k}C_{\ell}^{2k+1}|\rho|^{2k+1-\ell}E[|y_{n-1}|^{2k+1-\ell}]E[|u_{n}^{\ell}|]+E[|u_{n}|^{2k+1}]~{},$

and by the inductive hypothesis, we know
$E[|y_{n-1}|^{2k+1-\ell}]E[|u_{n}^{\ell}|]\leq C_{k-\ell/2}E[|u_{n}|^{2k+1}]$,
where we used that $E[|X|^{a}]E[|X|^{b}]\leq E[|X|^{a+b}]$. Thus, the
inductive hypothesis implies that

$E[y_{n}^{2k+1}]\leq|\rho|^{2k+1}E[|y_{n-1}|^{2k+1}]+E[|u_{n}|^{2k+1}]\left(\sum_{\ell=1}^{2k}C_{\ell}^{2k+1}|\rho|^{2k+1-\ell}C_{k-\ell/2}+1\right)~{},$

in a similar way as in the initial case, we conclude. Note that the final
constant $C$ only involves $a$ and $k$. The other case is analogous.

######  Lemma D.14.

Suppose Assumption 4.1 holds. For a given $h,k\in\mathbf{N}$ and $a\in(0,1)$.
Then, for any $|\rho|\leq 1-a$ and $h\in\mathbf{N}$, there exist a constant
$C=C(a,k,r,s,h,C_{\sigma})>0$ such that

  1. 1.

$P(~{}(n-h)^{1/2}|(n-h)^{-1}\sum_{i=1}^{n-h}u_{i}^{r}y_{i-1}^{s}-m_{r,s}|>\delta)\leq
C\delta^{-k}E\left[|u_{i}|^{(r+s)k}\right]$

  2. 2.

$P(~{}(n-h)^{1/2}|(n-h)^{-1}\sum_{i=1}^{n-h}\xi_{i}u_{i}^{r}y_{i-1}^{s}|>\delta)\leq
C\delta^{-k}E\left[|u_{i}|^{(1+r+s)k}\right]$

  3. 3.

$P(~{}(n-h)^{1/2}|(n-h)^{-1}\sum_{i=1}^{n-h}\xi_{i}^{2}u_{i}^{2}-V|>3\delta)\leq
C\delta^{-k}E[u_{i}^{4k}]$

  4. 4.

$P(~{}(n-h)^{1/2}|(n-h)^{-1}\sum_{i=1}^{n-h}\xi_{i}^{2}u_{i}y_{i-1}|>3\delta)\leq
C\delta^{-k}E[u_{i}^{4k}]$

  5. 5.

$P(~{}(n-h)^{1/2}|(n-h)^{-1}\sum_{i=1}^{n-h}\xi_{i}^{2}y_{i-1}^{2}-\sigma^{4}g(\rho,h)^{2}(1-\rho^{2})^{-1}|>5\delta)\leq
C\delta^{-k}E[u_{i}^{4k}]$

for any $\delta>0$ and any $n>h$, where
$\xi_{i}=\xi_{i}(\rho,h)=\sum_{\ell=1}^{h}\rho^{h-\ell}u_{i+\ell}$,
$g(\rho,h)=\left(\sum_{\ell=1}^{h}\rho^{2(h-\ell)}\right)^{1/2}$, and
$m_{r,s}=E\left[u_{i}^{r}\left(\sum_{j\geq
1}\rho^{j-1}u_{i-j}\right)^{s}\right]$.

###### Proof.

Define the filtration $\mathcal{F}_{j}=\sigma(u_{i}:i\leq j)$. In what
follows, we use Markov’s inequality, Lemmas D.8, D.9, and D.13. The constant
$C$ will replace other constants and will only depend on $a$, $k$, $r$, $s$,
$h$, $C_{\sigma}$, and the constants that appears in Lemmas D.9 and D.13.

Item 1: We prove this item by induction on $s$. First, consider $s=0$ and
$r\geq 0$. Note that $\\{(u_{i}^{r}-E[u_{i}^{r}],\mathcal{F}_{i}):1\leq i\leq
n-h\\}$ define a martingale difference sequence. Therefore, Markov’s
inequality and Lemma D.9 imply
$P(~{}(n-h)^{1/2}|(n-h)^{-1}\sum_{i=1}^{n-h}u_{i}^{r}-m_{r,0}|>\delta)\leq
C\delta^{-k}E\left[|u_{i}|^{rk}\right]$, since $m_{r,0}=E[u_{i}^{r}]$. Let us
suppose that item 1 holds for any $(r,s)$ such that $r\geq 0$ and $s\leq
s_{0}$ (this is a strong inductive hypothesis). Next, let us prove item 1 for
$(r,s_{0}+1)$. We write

$(n-h)^{-1}\sum_{i=1}^{n-h}u_{i}^{r}y_{i-1}^{s_{0}+1}-m_{r,s_{0}+1}=I_{1}+I_{2}~{},$

where
$I_{1}=(n-h)^{-1}\sum_{i=1}^{n-h}\left(u_{i}^{r}-m_{r,0}\right)y_{i-1}^{s_{0}+1}$
and
$I_{2}=(n-h)^{-1}\sum_{i=1}^{n-h}m_{r,0}\left(y_{i-1}^{s_{0}+1}-m_{0,s_{0}+1}\right)$.
Note that
$\\{(\left(u_{i}^{r}-m_{r,0}\right)y_{i-1}^{s_{0}+1},\mathcal{F}_{i}):1\leq
i\leq n-h\\}$ define a martingale difference sequence; therefore, we conclude
that $P((n-h)^{1/2}|I_{1}|>\delta)\leq C\delta^{-k}E[|u_{i}|^{k(r+s_{0}+1}]$
using Markov’s inequality and Lemmas D.9 and D.13. Now, let us write

$y_{i}^{s_{0}+1}=(\rho
y_{i-1}+u_{i})^{s_{0}+1}=\rho^{s_{0}+1}y_{i-1}^{s_{0}+1}+\sum_{j=1}^{s_{0}+1}\binom{s_{0}+1}{j}\rho^{s_{0}+1-j}u_{i}^{j}y_{i-1}^{s_{0}+1-j}~{},$

which implies the following identity

$(1-\rho^{s_{0}+1})(n-h)^{-1}\sum_{i=1}^{n-h}y_{i-1}^{s_{0}+1}=\frac{-y_{i}^{s_{0}+1}}{n-h}+\sum_{j=1}^{s_{0}+1}\binom{s_{0}+1}{j}\rho^{s_{0}+1-j}\left((n-h)^{-1}\sum_{i=1}^{n-h}u_{i}^{j}y_{i-1}^{s_{0}+1-j}\right)~{}.$

In a similar way, using $z_{i}=\sum_{j\geq 1}\rho^{j-1}u_{i-j}$ instead of
$y_{i}$, we can derive the following identity

$(1-\rho^{s_{0}+1})m_{0,s_{0}+1}=\sum_{j=1}^{s_{0}+1}\binom{s_{0}+1}{j}\rho^{s_{0}+1-j}m_{j,s_{0}+1-j}~{}.$

Using that $|m_{r,0}|^{k}=|E[u_{i}^{r}]|^{k}\leq E[|u_{i}|^{rk}]$, the
previous two identities, the inductive hypothesis to
$(n-h)^{-1}\sum_{i=1}^{n-h}u_{i}^{j}y_{i-1}^{s_{0}+1-j}-m_{j,s_{0}+1-j}$ for
$j=1,\dots,s_{0}+1$, and Lemmas D.13 and D.8, we conclude that
$P((n-h)^{1/2}|I_{2}|>\delta)\leq C\delta^{-k}E[|u_{i}|^{k(r+s_{0}+1}]$, which
completes the proof due to Lemma D.8.

Item 2: Consider the following derivation for a sequence of random variable
$f_{i}\in\mathcal{F}_{i}$:

$\displaystyle\sum_{i=1}^{n-h}\xi_{i}f_{i}$
$\displaystyle=\sum_{i=1}^{n-h}\sum_{\ell=1}^{h}\rho^{h-\ell}u_{i+\ell}~{}f_{i}=\sum_{i=1}^{n-h}\sum_{j=i+1}^{i+h}\rho^{i+h-j}u_{j}~{}f_{i}=\sum_{j=1}^{n}u_{j}b_{n,j}~{},$

where $b_{n,j}=\sum_{i=j-h}^{j-1}\rho^{i+h-j}~{}f_{i}I\\{1\leq i\leq n-h\\}$.
Note that $\\{u_{j}b_{n,j}\in\mathcal{F}_{i}:1\leq j\leq n-h\\}$ defines a
martingale difference sequence and

$\displaystyle E[|u_{j}b_{n,j}|^{k}]$ $\displaystyle\leq
h^{k-1}\sum_{i=j-h}^{j-1}|\rho|^{(i+h-j)k}~{}E[|u_{j}|^{k}]E[|f_{i}|^{k}]I\\{1\leq
i\leq n-h\\}$ $\displaystyle\leq h^{k-1}E[|u_{i}|^{k}]\max_{1\leq i\leq
n-h}E[|f_{i}|^{k}]\sum_{\ell=1}^{h}(1-a)^{(h-\ell)k}\leq
C_{k,h,a}E[|u_{i}|^{k}]\max_{1\leq i\leq n-h}E[|f_{i}|^{k}]~{},$

where $C_{k,h,a}=h^{k-1}\sum_{\ell=1}^{h}(1-a)^{(h-\ell)k}$. Finally, we take
$f_{i}=u_{i}^{r}y_{i-1}^{s}$ and use $E[|u_{i}^{r}y_{i-1}^{s}|^{k}]\leq
CE[|u_{i}|^{rk}]E[|u_{i}|^{sk}]$ due to Lemma D.13. We conclude by Jensen that
$E[|u_{i}|^{k}]E[|f_{i}|^{k}]\leq E[|u_{i}|^{(1+s+r)k}]$.

Item 3: Consider the following derivation for item 3 as we did in the proof of
Lemma D.4. The expression $\sum_{i=1}^{n-h}\xi_{i}^{2}u_{i}^{2}$ can be
decomposed into the sum of its expected value and another three zero mean
terms:

$(n-h)g(\rho,h)^{2}\sigma^{4}+\sum_{j=1}^{n}(u_{j}^{2}-\sigma^{2})b_{n,j}+\sum_{j=1}^{n}u_{j}d_{n,j}+\sum_{j=1}^{n-h}(u_{j}^{2}-\sigma^{2})g(\rho,h)^{2}\sigma_{n}^{2}~{},$

where the first term is equal to $(n-h)V$,
$b_{n,j}=\sum_{i=j-h}^{j-1}\rho^{2(h+i-j)}u_{i}^{2}I\\{1\leq i\leq n-h\\}$,
and

$\displaystyle
d_{n,j}=\sum_{i=j-h}^{j-1}\sum_{\ell_{2}=1}^{j-i-1}u_{i+\ell_{2}}\rho^{2h-j+i-\ell_{2}}u_{i}^{2}I\\{1\leq
i\leq n-h\\}~{}.$

Note that $b_{n,j}$ and $d_{n,j}$ are measurable with respect to
$\mathcal{F}_{j-1}$; therefore, $(u_{j}^{2}-\sigma^{2})b_{n,j}$ and
$u_{j}d_{n,j}$ define a martingale difference sequence with respect to
$\mathcal{F}_{j-1}$. Also, note that

$E[|(u_{j}^{2}-\sigma^{2})b_{n,j}|^{k}]\leq
E[|u_{j}^{2}-\sigma^{2}|^{k}]E[|b_{n,j}|^{k}]\leq
C_{k,h,a,1}E[u_{i}^{2k}]E[u_{i}^{2k}]~{},$

where $C_{k,h,a,1}=2^{k}h^{k-1}\sum_{\ell=1}^{h}(1-a)^{2(h-\ell)k}$ and

$\displaystyle E[|u_{n,j}d_{n,j}|^{k}]$ $\displaystyle\leq
E[|u_{i}|^{k}]h^{k-1}\sum_{i=j-h}^{j-1}E[|\sum_{\ell_{2}=1}^{j-i-1}u_{i+\ell_{2}}\rho^{2h-j+i-\ell_{2}}u_{i}^{2}|^{k}]I\\{1\leq
i\leq n-h\\}$ $\displaystyle\leq
E[|u_{i}|^{k}]h^{k-1}\sum_{\ell_{1}=1}^{h}E[|\sum_{\ell_{2}=1}^{\ell_{1}-1}u_{\ell_{1}+\ell_{2}}\rho^{2h-\ell_{1}-\ell_{2}}u_{\ell_{1}}^{2}|^{k}]$
$\displaystyle\leq
E[|u_{i}|^{k}]h^{2(k-1)}\sum_{\ell_{1}=1}^{h}\sum_{\ell_{2}=1}^{h}|\rho|^{(2h-\ell_{1}-\ell_{2})k}E[|u_{i+\ell_{2}}|^{k}|u_{\ell_{1}}|^{2k}]$
$\displaystyle\leq C_{k,h,a,2}E[|u_{i}|^{k}]^{2}E[u_{i}^{2k}]~{},$

where
$C_{k,h,a,2}=h^{2(k-1)}\sum_{\ell_{1}=1}^{h}\sum_{\ell_{2}=1}^{h}(1-a)^{(2h-\ell_{1}-\ell_{2})k}$.
We use the above decomposition and Bonferroni’s inequality to derive

$\displaystyle
P(~{}(n-h)^{1/2}|(n-h)^{-1}\sum_{i=1}^{n-h}\xi_{i}^{2}u_{i}^{2}-V|>3\delta)\leq$
$\displaystyle
P(~{}|(n-h)^{-1/2}\sum_{j=1}^{n}(u_{j}^{2}-\sigma^{2})b_{n,j}|>\delta)$
$\displaystyle+P(~{}|(n-h)^{-1/2}\sum_{j=1}^{n}u_{j}d_{n,j}|>\delta)$
$\displaystyle+P(~{}|(n-h)^{-1/2}\sum_{j=1}^{n-h}(u_{j}^{2}-\sigma^{2})g(\rho,h)^{2}\sigma^{2}|>\delta)~{}.$

By Markov’s inequality, Lemma D.9 and the previous derivation, we derive the
following inequalities for each of the terms that appear in the right-hand
side of the previous display.

$\displaystyle
P(~{}|(n-h)^{-1/2}\sum_{j=1}^{n}(u_{j}^{2}-\sigma^{2})b_{n,j}|>\delta)$
$\displaystyle\leq\delta^{-k}\left(\frac{n-h}{n}\right)^{k/2}E[|n^{-1/2}\sum_{j=1}^{n}(u_{j}^{2}-\sigma^{2})b_{n,j}|^{k}]$
$\displaystyle\leq\delta^{-k}d_{k}C_{k,h,a,1}E[u_{i}^{2k}]E[u_{i}^{2k}]~{}.$

In a similar way, we conclude

$\displaystyle P(~{}|(n-h)^{-1/2}\sum_{j=1}^{n}u_{j}d_{n,j}|>\delta)$
$\displaystyle\leq\delta^{-k}\left(\frac{n-h}{n}\right)^{k/2}E[|n^{-1/2}\sum_{j=1}^{n}u_{j}d_{n,j}|^{k}]$
$\displaystyle\leq\delta^{-k}d_{k}C_{k,h,a,2}E[|u_{i}|^{k}]^{2}E[u_{i}^{2k}]~{}.$

Finally,

$\displaystyle
P(~{}|(n-h)^{-1/2}\sum_{j=1}^{n-h}(u_{j}^{2}-\sigma^{2})g(\rho,h)^{2}\sigma^{2}|>\delta)$
$\displaystyle\leq\delta^{-k}E[|(n-h)^{-1/2}\sum_{j=1}^{n-h}(u_{j}^{2}-\sigma^{2})g(\rho,h)^{2}\sigma^{2}|^{k}]$
$\displaystyle\leq\delta^{-k}d_{k}{C}_{k,h,a,3}E[u_{i}^{2k}]E[u_{i}^{2k}]~{},$

where ${C}_{k,h,a,3}=2^{k}(\sum_{\ell=1}^{h}(1-a)^{2(h-\ell)})^{k}$, since
$|\rho|\leq 1-a$ and $\sigma^{2k}=E[u_{i}^{2}]^{k}\leq E[u_{i}^{2k}]$.

Item 4: Similarly, we express $\sum_{i=1}^{n-h}\xi_{i}^{2}u_{i}y_{i-1}$ as the
sum of three martingale differences:

$\sum_{j=1}^{n}(u_{j}^{2}-\sigma^{2})b_{n,j}+\sum_{j=1}^{n}u_{j}d_{n,j}+\sum_{j=1}^{n-h}g(\rho,h)^{2}\sigma_{n}^{2}u_{i}y_{i-1}~{},$

where $b_{n,j}=\sum_{i=j-h}^{j-1}\rho^{2(h+i-j)}u_{i}y_{i-1}I\\{1\leq i\leq
n-h\\}$ and

$d_{n,j}=\sum_{i=j-h}^{j-1}\sum_{\ell_{2}=1}^{j-i-1}u_{i+\ell_{2}}\rho^{2h-j+i-\ell_{2}}u_{i}y_{i-1}I\\{1\leq
i\leq n-h\\}~{}.$

Note that $b_{n,j}$ and $d_{n,j}$ are measurable with respect to
$\mathcal{F}_{j-1}$; therefore, $(u_{j}^{2}-\sigma^{2})b_{n,j}$ and
$u_{j}d_{n,j}$ define a martingale difference sequence with respect to
$\mathcal{F}_{j-1}$. Also, note that

$E[|(u_{j}^{2}-\sigma^{2})b_{n,j}|^{k}]\leq
E[|u_{j}^{2}-\sigma^{2}|^{k}]E[|b_{n,j}|^{k}]\leq
C_{k,h,a,1}E[u_{i}^{2k}]E[|u_{i}|^{k}]^{2}~{},$

where $C_{k,h,a,1}=2^{k}h^{k-1}\sum_{\ell=1}^{h}(1-a)^{2(h-\ell)k}$ and

$\displaystyle E[|u_{n,j}d_{n,j}|^{k}]$ $\displaystyle\leq
E[|u_{i}|^{k}]h^{k-1}\sum_{i=j-h}^{j-1}E[|\sum_{\ell_{2}=1}^{j-i-1}u_{i+\ell_{2}}\rho^{2h-j+i-\ell_{2}}u_{i}y_{i-1}|^{k}]I\\{1\leq
i\leq n-h\\}$ $\displaystyle\leq
E[|u_{i}|^{k}]h^{k-1}\sum_{\ell_{1}=1}^{h}E[|\sum_{\ell_{2}=1}^{\ell_{1}-1}u_{\ell_{1}+\ell_{2}}\rho^{2h-\ell_{1}-\ell_{2}}u_{\ell_{1}}y_{\ell_{1}-1}|^{k}]$
$\displaystyle\leq
E[|u_{i}|^{k}]h^{2(k-1)}\sum_{\ell_{1}=1}^{h}\sum_{\ell_{2}=1}^{h}|\rho|^{(2h-\ell_{1}-\ell_{2})k}E[|u_{i+\ell_{2}}|^{k}|u_{\ell_{1}}|^{k}|y_{\ell_{1}-1}|^{k}]$
$\displaystyle\leq C_{k,h,a,2}E[|u_{i}|^{k}]^{2}E[|u_{i}|^{k}]^{2}~{},$

where
$C_{k,h,a,2}=h^{2(k-1)}\sum_{\ell_{1}=1}^{h}\sum_{\ell_{2}=1}^{h}(1-a)^{(2h-\ell_{1}-\ell_{2})k}$.
Similar to item 3, we use the above decomposition to obtain
$P(~{}(n-h)^{1/2}|(n-h)^{-1}\sum_{i=1}^{n-h}\xi_{i}^{2}u_{i}y_{i-1}|>3\delta)\leq
I_{1}+I_{2}+I_{3}$ where

$\displaystyle I_{1}$
$\displaystyle=P(~{}|(n-h)^{-1/2}\sum_{j=1}^{n}(u_{j}^{2}-\sigma^{2})b_{n,j}|>\delta)\leq\delta^{-k}d_{k}C_{k,h,a,1}E[u_{i}^{2k}]E[|u_{i}|^{k}]^{2}$
$\displaystyle I_{2}$
$\displaystyle=P(~{}|(n-h)^{-1/2}\sum_{j=1}^{n}u_{j}d_{n,j}|>\delta)\leq\delta^{-k}d_{k}C_{k,h,a,2}E[|u_{i}|^{k}]^{2}E[|u_{i}|^{k}]^{2}$
$\displaystyle I_{3}$
$\displaystyle=P(~{}|(n-h)^{-1/2}\sum_{j=1}^{n-h}g(\rho,h)^{2}\sigma^{2}u_{i}y_{i-1}|>\delta)\leq\delta^{-k}d_{k}C_{k,h,a,3}{C}_{k,h,a,3}E[|u_{i}|^{k}]^{2}E[u_{i}^{2k}]$

where ${C}_{k,h,a,3}=2^{k}(\sum_{\ell=1}^{h}(1-a)^{2(h-\ell)})^{k}$, since
$|\rho|\leq 1-a$ and $\sigma^{2k}=E[u_{i}^{2}]^{k}\leq E[u_{i}^{2k}]$.

Item 5: We proceed in a similar (but not exactly) way as in the proof of items
3 and 4. Let us write
$\sum_{i=1}^{n-h}\xi_{i}^{2}y_{i-1}^{2}-\sigma^{4}g(\rho,h)^{2}(1-\rho^{2})^{-1}$
as the sum of three terms:

$\displaystyle\sum_{j=1}^{n}(u_{j}^{2}-\sigma^{2})b_{n,j}+\sum_{j=1}^{n}u_{j}d_{n,j}+g(\rho,h)^{2}\sigma^{4}(1-\rho^{2})^{-1}\sum_{j=1}^{n-h}(\sigma^{-2}(1-\rho^{2})y_{i-1}^{2}-1)~{},$

where $b_{n,j}=\sum_{i=j-h}^{j-1}\rho^{2(h+i-j)}y_{i-1}^{2}I\\{1\leq i\leq
n-h\\}$ and

$\displaystyle
d_{n,j}=\sum_{i=j-h}^{j-1}\sum_{\ell_{2}=1}^{j-i-1}u_{i+\ell_{2}}\rho^{2h-j+i-\ell_{2}}y_{i-1}^{2}I\\{1\leq
i\leq n-h\\}~{}.$

Note that as before that $(u_{j}^{2}-\sigma^{2})b_{n,j}$ and $u_{j}d_{n,j}$
define a martingale difference sequence with respect to $\mathcal{F}_{j}$, and
we can proceed as before. What is new is the third term, which can be
controlled using item 1 and using that $m_{0,2}=\sigma^{2}(1-\rho^{2})^{-1}$.
In particular, we obtain that

$P(~{}(n-h)^{1/2}|(n-h)^{-1}\sum_{i=1}^{n-h}\xi_{i}^{2}y_{i-1}^{2}-\sigma^{4}g(\rho,h)^{2}(1-\rho^{2})^{-1}|>5\delta)\leq
I_{1}+I_{2}+I_{3}~{},$

by Bonferroni’s inequality, is lower or equal to

$\displaystyle I_{1}$
$\displaystyle=P(~{}|(n-h)^{-1/2}\sum_{j=1}^{n}(u_{j}^{2}-\sigma^{2})b_{n,j}|>\delta)\leq\delta^{-k}C_{k,h,a,1}d_{k}E[u_{i}^{2k}]^{2}$
$\displaystyle I_{2}$
$\displaystyle=P(~{}|(n-h)^{-1/2}\sum_{j=1}^{n}u_{j}d_{n,j}|>\delta)\leq\delta^{-k}C_{k,h,a,2}d_{k}E[|u_{i}|^{k}]^{2}E[u_{i}^{2k}]^{2}$
$\displaystyle I_{3}$
$\displaystyle=P(~{}|(n-h)^{-1/2}\frac{g(\rho,h)^{2}\sigma^{4}}{1-\rho^{2}}\sum_{j=1}^{n-h}(\sigma^{-2}(1-\rho^{2})y_{i-1}^{2}-1)|>\delta)\leq\delta^{-k}d_{k}C_{k,h,a,3}E[u_{i}^{2k}]^{2}~{},$

where we compute the first and second constant as we did in the proof of items
3 and 4,
$C_{k,h,a,1}=2^{k}\tilde{C}_{2k}h^{k-1}\sum_{\ell=1}^{h}(1-a)^{2(h-\ell)k}$,$C_{k,h,a,2}=h^{2(k-1)}\tilde{C}_{2k}\sum_{\ell_{1}=1}^{h}\sum_{\ell_{2}=1}^{h}(1-a)^{(2h-\ell_{1}-\ell_{2})k}$,
while the third one follows by item 1 and using that
$m_{0,2}=\sigma^{2}(1-\rho^{2})^{-1}$,
$C_{k,h,a,2}=2^{k}(\tilde{C}_{k}+2)\left(\sum_{\ell=1}^{h}(1-a)^{2(h-\ell)}\right)^{k}$.
We conclude $I_{1}+I_{2}+I_{3}\leq C\delta^{-k}E[u_{i}^{4k}]$, where the
constant $C$ absorb all the previous constants.

######  Lemma D.15.

Suppose Assumption 4.1 holds. For fixed $h,k\in\mathbf{N}$ and $a\in(0,1)$.
Then, for any $|\rho|\leq 1-a$, there exist a constant
$C=C(a,k,h,C_{\sigma})>0$ such that

  1. 1.

$P\left((n-h)^{2/2}\left|\hat{\rho}_{n}(h)-\rho-(1-\rho^{2})\sigma^{-2}\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}\right|>\delta^{2}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]\right)$

  2. 2.

$P\left((n-h)^{2/2}\left|\hat{\beta}_{n}(h)-\beta(\rho,h)-\sigma^{-2}\frac{\sum_{i=1}^{n-h}\xi_{i}(\rho,h)u_{i}}{n-h}\right|>\delta^{2}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]\right)$

  3. 3.

$P\left((n-h)^{2/2}\left|\hat{\gamma}_{n}(h)-\frac{(1-\rho^{2})\sum_{i=1}^{n-h}\xi_{i}(\rho,h)y_{i-1}}{\sigma^{2}(n-h)}+\frac{\rho\sum_{i=1}^{n-h}\xi_{i}(\rho,h)u_{i}}{\sigma^{2}(n-h)}\right|>\delta^{2}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]\right)$

  4. 4.

$P\left((n-h)^{2/2}\left|\hat{\eta}_{n}(\rho,h)-\eta(\rho,h)-\frac{(1-\rho^{2})\sum_{i=1}^{n-h}\xi_{i}y_{i-1}}{\sigma^{2}(n-h)}\right|>\delta^{2}\right)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]\right)$

  5. 5.

$P(n^{1/2}|\hat{\rho}_{n}-\rho|>\delta)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]\right)$

  6. 6.

$P(n^{1/2}|\hat{\beta}_{n}(h)-\beta(\rho,h)|>\delta)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]\right)$

  7. 7.

$P(n^{1/2}|\hat{\gamma}_{n}(h)|>\delta)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]\right)$

  8. 8.

$P(n^{1/2}|\hat{\eta}_{n}-\eta|>\delta)\leq
C\delta^{-k}\left(E[|u_{i}|^{k}]+E[u_{i}^{2k}]\right)$

for any $\delta<n^{1/2}$, where $\hat{\rho}_{n}(h)$ is as in (6),
$(\hat{\beta}_{n}(h),\hat{\gamma}_{n}(h))$ is as in (4), and
$\xi_{i}(\rho,h)=\sum_{\ell=1}^{h}\rho^{h-\ell}u_{i+\ell}$.
$\hat{\eta}_{n}(\rho,h)=\rho\hat{\beta}_{n}(h)+\hat{\gamma}_{n}(h)$,
$\eta(\rho,h)=\rho\beta(\rho,h)$.

###### Proof.

In what follows, we use $\xi_{i}=\xi_{i}(\rho,h)$, and the constant $C$
accumulates previous constants $C$’s and includes $a$,$k$,$h$, and
$C_{\sigma}$. To prove item 1, we first use the definition of
$\hat{\rho}_{n}(h)$,

$\hat{\rho}_{n}(h)-\rho=\frac{(n-h)^{-1}\sum_{i=1}^{n-h}u_{i}y_{i-1}}{(n-h)^{-1}\sum_{i=1}^{n-h}y_{i-1}^{2}}=\frac{(1-\rho^{2})\sum
u_{i}y_{i-1}}{\sigma^{2}(n-h)}(1+W_{n})^{-1}~{},$

where

$W_{n}=\frac{(1-\rho^{2})\sum_{i=1}^{n-h}y_{i-1}^{2}}{\sigma^{2}(n-h)}-1~{}.$

Using this notation, we have

$\hat{\rho}_{n}(h)-\rho-\frac{(1-\rho^{2})\sum
u_{i}y_{i-1}}{\sigma^{2}(n-h)}=\frac{(1-\rho^{2})\sum
u_{i}y_{i-1}}{\sigma^{2}(n-h)}\left((1+W_{n})^{-1}-1\right)~{}.$

Since $P(n^{1/2}|(n-h)^{-1/2}\sum_{i=1}^{n-h}u_{i}y_{i-1}|>\delta)\leq
C\delta^{-k}E[u_{i}^{2k}]$ holds by Lemma D.14, it is sufficient to show that
$P(n^{1/2}|(1+W_{n})^{-1}-1|>\delta)\leq C\delta^{-k}E[u_{i}^{2k}]$ due to
Lemma D.8. To prove the last inequality we use $P(n^{1/2}|W_{n}|>\delta)\leq
C\delta^{-k}E[u_{i}^{2k}]$ (which holds by Lemma D.14) and part 5 in Lemma
D.8.

To prove item 2, we proceed similarly as we did above. We use the definition
of $\hat{\beta}_{n}(h)$,

$\displaystyle\hat{\beta}_{n}(h)-\beta$
$\displaystyle=\frac{\left(\frac{\sum_{i=1}^{n-h}\xi_{i}u_{i}}{n-h}\right)\left(\frac{\sum_{i=1}^{n-h}y_{i-1}^{2}}{n-h}\right)-\left(\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}\right)\left(\frac{\sum_{i=1}^{n-h}\xi_{i}y_{i-1}}{n-h}\right)}{\left(\frac{\sum_{i=1}^{n-h}u_{i}^{2}}{n-h}\right)\left(\frac{\sum_{i=1}^{n-h}y_{i-1}^{2}}{n-h}\right)-\left(\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}\right)^{2}}~{},$
$\displaystyle=\left(\frac{\sum_{i=1}^{n-h}\xi_{i}u_{i}}{\sigma^{2}(n-h)}\right)\frac{1+W_{n}}{1+Z_{n}}-\left(\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}\right)\left(\frac{(1-\rho^{2})\sum_{i=1}^{n-h}\xi_{i}y_{i-1}}{\sigma^{2}(n-h)}\right)\frac{1}{1+Z_{n}}~{},$

where $W_{n}$ is defined as in the proof of item 1 and

$Z_{n}=\left(\frac{\sum_{i=1}^{n-h}u_{i}^{2}-\sigma^{2}}{\sigma^{2}(n-h)}\right)\left(1+W_{n}\right)-\left(\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}\right)^{2}\frac{1-\rho^{2}}{\sigma^{4}}+W_{n}~{}.$

Note that Lemmas D.8 and D.14 imply $P(n^{1/2}|Z_{n}|>\delta)\leq
C\delta^{-k}E[u_{i}^{2k}]$. We can use Lemmas D.8 and D.14, and the argument
we used in the proof of item 1 by replacing $W_{n}$ by $Z_{n}$, to conclude
the proof of item 2.

To prove item 3, we proceed similarly as we did above. We use the definition
of $\hat{\gamma}_{n}(h)$,

$\displaystyle\hat{\gamma}_{n}(h)$
$\displaystyle=\frac{\left(\frac{\sum_{i=1}^{n-h}\xi_{i}y_{i-1}}{n-h}\right)\left(\frac{\sum_{i=1}^{n-h}u_{i}^{2}}{n-h}+\rho\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}\right)-\left(\frac{\sum_{i=1}^{n-h}\xi_{i}u_{i}}{n-h}\right)\left(\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}+\rho\frac{\sum_{i=1}^{n-h}y_{i-1}^{2}}{n-h}\right)}{\left(\frac{\sum_{i=1}^{n-h}u_{i}^{2}}{n-h}\right)\left(\frac{\sum_{i=1}^{n}y_{i-1}^{2}}{n-h}\right)-\left(\frac{\sum_{i=1}^{n-h}u_{i}y_{i-1}}{n-h}\right)^{2}}~{}.$

The proof follows using the same arguments (Lemmas D.8 and D.14) as before
since all the sample average that appears in the previous display are also
studied in Lemma D.14.

The proof of item 4 follows by the results of items 2 and 3, the definition of
$\hat{\eta}_{n}(\rho,h)$ and $\eta(\rho,h)$, and Bonferroni’s inequality.
Items 5-8 are implied by items 1-4, Bonferroni’s inequality, and Lemma D.14.