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\begin{document}
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\begin{center}
{\Large \bf
SUPPLEMENTARY MATERIAL ON ``SPECIFICATION TESTS FOR MULTIPLICATIVE ERROR MODELS"
}\\[1cm]
{\large Indeewara Perera} \\
{\em Department of Econometrics and Business Statistics, Monash University}\\
{indeewara.perera@monash.edu}\\[0.3cm]
{\large Mervyn J. Silvapulle} \\
{\em Department of Econometrics and Business Statistics, Monash University}\\
{mervyn.silvapulle@monash.edu}\\
\end{center}

\bigskip

This online Supplementary Material provides omitted proofs (Appendix~B),
simulation results for evaluating density forecasts (Appendix~C), and additional results for
 the empirical example in Section~\ref{sec:example} (Appendix~D).


\section*{APPENDIX B: Omitted proofs}
\label{appendixB}
\renewcommand{\theequation}{B.\arabic{equation}}
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\begin{proof}[\bf Proof of Lemma~\ref{lem:fix:x}]
Let $$K_{ni} := I[\eta_{ni} \le x(1+\rho_{ni})] - I[\eta_{ni} \le x] - H_{\vth_n}[x(1+\rho_{ni})] + H_{\vth_n}(x).$$
Then, for $\vth_n \in B$,
\benn
\frac{1}{n} \s \g_{ni}^2 \mathbb{E}[K_{ni}^2 | \cF_{ni}] \le \frac{1}{n}  \s \g_{ni}^2 \sup_{\vth_n \in B} \sup_{|z| \le b} |H_{\vth_n}\{x(1+z)\} - H_{\vth_n}(x)| \le \frac{a}{n}  \s \g_{ni}^2.
\eenn
Now, from Lemma~\ref{freedman} with $D_{ni} = n^{-1} \g_{ni} K_{ni}$, it follows that for any $\eta, c > 0$,
\benrr
\lefteqn{P_n\big([|\wti U_n(x) - U_n(x)| > \eta] \cap \Pi_n \big)}\\
&& \le P_n\Big(\Big[|n^{-1/2}\s \g_{ni} I\{|\g_{ni}| \le an^{-1/2}\}K_{ni}| > \eta\Big] \cap \Big[\s \mathbb{E}(D_{ni}^2 \mid \cF_{ni}) \le ac\Big]\Big)\\
&& \le \exp\{-\eta^2/2a(\eta+c)\}.
\eenrr
\end{proof}


\begin{proof}[\bf Proof of Lemma~\ref{lem:fix:x:2}]
Fix $x \ge 0$ and $\vep, \eta > 0$.
Choose $c > 0$ and a positive integer $n_1$ such that $P_n(n^{-1} \s \g_{ni}^2 > c) < \vep$ for all $n \ge n_1$.
Select  $a > 0$ to have $\exp\{-\eta^2/2a(\eta+c)\} < \vep$.
Further, choose $\delta > 0$ and $b_0 > 0$ such that $a - \delta > 0$ and
$
\L_0 := \sup_{|z| \le b_0} |H_{\vth_0} \{x(1+z)\} - H_{\vth_0}(x)| < \delta
$.
Then, $\L_n := \sup_{|z| \le b_0} |H_{\vth_n} \{x(1+z)\} - H_{\vth_n}(x)|$ forms a real sequence such that $\L_n \to \L_0$ as $n \to \iny$.
Therefore, there exists an $n_2 \in \mathbb{N}$ such that $\L_n < \L_0 + a - \delta < a$ for all $n \ge n_2$.
We also have that $\vth_n \in B$ for all $n \ge n_0$ for some $n_0$.
For each $n \in \{n_0, n_0+1, \ldots, \max(n_1, n_2)\}$, there exists a $b_n > 0$ such that $\sup_{|z| \le b_n} |H_{\vth_n} \{x(1+z)\} - H_{\vth_n}(x)| < a$.

Now, let $b := \min\{b_{n_0}, b_{n_0+1}, \ldots, b_{\max(n_1, n_2)}, b_0\}$.
Then, for each $\vth_n \in B$,
\benrr
&\sup_{|z| \le b} |H_{\vth_0} \{x(1+z)\} - H_{\vth_0}(x)| < a.&
\eenrr
Therefore,
$
\sup_{\vth_n \in B}\sup_{|z| \le b} |H_{\vth_0} \{x(1+z)\} - H_{\vth_0}(x)| < a.
$
Thus, it follows from Lemma~\ref{lem:fix:x} that for $\vth_n \in B$,
\benrr
P_n(|\wti U_n(x) - U_n(x)| > \eta)
&\le& P_n([|\wti U_n(x) - U_n(x)| > \eta] \cap \Pi_n) + P_n(\Pi_n^c)\\
&\le&
\exp\{-\eta^2/2a(\eta+c)\} + P_n(\max_i|\g_{ni}| > an^{-1/2}) \\
 && \ + P_n(n^{-1}\s \g_{ni}^2 > c) + P_n((\max_i|\rho_{ni}| > b).
\eenrr
The first term of the last upper bound  is less than $\vep$; the second and fourth terms are $o(1)$, and the third term is less than $\vep$ for all $n \ge n_1$.
Consequently,
\benrr
&\limsup_{n \to \iny} P_n(|\wti U_n(x) - U_n(x)| > \eta) < 2\vep.&
\eenrr
Since $\vep$ is arbitrary, we have $|\wti U_n(x) - U_n(x)| = o_{p_n}(1)$.
\end{proof}


\begin{proof}[\bf Proof of Lemma~\ref{lem-entropy}]
Fix $\delta$ and $b$ such that $0 < \delta < 1$ and $0 \le b <1$.
Choose $K_{1\delta}^B > 0$ such that
 $\sup_{\vth \in B} H_\vth (K_{1\delta}^B) \le \delta^2/2$.
Let
$x_0 = K_{1\delta}^B/(1+b)$,
and define
$\|h_\vth\|_{\infty}^B := \sup_{\vth \in B} \sup_{x\ge0} |h_\vth(x)|$, where $h_\vth$ is the density of $H_\vth$.
Choose an integer
$N_{\delta}^B > 0$ to have
$\delta^2/4 \le \sup_{\vth \in B} \{ 1- H_\vth (K_{2\delta}^B)\} \le \delta^2/2$,
where
$K_{2\delta}^B := (1-b)[x_0 + N_{\delta}^B \delta^2/ \{2\|h_{\vth}\|_{\infty}^B\}]$.

Now, partition $\R^+ \cup \{0\}$ as
\ben\label{partition}
[0, x_0] \cup (x_0, x_1] \cup \ldots \cup (x_{N_{\delta}^B}, \infty),
\een
where $x_{N_{\delta}^B} = K_{2\delta}^B/(1-b)>0$ and $x_k - x_{k-1} = \delta^2/ \{2 \|h_{\vth}\|_{\infty}^B\}$, $k = 1, \ldots, N_{\delta}^B$.
In this partition, there are $N_{\delta}^B$  subintervals of length $ \delta^2/ \{2 \|h_{\vth}\|_{\infty}^B\}$ covering
the interval $(x_0, x_{N_\delta^B}]$.
Because $x_0=K_{1 \delta}^B/(1+b)>0$,
\benrr
N_{\delta}^B
= (1-b^2)^{-1} \{(1+b)K_{2 \delta}^B - (1-b)K_{1 \delta}^B\} 2 \|h_{\vth}\|_{\infty}^B / \delta^2.
\eenrr
Since $x_0 > 0$, we also have
 $\{x(1+b)\} \leq K_{1\delta}^B$ for $x \in [0, x_0]$.
Therefore,
\benrr
 &\text{$[\mu_b^B(x, y)]^2 \leq 2\sup_{\vth \in B}  H_\vth(K_{1\delta}^B) \leq \delta^2$ for $x, y \in [0, x_0]$.}&
\eenrr
For $x , y \in (x_{N_\delta^B}, \infty)$, we have
 $\{x(1-b)\}\geq K_{2\delta}^B$ and $ \{y(1-b)\} \geq K_{2\delta}^B$,
 and hence,
\benrr
&\text{$[\mu_b^B(x, y)]^2  \le 2\sup_{\vth \in B}   |1- H_\vth(K_{2\delta}^B)|  \leq   \delta^2$ for $x, y \in (x_{N_\delta^B}, \infty)$.}&
\eenrr
Because $0 \le b < 1$, by applying the mean value theorem,
$$[\mu_b^B(x,y)]^2 \leq  \delta^2 \{2 \|h_\vth\|_\iny^B \}^{-1} \  (1+b)  \|h_\vth\|_\iny^B
\leq \delta^2, \text{ for $x, y \in (x_{k-1}, x_k]$, $k = 1, \ldots, N_{\delta}^B$.}$$
Thus, each interval in the partition~\eqref{partition}
has diameter less than $\delta$ with respect to
the pseudo-metric $\mu_b^B$.
Therefore, $\cN(\delta,b) \leq  2 + N_{\delta}^B$, and~hence,
\ben\label{en-eq-1}
\cN(\delta,b) \leq  2 + (1-b^2)^{-1} \{(1+b)K_{2 \delta}^B - (1-b)K_{1 \delta}^B\} 2 \|h_{\vth}\|_{\infty}^B / \delta^2.
\een


Now, let $\mu_\vth = \mathbb{E} (\epsilon_\vth)$  and $\mu_B = \sup_{\vth\in B} \mu_\vth$,
where $\epsilon_\vth \sim H_\vth$.
It follows by applying Markov's inequality that
\benrr
\delta^2/4 &\le& \sup_{\vth \in B} \{1- H_\vth(K_{2\delta}^B)\} = \sup_{\vth \in B} Pr(\epsilon_\vth \geq K_{2\delta}^B)
\leq \sup_{\vth \in B} \mathbb{E}(\epsilon_\vth)/K_{2\delta}^B \leq \mu_B/K_{2\delta}^B.
\eenrr
Thus, $K_{2\delta}^B \le 4 \mu_B / \delta^2$.
Further, $K_{1\delta}^B > 0$ and $0 < \delta <1$.
Therefore, it follows from~\eqref{en-eq-1} that
$\cN(\delta,b) \leq D(b)/\delta^4$,
where
$$D(b) := 2 (1-b^2)^{-1} [ (1-b^2) +  4(1+b) \mu_B \|h_{\vth}\|_{\infty}^B].$$
Because
$D(b)$ is increasing in $b$, we have that
 $I(b) = \int_0^1 [\log \cN(u, b)]^{1/2} du < \iny$  for $0 \le b < 1$.
\end{proof}


\begin{proof}[\bf Proof of Lemma~\ref{lem:lin}]
Fix $M < \iny$. From~\ref{A:4} and \ref{A:9} it follows that
$$\sup_{\{\phi \in \varPhi,~\sqrt{n} \|\phi -  \phi_n\| \le M\}}\max_{1 \le i  \le n} \{v_{ni}(\phi) - (\phi - \phi_n)^\top \lambda_i(\phi_n)\} =
o_{p_n}(n^{-1/2}).$$
Further, because $\max_{1 \leq i \leq n } n^{-1/2}\lambda_i(\phi_n) = o_{p_n}(1)$ and
$n^{-1} \sum_{i=1}^n  \|\lambda_i(\phi_n)\| = O_{p_n}(1)$,
one obtains that
$\sup_{\{\phi \in \varPhi, \sqrt{n} \|\phi -  \phi_n\| \le M\}}n^{-1/2}\sum_{i=1}^n |v_{ni}(\phi)| = O_{p_n}(1)$.

 Let $A_n = \{(x,\phi) : x \ge 0, \phi \in \varPhi \and \sqrt{n} \|\phi -  \phi_n\| \le M\}$.
 Let $a > 0$ be as in Condition~\ref{A:5}. Then, there exists an
$n_0>0$ such that for all $n > n_0$ and $(\phi, x) \in A_n$,
$F_{\theta_n}(x + x v_{ni}(\phi)) - F_{\theta_n}(x)
 = x v_{ni}(\phi) f_{\theta_n}(x) + 2^{-1}\{x v_{ni}(\phi)\}^2 f_{\theta_n}'(x(1+\delta_x^*))$,
 where $\delta_x^*$ is a real number satisfying $|\delta_x^*| < a$.

 Because $\sup_{\{\phi \in \varPhi,~\sqrt{n} \|\phi -  \phi_n\| \le M\}}n^{-1/2}\sum_{i=1}^n |v_{ni}(\phi)| = O_{p_n}(1)$,
 it follows from
 the assumptions in
 Condition~\ref{A:5} that
 $\sup_{x,\phi,M}n^{-1/2}\sum_{i=1}^n 2^{-1}\{x v_{ni}(\phi) \}^2 f_{\theta_n}'\{x(1+\delta_x^*)\} = o_{p_n}(1)$.
By Conditions~\ref{A:4}, \ref{A:5}, \ref{A:7}, and Assumption~\ref{A:9}, one also obtains that
 $\sup_{x,\phi,M} |\{n^{-1/2}\sum_{i=1}^n x v_{ni}(\phi) f_{\theta_n}(x)\} - B_n(x)| = o_{p_n}(1)$.
 Thus, the proof follows.
\end{proof}



\begin{proof}[\bf Proof of Lemma~\ref{lemma:th}]
We indicate only the main idea of the proof.
 Recall that $\dot g_\theta(t) = (\partial / \partial \theta) g_{\theta}(t) = [\dot g_{\theta 1}(t), \ldots, \dot g_{\theta q}(t)]^{\top}.$
Fix a $j \in \{1, \ldots, q\}$.
Because $\h \th = \arg\max_{\th \in \Th} \s g_\th(\wti \vep_i),$ under~\ref{A:6},
\benr\label{eq1:plem:th}
&0 = \s \dot g_{\h \th_j}(\wti \vep_i) = \s \dot g_{\th_{0j}}(\wti \vep_i) + (\hat{\theta}_j - \theta_{0j}) \s \big[(\partial / \partial \theta_j) \dot g_{\theta_j}(\wti \vep_i)\mid_{\th_j = \bar{\th}_j}\big], \qquad &
\eenr
for some $\bar{\th}_j$, where $|\bar{\th}_j - \th_{0j}| < |\h \th_j - \th_{0j}|$.
Let
\benn
\wti H_n(\th) = n^{-1}\s \ddot g_{\theta}(\wti \vep_i), \quad H_n(\th) = n^{-1}\s \ddot g_{\theta}(\vep_i).
\eenn
Suppose that $H_0$ holds. Then, under \ref{A:1},
\ref{A:3} and~\ref{A:6}, there exist an $\al_L > 0$, $0 < K < \iny$, and an open neighbourhood $B$ of $\theta_0$, such that
\benrr
\lefteqn{\sup_{\th \in B}\|\wti H_n(\th) - H_n(\th)\|}\\
&\le& K n^{-1}\s |\wti \vep_i - \vep_i|
\ = \ K n^{-1}\s Z_i \Big|\frac{1}{\wti \vsi_i(\h \phi)} - \frac{1}{\vsi_i(\h \phi)} + \frac{1}{\vsi_i(\h \phi)} - \frac{1}{\vsi_i(\phi_0)}\Big |\\
&\le& \frac{K}{\al_L n}\s  \vep_i \sup_{\phi \in \varPhi} |\wti \vsi_i(\phi) - \vsi_i(\phi)| +
\frac{K}{\al_L n}\s Z_i |\vsi_i(\h \phi) - \vsi_i(\phi_0)|\vsi_i^{-1}(\phi_0) + o_p(1)\\
&=& S_{n1} + S_{n2} + o_p(1), \quad \text{say.}
\eenrr
Since $\sup_{\phi \in \varPhi} |\wti \vsi_i(\phi) - \vsi_i(\phi)| \stackrel{e.a.s.}{\rightarrow} 0$ by~\ref{A:4} and $\{\vep_i\}$ are iid,
it follows from Lemma 2.1 of \cite{Straumann:Mikosch:06} that $S_{n1} = O_p(n^{-1})$.
By Condition~\ref{A:4},
\benrr
S_{n2} &\le& \|n^{1/2} (\h \phi - \phi_0)^\top \| \max_{1 \le i \le n} n^{-1/2}\|\lambda_{i}(\phi_0)\| \frac{K}{\al_L n}\s Z_i + o_p(n^{-1/2}).
\eenrr
Since $n^{1/2} (\h \phi - \phi_0) = O_p(1)$, it follows from Condition~\ref{A:7} and the Ergodic Theorem that the first term in the last upper bound is $o_p(1)$.
Hence, $\sup_{\th \in B}\|\wti H_n(\th) - H_n(\th)\| = o_p(1)$.
Therefore, one obtains by~\eqref{eq1:plem:th} that $\hat{\theta}_j - \theta_{0j} = n^{-1} \s h_{\theta_{0j}}(\wti \vep_i) + o_p(n^{-1/2})$.
Now, apply a two-term Taylor series expansion for $n^{-1} \s h_{\theta_{0j}}(\wti \vep_i)$
and verify that the remainder term is $o_p(n^{-1/2})$. This completes the proof of the first part of Lemma~\ref{lemma:th}.

For the second part, we establish the corresponding asymptotic expansion for $\hat{\theta}_j^* - \h \theta_{j}$, in probability. To this end, let~$(a_n)$ be a subsequence of~$(n)$.
Because $(\h \phi, \h \th)$ converges in probability to $(\phi_0, \th_0)$, the subsequence $(a_n)$ contains a further subsequence $(r_n)$ such that $(\h \phi_{r_n}, \h \th_{r_n}) \stackrel{a.s.}{\rightarrow} (\phi_0, \th_0)$.
Now, choose a sample path along which $(\h \phi_{r_n}, \h \th_{r_n}) \rightarrow (\phi_0, \th_0)$.
Then, it follows from~\ref{A:9} that
$\h \th_{r_n}^* - \h \th_{r_n} = o_{p_{r_n}^*}(1)$ and $\h \phi_{r_n}^* - \h \phi_{r_n} = O_{p_{r_n}^*}(r_n^{-1/2})$ along the chosen fixed sample path.

Further, because the bootstrap is carried out under $H_0$, it follows by proceeding as in the proof of the first part that
\benn
\hat{\theta}_{r_nj}^* - \h {\theta}_{r_nj}
= {r_n}^{-1}\sum_{i=1}^{r_n} \{h_{\h {\theta}_{r_nj}}^*(\varepsilon_i^*) -
(\hat{\phi}_{r_n}^* - \h \phi_{r_n})^\top \varepsilon_i^* \lambda_i^*(\h \phi_{r_n})h_{\h {\theta}_{r_nj}}^{*'}(\varepsilon_i^*) \} + o_{p_{r_n}^*}(r_n^{-1/2}).
\eenn
Because this holds true for almost all sample paths,
\eqref{Parameter:expansion:2:2} holds in probability.
\end{proof}

In the technical details involving the bootstrap method, we often need to show that certain terms are small, in the sense that they are $o_{p_n^*}(1)$, in probability.
To establish this,
it suffices to restrict the arguments to a subsequence $(r_n)$ for which
 $(\h \phi_{r_n}, \h \th_{r_n})$ converges almost surely to $(\phi_0,\theta_0)$, and to work along a fixed sample path for which  $(\h \phi_{r_n}, \h \th_{r_n}) \rightarrow (\phi_0, \th_0)$.
Hence, for showing that a quantity is $o_{p_n^*}(1)$ in probability, one may assume without loss of generality that $(\hat{\phi}, \hat{\theta}) \rightarrow (\phi_0,\theta_0)$ along almost all sample paths
 (see for example Theorem 5 of \citealp*{Salinetti:etal:86}).
In the following lemmas, we restrict attention to such a fixed sample path.
Thus, the terms `$e.a.s.$' and `$a.s.$' below may correspond to $P_n^*$ probability, along a fixed sample~path for which $(\hat{\phi}, \hat{\theta}) \rightarrow (\phi_0,\theta_0)$.
%Further, for brevity, a statement such as `conditional on the observed sample, $X_n^* = o_{p_n^*}(1)$, in probability', is abbreviated to `$X_n^* = o_{p_n^*}(1)$'.


\begin{proof}[\bf Proof of Lemma~\ref{lem:n3}]
Recall that
$L_n^{*(m)}(\phi) = \sum_{i=1}^n \{\log \vsi_i^{*(m)}(\phi) + Z_i^{*(m)}/ \vsi_i^{*(m)}(\phi)\}$
and
$\hat{\phi}^{*(m)} = \arg\min_{\phi \in \varPhi} L_n^{*(m)}(\phi)$.
Let $\Delta_n^*(\phi) := (\partial/\partial \phi)\{L_n^{*(m)}(\phi) - L_n^{*}(\phi) \}$,
and
\benrr
a_i(\phi) &:=& \{ \vsi_i^{*(m)}(\hat{\phi}) / \vsi_i^{*(m)}(\phi) \}\{ \lambda_i^{*(m)}(\phi) - \lambda_i^{*}(\phi) \}, \\
 b_i(\phi) &:=& \lambda_i^{*}(\phi) \{\vsi_i^{*(m)}(\hat{\phi}) - \vsi_i^{*}(\hat{\phi})\}/  \vsi_i^{*(m)}(\phi),\\
 c_i(\phi) &:=& \lambda_i^{*}(\phi) \{ \vsi_i^{*}(\hat{\phi}) /  \vsi_i^{*}(\phi) \}
\{ \vsi_i^{*}(\phi) - \vsi_i^{*(m)}(\phi)\}/ \vsi_i^{*(m)}(\phi).
\eenrr
Then,  $\Delta_n^*(\phi)= \sum_{i=1}^n  \{ \lambda_i^{*(m)}(\phi) - \lambda_i^{*}(\phi) \} - \{a_i(\phi) + b_i(\phi) +c_i(\phi) \}\varepsilon_i^*$.
Now, let $e_{i1} = \|A_i(\cdot)\|$ and $e_{i2} = \|B_i(\cdot)\|$, where
$
A_i(\phi) = \{\vsi_i^{*(m)}(\phi) - \vsi_i^{*}(\phi)\}$ and $B_i(\phi) = \{\dot{\vsi}_i^{*(m)}(\phi)- \dot{\vsi}_i^{*}(\phi)\}.
$
Then, it follows from Conditions~\ref{A:8} and \ref{A:9} that $e_{i1}, e_{i2} \stackrel{e.a.s.}{\rightarrow} 0$.
Further, %we have that
\benrr
\lambda_i^{*(m)}(\phi) - \lambda_i^*(\phi)
&=& \{\dot{\vsi}_i^{*(m)}(\phi)\}/\{\vsi_i^{*(m)}(\phi)\} - \{\dot{\vsi}_i^{*}(\phi)\}/\{\vsi_i^{*}(\phi)\}\\
&=& \frac{1}{\vsi_i^{*(m)}(\phi)} B_i(\phi) -
\lambda_i^*(\phi)\{\frac{1}{\vsi_i^{*(m)}(\phi)}\} A_i(\phi).
\eenrr
Thus, for some fixed $\alpha > 0$,
$
\|\lambda_i^{*(m)}(\cdot) - \lambda_i^*(\cdot)\|  \leq \alpha^{-1}e_{i2} +
 \alpha^{-1} \|\lambda_i^*(\cdot)\|  e_{i1}.
$

Because $\|\lambda_i^*(\cdot)\|$, $i = 1, 2 \dots$ are identically distributed
and $e_{i1}, e_{i2} \stackrel{e.a.s.}{\rightarrow} 0$ as $i \to \iny$, then it follows
from Lemma 2.1 of \cite{Straumann:Mikosch:06} that
\text{$ \|\lambda_i^{*(m)}(\cdot) - \lambda_i^*(\cdot)\| \stackrel{e.a.s.}{\rightarrow} 0,$ as $i \to \iny$.}
Therefore, $\sum_{i=1}^n \|\lambda_i^{*(m)}(\cdot) - \lambda_i^*(\cdot)\|$ converges to a random variable (a.s.).
Further,  for some fixed $\alpha > 0$, we have that
\benrr
&&\|a_i(\cdot)\| \le e_{i1} \al^{-1} |\vsi_i^*(\hat{\phi})|\| \lambda_i^{*(m)}(\cdot) - \lambda_i^{*}(\cdot) \|, \\
&& \|b_i(\cdot)\| \le e_{i1} \al^{-1} \|\lambda_i^{*}(\cdot)\|, \quad \|c_i(\cdot)\| \le e_{i1} \al^{-2} |\vsi_i^*(\hat{\phi})| \|\lambda_i^{*}(\cdot)\|.
\eenrr
Therefore, each of
$\sup_{\phi \in \varPhi} |a_i(\phi)|$, $\sup_{\phi \in \varPhi} |b_i(\phi)|$,  and
$\sup_{\phi \in \varPhi} |c_i(\phi)|$ is bounded by terms
equal to the product of an identically distributed random variable and another term that
 $\stackrel{e.a.s}{\rightarrow} 0$ as $i \rightarrow \infty$.
In view of Lemma 2.1 of \cite{Straumann:Mikosch:06}, if $\{v_i\}_{i \in \Z^+}$ is a sequence of identically distributed random elements with values in a separable Banach space (e.g. $\R^p$ with Euclidean norm) and $\zeta_i \stackrel{e.a.s}{\rightarrow} 0$, then
$
\sum_{i=1}^n \zeta_i\|v_i\| \text{ converges to a random variable (a.s.).}
$
Therefore, we obtain that
$$\s \sup_{\phi \in \varPhi} |a_i(\phi)+ b_i(\phi)+c_i(\phi)| \vep_i^* = O_{p_n^*}(1),$$ and hence,
$\sup_{\phi \in \varPhi} | \Delta_n^*(\phi) | = O_{p_n^*}(1).$
This completes the proof of part~(a).

We indicate the main idea for the proof for part~(b).
To this end, it suffices to consider the simpler case when $\phi$ is a scalar parameter.
Let $0.5 < \eta < 1$.
Note that for a given $\delta > 0$, the curve
$n^{-1} (\partial / \partial \phi) L_n^{*(m)} (\phi)$ lies  in the
 band $n^{-1} (\partial / \partial \phi) L_n^{*} (\phi) \pm \delta n^{-\eta}$
 with probability approaching 1 as $n \to \iny$.
Let
$S(\phi) = -(\partial/\partial \phi) n^{-1}L_n^{*}(\phi)$ and
$J(\phi) = -(\partial/\partial \phi) n^{-1} L_n^{*(m)}(\phi).$
Let $\phi_a$ and $\phi_b$ be chosen such that
$S(\phi_a) = n^{-\eta}\delta$ and $S(\phi_b) = -n^{-\eta}\delta.$
Then, there exists a $K > 0$ such that $n^{1/2}|\phi_a- \h \phi^{*}| < K$ and $n^{1/2}|\phi_b- \h \phi^{*}| < K$.
Let $B = \{\phi: n^{1/2}|\phi- \h \phi^{*}| < K\}$.
In view of Assumptions~\ref{A:3} and \ref{A:9}, there exists $c_0 > 0$, such that
$P_n^*[ \sup_{\phi \in B} |\dot S(\phi)| > c_0] \rightarrow 1$ $\text{as $n \to \infty$}.$
In view of the mean value theorem,
for some $\bar{\phi}_a$ and $\bar{\phi}_b$ satisfying $\mid\bar{\phi}_a - \hat{\phi}^{*}\mid \le \mid\phi_a - \hat{\phi}^{*}\mid$ and $|\bar{\phi}_b - \h \phi^{*}| \le |\phi_b - \h \phi^{*}|$,
\benrr
n^{-\eta}\delta &=& S(\phi_a)= S(\h \phi^{*}) + (\phi_a - \h \phi^{*}) \dot S(\bar{\phi}_a) = 0 + (\phi_a - \h \phi^{*}) \dot S(\bar{\phi}_a),\\
-n^{-\eta}\delta &=& S(\phi_b)= S(\h \phi^{*}) + (\phi_b - \h \phi^{*}) \dot S(\bar{\phi}_b) = 0 + (\phi_b - \h \phi^{*}) \dot S(\bar{\phi}_b).
\eenrr
Since $\h \phi^{*} - \h \phi = o_{p^*_n}(1)$, then
with $P_n^*$ probability $\to 1$ as $n \to \iny$,
$|\phi_a - \phi_b| \leq 2 n^{-\eta}\delta c_0^{-1}$.

Because $P_n^*[ -\delta < n^{\eta} \sup_{\phi \in \varPhi} (\partial/\partial \phi)\{ n^{-1}L_n^{*(m)}(\phi) - n^{-1}L_n^{*}(\phi) \} < \delta ] \rightarrow 1$, we have
$J(\phi_a) \ge S(\phi_a) - n^{-\eta} \delta =0$ and $
J(\phi_b)\le S(\phi_b) + n^{-\eta} \delta =0,$
with $P_n^*$ probability $\to 1$.
Thus, $P_n^*[\hat \phi^{*(m)} \in (\phi_a, \phi_b)] \to 1$, and hence,
$
|\hat \phi^{*(m)} - \hat{\phi}^*| \leq | \phi_a - \phi_b| \leq 2 n^{-\eta} \delta c_0^{-1},$
with $P_n^*$ probability $\to 1$.
Therefore, part~(b) follows.

To prove part~(c),
let $\ell(\theta) = \sum_{i=1}^n g_{\theta}(\wti{\varepsilon}_i)$.
 Then, the corresponding bootstrap terms are
 $\ell^*(\theta) = \sum_{i=1}^n g_{\theta}(\wti{\varepsilon}_i^*)$ and $\ell^{*(m)}(\theta) = \sum_{i=1}^n g_{\theta}(\wti{\varepsilon}_i^{*(m)})$.
Further, for some $\bar{\varepsilon}_i$ between $\wti{\varepsilon}^{*(m)}_i$ and $\wti{\varepsilon}^{*}_i$,
$|\ell^{*(m)}(\theta) - \ell^{*}(\theta) | =
|\sum_{i=1}^n (\wti{\varepsilon}^{*(m)}_i - \wti{\varepsilon}^{*}_i)  g_{\theta}^{\prime}(\bar{\varepsilon}_i; \theta)|$.
Hence, $|\ell^{*(m)}(\theta) - \ell^{*}(\theta) |$ is bounded from above by
\benrr
\lefteqn{K_0 \sum_{i=1}^n | Z_i^{*(m)} / {\wti \vsi}_i^{*(m)}(\hat{\phi}^{*(m)}) -
                            Z_i^{*} / {\wti \vsi}_i^{*}(\hat{\phi}^{*})|}\\
&=&  K_0 \sum_{i=1}^n \vep_i^*| \vsi_i^{*(m)}(\hat{\phi}) / {\wti \vsi}_i^{*(m)}(\hat{\phi}^{*(m)})
- \vsi_i^{*}(\hat{\phi}) / {\wti \vsi}_i^{*(m)}(\hat{\phi}^{*(m)})\\
&& \quad + \
\vsi_i^{*}(\hat{\phi}) / {\wti \vsi}_i^{*(m)}(\hat{\phi}^{*(m)}) -
                            \vsi_i^{*}(\hat{\phi}) / {\wti \vsi}_i^{*}(\hat{\phi}^{*})|\\
%\eenrr
%\benrr
&\le&  K_0 \sum_{i=1}^n \vep_i^*[K_1 |\vsi_i^{*(m)}(\hat{\phi})
- \vsi_i^{*}(\hat{\phi})| +
K_2 \vsi_i^{*}(\hat{\phi})|{\wti \vsi}_i^{*}(\hat{\phi}^{*}) - {\wti \vsi}_i^{*(m)}(\hat{\phi}^{*(m)})|,
\eenrr
where $K_0, K_1$ and $K_2$ are fixed constants.
In view of Assumption~\ref{A:9}, the terms $\sup_{\phi \in \varPhi}|\vsi_i^{*(m)}(\hat{\phi})
- \vsi_i^{*}(\hat{\phi})|$ and
$\sup_{\phi \in \varPhi}|{\wti \vsi}_i^{*}(\hat{\phi}^{*}) - {\wti \vsi}_i^{*(m)}(\hat{\phi}^{*(m)})|$
converge to zero
(e.a.s) as $i \rightarrow \infty$.
Therefore, by Lemma 2.1 of \cite{Straumann:Mikosch:06}, for every $\eta <1$,
$n^{\eta} \sup_{\theta \in \Theta} \{ n^{-1}\ell^{*(m)}(\theta) - n^{-1}\ell^{*}(\theta) \}
 \stackrel{a.s.}{\rightarrow} 0$ as $n \rightarrow \infty$.
Thus, one obtains  by proceeding as in the proof of part~(b)
that $\hat{\theta}^{*(m)} - \hat{\theta}^{*} = o_{p_n^*}(n^{-\eta})$ for every $0.5 < \eta <1$.
\end{proof}


\begin{proof}[\bf Proof of Lemma~\ref{lemma:6}]
Let $v_{i}^* = \{\varPsi_{i}^*(\hat{\phi})\}^{-1}\{\varPsi_{i}^*(\hat{\phi}^*) - \varPsi_{i}^*(\hat{\phi})\}$.
Because $n^{1/2}(\h \phi^* - \h \phi) = O_{p_n^*}(1)$, by Condition~\ref{A:4},
$\max_{1 \le i  \le n} \{v_{i}^* - (\h \phi^* - \h \phi)^\top \lambda_{i}^*(\h \phi)\} =
o_{p_n^*}(n^{-1/2})$.
Because $\max_{1 \leq i \leq n } n^{-1/2}\|\lambda_{i}^*(\h \phi)\| = o_{p_n^*}(1)$, one obtains that
$\max_{1 \leq i \leq n } |v_{i}^*| = o_{p_n^*}(1)$.
Let us write $\h F_n^*(x)= n^{-1} \s I(\vep_i^* \le x + xv_{i}^*)$.
Then, it follows from Lemma~\ref{Thm:3}
with $\gamma_{ni} = 1$ and $\rho_{ni} = v_{i}^*$ that
$\sup_{x \ge 0}| U_n^*(x) - n^{-1/2}\sum_{i=1}^n \{F_{\theta_n}(x + x v_{i}^*) - F_{\theta_n}(x)\}|  = o_{p_n^*}(1)$.
Now, apply~Lemma~\ref{lem:lin} to complete the~proof.
\end{proof}



\begin{proof}[\bf Proof of Lemma~\ref{lem:n6}]
Let $v_{ni}^{*(m)} = n^{1/2} [ \vsi_i^{*(m)} (\hat{\phi}^{*(m)}) - \vsi_i^{*(m)}(\hat{\phi})  ] / \vsi_i^{*(m)}(\hat{\phi}).$
Then, it follows from a one-term Taylor~expansion that
$
v_{ni}^{*(m)} = n^{1/2}( \hat{\phi}^{*(m)} - \hat{\phi} )^{\top} \lambda_i^{*(m)}(\hat{\phi}) + r_{ni}^{*(m)}, $
for some random array $\{r_{ni}^{*(m)}\}$ satisfying
$n^{-1} \sum_{i=1}^nr_{ni}^{*(m)} = O_{p_n^*}(n^{-1/2})$.

From the proof of Lemma~\ref{lem:n3}, for some $\eta > 1/2$,
$n^{\eta-1}  \sum_{i=1}^n \|\lambda_i^{*(m)}(\hat{\phi}) - \lambda_i^*(\hat{\phi})\| \rightarrow 0 $ (a.s.).
By assumption, $\max_{1 \le i \le n}n^{-1/2}\|\lambda_i^*(\hat{\phi})\| = o_{p_n^*}(1)$, and from
Lemma~\ref{lem:n3}, $n^{1/2}( \hat{\phi}^{*(m)} - \hat{\phi} ) = O_{p_n^*}(1)$. Hence, $\max_{1 \le i \le n} |n^{-1/2}v_{ni}^{*(m)}| = o_{p_n^*}(1)$.
Further, with $\g_{ni} = 1$ and $\rho_{ni} = n^{-1/2}v_{ni}^{*(m)}$, it follows from Lemma~\ref{Thm:3} that uniformly in~$y \ge 0$,
\benrr
\lefteqn{n^{1/2} \hat{F}_n^{*(m)}(y) \ = \ n^{-1/2} \sum_{i=1}^n I (\hat{\varepsilon}_i^{*(m)} \leq y)
\ = \ n^{-1/2} \sum_{i=1}^n I \{ \varepsilon_i^*  \leq y + yn^{-1/2}v_{ni}^{*(m)}\}}\\
 &&=  n^{1/2} F_n^{*(m)}(y) + n^{-1/2}\sum_{i=1}^n \{F_{\h \th}(y + yn^{-1/2}v_{ni}^{*(m)}) - F_{\h \th}(y)\} + o_{p_n^*}(1)\\
 &&=  n^{1/2} F_n^{*(m)}(y) + n^{-1}\sum_{i=1}^n v_{ni}^{*(m)}y f_{\hat{\theta}}(y) + o_{p_n^*}(1).
\eenrr
This equality continues to hold with $^{*(m)}$ replaced by $^{*}$.
Further,
because $F_n^{*(m)}  = F_n^*$,
by triangle inequality,
\ben\label{eq-b2}
 \sup_{y \ge 0} n^{1/2} | \hat{F}_n^{*(m)}(y) - \hat{F}_n^{*}(y) |
\le \sup_{y \ge 0} \Big | n^{-1} \sum_{i=1}^n (v_{ni}^{*(m)} - v_{ni}^{*})y f_{\hat{\theta}}(y) \Big | + o_{p_n^*}(1).
\een
By direct substitution, we obtain that
\benrr
\lefteqn{ v_{ni}^{*(m)} - v_{ni}^{*} \ = \
n^{1/2} (\hat{\phi}^{*(m)} - \hat{\phi})^{\top} \lambda_i^{*(m)}(\hat{\phi}) -
     n^{1/2} (\hat{\phi}^{*} - \hat{\phi})^{\top} \lambda_i^{*}(\hat{\phi}) + [r_{ni}^{*(m)} - r_{ni}^*]}\\
&=& \{n^{1/2} (\hat{\phi}^{*(m)} - \hat{\phi}^{*})^{\top} \lambda_i^{*(m)}(\hat{\phi}) +
     n^{1/2} (\hat{\phi}^{*} - \hat{\phi})^{\top} [\lambda_i^{*(m)}(\hat{\phi})- \lambda_i^{*}(\hat{\phi})]\} + [r_{ni}^{*(m)} - r_{ni}^*].
\eenrr
Because $n^{1/2} (\hat{\phi}^{*(m)} - \hat{\phi}^{*}) = o_{p_n^*}(1)$, $n^{-1}  \sum_{i=1}^n \|\lambda_i^{*(m)}(\hat{\phi}) - \lambda_i^*(\hat{\phi})\| = o_{p_n^*}(n^{-\eta}),$ and
$n^{-1} \s [r_{ni}^{*(m)} - r_{ni}^*] = O_{p_n^*}(n^{-1/2})$,
this yields that $n^{-1} \sum_{i=1}^n \{v_{ni}^{*(m)} - v_{ni}^{*}\} = o_{p_n^*}(1)$.
Since $\sup_{\th \in B, y \ge 0}(1 + y) f_{\th}(y) < \iny$ for some open neighbourhood $B$ of $\th_0$, the first part follows from (\ref{eq-b2}).
Because $\hat{\theta}^{*(m)} - \hat{\theta}^{*} = o_{p_n^*}(n^{-\eta})$,
the second part follows from a one-term Taylor~expansion.
\end{proof}


\begin{proof}[\bf Proof of Lemma~\ref{prop:loc:power:2}]

Let $f_{(n)}$, $f_{\th_0}$ and $\ti f$ denote the densities corresponding to $F_{(n)}$, $F_{\th_0}$ and $\ti F$, respectively.
Let $\ell_n := \s \log\{f_{(n)}(\vep_i) / f_{\th_0}(\vep_i)\}$.
It follows from Theorem~7.2 in \cite{VanderVaart:98} that
\benrr
&\ell_n = \delta n^{-1/2} \s  \{[\ti f(\vep_i) - f_{\th_0}(\vep_i)]f_{\th_0}^{-1}(\vep_i)\} -2^{-1}\delta^2\sigma^2 + o_p(1),&
\eenrr
where $\sigma^2 = \int_{x \ge 0} \{\ti f(x) - f_{\th_0}(x)\}^2f_{\th_0}^{-1}(x) dx$.
Hence, by the central limit theorem,
$\ell_n \stackrel{d}{\rightarrow} N(-2^{-1}\delta^2\sigma^2, \delta^2\sigma^2)$ under $H_0$.
Therefore, by Le Cam's first lemma (see \citealp{VanderVaart:Wellner:96}, Theorem~3.10.2)
$H_{an}$ is contiguous with respect to $H_0$.


%Now, let
%$\cQ_i(\phi_0, \varepsilon_{i}) = n^{-1} \sum_{j=1}^n h_{\th_0}^{'}(\vep_j)
%\vep_j\{\lambda_j( \phi_0)\}^\top\xi_{i}(\phi_0, \varepsilon_{i})$,
%where
%$\xi_{i}(\phi_0, \varepsilon_{i}) = \tau(\phi_0) \lambda_{i}(\phi_0)(1-\varepsilon_{i})$.
%Define
%\benn
%\label{gi:pwr}
%g_i(t) = a_i(t) - b_i(t) + c_i(t),
%\eenn
%%with
%\benrr
%a_i(t)  &=& I\{\varepsilon_{i} \le F_{\th_0}^{-1}(t)\} - t, \ \ c_i(t) = F_{\th_0}^{-1}(t)f_{\th_0}\{F_{\th_0}^{-1}(t)\} \mathbb{E}[\lambda_{1}(\phi_0)]^{\top}
%\xi_{i}(\phi_0, \varepsilon_{i}),\\
%b_i(t) &=&  \{h_{\theta_0}(\varepsilon_{i}) -
%\cQ_i(\phi_0, \varepsilon_{i})\}^\top \dot{F}_{\th_0}\{F_{\th_0}^{-1}(t)\}.
%\eenrr
Let $G_n(t) = n^{-1/2} \s g_i(t)$, where $g_i(t)$ is defined in equation (\ref{def_g_i}) of the main text as
$ g_i(t) = a_i(t) - b_i(t) + c_i(t)$
with functions $a_i(\cdot), b_i(\cdot),$ and $c_i(\cdot)$ as defined in Section~\ref{sec:asy:power}.
Then, $G_n(\cdot)$ is the same as $G_n^*(\cdot)$ in the proof of Lemma~\ref{lem:n4}, except that it is now defined for the original sample instead of the bootstrapped sample.

It may be verified that each of
 $n^{-1/2}\sum a_{i}(t), $ $n^{-1/2}\sum b_{i}(t)$, and  $n^{-1/2}\sum c_{i}(t)$ is asymptotically  equicontinuous, under~$H_0$,
 by applying Markov's inequality and using
Assumption~\ref{A:6} and Conditions~\ref{A:5} and \ref{A:7}.
Because $g_i(\cdot)$ forms a martingale difference sequence, by a martingale CLT, the finite dimensional distributions of $G_n(t)$ converge to those of the centered Gaussian process $G(\cdot)$ in Theorem~\ref{thm:1} [under $H_0$].
Consequently, $G_n(t)$ converges weakly to $G(\cdot)$,
$\sup_t |\wti W_n\{F_{\th_0}^{-1}(t)\} - G_n(t) | =o_p(1)$, and
$\mathbb{E}[G_n(t)\ell_n] = m(t, \th_0) + o(1)$  under~$H_0$, $t \in [0,1]$,
where
\benrr
m(t, \th) &=&
\delta \int \{I(\vep \le F_{\theta}^{-1}(t)) - t \} \ d\ti F(\vep)\\
 && \ +  \delta \big[\int \{(\partial / \partial \theta) \dot g_{\theta}(y)\}^{-1}\ d F_{\th}(y) \int \dot g_\theta(\vep) \ d\ti F(\vep)\big]^\top \dot{F}_{\theta}(F_{\theta}^{-1}(t))\\
 &=& \delta [\ti F(F_{\theta}^{-1}(t)) - t] + \delta \big[\int \{\ddot g_{\theta}(y)\}^{-1}\ d F_{\th}(y) \int \dot g_\theta(\vep) \ d\ti F(\vep)\big]^\top \dot{F}_{\theta}(F_{\theta}^{-1}(t)).
\eenrr
Thus, $m(\cdot, \th_0)$ is the same as $m_a(\cdot)$ in \eqref{m-a}.
By applying a general version of Le Cam's third lemma for sequences of probability measures in metric spaces (see \citealp{VanderVaart:Wellner:96}, Theorem 3.10.7),
we obtain that  $\wti W_n \circ F_{\th_0}^{-1}(\cdot)$ converges weakly to $W_a(\cdot)$
 in $D[0,1]$, under $H_{an}$, where $W_a(\cdot) = m_a(\cdot) + G(\cdot)$.
 By assumption, $\int \dot g_{\th_0}(\vep) \ d \ti F(\vep) = 0$ only if $\ti F = F_{\theta_0}$.
 Further, $\ti F \neq F_{\theta_0}$ and
 \benn
 [t- \ti F(F_{\th_0}^{-1}(t))] \neq \big[\int \{\ddot g_{\th_0}(y)\}^{-1}\ d F_{\th_0}(y) \int \dot g_{\th_0}(\vep) \ d\ti F(\vep)\big]^\top \dot{F}_{\th_0}(F_{\th_0}^{-1}(t)), \quad t \in [0,1].
 \eenn
 Hence, $m_a \not \equiv 0$ for~$\delta > 0$.

Under $H_{an}$, $\h \th \to \th_0$ in probability, and $\th_0$ is the true value satisfying $F^0 = F_{\th_0}$ under $H_0$. Therefore, one may proceed as in the proof of Lemma~\ref{lem:n4} and show that $\wh W_n^*\circ F_{\h \th}^{-1}(\cdot)$ converges weakly to $G(\cdot)$, under $H_{an}$ [in probability].
Hence, the proof follows from Lemmas~\ref{lem:n6} and~\ref{hF-tiF-boot}.
\end{proof}

\begin{proof}[\bf Proof of Proposition~\ref{prop:loc:power:n1}]
Let $$\ell_n = -\s \left[\log\{\vsi_i(\phi_0) + r_i/\sqrt n\} - \log\{\vsi_i(\phi_0)\} - Z_i/\{\vsi_i(\phi_0) + r_i/\sqrt n\} + Z_i/\vsi_i(\phi_0)\right].$$
By arguing as in the proof of Lemma~\ref{prop:loc:power:2} we obtain the following:
(a)~$H_{bn}$ is contiguous with respect to $H_0$,
(b)~$G_n(\cdot)$ converges weakly to $G(\cdot)$ and $\sup_t |\wti W_n\{F_{\th_0}^{-1}(t)\} - G_n(t) | =o_p(1)$  under~$H_0$, where $G_n = n^{-1/2} \s g_i$,
and (c)~$\mathbb{E}[G_n(t)\ell_n] = m_b(t) + o(1)$.

Therefore, part~(i) follows from
 Le Cam's third lemma (see \citealp{VanderVaart:Wellner:96}, Theorem 3.10.7)
 and the continuous mapping theorem.

To prove part~(ii), note that under $H_{bn}$, $\h \phi = \arg\min_{\phi \in \varPhi} \s \ell_i(\phi)$, where
\benrr
\ell_i(\phi) = \log \wti \vsi_i(\phi) + \frac{\left[\vsi_i(\phi_0)+n^{-1/2}r_i\right]}{\wti \vsi_i(\phi)}\vep_i = \log \wti \vsi_i(\phi) + \frac{\left[\vsi_i(\phi_0)\vep_i\right]}{\wti \vsi_i(\phi)} + \frac{n^{-1/2}r_i \vep_i}{\wti \vsi_i(\phi)}.
\eenrr
Let $\h \phi^{(1c)} = \arg\min_{\phi \in \varPhi} \s \kappa_i(\phi)$, where
 $\kappa_i(\phi) = \log \wti \vsi_i(\phi) + [\vsi_i(\phi_0)\vep_i]/\wti \vsi_i(\phi)$.
In view of Assumption~\ref{A:3}, $\h \phi^{(1c)}  \stackrel{p}{\rightarrow} \phi_0$ as $n \to \iny$.
Further, we have that
\benn
\s (\partial/\partial \phi) \{ \ell_i(\phi) - \kappa_i(\phi)\} = -\s n^{-1/2}r_i \vep_i\left[\wti \lambda_i(\phi)/\wti \vsi_i(\phi)\right].
\eenn
Therefore, under $H_{bn}$, for any $1/4 < \eta < 1/2$,
\benrr
&n^\eta \sup_{\phi \in \varPhi}\left \| (\partial/\partial \phi) \left\{ n^{-1}\s \ell_i(\phi) - n^{-1}\s \kappa_i(\phi)\right\} \right\| \stackrel{a.s.}{\rightarrow} 0 \ \ \text{as $n \to \iny$}.&
\eenrr
Consequently, $\h \phi - \h \phi^{(1c)}  \stackrel{p}{\rightarrow} 0$ as $n \to \iny$, and hence,
$\h \phi  \stackrel{p}{\rightarrow} \phi_0$.
Further, $\h \th \equiv \th_0 = 1$ for each $n$
because the error distribution is standard exponential and the parameter space $\Th = \{1\}$.
Therefore, by arguments similar to those of the proof of Lemma~\ref{lem:n4}, $\wh W_n^*\circ F_{\h \th}^{-1}(\cdot)$ converges weakly to $G(\cdot)$ under $H_{bn}$ [in probability].
Because $T_j = \mathfrak{h}_j(\wti W_n \circ F_{\theta_0}^{-1}) + o_{p}(1)$ and $T_j^{*(m)} = \mathfrak{h}_j(\wti W_n^{*(m)} \circ F_{\h{\theta}}^{-1}) + o_{p^*}(1)$ [in probability] $(j= 1, \ldots, 5)$, the proof follows from Lemmas~\ref{lem:n6} and~\ref{hF-tiF-boot}, and the continuous mapping theorem.
\end{proof}





\begin{proof}[\bf Proof of Proposition~\ref{prop:loc:power:3}]
Let
\benrr
&\ell_n := -\s \left[\log\{\vsi_i(\phi_0) + r_i/\sqrt n\} - \log\{\vsi_i(\phi_0)\} + \log\{f_{(n)}(\vep_i) / f_{\th_0}(\vep_i)\}\right].&
\eenrr
One obtains by arguing as in the proof of Lemma~\ref{prop:loc:power:2}
that $H_{cn}$ is contiguous with respect to $H_0$.
From the proof of Proposition~\ref{prop:loc:power:n1} we have that
 $G_n(\cdot)$ converges weakly to $G(\cdot)$ and that
 $\sup_t |\wti W_n\{F_{\th_0}^{-1}(t)\} - G_n(t) | =o_p(1)$  under $H_0$.
We also have that $\lim_{n\to \iny} \mathbb{E}[G_n(t)\ell_n] = m_c(t) + m_a(t) + o(1)$.
Therefore, part~(i) follows as in the proof of Proposition~\ref{prop:loc:power:n1}.
% from Le Cam's third lemma (see \citealp{VanderVaart:Wellner:96}, Theorem 3.10.7)
% and the continuous mapping theorem.

To prove part~(ii), note that under $H_{cn}$, $\h \phi = \arg\min_{\phi \in \varPhi} \s \ell_i(\phi)$, where
\benrr
\ell_i(\phi) = \log \wti \vsi_i(\phi) + \frac{\left[\vsi_i(\phi_0)+n^{-1/2}r_i\right]}{\wti \vsi_i(\phi)}\vep_i = \log \wti \vsi_i(\phi) + \frac{\left[\vsi_i(\phi_0)\vep_i\right]}{\wti \vsi_i(\phi)} + \frac{n^{-1/2}r_i \vep_i}{\wti \vsi_i(\phi)}.
\eenrr
Now, let $\h \phi^{(2c)} = \arg\min_{\phi \in \varPhi} \s \kappa_i(\phi)$, where
 $\kappa_i(\phi) = \log \wti \vsi_i(\phi) + [\vsi_i(\phi_0)\vep_i]/\wti \vsi_i(\phi)$.
In view of Assumption~\ref{A:3}, $\h \phi^{(2c)}  \stackrel{p}{\rightarrow} \phi_0$ as $n \to \iny$.
Further, we have that
\benn
\s (\partial/\partial \phi) \{ \ell_i(\phi) - \kappa_i(\phi)\} = -\s n^{-1/2}r_i \vep_i\left[\wti \lambda_i(\phi)/\wti \vsi_i(\phi)\right].
\eenn
In view of these, under $H_{cn}$, for any $1/4 < \eta < 1/2$,
\benn
n^\eta \sup_{\phi \in \varPhi}\left \| (\partial/\partial \phi) \left\{ n^{-1}\s \ell_i(\phi) - n^{-1}\s \kappa_i(\phi)\right\} \right\| \stackrel{a.s.}{\rightarrow} 0 \ \ \text{as $n \to \iny$}.
\eenn
Further, $\h \phi - \h \phi^{(2c)}  \stackrel{p}{\rightarrow} 0$ as $n \to \iny$, and hence,
$\h \phi  \stackrel{p}{\rightarrow} \phi_0$ as $n \to \iny$.
Note that under $H_{cn}$, $\h \th \to \th_0$ as $n \to \iny$, where $\th_0$ is the true value satisfying $F^0 = F_{\th_0}$ under $H_0$.
As in the proof of Lemma~\ref{prop:loc:power:2},
 $\wh W_n^*\circ F_{\h \th}^{-1}(\cdot)$ converges weakly to $G(\cdot)$ under $H_{cn}$ [in probability],
 and hence the rest of the arguments follow as in the proof of Proposition~\ref{prop:loc:power:n1}.
\end{proof}

\section*{APPENDIX C: Simulation study on the importance of the tests in density forecasting}
\label{appendixC}
\renewcommand{\theequation}{C.\arabic{equation}}
      \renewcommand{\thesection}{C}
      \setcounter{equation}{0}

One important area of application of the tests proposed in the paper is forecasting the conditional distribution and/or density of $Z_{i+1}$.
The role of these tests in forecasting is that they can be used for testing the goodness-of-fit of the specified parametric model.
To evaluate the potential contribution of these tests, this simulation study estimates different measures of  `loss' in using an incorrect parametric family when the tests have adequate power to reject the incorrect family.

\medskip

 \noindent\textit{Design of the simulation  study:}

Let  $\{G_{\beta}: \beta \in B\}$ and $\{ F_{\theta}: \theta \in \Theta\}$ denote two distinct families of cumulative distribution functions (cdfs). Let $\vsi_i(\phi) = \phi_1 + \phi_2 Z_{i-1} + \phi_3 \vsi_{i-1}(\phi),$ $\phi = (\phi_1, \phi_2, \phi_3)^\top \in \varPhi$,
denote the parametric specification for $\vsi_i$.
The design of the simulation is based on the following scenario: The true DGP is
 $ \vsi_i = \vsi_i(\phi_0) \mbox{ for some $\phi_0 \in \varPhi$  and } F^0 \in \{ G_{\beta}: \beta \in B\}$.
The parametric model being considered for use in forecasting, and hence defines the null hypothesis is
$H_0: \vsi_i \in \{\vsi_i(\phi): \phi \in \varPhi\} \mbox{ and } F^0 \in \{ F_{\theta}: \theta \in \Theta\}$.
We are interested in estimating some measures of `loss' in using the incorrect parametric family $\{ F_{\theta}: \theta \in \Theta\}$ for forecasting and the extent to which
the goodness-of-fit tests could be expected to help in reducing such losses.


We use three different measures to estimate the `loss'.
Let $G$ denote the true cdf $G_{\beta_0}(x/\varPsi_{i+1}(\phi_0))$ of $Z_{i+1}$,
  $\hat{G}(x)$ denote its forecast $G_{\hat{\beta}}(x/\wti \vsi_{i+1}(\hat{\phi}))$,
  $\hat{F}(x)$ denote the forecast $F_{\hat{\theta}}(x/\wti \vsi_{i+1}(\hat{\phi}))$
  when the model in $H_0$ is used.
Let $A1 = (1/2) \int|\wh g(x) - g(x)| dx$ and $A2 = (1/2) \int|\wh f(x) - g(x)| dx$.
By using the term `misallocation' in a broad sense,
we may interpret $ A2$  as the proportion of the total probability of 1 that is misallocated by
 $\wh f$ when the true target is $g$.
We use $L_P := \mathbb{E}(A2)/\mathbb{E}(A1)$ as the first measure of the loss resulting from using the incorrect parametric model specified by $H_0$.

We also estimated the following two measures of loss: (a) $L_O := \mathbb{E}(B2)/\mathbb{E}(B1)$, and (b)~$L_T := \mathbb{E}(C2)/\mathbb{E}(C1)$,
 where
\benrr
&&B1 := \max_{a_L \le x \le a_U}\frac{|\wh g(x) - g(x)|}{g(x)}, \quad B2 := \max_{a_L\le x \le a_U}\frac{|\wh f(x) -g(x)|}{g(x)},\\
&&C1 := \max_{b_L \le x \le b_U}\frac{|\wh G(x) - G(x)|}{\{1- G(x)\}},\quad C2 := \max_{b_L\le x \le b_U}\frac{|\wh F(x) - G(x)|}{\{1- G(x)\}},\\
&&  a_L = G^{-1}(0.025),  \ a_U = G^{-1}(0.975), \quad b_L = G^{-1}(0.9), \ b_U = G^{-1}(0.99).
\eenrr
The first quantity $L_O$ measures the extent to which the \textit{ordinate} of the forecast pdf $\wh f(x)$ deviates from the true pdf $g$, relative to the deviation $|\wh g(x) - g(x)|$, which is due to purely random error.
Similarly, the second quantity, $L_T$, measures the extent to which the forecast of the upper \textit{tail} quantiles of $\wh F(x)$ deviates from the true quantiles of $G$,
 relative to the deviation $|\wh G(x) - G(x)|$ in the upper tail of the distribution~$G$.


%\np

\noindent\textit{Results:}

Estimates of $L_P, L_O,$ and $L_T$ are given in Table~\ref{tbl:den:f-2}.
As an example, consider the first entry of 3.1 for $L_P$ in that table.
It says that on an average, the probability misallocated by the forecast density because the use of the incorrect
 parametric family is 3.1 times (= 310\%) of what would be incurred had the true parametric family been used.
Therefore, the loss in terms of $L_P$ is large.
Since the goodness-of-fit tests, for example, the  $A^2$ test, have nearly 100\% power, the  tests almost certainly point us to the fact that the use of the null model would result in loss.




Table~\ref{tbl:den:f-2} also shows that $L_P$ increases with the power of  $A^2$.
Consequently, if the power of the test is low, then the null and the true models are likely to be close,
and hence, the loss in terms of $L_P$ is also likely to be low.
The estimated values of $L_O$ and $L_T$ are also of the same order of magnitude as those of $L_P$.
Therefore, the use of these tests can be expected to reduce such losses in density/quantile forecasting.




\section*{APPENDIX D: Additional results for the
empirical example in Section~\ref{sec:example}
}
\label{appendixD}
\renewcommand{\theequation}{D.\arabic{equation}}
      \renewcommand{\thesection}{D}
      \setcounter{equation}{0}


Figure~\ref{fig:ecd:fc} contains empirical cumulative distribution functions of the probability integral transforms of density forecasts based on the six multiplicative error models for the UTX realized volatility series considered in the empirical example in Section~\ref{sec:example}.
Figure~\ref{fig:utx} provides the summary plots for the UTX realized volatility series. The first panel gives a plot of UTX realized volatility expressed on a
percent annualized scale.
An Autocorrelogram is in the second panel. The last panel displays a residual correlogram for the MEM(1,1) model.

%\np


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\bibitem[Salinetti {\em et~al.}(1986)Salinetti, Vervaat and Wets]{Salinetti:etal:86}
Salinetti, G., W. Vervaat, \& R.J.-B. Wets (1986)
\newblock On the convergence in probability of random sets (measurable
  multifunctions).
\newblock {\em Mathematics of Operations Research} 11, 420--422.

\bibitem[Straumann and Mikosch(2006)]{Straumann:Mikosch:06}
Straumann, D. \& T. Mikosch (2006)
\newblock Quasi-maximum-likelihood estimation in conditionally heteroscedastic
  time series: a stochastic recurrence equations approach.
\newblock {\em The Annals of Statistics} 34, 2449--2495.

\bibitem[van~der Vaart(1998)]{VanderVaart:98}
van~der Vaart, A.W. (1998)
\newblock {\em Asymptotic Statistics}, volume~3 of {\em Cambridge Series in
  Statistical and Probabilistic Mathematics}.
\newblock Cambridge University Press, Cambridge.

\bibitem[van~der Vaart and Wellner(1996)]{VanderVaart:Wellner:96}
van~der Vaart, A.W. \& J.A. Wellner (1996)
\newblock {\em Weak Convergence and Empirical Processes: With Applications to Statistics}.
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\end{thebibliography}


\begin{table}[h!]
\centering
\caption{Estimated losses of using an incorrect parametric MEM for forecasting.}
\begin{tabularx}{\textwidth}{l  l @{\extracolsep{\fill}}  c  c c  c   }
\hline \hline \\ [-.3cm]
$G_{\beta}$	& $F_{\theta}$		&	$L_P$	& $L_O$  &	$L_T$ & Power of $A^2$	 \\ [.3cm] %\cline{1-7}  \cline{9-13} \\[-.25cm]


Gen. Gamma	&	Gamma	  &	3.1	       &2.1	&	2.9	&	0.99		\\	
	           &	Weibull	  &	2.6	      &	2.0	&	2.3	&	0.96		 \\	
	           &	Exp	      &	3.4	      &	2.4	&	3.2	&	0.99		 \\	
	           &	Burr	  &	1.8	       &1.2	&	1.2	&	0.17	 \\\\	
																								
Burr	&	Gamma	&	2.7	&	2.5	&	3.1	&	0.99	\\	
	&	Weibull	&	    3.0	&	3.0	&	2.9	&	0.99		 \\	
	&	Exp	&	       3.0	&	3.0	&	2.8	&	0.98		 \\	
	&	Gen. Gamma	&  1.5	&	1.6	&	1.6	&	0.43		 \\[.1cm]	
\hline\hline\\ [-.3cm]
\end{tabularx}
\label{tbl:den:f-2}
\caption*{{\em Note:}
The results are based on 1000 Monte Carlo replications and the sample size was $n=1000$.
The true DGP is MEM(1,1) for the mean function and  $G_{\beta}$ for the error distribution.
The model under consideration for forecasting, and hence, $H_0$ is MEM(1,1) for the mean function and  $F_{\theta}$ for the error distribution.
}
\end{table}


\begin{figure}[ph]
\caption{Empirical cumulative distribution functions of the probability integral transforms of density forecasts of the UTX realized volatility series
for the MEM(1,1): $\vsi_i(\phi) = \phi_1 + \phi_2 Z_{i-1} + \phi_3 \vsi_{i-1}(\phi),$ $\phi = (\phi_1, \phi_2, \phi_3)^\top$, when $F_\th$ is  Weibull [\textbf{\textcolor[rgb]{0.3,0.3,0.9}{\Large $- -$}}], and  mixture of Burr and Generalized Gamma [\textbf{\textcolor[rgb]{0.3,0.3,0.9}{\huge $. . .$}}].
}
\includegraphics[width=\textwidth]{fig_mix_ecdfs_ini_1612_n3}
\label{fig:ecd:fc}
\end{figure}


\begin{figure}[ph]
\caption{Summary figures for the UTX realized volatility series
and the corresponding residuals estimated by the MEM(1,1) model.}
\includegraphics[width=\textwidth]{utx_corr_2_2}
\label{fig:utx}
\end{figure}


\end{document}
