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\begin{document}
\fancyfoot[l]{\color{blaub}{\scriptsize{Advanced Quantitative Methods}}}
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\begin{slide}

\begin{center}\vfill
{\bf {\Large Advanced Quantitative Methods:\\[1ex] High Frequency Financial Econometrics - A Selection}} \\
\vspace*{0.5cm}
{\large{
Ingmar Nolte \\
\small
Room: C48 \\
I.Nolte@lancaster.ac.uk
}} \\

 \vspace*{0.5cm}

 \textbf{Office Hours: whenever you like}\\

 \vspace*{0.5cm}

Webpage:\\ {\color{blaub}{http://www.lancaster.ac.uk/staff/nolte/}} \\


Password: {\color{red}42GARCH}



\end{center}
\end{slide}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\pagestyle{fancy} \fancyhead[l]{\color{blaub}{\vspace*{-2cm}\sc{High Frequency Finance}}}




\begin{slide}
\small

\begin{itemize}
\item data is available on higher frequencies than daily
\item Engle (2000): "ultra high frequency" all single events are recorded
\item more information
\item individuals become more and more important
\end{itemize}

$\Rightarrow$ field of high frequency finance and market microstructure analysis
\end{slide}


\begin{slide}
\small

\textbf{Why is it important?}

\begin{itemize}
\item Older "finance" models can be refined (volatility, asset pricing, ...)
\item intraday risk management, intraday trading models,
\item analysis of trading behavior,
\item design of markets and exchanges (e.g. tick size change NYSE $26^{th}$ January 2001)
\end{itemize}

\textbf{Aim:} Understanding the mechanisms at the microlevel

 \textbf{Requirement:} Good data quality and appropriate models



\end{slide}



\begin{slide}
\begin{center}\textbf{2 days of SPY trading}\end{center}
\small
\epsfig{angle=0,width=12cm, height=7cm,file=eps/spy-2-days.eps}
\end{slide}

\begin{slide}
\begin{center}\textbf{50 observations of SPY trading}\end{center}
\small
\epsfig{angle=0,width=12cm, height=7cm,file=eps/spy-50-obs.eps}
\end{slide}

\begin{slide}
\begin{center}\textbf{Autocorrelogram: SPY}\end{center}
\small
\epsfig{angle=0,width=12cm, height=7cm,file=eps/spy-acf.eps}
\end{slide}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pagestyle{fancy} \fancyhead[l]{\color{blaub}{\vspace*{-2cm}\sc{Point Processes}}}


\begin{slide}
\small

\begin{itemize}
\item $t\in\mathbb{R}^+$ continuous physical calendar time.
\item $\{t_i\}_{i=1,2,3, \ldots}$ random arrival times of events with $0\leq t_i \leq t_{i+1}$ on $\mathbb{R}^+$
\item $\{t_i\}_{i=1,2,3, \ldots}$ is called point process on $\mathbb{R}^+$
\item point process $\{t_i\}_{i=1,2,3, \ldots}$ is simple if  $t_i < t_{i+1}$ almost surely
\end{itemize}



\end{slide}



\begin{slide}
\small

\begin{itemize}
\item right continuous counting function for $\{t_i\}_{i=1,2,3, \ldots}$ is given by $N(t)=N([0,t])=\sum_i \1_{\{t_i\leq t\}}$
\item left continuous counting function is given by $\breve N(t)=N([0,t))=\sum_i \1_{\{t_i<t\}}$
\end{itemize}

\textbf{We assume:}\\ $\{t_i\}_{i=1,2,3, \ldots}$ is integrable, i.e. for all $t$: ${\rm{E}}[N(t)]< \infty$.


\end{slide}

\begin{slide}
\small

\textbf{Events?}

\begin{itemize}
\item $\{W_i\}_{i=1,2,3, \ldots}$, with $W_i\in\{1,\ldots,K\}$, determines whether the $t_i$ arrival time is induced by the event of type $k \in\{1,\ldots,K\}$
\item $\{t_i,W_i\}_{i=1,2,3, \ldots}$ is then called a $K$ variate marked point process
\item representation of $K$ univariate point processes $t_i^{k}$, with $N^k(t)=\sum_i \1_{\{t_i\leq t\}}\1_{\{W_i= k\}}$.
\item an event can be related to the vector of explanatory variables $Z_i$
\item natural
filtration of $N(t)$:  $\mathfrak{F}^N_t\equiv
\sigma(N(s)|0\leq s \leq t)$
\item broader filtration (covariate process $\{Z_i\}_{i\in
\{1,2, \ldots \}}$): $\mathfrak{F}_t$ with
$\mathfrak{F}^N_t \subseteq \mathfrak{F}_t$.
\end{itemize}



\end{slide}



\begin{slide}
\small

\textbf{Consider:} univariate point processes $\{t_i\}_{i=1,2,3, \ldots}$\\

\textbf{Intensities, Compensators and Durations}\\

\begin{itemize}
\item $N(t)$ is an $\mathfrak{F}_t$-submartingale, because ${\rm{E}}[|N(t)|]<\infty$ and since for all $s,t\geq 0$ with $s\leq t$ the
condition ${\rm{E}}[N(t)|\mathfrak{F}_s]\geq {\rm{E}}[N(s)]$
\end{itemize}


Doob-Meyer decomposition of $N(t)$:
\begin{eqnarray}
N(t)=M(t)+\tilde \Lambda(t).
\end{eqnarray}

with

\begin{itemize}
\item  $\tilde \Lambda(t)$ a $\mathfrak{F}_t$-predictable
increasing compensator process,
\item $M(t)$ a zero mean martingale process,
\end{itemize}

\end{slide}

\begin{slide}
\small

The \textbf{$\mathfrak{F}_t$-intensity process} of $N(t)$ can then
be defined as that positive $\mathfrak{F}_t$-predictable process
$\lambda(t)$ fulfilling
\begin{eqnarray}
\tilde \Lambda(t)=\int_{(0,t]}\lambda(u)du.
\end{eqnarray}


\textbf{Theorem of Aalen's (1976)}:\\
With $\{\lambda(t)\}$ is bounded from above
by an integrable random variable, we get denoting
$\lambda(t^+)\equiv \lim\limits_{\Delta \to 0}
\lambda(t+\Delta)$ that
\begin{eqnarray}\label{ch6_intensity}
\lambda(t^+)=\lim_{\Delta \to 0} \frac{1}{\Delta}
{\rm{E}}[N(t+\Delta)-N(t)|\mathfrak{F}_t].
\end{eqnarray}
Moreover, assuming that $\lambda(t)$ is continuous we naturally
obtain that $\lambda(t^+)=\lambda(t)$.

\end{slide}

\begin{slide}
\small

$x_i=t_i-t_{i-1}$: the \textbf{inter event duration}, i.e. the time between $t_{i-1}$ and $t_i$\\

\textbf{hazard function:}
\begin{eqnarray}
h(x_i)=\lim_{\Delta \to 0}\frac{1}{\Delta} {\rm{Pr}}[x_i \leq X_i < x_i+\Delta|X_i \geq x_i]=\frac{f(x_i)}{1-F(x_i)},
\end{eqnarray}
where $x_i$ is considered as a realization of the random variable $X_i$, which has the density function $f(x_i)$ and the distribution function $F(x_i)$.
\begin{eqnarray}
S(x_i)\equiv 1-F(x_i)=\exp\left(-\int_{0}^{x_i}h(s)ds\right),
\end{eqnarray}
denotes the survivor function.

\end{slide}

\begin{slide}
\small

\textbf{Example: Homogeneous Poisson Process}\\

Setting $\lambda(t)\equiv \lambda \in \mathbb{R}^+$ defines the
homogenous Poisson process:
$\Rightarrow$
\begin{eqnarray}
\tilde \Lambda(t)=\int_{0}^{t}\lambda(u)du=t\lambda.
\end{eqnarray}
\begin{itemize}
\item  the number of events in a certain time interval $(a,b]$, say, $N((a,b])=N(b)-N(a)$ is Poisson distributed with ${\rm{Pr}}[N((a,b])=k]=\exp(-\lambda(b-a))\lambda(b-a)^k/k!$
\item the inter-event durations $x_i=t_i-t_{i-1}$
are independently exponentially distributed with parameter
$\lambda$.
\end{itemize}

\textbf{Three equivalent representations of the same point process:}
\begin{itemize}
\item  intensity
\item duration
\item counts
\end{itemize}

\end{slide}

\begin{slide}
\small

\begin{theo}[Meyer (1971), [Random Time Change]]~\\
If the counting process
$N(t)$ representing the point
process $\{t_i\}_{i\in \{1,2, \ldots \}}$ has a continuous
compensator process $\tilde \Lambda(t)$, with $\tilde \Lambda(\infty)=\infty$,
then the point process formed from
the times $\{\tilde \Lambda(t_i)\}_{i\in \{1,2, \ldots
\}}$ is a homogenous Poisson process with a unit Poisson
rate, i.e. $\lambda=1$ \hfill{$\diamond$}
\end{theo}


\textbf{Construction of} $\tilde N(t)$ representing the
transformed homogenous Poisson process unit Poisson
rate:\\
Choose the (unique)
$\mathfrak{F}_t$-stopping times $\tau(t)$ through
$\int_{0}^{\tau(t)}\lambda(u)du=t$  and setting
$\tilde N(t)= N(\tau(t))$.

\newpage

\textbf{Most important implication}: \\
The integrated intensities, $\Lambda(\cdot,\cdot)$ say, which
are the increments of $\{\tilde \Lambda(t_i)\}_{i\in \{1,2,
\ldots \}}$, given by
\begin{eqnarray}
\Lambda(t_{i-1},t_i)\equiv \tilde \Lambda(t_i)-\tilde
\Lambda(t_{i-1})=\int_{t_{i-1}}^{t_{i}}\lambda(u)du
\end{eqnarray}
are independently unit exponentially distributed.\\



\end{slide}

\begin{slide}
\small

\subsubsection*{Autoregressive Conditional Duration (ACD) Models}
Engle \& Russell (1998) and Engle (2000)\\

\textbf{Consider:}

$x_i\equiv\frac{t_i-t_{i-1}}{s(t_i)}$, where $s(\cdot)$ denotes a seasonality function to capture time of day effects

\textbf{Aim:}

Conditional density of $ f(x_i|\mathfrak{F}_{t-1})$ which should have a positive support.\\


Decompose $x_i$ multiplicatively
\begin{eqnarray*}
x_i=\varphi_i\varepsilon_i.
\end{eqnarray*}


\begin{itemize}
\item conditional mean function $\varphi_i$, capturing all the conditioning information
\item i.i.d error term $\varepsilon_i$ with unit mean
\end{itemize}


\end{slide}

\begin{slide}
\small

\textbf{Playground 1: density of $\varepsilon_i$}\\
A standard choice is the unit exponential density\\
More flexible specifications:
Weibull, Burr,  generalized Gamma or generalized F density

\textbf{Playground 2: dynamics of $\varphi_i$}\\

\textbf{Linear Models:}

\textbf{ACD($p$,$q$):}
\begin{eqnarray} \label{ACD(p,q)}
(1-\beta_p(L))\varphi_i = \omega+\alpha_q(L) x_i,
\end{eqnarray}

Explanatory variables can be included e.g. statically in the following form
\begin{eqnarray*}
(1-\beta_p(L))\left(\varphi_i - \zeta Z_i \right) = \omega+\alpha_q(L) x_i,
\end{eqnarray*}

\newpage

For $p=q=1$ the ACD model is covariance stationary if
\begin{eqnarray*}
(\alpha_1+\beta_1)^2-\alpha^2_1\sigma^2<1,
\end{eqnarray*}
where $\sigma^2=\var[\varepsilon_i]<\infty$. The moments are the given by
\begin{eqnarray*}
\ew(x_i)&=&\mu_x=\frac{\omega}{1-\alpha-\beta}\\
\var(x_i)&=&\mu_x^2\sigma^2\frac{1-\beta^2-2\alpha\beta}{1-(\alpha+\beta)^2-\alpha^2\sigma^2}\\
\rho_1&=&\frac{\alpha(1-\beta^2-\alpha\beta)}{1-\beta^2-2\alpha\beta}\\
\rho_k&=&(\alpha+\beta)\rho_{k-1}, \text{ for }k\geq 2
\end{eqnarray*}


\end{slide}



\begin{slide}
\small
Fractionally Integrated ACD \textbf{FIACD($p$,$q$):}\\

A stationary process $x_i$, is called a stationary process with long memory if there
is a $d\in (0.5,1)$ and a constant $C$ such that
\begin{eqnarray}\label{def_longmemory}
\lim\limits_{k \to \infty}\frac{\rho_k}{Ck^{2d-1}}=1.
\end{eqnarray}
This causes the divergence of the series of absolute autocorrelations, i.e.~\linebreak
$\sum\limits_{k=-\infty}^{\infty}|\rho_k|=\infty$, where $\rho_k$ denotes the $k^{th}$ autocorrelation coefficient of
$x_i$.

\newpage

Rewriting
(\ref{ACD(p,q)}) as an ARMA(max(p,q),p) model, yields
\begin{eqnarray*}
(1-\alpha_q(L)-\beta_p(L))x_i= \omega+(1-\beta_p(L))v_i,
\end{eqnarray*}
where $v_i=x_i-\varphi_i$. The fractionally integrated model is then obtained by
\begin{eqnarray} \label{FIACD-ARMA-style}
(1-\alpha_q(L)-\beta_p(L))(1-L)^dx_i= \omega+(1-\beta_p(L))v_i,
\end{eqnarray} where $(1-L)^d$, $0<d<1$ is the
fractional differencing operator given by
\begin{eqnarray*}
(1-L)^d=\sum_{k=0}^\infty\varpi_kL^k,
\end{eqnarray*}
with
\begin{eqnarray*}
\varpi_k=\frac{\Gamma(k-d)}{\Gamma(k+1)\Gamma(-d)}=\prod_{0<j\leq k}\frac{j-1-d}{j},\quad k=0,1,2,\ldots
\end{eqnarray*}
and $\Gamma(\cdot)$ denotes the gamma function given by
\begin{eqnarray*}
\Gamma(x) \equiv \left\{\begin{array}{lcl} \int\limits_0^\infty t^{x-1} \exp(-t) dt & \text{if} & x>0, \\ \infty & \text{if} & x=0, \\
x^{-1}\Gamma(1+x) & \text{if} & x<0.
\end{array} \right.
\end{eqnarray*}
Rewriting (\ref{FIACD-ARMA-style}) the FIACD($p,d,q$) model is given by
\begin{eqnarray*}
(1-\beta_p(L))\varphi_i&=& \omega + \left[1-\beta_p(L)-[1-\alpha_q(L)-\beta_p(L)](1-L)^d\right]x_i\\
&\equiv& \omega + \gamma_\infty(L)x_i,
\end{eqnarray*}



\end{slide}



\begin{slide}
\small

\textbf{Non-Linear ACD Models}:\\
Example:
\begin{eqnarray*}
 && (\text{LACD}_1): \quad (1-\beta_p(L))\ln(\varphi_i) = \omega+\alpha_q(L) \ln(\varepsilon_i),  \\
 && (\text{LACD}_2): \quad (1-\beta_p(L))\ln(\varphi_i) = \omega+\alpha_q(L) \varepsilon_i, \\
 && (\text{LACD}_3): \quad (1-\beta_p(L))\ln(\varphi_i) = \omega+\alpha_q(L) \ln(x_i).
\end{eqnarray*}
In general:
\begin{eqnarray*}
B(\varphi_i)=A(\varepsilon_i)B(\varphi_{i-1})+C(\varepsilon_{i}),
\end{eqnarray*}
where $\varepsilon_i$ is a sequence of i.i.d.~variables fulfilling certain regularity conditions,  $B(\varphi_i)$ is
an $\mathbb{R}$-valued process, $B(\cdot)$ a Borel-function and $A(\cdot)$ and $C(\cdot)$ are polynomial functions.

\end{slide}



%\begin{slide}
%\tiny
%\hspace*{-1cm}
% \renewcommand{\arraystretch}{0.9}
%\begin{tabular}{l c c c l l}
%\hline
%Model for &  &  &  &  & Parameter \\
%$p=q=1$   & \raisebox{1.5ex}[-1.5ex]{Author}       & \raisebox{1.5ex}[-1.5ex]{$B(\varphi_i)$}     & \raisebox{1.5ex}[-1.5ex]{$A(\varepsilon_i)$}  & \raisebox{1.5ex}[-1.5ex]{$C(\varepsilon_i)$} & Constraints \\
%
%  \hline \hline  ACD              & {Engle/Russell1998} & $\varphi^x_i$              & $\beta_1+\alpha_1\varepsilon_{i-1}$    & $\omega$       & $\omega,\alpha_1,\beta_1 \geq 0$ \\
%
%   & & & & & \\
%
%  \hline  $\text{LACD}_1$  & {Bauwens/Giot2000}/   & $\ln(\varphi_i)$         & $\beta_1$    & $\omega+\alpha_1\ln(\varepsilon_{i-1})$  & none \\
%
%   (Log 1) &  {Lunde2000} & & & & \\
%
%  \hline  $\text{LACD}_2$ & {Bauwens/Giot2000}/    & $\ln(\varphi_i)$         & $\beta_1$    & $\omega+\alpha_1\varepsilon_{i-1}$  & none \\
%
%   (Log 2) & {Lunde2000} & & & & \\
%
%  \hline  $\text{LACD}_3$ & {Bauwens/Giot2000}/    & $\ln(\varphi_i)$         & $\beta_1+\alpha_1$    & $\omega+\alpha_1\ln(\varepsilon_{i-1})$  & none \\
%
%  (Log 3) & {Lunde2000} & & & & \\
%
%  \hline  BACD            & {Hautsch2001}         & $(\varphi_i)^{\delta_1}$ & $\beta_1$    & $\omega+\alpha_1(\varepsilon_{i-1})^{\delta_2}$  & $\delta_1,\delta_2 \geq 0$ \\
%
%  (Box-Cox) & & & & & \\
%
%  \hline  EXACD           & {Dufour/Engle2000a}   & $\ln(\varphi_i)$         & $\beta_1$    & $\omega+\alpha_1\varepsilon_{i-1}+c|\varepsilon_{i-1}-1|$ & $c \in \mathbb{R}$  \\
%
%  (EXponential) & & & & & \\
%
%  \hline  ABACD            & {Hautsch2003}        & $(\varphi_i)^{\delta_1}$ & $\beta_1$    & $\omega+\alpha_1(|\varepsilon_{i-1}-b|+c(\varepsilon_{i-1}-b))^{\delta_2} $  & $\delta_1,\delta_2 \geq 0$,  \\
%
%(Augm. Box-Cox)& & & & & $b,c \in \mathbb{R} $ \\
%
%  \hline  AACD              & {Hautsch2003}       & $\varphi_i$              & $\beta_1$    & $\omega+\alpha_1\varepsilon_{i-1}$       & $\omega,\alpha_1,\beta_1 \geq 0$ \\
%
%(Additive) & & & & & \\
%
%  \hline  AMACD             & {Hautsch2003}       & $\varphi_i$              & $\beta_1+\alpha_1\varepsilon_{i-1}$  & $\omega+c \varepsilon_{i-1}$       & $\omega,\alpha_1,\beta_1 \geq 0$,  \\
%
%(Add./Multipl.)& & & & & $c \in \mathbb{R}$ \\
%
%  \hline  HACD             & {Fernandes/Grammig2001}/ & $(\varphi_i)^{\delta_1}$              & $\beta_1+\alpha_1(|\varepsilon_{i-1}-b|$  & $\omega$       &  $\delta_1,\delta_2 \geq 0$,  \\
%
%(Hentschel) &  {Hentschel1995} & & $+c(\varepsilon_{i-1}-b))^{\delta_2}$ & & $b,c \in \mathbb{R} $ \\
%
%  \hline AGACD             & {Hautsch2003}           & $(\varphi^x_i)^{\delta_1}$              & $\beta_1+\alpha_1(|\varepsilon_{i-1}-b|$  & $\omega+d(|\varepsilon_{i-1}-b|+c(\varepsilon_{i-1}-b))^{\delta_2} $       &  $\delta_1,\delta_2 \geq 0$, \\
%
%(Augmented) & & & $+c(\varepsilon_{i-1}-b))^{\delta_2}$ & & $b,c,d \in \mathbb{R} $ \\
%
%  \hline NPACD             & {Hautsch2003}/           & $\varphi_i$ or     & $\beta_1$  & $\omega+\sum\limits_{k=0}^{K^+}\alpha_j^+ \1_{\{\varepsilon_{i-1}\geq \bar \varepsilon_k\}}(\varepsilon_{i-1}-\bar \varepsilon_k) $       &  $K^+,K^-\in \mathbb{N}$, \\
%
%  \raisebox{1ex}[-1.5ex]{\parbox{1cm}{(Nonparametric News Impact)} }         &  {Engle/Ng1991}          & $\ln(\varphi_i)$  &   & $+\sum\limits_{k=0}^{K^-}\alpha_j^- \1_{\{\varepsilon_{i-1} < \bar \varepsilon_k\}}(\varepsilon_{i-1}-\bar \varepsilon_k)  $       &   $\bar \varepsilon_k $ breakpoints\\
%
%\hline
%\end{tabular}
%
%
%\end{slide}



\begin{slide}
\small


\subsection*{Dynamic Intensity Models}


\subsubsection*{Autoregressive Conditional Intensity (ACI) Models}
Russell (1999)\\

The conditional intensity function $\lambda(t|\mathfrak{F}_t)$:
\begin{eqnarray}
\lambda(t|\mathfrak{F}_t)=\varphi(t)\lambda_0(t)s(t).
\end{eqnarray}

\begin{itemize}
\item $\varphi(t)$ to capture the dynamic structure
\item baseline intensity component $\lambda_0(t)$ independent of $\mathfrak{F}_t$
\item seasonality component $s(t)$ to capture diurnally seasonality effects
\end{itemize}

\textbf{Why?}\\
Easy incorporation of time varying covariates $Z_{t_i^0}$.

They are allowed to change at any time $t_i^0\in \mathbb{R^+}$.


\newpage

The dynamic component is specified as
\begin{eqnarray}
\varphi(t)=\exp\left(\tilde \varphi_{\breve N(t)+1}+Z_{\breve N^0(t)}'\tilde \gamma\right),
\end{eqnarray}
with
\begin{eqnarray}
\tilde \varphi_i=\omega + \sum_{j=1}^p\alpha_j \varepsilon_{i-j}+\sum_{j=1}^q\beta_j \tilde \varphi_{i-j},
\end{eqnarray}
where the innovation term can be either specified as
\begin{eqnarray}
\varepsilon_{i}=1-\Lambda(t_{i-1},t_i)
\end{eqnarray}
or
\begin{eqnarray}
\varepsilon_{i}=-0.5772-\ln\Lambda(t_{i-1},t_i).
\end{eqnarray}
Please note that $\varphi(t)$ only changes between $t_{i-1}$ and $t_i$ because of the time varying covariates. Thus, abstracting from time varying covariates $\lambda(t)$ changes between $t_{i-1}$ and $t_i$ only deterministically due to $\lambda_0(t)$ and $s(t)$.

\newpage

Baseline intensity $\lambda_0(t)$ $\leftrightsquigarrow$ error term density in the ACD model.\\

ACD model specification $x_i=\varphi_i\varepsilon_i$ can be rewritten in terms of intensities as
\begin{eqnarray}
\lambda(t|\mathfrak{F}_t)=\lambda_{\varepsilon}\left(\frac{x(t)}{\varphi_{\breve N(t)+1}}\right)\frac{1}{\varphi_{\breve N(t)+1}},
\end{eqnarray}
where $\lambda_{\varepsilon}$ denotes the intensity function of the ACD error term and  $x(t)=t-t_{\breve N(t)}$ the backward recurrence time, i.e. the time since the last event $t_{\breve N(t)}$.

$\lambda_0(t)=const.$ $\leftrightsquigarrow$ exponentially distributed ACD error term.

Weibull parametrization $\lambda_0(t)=\exp(\omega)x(t)^{a-1}$, with $a>0$: monotone baseline intensity shape

Burr specification $\lambda_0(t)=\exp(\omega)x\frac{(t)^{a-1}}{1+\eta x(t)^a}$, with $a >0$ and $\eta \geq 0$: non-monotone baseline intensity shape.

\end{slide}



\begin{slide}
\small


\subsubsection*{Hawkes Models}

A general class of Hawkes processes is given by
\begin{eqnarray}
\lambda(t)=\phi(\mu(t)+\sum_{t_i<t}w(t-t_i)),
\end{eqnarray}
where $\phi$ denotes a nonlinear function, $\mu(t)$ a deterministic function in $t$ and $w(\cdot)$ a weighting function.

Hawkes (1971) suggested to parameterize $w(\cdot)$ as
\begin{eqnarray}
w(t)=\sum_{j=1}^p\alpha_j\exp(-\beta_jt),
\end{eqnarray}
where $\alpha_j, \beta_j \geq 0$ for $j=1,\ldots,p$. The $\alpha_j$ are scale parameters, whereas the $\beta_j$ determine the decay and therefore the influence of past time.

For $p>1$, the intensity is a superposition of differently parameterized exponentially decaying weighted sums of the backward recurrence times to \textbf{all} previous points.\\

 The stationarity condition for this type of Hawkes process is given by $0<\int_{0}^{\infty}w(s)ds<1$, which implies $\sum_{j=1}^p\frac{\alpha_j}{\beta_j}<1$.\\

An alterative specification of the weighting functions which induces a hyperbolic, which is slower than an exponential decay is given by
\begin{eqnarray}
w(t)=\frac{H}{(t+\kappa)^b},
\end{eqnarray}
with parameters $H, \kappa$ and $b>1$.


\end{slide}



\begin{slide}
\small

\subsubsection*{Maximum Likelihood Estimation}

The likelihood function can be based solely on $\lambda(t)$ (Karr (1991)). The log likelihood for a dataset where the physical time $t\in(0,T]$ is given by
\begin{eqnarray}
\ln \mathcal{L}(\theta)&=&\int_{0}^T(1-\lambda(s))ds+\int_{(0,T]}\ln\lambda(s)dN(s)\\
&=& \sum_{i=1}^n-\Lambda(t_{i-1},t_i)+\ln(\lambda(t_i))+T,
\end{eqnarray}
where $n$ denotes the number of events in $(0,t]$ and thus $\{t_i\}_{i=1, \ldots,n}$.


\end{slide}



\begin{slide}
\small
\textbf{Specification of the seasonality function:}

\begin{itemize}
\item linear splines
\item cubic splines
\item non-parametrically (Nardaraya-Watson)
\item flexible Fourier specification
\end{itemize}


\end{slide}

\begin{slide}
\small
\textbf{Goodness-of-fit}

\begin{itemize}
\item assess the dynamic and the distributional properties
\item employ standard tools to the generalized residuals which should be iid Exp(1) distributed
\item standard tools to check the dynamics specification: (acf, pacf, LB-tests, BDS-tests, conditional moment tests)
\item standard tools to check the distributional assumptions: (QQ-plots, density forecast tests, D-test)

\end{itemize}

\newpage

\textbf{{Conditional-Moment-Test}}\\
Newey (1985) and Tauchen (1985)\\

One basically tests on the property of a martingale difference sequence (MDS) or an asymptotically equivalent form
in models with multiplicative error terms, like the ACD models. Thus:
\begin{eqnarray}\label{CM}
\E[\varrho_t|\mathfrak{F}_{t-1}]=0,
\end{eqnarray}
where $\mathfrak{F}_{t-1}$ denotes the information filtration and $\varrho_t\equiv \varrho_t(Y_t,Z_t,\theta_0)$ is a
restriction function\footnote{In the ACD model context $\varrho_t=Y_t-\varphi^Y_t$ (MDS) or
$\varrho_t=\frac{Y_t}{\varphi^Y_t}-1$.}, that depends on the time series under consideration, $Y_t$, some further
explanatory variables, $Z_t$, and the true $k$-dimensional parameter vector, $\theta_0$.\\

The null hypothesis is equation (\ref{CM}) and the idea to construct a test is:
\begin{eqnarray*}
\E[B(\mathfrak{F}_{t-1})\varrho_t]=0 \text{ for all Borel-functions $B(\cdot)$ implies that }
\E[\varrho_t|\mathfrak{F}_{t-1}]=0.
\end{eqnarray*}
Hence, the CM-test statistic is based on the empirical counterparts of these unconditional moments, namely on a vector
of $\hat m_j$ with
\begin{eqnarray*}
\hat m_j \equiv\frac{1}{T} \sum_{t=1}^T \hat \varrho_tB_j(\mathfrak{F}_{t-1})
\end{eqnarray*}
where $B_j(\cdot) \in \mathbb{R}^T$ is an element of a \textbf{finite} set ($J$ elements) of arbitrary Borel-functions
$\{B_j(\cdot)|j=1,\ldots,J\}$. \\


Denote $\hat M$ as the $T \times J$ dimensional matrix consisting of $J$ $\hat M_j$'s,
where
\begin{eqnarray*}
\hat M_j \equiv \hat \varrho_tB_j(\mathfrak{F}_{t-1}), \quad j=1,\ldots,J.
\end{eqnarray*}
Let $\hat S$ denote the $T \times k$ dimensional score matrix (resulting from a ML estimation) and $\hat G \equiv \hat M
| \hat S$ as the stacked $T \times (J+k)$ dimensional matrix of moments and scores, then the CM-test statistic is given
by
 \begin{eqnarray}
 CM \equiv \iota' \hat G [\hat G'\hat G]^{-1}\hat G'\iota \stackrel{a}{\sim} \chi^2_J,
 \end{eqnarray}
with $\iota$ a $T$ dimensional vector consisting of ones. In the above formulae, the dependence on the
$\sqrt{T}$-consistently estimated parameter vector $\hat \theta_0$ is suppressed for notational clarity.\\

Choose the vector of Borel-functions
$B(\mathfrak{F}_{t-1})\equiv \left(B_j(\mathfrak{F}_{t-1}) \right)_{j=1,\ldots,J} $ to be
\begin{multline}\label{CM-spec}
B(\mathfrak{F}_{t-1})=(Y_{t-1},Y_{t-1}^2,Y_{t-1}^3,Y_{t-2},Y_{t-2}^2,Y_{t-2}^3,\\
\varepsilon^Y_{t-1},(\varepsilon^Y_{t-1})^2,(\varepsilon^Y_{t-1})^3,\varepsilon^Y_{t-2},(\varepsilon^Y_{t-2})^2,(\varepsilon^Y_{t-2})^3)'.
\end{multline}

\newpage
\textbf{{Integrated Conditional-Moment-Test}}\\
Bierens (1990) and Jong (1996)\\

Approximation of the set $\{B(\mathfrak{F}_{t-1})\}$, via a set of exponential functions given by $\{\exp(s
\cdot \mathfrak{F}_{t-1})| s\in \mathbb{R}\}$ (same notational convention as above) \\

The test statistic is designed with $s$ being integrated out. The null hypothesis in the ICM-test has to be changed
from (\ref{CM}) to
 \begin{eqnarray}\label{ICM}
 \E[\varrho_t|\mathfrak{F}_{t-1}]=0\quad \mathbf{a.s.},
 \end{eqnarray}
since we allow for exceptions on Lebesgue zero sets.


\newpage




{\textbf{Density-Forecast}}\\
\small
Diebold, Gunther Tay (1998) and Bauwen, Giot, Grammig \& Veredas (2000)\\


Let $\{f^{Y_t}(y_t|\mathfrak{F}_{t-1})\}_{t=1,\ldots,T}$ denote the sequence of one step ahead density forecasts, which
is the conditional density of $Y_t$ given the past information $\mathfrak{F}_{t-1}$ in an ACD model and $\{p^{Y_t}(y_t|\mathfrak{F}_{t-1})\}_{t=1,\ldots,T}$ the sequence
of true densities assumed to generate the data generating process. What one actually wants to test, is the equality
\begin{eqnarray*}
\{f^{Y_t}(y_t|\mathfrak{F}_{t-1})\}_{t=1,\ldots,T} =\{p^{Y_t}(y_t|\mathfrak{F}_{t-1})\}_{t=1,\ldots,T}.
\end{eqnarray*}
As $p^{Y_t}(y_t|\mathfrak{F}_{t-1})$ cannot be observed, we use the probability integral transformation of Rosenblatt (1952) , which states that
\begin{eqnarray}\label{z-transformed}
z_t=\int_{-\infty}^{y_t} f^{Y_t}(\nu) d \nu \qquad t=1,\ldots,T
\end{eqnarray}
is an i.i.d.~uniform sequence, under the null hypothesis that $\{f^{Y_t}(y_t|\mathfrak{F}_{t-1})\}_{t=1,\ldots,T}
=\{p^{Y_t}(y_t|\mathfrak{F}_{t-1})\}_{t=1,\ldots,T}$. This hypothesis can be tested for example with the
$\chi^2$-independence test against the uniform distribution and/or with e.g.~the Ljung-Box- and BDS-test against the
i.i.d.ness. \\

\newpage

{\textbf{D-Test}}\\
Ait-Sahalia (1996) and Fernandes \& Grammig (2000)\\

The idea behind the D-test is to design a test statistic, that measures the proximity between
the parametric and the nonparametric estimates of the density functions of the residuals. The null hypothesis is, that
there exists a parameter vector such that the parametric density chosen in the ML estimation coincides with the true
density of the error terms, i.e.~\begin{eqnarray*} \exists \theta_0 \in \Theta : \quad
f_1^{Y_t}(\cdot,\theta_0)=f_1^{Y_t}(\cdot),
\end{eqnarray*}
where $\Theta$ denotes the parameter space and $f_1^{Y_t}(\cdot,\theta_0)$ the parametric density of the error terms and
$f_1^{Y_t}(\cdot)$ the true density of the error terms. Fernandes and Grammig (2000) propose the following
distance that measures the proximity
\begin{eqnarray*}
\Psi_{f_1^{Y_t}}=\int_0^\infty\1_{\{\nu
\in\mathcal{S}\}}\left(f_1^{Y_t}(\nu,\theta_0)-f_1^{Y_t}(\nu)\right)^2f_1^{Y_t}(\nu) d \nu,
\end{eqnarray*}
which has the following sample counterpart for the estimated residuals
\begin{eqnarray*}
\Psi_{\hat f_1^{Y_t}}=\sum_{t=1}^T \1_{\{\hat \varepsilon_t^Y \in\mathcal{S}\}}\left(f_1^{Y_t}(\hat \varepsilon_t^Y,\hat
\theta)-\widehat f_1^{Y_t}(\hat \varepsilon_t^Y)\right)^2,
\end{eqnarray*}
where $\hat \theta$ and $\widehat f_1^{Y_t}(\cdot)$ are consistent estimates of $\theta_0$ and $f_1^{Y_t}(\cdot)$ and
$\mathcal{S}$ is a compact set. The greater $\Psi_{\hat f_1^{Y_t}}$ the less likely is the null hypothesis. Then, under
the null and some regularity conditions,
the test statistic
\begin{eqnarray*}
\hat \tau_T^D \equiv \frac{Th_T^{0.5}\Psi_{\hat f_1^{Y_t}}-h_T^{-0.5}\hat \delta_D}{\hat \sigma_D} \stackrel{d}{\to}
N(0,1),
\end{eqnarray*}
where $h_T$ denotes the bandwidth in the nonparametric density estimator. $\hat \delta_D$ and $\hat \sigma_D$ are
consistent estimates of $\delta_D=e_K\E(\1_{\{\nu \in\mathcal{S}\}}f_1^{Y_t}(\nu))$ and $\sigma^2_D=v_K\E(\1_{\{\nu
\in\mathcal{S}\}}(f_1^{Y_t}(\nu))^3)$ with $e_K=\int_uK^2(u) du$, $v_K=\int_v\left(\int_uK(u)K(u+v) du\right)^2d v$ and
$K(\cdot)$ the kernel function.


\end{slide}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pagestyle{fancy}
\fancyhead[l]{\color{blaub}{\vspace*{-2cm}\sc{Applications -- Volatility Spillovers}}}



\begin{slide}
\small
\textbf{Readings:}\\
\begin{itemize}
\item  Bauwens, L. \& N. Hautsch (2009): \emph{Modelling Financial High Frequency Data Using
Point Processes}, in: Handbook of Financial Time Series, T. G. Andersen, R. A. Davis, J.-P. Kreiss and T. Mikosch (eds.), Springer, 953-979.
\item Engle, R. F. \& J. R. Russell (1998): \emph{Autoregressive Conditional Duration: A New
Model for Irregularly Spaced Transaction Data}, Econometrica, 66, 1127-1162.
\item Engle (2000): \emph{The Econometrics of Ultra-High-Frequency Data}, Econometrica, 68, 1, 1-22.
\item Hautsch (2004): \emph{Modelling Irregularly Spaced Financial Data -- Theory and Practice of Dynamic Duration Models}, Lecture Notes in Economics and Mathematical Systems, Vol. 539, Springer-Verlag, Berlin.
\item  Hawkes, A. G. (1971): \emph{Spectra of some self-exciting and mutually exciting point processes},
Biometrika, 58, 83-90.
\item Jasiak, J. (1999): Persistence in Intertrade Durations, Working Paper, York University.
\item Karr, A. F. (1991): Point Processes and their Statistical Inference, Dekker, New York.
\item Russell, J. R. (1999): \emph{Econometric Modelling of Multivariate Irregularly-Spaced High
Frequency Data}, Working Paper, University of Chicago.
\end{itemize}

\end{slide}

\end{document} 