Try to pay attention of the .log file. You'll notice, for example, the following message:
! Missing $ inserted.
<inserted text>
$
l.29 where k_
{n} = (k_{1t}, k_{2t},.....,m_{Kt)) for $k=1,....., K$ time ...
This is LaTeX telling you that you're missing a $ sign in line 29. As David suggested in the comment, you need to enclose inline math contents with $...$ or \(...\). So for the two instances that cause you trouble, you should do the following
where $k_{n} = (k_{1t}, k_{2t},.....,m_{Kt})$ for $k=1,....., K$ time series.
and
Where $\gamma_{i}=-(I-A_{1} - ...A_{i})$, $i= 2,.....,p-1$
$\Pi= -(I - A_{1}-....-A_{p})$ is a $N$-dimensional time series,
$A_{0}$ is the intercept term, $e_{t}$ is white noise.
There are also several other places where the codes are not considered "best practices":
- instead of
+ ... +, use + \cdots +
- instead of
, ... ,, use , \ldots , or , \dots ,
- instead of using
eqnarray, use align
Full Code
\documentclass[10pt]{article}
\usepackage{graphicx}
\usepackage{amsmath}
\begin{document}
\title{Case studies: Long run relationship}
\maketitle
\begin{abstract}
The cointegrating approach is proposed
\end{abstract}
\section{introduction}
Over the last century
\section{Methods and Results}
The 9 prices indexes from each province
A VAR(p) is written as
\begin{equation}
k_{t } = A_{0}+ A_1k_{t-1}+ A_{2}k_{t-2}+ A_{2} k_{t-2} +\cdots +A_nk_{t-n} + e_{t}
\end{equation}
% ####################################################################################
where $k_{n} = (k_{1t}, k_{2t},\ldots,m_{Kt})$ for $k=1,\ldots, K$ time series.
The key assumptions are in the VAR(p)
% ##################################################################################
\begin{align}
\Delta k_{t} & = \Delta\Gamma{1}k_{t-1}+\Delta\Gamma{2}k_{t-2}+\cdots+\Delta\Gamma{n}k_{t-n}
\end{align}
% ##############################################################################
Where $\gamma_{i}=-(I-A_{1} - \cdots -A_{i})$, $i= 2,\ldots,p-1$
$\Pi= -(I - A_{1}-\cdots-A_{p})$ is a $N$-dimensional time series,
$A_{0}$ is the intercept term, $e_{t}$ is white noise
% ##############################################################################
\end{document}
Output
Something like this using the matrices? I think that the 2nd image is written with equation editor of Word with the option Insert equation.
\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb}
\begin{document}
\[\begin{bmatrix}
Y_{1j}\\
Y_{2j}\\
\cdots\\
Y_{n_ij}
\end{bmatrix}=\begin{bmatrix}
1\\
1\\
\cdots\\
1
\end{bmatrix}[\gamma_{00}]+[u_{0j}]+\begin{bmatrix}
e_{1j}\\
e_{2j}\\
\cdots\\
e_{n_ij}
\end{bmatrix}\]
\end{document}
Best Answer
You might want to take a look at Jeff Tupper's self-referential formula (link) or its derivatives (link). Or you may consider throwing in a little String Theory (link). Strictly speaking it might not be mathematics but it's still nice.