The Random Walk Model

Consider the following special case AR(1) process:

\[\begin{eqnarray*} y_{t}&=&y_{t-1}+\varepsilon_{t}\\ \Delta y_{t}&=&\varepsilon_{t} \end{eqnarray*}\]

If $y_{0}$ is a given initial condition, its solution is

\[\begin{eqnarray*} y_{t}&=&y_{0}+\sum_{i=1}^{t}\varepsilon_{i} \end{eqnarray*}\]

Take expectation

\[\begin{eqnarray*} Ey_{t}&=& E\left(y_{0}+\sum_{i=1}^{t}\varepsilon_{i}\right)=y_{0} \end{eqnarray*}\]

Taking expectation and variance

\[\begin{eqnarray*} E_{t}y_{t+1}&=&E_{t}\left(y_{t}+\varepsilon_{t+1}\right)=y_{t}\\ E_{t}y_{t+s}&=&E_{t}\left(y_{t}+\sum_{i=1}^{s}\varepsilon_{t+i}\right)=y_{t}\\ Var(y_{t})&=&Var\left(\sum_{i=1}^{t}\varepsilon_{i}\right)=t\sigma^{2}\\ Var(y_{t-s})&=&Var\left(\sum_{i=1}^{t-s}\varepsilon_{i}\right)=(t-s)\sigma^{2}\\ E\left[(y_{t}-y_{0})(y_{t-s}-y_{0})\right]&=&E\left[(\sum_{i=1}^{t}\varepsilon_{i} )(\sum_{i=1}^{t+s}\varepsilon_{i})\right]\\ &=&E\left[(\sum_{i=1}^{t-s}\varepsilon_{i}^{2} )\right]\\ &=&(t-s)\sigma^{2} \end{eqnarray*}\]

The correlation coefficient $\rho_{s}$ is

\[\begin{eqnarray*} \rho_{s}&=&\frac{(t-s)\sigma^{2}}{\sqrt{(t-s)\sigma^{2}}\times \sqrt{t\sigma^{2}}}\\ &=&\sqrt{\frac{t-s}{t}} \end{eqnarray*}\]

When t is big relative to s, the $\rho_{s}$ are close to unity and decay very slowly.

The Random Walk plus Drift Model Adding a constant term $a_{0}$:

\[\begin{eqnarray*} y_{t}&=&y_{t-1}+a_{0}+\varepsilon_{t} \end{eqnarray*}\]

Giving the initial condition $y_{0}$, its solution is

\[\begin{eqnarray*} y_{t}&=&y_{0}+a_{0}t+\sum_{i=1}^{t}\varepsilon_{i} \end{eqnarray*}\]

The behavior of $y_{t}$ is governed by two nonstationary components: a linear deterministic trend and the stochastic trend.

Function Spaces

$x$: an element of $C$, that is, any continuous curve traversing the unit interval, be denoted.
Coordinates of $x$: $x(r)\in\mathbb{R}$ is the unique values of $x$ at points $r\in[0,1]$ are called the coordinates of $x$.

For two members of $C$, $x\in C$ and $y\in C$, we need to say how close together they are. Technically, $C$ must be assigned a metric. For example, we can define Euclidean metric for any pair of real numbers $x$ and $y$ as $$d_{E}(x,y)=

x-y

$$.

The pair $(\mathbb{R},d_{E})$ is known as the Euclidean space.

We also can define a metric called uniform metric as

\[\begin{eqnarray*} d_{U}(x,y)&=&\sup_{0\leq r\leq 1}|x(r)-y(r)| \end{eqnarray*}\]

This is just the \textit{largest vertical separation} between the pair of functions over the interval. $(C,d_{U})$ is a metric space.

Brownian Motion

A Brownian motion $B$ is a real random function on the unit interval, with the following properties:

$B\in C$ with probability 1.
$B(0)=0$ with probability 1.
for any set of subintervals defined by arbitrary $0\leq r_{1} < r_{2} < \dots < r_{k}\leq 1$, the increments $B(r_{1}$, $B(r_{2})-B(r_{1}$, $\cdots$, $B(r_{k})-B(r_{k-1})$ are independent.
$B(t)-B(s)\sim N(0,t-s)$ for $0\leq s< t \leq 1$.

The Functional Central Limit Theorem

We construct a variable $X_{T}(r)$ from the sample mean of the first $r$th fraction of the observations, $r\in[0,1]$, defined by

\[\begin{eqnarray*} X_{T}(r)&\equiv&\frac{1}{T}\sum_{t=1}^{[Tr]}u_{t} \end{eqnarray*}\]

Dickey-Fuller Tests

Subtracting $y_{t-1}$ from each side of the equation $y_{t}=a_{1}y_{t-1}+\varepsilon_{t}$, we get $\Delta y_{t}=\gamma y_{t-1}+\varepsilon_{t}$, where $\gamma=a_{1}-1$. Testing the hypothesis $a_{1}=1$ is equivalent to testing $\gamma=0$.

Dickey and Fuller consider three different regression equations

\[\begin{eqnarray*} &&\mbox{random walk model}\\ \Delta y_{t}&=&\gamma y_{t-1}+\varepsilon_{t} \\ && \mbox{add a drift}\\ \Delta y_{t}&=&a_{0}+\gamma y_{t-1}+\varepsilon_{t}\\ &&\mbox{add a drift and linear time trend}\\ \Delta y_{t}&=&a_{0}+\gamma y_{t-1}+a_{2}t+\varepsilon_{t} \end{eqnarray*}\]

Run the OLS and get the estimated value of $\gamma$ and associated standard error of these three models. However, the critical values of the t-statistics do depend on whether a drift and/or time trend is included in regression models. Note that the appropriate critical values depend on sample size. For any given level of significance, the critical values of the t-statistic decrease as sample size increases.

Augmented Dicker-Fuller test

Consider the pth-order autoregressive process:

\[\begin{eqnarray*} y_{t}&=&a_{0}+a_{1}y_{t-1}+a_{2}y_{t-2}+a_{3}y_{t-3}+\cdots+a_{p-2}y_{t-p+2}+a_{p-1}y_{t-p+1}+a_{p}y_{t-p}+\varepsilon_{t} \end{eqnarray*}\]

Add and subtract $a_{p}y_{t-p+1}$

\[\begin{eqnarray*} y_{t}&=&a_{0}+a_{1}y_{t-1}+a_{2}y_{t-2}+\cdots+a_{p-2}y_{t-p+2}+a_{p-1}y_{t-p+1}+a_{p}y_{t-p+1}+a_{p}y_{t-p}-a_{p}y_{t-p+1}+\varepsilon_{t}\\ &=&a_{0}+a_{1}y_{t-1}+a_{2}y_{t-2}+\cdots+a_{p-2}y_{t-p+2}+(a_{p-1}+a_{p})y_{t-p+1}-a_{p}\Delta y_{t-p+1}+\varepsilon_{t} \end{eqnarray*}\]

Add and subtract $(a_{p-1}+a_{p})y_{t-p+2}$

\[\begin{eqnarray*} y_{t}&=&a_{0}+\cdots+a_{p-2}y_{t-p+2}+(a_{p-1}+a_{p})y_{t-p+2}+(a_{p-1}+a_{p})y_{t-p+1}-(a_{p-1}+a_{p})y_{t-p+2}-a_{p}\Delta y_{t-p+1}+\varepsilon_{t}\\ &=&a_{0}+\cdots+(a_{p-2}+a_{p-1}+a_{p})y_{t-p+2}-(a_{p-1}+a_{p})\Delta y_{t-p+2}-a_{p}\Delta y_{t-p+1}+\varepsilon_{t} \end{eqnarray*}\]

Continuing in this fashion, we get

\[\begin{eqnarray*} \Delta y_{t}&=&a_{0}+\gamma y_{t-1}+\sum_{i=1}^{p}\beta_{i}\Delta y_{t-i+1}+\varepsilon_{t}\\ where\ \ \gamma&=& -\left( 1-\sum_{i=1}^{p}a_{i}\right)\\ \beta_{i}&=& \sum_{j=i}^{p}a_{j} \end{eqnarray*}\]

We can use the same Dickey-Fuller statistics which depends on the regression models and sample size.