## Auto-regression

An auto-regressive, **AR(1)**, process is one that is not always stationary and may contain a unit root. 

For an AR process to be stationary, $ |\rho| < 1 $: 

\begin{align}
    X_{t} &= X_{t-1} + \epsilon_{t}  \\
            &= X_{t-2} + \epsilon_{t-1} + \epsilon_{t} \\
            &= X_{0} + \sum_{i=0}^{t-1} \epsilon_{t-i} 
\end{align}
for $ \epsilon_{t} \sim iid(0,1) $ where $ X_{0}$ is a constant.

Further attributes are as follows:

\begin{align}
            E(X_{t}) &=  E(X_{0}) + E \sum_{i=0}^{t-1} \epsilon_{t-i} \\
                    &= E(X_{0})          
\end{align}
             
since $ \epsilon_{t} \sim iid(0,1) $ and $ \therefore  E(X_{t}) = 0 $, $ Var(\epsilon_{t}) = \sigma^2 $, 
so the random noise has zero mean.

\begin{align}
    Var(X_{t}) &=  \sum_{i=0}^{t-1} Var ( \epsilon_{t-i}) \\
                 &=  t \sigma^2   \\
                 &= f(t)
\end{align}
since $ Var(\epsilon_{t}) = \sigma^2$. Therefore, the variance is a function of time, so it is non-stationary.
\\
Therefore a random walk is a non-stationary time series, it is a function of time. 
There is no covarance ($\epsilon_{t}$ terms) since $\epsilon_{t}$ are 
independent of one another.    

\begin{align}
    Cov(X_{t}, X_{t+h}) &= Cov(X_{t}, X_{t} + \sum_{i=0}^{h-1} \epsilon_{t+h-i} ) \\
            &= Cov(X_{t}, X_{t})
            &= Var(X_{t})
\end{align}
given $X_{t+h} = X_{t} + \sum_{i=0}^{h-1} \epsilon_{t+h-i}$, and since $\epsilon_{t}$ are independent, then $Cov(X_{t}, \epsilon_{t_{k}}) = 0$.

Therefore, a random walk is non-stationary, since the variance 
increases with time, and the covariance also increases with time.

773a01ed69ce5e655b86088aae067ffc2380ec9d
ipynb
Jupyter Notebook
src/jupyter-notebooks/.ipynb_checkpoints/Auto-Regression-checkpoint.ipynb
philip-papasavvas/PythonSkills
7744779314245f23065a36ba0318e21f83b1019b
MIT
2019-09-19T15:36:17.000Z
2019-12-23T11:38:01.000Z
src/jupyter-notebooks/.ipynb_checkpoints/Auto-Regression-checkpoint.ipynb
philip-papasavvas/projects
7744779314245f23065a36ba0318e21f83b1019b
MIT
src/jupyter-notebooks/.ipynb_checkpoints/Auto-Regression-checkpoint.ipynb
philip-papasavvas/projects
7744779314245f23065a36ba0318e21f83b1019b
MIT
2020-11-26T00:00:19.000Z
2022-03-25T01:54:38.000Z
Qwen/Qwen-72B
1. YES
2. YES