First Order Autocorrelation
Consider the multiple regression model
$$Y_t=\beta_1+\beta_2 X_{2t}+\beta_3 X_{3t}+\cdots+\beta_k X_{kt}+u_t,$$
in which the current observation of the error term ($u_t$) is a function of the previous (lagged) observation of the error term ($u_{t-1}$). That is,
\begin{align*}
u_t = \rho u_{t-1} + \varepsilon_t, \tag*{eq 1}
\end{align*}
where $\rho$ is the parameter depicting the functional relationship among observations of the error term $u_t$ and $\varepsilon_t$ is a stochastic error term which is iid (identically independently distributed). It satisfy the standard OLS assumption:
\begin{align*}
E(\varepsilon) &=0\\
Var(\varepsilon) &=\sigma_t^2\\
Cov(\varepsilon_t, \varepsilon_{t+s} ) &=0
\end{align*}
Note if $\rho=1$, then all these assumptions are undefined.
The scheme (eq1) is known as a Markov first-order autoregressive scheme, usually denoted by AR(1). The eq1 is interpreted as the regression of $u_t$ on itself tagged on period. It is first-order because $u_t$ and its immediate past value are involved. Note the $Var(u_t)$ is still homoscedasticity under AR(1) scheme.
The coefficient $\rho$ is called the first-order autocorrelation coefficient (also called the coefficient of autocovariance) and takes values from -1 to 1 or ($|\rho|<1$). The size of $\rho$ determines the strength of autocorrelation (serial correlation). There are three different cases:
- If $\rho$ is zero, then there is no autocorrelation because $u_t=\varepsilon_t$.
- If $\rho$ approaches to 1, the value of the previous observation of the error ($u_t-1$) becomes more important in determining the value of the current error term ($u_t$) and therefore, greater positive autocorrelation exists. The negative error term will lead to negative and positive will lead to a positive error term.
- If $\rho$ approaches to -1, there is a very high degree of negative autocorrelation. The signs of the error term have a tendency to switch signs from negative to positive and vice versa in consecutive observations.
For first order autocorrelation AR(1)
\begin{align*}
u_t &= \rho u_{t-1}+\varepsilon_t\\
E(u_t) &= \rho E(u_{t-1})+ E(\varepsilon_t)=0\\
Var(u_t)&=\rho^2 Var(u_{t-1}+var(\varepsilon_t)\\
\text{Because $u$’s and $\varepsilon$’s are uncorrelated}\\
Var(u_t)&=\sigma^2\\
Var(u_{t-1}) &=\sigma^2\\
Var(\varepsilon_t)&=\sigma_t^2\\
\Rightarrow Var(u_t) &=\rho^2 \sigma^2+\sigma_t^2\\
\Rightarrow \sigma^2-\rho^2\sigma^2 &=\sigma_t^2\\
\Rightarrow \sigma^2(1-\rho^2)&=\sigma_t^2\\
\Rightarrow Var(u_t)&=\sigma^2=\frac{\sigma_t^2}{1-\rho^2}
\end{align*}
For covariance, multiply equation (eq1) by $u_{t-1}$ and taking the expectations on both sides
\begin{align*}
u_t\cdot u_{t-1} &= \rho u_{t-1} \cdot u_{t-1} + \varepsilon_t \cdot u_{t-1}\\
E(u_t u_{t-1}) &= E[\rho u_{t-1}^2 + u_{t-1}\varepsilon_t ]\\
cov(u_t, u_{t-1}) &= E(u_t u_{t-1}) = E[\rho u_{t-1}^2 + u_{t-1}\varepsilon_t ]\\
&=\rho \frac{\sigma_t^2}{1-\rho^2}\tag*{$\because Var(u_t) = \frac{\sigma_t^2}{1-\rho^2}$}
\end{align*}
Similarly,
\begin{align*}
cov(u_t,u_{t-2}) &=\rho^2 \frac{\sigma_t^2}{(1-\rho^2)}\\
cov(u_t,u_{t-2}) &= \rho^2 \frac{\sigma_t^2}{(1-\rho^2)}\\
cov(u_t, u_{t+s}) &= \rho^p
\end{align*}
