AR(1) model

Suppose we have following AR(1) model (autoregressive model of order 1) defined by: $$ X_{t+1}=aX_t + b+ cW_{t+1} $$ where

$X_0$ and $W_{t}$ are independent,
[a,b,c] are scalar-valued parameters,
the state space is $\mathbb{R}$,
the process $W_{t}$ is independently distributed and standard normal, and
initial condition $X_0$ is drawn from the normal distribution of mean $\mu_0$ and standard error of $\nu_0$.

Iterating backward from time $t$, we have $$ X_t = a^tX_0 + b \sum_{j=0}^{t-1}a^j + c\sum_{j=0}^{t-1}a^jW_{t-j}, $$ which implies that $X_t$ depends on parameter [a,b,c], initial condition $X_0$, and the sequence of shocks from period 1 throughout t represented by $W_1, \cdots, W_t$.

Let $\mu_t$ and $\nu_t$ denote the mean and variance of $X_t$. Then we can trace out the sequence of distributions ${\psi_t}$ corresponding to the time series $X_t$.

$$ \psi_t=N(\mu_t, \nu_t) $$ where $$ \mu_{t+1}=a\mu_t + b \newline \nu_{t+1} = a^2 \nu_t + c^2. $$

The following code excerpts are from julia.quantecon.org.

"""
    plot_density_seq
    in: 
        mu_0=-3.0
        v_0=0.6
        sim_length=60
    during: 
        update mu 
        update v
        plot normal distribution with updated mu and v 
    out: 
        accumulated plots
"""
function plot_density_seq(mu_0=-3.0, v_0=0.6; sim_length=60)
    mu = mu_0
    v = v_0
    plt = plot()
    for t in 1:sim_length
        mu = a * mu + b
        v = a^2 * v + c^2
        dist = Normal(mu, sqrt(v))
        plot!(plt, x_grid, pdf.(dist, x_grid), label=nothing, linealpha=0.5)
    end
    return plt
end
plot_density_seq()
plot_density_seq(3.0)

Note that regardless of the starting $\mu$, the $\psi$distribution seems to be converging to some limiting distribution. (stationarity) This is true when $|a|<1$.

Reference

https://julia.quantecon.org/introduction_dynamics/ar1_processes.html

Reference#

Reference