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