Deriving conditional distributions of AR(1) process with drift
I have an AR(1) process with drift:
$y_t=μ+ρ_{t-1}+ε_t$
with the errors following AR(1) process: $ε_t=φε_{t-1}+u_t$
for $t=1, ..., T; ε_0=0$; and $u_t$ are iid $N(0, σ^2)$.
We have these independent priors: $φ$ ~ $N(0, 1)$; $ρ$ ~ $N(0, 1)$; $μ$ ~ $N(0, 100)$ and $σ^2$ ~ $IG(5, 10)$.
The task is to derive the following conditional distributions:
$f(σ^2|y,μ,ρ,φ)$, $f(φ|y,μ,ρ,σ^2)$, and $f(μ,ρ|y,φ,σ^2)$.
What I know is that the posterior is proportional to likelihood*prior. I assume that for the last conditional distribution, $f(μ,ρ|y,φ,σ^2)$ I am supposed to use joint $p(μ,ρ)$ as a prior.
But with this many parameters, I am very confused what likelihoods should be employed for the calculation of each conditional distribution. Could somebody please clarify this to me? How do I know what likelihood to use? Also, does AR(1) process play any role in this calculation?
probability-distributions bayesian
asked Dec 2 at 0:59
minenivi
11
add a comment |
I have an AR(1) process with drift:
$y_t=μ+ρ_{t-1}+ε_t$
with the errors following AR(1) process: $ε_t=φε_{t-1}+u_t$
for $t=1, ..., T; ε_0=0$; and $u_t$ are iid $N(0, σ^2)$.
We have these independent priors: $φ$ ~ $N(0, 1)$; $ρ$ ~ $N(0, 1)$; $μ$ ~ $N(0, 100)$ and $σ^2$ ~ $IG(5, 10)$.
The task is to derive the following conditional distributions:
$f(σ^2|y,μ,ρ,φ)$, $f(φ|y,μ,ρ,σ^2)$, and $f(μ,ρ|y,φ,σ^2)$.
What I know is that the posterior is proportional to likelihood*prior. I assume that for the last conditional distribution, $f(μ,ρ|y,φ,σ^2)$ I am supposed to use joint $p(μ,ρ)$ as a prior.
But with this many parameters, I am very confused what likelihoods should be employed for the calculation of each conditional distribution. Could somebody please clarify this to me? How do I know what likelihood to use? Also, does AR(1) process play any role in this calculation?
probability-distributions bayesian
asked Dec 2 at 0:59
minenivi
11
add a comment |
I have an AR(1) process with drift:
$y_t=μ+ρ_{t-1}+ε_t$
with the errors following AR(1) process: $ε_t=φε_{t-1}+u_t$
for $t=1, ..., T; ε_0=0$; and $u_t$ are iid $N(0, σ^2)$.
We have these independent priors: $φ$ ~ $N(0, 1)$; $ρ$ ~ $N(0, 1)$; $μ$ ~ $N(0, 100)$ and $σ^2$ ~ $IG(5, 10)$.
The task is to derive the following conditional distributions:
$f(σ^2|y,μ,ρ,φ)$, $f(φ|y,μ,ρ,σ^2)$, and $f(μ,ρ|y,φ,σ^2)$.
What I know is that the posterior is proportional to likelihood*prior. I assume that for the last conditional distribution, $f(μ,ρ|y,φ,σ^2)$ I am supposed to use joint $p(μ,ρ)$ as a prior.
But with this many parameters, I am very confused what likelihoods should be employed for the calculation of each conditional distribution. Could somebody please clarify this to me? How do I know what likelihood to use? Also, does AR(1) process play any role in this calculation?
probability-distributions bayesian
asked Dec 2 at 0:59
minenivi
11
I have an AR(1) process with drift:
$y_t=μ+ρ_{t-1}+ε_t$
with the errors following AR(1) process: $ε_t=φε_{t-1}+u_t$
for $t=1, ..., T; ε_0=0$; and $u_t$ are iid $N(0, σ^2)$.
We have these independent priors: $φ$ ~ $N(0, 1)$; $ρ$ ~ $N(0, 1)$; $μ$ ~ $N(0, 100)$ and $σ^2$ ~ $IG(5, 10)$.
The task is to derive the following conditional distributions:
$f(σ^2|y,μ,ρ,φ)$, $f(φ|y,μ,ρ,σ^2)$, and $f(μ,ρ|y,φ,σ^2)$.
What I know is that the posterior is proportional to likelihood*prior. I assume that for the last conditional distribution, $f(μ,ρ|y,φ,σ^2)$ I am supposed to use joint $p(μ,ρ)$ as a prior.
But with this many parameters, I am very confused what likelihoods should be employed for the calculation of each conditional distribution. Could somebody please clarify this to me? How do I know what likelihood to use? Also, does AR(1) process play any role in this calculation?
probability-distributions bayesian
probability-distributions bayesian
asked Dec 2 at 0:59
minenivi
11
asked Dec 2 at 0:59
minenivi
11
asked Dec 2 at 0:59
minenivi
11
asked Dec 2 at 0:59
minenivi
11
asked Dec 2 at 0:59
minenivi
11
11
add a comment |
add a comment |
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