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Suppose I have the following stationary $AR(1)$ process:

$$ y_{t}=\alpha+\rho y_{t-1} + u_{t} $$

where $u_{t} \sim \mathbb{N}(0,\sigma^{2})$, with $\sigma^{2}$ known. Suppose I have an initial condition $y_{0}$ and terminal condition $y_{T}$ and I would like to simulate my process for the periods in the interim, i.e $t = 2,\dots,T-1$. Can someone tell me what is the right distribution from which I should draw the $u_{t}$ if I want to impose both the initial and the terminal condition?

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  • $\begingroup$ Is that so? I am not sure about that. If I start from $y_{1}$ and simulate forward drawing $u_{t} \sim \mathcal{N}(0,\sigma^{2})$, $y_{T} \neq y_{T}^{SIM}$ which is a condition I want to impose. $\endgroup$
    – econ_ugrad
    Jan 29 '20 at 15:55
  • $\begingroup$ I edited the question a little bit. The crucial thing is that I want to condition both on the initial and the terminal conditions. $\endgroup$
    – econ_ugrad
    Jan 29 '20 at 15:58
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    $\begingroup$ Have you considered drawing from the (multivariate normal) conditional distribution of $Y_2,\ldots,Y_{T-1}|{Y_1,Y_T}$? $\endgroup$ Jan 30 '20 at 11:34
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You cannot impose both an initial and a final condition on a standard AR(1) process, as the latter is the stochastic version of a first-order difference equation whose solution is uniquely pinned down by one condition.

However, you could look for discrete versions of the Brownian Bridge process, a continuous-time process that can be parametrized so as to simultaneously satisfy an initial and a final condition. A possible result would be an auto-regressive process like the one you describe, but with time-varying coefficients: in particular both the auto-regressive coefficient $\rho$ and the variance $\sigma$ need to be zero at time $T$.

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