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Let the Vasicek model to be

$$\Delta r_{t}=k(\theta - r_{t-1})\Delta t+\sigma\Delta z_{t}$$

Due to the fact that

$$\Delta r_{t}=r_{t}-r_{t-1}$$

if you let $\Delta t=1$, it is easy to see by replacement that

$$r_{t}=k\theta+(1-k)r_{t-1}+\sigma\Delta z_{t}=\alpha+\beta r_{t-1}+\epsilon _{t}$$

which is a simple $AR(1)$ model.

This is quite useful because it allows you to estimate the SDE's parameters value through an easy OLS regression, without involving QMLE.

Questions:

  1. Could you show me the same proceeding starting from the formulation of an $AR(2)$ model and coming to the related SDE?
  2. Could you show me the general case of an $AR(p)$ process whose $p$-order is greater than $2$?
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Very good question. But shouldn't we always start with an SDE and then get some kind of recursive scheme by discretization. This sounds most natural to me. So I would invert question 1. Which discretization of which SDE leads to an $AR(2)$ process (there are various discretization schemes for SDEs). –  Richard Oct 18 '13 at 15:26
    
Indeed this could be a good way to proceed. –  Lisa Ann Oct 18 '13 at 15:36
    
Are you sure you are going to get discretizations to $AR(p)$ schemes $p>2$ at all? They would seem to require differentials with respect to time, which of course are undefined for nontrivial stochastic processes. –  Brian B Oct 18 '13 at 17:28
    
I am really not sure, Brian B: in fact, the only certainty I have got regards the $AR(1)$ with respect to Vasicek. I would be satisfied to find the link with $AR(2)$, at least. –  Lisa Ann Oct 18 '13 at 18:53
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1 Answer 1

up vote 3 down vote accepted

Note that you can understand the $\Delta$ as an "operator" acting on $r$. So just act on $r$ twice:

$$\Delta^2 r_t = r_t - 2 r_{t-1} + r_{t-2}. $$

In fact if you write the $r$ as a vector, $r = (r_1, r_2, \ldots, r_N)$, then $\Delta$ is an $N\times N$ matrix with elements $\Delta_{i,j} = \delta_{i,j} - \delta_{i-1,j}$.

The AR(2) model can be written as

$$ r_t - \phi_1 r_{t-1} - \phi_2 r_{t-2} = \epsilon_t.$$

We can choose $a, b$ and $c$ such that

$$ a \Delta^2 r_t + b \Delta r_t +c r_{t-2} = r_t - \phi_1 r_{t-1} - \phi_2 r_{t-2} ,$$

which gives $a= 2-\phi_1$, $b=\phi_1 - 1$ and $c = \phi_1 - \phi_2 -2$. So the AR(2) equation can be written as

$$ (2-\phi_1) \Delta^2 r_t + (\phi_1 - 1) \Delta r_t = -(\phi_1 - \phi_2 -2) r_{t-2}+ \epsilon_t.$$

It's not quite trivial to take the limit to the SDE from here... that's because the coefficients of $\Delta^2 r_t$ and $\Delta r_t$ are of the same "dimension" (physicist's l33t sp33k), but the differentials are not, because they are of different order. In fact, I think the limit only exists (or rather, converges to a second order SDE) only if you replace the $\epsilon_t$ with e.g. $\epsilon_t - \epsilon_{t-1}$, i.e. consider an ARMA(2,2) model instead of AR(2), but it's getting late... I can maybe edit/ add something more here tomorrow, if there's interest(?)

EDIT: by the way, you can use OLS with SDEs without setting $\Delta t = 1$ and/or writing the SDE as an AR model etc. Just multiply the (original) equation by $r_t$, take the expectation $\mathbb E()$ and replace with sample mean :) That gives an estimate for the drift term. Do the same but multiply with $\Delta r_t$ to get an estimate for $\sigma$.

ADDITIONAL STUFF:

So as Richard commented on the question, it's probably easier to start with an SDE. SO let's think of the following equation:

$$ \frac{d^2 x(t)}{dt^2} + a \frac{d x(t)}{dt} + c x(t) = \sigma \eta(t)$$

with $\mathbb E (\eta(t) \eta(t')) = \delta (t-t')$ with the Dirac delta function $\delta(t)$. Although mathematicians won't like this, this is a well defined 2nd order SDE describing a harmonic oscillator (for $c>0$) with a linear drag (for $a>0$) and random kicks by $\eta(t)$. Now discretize by letting $dt \to \Delta t$ etc. and multiply the equation by $\Delta t ^2$ (and denote $X(t) = r_t$). We get

$$\Delta^2 r_t + a \Delta t \Delta r_t + b \Delta t^2 r_t = \sigma \Delta t^2 \eta(t).$$

The key to arriving at an AR(2) model is the observation that, formally speaking, $\delta(t-t') = \frac{1}{dt} \delta_{t,t'}$ (you can check this by the Dirac delta definition and Riemann sums). Then $\Delta W_t := \Delta t \eta(t)$ is a standard white noise, and by defining new variables $\tilde a = a \Delta t, \tilde b = b \Delta t^2$ and $\tilde \sigma = \sigma \Delta t$, we get

$$\Delta^2 r_t + \tilde a \Delta r_t + \tilde b r_t = \tilde \sigma \Delta W_t.$$

From this you can go to AR(2), as described above. So if you start from this eqution with the $\tilde a, \tilde b, \tilde \sigma$, if you don't do these rescalings, the SDE limit won't exist because the coefficients blow up! So e.g. $\tilde a$ has to approach zero linearly in $\Delta t$ when $\Delta t \to 0$.

EDIT: and forget about my blabberings about ARMA...

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Of course there's interest, and I bet this is not just for me but for all readers. Please, go on with the case you're thinking about, even if it's an $ARMA(p,q)$ process instead of a simple $AR(2)$. –  Lisa Ann Oct 18 '13 at 20:05
    
OK I'll do it tomorrow if I can find the time. –  H. Arponen Oct 18 '13 at 20:06
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