# Particle filter for likelihood evaluation

I struggle to implement a particle filter to evaluate the likelihood of a textbook example. I got the following process:

$$x_t = \alpha + \beta x_{t-1}/(1+x_{t-1}^2) + w_t$$ where $$w_t \sim \mathcal{N}(0, \sigma^2)$$ and

$$y_t = \delta x_t + v_t$$ where $$v_t \sim \mathop{t}(2)$$.

I want to know the likelihood $$p(y|\alpha, \beta, \sigma, \delta)$$ given some observations $$y$$ and a known $$x_0 = 1$$. To achieve that, I draw $$N$$ particles $$x_t^j$$ for every $$x_{t}$$ and calculate the density of the residuals $$\hat{v}_t^j = y_t - x_t^j$$. After that, I resample the particles with stratified resampling. Now $$\sum_{t=1}^n\ln \left(\sum_{j=1}^N \mathop{p}(\hat{v}_t^j)/N\right)$$ gives me my approximated log-likelihood.

The problem is: It works very poorly. If I deviate a little from the true parameters, the likelihood gets, as expected, smaller. However, there are always alternative constellations which give me a higher likelihood.

Let's suppose e.g. I have $$n=1000$$, $$N = 10000$$, $$\alpha = 0.5$$, $$\beta = 0.3$$, $$\sigma = 1.0$$ and $$\delta = 1.0$$. Then my likelihood is significantly lower for $$\delta = 1.1$$, but the overall estimate can be extremely poor (e.g. all paramters deviate by $$0.5$$ from their true state).

Does someone have an idea, what goes wrong in my example?