# Maximum likelihood for lognormal mixture

I have a collection of historical data that I want to fit to the following model

$$y_{t+1} - y_t = \alpha + (\rho + \sigma_2 Z_{t+1} )y_t + \sigma_1 Z_{t+1}$$

where everything except the y's are constant.

To my understanding this model is something like a normal / lognormal mixture. If I plot my data against a random normal sample it more or less fit a straight line. Had it not been for $\sigma_2$ I'd known what to do but now its a bit more involved.

Any ideas?

Thanks.

• I'm voting to close this question as off-topic because it belongs on "Cross Validated". – Richard Hardy Jan 4 '17 at 15:13
• Why would you say that? This model is a combination of two time series which both are very common in QF. – Nid Jan 4 '17 at 15:52
• The question is about statistical modelling, so it naturally fits on Cross Validated. Different branches of Stack Exchange should not duplicate each other, and Cross Validated is the one dealing with statistics. Furthermore, there is nothing specific to finance in your question, so it does not seem to fit here well. – Richard Hardy Jan 4 '17 at 21:24