# Regression with latent variable response

I have a dataset with the following structure: $(x_1,x_2,x_3,...,x_n, y)$ where $x_k$ are some categorical predictors and $y$ the numerical (integer) response.

Assuming that $x_1 \in \{a,b,c\}$, where $a,b,c$ correspond to different markets (auction houses) for similar products (the remaining $x_i$ are the product characteristics from historical data, such as colour, used/not used, dimensions etc), I have all the $y$-values, corresponding to prices paid at some previous auction for these products, for auction house $x_1=a$ (e.g. prices falling into $[1,10]$), but only observe prices in a smaller range for $x_1\in\{b,c\}$ (e.g. $[1,5]$), i.e. $y$ are right-censored for $x_1\in\{b,c\}$. This is because some auction houses disclose prices for all auctions (won/lost items) whereas others only reveal such prices only for auctions that are won. I would like to predict price $y$ for $x_1\in\{b,c\}$ in the range $[6,10]$ as well by using some regression model that treats these $y$ values as latent variables.

Would this be possible/does anyone have an idea of how it could be done? I am still a beginner so any pointers to literature or ideas would be highly appreciated.

• If prices in the markets b and c never fall into $[6,10]$ why would you want to predict them in this range? Please try to clarify your question further. – sheß Aug 23 '15 at 23:02
• Why do you want to use a latent variable? Is your problem one of truncation/censoring or do prices naturally fall into this band? – sheß Aug 23 '15 at 23:04
• Thank you for the comment. As you said, the problem is that y values are right-censored but I am hoping that these are correlated between the different markets. – lstavr Aug 23 '15 at 23:09
• Maybe a good start/"Pointer to literature" would be Jeff Wooldridge's book on cross-section and panel data – sheß Aug 24 '15 at 9:49

There are fine details in the type of censoring/truncation/etc. that have bearing on what model is the most appropriate. So you will need to provide more details regarding your data if you want a ready-made solution.

Without any more detail I can only recommend you to read on Tobit-type models or dealing with censoring in general. E.g. here. Maybe just describe what the reason for the censoring is or where your data comes from. For example very different models would be appropriate if (1) you have data where only transactions up to a certain price-threshold are recorded, (2) data where people for some reason would never engage in a transaction above a certain threshold [e.g. because they'd rather go for alternatives].

You also haven't told us what your unit of observation is: transactions? products at one point in time?

The "different markets" are similarly a little difficult to understand from the short description. Are you talking about similar products (e.g. ice cream and yoghurt?) or about different places to sell stuff? Could be that seemingly unrelated regressions could be useful for you, which can be combined with tobit-type models. But I'm just guessing here since I'm not sure about your data/model at hand.

• Thank you a lot for the valuable reply. I do not have a formal statistical background. What kind of details would be needed? The data is right-censored based on some fixed maximum value, here 5, in 2 of the three markets, $b,c$. The $y$ values are integers. – lstavr Aug 24 '15 at 9:05
• I meant economic detail, this is fundamental for proper model-design. I modified my answer to contain more questions you might want to answer. – sheß Aug 24 '15 at 9:18
• Thank you. The data comes from auctions for similar products, e.g. ipad with some characteristics (colour, used/not used etc), and each observation corresponds the price for one such product at each day. Some markets (auction houses) provide the prices for items lost (i.e. the price is higher than the bid) whereas some others do not. So, I guess (1) would be the right case. – lstavr Aug 24 '15 at 9:45
• Please edit you original question so that other's get this info too without having to scroll down and read through all of this. – sheß Aug 24 '15 at 9:47