# 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

• 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