Consistent estimation with observed values lower than actual values

Question

Assume an IID sample of the form $ \left\{ y^{r}_{i},\mathbf{x}_{i} \right\}$ (notice the superscript on $y$). The observed values $y^{r}_{i}$ are bounded from above by the actual, unobserved values of $y_{i}$. The values that both $y_{i}$ and $y_{i}^{r}$ obtain are in $\mathbb{Z}_{+}$ (consider the following for instance: the former represent crimes counts in some neighborhood and the latter, reported crimes counts). I postulate the following simple linear model: $$y_{i}=\mathbf{x}_{i}\cdot \beta +\epsilon_{i1}$$ and assume the observed values come from the following: $$y^{r}_{i}=\mathbf{x}_{i}\cdot \beta -a_{i}\cdot \gamma +\epsilon_{i2}$$ while $a_{i}$ is unobserved.

Is there any way to consistently estimate $\beta$? I have read about Heckman's correction but it does not seem applicable here since the observed values are not observed after some threshold. I have failed to develop anything useful by myself. Ideas and suggestions welcome.

Answer 1

Unless you have some extra information from somewhere on the mean event rate $\mu$ or the reporting proportion $\pi_i$ (e.g. in the form of prior information, known full information for this for a subset of the data, have situations where you know how $\pi_i$ as well as $\mu$ will change across the situations, or the like), you will not be able to separately estimate $\mu$ and $\pi_i$.

You can see this, if you e.g. think of it as a count outcome, let's say $Y_i \sim \text{NegBin}(\mu, \kappa)$ (possibly $\mu$ is the same for all $\forall i=1,\ldots,I$, or perhaps we have separate $\mu_i$ or they follow some regression equation such as $\log \mu_i = \mathbf{x\beta}$). For each occurrence there's a separate independent binary decision on whether it's observed or not so that $Y_i^r | Y_i = y_i \sim \text{Bin}(y_i, \pi_i)$ (possibly $\pi_i=\pi$ $\forall i=1,\ldots,I$, or perhaps not). I am not 100% sure, but this might result in $Y_i^r \sim \text{NegBin}(\mu \pi_i, \kappa)$ (or something close to it) and you can only estimate $\mu \times \pi_i$ and $\kappa$, but not $\mu$ and $\pi_i$ separately.