Is this identifiable? I am interested in the following model : ($1 \leq i \leq p$, $1 \leq j \leq n_{i}$)
$$y_{i,j} = A (1+a_{i})(t_{i,j}+\gamma_{i}) + \varepsilon_{i,j}$$
where $A \in \mathbb{R}$, $(a_{i})_{1 \leq i \leq p}$ is a $p$-random sample from $\mathcal{N}(0,\sigma_{a}^{2})$ and $(\gamma_{i})_{1 \leq i \leq p}$ is a $p$-random sample from the distribution $\mathcal{N}(0,\sigma_{\gamma}^{2})$, independent from $(a_{i})_{1 \leq i \leq p}$. $(\varepsilon_{i,j})_{i,j}$ is a random sample from $\mathcal{N}(0,\sigma^{2})$, independent from $(a_{i})_{i}$ and $(\gamma_{i})_{i}$.
Given some observations $(y_{i,j},t_{i,j})_{i,j}$, I would like to estimate the model parameters ($A,\sigma_{a}^{2},\sigma_{\gamma}^{2},\sigma^{2})$. I am wondering whether this is possible. I created some synthetical data using the model in Matlab and tried to estimate the parameters of the model using nlmefit(for nonlinear mixed-effects models) but the estimated values for $A$ and $\sigma_{a}$ were (respectively) three times smaller (and three times) bigger than the ones I used to create the data. This makes me wonder whether the model is identifiable. Can you see another reason to this ?
 A: Actually, with my understanding corrected, I think the model is strictly speaking algebraically identifiable, but in some circumstances may go close to unidentifiability.
If we look at $\log(E(y))$ we have $\log(A) +\log(1+a_{i})+\log(t_{i,j}+\gamma_{i})$.
In this form, $\log(A)$ has the role of a intercept and $\log(1+a_{i})$ and $\log(t_{i,j}+\gamma_{i})$ are terms that in some circumstances can be close to collinear. If $t_{i,j}$ doesn't vary much across $j$ compared to its variation across $i$ (e.g. if $t_{i,j}$ turns out to be approximately $r_i+s_j$ and $s_j$ doesn't vary much compared to $r_j$) then there'd be near-collinearity on the log scale.
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I'm leaving this here for the present for context. I'll probably delete it eventually.
Unless I missed something, your model seems to be not identifiable. 
For example, 
consider a second version of the model with $A^* = A/k$, $t^*_{i,j}=kt_{i,j}$ and $\gamma^*_{i}=k\gamma_{i}$.
Then $E(y)$ is the same under both models.
Since $k$ is arbitrary, the model won't be identifiable.
