I am considering a semiparametric regression with a multiplicative model as below:

$$Y_i=m_1(X_{1i})+m_2(X_{2i})+g_1(X_{1i})*g_2(X_{2i})+U_i, \ i=1,...,n, \quad (1)$$

where $\{Y_i,X_{1i},X_{2i}\}_{i=1}^{n}$ are IID sample, $U_i$ is error term satisfying $E(U|X_1,X_2)=0$ for all $i=1,...,n$, and the four functions $m_1$,$m_2$,$g_1$, and $g_2$ are unknown and smooth. My goal is to obtain consistent estimate of those four functions $m_1$,$m_2$,$g_1$, and $g_2$, individually. To uniquely represent such model, I need to impose the identification conditions such that $E(m_j)=E(g_j)=0$, $j=1,2$, to prevent constants floating around. There are many approaches to approximate each of those functions, and I am using B-spline estimator.

To be more specific, $m_1(x_{1i})\approx \phi(x_{1i})'\lambda_{m_1}$, with $\phi(x_{1i})'=[\phi_1(x_{1i}),...,\phi_L(x_{1i})]$ is a $1\times L$ vector of known basis function values evaluated at $X_{1i}=x_{1i}$, and $\lambda_{m_1}$ is the corresponding $L\times 1$ vector of unknown coefficients. In other words, the unknown smooth function is approximated by a linear combination of its random variables, weighted by the known basis functions. Hence, the regression $(1)$ can be approximated by

$$Y_i=\phi(X_{1i})'\lambda_{m_1}+\phi(X_{2i})'\lambda_{m_2}+(\phi(X_{1i})'\lambda_{g_1})*(\phi(X_{2i})'\lambda_{g_2})+U_i. \quad (2)$$

The model like this is certainly a nonlinear model due to the multiplicative function $g1*g2$, and the goal is to consistently estimate those unknown coefficient vectors $\lambda_{m_1}$, $\lambda_{m_2}$, $\lambda_{g_1}$, and $\lambda_{g_2}$.

Clearly, directly applying nonlinear least square estimator (NLS) on $(2)$ is infeasible, because the individual components of $\lambda_{g_1}$ and $\lambda_{g_2}$ cannot be identified (i.e., only the product coefficients across $\lambda_{g_1}$ and $\lambda_{g_2}$ can be estimated, say, $\lambda_{l,g_1}*\lambda_{k,g_2} \ l=k=1,...,L$). So I followed a paper to revise model $(1)$ to be:

$$Y_i=m_1(X_{1i})+m_2(X_{2i})+e^{g_1(X_{1i})}*g_2(X_{2i})+U_i, \ i=1,...,n, \quad (3)$$

which can be then approximated by


Question: Throughout simulations, the NLS from the model $(3)$ often fail to converge, making the function estimation highly deviate from the true functions, and its not consistent. My conjecture is that there is still some identification problems in model $(3)$, although the reason is not clear to me. Can anyone help me out with this issue?

P.S. If I drop $m_2(X_{2i})$ from model $(3)$ as

$$Y_i=m_1(X_{1i})+e^{g_1(X_{1i})}*g_2(X_{2i})+U_i, \ i=1,...,n, \quad (4)$$

then applying NLS to model $(4)$ has no issue anymore, and each functions $m_1, m_2, g_2$ can be consistently identified. In fact, specification in $(4)$ show up in several papers. If $(4)$ can be identified, what is the difference between model $(3)$ and $(4)$?

Thank you in advance.


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