# Is this semiparametric multiplicative regression model identifiable?

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

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

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 $$(4)$$?