# Testing a multifactor asset pricing model against another one

I have a sample of $$(Y_{i,t},X_{1,i,t},X_{2,i,t})$$ for $$i=1,\dots,N$$ and $$t=1,\dots,T$$. I want to figure out which data generating process (DGP) it comes from, DGP1 or DGP2.

DGP1: $$Y_{i,t}=\lambda_0+\lambda_1\beta_{1,i}+\varepsilon_{i,t} \tag{1}$$ with $$\lambda_0=0$$ and $$\varepsilon_{\cdot,t} \stackrel{\text{i.i.d.}}{\sim} N(\mathbf{0},\Sigma)$$. $$\beta_{1,i}$$ is a latent variable that is the slope coefficient from $$Y_{i,t}=\beta_0+\beta_{1,i} X_{1,i,t}+u_{i,t}. \tag{i}$$

DGP2: $$Y_{i,t}=\tilde\lambda_0+\tilde\lambda_1\tilde\beta_{1,i}+\tilde\lambda_2\tilde\beta_{2,i}+\tilde\varepsilon_{i,t} \tag{2}$$ with $$\tilde\lambda_0=0$$ and $$\tilde\varepsilon_{\cdot,t} \stackrel{\text{i.i.d.}}{\sim} N(\mathbf{0},\tilde\Sigma)$$. $$\tilde\beta_{1,i}$$ and $$\tilde\beta_{2,i}$$ are latent variables that are the slope coefficients from $$Y_{i,t}=\tilde\beta_0+\tilde\beta_{1,i} X_{1,i,t}+\tilde\beta_{2,i} X_{2,i,t}+\tilde u_{i,t}. \tag{ii}$$

Question: How could I construct a test of $$H_0$$: DGP1 is the truth against $$H_1$$: DGP2 is the truth? Ideally, the answer would also cover the case where $$X_{1,i,t}$$ and $$X_{2,i,t}$$ are vector-valued random variables. I am also not very comfortable assuming the errors to be multivariate Gaussian, but hopefully we can get rid of the assumption asymptotically.

The background is multifactor asset pricing models and the more specific, finance-oriented question I have is here. The setup is not identical but similar. There, we could perhaps fit $$(1)$$ and $$(2)$$ (without the restriction $$\lambda_0=0$$) and compare the statistic $$\boldsymbol{\hat\lambda}_0^{\top} \text{Cov}(\boldsymbol{\hat\lambda}_0,\boldsymbol{\hat\lambda}_0^{\top})^{-1}\boldsymbol{\hat\lambda}_0\sim\chi^2_{?}$$ with $$\boldsymbol{\hat{\tilde{\lambda}}}_0^{\top} \text{Cov}(\boldsymbol{\hat{\tilde{\lambda}}}_0,\boldsymbol{\hat{\tilde{\lambda}}}_0^{\top})^{-1}\boldsymbol{\hat{\tilde{\lambda}}}_0\sim\chi^2_{??}$$, and perhaps that would $$\sim\chi^2_{???}$$?

Update: For the bounty, I expect a detailed answer that would be straightforward to implement. Thank you!

• Just bringing in different perspectives: Are you bound to testing? Can't you check for predictive performance, e.g. with leave one out Cross-Validation? Then you could do forward or backward selection of features. Also sometimes dropping variables may not be beneficial for generalization (e.g. in the case of omitted variable bias). Maybe this gives you another perspective or you may find this comment off-topic Commented Dec 3, 2023 at 21:01
• @Ggjj11, your suggestions are logical, but I am bound to testing. Commented Dec 4, 2023 at 6:18