Question

Let C(k),k=1,…,K denotes fixed and known intervals partitioning (−[infinity],[infinity]). Consider a panel data on wages, where only interval data on wages are recorded: log wage yᵢₜ∗​ is not observed, but yᵢₜ​​∈{1,…,K}, such that y*ᵢₜ​​∈C(yᵢₜ), is observed. Assume a normal linear model for log wage with heterogeneous coefficients conditional on a d×1 vector of covariates xᵢₜ​ :

y*ᵢₜ​=x′​ᵢₜβᵢ​+ϵᵢₜ​,ϵᵢₜ∼N(0,h−1),t=1,…,T,i=1,…,n.

Use a hierarchical prior for the heterogeneous coefficients: βᵢ∼N(β,H−1),i= 1,…,n

Derive conditionally conjugate priors for β,h, and H.

Answer 1

To derive conditionally conjugate priors for β, h, and H in the given panel data model, we assume a normal distribution prior for β, a gamma distribution prior for h, and an inverse Wishart distribution prior for H.

Step-by-step explanation:

To derive conditionally conjugate priors for β, h, and H, we consider the following:

• For β, we assume a normal distribution prior, so we have βᵢ ~ N(β, H⁻¹)
• For h, we assume a gamma distribution prior, so we have h⁻¹ ~ Γ(a, b)
• For H, we assume an inverse Wishart distribution prior with hyperparameters V and ν, so we have H⁻¹ ~ IW(V, ν)

In summary, the prior distributions for β, h, and H are:

βᵢ ~ N(β, H⁻¹)

h⁻¹ ~ Γ(a, b)

H⁻¹ ~ IW(V, ν)