that proper priors when chosen to be vague lead to improper posterior in [@Qian2008]. Another choice is to use conjugate priors for parameters $\{ {\boldsymbol \beta}, {\boldsymbol \gamma}, {\boldsymbol \sigma}^2\}$ and leave the prior for ${\boldsymbol \phi}$ unspecified and work with a concentrated restricted likelihood; see [@Gratiet2013]. We will show that this concentrated restricted likelihood is not decreasing to zero and can be maximized either at zero or infinity; see Appendix \[app: Gratiet\] for detailed discussions. This problem has been studied in great details in [@Gu2018] for Gaussian processes. The conclusions in [@Gu2018] can be potentially generalized for autoregressive cokriging models. To overcome these problems, objective priors of the form  are developed and they are shown to yield proper posteriors. Although this article is not focused on robust estimation, the independent reference prior can lead to robust estimation with parameterization given in [@Gu2018].

Integrated Likelihood {#sec: integrated likelihood}
---------------------

The integration of the marginal likelihood  and the prior  with respect to the prior over $({\boldsymbol \beta}, {\boldsymbol \sigma}^2)$ yields $$\begin{aligned}
\begin{split}
\int L({{\mathbf{y}}}\mid{\boldsymbol \beta},{\boldsymbol \gamma},{\boldsymbol \sigma}^{2},{\boldsymbol \phi})\pi({\boldsymbol \beta},{\boldsymbol \gamma},{\boldsymbol \sigma}^{2},{\boldsymbol \phi})\,d({\boldsymbol \beta},{\boldsymbol \gamma},{\boldsymbol \sigma}^{2})
= L^I({\boldsymbol \phi}\mid {\mathbf{y}}) \pi({\boldsymbol \phi}),
\end{split}\end{aligned}$$ with $$\begin{aligned}
 \label{eqn: integrated likelihood}
\begin{split}
    L^I({\boldsymbol \phi}\mid {\mathbf{y}})     