in Theorem \[thm: predictive mean and variance\].

\[cor: cokriging and UK\] Let ${\mathbf{x}}_0$ be a new input in the domain $\mathcal{X}$. Let $\hat{y}_{t}^{K}({\mathbf{x}}_0):=E\{y_t({\mathbf{x}}_0) \mid {\mathbf{y}}_t, {\boldsymbol \phi}_t\}$ and $\hat{v}_t^{K}({\mathbf{x}}_0): = Var\{ y_t({\mathbf{x}}_0) \mid {\mathbf{y}}_t, {\boldsymbol \phi}_t\}$ be the kriging predictor and kriging variance based on data $\{{\mathbf{y}}_{t}, {{\cal X}}_{t}\}$. If ${\mathbf{x}}_0 \in {{\cal X}}_{t} \setminus {{\cal X}}_{t+1}$ and with $t=1, \ldots, s-1$, we have $\hat{y}_{\ell}({\mathbf{x}}_0) = \hat{y}_{\ell}^K({\mathbf{x}}_0)=y_t({\mathbf{x}}_0)$ and $\hat{v}_{\ell}({\mathbf{x}}_0) = \hat{v}_{\ell}^K({\mathbf{x}}_0) = 0$ for $\ell = 1, \ldots, t$. If ${\mathbf{x}}_0\notin {{\cal X}}_t$, we have $\hat{v}_t({\mathbf{x}}_0)>\hat{v}_t^K({\mathbf{x}}_0)$.

[Corollary]{} \[cor: cokriging and UK\] indicates that cokriging predictors can be interpolators as a kriging predictor. If ${\mathbf{x}}_0$ belongs to the design $\mathcal{X}_s$ in the highest fidelity code, the resulting predictive variances at ${\mathbf{x}}_0$ across all levels are zeros, i.e., $\hat{v}_t=0$ for $t=1, \ldots, s$. In other words, the cokriging predictors in  are interpolators at all levels. When prediction is made at new inputs, the cokriging predictor has larger variance than simply using the one associated with the kriging predictor. The extra uncertainty in cokriging variances comes from the uncertainty from previous level and the uncertainty to estimate the scale discrepancy parameter. However, without data coming from the lower code levels, one cannot estimate 