# How to calculate $\text{cov}(\hat{Y}_{ij}, \hat{Y}_{kj})$ if $Y_{ij} = \mu + a_i + b_j + e_{ij}$?

Let's assume we have the model following Two-Factor model without replications : $$Y_{ij} = \mu + a_i + b_j + e_{ij}, \; i=1,\dots,p \; \text{and} \; j=1,\dots, q$$ I am interested in calculating the covariance : $$\text{cov}(\hat{Y}_{ij}, \hat{Y}_{kj}), \quad i \neq k$$ I know that the estimators can be written as : $$\hat{Y}_{ij} = \overline{Y}_{i\cdot} + \overline{Y}_{\cdot j} - \overline{Y}_{\cdot \cdot} \quad\text{and}\quad \hat{Y}_{ik} = \overline{Y}_{i\cdot} + \overline{Y}_{\cdot k} - \overline{Y}_{\cdot \cdot}$$ That would make the covariance expression : $$\text{cov}(\hat{Y}_{ij}, \hat{Y}_{kj}) = \text{cov}(\overline{Y}_{i\cdot} + \overline{Y}_{\cdot j} - \overline{Y}_{\cdot \cdot}, \overline{Y}_{i\cdot} + \overline{Y}_{\cdot k} - \overline{Y}_{\cdot \cdot})$$ I know that I can break this covariance up in combinations, but I am having trouble calculating each one. Except from the term $$\text{cov}(\overline{Y}_{\cdot \cdot}, \overline{Y}_{\cdot \cdot}) = \sigma^2/pq$$, I seem to struggle to find the rest of them. Any tips on the calculations ?