Analyze omitted-variable bias in regression

Suppose the true data-generating process for building energy use is Y = β0 + β1·Temp + β2·Occupancy + ε with E[ε|Temp,Occupancy]=0 and Var(ε|Temp,Occupancy)=σ²(Temp), i.e., heteroscedastic. You mistakenly fit OLS on Y ~ Temp only. Derive the expected bias of the OLS estimator for β1 in terms of Cov(Temp,Occupancy) and Var(Temp), state the sign of the bias when hotter days are more occupied, and explain how weighted least squares or robust SEs affect estimation vs inference here. Finally, design a test to detect remaining heteroscedasticity (name the test and null) and propose a practical diagnostic you would include in a production training report.