Analyze omitted-variable bias in regression

Assume the actual data-generating mechanism for building energy consumption is $Y = \beta_0 + \beta_1 \cdot Temp + \beta_2 \cdot Occupancy + \varepsilon$, where $E[\varepsilon \mid Temp, Occupancy] = 0$ and $Var(\varepsilon \mid Temp, Occupancy) = \sigma^2(Temp)$, so the errors are heteroscedastic.

You mistakenly fit OLS with $Y$ regressed on $Temp$ only.

Derive the expected bias of the OLS estimator for $\beta_1$ as a function of $\mathrm{Cov}(Temp, Occupancy)$ and $\mathrm{Var}(Temp)$, give the sign of that bias when warmer days tend to have higher occupancy, and explain how weighted least squares or robust standard errors affect estimation versus inference in this setting.

Finally, specify a test for detecting leftover heteroscedasticity—name the test and its null hypothesis—and suggest a practical diagnostic to include in a production training report.

Overview: This question assesses understanding of omitted-variable bias and heteroscedasticity in linear regression, testing how omitting a relevant covariate and having non-constant error variance affect OLS point estimation and inference.