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Setup: Consider a random sample of size n with binary outcome $Y_i\in\{0,1\}$. Assume $Y_i\sim Bern(\pi_i)$. Use a linear probability model so that $\pi_i=X_i^\intercal\beta$, where $X_i$ is a predictor vector of length P. Here consider $\beta$ by maximum likelihood.

Question: Show that the error term from the linear regression of $Y_i$ on $X_i$ is always heteroskedastic. If you were to use OLS for this model, would this problem be corrected?

Comment: The error term is heteroskedastic if it depends on $X_i$, but I'm not sure what approach I should use to get a formula for it.

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Heteroskedasticity means: the variance of the error term depends on $X$. In our case $$ Var[\varepsilon_i] = Var[Y_i] = \pi_i(1 - \pi_i) = X_i^T\beta(1 - X_i^T\beta). $$ So we have dependence of $Var[\varepsilon_i] $ on $X_i$... In general, memorize definitions before attempting anything.

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  • $\begingroup$ Does $var(e_i)=var(Y_I)$ because $Y_i=f(X_i^\intercal\beta)+\epsilon_i$ for this model basically? But the $f$ isn't obviously derived in this case. $\endgroup$ – Stacker Jan 21 at 4:39
  • $\begingroup$ You said: "linear model". Therefore, $f(x) = x$. But that is not very important. $Var[\varepsilon_i] = Var[Y_i]$ because $X_i$ is assumed to be deterministic. If $X_i$ is random, we should be talking about $Var[\varepsilon_i | X_i] = Var[Y_i | X_i]$. $\endgroup$ – stans Jan 21 at 4:52

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