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In the Tobit model: Why do you divide by σ when deriving the probability that y = 0? Is it something to do with the assumption that ε is distributed N(0,σ^2) and not N(0,1)? If this is the case why does dividing by σ standardise it?

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You’re correct - Dividing by $\sigma$ here makes it so that $\epsilon/\sigma\sim N(0,1)$. So now $\phi$ is the cdf of a standard normal random variable. Had the division not be made, it would be a normal random variable with variance $\sigma^2$. Dividing by $\sigma$ standardizes it because if $X\sim N(0,\sigma^2), $ $$var(X/\sigma)=\frac{1}{\sigma^2}var(X)=\sigma^2/\sigma^2=1.$$

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