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I have the following linear equation:

Dummy dependent variable = dummy main independent variable + control variable 1, absolute value of changes (also between 0 and 1) + control variable 2, sigma (also between 0 and 1)

I want to know how to exactly write the logit and probit regression equations in a formal academic context.

Regarding the LHS, should this be written as logit(y)/probit(y), or P(y=1 given the RHS)?

If the main Linear Probability Model equation included a fixed effects term and an error term, should those be omitted for the logit and probit models? Thanks!

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Logit Linear Model:

$$\text{logit: }E(y) = \frac{e^{\beta_{0} + \beta_{\text{dummy main IV}}\text{dummy main IV} + \beta_{\text{CV1}}\text{CV1} + \beta_{\text{abs changes}}\text{abs changes} + \beta_{\text{CV2}}\text{CV2} + \beta_{\sigma}\sigma}}{1+e^{\beta_{0} + \beta_{\text{dummy main IV}}\text{dummy main IV} + \beta_{\text{CV1}}\text{CV1} + \beta_{\text{abs changes}}\text{abs changes} + \beta_{\text{CV2}}\text{CV2} + \beta_{\sigma}\sigma}}$$

Probit Linear Model:

$$\text{probit: }E(y) = \Phi\left(\beta_{0} + \beta_{\text{dummy main IV}}\text{dummy main IV} + \beta_{\text{CV1}}\text{CV1} + \beta_{\text{abs changes}}\text{abs changes} \\+ \beta_{\text{CV2}}\text{CV2} + \beta_{\sigma}\sigma\right)$$

BONUS Complementary Log-Log Linear Model:

$$\text{clog-log: }E(y) = 1 - e^{-e^{\beta_{0} + \beta_{\text{dummy main IV}}\text{dummy main IV} + \beta_{\text{CV1}}\text{CV1} + \beta_{\text{abs changes}}\text{abs changes} + \beta_{\text{CV2}}\text{CV2} + \beta_{\sigma}\sigma}}$$

Notes:

  1. Your dependent variable is named $y$.
  2. I am assuming that $\sigma$ is a variable that you are conditioning on.
  3. I am assuming you want an intercept term ($\beta_{0}$).
  4. $\Phi()$ is the probit function (i.e. the inverse normal CDF).
  5. The clog-log function is another binomial link function.

Please let me know if this serves for what you mean by "a formal academic context": if not I will try to edit to incorporate your feedback.

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    $\begingroup$ thank you so much for your reply! I have some questions please. Regarding the LFH, should this be written as logit(y) or P(y =1 given the RHS)? If the main Linear Probability Model equation included a fixed effects term and an error term, should those be omitted for the logit and probit models? Thanks! $\endgroup$ – David Oct 19 '17 at 15:59
  • $\begingroup$ @David Hey! I am not sure I follow you: what is LFH? What is RHS? You can certainly add something like $\text{logit}(y|\text{dummy main IV}, \text{CV1}, \text{abs changes}, \text{CV2}, \sigma)$. Finally, and with utter sincerity, what do you mean by "should?" $\endgroup$ – Alexis Oct 19 '17 at 21:02
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    $\begingroup$ What do you mean by "$\sigma$"? It must be a number, because the left hand side must refer to a parameter of a conditional distribution--neither the logit nor the probit is defined for $y=0$ or $y=1$! As a number, $\sigma$ is confounded with $\beta_0$. $\endgroup$ – whuber Oct 20 '17 at 21:28
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    $\begingroup$ Presumably $y$ is coded to take on just the values $0$ and $1$. Since neither the logit, probit, nor complementary log-log functions are defined for either of those values, something must be wrong with your formulation. I believe you are confusing $y$ with its expectation. $\endgroup$ – whuber Oct 23 '17 at 20:53
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    $\begingroup$ @whuber Edited to incorporate your feedback. $\endgroup$ – Alexis Oct 23 '17 at 21:51

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