How to write a logit and probit regression equation? 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! 
 A: 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:


*

*Your dependent variable is named $y$.

*I am assuming that $\sigma$ is a variable that you are conditioning on.

*I am assuming you want an intercept term ($\beta_{0}$).

*$\Phi()$ is the probit function (i.e. the inverse normal CDF).

*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.
