Solved – How to write a logit and probit regression equation

categorical datalogitprobitstatsmodels

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!

Best Answer

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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