Using OLS over Logit regression I was hoping someone could clarify this for me.
I have used OLS regressions using the PISA dataset where most of my outcome variables are categorical in nature and measured on 4 point Likert scales, ranging from "Strongly disagree to strongly agree" with observations spread across all answer categories.  with the exception of one variable that was just 'yes' or 'no'. My independent variables have been recoded as dummy variables including 'gender' and others such as a school being urban or rural.
Would I be correct to justify OLS as my outcome variables are categorical in nature, non-extreme in the value they take i.e. not just 0 or 1 and independent variables have been recoded as dummies, therefore, OLS can be used over logistic regressions and easily interpretable. All responses in PISA are scaled using item response theory. There has been no issue with my model also.
 A: If your outcome variable is a 4 point Likert scale you definitely can't use a binary logit model, which requires that the outcome variable has only two values: zero and one.
You could, however use am ordered logit model which is a generalization of the binary logit model and was designed to analyze just these sorts of outcome variables. Instead of predicting the likelihood that Y=1, the ordered logit model predicts the likelihood of moving up "one point" on the ordinal scale.
However, even an ordered logit model might not be appropriate if (for example) the factors that are associated moving from "not at all" to "a little" are different from the factors associated with moving from "somewhat" to "very much" (this is called the parallels regression assumption and can be tested with a Brant test).
If an ologit model was appropriate but instead you tried to use an OLS model on these data you will probably get answers that are "ok" (in terms of sign and significance) even though ordinal variables violate any number of OLS assumptions. But the estimates of the effect size may be very wrong.
