Treating ordinal variables as continuous for regression problems In the social sciences I have encountered that it is common to treat ordinal variables as continuous, for example variables originating from rating or Likert scales (strongly disagree, disagree, agree, strongly agree).
This topic has been discussed for example in this post from 2010:
Under what conditions should Likert scales be used as ordinal or interval data?
I am looking for a more formal comparison/evaluation especially in the context of regression modeling. Rhemtulla et al. (2012) examine the performance of treating ordinal variables as continuous and make recommendations for structural equation models (SEM). I am not very familiar with SEMs, so I'm not sure if their results would also apply to regression problems.
Does anyone know about similar studies/literature in the context of regression?
Edit:
Just to answer the question below: 
I'm mainly interested in the case where the outcome variable is ordinal (with possibly an ordinal covariate).
 A: It's important to distinguish, as pointed out by Nick Cox, between iV and dV. As far as dV is concerned, why not use a ordinal regression model, as discussed excellently e.g. by Agresti: http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470082895.html 
I am less sure about the iV case. Standard would perhaps use dummy coding. I suppose this is what Frank Harrell means. Maybe Agresti discusses this as well.
A: With the luxury of time we would use dummy variables as with nominal predictors, then penalize them (penalized MLE) towards ordinality of effects.  Something like that was discussed in a paper by Hans van Houwelingen some years ago.  Short of that, we often approximate the effect of ordinal variables by fitting a quadratic effect.  It would also not be ridiculous to use AIC to select between a regular nominal dummy variables model and a restricted model that assumed the ordinal predictor was continuous (like the quadratic).
I'm not sure that the SEM results would apply, but they might.
A: I have one source, Snijders and Bosker`s (2012) multilevel analysis book, page 310, saying:
"if the number of categories is small (3 or 4), or if it is between 5 and 10, and the distribution cannot well be approximated by a normal distribution, then statistical methods for ordered categorical outcomes can be useful"
My understanding is, if you have at least 10 categories and approximately normally distributed dependent variable, it is safe to treat it as a continuous variable. For a more concrete answer, I would run a small scale simulation analysis.
A: Liddell & Kruschke (2018) is another source which discusses problems associated with treating ordinal data as continuous.  The paper illustrates a number of the problems that can occur.  
They advocate using ordered-probit models to deal with ordinal data.  While they specifically advocate for a Bayesian approach, they note that Frequentist approaches may also work
A: If an outcome is ordinal, one should want a method of analysis that is invariant to the codes used to label the levels. For example, suppose the outcome has levels : SD, D, N, A and SA. Then one might label the levels with codes 1, 2, 3, 4, 5. If one analyzes this outcome with a t-test, the p-value is invariant only to location or scale changes. For example, -2, -1, 0, 1, 2 or -4, -2, 0, 2, 4. The p-value from the t-test is not invariant to other codings like -10, -1, 0, 1, 100 or any coding that does not preserve 'distance'. The proportional odds model and the multinomial model give p-values that are invariant to the coding selection. [maybe this point has been made earlier?]
