Is it reasonable to treat a three point Likert dependent variable as a continuous variable? I have a three point Likert scale question: 
How happy are you?
    1= low levels of happiness 
    2= medium levels 
    3= high levels 

I want to do multiple linear regression on the variable. I am making the assumption that  that it has the same difference between low and medium and medium and high. I want to treat it as a continuos dependent variable. 
I know some people would see this as inappropriate, and there is the age old issue  about whether to treat survey data as continuous.


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*But are there any further problems applicable to my dependent variable because it is only on a 3 point scale as opposed to 4 or 5? 

 A: What I like to do in situations like this is run ordinal logistic, multinomial logistic and linear regression and compare results.  I compare results by looking at predicted values from the different models. This is easy for multinomial vs. ordinal, because both yield probabilities. To compare with linear, I sometimes see what the highest probability response is from logistic, and how close it is to the predicted value from linear.
A: A bit on terms first. Your observable variable isn't continuous, it is descrete. Instead of "continuous" you should have better use word "scale" (or "metrical" or "interval"+"ratio"), as opposed to "ordinal". And you plan to treat the variable as scale rather than ordinal.
Formally logically you have the right to treat any Likert variable as scale variable if it has 3 or more levels. Unless you suspect that the intervals are psychometrically uneven. For example, in many countries they use 1 through 5 scale to assess performance at schools. This scale is not equiinterval because difference in knowledge appeares to be greater between 3 (satisfactory) and 4 (good) than between 4 and 5 (excellent). So, the school gauge is ordinal.
If you are right deciding to treat a Likert variable as metrical then the more it has levels the beter it is statistically. But if the variable is rather ordinal and has many levels it may be better to roughen it into less levels. One example is awful 10-point (1 to 10) scale frequently used in marketing to assess satisfaction. I usually recode it into 3-level "1 through 4", "5 through 8", "9 through 10".
A: The issue is not about "treating it as a continuous variable", but rather, as you point out, the assumption that 1v2 is the same as 2v3.
There's no real issue about the 3 categories (vs more).  If there were just two categories, it would be equivalent to a 0/1 binary variable.  All that would change (for different encodings) would be the scale of the coefficient.
A full model would contain two columns (say, indicators for medium and high). Using just the one column is perfectly reasonable but the assumption deserves to be checked.
