Can a variable that contains 'don't know' as a response be ordinal?
For example, say the variable corresponds to responses to the question 'Do you agree with ...?' where the responses are:
The first four answers would form an ordinal variable but adding 'don't know' seems to violate this. Would the variable now be categorical?
"Don't know"s are necessarily problematic here. If "Don't know" was explicitly presented as an intermediate category in the questionnaire, there is a case for treating it that way.
That would be, in my experience, highly unusual; and it doesn't appear to have been the case here.
I'd argue that the only defensible actions otherwise are
to regard "Don't know"s as missing and to exclude them from the model fitting
to retreat to a multinomial model that doesn't regard the response or outcome as ordinal (but don't be surprised if the "don't know"s don't fit easily into any model).
(I'd assert that "categorical" (wide sense) as including ordinal is more common in statistically flavoured literature than "categorical" (very narrow sense) as meaning nominal only. It is, I think, uncontroversial to underline that narrow and wide senses are both encountered.)
You could put don't know intermediate:
But I suggest making "Don't know" = missing and leave the other 4 as ordinal. This would be a stronger recommendation if you already had "neither agree nor disagree".
Also note that your scale is tilted toward agree; this may be a bad idea or a good one.
If anything "don't know" could be in the middle:
Here's a similar Likert scale discussion.
Also, as @PeterFlom noted, you have more "agree" answers than disagree. Likert himself recommended using equal number of right and left answers. The paper is rather fascinating, btw, I suggest reading it in full.
