ENH: Have inferred_freq set offset. Have freq always try to infer.
I'm on the fence about this. Having a property with side effects doesn't quite sit right to me. But I think it kinda makes sense here. If it didn't set `freq`, then it really doesn't make sense for `freq` to exist. I'd just use `inferred_freq` all of the time.

Now calling `freq` always tries to infer the frequency. Why not? Seems sensible to me. `inferred_freq` is kind of redundant now, but I left it anyway. I was never really sure of the rationale for its existence.

AFAICT this was not done for 2 reasons:
- performance, unless you are doing a time-series operation that needs to freq no need to 'compute' it (if it is not already specified)
- their is not an easy way to represent 'no freq' except by `None`. So you can have a freq even though it really shouldn't (I have seen this when you have 5 mon-fri days, it get's inferred to a business day freq), and thus the inferernce _can_ be a tad ambiguous (see above)

maybe @wesm has a different view

another possible reason (and the reason the tests fail) is that merge/join doesn't want this and can lead to spurios errors, e.g. joining a single day that happens to be a month-end to a 'daily' frequency, is allowed but if you already have frequency assigned then it will fail.
 I see your points, though the join thing seems more like a corner case implementation detail.

Ok, so pretty much I _should_ just always used `inferred_freq` in statsmodels and take the possible (one-time) performance hit. The reason for this is that `freq` is only defined if it's given when the Index is created, but even then you can't count on `freq` being defined because `freq` gets stepped on when you're resetting indices, etc. #6631.

This just seems so esoteric from a user standpoint. I.e., `freq` == `user_defined_freq` unless it's been stepped on, and `infer_freq` == always gives me something if it can. It seems like 98% of the use-cases want the latter. I'm biased to data that almost always has a sane frequency.
 I think the reset case is really the issue. Say you reset, then set the index, freq is lost (and must be re-inferred), because the meta-data is not 'kept' anywhere. I suppose you could have a flag in DatetimeIndex that would 'force' inference to happen. Might make sense for statsmodels.

Actually the reset/set case _could_ be dealt with by caching the indexes on a reset/set. hmm.
 Right now I'm just leaning towards always using `infer_freq` (lazily) and the status quo. All the dates magic we can do requires some kind of frequency (forecast, backcasting, re-sampling, etc.). Thinking about it, dates really aren't that useful to have around without frequencies (from where I sit). I just have to remember, because it's not all that intuitive. If `infer_freq` is your performance bottleneck, then you should likely be closer to the metal anyway. Plus the vast majority of the time users load in data rather than creating a DatetimeIndex, passing in a freq. It's best if users don't have to think about frequencies and periodicities at all.
 I don't think you will have an issue as long as you don't reset/set (or pay the penality if you do). The more I think about it the more that's the reason it doesn't do this automatically. reset/set are cheap operations and you want them to stay that way.
