S&P and Equity Correlation


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So, I've read multiple times that the S&P 500 (and other indices) have a close correlation to the stocks it consists of. This makes sense, and it's pretty easy to tell just by comparing some charts. But I've never really looked to deeply into this relationship. So now I have a couple of questions.

First, which way does the correlation respond? Does an index (like the S&P) affect the underlying stocks, or vice-versa? Like, when the market has a good day, is this because there are some general factors pushing all the stocks up at once (so as to say its because the market is bullish) or is it that most of the stocks are just doing good that day, so the market is doing good.

Second, is how important is it to follow these indices? I guess this answer is directly related to the question above, but this is where I'm confused kind of. If you can find reversal patterns in an index like the S&P, then does that mean that the S&P has an affect on the other stocks? And if so, how 'strong' is this affect. Like, I know you're suppose to only go long in an equity only if the market is bullish, but how important is this factor?

It could be possible that I have all my information wrong, but this is what I kind of picked up. And if this is true, then allow me to lay out this example:
The S&P 500 is currently oversold according to Williams %R, and if a reversal pattern shows up, then the market could turn bearish. So, assuming this is true, should I only be looking to go short on individual stocks?

I don't know. It doesn't seem too complicated, but its obviously an important aspect of trading, so I want to make sure I have it down pat.
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