Is the PE ratio a good indicator of when to move money in and out of the stock market?

Great question - one could write an entire book answering it. The P/E ratio is a reasonable (I don't agree with the subjective term "good") indicator of whether a particular company is over or undervalued compared to its past performance. Remember that P/E is a ratio - you get the same P/E value if the stock price doubles and the earnings also double, so it only tells you so much about the financial health of a company. Generally, if the P/E of a company is higher than its typical (mean reverting) value, then more people are interested in the company stock (higher demand drives up price; earnings being constant this means P/E goes up). But it could also mean that earnings have decreased and the price is constant, which also results in a higher P/E ratio.

I find the SMA (simple moving average) time constants somewhat arbitrary, but Professor Shiller had to pick something. All it really says is that company prices oscillate over time, and on timescales that are unique to a particular company or industry. Crossings like these are more often than not good indicators. However, as to your direct question about whether you should use them to "get in or get out" of a particular market, that depends on your time horizon and risk tolerance. Since no one can predict the future, who's to say that a market dip will last one week, one year, or one decade or how large the dip is. Whether to sell and horde cash during a dip or "ride it out" is completely an individual decision based on risk tolerance and time horizon. For example, if you need a particular stock in order to pay your mortgage next month (a short time horizon where risks need to be kept small), then use a sell indicator and get out. On the other hand, if you have an investment that you don't need to touch for 20 years, then riding through the dip is a reasonable strategy.

Professor Shiller's analysis is sound, but I advise some skepticism in applying the results to future markets. Using data from the past (where a certain set of market dynamics and government policies were in place) may not apply to future markets. Think of how the 2017 tax change is going to affect P/E ratios. Are they going up (further from their mean values) because price will accelerate based on expectations of a better business climate, or will they go down because earnings will advance? Tough call. Could go either way.

No. P/E is a reverse looking indicator. It shows the past. There is something called a forward projected P/E - that may be better.

Here are examples where it totally failed:

AMD

Couple of years ago AMD was doing heavy losses AND investing in a new chip, heavily. P/E was horrific - years over years of losses. The new Chip - gave it a LOT of market share. P/R went positive pretty much immediately (within a quarder). NOW - AMD is 8 times the price, the chips are in the 5th generation and the new announced Ryzen 5 is beating Intel in pretty much every aspect.

P/E - looking ONLY backward - would totally have missed the significant changes that where in the making, heck, even when they where announced. It would even have missed them when the first generation Ryzen chips where FLYING OFF THE SHELVES, being sold out everywhere, because it takes a quarter for quarterly P/E to show up and a year for the year numbers to change.

Same way you can have a profitable business being hit by a catastrophy (i.e. a global pandemic and look at airline industries or cinemas) and the P/E ratio will trail so much it is not funny while the company struggles to survive, showing a good P/E until it FINALLY gets updated.

P/E is way too slow. Projected (forward P/E is more appropriate as it basically takes "assumed development over the next year" into account.