ONE MAJOR THEORY OF THE STOCK MARKET FOCUSES ON THE ANAYLSIS OF PRICE
TIME CYCLES. IF AN INVESTOR IS INVESTING REGULARLY IN A EQUITY INDEX
FUND,
WHAT WOULD BE THE BEST DAY OF THE MONTH TO INVEST ON A MONTHLY BASIS?
IN OTHER WORDS WHAT IS, BASED ON DAILY CLOSING PRICE, ON AVERAGE THE
DAY OF THE MONTH THAT HAS THE LOWEST STOCK INDEX PRICE?

The comments already gave you two 'tips', so for the real money, you
deserve
 a real answer, describing the state of the art.

  Answer is: There is no 'best date'. All days of the month give you
equal chance to loose
 or win.

  More detailed explanation:

 Many people disagree, which is not surprising since there is a  whole
industry making
money by selling 'best strategies'.  Not all of the opponents have
vested interest
 (a profit motive) and not all are wrong.
  However,  the question you posed asks about a 'rigid cycle' - which
would be periodic.
  The cycles which are being studies are more complex. Excellent book
on
  the topic is: Business Cycles: From John Law to Chaos Theory
  http://www.amazon.com/exec/obidos/ASIN/9057020661/mindconnection/002-3409150-0998410
      If a simple rigid cycle would exist, if there would be a 'best
day in the
 month' to buy and to sell, that knowledge would spread. People would
start
  buying and selling on those days, end the effect would disappear. 
That's the "Efficient Market Theory" applied to you case.
   The market is not allays efficient. It does take time for knowledge
to
  spread and for market to 'correct' the madness of the crowd. But any
such
  inefficiency or secret knowledge, which may still exist, would be
more complex then day
 of the month.

Additional Links:

  The major concepts or theories are listed here:
 http://www.investopedia.com/university/concepts/concepts6.asp
  The "Efficient Market Theory" is applicable to your question.  It's
consequence is the
'random walk theory', described here:
 http://www.stockcharts.com/education/What/Overview/randomWalk.html
 which has been tested empirical:
 http://www.sacbee.com/content/opinion/story/2448032p-2892473c.html


  Search Strategy:

   Random Walk
   Efficient Market
   Business Cycles

   Hope this help. And, BTW, this may an excellent time to invest for
the 'long term'
   as we may be 'anytime soon' be emerging from 'irrational despair'
phase of the
   bussines cycle. Good luck and do not take this to be an advice to
buy. I am not
   a professional securities analyst and am only describing the
current state of knowledge.

   hedgie

If you look into the Stock Trader's Almanac, there is
a lot of information on such cycles.

There is monthly cycles, seasonal cycles, etc.  Stocks tend to raise
in the end and beginning of months, and tend to be lowest in the 3rd
week of the month.

Just from my observations, I'd say the lowest day is typically the 3rd
tuesday of each month, although it varies.

I can tell you that based on the past 40 years of the S&P 500 that the
best day to invest is the 1st day of the month. This day has
traditionally given a positive return 57.3% of the time. Compare that
with the 23rd day of the month, which has given a positive return only
44.2% of the time in the past 40 years. (Derived from Businessweek.com
market data)

However, CAUTION! It is a well taught story in Business Schools that
such rules are dangerous. For the following reasons:

(1) past results are no indicator of future results. Suppose you had
based your decision for investing based on the period 1960-1989. The
best day to invest according to that time period was the 15th of the
month, when you would see a positive return on your money 57.8% of the
time. However had you applied this rule to the 1990s and 2000s, so far
you would have made a positive return only 53.0% of the time (the 1st
of the month was far better over this period, offering   positive
returns 65.2% of the time). So even if you find a handy rule of thumb
that worked it the past, there is no scientific reason why it should
work in the future.

(2) If an anomaly exists, other people in the market will more than
likely beat you at it. Occasionally, over a short period of time,
there are anomalies in the stock market (like the internet boom -
hindsight is a wonderful thing!). But unless you spend all of your
time researching financial markets (and think about what you could be
doing with the time you are researching - you could be relaxing at
home for one) you are more than likely to be beaten to the post by
large and experienced investors. By the time that most run-of-the-mill
investors had figured out that internet stocks were the place to be,
most of the gains had already been swallowed up... and by the time
that they realised that a bust was on it's way, they had lost most of
their winnings anyway.

Don't get me wrong, you can make money on the stock market - even a
little bit of research can help a lot. But unless you plan on making a
career of it, your best bet is to pick a day of the week and just go
with it, and don't worry too much about whether it is the right one.
What matters far more, is the type of fund you invest in and how long
you stay in the fund.

Good luck!

Steve