January theory Statistics Question

Question

According to the "January theory", if the stock market is up at the end of January, it will be "up" for the year. If it is "down" at the end of January, it will be "down" for the year. Within the last 34 years, this theory proved to be true for 29 years. A different theory is that the market change at the end of January and the market change at the end of year are unrelated. Specifically, for any market change in January, the probability that the market is "up" or "down" at the end of year is equally likely - that is, the probability is 0.5.

a) Based on history, what is the probability that a year will end with an "up" market when January ends with an "up" market?

b) If the January market change and year-end market change are unrelated, the probability that the market is "up" with an "up" January is 0.5. Using 0.5, what is the probability that the market would be 29 or more years?

c) what would be the mean number of years that the market is "up"?

d) Based on the result in b) and c), what is your conclsion regarding the "January theory"?

finished down? finished the same? or finishes up?

Answer 1

The most accurate way to test this would be to use the binomial distribution and its conjugate prior, the beta distribution. With the null hypothesis that p(s) [where s represents the stock market being 'up' at the end of the year] is equal to .5. the chances that we would see this distribution (29 out of 34 ) would be 1.12 x 10 ^-5 . (math not shown as calculating the inverse beta distribution is non-trivial. There is software out there for this )

HOWEVER, chances are you are looking for the simplified normal approximation for testing the mean. Let. n = 34, s = 29, p (or p_a) is approximately estimated as 29/34 or .8529. q (or 1-p) is 5/34 or .1471. (note: p₀ = .5)

Variance = pq. or .1254

Standard deviation (sd or σ)= .3542

standard error (σ_mean) = σ/√34 = .0607

A) what is the chance (based on history) that in a year with the market 'up' at the end of January will be up at the end of the year. Answer - rho or p. (.8529)

B).what is the chance that 29 out of 34 years would be up if rho (p) is actually .5?

to answer this we need to compare means. (.5 with .8529) using a normal approximation.

Calculating the Z_test = (.8529-.5000)/.0607 [ (p_a - p₀)/σ_mean] = 5.8138

When using the normal approximation we use a standardized normal distribution or Z distribution to determine the answer by comparing the calculated Z value to the Z distribution.

Using the normal distribution (calculator, table, on-line calculator) we find that the p value for the hypothesis test (i.e. that P(Z) >= 5.8138. [or P(ρ) >= .8529] is equal to 3.06 x 10 ^-9

C) Assuming the question is what is the mean number of years the market is up assuming p = .5 and a time frame of 34. This would be 34 x .5 or 17years

D) based on the data given, the January theory would be accepted as true as the chances of seeing the observed results if Ho were true is <= .00000000309

The chances of a random market year starting with January up but finishing down at the end of the year would would be .1471, The chances of a random market year finishing up would be .8529, the chances for being the same are not calculable with given information. [unstated assumptions or definitions about what is meant by ending the same must be defined first].

note: as with all answers, the results depend on underlying assumptions. For example, if in question D we assume that what was being asked was : What are the chances , assuming a binary distribution with a p value of .8529, that a 34 year period would have a) less than 29 up years up, exactly 29 years up, and more than 29 years up the answers would be .3839, .1900, .4261. This is not the same as the assumptions made in the answer above. So, beware of your assumptions.