What is the distribution of gold prices? In particular what is the distribution of the daily percent changes of gold price?
The first question you may ask is why this question should interest you?
Well, if you have ever used basic or advanced statistical analysis on gold prices, including calculating averages, standard deviations, linear correlations and regressions, then it should interest you. Why? Because all these statistical tools are based on the premise that gold price daily percent changes has normal distribution.
By the way, we usually use daily percent changes over the actual gold price, because it’s more interesting to figure out the changes around the trend and not the trend itself, which, as seen in the chart below (gold average price between 1998 and 2011), was upward in recent years.
You may know that usually it’s better to consider a very large sample over a small sample. For one thing, when using a large sample it’s possible to apply the Central Limit Theorem – basically it means, under certain conditions, and given a large enough sample, the mean distribution for a given variable will be approximately normally distributed. This premise is paramount in making all the linear regressions, correlations etc.
So where is the problem? As usual, in the fine print… as stated above under certain conditions… What are the conditions? They include a finite mean and variance. But what if gold price (daily percent changes) doesn’t have a mean or a variance – what if the gold price (daily percent changes) is a Cauchy distribution? This distribution has “fatter” tails than the Normal distribution does and doesn’t have any mean, variance or any higher moments.
This means that even given a large sample, the distribution of the mean of daily percent changes of gold price won’t act like a Normal distribution (because there isn’t one), and the premise on which much of the statistical analysis is built on was faultily.
So the question can be boiled down to: is the gold price (daily percent change) distribution a Normal distribution or a Cauchy distribution?
I hope I have persuades you the importance of this issue, so let’s check it out:
Let’s first see the distribution of daily percent gold prices between 1998 and 2011 (over 3000 samples):
The chart shows the there are outliers outside the bulk of most of the samples, mainly in recent years.
Let’s see the Normal distribution (0,1) with 3,000 samples (this means the mean is 0 and variance is 1):
As see above the distribution is much more contained with very few to none outliers (I saw only one sample in the -3.5 mark)
Finally, let see the Cauchy distribution (0,1) with 3,000 samples (this means the mode/median is 0 and scale (the half-width at half-maximum/ i.e. the scatter of the distribution) is 1):
As seen above, the distribution is spread out with many outliers.
Up to now, it seems that gold price (daily percent change) distribution is a hybrid between the Cauchy distribution and the Normal distribution.
Now let’s examine the averages of these three distributions:
Normal distribution (0,1)
The chart shows the changes in the averages as the sample size gets bigger: e.g. for a small sample of the first 50 data points, the average is -0.06, for a sample of a 1,000 data points the average is 0.02, and for a sample of a 3,000 data points the average is nearly 0. This shows that the Normal distribution’s mean converges very rapidly to the distribution’s expected value of 0.
Cauchy distribution (0,1)
The chart shows there is no one mean for this distribution; e.g. for a small sample of the first 50 data points, the average is -0.4, for a sample of a 1,000 data points the average is 0, and for a sample of a 3,000 data points the average is nearly -0.8. This shows that the Cauchy distribution doesn’t have a mean because the average doesn’t converge to a certain value.
Gold price daily percent changes
The chart shows that the average of the gold price (daily percent changes) fluctuates and doesn’t converges in the same sharp manner as the Normal distribution’s mean did, but it doesn’t scatter as the average of the Cauchy distribution did.
So what is the bottom line?
I wish I could give you a clear cut answer, but there isn’t one, at least in this case. It does show that the premise of gold prices (daily percent change) acting like a Normal distribution is shaky; furthermore, it seems that gold price has some similarity to the Cauchy distribution as well as to the Normal distribution.
Therefore, it doesn’t mean we should necessarily throw out the window all the statistical analysis on gold price, but we should take it with a grain of salt and any forecasts that use regressions, correlations, averages, etc. should be taken very skeptically.
For further reading:
Lior Cohen, M.A. commodities analyst and blogger at Trading NRG.