I have gathered together 9 years of data for the 30-Year Treasury Yield Index and the Phila. XAU Index. Starting in August 1994 and ending in July 2003 I have recorded the data for the first trading day of each month. The correlation analysis appears in Table 1. The analysis of the 2 variables shows a correlation of .68. This means that about 46% of what goes into XAU differences can be accounted for, or predicted by knowing the 30-year bond index value. Assuming that the factors that produced this relationship remain constant over subsequent years (a large assumption), we have quite a bit of leverage in predicting XAU values.
It was Francis Galton, during his studies on the inheritance of genius, who first worked out the method of statistical correlation (even though Karl Pearson worked out the present mathematical formulation) and the attendant concept of regression to the mean. He demonstrated that when 2 variables have a zero correlation the first variable provides no leverage on predicting the second one. Thus shoe size should give us no leverage in predicting income. In such a situation the most accurate behavior would be to predict average income for each and every person in our sample regardless of their shoe size. Galton & Pearson demonstrated that there is no income value or set of values that produce as little overall error (although it is a large amount of error) as guessing the mean income for everyone. This is the ultimate regression to the mean!
Pearson's Regression formula expressed in standard deviation units
Zy = rxy zx
shows that if the correlation is zero, the predicted value y, from all x values, will be zero. That is, zero deviation units from the mean which is the mean value in standard deviation units.
Regardless of shoe size the best prediction for everyone's income is the mean income. As we find variables that correlate higher (e.g., approach 1) there is less regression to the mean. The points gather tighter and tighter around the regression line (line of best fit). So if we could find a variable that had a perfect correlation with the measure that we wish to predict there would be no error of prediction and no regression towards the mean. In the current example the points tend to gather around a line from the lower left to upper right of the diagram. If the relationship was perfect, all of the points would fall exactly on the straight line. Predicting XAU from 30-year bond yields would be perfect. Each bond yield value would correspond with one, and only one, XAU value right on the regression line.
Real world relationships tend to be much less than perfect. In our case, the correlation of .68 gives us slightly less than half of a reduction in the magnitude of errors of estimation, from that which we would obtain if we had no useful information allowing us to predict XAU (i.e., the case where we would just guess the mean value of the XAU distribution). The magnitude of 30 year Treasury bond yields and the XAU do move in tandem across the 9 year period. According to this relationship, if we see bond yields heading up then we can make a bet of moderate confidence that we can expect to see the XAU on the rise also.
From the beginning of June 2003 to the beginning of August 2003 the 30-year Treasury bond yield index rose from 45.66 to 52.48. As expected the XAU started to rise (from 78.45 to 86.44). According to Pearson's formulation 52.48 represents -.962 standard deviations below the mean of the yield index distribution [(52.48-60.38)/8.21]. If we regress this value to get the "best" prediction of XAU we arrive at a value of 65.36 [(.68*-.962) * 28.22 + 83.82]. Based on the historical trend of 9 years worth of monthly data, the actual XAU is moving upwards at a greater rate than the regression equation would lead us to expect. Based upon the 9 years of data the odds of getting an XAU value of 86.44 or more, 1.42 standard deviation units above the trend line [(86.44-65.36)/14.89], would be <8 in 100.
Although we don't want to dwell too much on a single data point, this is a rather rare event. It might indicate that the XAU, if indeed still related to 30-year Treasury bond yields, is moving ahead of those yields. Whether the yields will catch up, or the XAU will settle back, cannot be forecasted by these data alone. Remember also that correlation data do not imply causality. All they tell us is that one set of data tends to consistently move in the same or opposite direction to the other one. We do not know if the bond yields directly drive XAU values or they are both driven by other factors. If, as some suspect, the falling dollar puts further upward pressure on bond yields, this trend might just continue.
Harry J. Clawar Ph.D.
Hjc@angelfire.com
August 12, 2003