Back in the 1980s, I talked to an economics professor who made forecasts for a large bank based on simple correlations like the one in Figure 1. If he wanted to forecast consumer spending, he made a scatter plot of income and spending and used a transparent ruler to draw a line that seemed to fit the data. If the scatter looked like Figure 1, then when income went up, he predicted that spending would go up. The problem with his simple scatter plots is that the world is not simple. Income affects spending, but so does wealth. What if this professor happened to draw his scatter plot using data from a historical period in which income rose (increasing spending) but the stock market crashed (reducing spending) and the wealth effect was more powerful than the income effect, so that spending declined, as in Figure 2? The professor’s scatter plot of spending and income will indicate that an increase in income reduces spending. Then, when he tries to forecast spending for a period when income and wealth both increase, his prediction of a decline in spending will be disastrously wrong. Multiple regression to the rescue. Multiple regression models have multiple explanatory variables. For example, a model of consumer spending might be: C = a + bY + cW where C is consumer spending, Y is household income, and W is wealth. The order in which the explanatory variables are listed does not matter. What does matter is which variables are included in the model and which are left out. A large part of the art of regression analysis is choosing explanatory variables that are important and ignoring those that are unimportant. The coefficient b measures the effect on spending of an increase in income, holding wealth constant, and c measures the effect on spending of an increase in wealth, holding income constant. The math for estimating these coefficients is complicated but the principle is simple: choose the estimates that give the best predictions of consumer spending for the data used to estimate the model. In Chapter 4, we saw that spurious correlations can appear when we compare variables like spending, income, and wealth that all tend to increase over time.
Clustering Scatter Plots Using Data Depth Measures