You have read about the so-called catch-up theory by economic historians, whereby nations that are further behind in per capita income grow faster subsequently. If this is true systematically, then eventually laggards will reach the leader

To put the theory to the test, you collect data on relative (to the United States) per capita income for two years, 1960 and 1990, for 24 OECD countries. You think of these countries as a population you want to describe, rather than a sample from which you want to infer behavior of a larger population. The relevant data for this question is as follows:

where X1 and X2 are per capita income relative to the United States in 1960 and 1990 respectively, and Y is the average annual growth rate in X over the 1960-1990 period. Numbers in the last row represent sums of the columns above.
(a) Calculate the variance and standard deviation of X1 and X2. For a catch-up effect to be present, what relationship must the two standard deviations show? Is this the case here?
(b) Calculate the correlation between Y and . What sign must the correlation coefficient have for there to be evidence of a catch-up effect? Explain.
What will be an ideal response?

Answer:
(a) The variances of X1 and X2 are 0.0520 and 0.0298 respectively, with standard deviations of 0.2279 and 0.1726. For the catch-up effect to be present, the standard deviation would have to shrink over time. This is the case here.
(b) The correlation coefficient is –0.88. It has to be negative for there to be evidence of a catch-up effect. If countries that were relatively ahead in the initial period and in terms of per capita income grow by relatively less over time, then eventually the laggards will catch-up.