A recent report from Cover the Insured.org reviews the how health insurance trends have evolved over the last ten to fifteen years. The percent of uninsured individuals has increased from 16.0% of the population in 1995 to 17.5% of the population in 2006. The increase is entirely due to the increase in uninsurance among working aged adults (i.e., aged 18-64). Among children, uninsurance rates dropped from 13.7% of children in 1995 to 11.7% of of children in 2006. SCHIP expansions likely played a large role in the increase in childhood insurance coverage.
Why are more and more working-aged adults left without insurance? The reason is that health insurance costs are increasing faster than income. Let us look at the following table:
Between 1996 and 2006, overall health insurance premiums for employer-provided plans increased by 4.87% for single coverage and 5.98% for family coverage. These are average annual increases above secular inflation (i.e., CPI). However, median real income increased by only 0.76% per year between 1994 and 2006. Is there really a 5% gap in between income and health insurance premium growth?
The answer is yes and no. There is still a large gap between income and health insurance premium growth, but the gap is not 5%. Using salary as the only measure of workers compensation does not take into account the fact employer contributions towards health insurance plans has increased over time. While salary has increased only by 0.76%, employer health insurance contributions have increased by 4.62% (single) and 6.03% (family) per year. After taking into account employer health plan contributions, we see that the true increase in real worker compensation ranges between 1.01% and 2.02% per year.
Even after taking into account the increased employer contributions, we still see that real health insurance premiums have outpaced real labor compensation by about 3.9%. In order to decrease the number of uninsured, we need to bring down the cost of health insurance. This means either increased cost-sharing or enacting more limitations on medical services provided. If we do not want to increase cost-sharing or limit care coverage, premiums will continue to increase. Whether these premiums are paid by individuals, the employer or the government, they represent a real cost to society that needs to be spent efficiently.
- At the Brink: Trends in America’s Uninsured, March 2009, RWJ Foundation.
The History of Least Squares
March 23, 2009 in Books, Econometrics | 1 comment
Let us say you have 10 observations of 2 different variables. How do you determine which of the observations to use? Should you throw out the outliers? Should you only include the most similar values? Does more observations increase or decrease the amount of measurement error?
These problems can be answered by the discipline of Statistics. An interesting book by Stigler recounts The History of Statistics. Astronomers lead many of the statistical advances in the seventeenth and eighteenth centuries. Accurate measurement is very important to astronomers. Further, observations with respect to the circumference and oblateness of the earth were made at different times and places throughout history. This leaves a conundrum of how best to combine these observations.
Mayer, Boscovich, and others contributed to the development of the idea of least squares, but Stigler credits Legendre with the invention of least squares. Legendre came up with the idea in his attempt to measure the length of the median quadrant (the distance from the equator to the North Pole) through Paris.
To demonstrate some of his ideas, I will use a simpler example. Let us assume that a drug can have a dosage level between 0 and 5 and we want to find it’s impact on health (measured from a 0-10 scale). Let us look at the following data. The goal is to find the parameters m (slope) and b (intercept) that accurately measure the relationship between drug dosage and health (ignore any questions of endogeneity). Should we include all 10 observations?
Although Euler recognized that including more observations increases the maximum possible error, Legendre realized that adding more observations also greatly increased the probability of getting close to the true value of the parameters of interest.
In my example, we need to fit a line to measure the parameters m and b. How do we set up the errors so that we have the most accurate calculations. Laplace believed that the following two conditions would need to hold:
The first condition basically says that the errors are uncorrelated with the independent variables on average. The second condition hopes to minimize the errors. Legendre extended Laplace’s second condition to minimize the sum of the squared errors rather than just the absolute error level.
Another key point is that this regression line must go through the “center of gravity.” In my example, the average dosage for the ten observations is 2.2 and the average health level is 5.9. This means the center of gravity is at the coordinates (2.2, 5.9). In the solution in my example is to set m=1.1456 and b=3.3797. We see that if we plug 2.2 into the equation, the output is 5.9; thus, the regression line does indeed go through the center of gravity.
Understanding the historical development of modern statistical techniques is an interesting task, and Stigler’s book enlightens the reader with much detail.
Tags: Books, Econometrics, Least Squares, Statistics