My twitter feed all day has been filled with Harold Pollack and others arguing about Markov processes and the Obamacare data showing that most of the people signing up on the exchanges already had insurance of some kind. Some people seem to think that because most of the people signing up for the Obamacare exchanges already had insurance, Obamacare must not be working to increase the percent of the population that's insured. Pollack responds, correctly, that it's insane to jump to such conclusions.
To a certain extent, we're just witnessing an example of the prosecutor's fallacy--people are looking at the probability of an Obamacare enrollee was previously insured as evidence that it isn't working, without taking into account the prior probability of any individual, enrolling or not, having insurance. As Pollack points out nearly 80 percent of the population is previously insured, so the fact that most of the Obamacare enrollees are insured is, in fact, nothing out of the ordinary.
Pollack's actual rebuttal is a bit more mathematical, however:
If reporters would explain that individual coverage status is a continuous-time Markov process we would have a much better discussion.
I don't know why he jumps to continuous time here--in my experience almost any continuous-time model can be turned into a substantially more tractible discrete-time model (and by the way, data reports aren't generated in real time, so discrete time really is more realistic). So, let's illustrate Pollack's point with a discrete-time markov process. It's discrete time, so we will talk in terms of "periods" of time, which may correspond to weeks or months or years or whatever. In each period, each individual exists in exactly one of three possible states: they either have employer sponsored health insurance (employer), are enrolled in an obamacare exchange (exchange), or else they have no insurance (uninsured). In reality, other possibilities exist, but they don't really change anything here so lets stick to those for simplicity. Now, let's specify a transition matrix:
Period t State
Period t+1 State
Employer
Exchange
Uninsured
Employer
0.8
0.8
0.1
Exchange
0.2
0.1
0.1
Uninsured
0
0.1
0.8
The transition matrix specifies the probability that an individual in one particular state (shown in columns) will transition into a particular state in the next period (shown in rows). So the upper left cell entry has 0.8 meaning that 80 percent of those who currently have employer sponsored insurance will continue to have employer sponsored insurance next period. Any transition matrix is valid so long as it's entries are probabilities (between 1 and 0), and the columns each sum to 1 (all individuals must go somewhere next period). The numbers I have specified are just hypothetical of course, picked because they are at least somewhat realistic and capture the gist of what's going on in terms of "previously insured" people enrolling in the exchanges. The matrix is actually sufficient to compute the long run, but for illustration purposes I will plug in some starting data and simulate the transition period. Prior to the creation of the exchanges, roughly 80 percent of the population had insurance of some kind, mostly through their employer, so lets assume in our hypothetical that 80 percent start with employer-sponsored insurance while 20 percent start off uninsured. Then the government creates the exchanges, we get the transition matrix above, and this happens: The Markov process at work. It turns out that, with our calibration, almost all of the enrollment into the exchanges in the first two periods was by people who had previously had employer-sponsored insurance, much like our actual experience with the Obamacare exchanges. Yet, in our scenario we still see a long-run increase in the number of people with insurance on the order of 10 percentage points--roughly what the CBO projected Obamacare to do in reality.
So the bottom line is that Pollack is absolutely right: the high percentage of "previously insureds" isn't an aberration to be worried about--it's not only consistent with Obamacare's goal to increase insurance coverage, it's exactly what everyone should have expected in the first place.
