Playing around with inflation, part 2

Joe Tomlinson tipped me off to another way to model inflation and sent me the paper that explained it (The Role of Real Annuities and Indexed Bonds in an Individual Accounts Retirement Program [2001] Jeffrey R. Brown, Olivia S. Mitchell,and James M. Poterba).  Also, since the last post I realized I had made an error in the historical inflation model and I wanted to correct that here.  Since I'm goofing around on this and have so few readers, I'm hoping the error did not affect anything anyone does. Doubtful it did.



In the linked article, Brown et al. describe a simple AR(1) model for inflation that is easy enough to code. They describe it like this:

We consider two cases for the inflation process, corresponding to different assumptions about the degree of inflation persistence over time. The first case treats each annual inflation rate as an independent draw from our six-point distribution. This approach to modeling inflation tends to understate the long-run variance of the real value of fixed nominal payments and thus serves as a lower bound on the effect of inflation. Our empirical findings in the last section demonstrate clearly that inflation is a highly persistent process.

 

In the second case, we incorporate persistence by allowing inflation to follow a stylized AR(1) process. In the first period, inflation is drawn from the same six-point distribution as in the i.i.d. scenario. In later periods, however, there is a probability γ that π_t+1 will be equal to π_t and a probability 1- γ of taking a new draw from the six-point distribution. An attractive feature of this approach is that γ is equal to the AR(1) coefficient in a regression of inflation on its one-period lagged value, and thus can be parameterized using historical inflation data. Using U.S. historical data from the period 1926–97, the AR(1) coefficient for inflation is equal to 0.64, and this is the value of γ that we use in modeling a persistent inflation process.

With the difference here that I am sampling from the full historical distribution rather than from a six-point distribution.  I could change that at some point.  I borrowed, without reflection or evaluating changes in AR coefficients since 2001, their use of .64

I am less interested here in summary stats today than just seeing the shape of the outcomes.  The outcome, as before is defined as the end state of spending an initial 40k at t=30 periods.

First here is the density of spending comparison where red uses the non-persistent historical model and black uses the Brown model above.

This is the time series progression of the first 100 simulation iterations for spending over time using the historical non-persistent model. This corrects the last post which wildly overstated variability.

And this is the time series progression of the first 200 simulation iterations for spending over time using the historical AR(1) model. This is a little closer to what I generated (incorrectly) last time. Just for reference, 40k is in the 3rd percentile at t=30 and the median t=30 "spend divided by the initial 40k" is 2.48.