Choosing the number of regimes

How many regimes are enough?

Should the market have two regimes, three, four? You cannot decide by the fit alone. A model with more states always fits the data at least as well, because it has more freedom, and with enough states it would fit noise. We need a score that rewards a good fit but charges a price for each extra parameter. The two common ones are $\mathrm{AIC}=-2\ln L+2p$ and $\mathrm{BIC}=-2\ln L+p\ln n$, where $L$ is the likelihood of the data, $p$ the number of free parameters and $n$ the number of observations. Lower is better. BIC charges more per parameter when $n$ is large, so it prefers simpler models.

For a Gaussian HMM with $K$ states, the free parameters are the $K(K-1)$ free transition probabilities, the $K-1$ free start probabilities and a mean and a volatility for each state, which gives $p=K^{2}+2K-1$. Fitted to the $5{,}404$ daily returns of SPY, the log-likelihood is $16{,}252$ for one state, $17{,}521$ for two, $17{,}803$ for three and $17{,}877$ for four. The gain falls fast: $1{,}269$, then $282$, then only $75$. The BIC is lowest for four states, but its improvement over three is only $72$, and the extra state is hard to interpret. Interpretability, stability on new data and the size of the gain matter as much as the score.

• More states always fit better, so the fit alone cannot choose the number of regimes.
• AIC and BIC subtract a price per parameter; BIC charges $\ln n$ per parameter and so prefers simpler models.
• Look at the size of the gain and at interpretability, not only at the lowest score.

Lab · what's graded

• Parameters of three states10 pts
• Parameters of two states10 pts
• A BIC15 pts
• An AIC15 pts
• The gain from a fourth state10 pts
• Why not just the best fit10 pts
• Match the vocabulary15 pts
• Why keep three regimes15 pts