the same mean and correlation structure, but different volatilities.13 Most of the time returns are associated with a low-volatility regime, but every so often volatilities spike up and returns are drawn from a high-volatility regime. Variances in the high-volatility regime are a constant multiple of the variances in the low-volatility regime. In this case, the likelihood function measures the probability of our data being generated by a mixture of normal distributions. In addition to the decay rate, we now need to estimate the ratio between the volatilities in the two regimes, and the probability of being in one of the two volatility regimes. Using our sample, we find that with a mixture of normal distributions the optimal decay rate on a monthly basis is equal to 9 percent, which corresponds to a half-life of 7.3 months. We also find that the ratio between volatilities in the high and low regimes is equal to 3.23, and that the probability of being in the low-volatility regime is equal to 84 percent. One may ask whether the difference in decay rates between the likelihood that assumes normality (10 percent) and the likelihood that assumes a mixture of normal distributions (9 percent) is actually meaningful or, to use a more technical term, statistically significant. Econometricians use a simple technique to answer this question. They measure the likelihood function in the more general case (mixture of normal distributions in our exercise) and in the restricted case (normal distribution in our exercise) and then ask whether the change in the value of the likelihood function is sufficiently large to claim that the difference in the estimated parameters is significant from a statistical point of view. The intuition behind this procedure is relatively simple. The model with a normal distribution is obviously a special case of the model with a mixture of normal distributions. In fact, if the data were generated by a single volatility regime, then the estimated parameters when using a mixture of normal distributions would indicate that the ratio between volatilities in the two regimes is one and that the probability of being in the low-volatility regime is one. In other words, the likelihood functions in the two different specifications would coincide. However, if the model with two regimes is a better description of the data generating process, then the value of the likelihood function associated with it will be higher. In our case, the difference between the two likelihood functions leads to a strong rejection of the hypothesis that the data are generated by a normal distribution with a single volatility regime. Do Volatilities and Correlations Move at a Different Speed? Although there is a widespread consensus among academics and practitioners that volatilities and correlations change over time, opinions are less uniform when looking at the speed at which volatilities and correlations change through time. More specifically, volatility displays interesting regularities: First, it changes rather quickly in response to market shocks; second, it occurs in clusters so that periods of high (low) volatility tend to be followed by more periods of high (low) 13Clearly, these assumptions can be relaxed to accommodate even richer scenarios. We leave those extensions to future research.