PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

194 MODELING FINANCIAL SCENARIOSabove the mean reversion level, the first term is negative. Therefore,Formula (3.1) predicts that the expected change in inflationwill be negative; that is, inflation is expected to fall. The secondterm on the right-hand side of Formula (3.1) represents the uncertaintyin the process. The change in Brownian motion (dB t )can be likened to a draw from a standardized normal randomvariable (represented by " q in the discrete form of the model).The uncertainty is scaled by the parameter ¾ q , which affects themagnitude of the volatility associated with the inflation process.We can rearrange the last equation above to show that theOrstein-Uhlenbeck process is a continuous time version of a firstorderautoregressive process:q t+1 ¡ ¹ q = ¹ q·q¢t ¡ ¹ q +(1¡ ·q¢t) ¢ q t + " q ¾ qp¢t=(1¡ ·q¢t) ¢ q t ¡ (1 ¡ ·q¢t) ¢ ¹ q + " q ¾ qp¢t=(1¡ ·q¢t) ¢ (q t ¡ ¹ q )+" q ¾ qp¢t: (3.3)Using the last equation in (3.2), we can estimate the parametersof the inflation model using the following time-series regression:q t+1 = ® + ¯q t + " 0 qt : (3.4)Note that we have not run the regression using the changein inflation as the dependent variable since this would not allowus to simultaneously derive the mean reversion speed (·q)and the mean reversion level (¹ q ). To derive the parameters ofthe inflation process, we transform the regression coefficients in(3.4):¯ =1¡ ·q¢t, ·q = 1 ¡ ¯(3.5)¢t® = ·q¹ q ¢t = 1 ¡ ¯ ¹¢t q ¢t, ¹ q = ®1 ¡ ¯ : (3.6)We gathered inflation data from the Consumer Price Index(CPI) data collected by the Bureau of Labor statistics