Consider the Vasicek model of interest rates. Download daily data on the one-month T-bill rate from the Federal Reserve web site and use these data to estimate γ, r and σ. This estimation can be accomplished through a linear regression. Take literally dt = 1/252 = 1 day and "discretize" the process into

Where ɛt+dt ~ N (0, σ2dt). This looks similar to a regression
(rt+dt -rt) = α + βrt + ɛt + dt
(a) What are α and β in terms of the original parameters 7 and r? How can you estimate σ? Be careful with the "annualization" (your data are daily, but we need to be "annualized). Also, recall that rt in the Vasicek model is a continuously compounded rate.
(b) Given your parameter estimates and the current rate today, perform a forecast of the future interest rate as of today. Extend your forecast up to 5 years in the future.
(c) Use simulations to perform a scenario analysis about the future possible interest rates. Compute the histogram of the interest rates in 1, 3, and 5 years from now and compare it to your forecasted value.

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(rt+dt - ri) = 7(F-r)dt + €+dt