Estimating Drift and Diffusion Functions Gauss file(s) npr_chapman.g Matlab file(s) npr_chapman.m The true data generating process of the interest rate is given by the SDE dr = α(µ − r)dt + σr1/2 dW , where dW ∼ N(0, dt) and {α, µ, σ} are parameters.

(a) Generate a sample of daily observations of size T = 2500, by simulating the following model of the interest rate rt rt − rt−1 = α(µ − rt−1)∆ + σr 1/2 t−1∆1/2 et , where ∆ = 1/250, et ∼ N(0, 1) and the parameters are α = 0.21459 µ = 0.08571 σ = 0.07830 .

(b) Plot the following following series and discuss their properties y1,t = rt − rt−1 ∆ , y2,t = (rt − rt−1) 2 ∆ .

(c) Let the dependent and independent variables be respectively yt = rt − rt−1 ∆ xt = rt−1 . Estimate the conditional mean using the Nadaraya-Watson nonparametric estimator with a Gaussian kernel and a bandwidth of h = 0.023 over the grid of values x = r = {0.000, 0.002, 0.004, · · · , 0.200}. Compare the estimated values with the true value, α(µ − r).

(d) Let the dependent and independent variables be respectively yt = (rt − rt−1) 2 ∆ xt = rt−1 . Estimate the conditional mean using the Nadaraya-Watson nonparametric estimator with the same specification as for (c). Compare the estimated values with the true value, σ 2 r.