Sampling Data Gauss file(s) basic_sample.g Matlab file(s) basic_sample.m This exercise reproduces the simulation results in Figures 1.1 and 1.2. For each model, simulate T = 5 draws of yt and plot the corresponding distribution at each point in time. Where applicable the explanatory variable in these exercises is xt = {0, 1, 2, 3, 4} and wt are draws from a uniform distribution on the unit circle.

(a) Time invariant model yt = 2zt , zt ∼ iid N(0, 1).

(b) Count model f (y; 2) = 2 y exp[−2] y! , y = 1, 2, · · · .

(c) Linear regression model yt = 3xt + 2zt , zt ∼ iid N(0, 1).

(d) Exponential regression model f(y; θ) = 1 µt exp  − y µt  , µt = 1 + 2xt .

(e) Autoregressive model yt = 0.8yt−1 + 2zt , zt ∼ iid N(0, 1).

(f) Bilinear time series model yt = 0.8yt−1 + 0.4yt−1ut−1 + 2zt , zt ∼ iid N(0, 1). (g) Autoregressive model with heteroskedasticity yt = 0.8yt−1 + σtzt , zt ∼ iid N(0, 1) σ 2 t = 0.8 + 0.8wt . (h) The ARCH regression model yt = 3xt + ut ut = σtzt σ 2 t = 4 + 0.9u 2 t−1 zt ∼ iid N(0, 1).