Consider the zero-mean and stationary N 1 vector yt that contains N economic variables and another zero-mean and stationary 2 1 vector 2t that contains 2 climatic variables, where N and N2 are very large. Assuming that climatic variables in yt are driven by ft, an n1 x 1 vector of climatic factors, and assuming that economic variables in yt are driven by both f, andg, where gt is a n2 x 1 vector of economic factors. We only observe yt and y2t, t = 1, ..., T, whereas f and g are not observed. Furthermore, Ni >> nj, i, j = 1 or 2. We use a dynamic factor model to study how the climate affects economic movements. The model can be written as = Y1t Aggt+Toft +Tift-1 +2ft-2 + , ~N(0, g), Y2t = Afft + {, &{~ N(0,), for economic variables: for climatic variables: for economic factors: It = 19t1 + 291-2+ni, ni ~ N(0, 0), (5) (6) (7) for climatic factors: ft = 1t1+2t-2+ 3 ft3+4ft4 + ni, n{ ~ N(0, N). (8) - Based on your understanding of the model structure, answer questions Q5.1 Q5.5, using what have learnt in the subject. 1. (1 marks) Interpret Equation (5), by discussing how climate affects the economy in this model. 2. (22 marks) Put the model expressed in (5)-(8) in state space form, pointing out which one is the observation equation and which one is the state equation. Hint: there are only two equations in a state space model. [3. (2 marks) Suppose turns out to be a diagonal matrix. This means shocks driving the climatic factors are independent of each other, and thus can be considered as structural shocks. Based on the model structure, explain how a unit shock in n{t (i.e., the first climatic shock in n) affects economic variables in yt 4. (1+1 marks) Propose a multi-step estimation procedure for estimating Ag, Af, o, F1, F2, 1, 2, 1, 2, 3, 4, g, f, g, and f. Hint: Doz et al. two-step procedure introduced in Week 11. 5. (1 mark) After estimating all parameters in the model, we can use the Kalman filter to get E(ft Y11Y21 Yt, Y2t) or the Kalman smoother to get E(f|y11, 921, YT, Y2T), for all t = 1,..., T. How can we make use of these two quantities? .