calculate the equations please

Question 7. Point allocation: equal weights for each question. This question contains 6 short questions. I am looking for concise and specific answers and many do not even require that you use any math notation. Do not ramble, you do not have the time. (a) Briefly explain (in words) the concept of uniform convergence and its usefulness (i.e., in what way is it used in asymptotic theory). (b) In a regime-switching model with two states, say 1 and 2, you may remember that the auxiliary vector & is a 2 x 1 vector that is equal to the first column of the matrix , when the state is s, = 1, and is the second column of /2 when the state is s, = 2. This vector is used to formalize the conditional transition probabilities. Explain, in words, the differences and usefulness of each of the following entities: &; +1/; Eqr. Be sure to discuss how an empirical practitioner may use each quantity in practice. (c) What are the advantages and disadvantages of using the equal-weights matrix over the optimal weighting matrix in a minimum distance (e.g. GMM) type problem? (d) Consider the following state-space model yi+1 = Pyi + El+1 where E() = of; E(u;) = 02. Identify the state variable, the observed variable and hence the state equation and the observation equation. What is the mean-squared error of the one-period ahead forecast (i.e. P+1/: in the notation used in the Kalman filter) when there is no measurement error? Suppose you need to provide your best guess about the unobserved variable {yi)is to a policy maker (he may then use this information for some other analysis with other data). Assuming on * 0, what quantity (or quantities) from the Kalman filter should you provide him with (and why)? (e) Suppose you are interested in constructing a 95% simultaneous confidence region for the parameters 6 and 02. These parameters must meet the following restrictions: 0 > 01 > 1 and 62 > 0. Provide an algorithmic method based on simulation techniques to construct such a region. You may assume that the posterior density for each parameter individually is available and is easy to take draws from. (f) Briefly explain what is a particle filter, when and what is it used for, and give the intuition as to how it works.Question 6. Point allocation: (a) 10%; (b) 10%; (c) 20%; (d) 10%; (e) 10%; (f) 20%; (g) 20%. Consider the panel data model yit = Qu + Bra tun, i = 1, ..., N, t = 1, ..., T, where ua are independently distributed over i and t, Elualo,, zu] = 0, and Ta is a scalar regressor. If you need to make any additional assumptions in answering the following, state them clearly. (a) State how to obtain a consistent estimator of # using only an OLS program under the assumption that zy is uncorrelated with of. (b) State how to obtain a consistent estimator of & using only an OLS program under the assumption that zy is correlated with a;. (c) State in some detail how to determine whether or not zo is correlated with a;. (d) You are told that ry has small within variation and large between variation. What does this mean and how, if at all, will it effect your estimators in part (a) and part (b)? (e) Suppose zy is correlated with both a, and wy. State how to obtain a consistent estimator of B. (f) Suppose Fit = yi-1. State how to obtain a consistent and efficient estimator of B. (g) Suppose we write the above model in matrix notation as yi = ec; + 3x; + uj, where yi, x, and u, are T x 1 vectors with th entries ya, Ta and up, and e is an N x 1 vector of ones. Consider the estimator - [ExQx] EX Qy., where Q = (Ir - fee'). Obtain the variance of 3 under the assumption that wy is i.i.d. (0, a').Question 5. Point allocation: (a) 20%; (b) 20%; (c) 10%; (d) 10%; (e) 10%; (f) 30%. Consider the regression model yi = exp(X 0) + wi, where ( is a K x 1 vector and u, are independent over i with Eu;(x;] = 0 and Vu,(x] = of. Consider the estimator / that minimizes Q() = _ _(3 - exp (x, 3))?. (a) Give the Newton-Raphson iterations for computing the estimator, with the simplification that the expected value of the Hessian is used in place of the Hessian. (b) Give a formula to compute a consistent estimate of the variance matrix of A. (c) Provide a direct interpretation of the coefficient , in this model. (d) Give a complete formula that allows computation of the average marginal effect in this model of a change in an indicator regressor variable using the finite-difference method. (e) Define the residual to be u, = y; - exp(x,). Will ), u, = 0? Explain. (f) Suppose that Eux;] # 0. Instead there exist K + 2 variables z; that satisfy Eu; | z;] = 0. Explain in detail how to calculate a consistent and efficient estimator of B.Question 4. Point Allocation: (a) 20%, (b) 20%, (c) 20%, (d) 20%, (e) 20%. Let , be an iid sequence with &, ~ N (0, 1) and let u, be an iid sequence with Eu, = 0 and Eu, = 1. Assume that yr = at, + with a|