宏观计量经济学代考 Above: the βs and ρs are parameters; the & parameters are numbers strictly between zeroand 1; and ηt is white noise. Derive mathematically E (εt).
Problem 1 (25 points total) Suppose you are interested in running the following regres- sion:
yt = β0 + β1xt + βt，
βt = &0 + &1βt–1 + ηt,
Above: the βs and ρs are parameters; the & parameters are numbers strictly between zeroand 1; and ηt is white noise. Derive mathematically E (εt). Please show your work. Hints:
- Use the factthat
ρt = ρ0 + ρ1εt–1 + ηt
implies that 宏观计量经济学代考
εt-1 = ρ0 + ρ1εt-2 + ηt-1,
which implies that
εt-2 = ρ0 + ρ1εt-3 + εt-2,
and so on and so forth, in order to implement iterative backwards substitution.
- Section 2 of the Lesson 1 lecture notes and Section 3 of the Lecture 3 notes are the main references to get at the solution.
Problem 2 (25 points total). Continuing with problem 1, denote the variance of η byσ2η. Then, please address the following two parts. 宏观计量经济学代考
Part a (12.5 points). Suppose that &1 =0 but all other assumptions from problem 1 remain unchanged. What properties of white noise does “ satisfy, if any? Please show your work. Hints:
- Themain reference is Section 3 of the Lesson 3 lecture notes.
- Developthe work necessary to, given the process ρt = ρ0 + ηt, arrive at this processí mean, variance and kth order autocovariance, and compare these moments to the moments that a process needs to satisfy in order to be white noise.
Part b (12.5 points). Continuing with part (a), suppose that, as in part (a), in reality &1 = 0. However, you incorrectly assume that the equation for ” is instead exactly as in problem 1 meaning that, in particular, &1 (0; 1). Given this assumption, you decide to use the transformation strategy to try to solve the assumed (yet non-existent) issue of first order serial correlation for the purposes of running the regression noted in problem 1, that is,
yt = β0 + β1xt + εt.
Does this (ultimately incorrect) use of the transformation strategy generate any problematicissues for the purposes of estimating the transformed equation and getting at the parameters of interest? Please show your work in detail. Hint: work out the transformed equation and focus on the error term implied by the transformed equation.
Problem 3 (25 points total). 宏观计量经济学代考
Let β^ be a k × 1 vector of OLS estimates that result from running a regression of the vector y on the matrix X: Show that for the k × 1 vector β of true underlying parameters we can write the sum of squared residuals, denoted SSR (β), as
While we have generally worked with time subscripts in this class, for simplicity they are omitted in this case. Above, ^” is the vector of estimated residuals, and X is the matrix of regressors. Please show your work in detail. Hint: Use the fact that: 宏观计量经济学代考
and by the first-order conditions for OLS = 0. More generally, the Lesson 1 lecture materials are the main reference for this problem.
Problem 4 (25 points total). Consider the following regressions:
yt = 1.2yt-1 + εt
xt = 1.3xt−1 – 0.4xt–2 +ζt,
where the error terms ” and – are white noise. Assess whether the following statement as a whole is completely true, completely false, entirely uncertain, or partly true: “Given the regressions above, it follows trivially that the equation for y is explosive because the slope parameter on the first lag of this variable is greater than 1. 宏观计量经济学代考
Similarly, because in the equation for x the slope parameter on the first lag of this variable is greater than 1, then the process governing the behavior of x is explosive as well.” Please justify your answer mathematically and in detail. An answer without mathematical justification earns zero points. Hint: the key here is the material on homogeneous equations covered in Lesson 2. What you need to do is get at characteristic equations and assess the implications of the roots.