代考宏观计量经济学 Above: the βs and &s are parameters; the ρ parameters are numbers strictly between zero and 1; and ηt is white noise.
Problem 1 (25 points total)
Suppose you are interested in running the following regres- sion:
yt = β0 + β1xt + εt;
εt = ρ0 + ρ1εt–1 + ηt,
Above: the βs and &s are parameters; the ρ parameters are numbers strictly between zero and 1; and ηt is white noise. Derive mathematically E (εt). Please show your work.
Problem 2 (25 points total). 代考宏观计量经济学
Continuing with problem 1, denote the variance of η by Then, please address the following two parts.
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. 代考宏观计量经济学
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 Örst 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 problematic issues for the purposes of estimating the transformed equation and getting at the parameters of interest? Please show your work in detail.
Problem 3 (25 points total). 代考宏观计量经济学
Let 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.
Problem 4 (25 points total). Consider the following regressions:
yt = 1.2yt–1 + εt
xt = 1.3xt–1 – 0.4xt–2 + t, 代考宏观计量经济学
where the error terms ε 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. 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.