Macroeconometrics 2:
Difference Equations and Filtering of Time Series Data
代写宏观计量经济学 Relate your answer to problems (5) and (6) to the stability conditions for nth order difference equations highlighted in the lecture notes.
1.Consider the version of Samuelsonís (1939) classic Keynesian model discussed in this Lesson’s lecture notes:
yt = ct+ it; (1)
ct = αyt-1 + εct; (2)
and
it = β (ct – ct-1) +εit. (3)
Find a reduced-form equation for investment.
2.Consider again the framework in problem (1). Is the reduced-form equation in the solutions unique?
*These lecture notes closely and sometimes literally follow sections from: Becketti, Sean, Introduction to Time Series Analysis Using Stata, Stata Press, 2013; Dowling, Edward T. Introduction to Mathematical Economics. Shaum’s Outlines, 3rd ed., McGraw Hill, 2001; Enders, Walter, Applied Econometric Time Series, 4th ed., Wiley, 2015; and Hamilton, James D., Time Series Analysis, Princeton University Press,1994.
3.Consider again the framework in problem (1). Find a reduced-form equation for GDP. 代写宏观计量经济学
4.Consider the difference equation yt = a0 + a1yt-1 with the initial condition y0. Solving this equation by iterating backwards, the solution is:
In contrast, adding this equation’s homogeneous and particular solutions yields as a solution:
Show that these two solutions are identical for |a1| < 1.
5.Find the homogeneous solution to the following di§erence equation:
yt = a0 + 1.5yt-1 – 0.5yt-2 + εt.
6.Continuing with problem (5), how would you characterize the dynamic behavior of this time series? How can you tell?
7.Relate your answer to problems (5) and (6) to the stability conditions for nth order difference equations highlighted in the lecture notes.
8.Consider the equation 代写宏观计量经济学
yt = 1.5yt-1-0.5yt-2 + εt.
Set up a Stata .do file that generates a time path for y based on this difference equation and satisfying the following. There are 1,000 observations for y, and the first observa-tion occurs in period 0 and is equal to 10 as is the observation in period 1. All values of ε (the Greek letter “epsilon”) are equal to 0, except ε100, ε200, ε300, ε400, ε500, v600,ε700, ε800, and ε900, all of which are equal to 1. Plot the resulting time series for 1 and describe how visual inspection of this time series suggests that the time series has a unit root.
(a) A Stata code that would solve this problem is the Stata .do file called “Problem8” that is available for download on Blackboard. The resulting graph is:
Note that when a shock hits, i.e., when ε ≠ 0, the time series adjusts per its functional form, and as the shock dissipates the time series converges to some value. The time series stays at that value until another shock hits and the time series eventually converges to some other value. This means that the time series is neither explosive nor convergent: a shock to the time series has a permanent impact on the level of the time series. 代写宏观计量经济学
(An explosive time series is one that when shocked starts diverging forever from the value it took prior to the shock and never converges to a new value. A convergent time series is one that when shocked eventually returns to the value it took prior to the shock. A time series that has a unit root is one that when shocked diverges forever from the value it took prior to a shock but eventually converges steadily to some new value).
