Macroeconometrics 5: Introduction to ARMA Models (Graded HW Problems)
宏观计量经济学代写 Show that stated in growth rates this process is (Hint: take logs, and note that for any variable x, ln (ex) =x).
1.Consider the ARMA(1,1)process
yt = Ø1yt-1 + εt – θ1εt-1.
What conditions must “1 and $1 satisfy in order for this process to be invertible?
2.In this Lesson’s practice problems with solutions it was shown that given theprocessyt =α+ βt + εt, yt is not stationary. Show that the Örst dierence of yt, dyt ≡ yt yt–1, is indeed stationary. 宏观计量经济学代写
3.Consider theprocess 宏观计量经济学代写
yαt · yø t-1·eηt,
where εt is a random walk process such that
ηt = ηt-1 + εt,
where εt is white noise, and |ø| < 1.
(a)Show that stated in growth rates this process is (Hint: take logs, and note that for any variable x, ln (ex) =x).
4.Considerthe processes 宏观计量经济学代写
ln (yt) = αt + ø ln (yt–1)+ ηt
and
ηt = ηt-1 + εt
The Stata .dta file called “HW5AR1” contains data for the variable t, the parameter α, the parameter ø, and the white noise process εfor 1,000 observations. Create a Stata .do file called “HW5AR1control” to do the following.
(a)Load the Clear the memory and generate a variable called eta that generates theprocess
ηt = ηt-1 + εt
assuming that the Örst observation of this process is equal to the first observation of ε. Using these data, generate a variable called logy that generate the process
ln (yt) = εt + ø ln (yt–1)+ ηt
assuming that the Örst observation of this process is equal to the Örst observation of ε.
(b)Use the wntestq command to test η and assess whether it is white noise.
Write down your conclusions as comments in your Also, use the ac command to generate autocorrelation graphs for η and ln (yt). What do these graphs sug- gest about the stationarity of each of these variables? Please write down your conclusions as comments in yourcode.
(c)Generate a variable called dlogy that is equal to the growth rate of logy using theÖrst di§erence Use the ac command to generate an autocorrelation graph for dlogy and the pac command to generate a partial autocorrelations graph for dlogy. What do you infer about the stationarity of these data? Also, if you were to model these data as an ARMA process, what order might you begin from given the results from the graphs you generated? Please write down your answers as comments in your code. 宏观计量经济学代写
(d)Generate a variable called Clogy that is equal to the cyclical component oflogy obtained using an HP Ölter with smoothing parameter equal to Use the ac command to generate an autocorrelation graph for Clogy and the pac command to generate a partial autocorrelations graph for Clogy. What do you infer about the stationarity of these data? Also, if you were to model these data as an ARMA process, what order might you begin from given the results from the graphs you generated? Please write down your answers as comments in your code.
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