Math 5075, Spring 2019
homework | report代做 | math- 这道题目是math相关的代写任务, 涉及了report等代写方面, 这是值得参考的homework代写的题目
Version: Tuesday 15
January, 2019, 22:
Due: Thursday, Jan 31, 2019
- Letwtbe iid N(0,^2 ) random variables, i.e. a white noise. Define a new seriesXtby
Xt=. 2 wt 2 . 3 wt 1 +wt+. 7 wt+1+. 4 wt+3.
(a) Compute the autocovariance and autocorrelation functions ofXt. (b) Can you find a different weighted average of thewtthat has the same ACFs?
- Letwtbe iid N(0,^2 ) random variables, and define a new seriesXtby
Xt=^12 Xt 1 +wt.
Iterate this equation to writeXtas an infinite linear combination ofwt,wt 1 ,wt 2 ,.... Use that thewtare iid normal and the fact that linear combinations of normals are normal to show thatXtis normal. Compute its mean and variance. Do these quantities depend ont?
- In the last question, compute the covariance Cov(Xt,Xs), using the representation ofXtas a linear combination of thewt. Based on your answer, isXta weakly stationary series?
- A time series with a periodic component can be constructed from
Xt=U 1 sin(2 0 t) +U 2 cos(2 0 t).
HereU 1 andU 2 are independent random variables that are mean zero and have variance^2. The value 0 is a non-random constant, it determines the period of the sine and cosine. Show thatXtis weakly stationary and compute its autocovariance function.
- LetXtbe a stationary Gaussian process (i.e. the whole series is jointly Gaussian) with mean and autocovariance function(h). That is,
E[Xt] =, Cov(Xt,Xt+h) =(h).
Define a new time seriesYtbyYt=eXt.
(a) IsYta weakly stationary time series? (b) ComputeE[Yt]. (c) Compute Cov(Yt,Yt+h) =Y(h).
- For this question you will need to use R. Before you begin please issue the commandset.seed(6222016). This will make sure everyones output is the same so its easier for me to grade. Letwtbe iid N(0,1) random variables. Recall you can simulate them using the commandw <- rnorm(150). Simulate 150 observations from the process
Xt=. 5 Xt 1 . 4 Xt 2 +wt,
You can use theRcommandarima.sim(n=150, list(ar=c(.5,-.4)), n.start=50)to do this. Its a good idea to look at the help file for thearima.sim()function and understand all the parameters as well use it a lot. Then.start=50part of the command burns in the series; that is it produces an arbitrary starting pointX 0 , lets the series run for 50 iterations, and only then starts recording the next 150 observations and outputs them to you. This makes the output more stationary, meaning that it essentially forgets about the choice of X 0.
(a) Plot the observations and compute the sample ACF. (b) Now suppose were given the observations youve plotted, we strongly believe that its an AR process, and we want to estimate the coefficients of the process. That is, from the data we want to see if we can determine the.5, the.4, and the fact that theXdepends (explicitly) only on the values at the two previous times. Thus use the method of least squares (inRyou use the functionlm()) to find estimators for the unknown parameters , 1 , 2 , 3 in each of the models
Xt=+ 1 Xt 1 +wt, Xt=+ 1 Xt 1 + 2 Xt 2 +wt, Xt=+ 1 Xt 1 + 2 Xt 2 + 3 Xt 3 +wt.
Thelm()function will providep-values for the hypothesis test that each of the three given parameters are zero. On the first one, for example, if youre time series data is stored in an array calledmysimyou can use the R code
mylm <- lm(mysim[2:150] mysim[1:149]) summary.lm(mylm)
to get the fitted values ofand 1 and thep-value. Find those values for each of the three models and report back on them. Based on their values, which of the three models seems to be the best fit? Does it match the true model that you used to generate the data?