Time Series
A First Course with Bootstrap Starter
Exercise
时间序列代做 Exercise E:(i) Simulate a stretch of 100 data points from an AR(2) model with phi_1=0.5, phi_2=0.2 and standard normal errors;
Exercise E
(i) Simulate a stretch of 100 data points from an AR(2) model with phi_1=0.5, phi_2=0.2 and standard normal errors; give the time series plot.
(ii) Forget that it was you who generated the data, and use the 100 data points to estimate the acf and pacf for lags up to 20. Can you confirm that the pacf is negligible for lags bigger than two?
(iii) How would you fit an AR(1) model to your dataset, i.e., how would you estimate the single phi coefficient of an AR(1) model?
Exercise G 时间序列代做
Suppose we want to fit an AR(p) model to the Wolfer sunspots data as in Exercise D.
(i) Use the empirical rule of p. 335 to identify the order p.
(ii) Use the R function ar (choose option Yule-Walker) to fit AR(r) models to the data for r=1,…, 12. For each r, record the estimated variance of the white noise driving the AR model; denote it by sigma^2_r.
(iii) For each r, compute the AIC = n log (sigma^2_r) + 2r (see Eq. (10.9.2).
(iv) Plot AIC as a function of r=1,…, 12. Find the r that minimizes the AIC. Is this different from your answer to part (i)?
Exercise H 时间序列代做
(i) Use the R function arma to fit all ARMA(p,q) models for p=0,1,2 and q=0,1,2, i.e., 9 models, to the Wolfer sunspots data. Create a table that shows the estimated variance of the white noise driving the ARMA model for all combinations of p and q.
(ii) Use AIC to select the best ARMA model in this case.
Exercise I
Let X_t be the GARCH(p,q) model of Definition 11.4.1. Let p=1=q and show that X_t satisfies an ARCH(infinity) equation. Identify the ARCH(infinity) coefficients.
Exercise J 时间序列代做
Let X_t be the ARCH(p) process of Example 11.1.6. Let G_t denote the information set: (X_s for s < t).
(i) Show that X_t has mean zero, i.e., EX_t=0. [Hint: first show E(X_t|G_t) =0 and use the law of iterated expectations.]
(ii) Show that X_t is a weakly stationary white noise, and compute its variance.
(iii) Define Y_t=X_t^2, i.e., the square of X_t. Find the optimal predictor of Y_t given G_t, denote it by \hat Y_t, and show that it is linear in the variables (Y_s for s < t).
(iv) Let U_t= Y_t – \hat Y_t. Show that U_t is a mean zero, white noise. Use this to claim that that Y_t satisfies an AR model with U_t as input.
(v) Do the above suffice to prove that Y_t is a linear time series? [Hint: the answer is NO–but need to argue for it!]
Exercise K 时间序列代做
Let X_t denote the Wolfer sunspots data. Denote \mu=EX_t and f(w) the spectral density at point w.
(i) Estimate f(w) using the best fitted AR model from Exercice G.
(ii) Use the above, together with a CLT for the sample mean, to write down a 95% confidence interval for \mu.
(iii) Let Y_t = log (X_t + 1). Use the R function ar.yw (with aic = TRUE) to fit an AR model to the Y_t data. Does the chosen AR order for Y_t coincide with that of X_t (as done in Exercice G)?
(iv) What is the (nonlinear) model for X_t that is implied by the fitted AR model to the Y_t data?
更多代写：cs代上网课价格 Final exam代考推荐 英国代写机构 论文英语摘要代写 澳洲paper代写 数学模型建模代写
