If the time series of Xt = 2+ 3t, then what should be the variance of Xt?
If the time series of Xt = 2+ 3t, then what should be the variance of...
It should be clear from the discussion that a strictly stationary, finite variance, time series is also stationary. However, the converse may or may not be true, in general. A time series {Xti t 0,?1,?2, ) is said to be a Gaussian process, if the n-dimensional random vectors X = (Xti, Xt2, .. . , X.). for every collection of time points t1, t2,..., tn, and every positive integer n, have a Multivariate Normal distribution. } is a stationary Gaussian...
Consider the time series Consider the time series of Xt Xt-1 + Wt, where Wt ~ N(0, σ2) and Xi. Derive the general form for var(X). (Hint: i=1 5 ? -n(n+1)(2n+1)
Consider the time series of Xt = Xt−1 + Wt, whereWt are i.i.d and Wt ∼ N (0, σ2 ) and X0 = 0. Let X¯ = 1 n Pn i=1 Xi . Derive the general form for var(X¯). (Hint: Pn i=1 i 2 = n(n+1)(2n+1) 6 ) Consider the time series of X-Xi-1+Wi, whereW are i.i.d and W N(0, σ2) and Xo 0. Let X = n Σ, Derive the general form for var(X). (Hint: Σ i- n(n+1)(2n+1))
time series 3. Specify and interpret the model in terms of a time- series data Xt for the following multiplicative models ARIMA: (1, 0, 0)(1, 0, 0)4 ARIMA (0, 1, 0)(1, 1, 0)6 ARIMA (0, 1, 1)(0, 0, 1)12
The sample data x1,x2,...,xn sometimes represents a time series, where xt = the observed value of a response variable x at time t. Often the observed series shows a great deal of random variation, which makes it difficult to study longer-term behavior. In such situations, it is desirable to produce a smoothed version of the series. One technique for doing so involves exponential smoothing. The value of a smoothing constant α is chosen (0 < α < 1). Then with...
Suppose Zt = 2 + Xt -2Xt-1+Xt-2, where {Xt} is zero-mean stationary series with autocovariance function. Calculate the autocovariance of Zt
2. Suppose that Ya ut where the ut are iid Normal with mean zero and variance σ2, but that you mistakenly think Yt is difference stationary. You therefore construct a new series a) Are the Xt i.i.d.? Explain b) Is X stationary? Explain c) Calculate the mean, variance, and autocorrelation function of X d) How does the answer you obtained in (c) compare with the mean, variance and autocor- relation function of Y? 2. Suppose that Ya ut where the...
Exercise 2.31 Superposition [] Given two independent weakly stationary time series Xt and Yi) with autocovariance functions x(h) and y (h), show that Zt- Xt +Yt is also weakly stationary, with autocovariance function given by yz(h)-x(h)y(h).
Time series analysis 2. Set n 100 and generate and plot the time series xt 2 cos(2π.06t) + 3 sin(2π.06t) Ý,-4 cos(2n. 10t) + 5 sin(2m10) z, 6 cos(2π·40t) + 7 sin(2π·40t) (a) Use the periodogram function in R to plot the periodogram of Vi. Can you explain the spikes? (b) Now let wi ~ N(0, 25) be iid and plot the periodogram of the series V +w. Does it still pick out the periodic components? 2. Set n 100...
Let Wt de a (Gaussian) white noise with variance σ 2 . Then, let Xt = WtWt−1 + µ, where µ is a real constant. Determine the mean and autocovariance of (Xt)? Is this process stationary? Let W, de a (Gaussian) white noise with variance σ2. Then, let of where μ is a real constant. Determine the mean and (X)? Is this process stationary?