The time series {} is said to be an AR(2) process if , where {} is a white noise process with variance <
a) For what values of is the process weakly stationary?
b) Select in the range where the process is weakly stationary and plot the autocorrelation function for the chosen
The time series {} is said to be an AR(2) process if , where {} is...
Consider the random walk model where {} is a white noise process with variance a) How many parameters does this model have? b) calculate and c) Compute for d) Is this model weakly stationary? We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imagevar(yt) We were unable to transcribe this imageWe were unable to transcribe this image
Recall that the MA(q) process is defined as: = where {} is a white noise process with variance < a) Construct an expression for b) Construct an expression for c) Construct an expression for the autocovariance and autocorrelation functions. We were unable to transcribe this imagei t We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageVar(yt)
3. A white Gaussian noise signal W (t) with autocorrelation function it passes through a linear filter invariant in time h (t). Calculate the average power of the exit process Y (t) knowing that We were unable to transcribe this imager. h2 (t)dt = 1. r. h2 (t)dt = 1.
where at is a white noise process with unit variance. It is known that the above process is overestimated.(a) Suggest a parismony model ARMA(2,1) for the above process.(b) Hence, determine the stationarity and invertibility of the process.(c) Find the mean, the variance and the first two lags of the autocovariance function of theprocess.(d) Find the first three lags of the autocorrelation function (ACF) for the process.(e) Find the first three lags of the partial autocorrelation coefficients.
Recall that a time series {εt} is called a white noise process if i. E[εt] = 0 t ; ii. Cov(εs, εt) = 0 s ≠ t ; iii. Var(εt) = σ2 < ∞ Construct the autocorrelation function f(h), h=0,-+1,-+2,… for the white noise process.
2. Consider the time series X, = 2 + 0.5t +0.8X1-1 + W, where W N(0.1). (a) (8 points) Calculate E(X2) Is this process weakly stationary? Give reasons for your answer. Hint: Find the mean function of {X) and then substitute t = 20. (b) (3 points) Calculate Var(X20) Question 2 continues on the next page... Page 4 of 12 c)(4 points) Consider the first differences of the time series above, that is Is {%) a weakly stationary process. Prove...
Question 2 (a) The following table gives the sample autocorrelation coefficients and partial autocorrelation coefficients for a time series with 100 observations. 4 ,-0.55 -0.17 0.09 0.0.00.010.040.07 -0.55 | -0.4 0.29 | -0.22 -0.11- -0.13 -0.14 0,05 Suppose the sample mean of the time series is zero. Based on the above information, suggest an ARMA model for the data. Briefly explain your answer. (5 marks) (b) Let X, be a time series satisfying the following AR(2)model: X, = 0.3X,-1 +0.04X,-2...
: Assume Yt is a time series process and Et is a white noise process with mean zero and constant variance. (a). Write an equation for AR(4) process. (b). Write an equation for AR(5) process. (c). Write an equation for MA(3) process. (d). Write down an equation for MA(2) process. (e). Write an equation for ARMA (4,2) process. (f). Do more research and write an equation for ARIMA (4,0,2) proce
QUESTION 3 (a) Consider the ARMA (1, 1) process -Bat-1-where o and θ are model parame- are independent and identically distributed random variables with mean 0 z, oz,-1 ters, and a1, a2, and variance σ (i) Show that the variance of the process is γ,- (ii) Using (i) or otherwise, show that the autocorrelation function (ACF) of the process is: ifk=0. (b) Let Y be an AR(2) process of the special form Y-2Y-2e (i) Find the range of values of...
2.4 Let (e) be a zero mean white noise process. Suppose that the observed process is Y = e, + 0,-1, where is either 3 or 1/3. (a) Find the autocorrelation function for {Y} both when 0 = 3 and when 0 = 1/3. (b) You should have discovered that the time series is stationary regardless of the value of and that the autocorrelation functions are the same for 0 = 3 and 0 = 1/3. For simplicity, suppose that...