Determine which of the following processes are stationary and
invertible. In the following we always assume that ut is white
noise with mean zero and variance σ2, i.e. ut ∼ WN(0, σ2)
Determine which of the following processes are stationary and invertible. In the following we alw...
a) Consider the following moving average process, MA(2): Yt = ut + α1ut-1 + α2ut-2 where ut is a white noise process, with E(ut)=0, var(ut)=σ2 and cov(ut,us)=0 . Derive the mean, E(Yt), the variance, var(Yt), and the covariances cov( Yt,Yt+1 ) and cov(Yt,Yt+2 ), of this process. b) Give a definition of a (covariance) stationary time series process. Is the MA(2) process (covariance) stationary?
b) In what follows, we assume cc return r_t is a covariance stationary process. Prove the following statements: i. If r_t iid(0; sigma^2) (or independent white noise); then r_t mds(0; sigma^2). ii. If r_t mds(0; sigma^2); then r_t WN(0; sigma^2) (or weak white noise).
Time Series transformation Let an annual series Yt be stationary. However, the series transformed and differentiated Dt = ln(Yt) - ln(Yt-1) is stationary. Moreover, we suppose that it obeys the following theoretical model: Dt = -0.12 + 0.75 Dt-1 + et, in which the error term and is a white noise of variance σ2 = 0.012. How can I transform this model to get the original one before the transformation?
Let us consider the binary digital communication system in which bit 1 is represented by the waveform Acos(ωt) of bit duration T, where ω is the carrier radial frequency and A is the constant amplitude. On the hand, the bit 0 is represented by the following waveform instead (A/10)cos(ωt). During the transmission the channel has introduced the uniform random phase shift Φ and transmitted waveform is affected by zero-mean white Gaussian noise of variance σ2. To demodulate, we perform the...
Suppose that we believe a weakly stationary return sequence r following the model, where at ls the 1.1.d. noise sequence with mean 0 and variance σ. and at s independent of rt-1,Tt-2. (a) Express the mean μ of the return sequence rt using φο, φι, φ2 and σ (lag-0 autocovariance) of r) (d) Express the lag-1 autocorrelation ρι using φο, φι, φ2 and σ
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...
2. (a) Consider the following process: where {Z) is a white noise process with unit variance. [1 mark] ii. Find the infinite moving average representation of X,i.e., find the scquence [6 marks] i. Explain why the process is stationary. (6) such that Xt = Σ b,2-j. iii. Calculate the mean and the autocovariance "Yo, γι and 72 of the process. 7 marks iv. Given 40 = 0.1 and Xo = 1.8, find the 2-step ahead forecast of the time series...
1. Determine which of the following matrices are invertible. Use the Invertible Matrix Theorem (or other theorems) to justify why each matrix is invertible or not. Try to do as few computations as possible. (2) | 5 77 (a) 1-3 -6] [ 3 0 0 1 (c) -3 -4 0 | 8 5 -3 [ 30-37 (e) 2 0 4 [107] F-5 1 47 (d) 0 0 0 [1 4 9] ſi -3 -67 (d) 0 4 3 1-3 6...
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...
Suppose that we are given the following communication system described in Fig. 1 with the channel corrupted by an additive white Gaussian noise z with zero mean and variance 1 where the channel input.x is used for signal transmission to produce the channel output y,i.e., r- x . Then the channel is further passed through a hard limiter, i.e., sign detector described by Q2(r) in Fig.2 decisions 22(r) Figure 1. A channel with the input x and output r corrupted...