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use geometric series. !!!!
Consider the AR(1) model it-onrt-1 + ur where ur ~ NID(0, σ.)....
use geometric series. hrt-1 + ur where ur ~ NID(0, σ.). Show that for Consider the...
use geometric series.
hrt-1 + ur where ur ~ NID(0, σ.). Show that for Consider the AR(1) model zt øl<1, the auto-covariance is
Consider the 5 point running mean where ut ~ NID(0, σ ), and let σ -1....
Consider the 5 point running mean where ut ~ NID(0, σ ), and let σ -1. (i) Determine the theoretical auto-covariance (ACF) for v, and the theoretical cross-covariance function (CCF) between w and vt. (ii) Generate a realization of w of length 1000, compute the associated 5-point moving average v and plot these two time series on the same graph.i Calculate the corresponding sample versions for the ACF and CCF and remark on how these resemble and differ from the...
B. Consider the GARCH (1, 1) model Xt-σ.zt, σ -00 + α1XL1 + βισ -1 where Zt are iid N (0, 1) proc...
B. Consider the GARCH (1, 1) model Xt-σ.zt, σ -00 + α1XL1 + βισ -1 where Zt are iid N (0, 1) process, ao 0, α120, ai 1 > α1 + β1. Show that 0, A > 0 and 2IV2 t-1't-2 ..
B. Consider the GARCH (1, 1) model Xt-σ.zt, σ -00 + α1XL1 + βισ -1 where Zt are iid N (0, 1) process, ao 0, α120, ai 1 > α1 + β1. Show that 0, A > 0...
6. (13 marks) where {U, } ~ WN(0,00) is Consider two independent AR(1) series< independent of {K} ~ WN(0,OF). Does their sum Z,-X,-X necessarily follow an AR(1) series? Prove or disprove. (Hint: C...
6. (13 marks) where {U, } ~ WN(0,00) is Consider two independent AR(1) series< independent of {K} ~ WN(0,OF). Does their sum Z,-X,-X necessarily follow an AR(1) series? Prove or disprove. (Hint: Compare the causal representation of the sum to that of an AR(1) process)
6. (13 marks) where {U, } ~ WN(0,00) is Consider two independent AR(1) series
4. Consider two functions x(h),y (h), Vh> 0 and cross-covariance function 7xy(h), Vh e Z (a)...
4. Consider two functions x(h),y (h), Vh> 0 and cross-covariance function 7xy(h), Vh e Z (a) (7 points) Find the Best Linear Predictor (BLP) of Y given Xt, and its Mean Square Prediction Error (MSPE), expressed in terms of x(h). (h), xy (h) jointly stationary time series (X, Y), with individual auto-covariance (b) (13 points) Find the BLP of Y given Xt, X-1,Y-1, and its MSPE, expressed in terms of 7x(h)(h), x.y (h). (Note: you don't need to solve the...
In each of the following, use the gometrie series Σ's 28 , where-l < x < 1, to express the given function as a power series and indicate the interval of convergence. 1n(1 +r')= Σ T- ア=...
In each of the following, use the gometrie series Σ's 28 , where-l < x < 1, to express the given function as a power series and indicate the interval of convergence. 1n(1 +r')= Σ T- ア=Σ 2)2
In each of the following, use the gometrie series Σ's 28 , where-l
1. A simple regression model is given by Y81B2X+ e for t 1, (1) ,n errors e with Var (e) a follow AR(1) model where th...
1. A simple regression model is given by Y81B2X+ e for t 1, (1) ,n errors e with Var (e) a follow AR(1) model where the regression et pet-1 + , t=1...n where 's are uncorrelated random variables with constant variance, that is, E()0, Var (v) = , Cov (, ,) 0 for t Now given that Var (e) = Var (e1-1)= , and Cov (e-1, v)0 (a) Show that (b) Show that E (ee-1)= p. (c) What problem(s) will...
Consider the following model on a return series rt=t+ at +0.25at-1, where at riid N(0,02), t = 1, ... ,T. (a) What are...
Consider the following model on a return series rt=t+ at +0.25at-1, where at riid N(0,02), t = 1, ... ,T. (a) What are the mean function and autocovariance function for this return series? Is this return series {rt} weakly stationary? Justify your answer. (b) Consider first differences of the return series above, that is, consider wt = Vrt=rt – Pt-1. What are the mean function and autocovariance function for this time series? Is this time series {wt} weakly stationary? Justify...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solutio...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0
1. Consider the Partial Differential Equation ot u(0,t) =...
QUESTION 3 (a) Consider the ARMA (1, 1) process -Bat-1-where o and θ are model parame-...
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...
use geometric series.
hrt-1 + ur where ur ~ NID(0, σ.). Show that for Consider the AR(1) model zt øl<1, the auto-covariance is
Consider the 5 point running mean where ut ~ NID(0, σ ), and let σ -1. (i) Determine the theoretical auto-covariance (ACF) for v, and the theoretical cross-covariance function (CCF) between w and vt. (ii) Generate a realization of w of length 1000, compute the associated 5-point moving average v and plot these two time series on the same graph.i Calculate the corresponding sample versions for the ACF and CCF and remark on how these resemble and differ from the...
B. Consider the GARCH (1, 1) model Xt-σ.zt, σ -00 + α1XL1 + βισ -1 where Zt are iid N (0, 1) process, ao 0, α120, ai 1 > α1 + β1. Show that 0, A > 0 and 2IV2 t-1't-2 ..
B. Consider the GARCH (1, 1) model Xt-σ.zt, σ -00 + α1XL1 + βισ -1 where Zt are iid N (0, 1) process, ao 0, α120, ai 1 > α1 + β1. Show that 0, A > 0...
6. (13 marks) where {U, } ~ WN(0,00) is Consider two independent AR(1) series< independent of {K} ~ WN(0,OF). Does their sum Z,-X,-X necessarily follow an AR(1) series? Prove or disprove. (Hint: Compare the causal representation of the sum to that of an AR(1) process)
6. (13 marks) where {U, } ~ WN(0,00) is Consider two independent AR(1) series
4. Consider two functions x(h),y (h), Vh> 0 and cross-covariance function 7xy(h), Vh e Z (a) (7 points) Find the Best Linear Predictor (BLP) of Y given Xt, and its Mean Square Prediction Error (MSPE), expressed in terms of x(h). (h), xy (h) jointly stationary time series (X, Y), with individual auto-covariance (b) (13 points) Find the BLP of Y given Xt, X-1,Y-1, and its MSPE, expressed in terms of 7x(h)(h), x.y (h). (Note: you don't need to solve the...
In each of the following, use the gometrie series Σ's 28 , where-l < x < 1, to express the given function as a power series and indicate the interval of convergence. 1n(1 +r')= Σ T- ア=Σ 2)2
In each of the following, use the gometrie series Σ's 28 , where-l
1. A simple regression model is given by Y81B2X+ e for t 1, (1) ,n errors e with Var (e) a follow AR(1) model where the regression et pet-1 + , t=1...n where 's are uncorrelated random variables with constant variance, that is, E()0, Var (v) = , Cov (, ,) 0 for t Now given that Var (e) = Var (e1-1)= , and Cov (e-1, v)0 (a) Show that (b) Show that E (ee-1)= p. (c) What problem(s) will...
Consider the following model on a return series rt=t+ at +0.25at-1, where at riid N(0,02), t = 1, ... ,T. (a) What are the mean function and autocovariance function for this return series? Is this return series {rt} weakly stationary? Justify your answer. (b) Consider first differences of the return series above, that is, consider wt = Vrt=rt – Pt-1. What are the mean function and autocovariance function for this time series? Is this time series {wt} weakly stationary? Justify...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0
1. Consider the Partial Differential Equation ot u(0,t) =...
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...
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