If ut is the error in time t, ut-1 is the error in the previous time...
If ut is the error in time t, ut-1 is the error in the previous time period, ρ is the correlation coefficient, and vt a independent and identically distributed (iid) random variable, which of the following best represents an AR(2) second-order autoregressive model of autocorrelated behavior?
A. ut = ρ1ut-1 + ρ2ut-2 + vt
B. ut = ρ1ut-1 + vt
C. ut = ut-1 + ut-2 + vt
D. None of these are correct.
Homework Answers
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If ut is the error in time t, ut-1 is the error in the previous
time...
If ut is the error in time t, ut-1 is the error in the previous time...
If ut is the error in time t, ut-1 is the error in the previous time period, ρ is the correlation coefficient, and vt a independent and identically distributed (iid) random variable, which of the following is the first-order autoregressive model of autocorrelated behavior? A. ut = ρut-1 + vt B. ut = ρut-1 - vt C. ut = D. ut = ρut-1vt
Q3. [10 points [Serial Correlation Consider a simple linear regression model with time series dat...
Q3. [10 points [Serial Correlation Consider a simple linear regression model with time series data: Suppose the error ut is strictly exogenous. That is Moreover, the error term follows an AR(1) serial correlation model. That where et are uncorrelated, and have a zero mean and constant variance a. 2 points Will the OLS estimator of P be unbiased? Why or why not? b. [3 points Will the conventional estimator of the variance of the OLS estimator be unbiased? Why or...
9. Consider the following hidden Markov model (HMM) (This is the same HMM as in the previous HMM problem): ·X=(x, ,...
9. Consider the following hidden Markov model (HMM) (This is the same HMM as in the previous HMM problem): ·X=(x, ,x,Je {0,1)、[i.e., X is a binary sequence of length n] and Y-(Y Rt [i.e. Y is a sequence of n real numbers.) ·X1~" Bernoulli(1/2) ,%) E Ip is the switching probability; when p is small the Markov chain likes to stay in the same state] . conditioned on X, the random variables Yı , . . . , y, are...
Let X1,X2 be two independent exponential random variables with λ=1, compute the P(X1+X2<t) using the joint...
Let X1,X2 be two independent
exponential random variables with λ=1, compute the
P(X1+X2<t) using the joint density function. And let Z be gamma
random variable with parameters (2,1). Compute the probability that
P(Z < t). And what you can find by comparing P(X1+X2<t) and
P(Z < t)? And compare P(X1+X2+X3<t) Xi iid
(independent and identically distributed) ~Exp(1) and P(Z < t)
Z~Gamma(3,1) (You don’t have to compute)
(Hint: You can use the fact that Γ(2)=1,
Γ(3)=2)
Problem 2[10 points] Let...
Question 5 20 marks] The model for a discrete-time received signal is Xn,b+En n= 1,2, ......
Question 5 20 marks] The model for a discrete-time received signal is Xn,b+En n= 1,2, ... , N 0 or 1, and the random variables 1, @2,,En are whereb independent and identically distributed normally with means 0 and variances , where signal, while if b = 1, the signal is present 'detect' the signal: We set this out as the formal hypothesis test 2 is known. If b=0, there is no We wish to of HA b 1 (a) Show...
QUESTION4 (a) Let e be a zero-mean, unit-variance white noise process. Consider a process that begins...
QUESTION4 (a) Let e be a zero-mean, unit-variance white noise process. Consider a process that begins at time t = 0 and is defined recursively as follows. Let Y0 = ceo and Y1-CgY0-ei. Then let Y,-φ1Yt-it wt-1-et for t > ï as in an AR(2) process. Show that the process mean, E(Y.), is zero. (b) Suppose that (a is generated according to }.-10 e,-tet-+扣-1 with e,-N(0.) 0 Find the mean and covariance functions for (Y). Is (Y) stationary? Justify your...
QUESTION 2 (a) For each of the ARIMA models below, give the values for E(VY) and...
QUESTION 2 (a) For each of the ARIMA models below, give the values for E(VY) and Var(VY) 0.Tet-1 (ii) Yt = 10 + 1.25%-1-0.25Yt-2 et-0.14-i (b) Show that the function Z, a t-1 not stationary, but the first difference of Z, is stationary QUESTION 5 (a) From a series Y, of length 100, the sample autocorrelations at lags 1-3 are 0.8, 0.5 and 0.4, respectively. Furthermore, the respective sample mean and sample variance of the series are 2 and s-5....
1.Idiosyncratic error is the error that occurs due to _____. a. unobserved factors that affect the...
1.Idiosyncratic error is the error that occurs due to _____. a. unobserved factors that affect the dependent variable and change over time b. unobserved factors that affect the dependent variable and do not change over time c. incorrect measurement of an economic variable d. correlation between the independent variables 2.Which of the following is a reason for using independently pooled cross sections? a. To increase the sample size b. To select a sample based on the dependent variable c. To...
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...
5.This is a copy of the original series moved forward two time periods A. Lagged series...
5.This is a copy of the original series moved forward two time periods A. Lagged series with no lag B. Lagged series with lag-1 C. Lagged series with lag-2 D.None of the above 6. This reflects swings in the series, where high values are immediately followed by low values and vice versa. A.Negative lag-1 autocorrelation B. Strong autocorrelation C. Positive lag-1 autocorrelation D.All the above 7.This approach is generally taken to take advantage of auto correlation A. Directly build autocorrelation...
Q3. [10 points [Serial Correlation Consider a simple linear regression model with time series data: Suppose the error ut is strictly exogenous. That is Moreover, the error term follows an AR(1) serial correlation model. That where et are uncorrelated, and have a zero mean and constant variance a. 2 points Will the OLS estimator of P be unbiased? Why or why not? b. [3 points Will the conventional estimator of the variance of the OLS estimator be unbiased? Why or...
9. Consider the following hidden Markov model (HMM) (This is the same HMM as in the previous HMM problem): ·X=(x, ,x,Je {0,1)、[i.e., X is a binary sequence of length n] and Y-(Y Rt [i.e. Y is a sequence of n real numbers.) ·X1~" Bernoulli(1/2) ,%) E Ip is the switching probability; when p is small the Markov chain likes to stay in the same state] . conditioned on X, the random variables Yı , . . . , y, are...
Let X1,X2 be two independent
exponential random variables with λ=1, compute the
P(X1+X2<t) using the joint density function. And let Z be gamma
random variable with parameters (2,1). Compute the probability that
P(Z < t). And what you can find by comparing P(X1+X2<t) and
P(Z < t)? And compare P(X1+X2+X3<t) Xi iid
(independent and identically distributed) ~Exp(1) and P(Z < t)
Z~Gamma(3,1) (You don’t have to compute)
(Hint: You can use the fact that Γ(2)=1,
Γ(3)=2)
Problem 2[10 points] Let...
Question 5 20 marks] The model for a discrete-time received signal is Xn,b+En n= 1,2, ... , N 0 or 1, and the random variables 1, @2,,En are whereb independent and identically distributed normally with means 0 and variances , where signal, while if b = 1, the signal is present 'detect' the signal: We set this out as the formal hypothesis test 2 is known. If b=0, there is no We wish to of HA b 1 (a) Show...
QUESTION4 (a) Let e be a zero-mean, unit-variance white noise process. Consider a process that begins at time t = 0 and is defined recursively as follows. Let Y0 = ceo and Y1-CgY0-ei. Then let Y,-φ1Yt-it wt-1-et for t > ï as in an AR(2) process. Show that the process mean, E(Y.), is zero. (b) Suppose that (a is generated according to }.-10 e,-tet-+扣-1 with e,-N(0.) 0 Find the mean and covariance functions for (Y). Is (Y) stationary? Justify your...
QUESTION 2 (a) For each of the ARIMA models below, give the values for E(VY) and Var(VY) 0.Tet-1 (ii) Yt = 10 + 1.25%-1-0.25Yt-2 et-0.14-i (b) Show that the function Z, a t-1 not stationary, but the first difference of Z, is stationary QUESTION 5 (a) From a series Y, of length 100, the sample autocorrelations at lags 1-3 are 0.8, 0.5 and 0.4, respectively. Furthermore, the respective sample mean and sample variance of the series are 2 and s-5....
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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