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?
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A. ut = ρut-1 + vt
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B. ut = ρut-1 - vt
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C. ut =
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D. ut = ρut-1vt
Homework Answers
The first order auto regressive model is given by:
A. ut = ρut-1 + vt
Since 
observation depends upon 
and random error 
Add Answer to:
If ut is the error in time t,
ut-1 is the error in the previous
time...
First question:Second question: The model: Y4 = Bo + B2C+ + ut, t = 1,2,.....n, is...
First question:Second question:
The model: Y4 = Bo + B2C+ + ut, t = 1,2,.....n, is an example of a(n): a) Autoregressive conditional heteroscedasticity model. b) static model. c) finite distributed lag model. d) infinite distributed lag model. Consider a linear model to explain monthly beer consumption: | Beer=Bo +Biinc +B2 price +B3educ +B4 female +u E(ulinc,price,educ, female) =0 Var(uſinc,price,educ, female) = o zinc?. Write the transformed equation that has a homoskedastic error term.
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...
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...
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...
Power Spectral Density of Signal A signal s(t) can be expressed as the following equation: L-1...
Power Spectral Density of Signal
A signal s(t) can be expressed as the following equation: L-1 where L is a positive integer. {An}n=0 are independent and identically distributed (i.i.d.) discrete random variables. The probability mass function (PMF) of An is An() 0 otherwise, where A is a positive constant in volt. To is a uniformly distributed random variable with probability density function (PDF) defined by 0. otherwise. L-1 To and {An}n=d are independent. The signal p(t) is a pulse and...
t2s points). Let B(t) be Brownian motion and let zu) with drift u0. For any B()+ ut be Brownian motion t 0, let T be the first time Z(t) hits a. a) Show that T, is a random time for Z(t) and for...
t2s points). Let B(t) be Brownian motion and let zu) with drift u0. For any B()+ ut be Brownian motion t 0, let T be the first time Z(t) hits a. a) Show that T, is a random time for Z(t) and for B(t). b) Show P(T, 0o)1. c) Use martingale methods to compute E [e-m] for any θ > 0 r>
t2s points). Let B(t) be Brownian motion and let zu) with drift u0. For any B()+ ut be...
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...
1. Consider the following unobserved effects regression model: Suppose that the idiosyncratic error u, t 1,..,T...
1. Consider the following unobserved effects regression model: Suppose that the idiosyncratic error u, t 1,..,T are serially correlated with constant variance σ2. Show that the correlation between adjacent differences, Δυ" and Δυ,-l is -0.5. Therefore under the ideal FE assumptions, first differencing induces negative serial correlation of a known value. [Note:
The Black-Scholes-Merton model for stock pricing in discrete time Let So be the initial stock price...
The Black-Scholes-Merton model for stock pricing in discrete time Let So be the initial stock price at time t = 0. At time t = 1,2,-. ., the stock price is S,ett+σ Σ. 2. the drift where a 0 is known as the volatility and the independently and identically distributed standard Normal N(0,1) random 0 is known as Zi variables are (a) Show that S, = S¢_1e#+oZ¢ _ St St-1 (b) What is the distribution of ln (c) What is...
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....
First question:Second question:
The model: Y4 = Bo + B2C+ + ut, t = 1,2,.....n, is an example of a(n): a) Autoregressive conditional heteroscedasticity model. b) static model. c) finite distributed lag model. d) infinite distributed lag model. Consider a linear model to explain monthly beer consumption: | Beer=Bo +Biinc +B2 price +B3educ +B4 female +u E(ulinc,price,educ, female) =0 Var(uſinc,price,educ, female) = o zinc?. Write the transformed equation that has a homoskedastic error term.
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...
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...
Power Spectral Density of Signal
A signal s(t) can be expressed as the following equation: L-1 where L is a positive integer. {An}n=0 are independent and identically distributed (i.i.d.) discrete random variables. The probability mass function (PMF) of An is An() 0 otherwise, where A is a positive constant in volt. To is a uniformly distributed random variable with probability density function (PDF) defined by 0. otherwise. L-1 To and {An}n=d are independent. The signal p(t) is a pulse and...
t2s points). Let B(t) be Brownian motion and let zu) with drift u0. For any B()+ ut be Brownian motion t 0, let T be the first time Z(t) hits a. a) Show that T, is a random time for Z(t) and for B(t). b) Show P(T, 0o)1. c) Use martingale methods to compute E [e-m] for any θ > 0 r>
t2s points). Let B(t) be Brownian motion and let zu) with drift u0. For any B()+ ut be...
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...
1. Consider the following unobserved effects regression model: Suppose that the idiosyncratic error u, t 1,..,T are serially correlated with constant variance σ2. Show that the correlation between adjacent differences, Δυ" and Δυ,-l is -0.5. Therefore under the ideal FE assumptions, first differencing induces negative serial correlation of a known value. [Note:
The Black-Scholes-Merton model for stock pricing in discrete time Let So be the initial stock price at time t = 0. At time t = 1,2,-. ., the stock price is S,ett+σ Σ. 2. the drift where a 0 is known as the volatility and the independently and identically distributed standard Normal N(0,1) random 0 is known as Zi variables are (a) Show that S, = S¢_1e#+oZ¢ _ St St-1 (b) What is the distribution of ln (c) What is...
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....
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