Tutorial 01 (Updated @ 29/09)

• In exercise 2, the question assumes "a linear trend" instead of "a quadratic trend".

• In exercise 8b, the original solution mistakenly regards the upper index for the filter as 2 instead of s.

• In P.1, it has been updated that {N_t} could be characterized by E(N_t)=0 for all t. (To prevent identifiability issues among trends and noise)

• In exercise 6c, the order of the seasonal differencing operator and first-order differencing should be swapped.

• In section 1.1,  the last entry in the third row of the design matrix under seasonal effect X_{+s} is 3^p instead of 2^p.

 Tutorial 02 (Updated @ 24/10)

• In exercise 3, the original solution missed the multiplier sigma^2 in the evaluation of the covariance term.

• In exercise 2a, the wording in the solution should be "ACVF and ACF" instead of "ACF and PACF".

 Tutorial 03 (Updated @ 12/10)

• In the definition of SARIMA, the notation of "the seasonal  MA" polynomial should be the capital letter Theta instead of the small letter theta.

• In exercise 2b, the third term in the RHS of the identity should be Z_{t-1} instead of Z_t.

• In P.4, the AR representation should be Z_t = theta(B)^{-1}\phi(B)Y_t instead of Z_t = theta(B)^{-1}\phi(B)Z_t.

• Exercise 8 has been modified & Exercise 9 has been removed (postponed to the next tutorial).

 Tutorial 04 (Updated @ 20/10)

• In exercise 6a,  {Yt} ~ SARIMA(2,0,0)x(1,0,2)_3 instead of  SARIMA(3,0,1)x(1,0,2)_3.

• In exercise 6b,  {Yt} ~ SARIMA(1,2,1)x(2,0,0)_3 instead of SARIMA(1,2,1)x(1,0,0)_4.

• The solution to exercise 7b has been updated,  we could not comment on causality and stationarity based on P2.

• The evaluation of PACF is added to the solution to exercise 5.

• The solution to Exercise 3b is updated. There is some mistake in the original solution to the ACVF.

• The hints to Exercise 10 are updated, you should verify whether the sum is both bounded above and below, instead of only verifying the existence of the upper bound.

 Tutorial 05 (Updated @ 09/12)

• In remark 2.3, the terms in Cov(Yt,Y_{t-k}) should be theta instead of psi.

• In section 5.2.1, the first point of the box for "Alternative Computation Method", the index t should be running from t=p+1 to n instead to n-p.

• In exercise 8, it should be Y1=cZ1 instead of Yt=cZ1.

• The "additional exercise" section has been removed. More relevant exercises are included in Tutorial 06 instead.

• The numerical solution to Exercise 3 was wrong, the error has been fixed.

• In the solution to Exercise 5 is updated, the value of the estimated asymptotic variance and test was wrong. The final decision is "not reject H0".

• In Method II to find MLE, the original joint density is miswritten. It should be Sigma^{-1} instead of Sigma in the kernel part of density. Hence the solution to Exercise 10 have to be updated. Notice that you will NOT be asked to write the joint density in the examination. (I would update it later). 

• In  the solution to Exercise 9a, it should be (Yt- phi*Y_{t-1})^2 instead of (Yt- phi*Y_{t-1}^2) in the kernel of joint density.

 Tutorial 06 (Updated @ 09/12)

• In the definition of BIC, the term involved should be the sum of Yi^2 instead of Xi^2.

• In Figure 3, those model should be AR(1), AR(2) and AR(3) models instead of AR(1), AR(3) and AR(5).

• In Exercise 6, the original code for calculating asymptotic variance was wrong, the denominator should be sqrt(n) instead of n.

• In Exercise 9, the sign of the MA coefficient estimate should be reversed due to the discrepancy between the definition of the ARMA model in R and our course.

• In Exercise 9, the estimate of sigma^2 is missed. It could be found through the R-command "MLE_fit$sigma2".

• In the beginning of Section 6.2, L(hat{beta}, hat{sigma}^2) has a denominator in the kernel as 2hat{sigma}^2 instead of 2*pi*hat{sigma}^2.

• In exercise 7, you are asked to find the Yule-walker estimator instead of the Least-square estimator.

• In Exercise 4d, the original quantile considered chi-squared distribution with the degree of freedom n-p-q, but it should be h-p-q instead. The question and solution have been modified accordingly.

 Tutorial 07 (Updated @ 09/12)

• In Exercise 1, for the evaluation of e4(2), it is e4(2) = phi1*(Y5 - hat Y5) + Z6 - theta1*Z5 instead of e4(2) = Z6 - theta1*Z5, Hence the Var(e4(2)|Y1,...,Y4)) and the prediction interval for Y6 have to be adjusted accordingly.

• In  Exercise 2, for the evaluation of e{n,X}(2), it is e{n,X}(2) = phi1*e{n,X}(1) + Z{n+2} - theta1*Z[n+1] instead of e{n,X}(2) = Z{n+2} - theta1*Z[n+1], Hence the Var(e{n,Y}(2)|Y1,...,Yn)), Var(e{n,Y}(2)|Y1,...,Yn) and the prediction interval for Y{n+2} have to be adjusted accordingly.

• The notation for forecasted values is all changed to be hat Y_{n+h} instead of Y_{n+h}^n for consistency of notation.

• In Exercise 2, we have to assume {Xt} being causal and invertible.

• In Exercise 2(c), the original solution misregard Xt as Yt. Also, you should set Y0=0 by the invertibility assumption.

• In Exercise 1, the value of Y1 is changed from 2 to 1.

 Tutorial 08 (Updated @ 08/12)

• The definition of Heteroskedastic and conditional heteroskedastic has been updated.

• In Theorem 3 and Remark 6, the X_{t-i} should be replaced by X_{t-i}^2.

• In Remark 6, it should be ARCH(p)=GARCH(0,p) model instead of GARCH(p,0) model.

• The H0 for Lagrange Multiplier Test is alpha1=...=alpha_p=0 instead of only alpha1=...=alpha_p.

• In the solution of Exercise 1c and 3a, sigma_t^2=1.5 + 0.5sigma_{t-1}^2 + 0.2X_{t-1}^2 instead of 1 + 0.5sigma_{t-1}^2 + 0.2X_{t-1}^2.

• In the solution of Exercise 4b, the summation index should run from 1 to 4 instead of 3.

• In the solution of Exercise 4c, the standard deviation is sqrt(4.25) instead of 4.25.

Mock Final Exam (Updated @ 09/12)

• In Q1c, the model mentioned in the claim should be "AR(3)" instead of "MA(3)". 

• In the solution of Q2c, it should be Var(hat{phi1} - 2hat{phi2})=0.12 instead of Var(hat{phi1} - hat{phi2})=0.12

• In the solution of Q2d, the original solution misregard the variance of estimate as sd of the estimate.

• In the solution of Q4b, the evaluation of a3 and hence the other quantities were wrong.

• In the solution of Q5c, the value of gamma(0) and gamma(1) were miscalculated. (The original derivation is correct, just that I made careless mistake when solving the yule-walker equation)

• In the solution of Q6d, the original solution of E(Y_t^2) was wrong. Also, the form of estimated variance is allowed to be any sensible form (or just state it as the sample variance of estimated noise).