Reference no: EM132476963

MAT 3379 Introduction to Time Series Assignment - University of Ottawa, Canada

Q1. Theoretical Question

Durbin-Levinson algorithm and PACF for AR(p) models. Assume that Z_t are i.i.d random variables with mean 0 and variance σ_Z². Obtain the coefficients φ₁₁, φ₂₂, φ₃₃ for AR(2) model.

Q2. Theoretical Question

Yule-Walker procedure and Durbin-Levinson algorithm for MA(q) models. Consider the MA(1) model X_t = Z_t + θZ_t-1, θ ∈ R, where Z_t are i.i.d. random variables with mean 0 and variance σ_Z². Our goal is to find the best linear predictor P_nX_n+1 of X_n+1 based on X₁, . . . , X_n.

(a) Let n = 2. Apply the Durbin-Levinson algorithm to get P₂X₃ = φ₂₁X₂ + φ₂₂X₁. You should get the same answer as in the previous assignment; note however that there is a change of notation: (a₁, a₂) are changed into (φ₂₁, φ₂₂).

Q3. Theoretical Question - Maximum Likelihood Estimation for AR(p) models.

Consider AR(1) model X_t = φX_t-1 + Z_t, where Z_t are i.i.d. normal random variables with mean zero and variance σ_Z². Derive MLE for φ and σ_Z².

Q4. Practical Question

(a) We have already fitted AR(4) to US unemployment data. We estimated parameters using the Yule-Walker procedure.

(b) Calculate the residuals, plot their ACF and PACF. Is the chosen AR(4) model appropriate?

(c) Predict the next observation (remember about the mean!).

(d) "Predict" the past observations and verify quality of the "prediction" by plotting the original values and the "predicted" values on the same graph. Compute the squared error of that prediction.

(e) Now, pretend that the model is AR(1). Estimate parameters of the model. Repeat (b)-(d). State conclusions.

Q5. Practical/Theoretical Question

(a) Type

My.TS<-LakeHuron; help(LakeHuron); mean=mean(My.TS);

My.Centered.TS<-My.TS-mean(My.TS);

The first command loads data set LakeHuron which is in-built in R. The second command shows description of the data set. The third command centers your data set.

(b) Fit AR(2) model using the Yule-Walker estimator. Obtain φ^{^}₁, φ^{^}₂, σ^{^}_Z².

fit.ar<-ar(My.Centered.TS,method="yule-walker");

(c) Verify that the command ar leads to the correct Yule-Walker estimator.

- Type

ACF<-acf(LakeHuron)

and readb ρ^{^}_X(1) and ρ^{^}_X(2). Type var(LakeHuron)to get γ^{^}_X(0). Using these information, compute γ^{^}_X(1), γ^{^}_X(2).

- Create a vector (γ^{^}_X(1), γ^{^}_X(2)) and call it gamma.vector.

- Create a matrix Γ^{^}₂ and call it Gamma.matrix.

- Type

solve(Gamma.matrix)%*%gamma.vector;

and compare the obtained values with part (b).

Q6. Practical Question

The following exercise shows that it is hard to identify AR model with p ≥ 2.

Download BadData.txt. Denote by X your data set.

(a) Based on ACF and PACF argue that an AR(3) model can be chosen.

(b) Type

fit.ar<-ar(X, method="mle");

fit.ar;

What order has been selected? Denote this order by p.

(c) Use p from (b) and type

fit.arima<-arima(X, order=c(3,0,0));

fit.arima;

fit.arima1<-arima(X, order=c(p,0,0));

fit.arima1;

Why did MLE select p, not 3?

Attachment:- Introduction to Time Series Assignment File.rar