Time Series Analysis

Time Series Analysis Practice 1

Problems on ARMA models, forecasting, unit roots, seasonal models, and spectral analysis

8 Problems

Suggested: 2 hours

Instructions

• Try to solve each problem before viewing the solution
• Click "Show Solution" to reveal the answer and detailed explanation
• Focus on understanding the problem-solving methodology

1AR(1) Process Properties

Problem

Consider the AR(1) process: $X_t = 0.8X_{t-1} + \varepsilon_t$ where $\varepsilon_t \sim \text{WN}(0, \sigma^2)$ .

(1) Is the process stationary?

(2) Find the mean and variance of Xₜ.

(3) Find the autocorrelation function ρ(h).

(4) Calculate ρ(1), ρ(2), ρ(5).

2MA(2) Identification

Problem

A time series has the following sample ACF: ρ̂(0)=1, ρ̂(1)=0.6, ρ̂(2)=-0.3, ρ̂(h)≈0 for h≥3.

(1) Suggest an appropriate model.

(2) For MA(2): $X_t = \varepsilon_t + \theta_1\varepsilon_{t-1} + \theta_2\varepsilon_{t-2}$ , write the Yule-Walker equations.

(3) Discuss invertibility conditions.

3ARIMA Model Selection

Problem

A time series shows a linear trend and the ACF decays very slowly.

(1) What does this suggest about stationarity?

(2) After first differencing, the ACF cuts off after lag 1 and PACF decays exponentially. Suggest a model.

(3) Write the model equation in detail.

4Forecasting with ARMA

Problem

An AR(2) model is estimated: $X_t = 0.6X_{t-1} + 0.3X_{t-2} + \varepsilon_t$ with $\sigma^2 = 4$ .

Given X₁₀₀ = 5, X₉₉ = 3:

(1) Find the 1-step ahead forecast $\hat{X}_{101}$ .

(2) Find the forecast error variance.

(3) Construct a 95% prediction interval for X₁₀₁.

5Unit Root Testing

Problem

Consider testing for a unit root in the model: $X_t = \rho X_{t-1} + \varepsilon_t$ .

(1) State the null and alternative hypotheses for the Dickey-Fuller test.

(2) Why can't we use standard t-test?

(3) If the test statistic is -2.5 and the 5% critical value is -2.86, what is your conclusion?

6Seasonal ARIMA

Problem

Quarterly sales data shows both trend and seasonal pattern with period s=4.

(1) Suggest appropriate differencing.

(2) Write the general form of SARIMA(p,d,q)×(P,D,Q)ₛ.

(3) For SARIMA(1,1,1)×(1,1,1)₄, write the full model equation.

7Spectral Analysis

Problem

For an AR(1) process $X_t = \phi X_{t-1} + \varepsilon_t$ with |φ| < 1:

(1) Derive the spectral density function.

(2) At what frequency is the spectrum maximized when φ = 0.9?

(3) What does this imply about the process?

8ARCH/GARCH Models

Problem

A GARCH(1,1) model is specified as:

r_t = \mu + \varepsilon_t, \quad \varepsilon_t = \sigma_t z_t, \quad z_t \sim N(0,1)

\sigma_t^2 = \omega + \alpha\varepsilon_{t-1}^2 + \beta\sigma_{t-1}^2

(1) What does GARCH model capture?

(2) State the stationarity condition.

(3) If ω=0.1, α=0.15, β=0.8, is the process stationary?