Time Series Analysis

Time Series Analysis Practice 2

Advanced topics: VAR models, cointegration, state space, intervention analysis, and frequency domain

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

1Vector Autoregression (VAR)

Problem

A bivariate VAR(1) model for {Yₜ, Xₜ} is given by:

\begin{pmatrix} Y_t \\ X_t \end{pmatrix} = \begin{pmatrix} 0.5 & 0.2 \\ 0.3 & 0.6 \end{pmatrix} \begin{pmatrix} Y_{t-1} \\ X_{t-1} \end{pmatrix} + \begin{pmatrix} \varepsilon_{1t} \\ \varepsilon_{2t} \end{pmatrix}

(1) Check stability of the system.

(2) Find the forecast $\hat{Y}_{t+1}$ given Yₜ = 10, Xₜ = 8.

2Cointegration

Problem

Two I(1) series Yₜ and Xₜ satisfy: $Y_t - 2X_t \sim I(0)$ .

(1) What does this imply about the relationship between Y and X?

(2) Write the error correction model (ECM) representation.

(3) Why is cointegration important for modeling economic relationships?

3Exponential Smoothing

Problem

Simple exponential smoothing with parameter α = 0.3 is used.

Given: S₉ = 100, X₁₀ = 105.

(1) Calculate S₁₀.

(2) What is the 1-step ahead forecast for X₁₁?

(3) Express simple exponential smoothing as an ARIMA model.

4Box-Jenkins Methodology

Problem

Describe the Box-Jenkins approach to time series modeling:

(1) List the main steps in the methodology.

(2) What role does the ACF and PACF play in model identification?

(3) Describe diagnostic checking procedures.

5State Space Models

Problem

A local level model (random walk plus noise) is specified as:

Y_t = \mu_t + \varepsilon_t, \quad \varepsilon_t \sim N(0, \sigma_\varepsilon^2)

\mu_t = \mu_{t-1} + \eta_t, \quad \eta_t \sim N(0, \sigma_\eta^2)

(1) Write in state space form.

(2) What is the Kalman filter used for?

(3) How does the signal-to-noise ratio q = σ²_η/σ²_ε affect smoothing?

6Intervention Analysis

Problem

A time series experiences a policy intervention at time T = 50.

(1) Model an additive outlier (sudden temporary shock).

(2) Model a level shift (permanent change in mean).

(3) Model a gradual change with decay parameter δ.

7Multivariate Forecasting

Problem

For a VAR(1) model with 2 variables, given coefficient matrix A and error covariance Σ:

(1) Derive the h-step ahead forecast error variance.

(2) How do forecast error bounds expand with horizon h?

(3) Compare with univariate ARIMA forecasts.

8Frequency Domain Analysis

Problem

The periodogram of a time series shows a large peak at frequency ω = π/6.

(1) What does this suggest about the data?

(2) What is the corresponding period T?

(3) How would you test if this peak is significant?