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.

Answer Summary

Set up the multivariate lag system, then interpret each equation through cross-variable feedback and the joint stability condition.

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?

Answer Summary

Look for a stationary linear combination among nonstationary series so the long-run equilibrium can be separated from short-run drift.

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.

Answer Summary

Weight recent observations more heavily than older ones and interpret the smoothing constant as the memory of the forecast rule.

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.

Answer Summary

Work through identification, estimation, diagnostic checking, and refinement as one iterative workflow rather than isolated calculations.

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?

Answer Summary

Separate the hidden-state evolution equation from the observation equation, then reason through filtering or smoothing updates.

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.

Answer Summary

Model the external shock explicitly and test whether the intervention changes the series level, slope, or dynamic response.

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.

Answer Summary

Forecast jointly rather than series by series so cross-series dependence improves the final predictive equations.

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?

Answer Summary

Translate the time-domain behavior into spectral language and identify which frequencies explain most of the variation.

Related Theory

Review Statistical Inference and Estimation

Revisit the theory that supports these worked practice problems.

Review Stationary Processes

Revisit the theory that supports these worked practice problems.

Review ARMA Models

Revisit the theory that supports these worked practice problems.

Review Autoregressive Models

Revisit the theory that supports these worked practice problems.