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).

Answer Summary

Use the recursion repeatedly to derive the mean, variance, and autocorrelation pattern, and then check stationarity from the root condition.

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.

Answer Summary

Read the cutoff behavior of the autocorrelation function and match it to the short-memory structure of an MA model.

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.

Answer Summary

Difference only as needed for stationarity, then combine ACF/PACF diagnostics with parsimony when proposing the final model order.

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.

Answer Summary

Project future values onto the observed history and separate the predictable linear part from future shocks.

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?

Answer Summary

Focus on the null of a unit root, not the usual stationarity null, and interpret rejection as evidence for a stable mean-reverting process.

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.

Answer Summary

Track both ordinary and seasonal differencing and identify which patterns belong to the seasonal lag structure.

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?

Answer Summary

Move from autocovariance into the frequency domain so the answer emphasizes dominant cycles and power concentration by frequency.

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?

Answer Summary

Treat the conditional variance as a dynamic process and check how past shocks and past volatility feed into current risk.

Related Theory

Review Stationary Processes

Revisit the theory that supports these worked practice problems.

Review Autoregressive Models

Revisit the theory that supports these worked practice problems.

Review Moving Average Models

Revisit the theory that supports these worked practice problems.

Review ARMA Models

Revisit the theory that supports these worked practice problems.