ISYE 6402 / ISYE 6402 FINAL EXAM (LATEST UPDATE 2026 /
2027) TIME SERIES ANALYSIS | QUESTIONS & ANSWERS |
GRADE A | 100% CORRECT –
1. Stationarity Concepts
A time series {Xt} is said to be weakly stationary if certain statistical
properties remain constant over time.
Which of the following best describes this property?
A. All higher-order moments of the distribution remain constant for all t
B. The full joint distribution remains the same under any time shift
C. The mean is constant, and the autocovariance depends only on the lag
D. There is no autocorrelation present at any lag
Correct Answer: C
Explanation: Weak stationarity requires constant mean and autocovariances
depending only on lag.
2. AR(1) Variance Calculation
Consider the stationary AR(1) model
Xt = 0.6 Xt−1 + εt,
where εt is white noise with variance 1. What is the long-run variance of Xt?
A. 1.00
B. 1.25
C. 1.56
D. 2.50
Correct Answer: C
Explanation: Var(X) = σ² / (1 − φ²) = 1 / (1 − 0.36) = 1.5625.
,3. Identifying Model from ACF/PACF
A time series has an autocorrelation function (ACF) with large spikes at lags
1 and 2 and near zero afterward.
The partial autocorrelation function (PACF) decays slowly without a sharp
cutoff. Which model fits best?
A. AR(2)
B. MA(2)
C. ARMA(1,1)
D. AR(1)
Correct Answer: B
Explanation: MA(q) models have an ACF cutoff at lag q and a PACF that
tails off gradually.
4. Seasonality in Time Series
You are analyzing monthly data and the ACF shows strong peaks at lags 12,
24, and 36.
This indicates a repeating seasonal pattern. Which type of model is most
appropriate?
A. Non-seasonal ARIMA(1,1,1)
B. SARIMA(p,d,q)(P,D,Q)12
C. Linear regression with a time trend only
D. Random walk without drift
Correct Answer: B
Explanation: Seasonal ARIMA handles periodic patterns through seasonal
differencing and seasonal terms.
5. Ljung–Box Test Interpretation
,The Ljung–Box test is applied to a time series to assess its autocorrelation
structure.
What is the null hypothesis of this test?
A. The series has a unit root
B. The series is Gaussian
C. There is no autocorrelation up to a specified number of lags
D. The series is heteroskedastic
Correct Answer: C
Explanation: Ljung–Box tests whether autocorrelations jointly equal zero,
indicating white noise.
6. ARIMA Model Representation
Consider an ARIMA(1,1,1) model.
Which of the following correctly represents the model in first-difference
form?
A. Xt = φ Xt−1 + θ εt−1 + εt
B. (1 − φB) ΔXt = (1 + θB) εt
C. ΔXt = φ ΔXt−1 + εt
D. (1 − B) Xt = φ Xt−1 + εt
Correct Answer: B
Explanation: ARIMA(1,1,1) applies both differencing and ARMA structure
to ΔXt.
7. Model Identification from ACF
A time series shows an ACF that cuts off after lag 2 but a PACF that decays
gradually.
Which model is most likely appropriate?
A. AR(2)
B. MA(2)
, C. ARMA(2,2)
D. AR(1)
Correct Answer: B
Explanation: For MA(q), ACF cuts off at lag q, while PACF tails off.
8. ADF Unit Root Test
The Augmented Dickey–Fuller (ADF) test is commonly used to assess
stationarity.
What is the null hypothesis of this test?
A. The series is stationary
B. The series has a unit root (nonstationary)
C. The series is white noise
D. The series is heteroskedastic
Correct Answer: B
Explanation: ADF null hypothesis states the presence of a unit root.
9. One-Step Forecast for AR(1)
For Xt = 0.8 Xt−1 + εt with Var(εt) = 4 and X100 = 5,
what is the 1-step-ahead forecast and its mean squared error?
A. Forecast 4, MSE 1
B. Forecast 4, MSE 4
C. Forecast 0.8, MSE 4
D. Forecast 5, MSE 0.64
Correct Answer: B
Explanation: Forecast = 0.8 × 5 = 4, and MSE = Var(εt) = 4.
10. Forecast Interval Types
2027) TIME SERIES ANALYSIS | QUESTIONS & ANSWERS |
GRADE A | 100% CORRECT –
1. Stationarity Concepts
A time series {Xt} is said to be weakly stationary if certain statistical
properties remain constant over time.
Which of the following best describes this property?
A. All higher-order moments of the distribution remain constant for all t
B. The full joint distribution remains the same under any time shift
C. The mean is constant, and the autocovariance depends only on the lag
D. There is no autocorrelation present at any lag
Correct Answer: C
Explanation: Weak stationarity requires constant mean and autocovariances
depending only on lag.
2. AR(1) Variance Calculation
Consider the stationary AR(1) model
Xt = 0.6 Xt−1 + εt,
where εt is white noise with variance 1. What is the long-run variance of Xt?
A. 1.00
B. 1.25
C. 1.56
D. 2.50
Correct Answer: C
Explanation: Var(X) = σ² / (1 − φ²) = 1 / (1 − 0.36) = 1.5625.
,3. Identifying Model from ACF/PACF
A time series has an autocorrelation function (ACF) with large spikes at lags
1 and 2 and near zero afterward.
The partial autocorrelation function (PACF) decays slowly without a sharp
cutoff. Which model fits best?
A. AR(2)
B. MA(2)
C. ARMA(1,1)
D. AR(1)
Correct Answer: B
Explanation: MA(q) models have an ACF cutoff at lag q and a PACF that
tails off gradually.
4. Seasonality in Time Series
You are analyzing monthly data and the ACF shows strong peaks at lags 12,
24, and 36.
This indicates a repeating seasonal pattern. Which type of model is most
appropriate?
A. Non-seasonal ARIMA(1,1,1)
B. SARIMA(p,d,q)(P,D,Q)12
C. Linear regression with a time trend only
D. Random walk without drift
Correct Answer: B
Explanation: Seasonal ARIMA handles periodic patterns through seasonal
differencing and seasonal terms.
5. Ljung–Box Test Interpretation
,The Ljung–Box test is applied to a time series to assess its autocorrelation
structure.
What is the null hypothesis of this test?
A. The series has a unit root
B. The series is Gaussian
C. There is no autocorrelation up to a specified number of lags
D. The series is heteroskedastic
Correct Answer: C
Explanation: Ljung–Box tests whether autocorrelations jointly equal zero,
indicating white noise.
6. ARIMA Model Representation
Consider an ARIMA(1,1,1) model.
Which of the following correctly represents the model in first-difference
form?
A. Xt = φ Xt−1 + θ εt−1 + εt
B. (1 − φB) ΔXt = (1 + θB) εt
C. ΔXt = φ ΔXt−1 + εt
D. (1 − B) Xt = φ Xt−1 + εt
Correct Answer: B
Explanation: ARIMA(1,1,1) applies both differencing and ARMA structure
to ΔXt.
7. Model Identification from ACF
A time series shows an ACF that cuts off after lag 2 but a PACF that decays
gradually.
Which model is most likely appropriate?
A. AR(2)
B. MA(2)
, C. ARMA(2,2)
D. AR(1)
Correct Answer: B
Explanation: For MA(q), ACF cuts off at lag q, while PACF tails off.
8. ADF Unit Root Test
The Augmented Dickey–Fuller (ADF) test is commonly used to assess
stationarity.
What is the null hypothesis of this test?
A. The series is stationary
B. The series has a unit root (nonstationary)
C. The series is white noise
D. The series is heteroskedastic
Correct Answer: B
Explanation: ADF null hypothesis states the presence of a unit root.
9. One-Step Forecast for AR(1)
For Xt = 0.8 Xt−1 + εt with Var(εt) = 4 and X100 = 5,
what is the 1-step-ahead forecast and its mean squared error?
A. Forecast 4, MSE 1
B. Forecast 4, MSE 4
C. Forecast 0.8, MSE 4
D. Forecast 5, MSE 0.64
Correct Answer: B
Explanation: Forecast = 0.8 × 5 = 4, and MSE = Var(εt) = 4.
10. Forecast Interval Types
