ISYE 6402 Midterm | Georgia Tech ISYE 6402 Time Series Analysis Midterm
Study Guide & Exam Prep 2026–2027 | ISYE 6402 Midterm 1 & Midterm 2
Review | Georgia Tech Time Series Analysis, Time Series Fundamentals,
Stationarity, Weak & Strict Stationarity, Autocorrelation, Autocovariance,
Partial Autocorrelation, White Noise, Random Walks, Trend & Seasonality,
Time Series Decomposition, Forecasting, Moving Average, Exponential
Smoothing, Holt’s Method, Regression for Time Series, AR Models, MA
Models, ARMA Models, ARIMA Models, Differencing, Model Identification,
Parameter Estimation, Model Diagnostics, ACF & PACF, Residual Analysis,
GARCH Models, VAR Models, Cointegration, Multivariate Time Series,
Forecast Accuracy & Time Series Forecasting | Practice Questions, Data
Analysis, R/Python Exercises & Detailed Rationales
Question 1: In time series analysis, which condition is necessary for a process to
be considered weakly stationary?
A. The process must be strictly stationary with all moments time-invariant
B. The mean and variance are constant over time, and autocovariance depends
only on lag
C. The process must have no trend but may have time-varying variance
D. The autocorrelation function must decay exponentially to zero
CORRECT ANSWER: B. The mean and variance are constant over time, and
autocovariance depends only on lag
Rationale: Weak stationarity (covariance stationarity) requires constant mean,
constant variance, and autocovariance that depends only on the lag between
observations, not on time itself. Strict stationarity is a stronger condition requiring
the entire joint distribution to be time-invariant.
Question 2: A time series exhibits a sample ACF that decays slowly and remains
positive for many lags. What does this pattern most likely indicate?
A. The series is stationary with short memory
B. The series contains a unit root and requires differencing
C. The series follows a pure MA(1) process
D. The series has no autocorrelation and is white noise
CORRECT ANSWER: B. The series contains a unit root and requires differencing
,Rationale: Slow, linear decay in the ACF is a hallmark of non-stationary processes
such as random walks. Stationary ARMA processes typically show rapid
exponential decay or cutoff patterns. First differencing is commonly applied to
achieve stationarity.
Question 3: For an AR(1) process Yt=ϕYt−1+ϵtYt=ϕYt−1+ϵt, what condition
ensures the process is stationary?
A. ϕ=1ϕ=1
B. ∣ϕ∣<1∣ϕ∣<1
C. ϕ>0ϕ>0
D. ∣ϕ∣>1∣ϕ∣>1
CORRECT ANSWER: B. ∣ϕ∣<1∣ϕ∣<1
Rationale: The AR(1) process is stationary if and only if the absolute value of the
autoregressive parameter is less than one. When ∣ϕ∣<1∣ϕ∣<1, the effect of past
shocks decays over time. If ϕ=1ϕ=1, the process is a random walk (non-
stationary).
Question 4: In the method of moments estimation for AR parameters, which
system of equations is used to relate autocovariances to model parameters?
A. Newton-Raphson equations
B. Yule-Walker equations
C. Kalman recursion equations
D. Cramér-Rao equations
CORRECT ANSWER: B. Yule-Walker equations
Rationale: The Yule-Walker equations express the theoretical autocovariances of
an AR process in terms of the autoregressive parameters. Solving these equations
provides method-of-moments estimates for the AR coefficients.
Question 5: Which information criterion penalizes model complexity more
heavily, favoring more parsimonious models?
A. AIC
B. BIC
,C. Adjusted R-squared
D. Mean squared error
CORRECT ANSWER: B. BIC
Rationale: The Bayesian Information Criterion (BIC) includes a penalty term of
kln(n)kln(n), which grows with sample size. This penalty is larger than the AIC's
2k2k penalty when n>7n>7, making BIC more conservative in selecting
parsimonious models.
Question 6: After fitting an ARIMA model, a Ljung-Box test on residuals yields a
p-value of 0.02. What is the appropriate interpretation?
A. The residuals are white noise and the model is adequate
B. There is significant autocorrelation remaining, suggesting model inadequacy
C. The model is overparameterized and should be simplified
D. The residuals follow a normal distribution
CORRECT ANSWER: B. There is significant autocorrelation remaining, suggesting
model inadequacy
Rationale: The Ljung-Box test has a null hypothesis of no autocorrelation in
residuals up to a specified lag. A p-value below the significance level (e.g., 0.05)
leads to rejection of the null, indicating that the model has not captured all serial
dependence.
Question 7: For a random walk process St=∑j=1tXjSt=∑j=1tXj where
Xj∼IID(0,σ2)Xj∼IID(0,σ2), what is the variance of StSt?
A. σ2σ2
B. tσ2tσ2
C. σ2/tσ2/t
D. t2σ2t2σ2
CORRECT ANSWER: B. tσ2tσ2
Rationale: The variance of a random walk grows linearly with time. Each
additional step adds independent noise with variance σ2σ2, so Var(St)=tσ2Var(St
)=tσ2. This time-dependent variance violates the stationarity requirement.
, Question 8: Which transformation is most appropriate to stabilize variance
when the variability of a time series increases with its level?
A. First differencing
B. Logarithmic transformation
C. Moving average smoothing
D. Seasonal adjustment
CORRECT ANSWER: B. Logarithmic transformation
Rationale: When variance grows proportionally with the mean level, a logarithmic
(or Box-Cox) transformation stabilizes variance. Differencing addresses trends and
unit roots, not variance heterogeneity.
Question 9: A time series has an ACF that cuts off after lag 2 and a PACF that
decays gradually. Which model specification is most appropriate?
A. AR(2)
B. MA(2)
C. ARMA(2,2)
D. AR(1)
CORRECT ANSWER: B. MA(2)
Rationale: For a moving average process of order q, the theoretical ACF cuts off
after lag q, while the PACF decays gradually (tails off). An AR(2) would show the
opposite pattern: PACF cutoff after lag 2 with ACF tailing off.
Question 10: Which test is specifically designed to test the null hypothesis that a
time series has a unit root?
A. Ljung-Box test
B. Augmented Dickey-Fuller test
C. Jarque-Bera test
D. Breusch-Pagan test
CORRECT ANSWER: B. Augmented Dickey-Fuller test
Rationale: The Augmented Dickey-Fuller (ADF) test evaluates whether a time
series contains a unit root. The null hypothesis is that a unit root is present. Ljung-
Study Guide & Exam Prep 2026–2027 | ISYE 6402 Midterm 1 & Midterm 2
Review | Georgia Tech Time Series Analysis, Time Series Fundamentals,
Stationarity, Weak & Strict Stationarity, Autocorrelation, Autocovariance,
Partial Autocorrelation, White Noise, Random Walks, Trend & Seasonality,
Time Series Decomposition, Forecasting, Moving Average, Exponential
Smoothing, Holt’s Method, Regression for Time Series, AR Models, MA
Models, ARMA Models, ARIMA Models, Differencing, Model Identification,
Parameter Estimation, Model Diagnostics, ACF & PACF, Residual Analysis,
GARCH Models, VAR Models, Cointegration, Multivariate Time Series,
Forecast Accuracy & Time Series Forecasting | Practice Questions, Data
Analysis, R/Python Exercises & Detailed Rationales
Question 1: In time series analysis, which condition is necessary for a process to
be considered weakly stationary?
A. The process must be strictly stationary with all moments time-invariant
B. The mean and variance are constant over time, and autocovariance depends
only on lag
C. The process must have no trend but may have time-varying variance
D. The autocorrelation function must decay exponentially to zero
CORRECT ANSWER: B. The mean and variance are constant over time, and
autocovariance depends only on lag
Rationale: Weak stationarity (covariance stationarity) requires constant mean,
constant variance, and autocovariance that depends only on the lag between
observations, not on time itself. Strict stationarity is a stronger condition requiring
the entire joint distribution to be time-invariant.
Question 2: A time series exhibits a sample ACF that decays slowly and remains
positive for many lags. What does this pattern most likely indicate?
A. The series is stationary with short memory
B. The series contains a unit root and requires differencing
C. The series follows a pure MA(1) process
D. The series has no autocorrelation and is white noise
CORRECT ANSWER: B. The series contains a unit root and requires differencing
,Rationale: Slow, linear decay in the ACF is a hallmark of non-stationary processes
such as random walks. Stationary ARMA processes typically show rapid
exponential decay or cutoff patterns. First differencing is commonly applied to
achieve stationarity.
Question 3: For an AR(1) process Yt=ϕYt−1+ϵtYt=ϕYt−1+ϵt, what condition
ensures the process is stationary?
A. ϕ=1ϕ=1
B. ∣ϕ∣<1∣ϕ∣<1
C. ϕ>0ϕ>0
D. ∣ϕ∣>1∣ϕ∣>1
CORRECT ANSWER: B. ∣ϕ∣<1∣ϕ∣<1
Rationale: The AR(1) process is stationary if and only if the absolute value of the
autoregressive parameter is less than one. When ∣ϕ∣<1∣ϕ∣<1, the effect of past
shocks decays over time. If ϕ=1ϕ=1, the process is a random walk (non-
stationary).
Question 4: In the method of moments estimation for AR parameters, which
system of equations is used to relate autocovariances to model parameters?
A. Newton-Raphson equations
B. Yule-Walker equations
C. Kalman recursion equations
D. Cramér-Rao equations
CORRECT ANSWER: B. Yule-Walker equations
Rationale: The Yule-Walker equations express the theoretical autocovariances of
an AR process in terms of the autoregressive parameters. Solving these equations
provides method-of-moments estimates for the AR coefficients.
Question 5: Which information criterion penalizes model complexity more
heavily, favoring more parsimonious models?
A. AIC
B. BIC
,C. Adjusted R-squared
D. Mean squared error
CORRECT ANSWER: B. BIC
Rationale: The Bayesian Information Criterion (BIC) includes a penalty term of
kln(n)kln(n), which grows with sample size. This penalty is larger than the AIC's
2k2k penalty when n>7n>7, making BIC more conservative in selecting
parsimonious models.
Question 6: After fitting an ARIMA model, a Ljung-Box test on residuals yields a
p-value of 0.02. What is the appropriate interpretation?
A. The residuals are white noise and the model is adequate
B. There is significant autocorrelation remaining, suggesting model inadequacy
C. The model is overparameterized and should be simplified
D. The residuals follow a normal distribution
CORRECT ANSWER: B. There is significant autocorrelation remaining, suggesting
model inadequacy
Rationale: The Ljung-Box test has a null hypothesis of no autocorrelation in
residuals up to a specified lag. A p-value below the significance level (e.g., 0.05)
leads to rejection of the null, indicating that the model has not captured all serial
dependence.
Question 7: For a random walk process St=∑j=1tXjSt=∑j=1tXj where
Xj∼IID(0,σ2)Xj∼IID(0,σ2), what is the variance of StSt?
A. σ2σ2
B. tσ2tσ2
C. σ2/tσ2/t
D. t2σ2t2σ2
CORRECT ANSWER: B. tσ2tσ2
Rationale: The variance of a random walk grows linearly with time. Each
additional step adds independent noise with variance σ2σ2, so Var(St)=tσ2Var(St
)=tσ2. This time-dependent variance violates the stationarity requirement.
, Question 8: Which transformation is most appropriate to stabilize variance
when the variability of a time series increases with its level?
A. First differencing
B. Logarithmic transformation
C. Moving average smoothing
D. Seasonal adjustment
CORRECT ANSWER: B. Logarithmic transformation
Rationale: When variance grows proportionally with the mean level, a logarithmic
(or Box-Cox) transformation stabilizes variance. Differencing addresses trends and
unit roots, not variance heterogeneity.
Question 9: A time series has an ACF that cuts off after lag 2 and a PACF that
decays gradually. Which model specification is most appropriate?
A. AR(2)
B. MA(2)
C. ARMA(2,2)
D. AR(1)
CORRECT ANSWER: B. MA(2)
Rationale: For a moving average process of order q, the theoretical ACF cuts off
after lag q, while the PACF decays gradually (tails off). An AR(2) would show the
opposite pattern: PACF cutoff after lag 2 with ACF tailing off.
Question 10: Which test is specifically designed to test the null hypothesis that a
time series has a unit root?
A. Ljung-Box test
B. Augmented Dickey-Fuller test
C. Jarque-Bera test
D. Breusch-Pagan test
CORRECT ANSWER: B. Augmented Dickey-Fuller test
Rationale: The Augmented Dickey-Fuller (ADF) test evaluates whether a time
series contains a unit root. The null hypothesis is that a unit root is present. Ljung-
