ISYE 6402 Exam 1: Time Series Analysis
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1. Which of the following is a requirement for a time series to be strictly
stationary?
A. The mean and variance must be constant over time.
B. The autocovariance function depends only on the time lag h.
C. The joint distribution of any set of observations is invariant to time shifts.
D. The series must follow a normal distribution.
Answer: C
Explanation: Strict stationarity requires the entire joint distribution to remain unchanged
when shifted in time, whereas weak stationarity only requires the first two moments
(mean and autocovariance) to be time-invariant.
2. In a weakly stationary time series, the autocovariance function gamma(h) is
defined as:
A. Cov(X_t, X_{t+h})
B. Var(X_t) / Var(X_{t+h})
C. E[X_t] * E[X_{t+h}]
D. Corr(X_t, X_{t+h})
Answer: A
Explanation: The autocovariance at lag h is the covariance between the values of the series
at time t and time t+h.
,3. If a time series is a White Noise process, its Autocorrelation Function (ACF) at
lag h > 0 is:
A. 0
B. 1
C. Decreasing exponentially
D. Constant and equal to the variance
Answer: A
Explanation: White noise consists of uncorrelated random variables, so the correlation
between observations at different time points (h > 0) is zero.
4. Which transformation is most appropriate for stabilizing the variance of a
time series that increases with the level of the series?
A. First-order differencing
B. Moving average smoothing
C. Logarithmic or Box-Cox transformation
D. Seasonal adjustment
Answer: C
Explanation: Logarithmic and Box-Cox transformations are used to stabilize non-constant
variance (heteroscedasticity), while differencing is used to address non-stationarity in the
mean (trends).
5. What is the primary purpose of applying a first-order difference (delta X_t =
X_t - X_{t-1}) to a time series?
A. To remove seasonality
B. To stabilize the variance
C. To remove a linear trend
D. To reduce the noise in the series
Answer: C
, Explanation: First-order differencing is the standard method for removing a linear trend
and making the mean of the series constant.
6. Consider an AR(1) process: X_t = 0.8 * X_{t-1} + w_t. What is the shape of its
ACF?
A. Truncated after lag 1
B. Decays exponentially towards zero
C. Constant at all lags
D. Shows a single spike at lag 1 and zero elsewhere
Answer: B
Explanation: The ACF of an AR(1) process decays exponentially as rho(h) = phi^h. A single
spike at lag 1 in the ACF is characteristic of an MA(1) process.
7. Which plot is used to identify the order ‘p’ of an AR(p) model?
A. Autocorrelation Function (ACF)
B. Normal Q-Q plot
C. Time plot
D. Partial Autocorrelation Function (PACF)
Answer: D
Explanation: For an AR(p) process, the PACF ‘cuts off’ after lag p, meaning it is zero for all
lags greater than p, making it the primary tool for identifying the order p.
8. What characterizes the ACF and PACF of an MA(q) process?
A. ACF cuts off after lag q; PACF decays exponentially
B. ACF decays exponentially; PACF cuts off after lag q
C. Both ACF and PACF cut off after lag q
D. Both ACF and PACF decay exponentially
Answer: A
Explanation: An MA(q) process has an ACF that is zero for lags > q and a PACF that decays
exponentially (possibly with oscillations).
Verified and Latest Questions and Answers -
Georgia Tech
1. Which of the following is a requirement for a time series to be strictly
stationary?
A. The mean and variance must be constant over time.
B. The autocovariance function depends only on the time lag h.
C. The joint distribution of any set of observations is invariant to time shifts.
D. The series must follow a normal distribution.
Answer: C
Explanation: Strict stationarity requires the entire joint distribution to remain unchanged
when shifted in time, whereas weak stationarity only requires the first two moments
(mean and autocovariance) to be time-invariant.
2. In a weakly stationary time series, the autocovariance function gamma(h) is
defined as:
A. Cov(X_t, X_{t+h})
B. Var(X_t) / Var(X_{t+h})
C. E[X_t] * E[X_{t+h}]
D. Corr(X_t, X_{t+h})
Answer: A
Explanation: The autocovariance at lag h is the covariance between the values of the series
at time t and time t+h.
,3. If a time series is a White Noise process, its Autocorrelation Function (ACF) at
lag h > 0 is:
A. 0
B. 1
C. Decreasing exponentially
D. Constant and equal to the variance
Answer: A
Explanation: White noise consists of uncorrelated random variables, so the correlation
between observations at different time points (h > 0) is zero.
4. Which transformation is most appropriate for stabilizing the variance of a
time series that increases with the level of the series?
A. First-order differencing
B. Moving average smoothing
C. Logarithmic or Box-Cox transformation
D. Seasonal adjustment
Answer: C
Explanation: Logarithmic and Box-Cox transformations are used to stabilize non-constant
variance (heteroscedasticity), while differencing is used to address non-stationarity in the
mean (trends).
5. What is the primary purpose of applying a first-order difference (delta X_t =
X_t - X_{t-1}) to a time series?
A. To remove seasonality
B. To stabilize the variance
C. To remove a linear trend
D. To reduce the noise in the series
Answer: C
, Explanation: First-order differencing is the standard method for removing a linear trend
and making the mean of the series constant.
6. Consider an AR(1) process: X_t = 0.8 * X_{t-1} + w_t. What is the shape of its
ACF?
A. Truncated after lag 1
B. Decays exponentially towards zero
C. Constant at all lags
D. Shows a single spike at lag 1 and zero elsewhere
Answer: B
Explanation: The ACF of an AR(1) process decays exponentially as rho(h) = phi^h. A single
spike at lag 1 in the ACF is characteristic of an MA(1) process.
7. Which plot is used to identify the order ‘p’ of an AR(p) model?
A. Autocorrelation Function (ACF)
B. Normal Q-Q plot
C. Time plot
D. Partial Autocorrelation Function (PACF)
Answer: D
Explanation: For an AR(p) process, the PACF ‘cuts off’ after lag p, meaning it is zero for all
lags greater than p, making it the primary tool for identifying the order p.
8. What characterizes the ACF and PACF of an MA(q) process?
A. ACF cuts off after lag q; PACF decays exponentially
B. ACF decays exponentially; PACF cuts off after lag q
C. Both ACF and PACF cut off after lag q
D. Both ACF and PACF decay exponentially
Answer: A
Explanation: An MA(q) process has an ACF that is zero for lags > q and a PACF that decays
exponentially (possibly with oscillations).
