ISYE 6402 Time Series Analysis Midterm
1 Practice Exam with Complete Solutions
| Mt. Everest Temperature Data |
Questions and Answers | 2026 Update |
100% Correct - GT.INSTANT DOWNLOAD
Q1: Stationarity Definition
Question: If the mean of a time series does not depend on time t, is the
time series stationary?
Answer: False
Rationale: While a constant mean is a necessary condition for stationarity,
it is not sufficient. Weak stationarity requires:
Constant mean
Constant variance
Autocovariance that depends only on lag (not time)
Q2: Random Walk Properties
Question: For a random walk process St=∑j=1tXjSt=∑j=1tXj
where Xt∼IID(0,σ2)Xt∼IID(0,σ2), is Var(St)>Var(St−1)Var(St)>Var(St−1
)?
Answer: True
,Rationale: For a random walk, Var(St)=tσ2Var(St)=tσ2, which is greater
than Var(St−1)=(t−1)σ2Var(St−1)=(t−1)σ2. The variance grows linearly
with time, which violates stationarity.
Q3: White Noise Properties
Question: Are consecutive observations in a white noise process always
independent?
Answer: False
Rationale: White noise observations are uncorrelated but not
necessarily independent. Independence is a stronger condition that only
holds if the process is Gaussian.
ARIMA Model Concepts
Q4: The 'I' in ARIMA
Question: What does the 'I' component represent in ARIMA modeling?
Answer: The differencing needed to make the time series stationary
Rationale: 'I' stands for "Integrated" and refers to differencing operations
to stabilize the mean of the time series. The number of differences required
is denoted as 'd'.
Q5: AR(1) Properties
Question: For an AR(1) model Yt=0.8Yt−1+ϵtYt=0.8Yt−1+ϵt, is the process
stationary?
Answer: True
,Rationale: The stationarity condition for an AR(1) model is ∣ϕ1∣<1∣ϕ1∣<1.
Since ∣0.8∣=0.8<1∣0.8∣=0.8<1, the process is stationary.
Additional Property: The ACF of an AR(1) model decays exponentially to
zero: ρk=ϕkρk =ϕk .
Q6: AR(1) Causality Condition
Question: Is the AR(1) process causal if and only if the autoregressive
parameter φ is between 0 and 1?
Answer: False
Rationale: The AR(1) process is causal if ∣ϕ∣<1∣ϕ∣<1 (absolute value less
than 1), not just between 0 and 1. This allows for negative values as well.
MA and ARMA Models
Q7: MA(q) ACF Properties
Question: What is the ACF pattern for an MA(1) process?
Answer: The ACF cuts off after lag 1.
Rationale: For an MA(q) model, the ACF is zero for all lags greater than q.
For q=1, ρk=0ρk =0 for k>1k>1. This is the key identifying feature of MA
models.
Q8: MA(q) Stationarity
Question: Is an MA(q) process always stationary?
Answer: True
, Rationale: MA models are finite linear combinations of white noise. As long
as the MA coefficients are finite, the process is always stationary regardless
of coefficient values.
Q9: MA(1) Invertibility
Question: Is an MA(1) model with θ = 1.5 invertible?
Answer: No
Rationale: Invertibility requires ∣θ∣<1∣θ∣<1. Since 1.5 > 1, this model is not
invertible.
Q10: ARMA Forecasting Properties
Question: What happens to forecast error for an MA(1) model after 1 step
ahead?
Answer: The forecast error variance becomes constant after 1 step ahead.
Rationale: For an MA(q) model, the process has no "memory" beyond q.
For horizons h > q, the forecast is just the mean, and the forecast error
variance becomes constant.
Model Diagnostics
Q11: Ljung-Box Test
Question: What is the null hypothesis of the Ljung-Box test?
Answer: The residuals are uncorrelated (white noise).
1 Practice Exam with Complete Solutions
| Mt. Everest Temperature Data |
Questions and Answers | 2026 Update |
100% Correct - GT.INSTANT DOWNLOAD
Q1: Stationarity Definition
Question: If the mean of a time series does not depend on time t, is the
time series stationary?
Answer: False
Rationale: While a constant mean is a necessary condition for stationarity,
it is not sufficient. Weak stationarity requires:
Constant mean
Constant variance
Autocovariance that depends only on lag (not time)
Q2: Random Walk Properties
Question: For a random walk process St=∑j=1tXjSt=∑j=1tXj
where Xt∼IID(0,σ2)Xt∼IID(0,σ2), is Var(St)>Var(St−1)Var(St)>Var(St−1
)?
Answer: True
,Rationale: For a random walk, Var(St)=tσ2Var(St)=tσ2, which is greater
than Var(St−1)=(t−1)σ2Var(St−1)=(t−1)σ2. The variance grows linearly
with time, which violates stationarity.
Q3: White Noise Properties
Question: Are consecutive observations in a white noise process always
independent?
Answer: False
Rationale: White noise observations are uncorrelated but not
necessarily independent. Independence is a stronger condition that only
holds if the process is Gaussian.
ARIMA Model Concepts
Q4: The 'I' in ARIMA
Question: What does the 'I' component represent in ARIMA modeling?
Answer: The differencing needed to make the time series stationary
Rationale: 'I' stands for "Integrated" and refers to differencing operations
to stabilize the mean of the time series. The number of differences required
is denoted as 'd'.
Q5: AR(1) Properties
Question: For an AR(1) model Yt=0.8Yt−1+ϵtYt=0.8Yt−1+ϵt, is the process
stationary?
Answer: True
,Rationale: The stationarity condition for an AR(1) model is ∣ϕ1∣<1∣ϕ1∣<1.
Since ∣0.8∣=0.8<1∣0.8∣=0.8<1, the process is stationary.
Additional Property: The ACF of an AR(1) model decays exponentially to
zero: ρk=ϕkρk =ϕk .
Q6: AR(1) Causality Condition
Question: Is the AR(1) process causal if and only if the autoregressive
parameter φ is between 0 and 1?
Answer: False
Rationale: The AR(1) process is causal if ∣ϕ∣<1∣ϕ∣<1 (absolute value less
than 1), not just between 0 and 1. This allows for negative values as well.
MA and ARMA Models
Q7: MA(q) ACF Properties
Question: What is the ACF pattern for an MA(1) process?
Answer: The ACF cuts off after lag 1.
Rationale: For an MA(q) model, the ACF is zero for all lags greater than q.
For q=1, ρk=0ρk =0 for k>1k>1. This is the key identifying feature of MA
models.
Q8: MA(q) Stationarity
Question: Is an MA(q) process always stationary?
Answer: True
, Rationale: MA models are finite linear combinations of white noise. As long
as the MA coefficients are finite, the process is always stationary regardless
of coefficient values.
Q9: MA(1) Invertibility
Question: Is an MA(1) model with θ = 1.5 invertible?
Answer: No
Rationale: Invertibility requires ∣θ∣<1∣θ∣<1. Since 1.5 > 1, this model is not
invertible.
Q10: ARMA Forecasting Properties
Question: What happens to forecast error for an MA(1) model after 1 step
ahead?
Answer: The forecast error variance becomes constant after 1 step ahead.
Rationale: For an MA(q) model, the process has no "memory" beyond q.
For horizons h > q, the forecast is just the mean, and the forecast error
variance becomes constant.
Model Diagnostics
Q11: Ljung-Box Test
Question: What is the null hypothesis of the Ljung-Box test?
Answer: The residuals are uncorrelated (white noise).
