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Questions and Answers
Which property must hold for the AR(1) model to be considered stationary?
Which property must hold for the AR(1) model to be considered stationary?
In a moving average process MA(q), what does the model utilize to achieve forecasts?
In a moving average process MA(q), what does the model utilize to achieve forecasts?
What formula represents the random walk model?
What formula represents the random walk model?
Under what condition is the AR(2) model stationary?
Under what condition is the AR(2) model stationary?
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Which statement about the linear trend model is true?
Which statement about the linear trend model is true?
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What does the MA(q) model output represent?
What does the MA(q) model output represent?
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Which process is characterized primarily by its random shocks or errors?
Which process is characterized primarily by its random shocks or errors?
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What does $Y_t = c + heta_1 Y_{t-1} + heta_2 Y_{t-2} + e_t$ represent?
What does $Y_t = c + heta_1 Y_{t-1} + heta_2 Y_{t-2} + e_t$ represent?
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Study Notes
White Noise Process
- A sequence of random errors (et) is independent and identically distributed (iid)
- Each error has a mean of 0
- Each error has a variance of σ2
- The time series (Yt) is simply the error: Yt = et
Moving Average Process
- A sequence of random errors (et) is iid
- Each error has a mean of 0
- Each error has a variance of σ2
- The time series (Yt) is a combination of the current error and a weighted previous error: Yt = et + 0.5et-1
Random Walk
- A sequence of random errors (et) is iid
- Each error has a mean of 0
- Each error has a variance of σ2
- The time series (Yt) is the sum of all past errors: Yt = e1 + e2 + ... + et which equals Yt = Yt-1 + et
Linear Trend
- A sequence of random errors (et) is iid
- Each error has a mean of 0
- Each error has a variance of σ2
- The time series (Yt) is a linear function of time with added noise: Yt = a + bt + et
- a and b are constants
Autoregressive Process (AR(p))
- Forecasts the variable of interest using a linear combination of past values.
- If the model uses the last p values, it's called an AR(p) model.
- The time series (Yt) is a linear combination of past values plus an error: Yt = c + Φ1Yt-1 + Φ2Yt-2 + ... + ΦpYt-p + et
AR(1)
- A special case of AR(p) where p=1
- Yt = c + Φ1Yt-1 + et
- The AR(1) model is stationary if -1 < Φ1 < 1
AR(2)
- A special case of AR(p) where p=2
- Yt = c + Φ1Yt-1 + Φ2Yt-2 + et
- The AR(2) model is stationary if: -1 < Φ2 < 1, Φ1 + Φ2 < 1, Φ2 - Φ1 < 1
Moving Average Process (MA(q))
- Rather than using past variable values, a MA model uses past errors to forecast
- If the model uses the last q errors, it's called an MA(q) model
- The time series (Yt) is a weighted moving average of past forecast errors: Yt = c + et + θ1et-1 + ... + θqet-q
Autoregressive Moving Average Process (ARMA(p,q))
- A combination of AR(p) and MA(q) models
- The time series (Yt) is a combination of past values and past errors: Yt = Φ1Yt-1 + ... + ΦpYt-p + et - θ1et-1 - ... - θqet-q
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Description
Explore the concepts of White Noise, Moving Average, Random Walk, and Linear Trend in time series analysis. This quiz will test your understanding of these foundational time series models, their characteristics, and mathematical formulations. Get ready to apply your knowledge in this engaging quiz!
