L2 - Time series Flashcards
What is a stochastic process and its associated notiation?
A stochastic process is a sequence of random variables
indexed by time.
We will use the notation {yₜ: 1, 2, . . .}, or {yₜ} to denote this sequence
How do we represent the mean (µ)?
E[yₜ]
How do we represent the variance (γ₀)?
E[(yₜ - µ)²]
How do we represent the autocovariance (γⱼ)?
E[(yₜ - µ)(yₜ ₋ ⱼ - µ)]
In order to ‘operationalise’ the assumption that the future behaves like the past, what do we need to do?
We impose restrictions on autocovariances (Stationarity).
What is Stationarity?
A stationary time series process is a process whose probability distributions are stable over time
What does it mean to be weekly stationary?
mean, variance and covariances are stable:
E[yₜ] = µ < ∞
Var(yₜ) = γ₀ < ∞
Cov(γⱼ, yₜ ₋ ⱼ) = γⱼ
Why do we need to impose stability in a stationary process?
If the relationship between y and x keeps changing randomly over time, we won’t be able to understand how a change in x affects y
What is the equation of standardised autocovariances (ACF/correlogram)
ρⱼ = Cov(γⱼ, yₜ ₋ ⱼ) / Var(yₜ) = γⱼ / γ₀
−1 ≤ ρⱼ ≤ 1
What is a ‘White Noise’ process
A type of time series that is not predictable. It must:
E[εₜ] = 0 for all t
E[εₜεₛ] = σ² for t = s
= 0 for t ≠ s
What does it mean to be ‘serially uncorrelated’
A stochastic process with zero correlation across time periods
One examples of a stationary process is the ‘AR’ model.
Model an AR(1) process.
yₜ = θyₜ₋₁ + εₜ
where εₜ is white noise
Model an AR(3) process
yₜ = θ₁yₜ₋₁ + θ₂yₜ₋₂ + θ₃yₜ₋₃ + εₜ
where εₜ is white noise
One examples of a stationary process is the ‘MA’ model.
Model an MA(1) process.
yₜ = εₜ + αεₜ₋₁
where εₜ is white noise
Model an MA(3) process.
yₜ = εₜ + αεₜ₋₁ + αεₜ₋₂ + αεₜ₋₃
where εₜ is white noise
