QA10 - Stationary Time Series

Question 1

Describe the requirements for a series to be covariance-stationary

Answer 1

First two moments must satisfy all three conditions:
1. E(Yt) = mu for all t
2. Var(Yt) = γ0 < infinity for all t
3. Cov(Yt, Y_(t-h)) = γh for all t

Question 2

Describe the autocovariance function and the autocorrelation function

Answer 2

Autocovariance γ_t,y = E[(Yt - E(Yt))(Y_(t-h) - E(Y_(t-h))]

Autocorrelation rho_h = γh / γ0

Question 3

Define white noise, and describe independent white noise and normal (Gaussian) white noise

Answer 3

White noise comes from a distribution with mean zero, independent when iid

Normal white noise is when random samples come from a normal distribution

Question 4

Define and describe the properties of autoregressive (AR) processes

Answer 4

Autoregressive models relate value of stochastic process to the previous one

AR(1): Yt = d + φY_(t-1) + e_t

Covariance stationary when |φ| < 1, non-stationary when φ = 1

mu = d / (1 - φ)

variance = sigma^2 / (1 - φ^2)

Question 5

Define and describe the properties of moving average (MA) processes

Answer 5

Relates value to the shock of the previous values

MA(1) = Yt = mu + θe_(t-1) + e_t

E(Yt) = mu
Var(Yt) = (1 + θ^2) * simga^2

Question 6

Explain mean reversion and calculate a mean-reverting level

Answer 6

Mean reversion is where a stochastic process returns to its mean eventually

Question 7

Define and describe the properties of autoregressive moving average (ARMA) processes

Answer 7

ARMA(1,1): Yt = d + φY_(t-1) + θe_(t-1) + e_t

mu = d / (1 - φ)

Question 8

Describe sample autocorrelation and partial autocorrelation

Answer 8

Applied to ARMA models to understand dependence structure and select candidate ARMA models

Question 9

Describe the Box-Pierce Q statistic and the Ljung-Box Q statistic

Answer 9

Used to do joint tests of joint auto correlations when validating a model, the are roughly equal for large t

Question 10

Explain how forecasts are generated from ARMA models

Answer 10

Expectation of future value requires taking expectation of previous observations

Question 11

Explain how seasonality is modelled in a covariance-stationary ARMA

Answer 11

Pure seasonal model only uses lags at seasonal frequency, for example with quarterly seasonality:
AR(1): Yt = d + φY_(t-4) + e_t