Module 17: Time series analysis Flashcards
Strict Stationarity
A process is strictly stationary if its characteristics do not change over time,
ie f(X_{r}, X_{r+1}, ..., X_{s}) = f(X_{r+k}, X_{r+1+k}, ..., X_{s+k})
Strict stationarity is restrictive and unlikely in real-world data.
Weak Stationarity
A process is said to be weakly stationary of order n if the moments of subsets of the process are equal (and finite) up to the nth moment.
Covariance Stationarity
A process is covariance stationary if it is weakly stationary of order n if the moments of subsets of the process are equal (and finite) up to the nth moment.
White noise
A process {εₜ} that has a mean of 0 and a variance of σ².
Each observation is uncorrelated with previous observations.
White noise is covariance stationary.
If, in addition, the process is iid, then the series is strictly stationary.
Trend-Stationarity
A trend-stationary process is one where the observations oscillate randomly around a trend line α₀ + α₁ t that is a function of time only.
Integrated process of order d
An I(d) process is one where the process needs to be differenced d times before the result, Δᵈ Xₜ, is covariance stationary.
Difference stationary process
An I(1) process, i.e. ΔXₜ is covariance stationary.
Autoregressive process of order p
An AR(p) process is one where each observation is a linear combination of the p previous values plus a random error.
Moving average process of oder p
An MA(q) process is one where each observation is a linear combination of the q previous error terms plus a current random error term.
An autogregressive process can be defined in moving average terms and vice versa.
Durbin-Watson statistic
The Durbin-Watson statistic can be used to test for serial correlation in the observations. There is no such correlation in a true moving average process.
Autoregressive moving average time series
Combining an AR(p) process and an MA(q) process results in an autoregressive moving average process ARMA(p,q).
Integrated autoregressive moving average process
An integrated autogregressive moving average process, or ARIMA(p,d,q) process, is one where the dth difference is a stationary ARMA(p,q) process.
It is possible to model 3 specific features of the data
- seasonality (using indicator variables)
- step changes in the value of the process (using a Poisson variable)
- altered rates of change (eg of the drift, rate of mean reversion, time trend).
Heteroskedasticity
A heteroskedastic time series is one where the variance changes over time.
Autoregressive conditional heteroskedastic process
An ARCH process is constructed so that the volatility varies over time.
A large change in the previous values of the process is often followed by a period of high volatility (volatility clustering).
