Time series are used to model processes that are discrete. These processes can be genuinely discrete or they can be continuous, but we observe them periodically. 

For instance, we can use time series to model corporation cuμlative income. The income of a corporation is continuous, but we may only observe them quarterly in their quarter reports.

The model is solved when we can describe or forecast the distribution of values.

Examples of Linear Time Series

MA(1) moving average model

r_t = μ + σ*z_t + ɸ*z_t−1, where z_t ~ iid(0, σ²)

A MA(1) model depends on both the current and previous z_t. The video below elaborates what a MA(1) process is.

GARCH model

r_t = μ + σ_tz_t

For a GARCH model, the distribution itself is time−varying as σ is time-dependent.

AR(p) autoregressive model of order p

R_t = c₀ + c₁R_t−1 + … + c_pR_t−p + σ*z_t, where z_t ~ IID(0, 1)

Here, the output variable R_t depends linearly on its own previous values R_t-p and on a stochastic term z_t.

ARMA(p, q)

R_t = c₀ + c₁R_t−1 + … + c_pR_t−p + σ*z_t + ɸ*z_t−1 + … + ɸ_q*z_t−q

This model combines the AR and MA models.

Similarities between the 4 Models

• They all have a linear structure
• Past observations determine likelihood of future outcomes
• Their coefficients are to be determined