Week 2.2 Time Series Models PDF
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This document provides an overview of basic time series models such as white noise, moving average, random walk, linear trend, autoregressive (AR) processes, and autoregressive moving average (ARMA) processes. It includes descriptions and explanations of each model and discusses their characteristics and applications.
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Basic Time Series Models White noise process Let 𝑒1 , 𝑒2 , … be a sequence of random errors (iid) with 0 mean and variance 𝜎𝑒2. Time series 𝑌𝑡 is obtained as: 𝑌𝑡 = 𝑒𝑡 1 Basic Time Series M...
Basic Time Series Models White noise process Let 𝑒1 , 𝑒2 , … be a sequence of random errors (iid) with 0 mean and variance 𝜎𝑒2. Time series 𝑌𝑡 is obtained as: 𝑌𝑡 = 𝑒𝑡 1 Basic Time Series Models Moving average process Let 𝑒1 , 𝑒2 , … be a sequence of random errors (iid) with 0 mean and variance 𝜎𝑒2. Time series 𝑌𝑡 is obtained as: 𝑌𝑡 = 𝑒𝑡 + 0.5𝑒𝑡−1 2 Basic Time Series Models Random walk Let 𝑒1 , 𝑒2 , … be a sequence of random errors (iid) with 0 mean and variance 𝜎𝑒2. Time series 𝑌𝑡 is obtained as: 𝑌1 = 𝑒1 𝑌2 = 𝑒1 + 𝑒2 ⇒ 𝑌2 = 𝑌1 + 𝑒2 𝑌3 = 𝑒1 + 𝑒2 + 𝑒3 ⇒ 𝑌3 = 𝑌2 + 𝑒3.. 𝑌𝑡 = 𝑒1 + ⋯ + 𝑒𝑡 ⇒ 𝑌𝑡 = 𝑌𝑡−1 + 𝑒𝑡 3 Basic Time Series Models Linear trend Let 𝑒1 , 𝑒2 , … be a sequence of random errors (iid) with 0 mean and variance 𝜎𝑒2. Time series 𝑌𝑡 is obtained as: 𝑌1 = 𝑎 + 𝑏𝑡 + 𝑒𝑡 Where 𝑎, 𝑏 are constants. 4 Autoregressive Process: AR(p) We forecast the variable of interest using a linear combination of past values of the variable. If the model uses last 𝑝 values, it’s called as an AR(p) model. 𝑌𝑡 = 𝑐 + 𝜙1 𝑌𝑡−1 + 𝜙2 𝑌𝑡−2 + ⋯ + 𝜙𝑝 𝑌𝑡−𝑝 + 𝑒𝑡 Why is it called as auto-regressive? 5 Basic Autoregressive Models AR(1) 𝑌𝑡 = 𝑐 + 𝜙1 𝑌𝑡−1 + 𝑒𝑡 AR(1) model is stationary if: −1 < 𝜙1 < 1 6 Basic Autoregressive Models AR(2) 𝑌𝑡 = 𝑐 + 𝜙1 𝑌𝑡−1 + 𝜙2 𝑌𝑡−2 + 𝑒𝑡 AR(2) model is stationary if: −1 < 𝜙2 < 1, 𝜙1 + 𝜙2 < 1, 𝜙2 − 𝜙1 < 1 7 Moving Average Process: MA(q) Rather than using the past values of the forecast variable in a regression, a MA model uses the past errors in a regression like model. If the model uses last q errors, it’s called as a MA(q) model. 𝑌𝑡 = 𝑐 + 𝑒𝑡 + 𝜃1 𝑒𝑡−1 + ⋯ + 𝜃𝑞 𝑒𝑡−𝑞 Each value of 𝑌𝑡 can be considered as a weighted moving average of past forecast errors. A MA model is suitable for an irregular component 8 Autoregressive Moving Average Process: ARMA(p,q) Combination of AR(p) and MA(q) models 𝑌𝑡 = 𝜙1 𝑌𝑡−1 + ⋯ + 𝜙𝑝 𝑌𝑡−𝑝 + 𝑒𝑡 − 𝜃1 𝑒𝑡−1 − ⋯ − 𝜃𝑞 𝑒𝑡−𝑞 9 Simulated Processes White Noise 10 Simulated Processes Stationary AR(1) Coefficient is 0.7 11 Simulated Processes Random Walk 12
