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Time Series Decomposition

Time Series Decomposition. Goals. More practice working with data Regressions and their residuals Seasonal Adjustment. Modeling Time Series. Goal is to distinguish between the deterministic (or predictable) and stochastic (or random) parts Y t = m t + u t

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Time Series Decomposition

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  1. Time Series Decomposition

  2. Goals • More practice working with data • Regressions and their residuals • Seasonal Adjustment

  3. Modeling Time Series • Goal is to distinguish between the deterministic (or predictable) and stochastic (or random) parts Yt = mt + ut • mt is the deterministic component – secular trend, seasonal and cyclical movements • ut is the stochastic component Yt = Tt + Ct + St + ut

  4. Assumptions: Random Component • Typically make three assumptions about ut • Mean zero: E(ut) = 0 • Constant variance/no covariance E(ut ut+i) = s2u if i=0 (Constant variance) E(ut ut+i) = 0 if i0 (Zero covariance) • Normally distributed ut ~ N(0, s2u)

  5. Autocovariances • Covariance between two observations • Example: kth-order autocovariance is the covariance between observations of a time series k periods apart (or lagged k periods) • Cov(Yt Yt-k) • If the autocovariances of a time series are stationary (do not change over time) then they can be used to forecast a series • Autocovariances are a measure of predictability

  6. Autocorrelations • Closely related to autocovariances • Just the correlation between any two observations of a time series • If Cov(Yt Yt-k) is the autocovariance, then cor(Yt Yt-k) = Cov(Yt Yt-k)/var(Yt)

  7. Dummy Variables: Trends • Uses a time variable T (=1,2,3,…) and extrapolates X along its time path Linear: Xt = a + bTt Exponential: X = ea + bTt Reciprocal: X = 1/[a + bTt] Parabolic: X = b0 + b1 Tt,+ b2T2t

  8. Dummy Variables: Seasonal • These are “Intercept shifters” - they allow the intercept term b0 to vary systematically • Single Equation Model with Quarterly Dummies: Yt = g1Q1+g2Q2+g3Q3+g4Q4+b1X1t+…+bkX1k+ut • Can also use monthly dummies if Y is monthly • Get a different forecast for each quarter

  9. Moving Averages • MA(q): Moving average of order q • Generally, for variable X MAqt = (Xt + Xt-1+ … +Xt-q)/q • Can also have centered moving average

  10. Data

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