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Ec 178 – ECON/BUSINESS FORECASTING LECTURE NOTES Foster, UCSD February 25, 2010 TOPIC 3 - SEASONAL MODELS A. Seasonal Data 1. Seasonal Patterns: a) Seasonal pattern -- regular fluctuation in a data series which repeats every M periods. Annual pattern in quarterly data (M = 4) or monthly data (M = 12) Daily pattern in hourly data (M = 24) b) Seasonal pattern is typically only one component of raw series y t , along with trend, cycle and random components. c) Notation and illustration. Symb ol Definition M T t A(j) A t y(sa) t C t u t Length of seasonality Trend component at time t j th seasonal term, j = 1. ..M Seasonal component at time t Seasonally-adjusted y t series Cyclical component at time t Random error at time t 2. Additive v. Multiplicative Seasonality: a) Additive seasonal model: y t = T t + A t + C t + u t Seasonal term is added to trend Constant amplitude Average of A(j) = 0, j = 1...M. E(u t ) = 0 b) Multiplicative seasonal model: y t = T t × A t × C t × ε t Seasonal factor multiplies trend Changing amplitude because seasonal multiplies trend Average of A(j) = 1, j = 1. ..M. Fig. 1 – Trend and Seasonal (M = 4) Season 1 Trend Yr 1 Yr 2 Additive Multiplicative Fig. 2
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Ec 178 SEASONAL MODELS p. 2 E( ε t ) = 1 3. Overview of Decomposition: a) Assume a multiplicative seasonal model: 1 ) ( ; ; = = × × × = t u t t t t t t E e C A T y t ε b) Objective -- separate y t into its various components for a number of reasons. 1) Forecasting -- knowledge of the various components gives us a better understanding of the generating process. 2) Seasonal adjustment -- by removing the seasonal component we get a seasonally- adjusted series that is more useful for judging long-run developments or trends. c) First step -- separate trend-cycle from seasonal-error. 1) Calculate the M-period CMA t . The seasonal and error components average out, leaving a series containing mostly just trend and cycle: 2) Then compute the raw-to-MA ratio. This separates the series into two subcomponents. One is T t × C t ; the other is A t × ε t . d) Second Step -- separate seasonal from error component. 1) Sub-component series y t /CMA t = A t × ε t . 2) Take average within each season j = 1...M. Error component ε t averages out, leaving seasonal factor A(j). e) Third Step -- calculate seasonally adjusted series y(sa) which is the original y series with only the seasonal component removed: f) Fourth Step -- Separate trend from cycle and error components ( i.e. estimate trend). 1) Examine CMA t or y(sa) t series plot to determine the trend pattern. 2) Use y(sa) series to estimate the trend line. For a linear trend, y(sa) t = β 1 + β 2 t + u t . Trend component is t T t 2 1 ˆ ˆ β + = . g) If there is a cycle component, it can be measured as t t t t t t T C T T CMA C × = = . h) Possible macroeconomic decomposition. For y = GDP, fit exponential growth trend (constant LR real growth rate).
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This note was uploaded on 02/25/2010 for the course ECON econ178 taught by Professor Foster during the Spring '10 term at UCSD.

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