a425_trend

a425_trend - Time Series - Trends & Seasonality APS 425...

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Time Series - Trends & Seasonality APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2002-2010 1 APS 425 – Winter 2010 Time Series Analysis: Trend & Seasonality Instructor: G William Schwer Instructor: G. William Schwert 585-275-2470 schwert@schwert.simon.rochester.edu Topics • Deterministic trends & seasonality • Stochastic trends & seasonality • Differencing & seasonal differencing
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Time Series - Trends & Seasonality APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2002-2010 2 Time Trends Linear Z t = a + b t + e t • do you really expect such a pattern to persist forever? • e.g., accumulated horse manure in the streets of Philadelphia in the late 18 th century exponential (linear in logs) log(Z t ) = a + b t + e t Z t = exp{a + b t + e t } In Eviews, time = @trend(K) creates a time trend equal to 0 in observation K 20 25 Linear Trend Example Actual Fitted 5 10 15 0 0 1 02 03 04 05 06 07 08 09 0 1 0 0
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Time Series - Trends & Seasonality APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2002-2010 3 140 160 180 Exponential Trend Example Actual Fitted 40 60 80 100 120 0 20 0 1 02 03 04 05 06 07 08 09 0 1 0 0 Polynomial Trends Z t = a + b 1 t + b 2 t 2 + e t • parabola •i f b 2 < 0, then it eventually turns negative f b 2 > 0, then it eventually sky-rockets up • you can approximate most samples with polynomials, but you wouldn't expect to do well forecasting out-of- sample
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Time Series - Trends & Seasonality APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2002-2010 4 14 16 18 Quadratic Trend Example 4 6 8 10 12 Actual Fitted 0 2 0 1 02 03 04 05 06 07 08 09 0 1 0 0 Other Functions of Time Cosine or Sine represent fixed cycles S-curves (logistic) -- new product sales log((L - Z t )/Z t ) = a + b t + e t Z = L / {1 + exp[ a + b t + ]} Z t = L / {1 + exp[ a + b t + e t • L is the upper limit (asymptote) for sales • can iterate on this by trying different values to see which fits best
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Time Series - Trends & Seasonality APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2002-2010 5 10000 12000 S-Curve Trend Example Actual Fitted 4000 6000 8000 0 2000 0 1 02 03 04 05 06 07 08 09 0 1 0 0 Other Types of Stationarity-inducing Transformations it •per capita • divide by population (if it is growing) •real • divide by price index (e.g., CPI) •market share • divide by industry sales
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a425_trend - Time Series - Trends &amp; Seasonality APS 425...

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