SEASONAL TIME SERIES MODELS

[Pages:65]SEASONAL TIME SERIES MODELS

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SEASONAL TIME SERIES

? A time series repeats itself after a regular period of time.

? "Business cycle" plays an important role in economics. In time series analysis, business cycle is typically represented by a seasonal (or periodic) model.

? A smallest time period for this repetitive phenomenon is called a seasonal period, s.

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SEASONAL TIME SERIES

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SEASONAL TIME SERIES

Seasonality

Stochastic SARIMA

Deterministic

Seasonal means (dummies) + linear time trend Sums of cosine curves at various frequencies + linear time trend

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SEASONAL TIME SERIES

? For deterministic function f(.), we say that f(.) is periodic with a periodicity s if

f t f t k s,k 0,1,2,

? A typical example of a deterministic periodic function is a trigonometric series,

e.g. sin() = sin(+2k) or cos() = cos(+2k).

? The trigonometric series are often used in econometrics to model time series with strong seasonality. [In some cases, seasonal dummy variables are used.]

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SEASONAL TIME SERIES

?

For stochastic process seasonal (or periodic)

tYimt, we esesraieysthwaitthitpiseraiodicity

s

if Yt and Yt+ks have the same distribution.

? For instance, the series of monthly sales of a

department store in the U.S. tends to peak at

December and to be periodic with a period 12.

OR quarterly ice cream sales is seasonal with period 4.

? In what follows, we shall use s to denote periodicity of a seasonal time series. Often s = 4 and 12 are used for quarterly and monthly time series, respectively.

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MODELLING SEASONALITY BY SEASONAL DUMMIES

? One approach to model seasonality is regression on seasonal dummies. It is a simple application of dummy variables defined to reflect movement across the "seasons" of the year.

? For quarterly data, s = 4, ? For monthly data, s = 12, ? For weekly data, s=52. ? For daily data, s=7.

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SEASONAL DUMMY

? Then we construct s seasonal dummy variables to indicate the season. So, if we have quarterly data and assuming the first observation we have is in the first quarter, we create: D1 = (1,0,0,0, 1,0,0,0, 1,0,0,0,...) D2 = (0,1,0,0, 0,1,0,0, 0,1,0,0,...) D3 = (0,0,1,0, 0,0,1,0, 0,0,1,0,...) D4 = (0,0,0,1, 0,0,0,1, 0,0,0,1,...)

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