lec6 - Financial Data Analysis Professor S. Kou Department...

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Financial Data Analysis Professor S. Kou Department of IEOR, Columbia University Lecture 6. Seasonality, Unit Root Test, and Tests of Stationary 1S e a s o n a l i t y Last time we learned MA, ARMA, and ARIMA models, which works for stationary time series without any seasonality. However, many time series data display seasonality. For example, we can look at the quarterly earnings per share of Johnson and Johnson from the f rst quarter of 1960 to the last quarter of 1980. Although the e f ects of earnings announcement to stock prices are not very clear, earnings may a f ect the short term prices, partly due to self-ful f lling prophecies. In other words, if many analysts focus on earnings, then the stock prices will be a f ected by earnings in the short term. #First, we will create a time series object. jjEarning$date - timeSeq(from="1/1/1960", to="10/1/1980", by="quarters", format="%Y:%Q") jjEarning.ts - timeSeries (data=jjEarning$jj, pos=jjEarning$date) #Then we plot the data plot(jjEarning.ts) The seasonality is very strong. In fact, it is immediately clear that the period of the seasonality is about 4 (which means one year). Now just like we take log of the stock prices to get returns, we can look at the log transform of the earnings. #Do the log transform logJJ - log(jjEarning.ts) plot(logJJ) We shall try to f t a seasonal ARIMA model to the log earnings. A standard way to get ride of seasonality is by looking at the di f erence. In our example, since we are quite certain the seasonality period is 4, we can simply take the transform 1  =4 1
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where is the back-shifting operator. In other words, we shall only compare the current quarter earning with the earnings in the same quarter of the previous years. After the transform, then we can f tanARIMAmode l . 1.1 Multiplicative Seasonal Model (Airline Model) A popular way to generalize the above idea is to f t the Multiplicative Sea- sonal Model (also called the “airline model”): (1 )(1 ) =(1  )(1 Θ ) (1) where is the period. Note that (1  )(1 Θ ) =( 1  Θ + Θ +1 ) =  1 Θ + Θ 1 and (1 )(1 ) =( 1 + +1 ) = 1 + 1 Therefore, the airline model is 1 + 1 =  1 Θ + Θ 1 (2) The motivation of the airline model is as follows. First, we expect the value is related to . In the airline model, this is modeled as a ARMA(0,1,1) with period ,i .e .aMA(1)mode lforthe f rst order di f er- ence, except with period . More precisely,
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lec6 - Financial Data Analysis Professor S. Kou Department...

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