1a - Exploratory analysis

Question 1

What is the classical time series decomposition?

Answer 1

Classical time series decomposition describes a univariate time series (Yt) with t≥1 as composed by structural components such as a trend, a seasonal compoment, a cycle, and an erratic component.

Question 2

How can a classical decomposition be?

Answer 2

• additive: Yt =Tt +St +Ct +Et
• multiplicative: Yt = Tt ∗ St ∗ Ct ∗ Et

Notice that a log transforms a multiplicative decomposition into an additive one: logYt =logTt +logSt +logEt

Question 3

How do we fit a trend in a series that just shows a trend and no seasonal component? Like Yt = Tt + Et

Answer 3

1. Fit a smooth function of t
2. Use moving averages

Question 4

How do we fit a smooth function of t?

Answer 4

We can use for example a linear trend or we can use more complex functions such as a quadratic trend or an exponential trend.

It is important to notice that bein too flexible makes it difficult to identify a trend or a cycle.

Question 5

How do we use a moving average?

Answer 5

This gives a smoothed version of the observed series, intrpreted as the trend.
- The higher the k, the smoother the fitted trend.

Question 6

What happens if k is even when we use the moving average?

Answer 6

Question 7

How do we detrend?

Answer 7

• Having fitted the trend, we can remove it to obtain a detrended time series.
• In this way we get just the erratic component which is possibly stationary

Question 8

How do we fit a seasonal component in a series that just shows a seasonal component and no trend? Like Yt = St + Et

Answer 8

1. Seasonal factors
2. Moving averages

Question 9

How do we use the season factors to fit the seasonal component?

Answer 9

• Suppose you have monthly data y , with mean y ̄ = 0.
• A simple way to proceed is to introduce seasonal factors α(jan), α(feb), …, α(dec) and describe:
  yt = αmonth(t) + Et
  where, if t corresponds to January, αmonth(t) = α(jan), and so on.
• For identifiability, we assume that the sum of the seasonal factors is zero.
• Remeber we are only considering additive decompostition.

Question 10

How do we use the moving average to fit the seasonal component?

Answer 10

For example, if we have monthly data, we would use a MA(12).

Question 11

What if we have a time series that has both the trend and the seasonal component? Like yt =Tt +St +Et

Answer 11

We can proceed in two ways:
1. we first detrend, and the we fit the seasonal component
2. we first fit the seasonal component, and then on the deseasonalized series we fit the trend