Components of Time Series Su, Chapter 2, section II.

Four Primary Components of a Time Series: Secular Trend Seasonal Trend Cyclical Movements Irregular Components

Example: Secular Trend

Example: Seasonal Component

Example: Cyclical Component

Example: Random/Irregular

Mathematical Representations Additive: Y = T + S + C + I Multiplicative: Y = T x S x C x I

Observations: Traditional time series analysis is “atheoretic”. No economic theory guides us in writing down this decomposition. Typically, one of these components will dominate and this will affect the behavior of the series.

Secular Trends Often called “Time Trends” Visual representation is called “Time Path” or “Time Shape” A continuous set of integers is used to represent time in these models. Linear Time Trend Model Y t =  0 +  1 T t

How predictable is the secular trend in a series?

Removing Time Trends: Detrending Often, the trend component of a time series dominates, but the interesting part of the series is another component. Example: Civilian Employment

Civilian Employment Time Series The trend component is dominant. Why? Suppose we’re interested in changes in employment over the business cycle. –Problem: The cyclical component subordinate to the secular trend. –Solution: Detrend

Detrending Step 1: Estimate the Secular Trend using regression model Step 2: Subtract the estimated secular trend from the original series. Note: This is also the “Residual Approach” to analyzing cyclical data in C1.

Regression Output

Plot of Detrended Data

Seasonal Component Found in High Frequency data (Quarterly, monthly) Caused by natural or budget calendars –Retail Sales higher during holidays –Travel more frequent in summer –Weather Want to quantify or remove in forecasting How predictable is this component?

“Unadjusted Data”

Same series after adjustment

Seasonal Adjustment Methods Dummy Variables Ratio-to-moving-average X-11 - Asymmetric Moving Averages

A (Partial) Example: Income Tax Revenues

Moving Average Computation

Regression Method

Irregular or Random Components Special events that pull macro variables off their usual paths. Can be expected or unexpected.

Modeling Random / Irregular Components May stem from randomness or irregularities in human behavior. Keynes: “Animal Spirits” Some irregular events are not random, they are caused by specific factors If they are truly random, there is no way to predict them. If they are just irregular, they can be handled by dummy variables.

Modeling Time Series Goal is to distinguish between the deterministic (or predictable) and stochastic (or random) parts Y t =  t + u t  t is the deterministic component – secular trend, seasonal and cyclical movements u t is the stochastic component Y t = T t + C t + S t + u t

Assumptions: Random Component Typically make three assumptions about u t Mean zero: E(u t ) = 0 Constant variance/no covariance E(u t u t+i ) =  2 u if i=0 (Constant variance) E(u t u t+i ) = 0 if i  0 (Zero covariance) Normally distributed u t ~ N(0,  2 u )

Stationarity Refers to the idea that a time series should be stable over time – returns to an equilibrium level Stationarity is an important concept for forecasting because only stationary time series are predictable A stationary time series has a mean, variance and autocovariances that do not change over time Many economic time series are not stationary – alternative is “random walk” Tests for stationarity exist – later in semester

Transformations A nonstationary series must be transformed to make it stationary (First) Differencing:  Y t = Y t – Y t-1 Detrending

Stationary Time Path t YtYt Equilibrium Time Path Shock

Autocovariances Covariance between two observations Example: kth-order autocovariance is the covariance between observations of a time series k periods apart (or lagged k periods) Cov(Y t Y t-k ) If the autocovariances of a time series are stationary (do not change over time) then they can be used to forecast a series Autocovariances are a measure of predictability

Autocorrelations Closely related to autocovariances Just the correlation between any two observations of a time series If Cov(Y t Y t-k ) is the autocovariance, then cor(Y t Y t-k ) = Cov(Y t Y t-k )/var(Y t )