Econ 424
Time Series Concepts

Eric Zivot

January 20 2015

Time Series Processes
Stochastic (Random) Process{    1 2      +1   } = {}∞=−∞
sequence of random variables indexed by time

Observed time series of length {1 = 1 2 = 2      =  } = {}
=1

Stationary Processes• Intuition: {} is stationary if all aspects of its behavior are unchanged byshifts in time
• A stochastic process {}∞=1 is strictly stationary if, for any given finite
integer  and for any set of subscripts 1 2      the joint distributionof(1  2       )depends only on 1 −  2 −       −  but not on Remarks

1. For example, the distribution of (1 5) is the same as the distribution of(12 16)

2. For a strictly stationary process,  has the same mean, variance (moments) for all 

3. Any function/transformation (·) of a strictly stationary process, {()}is also strictly stationary. E.g., if {} is strictly then {2} is strictlystationary.

Covariance (Weakly) Stationary Processes {} :• [] =  for all • var() =  2 for all • cov( − ) =  depends on  and not on 
Note 1: cov( − ) =  is called the j-lag autocovariance and measuresthe direction of linear time dependence
Note 2: A stationary process is covariance stationary if var()  ∞ andcov( − )  ∞Autocorrelationscov( − )corr( − ) =  = q= 2
var()var(− )Note 1: corr( − ) =  is called the j-lag autocorrelation and measuresthe direction and strength of linear time dependence
Note 2: By stationarity var() = var(− ) =  2
Autocorrelation Function (ACF): Plot of  against Example: Gaussian White Noise Process ∼ iid (0  2) or  ∼  (0  2)
[] = 0 var() = 2

 independent of  for  6= ⇒ cov( −) = 0 for  6= Note: “iid” = “independent and identically distributed”.
Here, {} represents random draws from the same (0  2) distribution

Example: Independent White Noise Process ∼ iid (0  2) or  ∼  (0  2)
[] = 0 var() = 2

 independent of  for  6= Here, {} represents random draws from the same distribution. However, wedon’t specify exactly what the distribution is - only that it has mean zero andvariance 2 For example,  could be iid Student’s t with variance equal to2 This is like GWN but with fatter tails (i.e., more extreme observations).

Example: Weak White Noise Process ∼  (0  2)[] = 0 var() = 2cov( ) = 0 for  6= Here, {} represents an uncorrelated stochastic process with mean zero andvariance 2 Recall, the uncorrelated assumption does not imply independence.
Hence,  and  can exhibit non-linear dependence (e.g. 2 can be correlatedwith 2)

Nonstationary Processes
Defn: A nonstationary stochastic process is a stochastic process that is notcovariance stationary.
Note: A non-stationary process violates one or more of the properties of covariance stationarity.
Example: Deterministically trending process = 0 + 1 +   ∼  (0 2)[] = 0 + 1 depends on Note: A simple detrending transformation yield a stationary process: =  − 0 − 1 = Example: Random Walk = −1 +   ∼  (0 2) 0 is fixed= 0 +

X

=1 ⇒ var() = 2 ×  depends on Note: A simple detrending transformation yield a stationary process:∆ =  − −1 = Time Series Models
Defn: A time series model is a probability model to describe the behavior of astochastic process {}
Note: Typically, a time series model is a simple probability model that describes
the time dependence in the stochastic process {}

Moving Average (MA) Processes
Idea: Create a stochastic process that only exhibits one period linear timedependence
MA(1) Model =  +  + −1 − ∞    ∞ ∼  (0 2) (i.e.,  ∼  (0 2))

 determines the magnitude of time dependence

Properties
[] =  + [] + [−1]=+0+0=var() =  2 = [( − )2]= [( + −1)2]= [2 ] + 2[−1] + 2[2−1]= 2 + 0 + 22 = 2(1 + 2)
cov( −1) = 1 = [( + −1)(−1 + −2)]= [−1] + [−2]+ [2−1] + 2[−1−2]= 0 + 0 + 2 + 0 = 2

Furthermore,cov( −2) = 2 = [( + −1)(−2 + −3)]= [−2] + [−3]+ [−1−2] + 2[−1−3]=0+0+0+0=0
Similar calculation show thatcov( − ) =  = 0 for   1Autocorrelations
1
2

1 = 2 = 2=
 (1 + 2)(1 + 2) = 2 = 0 for   1Note:
1 = 0 if  = 0
1  0 if   0
1  0 if   0Result: MA(1) is covariance stationary for any value of 

Example: MA(1) model for overlapping returns
Let  denote the 1−month cc return and assume that ∼ iid (  2)
Consider creating a monthly time series of 2−month cc returns using
(2) =  + −1
These 2−month returns observed monthly overlap by 1 month(2) =  + −1
−1(2) = −1 + −2
−2(2) = −2 + −3..
Claim: The stochastic process {(2)} follows a MA(1) process

Autoregressive (AR) Processes
Idea: Create a stochastic process that exhibits multi-period geometrically decaying linear time dependence
AR(1) Model (mean-adjusted form) −  = (−1 − ) +  − 1    1 ∼ iid (0 2)

Result: AR(1) model is covariance stationary provided −1    1

Properties
[] = var() = 2 = 2(1 − 2)cov( −1) = 1 = 2

corr( −1) = 1 = 12 = cov( − ) =  = 2corr( − ) =  =  2 = Note: Since ||  1lim  =  = 0→∞AR(1) Model (regression model form) −  = (−1 − ) +  ⇒ =  −  + −1 + =  + −1 + 

whereRemarks:

 = (1 − ) ⇒  =1−• Regression model form is convenient for estimation by linear regression

The AR(1) model and Economic and Financial Time Series
The AR(1) model is a good description for the following time series• Interest rates on U.S. Treasury securities, dividend yields, unemployment• Growth rate of macroeconomic variables— Real GDP, industrial production, productivity— Money, velocity, consumer prices— Real and nominal wages