30C00200 Econometrics

10 Time series econometrics

a) AR(1) process

Timo KuosmanenProfessor, Ph.D.

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Examples of time series

• Macroeconomic data:

– GDP

– Inflation rate

• Financial market data:

– Asset returns

– Stock prices

• Business Analytics: ”Big data” (new types of data from internet,

social media, etc.):

– Google Trends

– Facebook ”likes”2

How time series and cross sections differ?

• Cross section i =1,…,n represents a randomly drawn sample from

the population

• Time series t = 1,…,T describes a (random) path of a variable in a

time window [1, T]

• Time series have a natural chronological order, whereas in a cross

section the ordering of observations does not matter

• Path dependence: correlations across time periods (t, t+1)

– Autocorrelation (serial correlation)

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Static vs. Dynamic time series models

• Static model

yt = β1 + β2xt + εt , t=1,…,T.

Allow for autocorrelation (Cov[εt, εs] ≠ 0) using AR(1) process.

• Dynamic models

– Lagged explanatory variable(s)

yt = β1 + β2xt + β3xt-1 + εt, t=1,…,T.

– Lagged dependent variable(s) forming AR(1) process

yt = γyt-1 + β1 + β2xt + εt , t=1,…,T.

Autoregressive (AR) process

• First-order autoregressive AR(1) process of random variable v

vt = ρvt-1 + ut

• ρ is autocorrelation coefficient and

• ut is stationary, nonautocorrelated white noise process with

E[ut]=0, Cov[ut , ut-1]=0, Var[ut]=σu2

Simulated AR(1) process, t = 1,…, 500

ρ = 0 (white noise)

Simulated AR(1) process, t = 1,…, 500

ρ = 0.8 (positive autocorrelation)

Simulated AR(1) process, t = 1,…, 500

ρ = -0.8 (negative autocorrelation)

Higher-order autoregressive (AR) processes

• Second-order autoregression AR(2) process

vt = ρ1vt-1 + ρ2vt-2 + ut

• Pth-order autoregression AR(P) process

vt = ρ1vt-1 + ρ2vt-2 + … + ρPvt-P + ut

AR(1) process

• Each variable vt embodies the entire past history of u

vt = ρvt-1 + ut

= ut + ρut-1 + ρ2ut-2 + ρ3ut-3 … + ρt-1u1

• Hence,

Var[vt]= σu2 + ρ2σu

2 + ρ4σu2 + ρ6σu

2 … + ρ(t -1)(t -1)σu2

• Process is stationary if ρ satisfy

|ρ| < 1

Unit root

• The case where ρ = 1 is called the unit root

vt = vt-1 + ut

= ut + ut-1 + ut-2 + ut-3 … + u1

Simulated AR(1) process, t = 1,…, 500

ρ = 1 (unit root)

Simulated AR(1) process, t = 1,…, 500

ρ = 1.05

Next topic

b) Static model with autocorrelation

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