Econometría 2: Análisis de series de TiempoTime Series Models Stationary Models Stationary conditions for AR(1) using operator B: I An equivalent condition for stationarity, is that

Econometrıa 2: Analisis de series de Tiempo

Karoll [email protected]

http://karollgomez.wordpress.com

Segundo semestre 2016

III. Stationary models

Time Series ModelsStationary Models

1 Purely random process2 Random walk (non-stationary)3 Moving average process MA(q)4 Autoregressive process AR(p)5 Mixed ARMA(p,q)


For the stationary stochastic process





R code to generate a Random walk




NOTE: They were introduced by Slutsky (1937).


Alternative notation for MA(q) processes is:

Yt = εt + θ1εt−1 + ...+ θqεt−q

with εt white noise.

Autocorrelation function (ACF):





R code to generate a MA(1)


Other example: Yt = εt + 0,9εt−1x = arima.sim(model = list(ma=.9), n = 100)plot(x)plot(0:14, ARMAacf(ma=.9, lag=14), type=”h”, xlab = ”Lag”,ylab = .ACF”)abline(h = 0)


Partial autocorrelation function (PACF):

In addition to the autocorrelation between Xt and Xt−k , we maywant to investigate the conditional correlation between:

corr(Xt ,Xt−k | Xt−1, ...,Xt−(k−1))

which is referred as PACF in time series analysis.

REMARKS:

I This represent the correlation between Xt and Xt−k after theirlinear dependency on the intervening variablesXt−1, ...,Xt−(k−1) has been removed.

I in terms of a regression model, where the dependent variableis Xt−k we have:


Xt−k = φk,1Xt−(k−1) + φk,2Xt−(k−2) + ....+ φk,kXt + et−k

where φk,i represent the i-esimo regression parameter and et−k is aerror term with mean 0, variance σ2e , and uncorrelated withXt−(k−j) with j = 1, 2, 3, ..., k.

Multipliying both sides by Xt−(k−j) and taking expectations:

γj = φk,1γj−1 + φk,2γj−2 + ....+ φk,kγj−k

hence,ρj = φk,1ρj−1 + φk,2ρj−2 + ....+ φk,kρj−k

ACF and PCAF of MA(1): Yt = εt + θ1εt−1

ACF and PACF of MA(2): Yt = εt + θ1εt−1 + θ2εt−2

ACF and PACF of MA(2): continued

Example 1: Table, ACF and PACF for a simulated MA(1)Yt = εt + 0,5εt−1

Example 2: Table, ACF and PACF for a simulated MA(2)Yt = εt + 0,65εt−1 + 0,24εt−2


Alternative notation for AR(p) processes is:

Yt = φ1Yt−1 + ...+ φpYt−p + εt

with εt white noise.

Autoregressive series are important because:

I They have a natural interpretation: the next value observed is a slightperturbation of a simple function of the most recent observations.

I It is easy to estimate their parameters and it can be done with standardregression software.

I They are easy to forecast and standard regression software will do the job.

I They were introduced by Yule (1926)


4.0 Mean and Variance of AR processes


4.1 Yule walker equations (autocorrelation function)



ACF and PCAF of AR(1): Yt = φ1Yt−1 + εt


ACF and PACF of AR(2): Yt = φ1Yt−1 + φ2Yt−2 + εt

ACF and PCAF of AR(2): continued


R code to generate a AR(1)

Example 1: Table, ACF and PACF for a simulated AR(1)Yt = 0,65Yt−1 + εt

Example 2: Table, ACF and PACF for a simulated AR(2)Yt = 0,5Yt−1 + 0,3Yt−2 + εt


Summary I


Summary I (continued)

4 Autoregresive porcesses AR(p):


4.2 Stationarity for AR(p) processes

4.2.1 Auxiliary vs Characteristic equation


I An alternative solution is possible using the lag (orbackshift) operator:

ρ(k)− α1ρ(k − 1)− · · · − αpρ(k − p) = 0

I The general solution is

ρ(k) = A1π|k|1 + · · ·+ Apπ

|k|p

where α are the solution of the characteristic polynomial

(1− α1B − · · · − αpBp)ρ(k) = 0

orφ(B)ρ(k) = 0

I The necessary an sufficient condition for Xt be stationary isthat all |πi | > 1, i.e. the modulus of all the roots ofcharacteristic polynomial are greater than 1.




4.2.2 Back shift Operator B (or L)


4.2.2.1 Alternative view of the models using the operator B


Remarks:


4.2.2.2 Autocovariance and autocorrelation using the operator B


4.2.3 Stationarity Conditions

4.2.3.1 Stationary conditions for AR(1):

Xt =αXt−1 + Zt

(1− αB)Xt =Zt

Xt =∞∑i=0

αiZt−i

I If |α| < 1 then there is a stationary solution to AR(1) processI If |α| = 1 there is no stationary solution to AR(1) process.I If |α| > 1 there is no stationary solution to AR(1) process.I This means that for the purposes of modelling and forecasting

stationary time series, we must restrict our attention to series forwhich |α| < 1.


Stationary conditions for AR(1) using operator B:

I An equivalent condition for stationarity, is that the root of theequation 1− αB = 0 lies outside the unit circle in thecomplex plane.

I This means that for the purposes of modelling and forecastingstationary time series, we must restrict our attention to seriesfor which |α| < 1 or, equivalently, to series for which the rootof the polynomial 1− αB = 0 lies outside the unit circle inthe complex plane.


Example AR(1):



Example AR(2):







Real roots: Complex roots:

REMARK: The pattern showed by ACF and PACF also depends onthe solution of the characteristic polynomial.


R code for an AR(2) process

To generate the processlibrary(ts)x = arima.sim(model = list(ar=c(1.5,-.75)), n = 100)plot(x)

To find the rootsMod(polyroot(c(1,-2.5,1)))


To compute and plot the ACF


4.2.3.2 Stationarity for AR(p) processes


4.3 Invertable process: duality between AR and MA



Summary II


5. Mixed ARMA(p,q)

They were introduced by Wold (1938)




Example


ACF and PACF of ARMA(1,1): Yt = φ1Yt−1 + εt + θ1εt−1

ACF and PCAF of ARMA(1,1): continued

ACF and PCAF of ARMA(1,1): continued


R code for an ARMA(1,1) process

Consider the model: Yt = −0,5Yt−1 + εt + 0,3Yt−1x = arima.sim(model = list(ar=-.5,ma=.3), n = 100)plot(x)plot(0:14, ARMAacf(ar=-.5, ma=.3, lag=14), type=”h”, xlab =”Lag”, ylab = .ACF”)abline(h = 0)

Example 1: Table, ACF and PACF for a simulated ARMA(1,1)Yt = 0,9Yt−1 + εt + 0,5εt−1

Example 2: Table, ACF and PACF for a simulated ARMA(1,1)Yt = 0,6Yt−1 + εt + 0,5εt−1


5.1 Causality


Summary III


Characteristics of time series processes


Characteristics of theoretical ACF and PACF for stationaryprocesses

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Econometría 2: Análisis de series de TiempoTime Series Models Stationary Models Stationary conditions for AR(1) using operator B: I An equivalent condition for stationarity, is that