Econometría 2: Análisis de series de TiempoTime Series Models Stationary Models Stationary...
Transcript of Econometría 2: Análisis de series de TiempoTime Series Models Stationary Models Stationary...
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Econometrıa 2: Analisis de series de Tiempo
Karoll [email protected]
http://karollgomez.wordpress.com
Segundo semestre 2016
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III. Stationary models
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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)
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For the stationary stochastic process
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R code to generate a Random walk
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NOTE: They were introduced by Slutsky (1937).
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Alternative notation for MA(q) processes is:
Yt = εt + θ1εt−1 + ...+ θqεt−q
with εt white noise.
Autocorrelation function (ACF):
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R code to generate a MA(1)
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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)
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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:
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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
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ACF and PCAF of MA(1): Yt = εt + θ1εt−1
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ACF and PACF of MA(2): Yt = εt + θ1εt−1 + θ2εt−2
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ACF and PACF of MA(2): continued
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Example 1: Table, ACF and PACF for a simulated MA(1)Yt = εt + 0,5εt−1
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Example 2: Table, ACF and PACF for a simulated MA(2)Yt = εt + 0,65εt−1 + 0,24εt−2
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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)
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4.0 Mean and Variance of AR processes
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4.1 Yule walker equations (autocorrelation function)
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ACF and PCAF of AR(1): Yt = φ1Yt−1 + εt
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ACF and PACF of AR(2): Yt = φ1Yt−1 + φ2Yt−2 + εt
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ACF and PCAF of AR(2): continued
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R code to generate a AR(1)
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Example 1: Table, ACF and PACF for a simulated AR(1)Yt = 0,65Yt−1 + εt
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Example 2: Table, ACF and PACF for a simulated AR(2)Yt = 0,5Yt−1 + 0,3Yt−2 + εt
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Summary I
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Summary I (continued)
4 Autoregresive porcesses AR(p):
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4.2 Stationarity for AR(p) processes
4.2.1 Auxiliary vs Characteristic equation
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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.
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4.2.2 Back shift Operator B (or L)
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4.2.2.1 Alternative view of the models using the operator B
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Remarks:
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4.2.2.2 Autocovariance and autocorrelation using the operator B
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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.
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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.
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Example AR(1):
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Example AR(2):
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Real roots: Complex roots:
REMARK: The pattern showed by ACF and PACF also depends onthe solution of the characteristic polynomial.
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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)))
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To compute and plot the ACF
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4.2.3.2 Stationarity for AR(p) processes
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4.3 Invertable process: duality between AR and MA
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Summary II
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5. Mixed ARMA(p,q)
They were introduced by Wold (1938)
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Example
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ACF and PACF of ARMA(1,1): Yt = φ1Yt−1 + εt + θ1εt−1
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ACF and PCAF of ARMA(1,1): continued
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ACF and PCAF of ARMA(1,1): continued
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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)
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Example 1: Table, ACF and PACF for a simulated ARMA(1,1)Yt = 0,9Yt−1 + εt + 0,5εt−1
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Example 2: Table, ACF and PACF for a simulated ARMA(1,1)Yt = 0,6Yt−1 + εt + 0,5εt−1
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5.1 Causality
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Summary III
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Characteristics of time series processes
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Characteristics of theoretical ACF and PACF for stationaryprocesses