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TIME SERIES ANALYSISTIME SERIES ANALYSISTIME SERIES ANALYSISTIME SERIES ANALYSIS.

LECTURE 8: LECTURE 8: LECTURE 8: LECTURE 8: ARIMAARIMAARIMAARIMA MODELS: MODELS: MODELS: MODELS: SEASONAL MODELS (1).SEASONAL MODELS (1).SEASONAL MODELS (1).SEASONAL MODELS (1).

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Seasonal models.

�Concepts recap.

�Which type of information suggests the building of a seasonal model?

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Seasonal Processes.

�Univariate Time Series Analysis (TSA).

�Objective: Forecasting.

�Wold (1938):

◦ Any stationary process can be uniquely represented by

Y(t) ≡ Yt = Dt + Xt

◦ D: linear deterministic (usually the mean) ; X is

stochastic MA(∞), uncorrelated. B

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Seasonal information.

�Real TS socioeconomic data collected on a lower than annual basis. It captures the information of annual TS variables but for shorter periods. Unemployment (EPA vs INEM).

�Seasonal TS show a seasonality pattern, with variations in the level of the series which recur, at the same period, every year. A repeated pattern dependent on the seasons.

�Examples: quarterly; monthly.

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Inbound Tourism (Spanish monthly data)

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2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

8,000,000

9,000,000

10,000,000

11,000,000

I II III IV I II III IV I II III IV I II III IV I II III IV I II III IV

1996 1997 1998 1999 2000 2001

Inbound Tourism (Spain)

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8.1. ARIMA(p,d,q)*(P,D,Q)s models.

�TS models for quarterly or monthly data present non-seasonal (regular) and seasonal components.

�We will utilize multiplicative processes.

�For the sake of simplicity in the exposition we will first analyze “pure” seasonal processes.

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Formal expression:

Φs(L)[1-Ls]D Yt= Θs(L)εt

where:

Φs(L)=1-φ1Ls-φ2L

2s-......-φp LPs

Θs(L)=1+θ1Ls+θ2L

2s+.....+θq LQs

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8.1.1. Modelling steps.

�Box-Jenkins (1970).

�Steps:

◦ Identification;

◦estimation;

◦Diagnostic checking (validation) and

◦forecasting.

F

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Modelling strategy (steps)

Identification

Estimation

Diagnostic

checking

Forecasting

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Yes

No

BOX-JENKINS (1970).

B

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Identification

�Is the time series mean stationary in its seasonal component (D value)?

�Is the time series variance stationary?

�Which is the seasonal autoregresive process order (P value)?

�Which is the seasonal moving average process order (Q value)?

B

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8.1.2. ARMA(P,Q)s models.

�Some simple examples.

�AR(p)s.

�MA(q)s.

�Mixed, ARMA(p,q)s

◦ F

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Some simple examples.

AR(1)4 with parameter equal to 0.7.

AR(1)4 with parameter equal to -0.7.

AR(1)4 with parameter equal to 1.

AR(1)12 with parameter equal to 0.8.

AR(1)12 with parameter equal to -0.8.

AR(1)12 with parameter equal to 1.

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Some simple examples.

MA(1)4 with parameter equal to 0.5.

MA(1)4 with parameter equal to -0.5.

MA(1)12 with parameter equal to - 0.6.

MA(1)12 with parameter equal to 0.6.

ARMA(1,1)4 with parameters 0.8 y 0.5, respectively.

ARMA(1,1)12 with parameters 0.7 y 0.6, respectively.

B

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AR(P)s processes.

�Stationarity.

�Autocorrelation function.

�Partial autocorrelation function.

�Examples.

◦ B

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MA(Q)s processes.

�Invertibility.

�Autocorrelation function.

�Partial autocorrelation function.

�Examples.

◦ B

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ARMA(P,Q)s processes.

�Stationarity and invertibility.

�Autocorrelation function.

�Partial autocorrelation function.

�Examples.

◦B

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AR(P)s: Stationarity.

�Absence of seasonal unit roots.

�The seasonal polynomial have many roots.

�In particular.

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S=4 1-L4= 0; implies:

L=1; L= -1; L= i; L= -i

s=12 1-L12= 0 implies:

L=1; L= -1; and the roots of the polynomials

1+ L2; 1-L + L2; 1+L + L2

2

2

31

31

LL

LL

++

+−

B

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AR(P)s: Autocorrelation Function.

�Sample ACF has spikes at lag s and its multiples: 2s, 3s, 4s.....

�If you only consider the above lags (spikes) the ACF tails off.

B

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AR(P)s: Partial Autocorrelation Function.

�Sample PACF cuts off at lag sP (has no spikes for lags greater than sP ).

�PACF values for lags less than or equal to sP, but s multiples, depend on the parameters of the process values.

◦ B

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MA(Q)s: Autocorrelation Function.

�Sample ACF cuts off at lag sQ (has no spikes for lags greater than sQ ).

�ACF values for lags less than or equal to sQ, but s multiples, depend on the parameters of the process values.

◦B

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MA(Q)s: Partial Autocorrelation Function.

�Sample PACF has spikes at lag s and its multiples: 2s, 3s, 4s.....

�If you only consider the above lags (spikes) the PACF tails off (exponential or tapering sinusoidal, tapering oscillating, decay).

◦ B

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ARMA(P,Q)s: Autocorrelation Function.

�ACF has no definite pattern until the lag sQ.For lags greater than sQ it behaves like an AR(P)s process.

◦B

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ARMA(P,Q)s: Partial Autocorrelation Function.

�PACF has no definite pattern until the lag sP.For lags greater than sP it behaves like an MA(Q)s process.

◦B

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