Forecasting using ARIMAmodels

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Forecasting using ARIMA-models

• Step 1. Assess the stationarity of the given
  time series of data and form differences if
  necessary
• Step 2. Estimate auto-correlations and partial
  auto-correlations, and select a suitable
  ARMA-model
• Step 3. Compute forecasts according to the
  estimated model

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The general integrated auto-regressive-moving-aver
age model ARIMA(p, q)
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Weekly SEK/EUR exchange rate Jan 2004 - Oct 2007
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Weekly SEK/EUR exchange rate Jan 2004 - Oct
2007AR(2) model
Final Estimates of Parameters Type Coef
SE Coef T P AR 1 1.2170
0.0685 17.75 0.000 AR 2 -0.2767
0.0684 -4.05 0.000 Constant 0.549902
0.002726 201.69 0.000 Mean 9.21589
0.04569
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Consumer price index and its first order
differences
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Consumer price index - first order differences
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Consumer price index predictions using an
ARI(1) model
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Seasonal differencing

• Form
• where S depicts the seasonal length

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Consumer price index and its seasonal differences
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Consumer price index- seasonally differenced data
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Consumer price index- differenced and seasonally
differenced data
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The purely seasonal auto-regressive-moving-average
model ARMA(P,Q) with period S

• Yt is said to form a seasonal ARMA(P,Q)
  sequence with period S if
• where the error terms ?t are independent and
  N(0?)

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Typical auto-correlation functions of purely
seasonal ARMA(P,Q) sequences with period S

• Auto-correlations are non-zero only at lags S,
  2S, 3S,
• In addition
• AR(P) Autocorrelations tail off gradually with
  increasing time-lags
• MA(Q) Auto-correlations are zero for time lags
  greater than qS
• ARMA(P,Q) Auto-correlations tail off gradually
  with time-lags greater than qS

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No. air passengers by week in Sweden-original
series and seasonally differenced data
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No. air passengers by week in Sweden- seasonally
differenced data
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No. air passengers by week in Sweden-
differenced and seasonally differenced data
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The general seasonal auto-regressive-moving-averag
e model ARMA(p, q, P, Q) with period S

• Yt is said to form a seasonal ARMA(p, q, P, Q)
  sequence with period S if
• where the error terms ?t are independent and
  N(0?)
• Example p P 0, q Q 1, S 12
• .

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The general seasonal integrated
auto-regressive-moving-average model ARMA(p, q,
P, Q) with period S

• Yt is said to form a seasonal ARIMA(p, q, d,
  P, Q, D) sequence with period S if
• where the error terms ?t are independent and
  N(0?)

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Forecasting using Seasonal ARIMA-models

• Step 1. Assess the stationarity of the given
  time series of data and form differences and
  seasonal differences if necessary
• Step 2. Estimate auto-correlations and partial
  auto-correlations, and select a suitable
  ARMA-model of the short-term dependence
• Step 3. Estimate auto-correlations and partial
  auto-correlations, and select a suitable seasonal
  ARMA-model of the variation by season
• Step 4. Compute forecasts according to the
  estimated model

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Consumer price index- differenced and seasonally
differenced data
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No. registered cars and its first order
differences
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No. registered cars- first order differences