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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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## Forecasting using ARIMA-models

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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 Time series analysis - lecture 3**The general integrated auto-regressive-moving-average model**ARIMA(p, q) Time series analysis - lecture 3**Weekly SEK/EUR exchange rate Jan 2004 - Oct 2007**Time series analysis - lecture 3**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 Time series analysis - lecture 3**Consumer price index and its first order differences**Time series analysis - lecture 3**Consumer price index - first order differences**Time series analysis - lecture 3**Consumer price index – predictions using an ARI(1) model**Time series analysis - lecture 3**Seasonal differencing**Form where S depicts the seasonal length Time series analysis - lecture 3**Consumer price index and its seasonal differences**Time series analysis - lecture 3**Consumer price index- seasonally differenced data**Time series analysis - lecture 3**Consumer price index- differenced and seasonally differenced**data Time series analysis - lecture 3**The purely seasonal auto-regressive-moving-average model**ARMA(P,Q) with period S {Yt} is said to form a seasonalARMA(P,Q) sequence with period S if where the error terms t are independent andN(0;) Time series analysis - lecture 3**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 q*S ARMA(P,Q): Auto-correlations tail off gradually with time-lags greater than q*S Time series analysis - lecture 3**No. air passengers by week in Sweden-original series and**seasonally differenced data Time series analysis - lecture 3**No. air passengers by week in Sweden- seasonally differenced**data Time series analysis - lecture 3**No. air passengers by week in Sweden- differenced and**seasonally differenced data Time series analysis - lecture 3**The general seasonal auto-regressive-moving-average model**ARMA(p, q, P, Q) with period S {Yt} is said to form a seasonalARMA(p, q, P, Q) sequence with period S if where the error terms t are independent andN(0;) Example:p = P = 0, q = Q = 1, S = 12 . Time series analysis - lecture 3**The general seasonal integrated**auto-regressive-moving-average model ARMA(p, q, P, Q) with period S {Yt} is said to form a seasonalARIMA(p, q, d, P, Q, D) sequence with period S if where the error terms t are independent andN(0;) Time series analysis - lecture 3**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 Time series analysis - lecture 3**Consumer price index- differenced and seasonally differenced**data Time series analysis - lecture 3**No. registered cars and its first order differences**Time series analysis - lecture 3**No. registered cars- first order differences**Time series analysis - lecture 3

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