The unece organizes with the financial support of the World Bank

θ(B) = (1+ θ1B + … + θpBp)(1+ θ1Bs + …+ θPBs×P)

A seasonal ARIMA model is identied by the order of its polynomials: (p;d;q)(P;D;Q)

TRAMO / Reg-ARIMA

• Program for estimation, forecasting, and interpolation of regression models with missing observations and ARIMA errors, with possibly several types of outliers

• The program is aimed at monthly or lower frequency data (quarterly, semester, 4-month, bimonth, semester, year)

• Performs a pretesting to decide between a log transformation and no transformation

TRAMO / Reg-ARIMA

• Identifies the ARIMA model through an Automatic Model Identification (AMI) procedure

• Interpolates missing values

• Detects outliers

• Estimates the REG-Arima model

• Computes forecasts

Automatic model identification

• The ARIMA model can be automatically identified by the program

• Two steps

• Obtains the order of differencing
• max order ∆2 ∆s
• Obtains the multiplicative stationary ARMA model

• 0<=(p;q)<=3

• 0 <=(ps;qs )<=1

• Chosen with the BIC criterion, favors balanced model (similar orders of AR and MA parts)

• Otherwise, it can be input by the user (parameters P,D,Q, BP,BD, BQ)

• It works jointly with the Automatic Outlier Detection and Correction (AODC)

Outliers

• They represent the effect on the time series of some special events (new regulation, major political or economical reform, strike, natural disaster). Three possible forms of outliers:

• Additive outliers (AO)
• Level Shift (LS)
• Transitory Changes (TC)

Outliers

Calendar effects

• Calendar adjustment removes those non-seasonal calendar effects from the series, for which there is statistical evidence and an economic explanation. Four possibilities in TS:

• Trading days (working/non-working, 6 regressors))
• National and moving holidays ((provided by the user))
• Leap-year (TS versus X-12-ARIMA)
• Easter

• A pre-testing on the presence of these effects.

• If trading days are significant, adding the holidays variable improves significantly the results!

Examples of calendar effects

Trading/Working Day Adjustment

• Aims at a series independent of the length and the composition in days

• Length of month, number of working days and weekend days, composition of working days (Monday/Friday)

• Working or trading-day adjustment is recommended for series with such effects

• If effects not present –Regressors should not be applied

• Compile, maintain and update national calendars!

• A historical list of public holidays including information on compensation holidays

Correction for Moving Holidays

• Occur irregularly in the course of the year

• Correct for detected moving holidays in series

• Not removed by standard filters
• If effects not present –Regressors should not be applied

• These effects may be partly seasonal:

• The Catholic Easter, for example, falls more often in April than in March

• Since the seasonal part is captured by seasonal adjustment filters, it should not be removed during the calendar adjustment

An illustrative example for national calendar regressor

Original vs. Linearized series

Model Selection, Seasonal Adjustment, Analyzing Results

• Necmettin Alpay KOÇAK

• UNECE Workshop on Short-Term Statistics (STS) and Seasonal Adjustment

• 14 – 17 March 2011

• Astana, Kazakhstan

Model Selection

• Pre-treatment is the most important stage of the seasonal adjustment

• X-12-ARIMA and TRAMO&SEATS methods use very similar (nearly same) approaches to obtain the linearized (pre-treated) series.

• Both method use ARIMA model for pre-treatment.

• The most appropriate ARIMA model → Linearized series of top quality

ARIMA Model selection

• zt = ytβ+xt

• Φ(B)δ(B)xt=θ(B)at

• (p,d,q)(P,D,Q)s → Structure of ARIMA

• (0,1,1)(0,1,1)4,12

• For the model

• Parsimonious
• Significance of parameters
• Smallest BIC or AIC

• For the residuals

• Normality
• Lack of auto-correlation
• Linearity
• Randomness

Diagnostics

• Are there really any seasonal fluctations in the series ?

• Seasonality test

• If, yes

• Diagnostics based on residuals are the core of the analysis.
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