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.
