1 Why seasonal adjustment and trend-cycle are required?

Seasonal adjustment and trend-cycle estimation of the original economic time series are the primary input to meaningful business cycle study, turning points detection and forecasting.

Seasonally adjusted data helps to better understand the short term movements of the original time series by removing the seasonal effects.

And trend-cycle estimation captures the long term trend and cyclical movements of the original series. It is estimated by removing and/or adjusting short term volatility due to presence of seasonal effects, irregular components and outliers in the original series.

2 Use of software in seasonal adjustment and trend-cycle estimation

Seasonal adjustment and trend-cycle estimation requires “decomposition” of the overall time series (TS) data to isolate the seasonal (S), irregular (I) and trend-cycle (C) components in the time series.

This can be done accurately by the advance statistical software.

The most commonly used software seasonal adjustment and trend-cycle estimation is X-13ARIMA-SEATS developed by US Economic Bureau. Though the software is developed to handle US economic data but it provides some levers for customization.

Using this feature of the software, Christoper Sax has developed a [package “seasonal”] for R. R is a powerful statistical software used by economists, data scientists and others for econometric modelling and big data analysis. The package “seasonal” is R-interface to X-13ARIMA-SEATS. The historical and future dates of Indian festival like Diwali are in-built in the package. It provides standard functions for Indian data. It automatically select econometric model and generate seasonally adjusted series and trend cycle estimates from the original data.

Absolute clarity is required while working with automatic model selection. The generated outputs need to be validated separately. The outputs should also pass model robustness. If the output does not clear these checks then manual intervention for model specification is necessary.

3 Developing a standard econometric framework for seasonal adjustment and trend-cycle estimation using the domestic passenger vehicle sales

The framework is developed by performing the following steps:-

3.1 Seasonality identification

Quantitative identification: Econometrics approach

TBATS model in the “forecast” package in R is used for identification of seasonal pattern ( following Dr. Hyndman, Professor of Statistics at Monash University and Editor-in-Chief of the International Journal of Forecasting). It can handle quarterly, monthly and even weekly seasonality. The package will automatically determine existence of seasonality in the given time series.

TBATS model output for the given times series is as follows:-

[1] "Seasonality = TRUE"

Intuitive identification: Graphical approach

The volatility is very high specially in October, November and December over the years.It may be due to moving seasonal effects from festive period.

3.2 Automatic model selection by software and decomposition: Seasonal adjustment and trend cycle estimation

The following output shows sesonally adjusted passenger vehicle sales along with the long term trend. The magnitude of the seasonal and irregular components (in the graph below) suggests their influence on the original car sales over time

3.3 Validation

Two econometric models are applier to check the presence of seasonality in the final output. If seasonality is present in the final model then the above model need to be improved.

TBATS: TBATS model output for the seasonal adjusted series is as follows:

[1] "Seasonality = FALSE"

QS Statistics: QS is a statistic that provides a test of the hypothesis of no seasonality. It finds if seasonality exists in the original series and the final series. The output is as follows:-

                                  qs   p-val
Original Series            219.53534 0.00000
Seasonally Adjusted Series   1.54191 0.46257

Both the tests confirm that the automatic selected models successfully removed the seasonal effects from the domestic passenger vehicle series.

3.4 Outlier detection

Outliers are data which do not fit in the tendency of the time series observed, which fall outside the range expected on the basis of the typical pattern of the trend and seasonal components.

The types of outliers are as follows:-

  • Additive Outliers (AOs) affect only one observation in the time series.
  • Level Shifts(LSs) increase or decrease all observations from a certain time point onward by some constant amount.
  • Temporary Changes (TCs) allow for an abrupt increase or decrease in the level of the series that returns rapidly to its previous level.
  • Seasonal Outliers (SOs) allow for an abrupt increase or decrease in the level of the seasonal pattern that is compensated for in the other months or quarters.
  • Ramps allow for a linear increase or decrease in the level of the series over a specified time interval , and temporary level changes increase or decrease all observations for a specific time span contained within the series by some constant amount.

Is outlier present in the domestic passenger vehicle series: Yes we see Level shift (LS) in 1999. The level shift may be attributed to new launch of Maruti Suzuki Wagon R and Hyundai Santro. These launches bring in new revolutionary in modern Indian automotive market.

3.5 Model robustness check

Thus no autocorrelation exists in the residuals and they follow normal distribution.

4 References