Box-Cox Transformation and the Estimation of Lambda Parameter

Authors and guest post by Eren Ocakverdi

This blog piece intends to introduce a new add-in (i.e. BOXCOX) that can be used in applying power transformations to the series of interest and provides alternative methods to estimate the optimal lambda parameter to be used in transformation.

Table of Contents

1. Introduction
2. Box-Cox family of transformations
3. Application to Turkey’s tourism data
4. Files
5. References

Introduction

A stationary time series requires stable mean and variance, which can then be modelled through ARMA-type models. If a series does not have a finite variance, it violates this condition and will lead to ill-defined models. Common practice in dealing with time varying volatility is modeling the variance explicitly through GARCH-type models. However, when the variance of a given series changes with respect to the level, then there is a practical alternative: transforming the original series so as to scale down (up) the large (small) values.

Box-Cox family of transformations

Box and Cox (1964) proposed a family of power transformations, which later became a popular tool in time series analysis to deal with skewness in the data:

$$ \tilde{y}_t = \begin{cases} \frac{y_t^\lambda - 1}{\lambda} & \text{if } \lambda \neq 0\\ log(y_t) & \text{if } \lambda = 0 \end{cases} $$ Transformation of a series is straightforward once the value of $\lambda$ is known. One way to determine the value of $\lambda$ is to maximize the (regular or profile) log likelihood of a linear regression model fitted to data. For trending and/or seasonal data, appropriate dummy variables are added to regressions to capture such effects. Guerrero (1993) proposed a model-independent method to select $\lambda$ that minimizes the coefficient of variation for the subsets of series.

Application to Turkey’s Tourism Data

With its pervasive trend and seasonal components, monthly tourism statistics emerge as a natural candidate for implementation (TOURISM.WF1). Suppose that we want to carry out a counterfactual analysis to estimate the potential loss of visitors to Turkey in 2020 due to the COVID-19 pandemic. First, set the training sample to cover the period until the end of 2019.

Figure 1: Visitors

Next, run the add-in. The following dialog pops up.

Figure 2: Box-Cox Dialog

The add-in would compute the optimal value of lambda to be 0.106.

Figure 3: Visitors (Box-Cox Transformation)

We can then apply Auto ARIMA method to original series and supply the value of estimated lambda to Box-Cox transformation as the power parameter. Forecasts produced by Auto ARIMA method can also be combined via Bayesian Model Averaging.

Figure 4: ARIMA Forecasting Dialog

As an alternative approach, one can also perform ETS Exponential Smoothing method on the transformed series to select for the best model and then back transform the forecasted values.

Figure 5: Visitors Loss

ARIMA model results imply that the number of visitors to Turkey might have decreased by 29 million during 2020. ETS model portrays an even worse picture by estimating a potential loss of 42 million visitors!

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Files

• TOURISM.WF1
• BOXCOX_EXAMPLE.PRG

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References

1. Box, G.E.P., and Cox, D.R. (1964), "An analysis of transformations", Journal of the Royal Statistical Society, Series B, vol. 26, no. 2, pp. 211-246.
2. Guerrero V.M. (1993), "Time-series analysis supported by power transformations", Journal of Forecasting, vol. 12, pp. 37-48.