Wavelet Transform as an Alternative to Power Transformation in Time Series Analysis

Abstract

This study examines the discrete wavelet transform as a transformation technique in the analysis of non-stationary time series while comparing it with power transformation. A test for constant variance and choice of appropriate transformation is made using Bartlett’s test for constant variance while the Daubechies 4 (D4) Maximal Overlap Discrete Wavelet Transform (DWT) is used for wavelet transform. The stationarity of the transformed (power and wavelet) series is examined with Augmented Dickey-Fuller Unit Root Test (ADF). The stationary series is modeled with Autoregressive Moving Average (ARMA) Model technique. The model precision in terms of goodness of fit is ascertained using information criteria (AIC, BIC and SBC) while the forecast performance is evaluated with RMSE, MAD, and MAPE. The study data are the Nigeria Exchange Rate (2004-2014) and the Nigeria External Reserve (1995-2010). The results of the analysis show that the power transformed series of the exchange rate data admits a random walk (ARIMA (0, 1, 0)) model while its wavelet equivalent is adequately fitted to ARIMA (1,1,0). Similarly, the power transformed version of the External Reserve is adequately fitted to ARIMA (3, 1, 0) while its wavelet transform equivalent is adequately fitted to ARIMA (0, 1, 3). In terms of model precision (goodness - of - fit), the model for the power transformed series is found to have better fit for exchange rate data while model for wavelet transformed series is found to have better fit for external reserve data. In forecast performance, the model for wavelet transformed series outperformed the model for power transformed series. Therefore, we recommend that wavelet transform be used when time series data is non-stationary in variance and our interest is majorly on forecast.