آیاتحولات Box-Cox به پیش بینی سری های زمانی اقتصاد کلان کمک می کند ؟

The paper investigates whether transforming a time series leads to an improvement in forecasting accuracy. The class of transformations that is considered is the Box–Cox power transformation, which applies to series measured on a ratio scale. We propose a nonparametric approach for estimating the optimal transformation parameter based on the frequency domain estimation of the prediction error variance, and also conduct an extensive recursive forecast experiment on a large set of seasonal monthly macroeconomic time series related to industrial production and retail turnover. In about a fifth of the series considered, the Box–Cox transformation produces forecasts which are significantly better than the untransformed data at the one-step-ahead horizon; in most cases, the logarithmic transformation is the relevant one. As the forecast horizon increases, the evidence in favour of a transformation becomes less strong. Typically, the naïve predictor that just reverses the transformation leads to a lower mean square error than the optimal predictor at short forecast lead times. We also discuss whether the preliminary in-sample frequency domain assessment conducted here provides reliable guidance as to which series should be transformed in order to improve the predictive performance significantly.

Transformations aim to improve the statistical analysis of time series, by finding a suitable scale for which a model belonging to a simple and well known class, e.g. the normal regression model, has the best performance. An important class of transformations which are suitable for time series measured on a ratio scale with strictly positive support is the power transformation; originally proposed by Tukey (1957) as a device for achieving a model with a simple structure, normal errors and a constant error variance, it was subsequently modified by Box and Cox (1964). The objective of this paper is to assess whether transforming a variable leads to an improvement in the forecasting accuracy. This issue has already been debated in the time series literature. The use of the Box–Cox transformation as a preliminary specification step before fitting an ARIMA model was recommended in the book by Box and Jenkins (1970). In his discussion of the paper by Chatfield and Prothero (1973), Tunnicliffe Wilson (1973) advocated its use and showed that for the particular case study considered in the paper, the monthly sales of an engineering company, maximum likelihood estimation of the power transformation parameter could lead to superior forecasts. This point was elaborated further by Box and Jenkins (1973). A more extensive investigation was carried out by Nelson and Granger (1979), who considered a dataset consisting of twenty-one time series. After fitting a linear ARIMA model to the power transformed series and using 20 observations for a post-sample evaluation, they concluded that the Box–Cox transformation does not lead to an improvement in the forecasting performance. Another important conclusion, which is also supported by simulation evidence, is that the naïve forecasts, which are obtained by simply reversing the power transformation, perform better than the optimal forecasts based on the conditional expectation. The explanation for this is that the conditional expectation underlying the optimal forecast assumes that the transformed series is normally distributed. However, this assumption may not be realistic. In contrast to Nelson and Granger’s results, Hopwood, McKeown, and Newbold (1981) find, for a range of quarterly earnings-per-share series, that the Box–Cox transformation can improve forecast efficiency. In related work, Lütkepohl and Xu (2011) have investigated whether the logarithmic transformation (as a special case of a power transformation) leads to an improved forecasting accuracy relative to the untransformed series; the target variables are annual inflation rates, computed from seasonally unadjusted price series. The overall conclusion is that forecasts based on the original variables are characterized by a lower mean square forecast error. On the other hand, based on data on a range of monthly stock price indices, as well as quarterly consumption series, Lütkepohl and Xu (2012) conclude that using logs can be quite beneficial for forecasting. They also point out that there does not appear to be any reliable criterion for deciding between logs and levels for the purpose of maximizing the forecast accuracy. From a theoretical standpoint, Granger and Newbold (1976) provided a general analytical approach to forecasting transformed series, based on the Hermite polynomials series expansion. Analytical expressions for the minimum mean squared error predictors were provided by Pankratz and Dudley (1987) for specific values of the Box–Cox power transformation parameter. Guerrero (1993) suggested a simple approximate method for obtaining bias-corrected forecasts in the original measurement scale. Carroll and Ruppert (1981) dealt with the contribution of transformation parameter estimation to the overall mean square forecast error, concluding that it is usually small. Collins (1991) discussed and compared different methods for the interval forecasting of transformed series. More recently, Pascual, Romo, and Ruiz (2005) proposed a bootstrap procedure for constructing prediction intervals for a series when an ARIMA model is fitted to its power transformation. De Bruin and Franses (1999) considered forecasting power transformed series using a class of nonlinear time series models (smooth transition autoregressive models). Finally, the Box–Cox transformation is popular in financial time series analysis and has been considered for forecasting volatility (see e.g. Goncalves & Meddahi, 2011; Higgins & Bera, 1992; and Sadorsky & McKenzie, 2008) and price durations (Fernandes & Grammig, 2006), for example. This paper contributes to the debate in two ways: first, we propose a fast nonparametric method, based on the estimation of the prediction error variance (p.e.v.) of the normalized Box–Cox power transformation, which can be used to estimate the transformation parameter and to decide whether or not to use the power transformation if forecasting is the objective. Our procedure has the advantage that it does not require the normality assumptions which are used in maximum likelihood procedures. Hence, it circumvents the problem observed by Nelson and Granger (1979). Our second contribution is to assess the empirical relevance of the choice of the transformation parameter by performing a large scale recursive forecast exercise, on a dataset consisting of 530 seasonal monthly time series. In the previous studies, only much more limited datasets were used, and by considering such a large dataset we hope to get a better overall picture of the situation, and may be able to explain some of the previous discrepancies in results. A side issue is whether the naïve predictor outperforms the optimal predictor in terms of the mean square forecast error. We find that there is a certain percentage of series where significant forecast improvements are obtained by a power transformation. The challenge is then to identify the series for which a power transformation may help. The plan of the paper is as follows. In Section 2, after reviewing the Box–Cox transformation, we discuss the predictors of interest. In Section 3, we present the nonparametric procedure for estimating the p.e.v. and the transformation parameter. Section 4 provides a detailed discussion of the advantages and limitations of the method, in the light of the assumptions underlying the analysis. Section 5 discusses the estimation results on the dataset. In Section 6 we judge the relevance of the transformation for out-of-sample forecasting by conducting a rolling forecasting experiment; and conclusions are drawn in Section 7.

It is argued that previous studies of the Box–Cox transformation as a means of improving the forecast accuracy may be distorted by the assumption of a normally distributed transformed series. The latter assumption is often adopted when estimating the transformation parameter and computing the optimal forecast of the original series. We propose a distribution-free, nonparametric method for estimating the Box–Cox transformation parameter, and perform a large-scale forecast comparison based on a much larger set of time series than previous studies. More precisely, we consider a set of 530 monthly, seasonal time series which are related to industrial production and retail turnover in a large number of countries. We find three main results. First, a 95% confidence interval around the estimated transformation parameter only fails to contain the value of one, which corresponds to no transformation, in about 20% of cases. Thus, using a transformation parameter which is significantly different from one as a criterion for considering the Box–Cox transformation, a transformation is indicated for roughly 20% of the series. Clearly, if forecast improvements are possible in one fifth of the series, this is too large a set of series to be ignored if one is seriously interested in improving the forecast accuracy. Out of the series for which transformations are indicated, only a very small fraction (about 5%) has a parameter which is significantly different from 0, which corresponds to the log transformation. Thus, in most cases where a transformation is indicated, a log transformation may be a good choice. This result is well in line with those of Nelson and Granger (1979), who perform a forecast comparison for a much smaller set of time series. As a second main result, our forecast comparison shows that transformations can indeed improve the forecast accuracy at short horizons. For about 20% of the series, the one-step-ahead forecasts are significantly improved by using a Box–Cox transformation. However, the advantage of the transformation diminishes for longer-term forecasts. It turns out that, although the log transformation is indeed very successful at providing the best forecasts when a transformation is needed, estimating the transformation parameter by our method results in the best forecasts in about one third of the cases where a significant improvement is found, at least for short horizons.7 Moreover, the naïve predictor obtained by just inverting the transformation performs better overall than the optimal predictor which is based on the conditional expectation and uses normality assumptions. Again, the latter result conforms well with those of previous related studies. Our third main result is that our nonparametric approach to estimating the p.e.v. can provide good guidance, and can be a useful substitute for a computationally more demanding rolling forecast exercise for deciding on the need for a Box–Cox transformation. In general, when the approximate 95% nonparametric interval estimate does not cover λ=1λ=1, the rolling forecast experiment also points to the need to transform the series in order to get significantly better forecasts (judging significance by a modified DM test). The numbers of false positive and false negative decisions are typically small. The false decisions may be due to structural change and inhomogeneity in the series. In fact, the rolling forecasting experiment is able to accommodate structural change, such as the 2008 international financial crisis, as the parameters are re-estimated using a time window of six or ten years of monthly data. Hence, for a given value of λλ, forecasting is based effectively on a time-varying autoregressive model. On the other hand, the estimation of the transformation parameter based on the minimization of the p.e.v. uses the entire sample, and is subject to the effects of outliers and structural breaks.