An application of vector GARCH model in semiconductor demand planning

Abstract

Demand planning has been the key to supply chain management in semiconductor industry. With an appropriate weight assignment scheme, the planning accuracy resulting from combinational forecasts can be improved by merging several individual candidate methods. In this paper we discuss the applicability of vector generalized autoregressive conditional heteroskedasticity (GARCH) model to determine the optimal combinational weights of component forecasts, where the conditional variances and correlations of forecast errors from candidate methods are represented and estimated by a maximum-likelihood procedure. The asymptotical properties of parameter estimators for GARCH model are investigated by simulation experiments. An example of the proposed method to real time series of electronic products demonstrates that this weight-varying combinational method produces less prediction errors, compared to other commonly used forecasting approaches that are based on single model selection criteria or fixed weights.

Introduction

Statistical models for time series analysis and forecasting, such as exponential smoothing, autoregressive moving average (ARMA) (Box and Jenkins, 1976), have been widely employed in supply chain management, where accurate projections on product demand are crucial to subsequent operations planning (e.g., capacity allocation, inventory management, etc.). In practical applications, however, it is usually a tough decision to make in regard to choosing the “best” forecasting model among a variety of candidates. For instance, hypothetical testing has been discussed in literature as a formal model selection technique to pick up a specific forecasting model. One major implementation drawback with statistical testing is the lack of objective guideline in choosing the size of individual test or the quantitative relationship between test and forecasting accuracy (Clemen, 1989, Yang, 2001).

Another way to choose a forecasting model is based on certain pre-defined model selection criterion. For instance, Akaike information criterion (AIC) is proposed by considering the discrepancy measure between the true model and a candidate (Akaike, 1973), and Bayesian information criterion (BIC) adopts an approximation on posterior model probabilities to determine the appropriate individual model (Schwarz, 1978). While the aforementioned model selection criteria are simple to use, instability have prevented their broad application. When the sample size of demand observations is small or moderate, several forecasting models may tend to yield similar performances and thereby hard to be distinguished. That is, the selection of the final model given by the smallest historical criterion value is not stable or guaranteed to be always the best for use. For instance, a slight structural change on time series may result in another model selection. As such, the demand forecast based on single model selection criterion has high variability (Yang, 2001, Armstrong, 1989).

On the other hand, as stated in forecasting literature (Clemen, 1989, Clement and Hendry, 1998, Bates and Granger, 1969, Makarides and Hibon, 2000, Chan et al., 2004), a combinational forecast can increase forecasting accuracy by integrating several separate forecast models when difficulties arise in identifying a single model. The intention of combinational forecasts is to average forecasts in hope that the biases among individual models will compensate for one another. As a result, predictions obtained from different forecasting models are expected to be more useful in cases of high uncertainty. Combinational forecast has been studied in the past decades as a potential to reduce variability that appears in the forced action of single model selection. When multiple models are considered for demand planning, as discussed above, the instability of single model selection does not necessarily lead to the best model. The combinational forecast method, however, is capable of integrating model uncertainty or statistical information from distinct models and yields less prediction error. The weights of candidate models also serve as an indicator of detecting the need for intervention when the assumed structure of underlying process alters (Chan et al., 2004, Chan et al., 1999, Yang, 2001).

Let x₁, … , x_t, … be the time series of a scalar product demand. At period t, we need to predict the new demand x_t+1 based on the observation history up to x_t. Suppose there are K different forecasting models such that each produces an individual one-step-ahead forecast y_i,t for x_t+1 (i = 1, … , K). Much work on combinational forecasting has been devoted to finding the optimal weights α_i and then producing a linear combination of y_i,t with fixed α_i, which relies on the assumption that product demand series {x_t} is stationary.

In the fixed weights (FW) based combinational forecasts, weights α_i can be obtained by solving a least-square problem, where demand is treated as response variable and the individual forecasts as independent predictor variables. Thus, the combinational weights are the optimal regression coefficients that minimize the sum of squared forecast errors, i.e., ${\sum }_{t=1}(X_t-{\mathit{\alpha }}^{\text{T}}{\mathbf{y}}_t{)}^2$, where α = [α₁, … , α_K]^T and y_t = [y_1,t, … , y_K,t]^T.

The scheme of assuming fixed weights in combinational forecasts is reasonable when the underlying process of product demand is stable. In such circumstances, the optimal weights used for one-step prediction ${\hat{x}}_{t+1}={\sum }_i{\alpha }_i{y}_{i\text{,}t}$ given at current period t will keep unchanged for computing ${\hat{x}}_{t+2}\text{,}{\hat{x}}_{t+3}\text{,}\dots$ However, the inspection on many product demand histories demonstrates that these processes consist of highly volatile periods as well as periods of relative tranquility. Therefore, the assumption of constant variance (homoskedasticity) or stationary process for fixed weights scheme in combinational forecast is inappropriate. A challenging problem in integrating candidate forecasts becomes how to assign proper weights to accommodate the unknown volatile demand process, while the research on combinational forecasts so far provides little guidance (Armstrong, 1989, Yang, 2001). To tackle this issue, a multivariate GARCH model is addressed in this paper for determining the weights of individual models when the underlying time series deviates from their stationary assumption, which is commonly observed in unsteady semiconductor product demand process. The ultimate objective of the research is then to improve the accuracy of demand planning and the overall performance of semiconductor or other innovative products based high-tech industrial supply chain management.

The plan of the remainder of this paper is as follows. Section 2 introduces some concepts of combinational forecasts relevant to the present work. In Section 3 we describe the construction of varying-variance GARCH model and its learning through the maximum-likelihood estimation (MLE) approach, as demonstrated by the simulation study. Some illustrative examples are given in Section 4 when the proposed model is applied to real product demand planning problems. The consequent quantitative comparison results against the fixed weight combinational forecast method are presented to illustrate the advantage of combining multiple prediction models. Section 5 concludes this paper with some final remarks.