1- Introduction

2- Formulation of calibration with varying window sizes: How to endogenise and make different window sizes comparable

3- Application of the Lagrange regularisation method to a simple linear-regression problem

4- Using the Lagrange regularisation method for detecting the beginning of financial bubbles

5- Conclusion

References

Abstract

Motivated by the question of identifying the start time τ of financial bubbles, we propose an improved calibration approach for time series in which the inception of the latest regime of interest is unknown. By taking into account the tendency of a given model to overfit data, we introduce the Lagrange regularisation of the normalised sum of the squared residuals, χ 2 np(Φ), to endogenously detect the optimal fitting window size := w∗ ∈ [τ : t¯ 2] that should be used for calibration, assuming a fixed pseudo present time t¯ 2. The Lagrange regularisation of χ 2 np(Φ) defines the Lagrange regularised sum of the squared residuals, χ 2 λ (Φ). Its performance is exemplified on a simple Linear Regression problem with a change point and compared against the performances of the Residual Sum of Squares (RSS) := χ 2 (Φ) and RSS/(N-p) := χ 2 np(Φ), where N is the sample size, p is the number of degrees of freedom and Φ is the parameter vector. Applied to synthetic models of financial bubbles with a well-defined transition regime and to a number of financial time series (US S&P500, Brazil IBovespa and China SSEC Indices), χ 2 λ (Φ) is found to provide well-defined reasonable determinations of the starting times for major bubbles such as the bubbles ending with the 1987 Black-Monday, the 2008 Sub-prime crisis and minor speculative bubbles on other Indexes, without any further exogenous information. The application of the method thus allows one to endogenise the determination of the starting time of bubbles, a problem that has yet not received a systematic objective solution. Moreover, the technique appears as a practical solution for comparing goodness-of-fit across unbalanced sample sizes.

Introduction

There is an inverse relationship between the tendency of a model to overfit data and the sample size under consideration. In other words, the smaller the sample size, the larger the flexibility for the model with a fixed number of parameters to overfit [1]. Due this characteristic feature, one cannot directly compare goodness-of-fit metrics of statistical models arbitrarily parametrised by the vector Φ of parameters such as the Residual Sum of Squares, RSS := χ 2 (Φ) or its normalised version RSS (N−p) := χ 2 np(Φ), over unequal sized samples. Here, N denotes the sample size while p is the number of degrees of freedom of a model. This has particular relevance when one is interested in selecting the optimal sub-sample of a data set to calibrate a model, a recurrent issue when estimating time series models in a moving window or/and when the model is only valid in a specific time window, which is unknown a priori.

مقاله سال 2016

Airports located in specific locations serve China’s transportation network as feeders of a hub-and-spoke system and as origins or destinations of point-to-point services (Button, 2002). The airport network service contributes to China’s economic growth and regional development (Wells and Young, 2004). Research conducted on China’s airports has generally focused on efficiency and productivity, while neglecting the role of space in the study of airport activity (Fung, 2008; Fung et al., 2008a, 2008b; Fan et al., 2014). Therefore, the locations of airports and its spatial effects are needed to be included into consideration to study airport benchmarking and policy review (Lian and Rønnevik, 2010; Frohlich and Niemeier, 2011; Fan et al., 2014; Pavlyuk, 2016 € ). The present research aims at filling this gap and analyze Chinese airports using spatial models; a cost function is estimated for the period from 2002 to 2012 that takes into account the spatial effects measured by latitude and longitude of the airports analyzed.

The motivations behind the present research are the following. First, China is an extensive country, and airports are a core asset for connecting the country and regions; therefore, an understanding ofthe spatial distribution of airports around the country is important. As the connectivity of China’s airports is complementary rather than competitive in nature, establishing a balance of competition and cooperation (Derudder et al., 2010), it is important to understand the spatial correlations that may exist on airports (Miller, 1999). Second, China’s airports have recently undergone several reforms since Deng Xiaoping’s policy changes. In this context, an analysis of the cost of China’s airports, as well as of their observed spatial characteristics, is justified, as it may help clarify the role that transportation networks play in cost, and thus allow for analysis that proposes optimal cost controls. Third, built on the deregulation of China’s air transportation, the hub airports also faced increasing competition for hub passengers because of the adoption of huband-spoke models for organizing route structures (Button, 2002). This factor justifies the focus on cost efficiency and the adoption of spatial analysis to identify spatial correlations. Finally, while research has noted the significant effects of location on an airport’s performance or productivity, no study has used spatial models in its methodology. Rather, the existing literature mainly focuses on the technical efficiency or productivity of Chinese civil airports with traditional SFA (Stochastic Frontier Analysis), which ignore independence between observations. This paper is innovative and make use of the spatial models found in the SFA to examine the cost ef- * Corresponding author. ficiency of Chinese airports.

The present research analyzes the cost functions of Chinese airports during the period from 2002 to 2012 using five alternative spatial models, which (to reiterate) no study on Chinese airports had done before. It also provides certain policy implications for improving cost efficiency that should be considered before a new round of construction of civil airports begins (Hyard, 2013). This research is of particular interest at this time (Chang et al., 2013; Fan et al., 2014; Jiang and Zhang, 2014; Wang et al., 2015).