Tests for regression neglected nonlinearity based on artificial neural networks (ANNs) have so far been studied by separately analyzing the two ways in which the null of regression linearity can hold. This implies that the asymptotic behavior of general ANN-based tests for neglected nonlinearity is still an open question. Here we analyze a convenient ANN-based quasi-likelihood ratio statistic for testing neglected nonlinearity, paying careful attention to both components of the null. We derive the asymptotic null distribution under each component separately and analyze their interaction. Somewhat remarkably, it turns out that the previously known asymptotic null distribution for the type 1 case still applies, but under somewhat stronger conditions than previously recognized. We present Monte Carlo experiments corroborating our theoretical results and showing that standard methods can yield misleading inference when our new, stronger regularity conditions are violated.

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We propose a new method for approximating the expected quadratic variation of an asset based on its option prices. The quadratic variation of an asset price is often regarded as a measure of its volatility, and its expected value under pricing measure can be understood as the market's expectation of future volatility. We utilize the relation between the asset variance and the Black-Scholes implied volatility surface, and discuss the merits of this new model-free approach compared to the CBOE procedure underlying the VIX index. The interpolation scheme for the volatility surface we introduce is designed to be consistent with arbitrage bounds. We show numerically under the Heston stochastic volatility model that this approach significantly reduces the approximation errors, and we further provide empirical evidence from the Nikkei 225 options that the new implied volatility index is more accurate in predicting future volatility. Keywords: Model-free implied volatility index; volatility forecasting; volatility surface; variance swaps. 433 Int. J. Theor. Appl. Finan. 2011.14:433-463. Downloaded from www.worldscientific.com by FLINDERS UNIVERSITY LIBRARY on 01/05/15. For personal use only. 434 M. Fukasawa et al.

We revisit the twofold identification problem discussed by Cho, Ishida, and White (2011), which arises when testing for neglected nonlinearity by artificial neural networks. We do not use the so-called “no-zero” condition and employ a sixth-order expansion to obtain the asymptotic null distribution of the quasi-likelihood ratio (QLR) test. In particular, we avoid restricting the number of explanatory variables in the activation function by using the distance and direction method discussed in Cho and White (2012). We find that the QLR test statistic can still be used to handle the twofold identification problem appropriately under the set of mild regularity conditions provided here, so that the asymptotic null distribution can be obtained in a manner similar to that in Cho, Ishida, and White ( 2011). This also implies that the weighted bootstrap in Hansen (1996) can be successfully exploited when testing the linearity hypothesis using the QLR test.