Testing the Markov property with high frequency data

This paper develops a framework to nonparametrically test whether discrete-valued irregularly spaced financial transactions data follow a Markov process. For that purpose, we consider a specific optional sampling in which a continuous-time Markov process is observed only when it crosses some discrete level. This framework is convenient for it accommodates the irregular spacing that characterizes transactions data. Under such an observation rule, the current price duration is independent of a previous price duration given the previous price realization. A simple nonparametric test then follows by examining whether this conditional independence property holds. Monte Carlo simulations suggest that the asymptotic test has huge size distortions, though a bootstrap-based variant entails reasonable size and power properties in finite samples. As for an empirical illustration, we investigate whether bid–ask spreads follow Markov processes using transactions data from the New York Stock Exchange. The motivation lies on the fact that asymmetric information models of market microstructures predict that the Markov property does not hold for the bid–ask spread. We robustly reject the Markov assumption for two out of the five stocks under scrutiny. Finally, it is reassuring that our results are consistent with two alternative measures of asymmetric information.     

A consistent model specification test with mixed discrete and continuous data

In this paper we propose a nonparametric kernel-based model specification test that can be used when the regression model contains both discrete and continuous regressors. We employ discrete variable kernel functions and we smooth both the discrete and continuous regressors using least squares cross-validation (CV) methods. The test statistic is shown to have an asymptotic normal null distribution. We also prove the validity of using the wild bootstrap method to approximate the null distribution of the test statistic, the bootstrap being our preferred method for obtaining the null distribution in practice. Simulations show that the proposed test has significant power advantages over conventional kernel tests which rely upon frequency-based nonparametric estimators that require sample splitting to handle the presence of discrete regressors.     

基于注意力机制的LSTM股价趋势预测研究

针对中国股票市场,提出了一种基于注意力机制的LSTM股价趋势预测模型。选取42只中国上证50从2009年到2017年的股票数据为实验对象,根据股票市场普遍认可的经验规则,分别对每个技术指标进行量化处理得到股票涨跌的趋势数据,并和交易数据混合作为预测模型的输入,然后使用基于注意力机制的LSTM模型提取股价趋势特征进行预测。实验结果表明:引入股票离散型趋势数据到预测模型中,能够在已有交易数据和技术指标的基础上提升预测精确度,与传统的机器学习模型SVM和单一的LSTM模型相比,基于注意力机制的LSTM模型具有更好的预测能力。     

Non-Walrasian decentralization of the core

We show that in large finite economies, core allocations can be approximately decentralized as Nash (rather than Walras) equilibrium. We argue that this exercise is an essential complement to asymptotic core equivalence results, because it implies that in some approximate sense individual attempts to manipulate the decentralizing prices cannot be beneficial, which fits precisely the interpretation of asymptotic core convergence, namely the emergence of price taking.     

Applications of the Likelihood Theory in Finance: Modelling and Pricing

This paper discusses the connection between mathematical finance and statistical modelling which turns out to be more than a formal mathematical correspondence. We like to figure out how common results and notions in statistics and their meaning can be translated to the world of mathematical finance and vice versa. A lot of similarities can be expressed in terms of LeCam’s theory for statistical experiments which is the theory of the behaviour of likelihood processes. For positive prices the arbitrage free financial assets fit into statistical experiments. It is shown that they are given by filtered likelihood ratio processes. From the statistical point of view, martingale measures, completeness, and pricing formulas are revisited. The pricing formulas for various options are connected with the power functions of tests. For instance the Black–Scholes price of a European option is related to Neyman–Pearson tests and it has an interpretation as Bayes risk. Under contiguity the convergence of financial experiments and option prices are obtained. In particular, the approximation of Itô type price processes by discrete models and the convergence of associated option prices is studied. The result relies on the central limit theorem for statistical experiments, which is well known in statistics in connection with local asymptotic normal (LAN) families. As application certain continuous time option prices can be approximated by related discrete time pricing formulas.     

Real Asset Valuation under Imperfect Competition: Can We Forget About Market Fundamentals?

Real assets are usually valued by computing the stream of profits they can bring to a price‐taking firm in a liquid market. This method ignores market fundamentals by assuming that all the relevant information is included in the spot price. Our article analyses the bias resulting from such an approach when the market is imperfectly competitive. We propose a stylised two‐period model of the natural gas market with no uncertainty, focusing on strategic interactions between two types of oligopolistic players—pure traders and suppliers with downstream customers—who have access to storage. We show that the true value of storage capacity is not the same for traders and for suppliers. Comparing the latter value with the traditional price‐taking valuation reveals a systematic bias that tends to induce underinvestment.     

On specification testing of ordered discrete choice models

We discuss how to test the specification of an ordered discrete choice model against a general alternative. Two main approaches can be followed: tests based on moment conditions and tests based on comparisons between parametric and nonparametric estimations. Following these approaches, various statistics are proposed and their asymptotic properties are discussed. The performance of the statistics is compared by means of simulations. An easy-to-compute variant of the standard moment-based statistic yields the best results in models with a single explanatory variable. In models with various explanatory variables the results are less conclusive, since the relative performance of the statistics depends on both the fit of the model and the type of misspecification that is considered.     

Bootstrap testing for the null of no cointegration in a threshold vector error correction model

We develop a test for the linear no cointegration null hypothesis in a threshold vector error correction model. We adopt a sup-Wald type test and derive its null asymptotic distribution. A residual-based bootstrap is proposed, and the first-order consistency of the bootstrap is established. A set of Monte Carlo simulations shows that the bootstrap corrects size distortion of asymptotic distribution in finite samples, and that its power against the threshold cointegration alternative is significantly greater than that of conventional cointegration tests. Our method is illustrated with used car price indexes.     

基于多项Logit选择模型的在线渠道价格促销效应实证研究

在线渠道下,不同品牌的价格折扣促销的差异性会引起在线消费者品牌选择行为变化。同时,在线的价格比较机制也增加了消费者价格敏感度,如果不分离出这一重要的促销效应,那么会产生估计偏差,影响模型的预测能力。本文首先回顾品牌选择的离散模型中价格促销的研究脉络;然后构建在线渠道消费者品牌选择离散Logit模型;接着在对中国某电子商务公司的消费者实际购买可乐的面板数据分析的基础上,研究价格折扣对于该公司消费者品牌选择行为的影响效应,研究发现：（1）如果不引入价格折扣促销因素,将会过高估计价格的影响;（2）价格折扣促销对于消费者网上品牌选择有显著的影响,且影响为正效应。文章最后剖析价格折扣对在线渠道的消费者品牌选择的行为影响机理,并提出未来的研究方向。     

Estimating the determinants of stock price changes

I develop an approach for estimating the determinants of stock price changes that uses all eligible trade data and other observable parameters of market activity. This approach backs out the unobserved continuous price change distribution from the observable discrete price changes, and does not constrain the determinants to be proportions of the traded bid-ask spread. I show that theoretically impermissible results and skewed estimates of cost components are obtained when the model used for estimating the determinants of stock price changes does not attempt to uncover the mapping between the observed price changes and the underlying unobserved continuous price change process, and does not effectively use all eligible trade data.     

Testing a discrete switching disequilibrium model of the UK labour market

This paper develops a test procedure for serial correlation for discrete switching disequilibrium models which include both an endogenous price adjustment equation and lagged dependent variables. The tests are applied to a model of the UK labour market and the model is respecified in the light of the test results.     

Testing Part of a DSGE Model by Indirect Inference

We propose a way of testing a subset of equations of a DSGE model. The test draws on statistical inference for limited information models and the use of indirect inference to test DSGE models. Using the numerical small sample distribution of our test for two subsets of equations of the Smets–Wouters model we show that the test has accurate size and good power in small samples, and better power than using asymptotic distribution theory. In a test of the Smets–Wouters model on US Great Moderation data, we reject the specification of the wage‐price but not the expenditure sector. This points to the wage‐price sector as the source of overall model rejection.     

Frequency domain estimation of temporally aggregated Gaussian cointegrated systems

This paper considers joint estimation of long run equilibrium coefficients and parameters governing the short run dynamics of a fully parametric Gaussian cointegrated system formulated in continuous time. The model allows the stationary disturbances to be generated by a stochastic differential equation system and for the variables to be a mixture of stocks and flows. We derive a precise form for the exact discrete analogue of the continuous time model in triangular error correction form, which acts as the basis for frequency domain estimation of the unknown parameters using discrete time data. We formally establish the order of consistency and the asymptotic sampling properties of such an estimator. The estimator of the cointegrating parameters is shown to converge at the rate of the sample size to a mixed normal distribution, while that of the short run parameters converges at the rate of the square root of the sample size to a limiting normal distribution.     

Itchy feet vs cool heads: Flow of funds in an agent-based financial market

Investors tend to move funds when they are unhappy with their current portfolio managers׳ performance. We study the effect of the size of this flow of funds in an agent-based model of the financial market. The model combines the discrete choice approach from agent-based modelling, where all capital is mobile, with the evolutionary finance framework where all growth is endogenous. Our results show that, if investors exhibit recency bias in evaluating portfolio managers׳ performance, even a small amount of freely flowing capital has a huge impact on the market dynamics and the survival of noise traders. We also find that investors׳ intensity of choice is a driving force for excess volatility and extreme price movements when the size of the flow of funds is large.     

Activity signature functions for high-frequency data analysis

We define a new concept termed activity signature function, which is constructed from discrete observations of a continuous-time process, and derive its asymptotic properties as the sampling frequency increases. We show that the function is a useful device for estimating the activity level of the underlying process and in particular for deciding whether the process contains a continuous martingale. An application to $ /DM exchange rate over 1986–1999 indicates that a jump-diffusion model is more plausible than a pure-jump model. A second application to internet traffic at NASA servers shows that an infinite variation pure-jump model is appropriate for its modeling.     

The adequacy of asymptotic approximations in the near-integrated autoregressive model with dependent errors

We consider the normalized least squares estimator of the parameter in a nearly integrated first-order autoregressive model with dependent errors. In a first step we consider its asymptotic distribution as well as asymptotic expansion up to order O_p(T⁻¹). We derive a limiting moment generating function which enables us to calculate various distributional quantities by numerical integration. A simulation study is performed to assess the adequacy of the asymptotic distribution when the errors are correlated. We focus our attention on two leading cases: MA(1) errors and AR(1) errors. The asymptotic approximations are shown to be inadequate as the MA root gets close to −1 and as the AR root approaches either −1 or 1. Our theoretical analysis helps to explain and understand the simulation results of Schwert (1989) and DeJong, Nankervis, Savin, and Whiteman (1992) concerning the size and power of Phillips and Perron's (1988) unit root test. A companion paper, Nabeya and Perron (1994), presents alternative asymptotic frameworks in the cases where the usual asymptotic distribution fails to provide an adequate approximation to the finite-sample distribution.     

Information structures and viable price systems

A dynamic model of capital/financial markets is developed. A surprise is a stopping time that is not foretellable. We show that if agents' preferences exhibit a kind of time complementarity, then between ex-dividend dates a viable price system can make discrete changes only at surprises. Under the same preference condition, when the information in the economy can be modeled by a Brownian motion, a viable price system is an Itô process between ex-dividend dates. The martingle characterization of a viable price system originated by Harrison and Kreps (1979) is extended to our economy. This martingale result is independent of the time complementarity of preferences alluded to above.     

Combining kernel estimators in the uniform deconvolution problem

We construct a density estimator and an estimator of the distribution function in the uniform deconvolution model. The estimators are based on inversion formulas and kernel estimators of the density of the observations and its derivative. Initially the inversions yield two different estimators of the density and two estimators of the distribution function. We construct asymptotically optimal convex combinations of these two estimators. We also derive pointwise asymptotic normality of the resulting estimators, the pointwise asymptotic biases and an expansion of the mean integrated squared error of the density estimator. It turns out that the pointwise limit distribution of the density estimator is the same as the pointwise limit distribution of the density estimator introduced by Groeneboom and Jongbloed (Neerlandica, 57, 2003, 136), a kernel smoothed nonparametric maximum likelihood estimator of the distribution function.     

Local GMM estimation of time series models with conditional moment restrictions

This paper investigates statistical properties of the local generalized method of moments (LGMM) estimator for some time series models defined by conditional moment restrictions. First, we consider Markov processes with possible conditional heteroskedasticity of unknown forms and establish the consistency, asymptotic normality, and semi-parametric efficiency of the LGMM estimator. Second, we undertake a higher-order asymptotic expansion and demonstrate that the LGMM estimator possesses some appealing bias reduction properties for positively autocorrelated processes. Our analysis of the asymptotic expansion of the LGMM estimator reveals an interesting contrast with the OLS estimator that helps to shed light on the nature of the bias correction performed by the LGMM estimator. The practical importance of these findings is evaluated in terms of a bond and option pricing exercise based on a diffusion model for spot interest rate.     

Volatility swaps and volatility options on discretely sampled realized variance

Volatility swaps and volatility options are financial products written on discretely sampled realized variance. Actively traded in over-the-counter markets, these products are often priced by continuously sampled approximations to simplify the computations. This paper presents an analytical approach to efficiently and accurately price discretely sampled volatility derivatives, under a general stochastic volatility model. We first obtain an accurate approximation for the characteristic function of the discretely sampled realized variance. This characteristic function is then applied to price discrete volatility derivatives through either semi-analytical pricing formulae (up to an inverse Fourier transform) or an efficient Fourier-cosine series method. Numerical experiments show that our approximation is more accurate in comparison to the approximations in the literature. We remark that although discretely sampled variance swaps and options are usually more expensive than their continuously sampled counterparts, discretely sampled volatility swaps are more prone to be cheaper than the continuously sampled counterparts. An analysis is then provided to explain why this is the case in general for realistic contract specifications and reasonable model parameters.