Martingale Property Research Articles

A Semiparametric Analysis of the Life Cycle-Permanent Income Hypothesis

This paper considers a semiparametric regression model to test the various implications of the Life Cycle-Permanent Income (LCP) hypothesis proposed by Hall (1978). The semiparametric regression model does not require any parametric assumption on the functional form of the unknown utility function in our analysis. In contrast, the linear regression models frequently used in the literature are justified under specific parametric forms of the utility function and may lead to a misleading conclusion on the LCP hypothesis if the parameterization is incorrect. Using both linear and semiparametric regression models, tests of the martingale property of consumption along with several specification tests are performed on the U. S quarterly data from 1947 to 1990. The results from the semiparametric model do not differ significantly from those from the linear model and suggest some evidences against the implications of the LCP hypothesis. [C14]

A SEMIPARAMETRIC ANALYSIS OF THE LIFE CYCLE-PERMANENT INCOME HYPOTHESIS

This paper considers a semiparametric regression model to test the various implications of the Life Cycle-Permanent Income (LCP) hypothesis proposed by Hall (1978). The semiparametric regression model does not require any parametric assumption on the functional form of the unknown utility function in our analysis. In contrast, the linear regression models frequently used in the literature are justified under specific parametric forms of the utility function and may lead to a misleading conclusion on the LCP hypothesis if the parameterization is incorrect. Using both linear and semiparametric regression models, tests of the martingale property of consumption along with several specification tests are performed on the U. S quarterly data from 1947 to 1990. The results from the semiparametric model do not differ significantly from those from the linear model and suggest some evidences against the implications of the LCP hypothesis. [C14]

Interacting measure branching processes. Some bounds for the support

We prove a tightness result for a sequence of interacting branching-diffusion processes. The limit points are continuous measure-valued processes and satisfy a martingale property. We state that th...

A martingale representation theorem for two independent semimartingales

Suppose that X 1and X 1 are independent semimartingales and both filtrations and have the martingale representation property, then the filtration also has the martingale representation property, wh...

A note on dilations and martingales

The purpose of this note is to investigate the effect of dilations on martingales and to give conditions under which a dilated martingale will retain the martingale property. The technique of dilating the joint distribution of a sequence of random variables has applications in optimal stopping theory and the study of “prophet inequalities” where martingales play a significant role.

A counterexample to a martingale characterization of a Wiener process

We show that a Wiener process is not characterized by a martingale and a reverse martingale property of some linear and quadratic transformations of the process.

A general risk process and its properties

We construct a risk process, where the law of the next jump time or jump size can depend on the past through earlier jump times and jump sizes. Some distributional properties of this process are established. The compensator is found and some martingale properties are discussed.

A general risk process and its properties

We construct a risk process, where the law of the next jump time or jump size can depend on the past through earlier jump times and jump sizes. Some distributional properties of this process are established. The compensator is found and some martingale properties are discussed.

Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances

We study the properties of the quasi-maximum likelihood estimator (QMLE) and related test statistics in dynamic models that jointly parameterize conditional means and conditional covariances, when a normal log-likelihood os maximized but the assumption of normality is violated. Because the score of the normal log-likelihood has the martingale difference property when the forst two conditional moments are correctly specified, the QMLE is generally Consistent and has a limiting normal destribution. We provide easily computable formulas for asymptotic standard errors that are valid under nonnormality. Further, we show how robust LM tests for the adequacy of the jointly parameterized mean and variance can be computed from simple auxiliary regressions. An appealing feature of these robyst inference procedures is that only first derivatives of the conditional mean and variance functions are needed. A monte Carlo study indicates that the asymptotic results carry over to finite samples. Estimation of several AR and AR-GARCH time series models reveals that in most sotuations the robust test statistics compare favorably to the two standard (nonrobust) formulations of the Wald and IM tests. Also, for the GARCH models and the sample sizes analyzed here, the bias in the QMLE appears to be relatively small. An empirical application to stock return volatility illustrates the potential imprtance of computing robust statistics in practice.

Complemented Subspaces of Spaces Obtained by Interpolation

If Z is a quotient of a subspace of a separable Banach space X, and V is any separable Banach space, then there is a Banach couple (A0, A1) such that A0 and A1 are isometric to X ⊕ V , and any intermediate space obtained using the real or complex interpolation method contains a complemented subspace isomorphic to Z. Thus many properties of Banach spaces, including having non-trivial cotype, having the Radon-Nikodym property, and having the analytic unconditional martingale difference sequence property, do not pass to intermediate spaces.

Convergence in distribution and Skorokhod Convergence for the general theory of processes

This paper proves some Skorokhod Convergence Theorems for processes with filtration. Roughly, these are theorems which say that if a family of processes with filtration (X n ,ℱ n ),n∈ℕ, converges in distribution in a suitable sense, then there exists a family of equivalent processes (Y n ,ℊ n ),n∈ℕ, which converges almost surely. The notion of equivalence used is that of adapted distribution, which guarantees that each (X n ,ℊ n ) has the same stochastic properties as (X n ,ℱ n ), with respect to its filtration, such as the martingale property or the Markov property. The appropriate notion of convergence in distribution is convergence in adapted distribution, which is developed in the paper. Fortunately, any tight sequence of processes has a subsequence which converges in adapted distribution. For discrete time processes, (Y n ,ℊ n ),n∈ℕ, and their limit (Y, ℊ) may be taken as all having the same fixed filtrationℊ n =ℊ. In the continuous time case, theY n ,ℊ n may require different filtrationsℊ n , which converge toℊ. To handle this, convergence of filtrations is defined and its theory developed.

Propriétés de martingales, explosion et représentation de Lévy—Khintchine d'une classe de processus de branchement à valeurs mesures

We study here by stochastic calculus methods some martingale properties of a general class of measure-valued branching processes. The form of the cumulant semigroup determines their local characteristics and the explosion time. Finally, by the infinite divisibility property of these processes, we obtain a Lévy-Khintchine representation on the paths space and we propose an interpretation of the canonical measures in terms of entrance laws.

Structure selection for bounded-parameter models: consistency conditions and selection criterion

Strong-consistency conditions for structure selection in bounded-parameter models are studied. A certain robust selection criterion, based on the volume of the exact parameter-bounding polytope, is proposed for linear regression models. The effectiveness of the polytope volume criterion is demonstrated on a model nonlinear in its variables but linear in its parameters. Its strong consistency is proved for a large class of noise distributions. The usual assumptions on the noise, namely independence, constant variance, or martingale difference properties, need not be made, but asymptotic independence is assumed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Some properties of the multitype measure branching process

Qualitative properties of the multitype measure branching process and its occupation time process are investigated, including martingale properties, Hausdorff dimension of supports, existence of densities and stochastic equations.

A martingale characterization of quantum Poisson processes

We study the characterization problem of quantum Poisson processes using the martingale property and some conditions on moments up to fourth order. These conditions are inspired by an assumption on the quadratic variation that is equivalent, in classical cases, to the fact that the process has purely discontinuous sample paths with unit jumps.

Forward rates and the expectations theory of the term structure: Tests for the Federal Republic of Germany

The expectations theory of the term structure was tested using data from West German capital and money markets for the period 1975:01–1986:12. If forward rates implicit in the term structure are market expectations of future spot rates, and if term premia are not time-dependent, then forward rates should follow a martingale sequence. This hypothesis was tested with the aid of standard time series techniques (autocorrelation functions, Box-Pierce, unit-roots,F-tests and co-integration). The expectations theory was for the most part rejected, although the martingale property, or equivalently, weak form efficiency, held for the latter part of the test period. The rejection of the simple expectations theory is consistent with the hypothesis of time-varying term premia.

Exchange Rate Volatility in an Equilibrium Asset Pricing Model

This paper analyzes a stochastic model of exchange rate determination with unrestricted access to capital and currency markets. It is shown that the only restriction imposed by the model on the equilibrium exchange rate is that it satisfy a martingale property. This implies that, for a given real equilibrium allocation (which is optimal), the model can display varying amounts of exchange rate volatility, and that volatility is unrelated to the welfare properties of the equilibrium allocation. Whether or not volatility is excessive therefore depends on which equilibrium is chosen as the basis for comparison. Copyright 1990 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

On Some Sample Path Properties of Intra-Day Futures Prices

This paper develops a time-series model for continuous time asset prices and then uses tick-by-tick data from Treasury bill futures to develop both a definition and test for efficiency in the continuous time case. The results suggest that intra-day data on futures prices do not behave like a Markov Renewal process; rather, lagged values of futures prices do have some predictive power. In addition, trading times are not useful in predicting futures prices. Finally, we estimate the bid-ask spread and show that even after adjusting for this spread, the serial dependence between current and lagged returns remains. The multitude of studies concerning efficiency in futures markets support the proposition that a Martingale approximation is reasonable for most commodity and capital asset markets, while the same data reject Gaussian processes as an appropriate model.1 All of these studies, however, use either close to close prices or open to open prices in the estimation process. The choice of daily data is arbitrary; a natural question concerns whether, based on continuous time data, futures prices can be shown to be realizations of continuous time Markov processes and whether they can be represented as stochastically linear processes. In this paper, we use intra-day, tick-by-tick, data on Treasury bills futures and develop both a definition and a test for efficiency in the continuous time case. Observations on continuous time prices yield two separate time series: the trading prices and times. As a result, we claim that the standard procedures for tests of market efficiency must be replaced by two separate necessary conditions; the first is the usual condition that successive price changes are independent; the second requires that the recurrence times between trades obey a Poisson process. We apply the above definitions to intra-day futures prices for Treasury bills for a 57 day period in 1983. The results indicate that the Markov model does not hold for intra-day futures prices but the trading times do seem to behave approximately as a Poisson process. Received for publication January 29, 1988. Revision accepted for publication July 19, 1989. * City University of New York and State University of New York at Stony Brook, respectively. We gratefully acknowledge financial support for this research from the Center for the Study of Futures Markets at Columbia University. Stephanie Dieringer provided invaluable help as a research assistant for this project. 1 Several studies have examined the martingale property. For a summary of these results see, for example, Kamara (1982). 2 See, for example, Fama (1965), Mandlebrot (1963), Stevenson and Bear (1970), and Neftci and Policano (1984). Some studies that do analyze intraday data include Feinstone (1985) and Hinich and Patterson (1985).

A martingale characterization of the Wiener process

The Wiener process is characterized by martingale and reverse martingale property of some linear and quadratic transformation of the process.

Conditions for optimality in the infinite-horizon portfolio-cum-saving problem with semimartingale investments

A model of optimal accumulation of capital and portfolio choice over an infinite horizon in continuous time is formulated in which the vector process representing returns to investments isa general semimartingale. Methods of stochastic calculus and calculus of variations are used to obtain necessary and sufficient conditions for optimality involving martingale properties ofthe shadow price processes associated with alternative portfolio cum saving plans.The relationship between such conditions and portfolio equations is investigated.The results are appliedtospecial cases where the returns process has stationary independent increments and the utility function has the discounted relative risk aversion form