Martingale Difference Sequence Research Articles

LIQUIDITY AND THE INFORMATIONAL EFFICIENCY OF AFRICAN STOCK MARKETS

AbstractThe hypothesis that a stock market price index follows a random walk is tested for 11 African stock markets, Botswana, Côte d’Ivoire, Egypt, Ghana, Kenya, Mauritius, Morocco, Nigeria, South Africa, Tunisia and Zimbabwe using joint variance ratio tests with finite‐sample critical values, over the period beginning in January 2000 and ending in September 2006. The iid random walk hypothesis is rejected in all 11 markets. In four stock markets, Egypt, Nigeria, Tunisia and South Africa, weekly returns are a martingale difference sequence. Liquidity is an important factor which contributes to whether a stock market follows a random walk.

Empirical likelihood inference for partial linear models under martingale difference sequence

We consider the application of the empirical likelihood method to a partially linear model with martingale difference error. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. Some simulations are conducted to illustrate the proposed method.

Large Deviations for Some Dependent Sequences

Let ( X i ) be a martingale difference sequence and S n = ∑ i = 1 n X i . Suppose ( X i ) is bounded in L p . In the case p≥ 2, Lesigne and Volny (Stochastic Process. Appl. 96 (2001) 143) obtained the estimation μ ( S n > n ) ≤ c n - p / 2 , Yulin Li (Statist. Probab. Lett. 62 (2003) 317) generalized the result to the case when p∈(1, 2] and obtained μ ( S n > n ) ≤ c n 1 - p , these are optimal in a certain sense. In this article, the authors study the large deviation of S n for some dependent sequences and obtain the same order optimal upper bounds for μ( S n>n ) as those for martingale difference sequence.

Cesaro mean convergence of martingale differences in rearrangement invariant spaces

We study the class of r.i. spaces in which Cesaro means of any weakly null martingale difference sequence is strongly null. This property is related to the Banach-Saks property. We show that in classical (separable) r.i. spaces (such as Orlicz, Lorentz and Marcinkiewicz spaces) these proper- ties coincide but this is no longer true for general r.i. spaces. We locate also a class of r.i. spaces having this property where an analogue of the classical Dunford-Pettis characterization of relatively weakly compact subsets in L1 holds. Mathematics Subject Classification (2000). 46E30 (primary); 46B20.

Rates of convergence for minimal distances in the central limit theorem under projective criteria

In this paper, we give estimates of ideal or minimal distances between the distribution of the normalized partial sum and the limiting Gaussian distribution for stationary martingale difference sequences or stationary sequences satisfying projective criteria. Applications to functions of linear processes and to functions of expanding maps of the interval are given.

Diagnostic checking of multivariate nonlinear time series models with martingale difference errors

In this article, we derive the asymptotic distribution of residual autocovariance and autocorrelation matrices for a general class of multivariate nonlinear time series models by assuming only that the error term is a martingale difference sequence. Two types of applications are developed: global test statistics of the portmanteau type and one-lag test statistics, which describe the residual correlation at individual lags. To illustrate the proposed methodology, simulation results are reported for diagnosing multivariate threshold time series models. The following test statistics are compared: the classical test statistics presuming independent errors and the proposed methodology which supposes only martingale difference errors.

Exact convergence rates in the central limit theorem for a class of martingales

We give optimal convergence rates in the central limit theorem for a large class of martingale difference sequences with bounded third moments. The rates depend on the behaviour of the conditional variances and, for stationary sequences, the rate n−1/2log n is reached.

Nonlinearity, nonstationarity, and spurious forecasts

Abstract Implications of nonlinearity, nonstationarity, and misspecification are considered from a forecasting perspective. Our model allows for small departures from the martingale difference sequence hypothesis by including a nonlinear component, formulated as a general, integrable transformation of the I ( 1 ) predictor. We assume that the true generating mechanism is unknown to the econometrician and he is therefore forced to use some approximating functions. It is shown that in this framework the linear regression techniques lead to spurious forecasts. Improvements of the forecast accuracy are possible with properly chosen nonlinear transformations of the predictor. The paper derives the limiting distribution of the forecasts’ mean squared error (MSE). In the case of square integrable approximants, it depends on the L 2 -distance between the nonlinear component and approximating function. Optimal forecasts are available for a given class of approximants.

Baum–Katz–Nagaev type results for martingales

We obtain a Baum–Katz–Nagaev type theorem for bounded martingale difference sequences that have more than a second moment, and prove that the celebrated Hsu–Robbins–Erdős theorem fails for martingales.

Some remarks on tangent martingale difference sequences in $L^1$-spaces

Let $X$ be a Banach space. Suppose that for all $p\in $ a constant $C_{p,X}$ depending only on $X$ and $p$ exists such that for any two $X$-valued martingales $f$ and $g$ with tangent martingale difference sequences one has $$\mathbb{E}\|f\|^p \leq C_{p,X} \mathbb{E}\|g\|^p \qquad (*).$$ This property is equivalent to the UMD condition. In fact, it is still equivalent to the UMD condition if in addition one demands that either $f$ or $g$ satisfy the so-called (CI) condition. However, for some applications it suffices to assume that $(*)$ holds whenever $g$ satisfies the (CI) condition. We show that the class of Banach spaces for which $(*)$ holds whenever only $g$ satisfies the (CI) condition is more general than the class of UMD spaces, in particular it includes the space $L^1$. We state several problems related to $(*)$ and other decoupling inequalities.

Change‐point monitoring in linear models

Summary We consider a linear regression model with errors modelled by martingale difference sequences, which include heteroskedastic augmented GARCH processes. We develop asymptotic theory for two monitoring schemes aimed at detecting a change in the regression parameters. The first method is based on the CUSUM of the residuals and was studied earlier in the context of independent identically distributed errors. The second method is new and is based on the squares of prediction errors. Both methods use a training sample of size m. We show that, as m∞, both methods have correct asymptotic size and detect a change with probability approaching unity. The methods are illustrated and compared in a small simulation study.

Nonparametric estimation of volatility models with serially dependent innovations

We are interested in modelling the time series process y t = σ ( x t ) ε t , where ε t = φ 0 ε t - 1 + v t . This model is of interest as it provides a plausible linkage between risk and expected return of financial assets. Further, the model can serve as a vehicle for testing the martingale difference sequence hypothesis, which is typically uncritically adopted in financial time series models. When x t has a fixed design, we provide a novel nonparametric estimator of the variance function based on the difference approach and establish its limiting properties. When x t is strictly stationary on a strongly mixing base (hereby allowing for ARCH effects) the nonparametric variance function estimator by Fan and Yao [1998. Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85, 645–660] can be applied and seems very promising. We propose a semiparametric estimator of φ 0 that is T -consistent, adaptive, and asymptotic normally distributed under very general conditions on x t .

GUNDY'S DECOMPOSITION FOR NON-COMMUTATIVE MARTINGALES AND APPLICATIONS

We provide an analogue of Gundy's decomposition for $L_1$-bounded non-commutative martingales. An important difference from the classical case is that for any $L_1$-bounded non-commutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column nature of non-commutative Hardy spaces of martingales. As applications, we obtain simpler proofs of the weak type $(1,1)$ boundedness for non-commutative martingale transforms and the non-commutative analogue of Burkholder's weak type inequality for square functions. A sequence $(x_n)_{n \ge 1}$ in a normed space $\mathrm{X}$ is called 2-co-lacunary if there exists a bounded linear map from the closed linear span of $(x_n)_{n \ge 1}$ to $l_2$ taking each $x_n$ to the $n$th vector basis of $l_2$. We prove (using our decomposition) that any relatively weakly compact martingale difference sequence in $L_1 (\mathcal{M}, \tau)$ whose sequence of norms is bounded away from zero is 2-co-lacunary, generalizing a result of Aldous and Fremlin to non-commutative $L_1$-spaces.

Stochastic mean-square performance analysis of an adaptive Hammerstein filter

This paper presents an almost sure mean-square performance analysis of an adaptive Hammerstein filter for the case when the measurement noise in the desired response signal is a martingale difference sequence. The system model consists of a series connection of a memoryless nonlinearity followed by a recursive linear filter. A bound for the long-term time average of the squared a posteriori estimation error of the adaptive filter is derived using a basic set of assumptions on the operating environment. This bound consists of two terms, one of which is proportional to a parameter that depends on the step size sequences of the algorithm and the other that is inversely proportional to the maximum value of the increment process associated with the coefficients of the underlying system. One consequence of this result is that the long-term time average of the squared a posteriori estimation error can be made arbitrarily close to its minimum possible value when the underlying system is time-invariant.

A note on strong law of large numbers of random variables

In this paper, the Chung’s strong law of large numbers is generalized to the random variables which do not need the condition of independence, while the sequence of Borel functions verifies some conditions weaker than that in Chung’s theorem. Some convergence theorems for martingale difference sequence such as Lp martingale difference sequence are the particular cases of results achieved in this paper. Finally, the convergence theorem for A-summability of sequence of random variables is proved, where A is a suitable real infinite matrix.

Unconditional martingale difference sequences in Banach spaces

For a Banach space Y, the question of whether L p ( μ , Y ) has an unconditional basis if 1 < p < ∞ and Y has unconditional basis, stood unsolved for a long time and was answered in the negative by Aldous. In this work we prove a weaker, positive result related to this question. We show that if ( y j ) is a basis of Y and ( d i ) is a martingale difference sequence spanning L p ( μ ) then the sequence ( d i ⊗ y j ) is a basis of L p ( μ , Y ) for 1 ⩽ p < ∞ . Moreover, if 1 < p < ∞ and ( y j ) is unconditional then ( d i ⊗ y j ) is strictly dominated by an unconditional tensor product basis. In addition, for 1 < p < ∞ , we show that if ( d i ) ⊂ L p ( μ ) is a martingale difference sequence then there exists a constant K > 0 so that ‖ ∑ i , j ∈ N ( α i j y j ) d i ‖ L p ( μ , Y ) ⩽ K ‖ ∑ i , j ∈ N ‖ α i j y j ‖ d i ‖ L p ( μ ) holds for every sequence ( y j ) ⊂ Y and every choice of finitely supported scalars ( α i j ) .

Stable limits of martingale transforms with application to the estimation of GARCH parameters

In this paper we study the asymptotic behavior of the Gaussian quasi maximum likelihood estimator of a stationary GARCH process with heavy-tailed innovations. This means that the innovations are regularly varying with index α∈(2,4). Then, in particular, the marginal distribution of the GARCH process has infinite fourth moment and standard asymptotic theory with normal limits and $\sqrt{n}$-rates breaks down. This was recently observed by Hall and Yao [Econometrica 71 (2003) 285–317]. It is the aim of this paper to indicate that the limit theory for the parameter estimators in the heavy-tailed case nevertheless very much parallels the normal asymptotic theory. In the light-tailed case, the limit theory is based on the CLT for stationary ergodic finite variance martingale difference sequences. In the heavy-tailed case such a general result does not exist, but an analogous result with infinite variance stable limits can be shown to hold under certain mixing conditions which are satisfied for GARCH processes. It is the aim of the paper to give a general structural result for infinite variance limits which can also be applied in situations more general than GARCH.

Martingales in Banach Lattices

In this article, we present a version of martingale theory in terms of Banach lattices. A sequence of contractive positive projections (En) on a Banach lattice F is said to be a filtration if EnEm = En∧m. A sequence (xn )i nF is a martingale if Enxm = xn whenever n m. Denote by M = M(F,(E n)) the Banach space of all norm uniformly bounded martingales. It is shown that if F doesn't contain a copy of c0 or if every En is of finite rank then M is itself a Banach lattice. Convergence of martingales is inves- tigated and a generalization of Doob Convergence Theorem is established. It is proved that under certain conditions one has isometric embeddings F� → M� → F ∗∗ . Finally, it is shown that every martingale difference sequence is a monotone basic sequence. In this paper, we define a martingale in terms of Banach lattices. Rephras- ing the title of (29), this paper could be entitled Martingales without prob- ability. We start with a brief review of classical martingale theory and Banach lattices.

Hoeffding's Inequality for Stopped Martingales and Semi-Markov Processes

In this article, we give a Hoeffding's inequality for a martingale difference sequence stopped by a counting process and some applications to semi-Markov, Poisson and compound Poisson processes.

Davis‐type theorems for martingale difference sequences

We study Davis‐type theorems on the optimal rate of convergence of moderate deviation probabilities. In the case of martingale difference sequences, under the finite pth moments hypothesis (1 ≤ p &lt; ∞), and depending on the normalization factor, our results show that Davis′ theorems either hold if and only if p &gt; 2 or fail for all p ≥ 1. This is in sharp contrast with the classical case of i.i.d. centered sequences, where both Davis′ theorems hold under the finite second moment hypothesis (or less).