On the structure of moving average processes

This paper uses Hilbert space methods to develop a rigorous proof that the sum of two uncorrelated moving average processes of order q₁ and q₂ is an MA process of order q ≦ max (q₁, q₂). The methods establish the existence of suitable random shocks for the summed process, they illuminate relationships between the coefficients of such processes and their random shocks, and they provide means for proving that the random shocks of the summed processes are normal when the shocks of the underlying processes are normal. The role of the Wold decomposition is examined in terms of multiple representations of an MA process.     

Exact moments of the sample autocorrelations from series generated by general arima processes of order (p, d, q), d=0 or 1

Formulae for the numerical computation of the first four exact moments of the sample autocorrelations, given a time series realisation from a general autoregressive moving average process of order (p, d, q) with d=0 or 1, are presented. The exact mean and variance of the sample autocorrelations are computed for various sample sizes and several time series models. The evaluated results are compared with those obtained from approximate formulae for the mean and variance of the sample autocorrelations. A specification of the numerical accuracy of the first two exact moments is included.     

Bayesian robust designs for linear models with possible bias and correlated errors

Consider the design problem for the approximately linear model with serially correlated errors. The correlated structure is the qth degree moving average process, MA(q), especially for q = 1, 2. The optimal design is derived by using Bayesian approach. The Bayesian designs derived with various priors are compared with the classical designs with respect to some specific correlated structures. The results show that any prior knowledge about the sign of the MA(q) process parameters leads to designs that are considerately more efficient than the classical ones based on homoscedastic assumptions.     

Forecasting contemporal aggregates of multiple time series

The paper considers forecasting a contemporal linear aggregate y_t of a vector time series Z^'_t =(Z_1t,...,Z_kt). We first disciss the case where Z_t follows a stationary multiple moving average process and propose a measure of the efficiency of aggregation. A necessary and sufficient condition is given for the case of no gain by employing the component series. Extension of the results to stationary multiple autoregressive process and some non-stationarity processes is discussed, and an illustrative example is given.     

A proof of a relationship between the generalized variances for associated autoregressive and moving average processes

Summary In this paper we give a simple proof of the result that, for any integer,r, given two processes of orderr, one autoregressive and the other moving average but both with the same parameters, then the generalized variance of all ordersk2r, for the autoregressive process, is exactly equal to the infinite order generalized variance for the moving average process.     

Existence of equilibria with infinitely many commodities: Banach lattices reconsidered

The purpose of this paper is twofold. The first aim is to present an extension of the results on the existence of Walrasian equilibrium to the infinite dimensional setting. The result depends on two crucial assumptions. These are the compactness of the collection of feasible allocations and the non-emptiness of the interior of the production set. The proof is a direct generalization of Bewley's (1972) proof for the L_∞ case. The second purpose of this paper is to show that the recent result of Mas-Colell (1986) on the existence of equilibrium for exchange economies on Banach lattices can be obtained through an argument based on the result outlined above. That is, exchange economies on Banach lattices with ‘uniformly proper’ preferences behave as though they were production economies in which the production sets have non-empty interior.     

A remark on the recursive estimation of ARMA order

Summary It is well known how, for an ARMA process of order (p ₀,q ₀), max (p ₀,q ₀) may be recursively estimatedHannan/Rissanen. Assuming max (p ₀,q ₀) to be known and, in addition,p ₀q ₀, a simple procedure for the recursive estimation of (p ₀,q ₀) is presented.     

On the individual moving average inequality

Summary A very simple deduction of a recently treated inequality is presented and some variations on this proof, drawing together various properties, are discussed. Closure of moving average processes undermultiplication is also mentioned.     

Multimodality in GARCH regression models

It is shown empirically that mixed autoregressive moving average regression models with generalized autoregressive conditional heteroskedasticity (Reg-ARMA-GARCH models) can have multimodality in the likelihood that is caused by a dummy variable in the conditional mean. Maximum likelihood estimates at the local and global modes are investigated and turn out to be qualitatively different, leading to different model-based forecast intervals. In the simpler GARCH(p,q) regression model, we derive analytical conditions for bimodality of the corresponding likelihood. In that case, the likelihood is symmetrical around a local minimum. We propose a solution to avoid this bimodality.     

Fully Bayesian analysis of ARMA time series models

Statistical analysis of autoregressive-moving average (ARMA) models is an important non-standard problem. No classical approach is widely accepted; legitimacy for most classical approaches is based solely on asymptotic grounds, while small sample sizes are common. The only obstacle to the Bayesian approach are designing a structure through which prior information can be incorporated and designing a practical computational method. The objective of this work is to overcome these two obstacles. In addition to the standard results, the Bayesian approach gives a different method of determining the order of the ARMA model, that is (p, q).     

On the structure of the equilibrium manifold

An elementary proof is given of Y. Balasko's theorem that the equilibrium manifold is diffeomorphic to R^lm. Our proof includes the case (l,m)=(2,2), which the original proof does not cover.     

Games of chance with multiple objectives

We consider games of chance between two players: Player M can win only by amassing point totals in several categories before player N scores a prescribed total number n of points. Let M have k objectives, with m _i points required in category i and probability q _i of scoring a point in that category. We resolve certain special cases: (a) For all m _i equal, the probabilities of M winning are ordered by majorization of the vectors (q ₁,...,q _k ). (b) For all q _i equal, the probabilities of M winning are ordered by majorization of the vectors (m ₁,...,m _k ). (c) For all m _i equal and all q _i equal, the probability of M winning approaches 0 as n → ∞ or as k → ∞. The results, which follow from inequalities of majorization and Schur convexity, are in accord with intuition.      

A new non-linear AR(1) time series model having approximate beta marginals

We consider the mixed AR(1) time series model $$X_t=\left\{\begin{array}{ll}\alpha X_{t-1}+ \xi_t \quad {\rm w.p.} \qquad \frac{\alpha^p}{\alpha^p-\beta ^p},\\ \beta X_{t-1} + \xi_{t} \quad {\rm w.p.} \quad -\frac{\beta^p}{\alpha^p-\beta ^p} \end{array}\right.$$ for ?1 < β ^p ≤ 0 ≤ α ^p  < 1 and α ^p ? β ^p  > 0 when X _t has the two-parameter beta distribution B₂(p, q) with parameters q > 1 and ${p \in \mathcal P(u,v)}$ , where $$\mathcal P(u,v) = \left\{u/v : u < v,\,u,v\,{\rm odd\,positive\,integers} \right\}.$$ Special attention is given to the case p = 1. Using Laplace transform and suitable approximation procedures, we prove that the distribution of innovation sequence for p = 1 can be approximated by the uniform discrete distribution and that for ${p \in \mathcal P(u,v)}$ can be approximated by a continuous distribution. We also consider estimation issues of the model.     

Asymptotic distributions of some scale estimators in nonlinear models

Often in the robust analysis of regression and time series models there is a need for having a robust scale estimator of a scale parameter of the errors. One often used scale estimator is the median of the absolute residuals s ₁. It is of interest to know its limiting distribution and the consistency rate. Its limiting distribution generally depends on the estimator of the regression and/or autoregressive parameter vector unless the errors are symmetrically distributed around zero. To overcome this difficulty it is then natural to use the median of the absolute differences of pairwise residuals, s ₂, as a scale estimator. This paper derives the asymptotic distributions of these two estimators for a large class of nonlinear regression and autoregressive models when the errors are independent and identically distributed. It is found that the asymptotic distribution of a suitably standardizes s ₂ is free of the initial estimator of the regression/autoregressive parameters. A similar conclusion also holds for s ₁ in linear regression models through the origin and with centered designs, and in linear autoregressive models with zero mean errors.  This paper also investigates the limiting distributions of these estimators in nonlinear regression models with long memory moving average errors. An interesting finding is that if the errors are symmetric around zero, then not only is the limiting distribution of a suitably standardized s ₁ free of the regression estimator, but it is degenerate at zero. On the other hand a similarly standardized s ₂ converges in distribution to a normal distribution, regardless of the errors being symmetric or not. One clear conclusion is that under the symmetry of the long memory moving average errors, the rate of consistency for s ₁ is faster than that of s ₂.     

Exit, selection, and the value of firms

This paper studies a competitive dynamic model with firm level uncertainty and derives implications for the distribution of firm values and Tobin's q. Allowing for entry and exit, the model determines endogenously the degree of selection. A consequence of this selection is that average industry q values are biased above one. As parameters describing the technology and firm level uncertainty are changed, the equilibrium distribution for q values changes. This comparative statics is developed in the paper.     

Hypothesis testing based on goodness-of-fit in the moving average time series model

Small sample properties of t-tests are compared with those of tests based on relative goodness- of-fit in the context of the first order moving average time series model. Monte Carlo experiments reported in the paper suggest that the actual size of these t-tests greatly exceeds theoretical large sample significance levels, while conformity of goodness-of-fit statistics to the appropriate chi-square or F-distributions is much closer. The evidence presented suggests that practitioners are well advised to employ goodness-of-fit tests as a check on results of t-tests particularly when the latter indicate ‘significance’.     

The profitability of moving average trading rules in BRICS and emerging stock markets

Technical analysis and trading systems have been widely used by practitioners in financial markets. Since some academic studies have highlighted that these tools can generate positive alphas when compared with a buy-and-hold strategy, we studied the main stocks of the BRICS and emerging markets. We considered the period from 2000 to 2015 and observed different combinations of moving average strategies and periods. The main results indicate that, for some countries, there is a combination of periods for moving averages producing better outcomes.     

A note on compact differentiability and theδ-method

Letη _n ,n ∈ ?, be arbitrary functions defined on a probability space (ω,A,P) with values in a normed vector spaceB ₁ ,μ ∈ B ₁ andξ ₀ a separable random element inB ₁ such thatξ _n :=√n(η _n ?μ) converges weakly toξ ₀ in the sense of Hoffmann-Jørgensen. Then with (B ₂, ∥·∥₂) being another normed vector space andφ:B ₁→B ₂ compactly differentiable atμ with derivateD _μ, the random variable $\parallel \sqrt n (\phi (n_n ) - \phi (\mu )) - D_\mu (\sqrt n (n_n - \mu ))\parallel 2*$ converges to 0P-stochastically where “*” denotes the measurable cover. We show that the classicalδ — method extends to the non-measurable case where in the proof we shall not make use of any representation theorems but only of a slight refinement of the usual characterisation of compact differentiability, due to the fact that we will not assume {ξ _n :n ∈ ?} being tight.     

Panel cointegration,quantile regressions,asymmetric adjustments and crises: The case of EU current accounts

This paper investigates the relationships between the current account and several fundamentals, including the real exchange rate, government consumption, investment, openness, terms of trade and real income in the EU28 group of countries. A main feature of the study is that we also assess the relationships for two subgroups, the EU15 + Cyprus and Malta, and the CEECs. Using data spanning the period between 1995q1 and 2019q2, we identify similarities and differences between the responses in these two subgroups, which are obscured when an aggregate study of the EU28 is conducted, rather than sub-groups. Our results suggest that, in assessing the current account for economic blocs, an a priori assumption of similar relationships for member countries may be misplaced.