拉格朗日乘数法中方程组的解法

拉格朗日乘数法依极值必要条件得到的方程组一般都是非线性的。本文主要介绍此类方程组的解法。     

Testing constancy of the error covariance matrix in vector models

In this paper a Lagrange multiplier test of the hypothesis that the covariance matrix of a multivariate time series model is constant over time is considered. It is assumed that under the alternative, the error variances are time-varying, whereas the correlations remain constant over time. Under the parameterized alternative hypothesis the variances may change continuously as a function of time or some observable stochastic variables. Small-sample properties of the test statistic are investigated by simulation. The assumption of constant correlations does not appear overly restrictive.     

A lagrange multiplier interpretation of disturbance estimators with an application to testing for nonlinearity

This paper describes a Lagrange multiplier interpretation of LUF disturbance estimators and an associated means of constructing residuals and procedures for testing for certain kinds of misspecification. Together with a recent technique for partially ordering multivariate data, the methods are used to devise an F test for nonlinearity in some or all of the explanatory variables in multiple regression. Unlike several alternatives, the proposed test does not require all misspecified variables to be positively correlated. Power computations, and comparisons with other procedures, yield encouraging results.     

Joint tests for regularity and autocorrelation in allocation systems

In the context of allocation models with vector autoregressive errors we propose a convenient procedure, based on the Lagrange multiplier principle, for testing any possible combination of absence of serial correlation, homogeneity, and symmetry against any possible alternative which specifies autocorrelation of an arbitrary given order. We also derive generic expressions for the maximum likelihood estimation of the models under six possible combinations of constraints. The methodology is illustrated with the Rotterdam model and the differential AIDS model, both estimated from the same quarterly British data.     

Evaluating GARCH models

In this paper, a unified framework for testing the adequacy of an estimated GARCH model is presented. Parametric Lagrange multiplier (LM) or LM type tests of no ARCH in standardized errors, linearity, and parameter constancy are proposed. The asymptotic null distributions of the tests are standard, which makes application easy. Versions of the tests that are robust against nonnormal errors are provided. The finite sample properties of the test statistics are investigated by simulation. The robust tests prove superior to the nonrobust ones when the errors are nonnormal. They also compare favourably in terms of power with misspecification tests previously proposed in the literature.     

Detection of Structural Change in the Long‐run Persistence in a Univariate Time Series*

In this paper, we investigate a test for structural change in the long‐run persistence in a univariate time series. Our model has a unit root with no structural change under the null hypothesis, while under the alternative it changes from a unit‐root process to a stationary one or vice versa. We propose a Lagrange multiplier‐type test, a test with the quasi‐differencing method, and ‘demeaned versions’ of these tests. We find that the demeaned versions of these tests have better finite‐sample properties, although they are not necessarily superior in asymptotics to the other tests.     

Testing with many weak instruments

This paper establishes the asymptotic distributions of the likelihood ratio (LR), Anderson–Rubin (AR), and Lagrange multiplier (LM) test statistics under “many weak IV asymptotics.” These asymptotics are relevant when the number of IVs is large and the coefficients on the IVs are relatively small. The asymptotic results hold under the null and under suitable alternatives. Hence, power comparisons can be made.     

On the smoothness of optimal paths

Abstract The aim of this paper is to study the differentiability property of optimal paths in dynamic economic models. We address this problem from the point of view of the differential calculus in sequence spaces which are infinite-dimensional Banach spaces. We assume that the return or utility function is concave, and that optimal paths are interior and bounded. We study the C ^r differentiability of optimal paths vis-à-vis different parameters. These parameters are: the initial vector of capital stock, the discount rate and a parameter which lies in a Banach space (which could be the utility function itself). The method consists of applying an implicit function theorem on the Euler–Lagrange equation. In order to do this, we make use of classical conditions (i.e., the dominant diagonal block assumption) and we provide new ones. Mathematics Subject Classification (2000): 90A16, 49K40, 93C55 Journal of Economic Literature Classification: C161, D99, O41     

Testing for Panel Cointegration with Multiple Structural Breaks*

This paper proposes a Lagrange multiplier (LM) test for the null hypothesis of cointegration that allows for the possibility of multiple structural breaks in both the level and trend of a cointegrated panel regression. The test is general enough to allow for endogenous regressors, serial correlation and an unknown number of breaks that may be located at different dates for different individuals. We derive the limiting distribution of the test and conduct a small Monte Carlo study to investigate its finite sample properties. In our empirical application to the solvency of the current account, we find evidence of cointegration between saving and investment once a level break is accommodated.     

The impact of purchasing power parity on auto foreign trade

In this study, the validity of the assumption saying that the import and export are a function of prices as in the classical, neo-classical approaches is studied within the framework of the import and export of automobile vehicles between 1997 and 2003 in Turkey and the EU countries which are automobile manufacturers. The price here is considered as the purchasing power parity. The effect of the purchasing power parity on the automobile import and export is determined by using classical models with constant coefficients, and fixed and random effects models with constant slope coefficients and a constant term differing according to units and/or time. The models comprise balanced linear panel data models. The likelihood ratio test and F-test are used in the selection of fixed effects and classical models; and the Lagrange multiplier test is used in the selection of random effects and classical models. As for the selection of fixed and random effects models, the Hausman test is used. As a result of these tests, the fixed effects models covering both individual and time effects are selected as the most appropriate import and export models.     

A Wald test for the cointegration rank in nonstationary fractional systems

This paper develops new methods for determining the cointegration rank in a nonstationary fractionally integrated system, extending univariate optimal methods for testing the degree of integration. We propose a simple Wald test based on the singular value decomposition of the unrestricted estimate of the long run multiplier matrix. When the “strength” of the cointegrating relationship is less than 1/2, the test statistic has a standard asymptotic distribution, like Lagrange Multiplier tests exploiting local properties. We consider the behavior of our test under estimation of short run parameters and local alternatives. We compare our procedure with other cointegration tests based on different principles and find that the new method has better properties in a range of situations by using information on the alternative obtained through a preliminary estimate of the cointegration strength.     

Maximum likelihood estimation and Lagrange multiplier tests for panel seemingly unrelated regressions with spatial lag and spatial errors: An application to hedonic housing prices in Paris

This paper proposes maximum likelihood estimators for panel seemingly unrelated regressions with both spatial lag and spatial error components. We study the general case where spatial effects are incorporated via spatial errors terms and via a spatial lag dependent variable and where the heterogeneity in the panel is incorporated via an error component specification. We generalize the approach of Wang and Kockelman (2007) and propose joint and conditional Lagrange multiplier tests for spatial autocorrelation and random effects for this spatial SUR panel model. The small sample performance of the proposed estimators and tests are examined using Monte Carlo experiments. An empirical application to hedonic housing prices in Paris illustrate these methods. The proposed specification uses a system of three SUR equations corresponding to three types of flats within 80 districts of Paris over the period 1990-2003. We test for spatial effects and heterogeneity and find reasonable estimates of the shadow prices for housing characteristics.     

Residual autocorrelation testing for vector error correction models

In applied time series analysis, checking for autocorrelation in a fitted model is a routine diagnostic tool. Therefore it is useful to know the asymptotic and small sample properties of the standard tests for the case when some of the variables are cointegrated. The properties of residual autocorrelations of vector error correction models (VECMs) and tests for residual autocorrelation are derived. In particular, the asymptotic distributions of Lagrange multiplier (LM) and portmanteau tests are given. Monte Carlo simulations show that the LM tests have satisfactory size properties only if autocorrelation of small order is tested in systems of small dimension. In contrast, portmanteau tests have roughly correct size in small samples only if higher order residual autocorrelation is tested. Their critical values have to be adjusted for the cointegration rank of the system, however.     

Income Distributive Effects in the Brazilian Economy

Studies aiming to evaluate the structural distribution of economic impacts usually treat consumption demand as an exogenous variable. In this case, the Leontief matrix multiplier lacks the multiplier process via the consumption function that one customarily finds in a Keynesian model. To regard the consumption as a fictitious production activity is not the appropriate procedure. Instead, the Keynesian consumption function is introduced at a disaggregated level. For that, a matrix multiplier was formulated in order to combine Leontief's propagation process with the Keynesian propagation process. This matrix includes the effects of endogenous changes in consumption demand. Based on the present production structure in Brazil, the results show how the propagation effect directs the induced income towards capitalists, depriving wage earners. The model also allows for evaluation of diverse effects of the propagation process according to income and consumption coefficients by sector.     

Nonlinearities and Fractional Integration in the US Unemployment Rate*

This paper proposes a model of the US unemployment rate which accounts for both its asymmetry and its long memory. Our approach introduces fractional integration and nonlinearities simultaneously into the same framework, using a Lagrange multiplier procedure with a standard null‐limit distribution. The empirical results suggest that the US unemployment rate can be specified in terms of a fractionally integrated process, which interacts with some nonlinear functions of labour‐demand variables such as real oil prices and real interest rates. We also find evidence of a long‐memory component. Our results are consistent with a hysteresis model with path dependency rather than a non‐accelerating inflation rate of unemployment (NAIRU) model with an underlying unemployment equilibrium rate, thereby giving support to more activist stabilization policies. However, any suitable model should also include business cycle asymmetries, with implications for both forecasting and policy‐making.     

Granger-causality in multiple time series

The concept of Granger-causality is formulated for a finite-dimensional multiple time series. Special attention is given to causality patterns in autoregressive series, and it is shown how these patterns can be tested under quite general assumptions using a χ² statistic. The power of the test is discussed, and it is shown that the χ² statistic results from a Lagrange multiplier test in the Gaussian case. The causality test is tried both on artificial data and some economic time series. Finally we consider the problem of constrained estimation in models with a known causality structure.     

A Smooth Transition Logit Model of The Effects of Deregulation in the Electricity Market

This paper introduces the smooth transition logit (STL) model that is designed to detect and model situations in which there is structural change in the behaviour underlying the latent index from which the binary dependent variable is constructed. The maximum likelihood estimators of the parameters of the model are derived along with their asymptotic properties, together with a Lagrange multiplier test of the null hypothesis of linearity in the underlying latent index. The development of the STL model is motivated by the desire to assess the impact of deregulation in the Queensland electricity market and ascertain whether increased competition has resulted in significant changes in the behaviour of the spot price of electricity, specifically with respect to the occurrence of periodic abnormally high prices. The model allows the timing of any change to be endogenously determined and also market participants' behaviour to change gradually over time. The main results provide clear evidence in support of a structural change in the nature of price events, and the endogenously determined timing of the change is consistent with the process of deregulation in Queensland. Copyright © 2015 John Wiley & Sons, Ltd.     

SMOOTH DYNAMIC FACTOR ANALYSIS WITH APPLICATION TO THE US TERM STRUCTURE OF INTEREST RATES

We consider the dynamic factor model and show how smoothness restrictions can be imposed on factor loadings by using cubic spline functions. We develop statistical procedures based on Wald, Lagrange multiplier and likelihood ratio tests for this purpose. The methodology is illustrated by analyzing a newly updated monthly time series panel of US term structure of interest rates. Dynamic factor models with and without smooth loadings are compared with dynamic models based on Nelson–Siegel and cubic spline yield curves. We conclude that smoothness restrictions on factor loadings are supported by the interest rate data and can lead to more accurate forecasts. Copyright © 2013 John Wiley & Sons, Ltd.     

企业估值中价值乘数的选择：基于行业差异性的研究

基于行业差异性理论分析不同行业在经济结构、经济周期、要素集约度、资本结构等方面的差异,目的在于论证行业的差异性是否会导致价值乘数的非对称性。探求不同价值乘数的行业适用性,以形成不同行业与价值乘数之间的双向匹配关系。根据要素集约度选取采掘业、建筑业、交通运输业以及信息技术业作为研究对象,在实证研究的基础上,确定价值乘数选择条件及优先等级,进而确定行业乘数。     

On input-output type output multipliers from incomplete information

Incomplete knowledge of data usually hinders the establishment of detailed input-output tables. It is for this reason that up-dating procedures (RAS) as well as short-cut methods have been developed. In this paper the short-cut output multiplier formula by Drake (1976) is compared with the output multiplier involved in the RAS procedures. It will turn out that both these output multiplier estimates are two special cases of a more general class of estimating procedures. It is demonstrated by an empirical example how this generalized procedure should be applied in practice. The kind of reasoning underlying this paper can be extended to other problems of input-output analysis as well.