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本文为丹麦奥尔胡斯大学（作者：Yukai Yang）的博士论文，共212页。

本文主要研究非线性向量经济的时间序列模型。众所周知，**许多经济变量之间的关系是非线性的**，非线性模型在经济理论中大量存在。有些市场并不明朗，因为有些变量（通常是价格）具有粘性，从而导致数量和价格之间的非线性。中央银行可能会为汇率设定边界，这意味着汇率与决定其价值的基础货币之间的关系是非线性的。在劳动力市场上，许多关于雇佣工人的企业行为的经济理论表明，**在宏观经济层面上，就业存在不对称波动**。许多这样的经济现象和理论已经产生了非线性经济计量模型。经济理论可以为研究者用来描述许多经济变量的动态行为关系的函数形式提供指导。然而，情况并非总是如此，许多非线性关系必须使用数据来发现。这篇论文由三个独立的章节组成，其中一章由作者独立完成，另外两章是合著完成的。

This thesis is concerned with nonlinearvector economic time series modelling. It is well known that relationshipsbetween many economic variables are nonlinear, and that nonlinear models aboundin economic theory. There are markets that do not clear because some variables,often prices, are sticky, which yields nonlinearity between quantity and price.Central banks may set bounds for the exchange rate, which implies that therelationship between the exchange rate and the underlying that determines itsvalue is nonlinear. In labour markets, many economic theories about thebehaviour of firms employing workers suggest asymmetric fluctuations inemployment on the macroeconomic level. A number of these economic phenomena andtheories have given rise to nonlinear econometric models. Economic theory mayoffer guidance as to the functional form of the relationship that theresearcher wants to use for characterizing dynamic behaviour of a number ofeconomic variables. However, this is not always the case, and many nonlinearrelationships have to be found using the data. The thesis consists of threeself-contained chapters, one-single-authored, and two written with a co-author.In chapter 1, together with Timo Ter¨asvirta, we concentrate on the VectorSmooth Transition Regression model. Smooth transition autoregressive modelshave become popular and are frequently applied to economic time series data.Ter¨asvirta (1994) devised a strategy for building univariate STAR models,consisting of specification, estimation and evaluation of the model. Camacho(2004) proposed a modelling strategy for a bivariate STAR model and applied itto joint modelling of the US GDP and a leading indicator by the ConferenceBoard. In this thesis, Camacho’s model is generalized into a multivariate STARmodel and extended in various ways. A reliable modelling strategy fordetermining whether a given relationship is nonlinear, what the nonlinearitylooks like, and whether it is adequately described by a particular parametricmodel, is needed. We devise a modelling strategy consisting of specification,including testing linearity, estimation and evaluation of these models. Iconsider the case where each equation can have its own transition variablecontrolling the nonlinear behavior. Linearity testing is then carried outequation by equation and, as in Camacho (2004), the classical tests forunivariate time series models can be used. But then, I also consider theinteresting special case in which the model only contains a single transitionvariable. This means that the existing univariate linearity tests aregeneralized into a multivariate joint tests. Another extension is that I allowfor multiple regimes in my STAR model. Modelling proceeds from specific togeneral. First I test linearity. If it is rejected, I estimate a standardmultivariate STAR model with a single transition. I evaluate the model usingmisspecification tests. If the results suggest adding another transition, I dothat, estimate the extended model and evaluate it as before. The fact that mymodel can have more than one transition is another extension to the bivariatemodel introduced by Camacho (2004). Maximum likelihood estimation of theparameters of the model is discussed, and the selection of starting-values fornonlinear estimation receives attention. Checking the stability of the systemis a necessary but sometimes neglected step in the evaluation of the STAR modelin empirical applications, when weak stationarity is assumed. I have devised acomputational method for checking stability of the estimated vector STAR model,which generalizes the method used in the univariate case. Two applicationsdemonstrate vector smooth transition modelling in practice. The first one isbased on monthly U.S. gasoline price and consumption time series. The issue ispossible asymmetry in the response of consumers to a price change. This hasbeen considered in many articles. Using a vector STAR model, I find that thepriceconsumption relationship is nonlinear and illustrate the behaviour of theVLSTAR model using generalized impulse response functions that can be appliedto studying dynamic properties of these models. This method has not been usedin previous studies of the problem. The second example is based on the Icelandicriver flow data that Tsay (1998) analyzed using the multivariate thresholdmodel. I re-analyzed the same data time series using my vector smoothtransition model. The model successfully captures the nonsynchronization inswitching of regimes. My results show that the regime-switching of the tworiver flows, which are controlled by the temperature, behaves differently notonly in smoothness but in turning-points as well. These are new findings whichcannot be found using other modelling procedures. Chapter 2, together with TimoTer¨asvirta, is concerned with the linearity and misspecification tests invector smooth transition autoregressive models. First, I developLagrange-multiplier type test statistics for the null hypothesis of a linearVAR model against the alternative hypothesis of a vector smooth transitionmodel. These new tests are likely to suffer from the problem that the number ofobservations available does not suffice for asymptotic inference, which canlead to significant size distortion of the tests.

This means that they are not reliable insmall and moderate samples. For this reason, small-sample properties of thetests have to be carefully studied. We suggest and investigate two teststatistics with improved size properties: Wilks’s Λ and Rao’s F-statistic. Myresults show that they alleviate the problem of size distortion in standardtest statistics. There are many ways for an estimated vector STAR model to bemisspecified. Finding out whether the model satisfies the assumptions under whichit was estimated should be an integral part of a normal modelling procedure. Weconsider three misspecification tests for possible model extensions: the testof no serial correlation, the test of no additive nonlinearity and theparameter constancy test. They are either Lagrange multiplier or Lagrangemultiplier type tests. We generalize the univariate misspecification tests inEitrheim and Ter¨asvirta (1996) to multivariate joint tests. As alreadymentioned, small-sample properties of the tests should be carefully studied dueto the dimension of the models. It turns out in my simulations that Wilks’s Λand Rao’s statistic efficiently remedy the size distortion problem present instandard LM and LM-type tests. In Chapter 3, the nonlinearity in error covariancematrix is investigated. I consider multivariate (vector) time series models inwhich the error covariance matrix may be time-varying. A recent statistic fortesting the constancy of the error covariance matrix can be found in Eklund andTer¨asvirta (2007). They derive a family of test statistics against varioustypes of misspecifications making the use of the constant conditionalcorrelation framework of Bollerslev (1990). The motivation of this chapter isto develop a new multivariate heteroskedasticity test as an alternative to theone proposed in Eklund and Ter¨asvirta (2007). Tests of constancy of the errorcovariance matrix against the alternative that the covariance matrix changesover time are developed based on the spectral decomposition of the errorcovariance matrix. The idea with this decomposition is to obtain tests againstparsimoniously parameterised alternatives such that the resulting tests wouldbe powerful against many kinds of departure from parameter constancy. A newfamily of Lagrange-multiplier type tests which allow for various types ofmisspecifications under this decomposition is developed. As in Eklund andTer¨asvirta (2007), three types of alternatives to constancy are considered inthis chapter. The first one may be viewed as a multivariate generalization ofthe heteroskedasticity test of White (1980), and the second one generalizes thetest against autoregressive conditional heteroskedasticity of Engle (1982). Thethird variant of the test generalizes the univariate constant variance test ofMedeiros and Veiga (2003), in which it is assumed that under the alternativehypothesis the variance changes smoothly over time. It can be seen that thejoint constancy test for error covariance matrix against multivariateheteroskedasticity is very easy to implement and use. It has satisfactory sizeand power properties even in high-dimensional vector models. Furthermore, thetest is still robust when the vector model is misspecified.

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