TAILIEUCHUNG - Econometric theory and methods, Russell Davidson - Chapter 6

Chapter 6 Nonlinear Regression Introduction Up to this point, we have discussed only linear regression models. For each observation t of any regression model, there is an information set Ωt and a suitably chosen vector Xt of explanatory variables that belong to Ωt . | Chapter 6 Nonlinear Regression Introduction Up to this point we have discussed only linear regression models. For each observation t of any regression model there is an information set Qt and a suitably chosen vector Xt of explanatory variables that belong to Qt. A linear regression model consists of all DGPs for which the expectation of the dependent variable yt conditional on Qt can be expressed as a linear combination Xt 3 of the components of Xt and for which the error terms satisfy suitable requirements such as being IID. Since as we saw in Section the elements of Xt may be nonlinear functions of the variables originally used to define Qt many types of nonlinearity can be handled within the framework of the linear regression model. However many other types of nonlinearity cannot be handled within this framework. In order to deal with them we often need to estimate nonlinear regression models. These are models for which E yt Qt is a nonlinear function of the parameters. A typical nonlinear regression model can be written as yt xt 3 ut ut IID 0 a2 t 1 . n where just as for the linear regression model yt is the tth observation on the dependent variable and 3 is a k-vector of parameters to be estimated. The scalar function xt 3 is a nonlinear regression function. It determines the mean value of yt conditional on Qt which is made up of some set of explanatory variables. These explanatory variables which may include lagged values of yt as well as exogenous variables are not shown explicitly in . However the t subscript of xt 3 indicates that the regression function varies from observation to observation. This variation usually occurs because xt 3 depends on explanatory variables but it can also occur because the functional form of the regression function actually changes over time. The number of explanatory variables all of which must belong to Qt need not be equal to k. The error terms in are specified to be IID. By this we mean something .

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