Non-linear Methods in Econometrics

Economic theory guides empirical research primarily by suggesting which variables ought to enter a relationship. But as to the functional form that this relationship ought to take, it only gives general information such as stating that certain first and second partial derivatives of a relationship must be positive or such as ruling out certain functional forms. In some applications, notably consumer demand systems, the theory rules out models that are linear in the parameters such as <m:math display='block'> <m:mrow> <m:mi>y</m:mi><m:mo>=</m:mo><m:mstyle displaystyle='true'> <m:mo>&#x2211;</m:mo> <m:mrow> <m:msub> <m:mi>x</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:mstyle><m:msub> <m:mi>&#x03B2;</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>+</m:mo><m:mi>e</m:mi></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$y = \sum {{x_i}} {\beta _i} + e$$ and thus provides a natural impetus to the development of statistical methods for models that are non-linear in the parameters such as <m:math display='block'> <m:mrow> <m:mi>y</m:mi><m:mo>=</m:mo><m:mo stretchy='false'>(</m:mo><m:mstyle displaystyle='true'> <m:mo>&#x2211;</m:mo> <m:mrow> <m:msub> <m:mi>x</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:mstyle><m:msub> <m:mi>&#x03B2;</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo stretchy='false'>)</m:mo><m:mo>/</m:mo><m:mo stretchy='false'>(</m:mo><m:mstyle displaystyle='true'> <m:mo>&#x2211;</m:mo> <m:mrow> <m:msub> <m:mi>x</m:mi> <m:mi>i</m:mi> </m:msub> <m:msub> <m:mi>y</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>&#x2212;</m:mo><m:mn>1</m:mn></m:mrow> </m:mstyle><m:mo stretchy='false'>)</m:mo><m:mo>+</m:mo><m:mi>e</m:mi></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$y = (\sum {{x_i}} {\beta _i})/(\sum {{x_i}{y_i} - 1} ) + e$$