Nonlinear Regression Models

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Chapter 9

• Nonlinear Regression Models

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What is the N.R.M.?

• Linear
• Nonlinear
• h(.) cannot be transformed into a linear function
• A nonlinear regression model gt first-order
  conditions for LSE of the parameters
  are nonlinear functions of the parameters

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Assumptions of the NRM (1)

• A1 Eyixi h(xi, ß), i 1,,n
• A2 Identifiability of the model parameters
• no nonzero ß0? ß s.t. h(xi, ß0) h(xi, ß)
• for all xi.
• A3 Zero mean of the disturbance
• yi h(xi, ß) ei
• Eeih(xi, ß) 0

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Assumptions of the NRM (2)

• A4 Homoscedasticity Nonautocorrelation
• A5 Well behaved population.
• A6 There is a well defined probability
  distribution generating ei

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The Linearized Regression

• Linear Taylor series approximation
• y h(x,ß) e

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Linearized Regression
The regressors
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Large Sample Properties

• Asymptotic properties essentially the same as
  for the linear model
• One exception derivatives of the linearized
  function evaluated at ß, X0 play the role of
  regressors
• pseudoregressors

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F-statistic

• y h(x,ß) e
• H0 r(ß)q
• b - nonrestricted NLS estimator
• b - restricted NLS estimator

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Computing the NLSE

• Gauss-Newton
• If is available, then linear least squares
• New parameter vector is the new
• Iteration continues until the difference between
  the parameter vectors is small enough to assume
  convergence
• is correct estimate of the ACM for the
  parameter estimator

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Computing the NLSE
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Computing the NLSE

• Convergence if
• Choice of starting values highly important
• Consistent estimator of
• ACM follows from

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Computing the NLSE

• R is no longer between 0 and 1
• However, it is a useful descriptive mea-sure

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ApplicationsNonlinear consumption function

• ? linear
• Linearised model
• NLSE procedure ? on

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ApplicationsNonlinear consumption function

• Starting point LLSE
• For hypothesis testing and constructing
  confidence intervals the usual procedures can be
  used
• Examples in book
• Testing whether or not
• Testing whether or not MPC1
• Note standard errors are estimated by the
  delta method

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ApplicationsThe Box-Cox Transformation

• Transformation
• For a given the model is a linear regression
  ?
• Normally al are assumed to be the same for
  every variable in the model
• Range between 2 and 2 is the range for
  econometric applications

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ApplicationsThe Box-Cox Transformation

• The logarithmic transformation is the limiting
  case that l -gt 0 (by L'Hôspital's rule)
• Optimum value of l ? LSE, mean squa-red residuals
  and l constitute the NLSE of the estimates of the
  parameters

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ApplicationsThe Box-Cox Transformation

• Remember l is an estimate not the true value, it
  will underestimate the correct asymptotic
  standard errors
• To get the appropriate values

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ApplicationsThe Box-Cox Transformation

• To estimate the asymptotic covariance matrix use
• Except for

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ApplicationsThe Box-Cox Transformation

• Delta method ?

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Questions?