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  Chapter 10 Optimization functionslibrary The  optimization function library  contains routines for maximum like-lihood estimation of econometric models. Given a likelihood function whichdepends on the data as well as the parameters of the estimation problem,these routines will find a set of parameter values that maximize the likelihoodfunction. In addition to discussing and illustrating use of the functions in the optimization function library , we also show how to use the MATLABsimplex optimization algorithms  fmin  and  fmins  as well as Em algorithmsto solve for maximum likelihood estimates in econometric models. An ex-ample that demonstrates how to incorporate a new optimization algorithmobtained from the Internet into the library is also provided.Unlike Gauss software, MATLAB does not contain a single ‘maxlik’ func-tion that serves as an ‘all purpose’ routine for solving econometric maximumlikelihood optimization problems. In an attempt to remedy this, a function maxlik  was devised for MATLAB. Unfortunately, this function is nowherenear as robust or professionally crafted as the Gauss function by the samename. This function as well as univariate and multivariate simplex routines,algorithms from  Numerical Recipes   that have been re-coded for MATLAB,some Internet optimization functions written for MATLAB and a generalpurpose function  minz  contributed by Mike Cliff form the basis of the op-timization library,.The MathWorks sells an optimization toolbox that contains a host of al-gorithms for solving constrained and unconstrained optimization problems.These functions are not oriented toward econometric models, but rathersolution of very general optimization problems. To use these routines in260  CHAPTER 10. OPTIMIZATION FUNCTIONS LIBRARY   261solving econometric optimization problems, an interface similar to that de-scribed in Section 10.3 would need to be developed to adapt the algorithmsfor use in an econometric likelihood function setting.Section 10.1 illustrates the use of a univariate simplex algorithm  fmin from the MATLAB toolbox. This type of algorithm is useful for cases wherethe optimization problem involves a single parameter, a case often encoun-tered with concentrated likelihood functions. It also demonstrates the useof a multivariate simplex optimization function  fmins  from the MATLABtoolbox.Section 10.2 demonstrates use of the EM-algorithm approach to solvingmaximum likelihood estimation problems. Finally, Section 10.3 deals withthe most general case where the likelihood function is optimized with respectto a multivariate vector of parameters using numerical gradient methods. 10.1 Simplex optimization Simplex optimization has an advantage in that it does not require com-putation of the derivatives of the function being optimized. It also has adisadvantage in that the usual estimates of dispersion for the parametersin the model obtained from the numerical hessian matrix in gradient meth-ods are not available. The MATLAB toolbox contains both a univariatesimplex algorithm  fmin  and a multivariate function  fmins . The use of theunivariate function is taken up in Section 10.1.1 and  fmins  is illustrated inSection 10.1.2. 10.1.1 Univariate simplex optimization As an example of using the MATLAB  fmin  univariate simplex optimiza-tion function to solve an econometric maximum likelihood estimation prob-lem, consider the Box-Cox model. This model relies on a transformationof the sample data vector  y ( λ ) = ( y λ −  1) /λ  for the case where  λ   = 0and  y ( λ ) =  ln ( y ) when  λ  = 0. The model may also involve an identicaltransformation of the explanatory variables matrix, which we denote  X  ( λ ) .A more general model involves different values,  λ 1 ,λ 2  associated with the y ( λ 1 ) -transformation and  X  ( λ 2 ) -transformations, and a very general modelallows for individual  λ xi ,i  = 1 ,...,k  for each explanatory variable vector inthe sample data matrix  X  .We focus initially on the simple model involving a single parameter  λ .This model produces a log-linear model when  λ 1  =  λ 2  = 0, a semi-logmodel when  λ 1  = 1 ,λ 2  = 0, as well as more general flexible functional forms  CHAPTER 10. OPTIMIZATION FUNCTIONS LIBRARY   262associated with values like,  λ  = 1 / 2, or  λ  = √  2. Some of the functionalforms associated with values for  λ 1 ,λ 2  are: ln y i  =  β  0  + ln X  i β   + ε i , λ 1  = 0 ,λ 2  = 0 (10.1) y i  =  β  0  + ln X  i β   + ε i , λ 1  = 1 ,λ 2  = 0 (10.2) y i  =  β  0  + X  i β   + ε i , λ 1  = 1 ,λ 2  = 1 (10.3)The log likelihood function is concentrated with respect to  β  ( λ ) ,σ ( λ ),producing a function of only a single parameter  λ : L ( λ | y,X  ) = const + ( λ − 1) n  i =1 ln y i  − ( n/ 2)  ln  ˆ σ 2 ( λ ) (10.4)where:ˆ σ 2 ( λ ) = [ y ( λ ) − X  ( λ ) ˆ β  ( λ )  ( y ( λ ) − X  ( λ ) ˆ β  ( λ )] /n  (10.5)ˆ β  ( λ ) = ( X  ( λ )  X  ( λ ) ) − 1 X  ( λ )  y ( λ ) To minimize the log of this likelihood function with respect to the param-eter  λ  we can use the MATLAB function  fmin  that implements a simplexoptimization procedure. This function also allows us to set lower and upperlimits on the parameter  λ  to which the simplex search will be constrained.It is often the case that, values of   − 2  ≤  λ  ≤  2 are thought to representa reasonable range of feasible values for the parameter  λ  in the Box-Coxmodel.Our first task in using  fmin  is to write a log-likelihood function thatevaluates the concentrated log-likelihood for any value of the parameter  λ and returns a scalar value equal to the negative of the log-likelihood function.(Minimizing the negative log-likelihood is equivalent to maximizing the log-likelihood.) This function is shown below: function like = box_lik(lam,y,x,model);% PURPOSE: evaluate Box-Cox model concentrated likelihood function%----------------------------------------------------% USAGE: like = box_lik(lam,y,x,model)% where: lam = box-cox parameter (scalar)% y = dependent variable vector (un-transformed)% x = explanatory variables matrix (un-transformed)% model = 0 for y-transform only, 1 for y,x both transformed  CHAPTER 10. OPTIMIZATION FUNCTIONS LIBRARY   263 % NOTE: x should contain intercept vector in 1st column (if desired)%----------------------------------------------------% RETURNS: like = (a scalar) = -log likelihood function%----------------------------------------------------[n k] = size(x); ys = boxc_trans(y,lam);if model == 1 % user wants to transform both y,x% see if an intercept term exists in the modeliota = x(:,1); ifind = find(iota == 1);if isempty(ifind), xs = boxc_trans(x,lam); % no interceptelse, if length(ifind) == n, % an interceptxtrans = boxc_trans(x(:,2:k),lam);xs = [ones(n,1) xtrans];else, xs = boxc_trans(x,lam); % no interceptend;end;elseif model == 0, xs = x; % transform only y-vectorend;bhat = inv(xs’*xs)*xs’*ys; e = ys - xs*bhat; sige = (e’*e)/n;like = (lam - 1)*sum(log(y)) -(n/2)*log(sige); like = -like; The function relies on another function  boxc trans  to carry out theBox-Cox data transformation. It also contains an argument ‘model’ thatallows for a case (‘model=0’) where the  y  variable alone is transformed andanother case (‘model=1’) where both  y  and the  X   variables are transformed.The function  boxc trans  is: function z = boxc_trans(x,lam)% PURPOSE: compute box-cox transformation%----------------------------------------------------% USAGE: bdata = boxc_trans(data,lam)% where: lam = scalar transformation parameter% data = matrix nobs x k%----------------------------------------------------% RETURNS: bdata = data matrix box-cox transformed[n k] = size(x); z = zeros(n,k); iota = ones(n,1);for i=1:k;if lam ~= 0, z(:,i) = (x(:,i).^lam - iota)/lam;else, z(:,i) = log(x(:,i)); end;end; Now we can turn attention to use of the MATLAB simplex optimizationfunction  fmin , which can be called with: a string containing the functionname ‘box lik’, an upper and lower limit for the simplex optimization searchover values of   λ  (‘lamlo’,‘lamup’), a 4x1 vector of optimization options (‘fop-tions’), and arguments that will be passed along to our likelihood function‘box lik’. In this example, we wish to pass ‘y,x,model’ as arguments to the‘box lik’ function. An example of the call is:

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Jul 23, 2017


Jul 23, 2017
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