production function CES, aproximation Kmenta

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This problem may be for mathematicians or programmers,I'm not sure. I estimate production function CES, approximation by Kmenta.
Original PF CES is




$Y=γ [δ* K^-ρ + (1-δ)*L^-ρ ] ^ -v/ρ$




After using logarithm and Taylor series, Kmenta got




$ln⁡Y = ln⁡ c+rγ ln⁡ K + r(1-γ)ln⁡P- frac12$ ρrγ(1-γ) * $(ln⁡K - ln⁡P)^2$




and after aproximation...




$ln⁡Y= β_0 +β_1 ln⁡K+ β_2 ln⁡P + β_3$[ln⁡ (K/P)$]^2$




we got function which can be estimated by OLS method.



My problem: Because third model faild in assumptions of OLS, I need differentiate it. I use program R, so I just add diff to parameter diff(log(K)) and diff(log(P)). But how to edit the last expression...




$[ln⁡ (K/P)]$^2




maybe...



  • $(diff[ln(K)]-diff[ln(P)])^2$


  • $diff [ln(K)-ln(P)]^2$


  • $diff ((ln(K)/ln(P))^2$


.. but realy I don't know, which expression is right. If someone need more detailed description of function approximation, I'll write it. Thanks;)




derivatives




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    up vote
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    down vote

    favorite












    This problem may be for mathematicians or programmers,I'm not sure. I estimate production function CES, approximation by Kmenta.
    Original PF CES is




    $Y=γ [δ* K^-ρ + (1-δ)*L^-ρ ] ^ -v/ρ$




    After using logarithm and Taylor series, Kmenta got




    $ln⁡Y = ln⁡ c+rγ ln⁡ K + r(1-γ)ln⁡P- frac12$ ρrγ(1-γ) * $(ln⁡K - ln⁡P)^2$




    and after aproximation...




    $ln⁡Y= β_0 +β_1 ln⁡K+ β_2 ln⁡P + β_3$[ln⁡ (K/P)$]^2$




    we got function which can be estimated by OLS method.



    My problem: Because third model faild in assumptions of OLS, I need differentiate it. I use program R, so I just add diff to parameter diff(log(K)) and diff(log(P)). But how to edit the last expression...




    $[ln⁡ (K/P)]$^2




    maybe...



    • $(diff[ln(K)]-diff[ln(P)])^2$


    • $diff [ln(K)-ln(P)]^2$


    • $diff ((ln(K)/ln(P))^2$


    .. but realy I don't know, which expression is right. If someone need more detailed description of function approximation, I'll write it. Thanks;)




    derivatives




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      This problem may be for mathematicians or programmers,I'm not sure. I estimate production function CES, approximation by Kmenta.
      Original PF CES is




      $Y=γ [δ* K^-ρ + (1-δ)*L^-ρ ] ^ -v/ρ$




      After using logarithm and Taylor series, Kmenta got




      $ln⁡Y = ln⁡ c+rγ ln⁡ K + r(1-γ)ln⁡P- frac12$ ρrγ(1-γ) * $(ln⁡K - ln⁡P)^2$




      and after aproximation...




      $ln⁡Y= β_0 +β_1 ln⁡K+ β_2 ln⁡P + β_3$[ln⁡ (K/P)$]^2$




      we got function which can be estimated by OLS method.



      My problem: Because third model faild in assumptions of OLS, I need differentiate it. I use program R, so I just add diff to parameter diff(log(K)) and diff(log(P)). But how to edit the last expression...




      $[ln⁡ (K/P)]$^2




      maybe...



      • $(diff[ln(K)]-diff[ln(P)])^2$


      • $diff [ln(K)-ln(P)]^2$


      • $diff ((ln(K)/ln(P))^2$


      .. but realy I don't know, which expression is right. If someone need more detailed description of function approximation, I'll write it. Thanks;)




      derivatives




      share|cite|improve this question











      This problem may be for mathematicians or programmers,I'm not sure. I estimate production function CES, approximation by Kmenta.
      Original PF CES is




      $Y=γ [δ* K^-ρ + (1-δ)*L^-ρ ] ^ -v/ρ$




      After using logarithm and Taylor series, Kmenta got




      $ln⁡Y = ln⁡ c+rγ ln⁡ K + r(1-γ)ln⁡P- frac12$ ρrγ(1-γ) * $(ln⁡K - ln⁡P)^2$




      and after aproximation...




      $ln⁡Y= β_0 +β_1 ln⁡K+ β_2 ln⁡P + β_3$[ln⁡ (K/P)$]^2$




      we got function which can be estimated by OLS method.



      My problem: Because third model faild in assumptions of OLS, I need differentiate it. I use program R, so I just add diff to parameter diff(log(K)) and diff(log(P)). But how to edit the last expression...




      $[ln⁡ (K/P)]$^2




      maybe...



      • $(diff[ln(K)]-diff[ln(P)])^2$


      • $diff [ln(K)-ln(P)]^2$


      • $diff ((ln(K)/ln(P))^2$


      .. but realy I don't know, which expression is right. If someone need more detailed description of function approximation, I'll write it. Thanks;)





      derivatives





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      asked Jul 26 at 6:49









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