Constrained regression with constrains on the slopes

I was the following regeression specification

ln Q = ln A + alpha * ln K + beta * ln L

I want to fins alpha and beta such that alpha + beta < 1 and alpha > 0 and beta > 0.

I am trying to use lsqlin but I don't understand how to write the specification.

[Torsten]

Why don't you use directly lsqnonlin and fit

Q = A * K^alpha * L^beta

?

[Prerna Mishra]

So will it be like:

fun = @(x)A*K^x(1)*L^x(2)
lb = [0 0]
ub = [1 1]
x0 = [0.1 0.1]
x = lsqnonlin(fun,x0,lb,ub)

[Prerna Mishra]

I did the following:

A = [97.167 97.821 103.409 101.346 101.5943333 100.0346667 99.97166667 100 103.0323333 106.3403333 106.771 107.6006667 113.092 119.3453333 122.0993333];
K = [755193.877 987963.218 1294402.425 1361879.589 1445084.001 1681364.633 1857769.88 1946306.617 2017371.005 2065104.433 1744891.733 1695633.052 1724777.265 1729554.563 1825513.804];
L = [562 619.875 663.2916667 709.1333333 643.4916667 654.8333333 739.3666667 796.6333333 810.7583333 838.3166667 760.2333333 616.8 626.325 677.7166667 677.9416667];
fun = @(x)A*K^x(1)*L^x(2);
lb = [0, 0];
ub = [1, 1];
%x0 = [0 0];
%x = lsqnonlin(fun,x0,lb,ub);
x0 = [0.1, 0.1];
options = optimoptions(@lsqnonlin,'Algorithm','trust-region-reflective');
x = lsqnonlin(fun,x0,ub,lb,options);

And I get the following error:

ProdFuncEval

Exiting due to infeasibility: at least one lower bound exceeds the corresponding upper bound.

[Torsten]

But you want to fit Q against A*K^x(1)*L^x(2).

In the above code, you try to make A*K^x(1)*L^x(2) = 0.

[Alex Sha]

Hi, Prerna Mishra, where are the data of Q?

Answer 1

[Matt J]

AA=log(K(:)./L(:));
bb=log(Q(:)./A(:)./L(:));

alpha=min(   max(AA\bb,0) ,1   );
beta=1-alpha;

[Torsten]

alpha + beta < 1, not = 1.

[Matt J]

Darn it. Well then, why not with lsqlin as originally proposed,

C=[log(K(:)) log(L(:));
d=log(Q(:)./A(:));

x0=lsqlin(C,d,[1,1],1,[],[],[0 0],[1,1]);
alpha=x0(1);
beta=x0(2);

If desired, you can use x0 to initialize a a nonlinear least squares fit to the original non-logged model,

fun = @(x,KL) A(:).'*KL.^x;
lb = [0, 0];
ub = [1, 1];

x=lsqcurvefit(fun,x0(:)',[K(:),L(:)], Q(:), lb,ub);
alpha=x(1);
beta=x(2);