How to get rid of high error linear regression for log-log model

I have the function D = alpha*(Pop)^b1*(Gdp)^b2*d^b3. I turn this into a log-log model: ln(D)= ln(alpha)+ b1*ln(Pop)+b2*ln(Gdp)+b3*ln(d). Now I would like to find the b coefficients and the intercept(ln(alpha)). I have the values for D and the other variables. I take the natural log of these vectors and try to find the coefficients by regression. This works only the errors are way too large. Especially when turned back to original value with exp().Any help to minimize this error or another appraoch would be greatly appreciated.

distanceln  = log(distancevec); POPln = log(POPvec); GDPln = log(GDPvec);demandln = log(Demandvec);
X = [ones(size(POPln)),POPln,GDPln,distanceln];
y = demandln;
b = X\y;
Y = X*b;
err = max(abs(y-Y));

Answer 1

[Star Strider]

This might be solvable with a simple nonlinear regression using fminsearch, avoiding the need to do the log transformation (considered to be inadvisable for the reason you have discovered).

Your ‘data.mat’ file has: ‘Demandvec’, ‘GDPvec’, ‘POPvec’, and ‘distancevec’. I assume that ‘d’ is ‘distancevec’, and you want to fit to ‘Demandvec’.

What is ‘alpha’? Are we estimating it as well?

[JJP van den Berg]

Thank you for your answer!! I will give that a try. Yes, I would like to fit the demand vector. Sorry for the bad description. It is a demand function. So I have for a reference year the demand(D), the gross domestic product(Gdp), the population(Pop) and the distance between airports(d). The data vectors for these variables are in the data.mat file. Alpha is a market size scaling constant I would als like to estimate alpha, as well as the other coefficients.

[Star Strider]

My pleasure!

You will probably have to provide more appropriate starting values for ‘B0’, since I just used a random vector that may not reflect the actual values.

The Code —

Dfun = @(b,Pop, Gdp, d) b(4).*(Pop).^b(1).*(Gdp).^b(2).*d.^b(3);

In = load('JJP van den Berg data.mat');
D = In.Demandvec;
Gdp = In.GDPvec;
Pop = In.POPvec;
d = In.distancevec;

NECF = @(b) norm(D - Dfun(b, Pop, Gdp, d));
B0 = rand(4,1);
opts = optimset('MaxFunEvals',1E+4, 'MaxIter',1E+3);
Be = fminsearch(NECF, B0, opts);

fprintf(1, '\n\tEstimated Parameters:\n\t\tb1    = %10.4f\n\t\tb2    = %10.4f\n\t\tb3    = %10.4f\n\t\talpha = %10.4f\n\n', Be)

  Estimated Parameters:
    b1    =     0.9006
    b2    =     0.1910
    b3    =    -0.9765
    alpha =   554.4910

The code is straightforward. It codes your ‘D’ equation as an anonymous function, calculates the norm of the difference between the estimated and actual values for ‘D’ in the ‘NECF’ (Norm Error Cost Function) anonymous function, then uses that as the function argument to fminsearch to estimate the parameters. Nonlinear regression is very sensitive to the initial parameter estimates, so bear that in mind when you choose them. (I chose the random vector to test the code.) The fprintf call just prints out the estimated parameters to the Command Window.

I renamed your ‘data.mat’ file to be unique to your Question because that is a common name for attached files, and I create unique names for each one.

[JJP van den Berg]

Thank you so much!!

[Star Strider]

My pleasure!