I have 500 observed data of variable $ x $ and corresponding $ y $. The functional model is
where
Is it possible to find suitable constants $ A , B $ ,$ \alpha , \beta $ so that the observed data fits the functional form ? If yes please tell me how.
[note: The variable $ x $ is random in nature and all values of $ x , y $ are positive integers ; however I don't know if it matters or not ]
Thanks and regards
Marcella
To do maximum likelihood you would need to make an assumption about the distribution of $x$.
But in any case your system is underidentified, and you cannot obtain a unique in some optimal sense set of values for the four coefficients. To see this clearly, take the natural logarithms (you can do this since you assume $\beta >\alpha$):
$$\ln y = \ln\left( AB^{\frac {\alpha}{\beta}}\frac {x^{1-\frac{\alpha}{\beta}}}{1-\frac{\alpha}{\beta}}\right) \Rightarrow \ln y = \ln\left(AB^{\frac {\alpha}{\beta}}\right) + \left(1-\frac{\alpha}{\beta}\right)\ln x-\ln \left(1-\frac{\alpha}{\beta}\right) \qquad $$
$$\Rightarrow \ln y = \ln A + \frac{\alpha}{\beta}\ln B - \ln \left(1-\frac{\alpha}{\beta}\right) + \left(1-\frac{\alpha}{\beta}\right)\ln x \qquad [1]$$
which can be compactly written as $$ \ln y = \gamma_0 + \gamma_1\ln x\;, \qquad \gamma_0 = \ln A + (1-\gamma_1)\ln B - \ln \gamma_1\;,\qquad \gamma_1 = 1-\frac{\alpha}{\beta} \qquad [2]$$
You have two regressors ($x$ and a series of ones corresponding to the $\gamma_0$ coefficient), and by a least-squares regression you will be able to obtain only two estimates, $\hat \gamma_0$ and $\hat \gamma_1$, against four underlying unknowns: under-identified. You can only derive two equations, the one linking $A$ and $B$, the other linking $\alpha$ and $\beta$ : $$ \alpha = (1-\hat \gamma_1)\beta\;,\qquad \ln A = \hat \gamma_0 - (1-\hat\gamma_1)\ln B + \ln \hat \gamma_1$$
The first gives you also estimated bounds for $\beta$. But you will have two free coefficients, and so no unique set of values.
(By the way, note that you have inequality (not equality) constraints on the coefficients, and so the regression should actually be "Inequality Constrained Least Squares" which some software support, but not necessarily all...)