# Estimate the coefficients of an equation system - can I use OLS?

#### Coffeedrinker

##### New Member
Hi!

I have not worked so much with this type of regressions and have tried to find the answer in my econometric books and on internet, but have unfortunately not had any luck. I appreciate all help I can get with this!

Im using time series data to measure the rate of return of the inputs Xi on the outputs Yi.

I need to find the values of the three coefficients a1, a2 and b, which are all appering in three related equations:

ln(Y1/Y3)=ln((1-a1+a2)/a2) +b*ln(X1/X3) + e
ln(Y2/Y3)=ln((1-a1)/a2)) +b*ln(X2/X3) + e
ln(Y1/Y2)=ln((a1+a2)/a1) +b*ln(X1/X2) +e

where X1, X2 and X3 are the independent variables and e the error term.

Is it correct to estimate this using OLS and then solve the equation system for a1 and a2? Or will the estimates be biased?

I only want one value of b, can I get that using this method?

Thank you so much for your time!
/C

#### staassis

##### Active Member
You did not describe the problem accurately. The error term e is a different random variable in each of the 3 equations, right?

#### Coffeedrinker

##### New Member
You did not describe the problem accurately. The error term e is a different random variable in each of the 3 equations, right?
Yes, I'm sorry, that is correct. The error term is a different random variable in each equation.

#### BGM

##### TS Contributor
Is there any typo in the system? If you sum the latter two equations, you will obtain the same variables in equation 1.

#### Coffeedrinker

##### New Member
Is there any typo in the system? If you sum the latter two equations, you will obtain the same variables in equation 1.
Yes, Im sorry, there was in fact a typo. Equation 3 is now corrected:

ln(Y1/Y3)=ln((1-a1+a2)/a2) +b*ln(X1/X3) + e
ln(Y2/Y3)=ln((1-a1)/a2)) +b*ln(X2/X3) + e
ln(Y1/Y2)=ln((1 -a1+a2)/a1) +b*ln(X1/X2) +e

Sorry for the misstake!