Why can we linearize the exponential regression?

calculus statistics optimization regression nonlinear-optimization

Solution 1:

You will not obtain the same results at all beacause the residues are totally different. In the first case, they are $$r_i=\hat y_i-y_i$$ while in the second case, they are $$r_i=\log(\hat y_i)-\log( y_i)$$

If you look at my answer to this question, I explain that the first case corresponds to the usual minimization of the sum of the squares of absolute errors while the second corresponds more or less to the minimization of the sum of the squares of relative errors which is totally different.

Solution 2:

You can try the following.

Suppose your data has just two points $(x_1,y_1)=(1,1), (x_2,y_2) = (2,2)$. Now fix one of the parameters in both of the problems, say $ a_2 = 1 $. Use calculus to determine what the corresponding 'minimizing' $ a_1 $ in each of the problem will be. Do they match? This should be easy to check due to the small size of data.

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