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A model of the form $y=a\cdot x^b$ can be linearly fitted by taking logs on both sides - giving $\ln(y)=\ln(a)+b\cdot\ln(x)$, where $\ln(y)$ is regressed against $\ln(x)$. This is a standard textbook approach.

What if there was an additive linear term included - e.g. $y=a\cdot x_1^b + c\cdot x_2$, how would you fit this for $a$, $b$, and $c$? Taking logs doesn't seem to do the trick, what other alternative methods are there?

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  • $\begingroup$ "A model of the form $y=a⋅x^b$ can be linearly fitted by taking logs on both sides" ... such an argument only works when $y$ is strictly positive, and the transformation of the model without regard for what happens to the error term generally only works when the errors are very small. $\endgroup$
    – Glen_b
    Commented Mar 28, 2023 at 23:34

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This is a job for something like nonlinear least squares, e.g., nls in R. Here is a snippet.

nl.eq <- function(x1, x2, a, b, d) {
  (a*x1^b+d*x2)
}

n <- 10000
x1 <- runif(n)
x2 <- rnorm(n)
a <- 2
b <- 3
d <- 4
y <- a*x1^b + d*x2 + rnorm(n)

nlsreg <- nls(y ~ nl.eq(x1, x2, a, b, d), start = list(a = 1, b = 1, d = 1))
summary(nlsreg)
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  • $\begingroup$ You mean a real-world example (my code already has a numerical example)? Many variables exhibit polynomial relationships, see e.g. wage returns to experience which may not increase linearly but flatten off for experienced employees. $\endgroup$ Commented May 22, 2023 at 9:22

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