Mohammad Ehsanul Karim wrote:
Dear Sundar Dorai-Raj,
Thank you very much for mentioning to exponentiate ALPHA.
However, so far i understand that the parameters in the non-linear equation
Y = ALPHA * (L^(BETA1)) * (K^(BETA2))
and the coefficients of log(L) and log(K) of the following equation (after linearizing)
log(Y) = log(ALPHA) +(BETA1)*log(L) + (BETA2)*log(K)
should be the same when estimated from either equation. Is it true? If it is, then why the estimates of the two procedure (see below) are different? Can you please explain it?
-----------------------------
> coef(lm(log(Y)~log(L)+log(K), data=klein.data))
(Intercept) log(L) log(K)
-3.6529493 1.0376775 0.7187662
-----------------------------
> nls(Y~ALPHA * (L^(BETA1)) * (K^(BETA2)), data=klein.data, start = c(ALPHA=exp(-3.6529493),BETA1=1.0376775,BETA2 = 0.7187662), trace = TRUE)
Nonlinear regression model model: Y ~ ALPHA * (L^(BETA1)) * (K^(BETA2)) data: klein.data ALPHA BETA1 BETA2 0.003120991 0.414100040 1.513546235 residual sum-of-squares: 3128.245 -----------------------------
Not necessarily. In the first model, you're minimizing:
sum((log(Y) - log(Yhat))^2)
because the nonlinear model you're fitting is:
Y = ALPHA * L^BETA1 * K^BETA2 * ERROR log(Y) = log(ALPHA) + BETA1 * log(L) + BETA2 * log(K) + log(ERROR)
Note the multiplicative error structure. In the second model you're mininmizing
sum((Y - Yhat)^2)
because the nonlinear model you're fitting is
Y = ALPHA * L^BETA1 * K^BETA2 + ERROR
Note the additive error structure. Different error structures, different parameter estimates.
Also, the residual sums of squares for the nls fit is smaller, although I'm not sure whether this comparison is really fair:
klein.lm <- lm(log(Y) ~ log(L) + log(K))
# `start' is not shown here but can be copied from above
klein.nls <- nls(Y ~ ALPHA * L^BETA1 * K^BETA2, data = klein.data,
start = start, trace = TRUE)
rss.lm <- sum((Y - exp(fitted(klein.lm)))^2) # 3861.147
rss.nls <- sum((Y - fitted(klein.nls))^2) # 3128.245Now, which one do you use? Depends on whether you believe you have multiplicative errors (use lm) or additive errors (use nls).
--sundar
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