I have data $y_i,x_i$ and the model $y_i=\alpha x_i^\beta + \epsilon_i$ where $\log{\epsilon_i} \in N(0,\sigma^2)$. Thus, I define $z_i=\log{y_i}$ and $w_i=\log{x_i}$ and my normal linear model becomes $z_i=\log{\alpha}+\beta w_i+\log{\epsilon_i}$.
The LSE and MLE of $\log{\alpha}$ should be $\log{\hat{\alpha}}=\bar{z}$. This is roughly 1.5 with the data below. However, the intercept, assuming that is $\log{\hat{\alpha}}$, is around 0.6. Am I interpreting this correctly?
x<-c(2,3,4,5,6,7,8,9,10,11)
y<-c(2.1,4,3.7,4.5,5,4.8,5.1,5.7,5.7,5.6)
w<-log(x)
z<-log(y)
mean(z)
fit<-lm(z~w)
summary(fit)
[1] 1.495164
Call:
lm(formula = z ~ w)
Residuals:
Min 1Q Median 3Q Max
-0.22232 -0.03190 -0.01444 0.06492 0.21840
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.61614 0.13723 4.490 0.002029 **
w 0.50223 0.07509 6.689 0.000154 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1249 on 8 degrees of freedom
Multiple R-squared: 0.8483, Adjusted R-squared: 0.8293
F-statistic: 44.74 on 1 and 8 DF, p-value: 0.0001545