Question

A random sampleY1 , Y 2 , ⋯,Yn is drawn from a distribution whose probability density function is given by: f (Y ) = βe − βY , Y  0 & β > 0 a). Obtain the maximum likelihood estimator (MLE) of β. (3 points) 1Econometrics- Assignment I b). Given that ∑ n Y i = 25 , ∑n Yi 2 = 50 , n = 50 calculate the maximum likelihood i =1 i =1 estimate of β. (3 point) c). Using the same data as in part (b), test the null hypothesis that β =1against the alternative hypothesis that β ≠1at 5% level of significance. (3 points) Problem 4 Suppose the production(Y) is determined as a function of labour input in hours (L) and capital input in machine hours (K). Using the Cobb-Douglas function

Answer 1

(a) maximum likelihood estimator.

$\frac{df}{dy} = \beta$$(\beta e$$-\beta$$Y)-$$\beta$($\beta$$e-\beta$$Y)$

(b)

M= $\frac{1}{n}\sum$ X_i (from 1to 50)

n= 50

X_i= 1

M=$\frac{1}{50}= 0.02$

(c)T²$=\frac{1}{n}\sum$(Xi -M)²

$=0.02(1-0.02)= 0.0196$

 Problem 4

the complete question: Suppose the production(Y) is determined as a function of labor input in hours (L) and capital input in machine hours (K). Using the Cobb-Douglas function:  

Y= β0 +  β1K β1 + ek Write the procedure to estimate the coefficients of this function 

fY(y)=c(β)y3(1−y)β10<y<1