Relationship Of Class Attendance And Exam Performance (Essay Sample)

Econometrics
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ECOM30002 Econometrics
1. Consider two random variables X1 and X2 with means µ1 = E(X1), µ 2 = E(X2), variance- covariance matrix
Solution
From S2=e1Pe/N-2
Variance=X1X2=δ12δ12δ12δ22
=S2 (X1PX) 1
The estimate of the variance predicted by the mean of Y given a value X, is given by X0
S2Yn/X0=S21,X0X1px-11X0
An estimate of the variance of the predicted value of Y for an individual for a specific value of X, say , is given by X0
To obtain the estimate of the predicted value that is Y given X in this case we use X0.
S2Y1/X0=S2+S2Yn/X0
But r=K=1NPjXJ-XYj-Y K=1NPjXJ-X2 k=1NYj-Y2
=SxySxxSyy
=±rxxrxy
=sing±rxxR2
Where Sxy is the covariance between X and Y, rxx is the slope from the regression of X on Y, and ryx is the slope from the regression of Y on X
QUESTION 2
* For conditional distribution
Suppose x~Npμ,σ
Let x be portioned as x=x1x2 then the conditional distribution of xi given x1 is
Nμ2+δ21δ-111X1-μ1, δ22δ21δ-111δ12
From FX2,X1=F(X)F(X1)
FX=12πPδ-21e-21(x-μ)1δ-1(x-μ )
x~Nμ1,δ11
fx1=1(2π)r2δ11-12e-21(x-μ1)1δ11-1(x1-μ1)
Let Q=x -μ1δ-1x-μ
Using the partitioning of x, we have
Q=δ11122πr2e-21Q*
Where Q*=Q-Q
Therefore x2/x1~μ +δ21δ11x1-μ1, δ22-δ21δ11-1δ12
* The n = 1000.
Compare the sample means and their variances.
Mu (µ) <-2
Sigma (δ1) <-2 sigma (δ2)<0.4
n<-10
sim<-1000 β1=2 β2=1.2 β0=0.222

Mean

variance

ρ=0

0.71

0.4

ρ=0.5

1.6

0.005

The condition of unbiased estimator requires that Ax=0
Implying that AC1=0 and CA1=0
Var (b) =δ32x1x-1+AC1+CA1+CC1
Var (b) =δ32x1x-1+CC1
Varb=δ32x1x-1+δ32CC1
Both this is equal to varβ
varβ=δ32x1x-1
Varb=varβ+δ32CC1 Meaning that Varb=varβ hence unbiased estimator
* Omitted variable bias occurs when the estimate is higher than the true parameter
For exa...