# Practical Assignment - Econometrics

```ECON4003: Practical Assignment
Part I
(a) π₯π − π = πΌππ E
π₯π − π
ππ =
πΌ
π₯π ππ
ππ = −
πΌ πΌ
π₯π ππ
− = π½π + π½1 π₯π + π’π
πΌ πΌ
π₯π ππ
π’π = − − π½0 − π½1 π₯π
πΌ πΌ
1
ππ
= ( − π½1 ) π₯π − − π½0
πΌ
πΌ
1 − πΌπ½1
ππ
=(
) π₯π − − π½0
πΌ
πΌ
1 − πΌπ½1
ππ
=
π₯π − π½0 −
πΌ
πΌ
1−πΌπ½1
(b) πΈ(π’π |π) = πΈ (
πΌ
π
ππ − π½0 − πΌπ |π)
1 − πΌπ½
π
= πΈ ( πΌ 1 ππ |π) − πΈ(π½0|π) − πΈ ( πΌπ |π)
1 − πΌπ½1
1
(πΈ(ππ |π)) − πΈ(π½0 |π) − πΈ(ππ |π)
πΌ
πΌ
1 − πΌπ½1
1
=
ππ − π½0 − &times; 0
πΌ
πΌ
1 − πΌπ½1
=
ππ − π½0
πΌ
=
1
(c) Assumption SR.3 confirms that the sample outcomes of ππ , π = 1, … , π , must take at
least two different values
π
Μ1 = ∑ π€π ππ
π½
π=1
π
= ∑ π€π (π½0 + π½1 ππ + π’π )
π=1
This substitution is possible by inputting the true model expression derived from the
linearity assumption
π
π
π
π½Μ1 = π½0 ∑ π€π + π½1 ∑ π€π ππ + ∑ π€π π’π
π=1
π=1
π=1
π
= π½1 + ∑ π€π π’π
π=1
since
π
∑ π€ = 0,
π=1
π
∑ π€ππ = 1
π=1
Taking the expectations of π½Μ1 conditional on the sample values of regressor X, where
π€π is treated as non-random in this case since it is a function only of X
π
πΈ(π½Μ1|π) = πΈ(π½1 + ∑ π€π π’π |π)
π=1
π
= πΈ(π½1 |π) + πΈ(∑ π€π π’π |π)
π=1
π
= π½1 + ∑ π€π πΈ(π’π |π)
π=1
2
π½1 is a constant and thus is its own
expected value and π€π can be
removed from the conditional
expectation expression as it is non
random
3
(d)
4
```