The true model is
y = ßx + ε
Stata calculates
ß = Σxy / Σxx
Substitution shows
ß = ß + Σxε / Σxx
Assume E(xε) = 0
E(ß) = ß
V(ß) = V(Σxε) / (Σxx)
2
=
σ
2
/ Σxx
V(Σxε) = V(x
1
ε
1
+ ... + x
n
ε
n
)
= x
1
2
V(ε
1
) + ... + x
n
2
V(ε
n
)
... + cov(x
i
ε
i
,x
j
ε
j
) + ... for all ij
= x
1
2
V(ε
1
) + ... + x
n
2
V(ε
n
)
= σ
2
Σxx
The true model is
y = ßx + δz + ε
Stata calculates
ß = Σxy / Σxx
Substitution shows
ß = ß + Σxz / Σxx + Σxε / Σxx
E(ß) = ß + Σxz / Σxx