ECON60622: Further Econometrics Semester 2, 2021-22
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ECON60622:
Further Econometrics
Semester 2, 2021-22
The IV estimator with a binary instrument
Consider the case where we estimate the simple linear regression model
y = β0 + β 1x + u
via Instrumental Variables using z as Instrument for x. We assume that E[u|z] = 0 which (as shown in lectures) implies the instrument exogeneity condition (Cov[z,u] = 0). We also assume the instrument relevance condition (Cov[z,x] 
0) holds. Under these conditions, it can be shown that (check for yourselves)1
Cov[z,y]
Cov[z,x] .
Now suppose that z is a binary random variable with sample space {0, 1} and P(z = 1) = T . In the case, it is stated in lectures that
E[y|z = 1] − E[y|z = 0]
β 1 =
In this handout, we show that (1) and (2) imply the same solution for β 1 .
We begin by showing that (2) holds. From the definition of the model, we have: E[y |z] = β0 + β 1E[x|z] + E[u|z] = β0 + β 1E[x|z]
where the second equality uses the given information that E[u|z] = 0. Evaluating this equation for the two possible outcomes for z, we obtain:
E[y | z = 0] = β0 + β 1E[x|z = 0]. (4)
Subtracting (4) from (3) and rearranging, we obtain:
E[y|z = 1] − E[y|z = 0]
β 1 =
which establishes that (2) holds.
We now establish that the right hand-side of (1) is equal to the right-hand side of (2). First consider Cov[z,y]. By definition, we have
= E[zy] − E[z]E[y]. (5)
Recalling that E[z] = ⇡, now consider E[zy]. Given the definition of z, there are only two states of the world: y occurs when z = 0 or when z = 1. Thus, we have
E[zy] = E[zy |z = 1]P(z = 1) + E[zy |z = 0]P(z = 0). Since E[zy | z = 0] = 0 and E[zy |z = 1] = E[y | z = 1] by definition, it follows that
E[zy] = E[y |z = 1]⇡ = E1⇡ , say, (6)
where we have introduced the notation Ek = E[y | z = k] (for k = 0, 1) for ease of presentation. Now consider E[y]. By similar logic, we have:
E[y] = E[y |z = 1]P(z = 1) + E[y |z = 0]P(z = 0) = E1⇡ + E0(1 − ⇡). (7)
Substituting (6)-(7) into (5) and using E[z] = ⇡, we obtain:
Cov[z,y] = E1⇡ − ⇡{E1⇡ + E0(1 − ⇡)} = ⇡(1 − ⇡){E1 − E0}
= ⇡(1 − ⇡){E[y |z = 1] − E[y |z = 0]}. (8)
To evaluate Cov[z,x], we can just repeat the argument above only with x replacing y to obtain
Cov[z,x] = ⇡(1 − ⇡){E[x|z = 1] − E[x|z = 0]}. (9)
Taking the ratio of (8) to (9), we obtain:
Cov[z,y] ⇡(1 − ⇡){E[y |z = 1] − E[y |z = 0]} E[y |z = 1] − E[y |z = 0]
= =
Cov[z,x] ⇡(1 − ⇡){E[x|z = 1] − E[x|z = 0]} E[x|z = 1] − E[x|z = 0] .
2022-08-29
The IV estimator with a binary instrument