Errors in Measurement and the Wald (1940) Estimator. This is based on Farebrother (1985). Let y∗

i be permanent consumption and X∗ be permanent income, both are measured with error:

y

∗

i = βx

∗

i where yi = y

∗

i + i and xi = x

∗

i + ui for i = 1, 2, . . . , n.

Let x∗

i , i and ui be independent normal random variables with zero means and variances σ2

∗, σ2 and σ2 u, respectively. Wald (1940) suggested the following estimator of β: Order the sample by the xi’s and split the sample into two. Let (¯y1, ¯x1) be the sample mean of the first half of the sample and (¯y2, ¯x2) be the sample mean of the second half of this sample. Wald’s estimator of β is

βW = (¯y2 − ¯y1)/(¯x2 − ¯x1). It is the slope of the line joining these two sample mean observations.

(a) Show that βW can be interpreted as a simple IV estimator with instrument zi = 1 for xi ≥ median(x)

= −1 for xi < median(x)

where median(x) is the sample median of x1, x2, . . . , xn.

(b) Define wi = ρ2x∗

i

− τ2ui where ρ2 = σ2 u/(σ2 u + σ2

∗) and τ 2 = σ2

∗/(σ2 u + σ2

∗). Show that E(xiwi) = 0 and that wi ∼ N(0, σ2

∗σ2 u/(σ2

∗ + σ2 u)).

(c) Show that x∗

i = τ2xi + wi and use it to show that E(βW/x1, . . . , xn) = E(βOLS/x1, . . . , xn) = βτ2.

Conclude that the exact small sample bias of βOLS and βW are the same.