Pop Quiz: If (D^*) and (D) are binary random variables and (D) is a loud measure of (D^*), is it attainable for the measurement error (W equiv D – D^*) to be classical? Clarify why or why not. (Reply beneath)
Classical measurement error is an issue that’s straightforward to know and comparatively straightforward to deal with.
Roughly talking, classical measurement error refers to a scenario during which the variable we observe equals the reality plus noise
[
text{Observed} = text{Truth} + text{Noise}
]
the place the noise is unrelated to the reality and “every little thing else.”
(I’ll be exact concerning the that means of “unrelated” and “every little thing else” in a second.)
Mis-measuring a regressor (X) on this manner biases the OLS slope estimator in the direction of zero (attenuation bias) however we are able to right for this with a legitimate instrument. Mis-measuring the result (Y) will increase normal errors however doesn’t bias the OLS estimator. You’ll find all the small print in your favourite introductory econometrics textbook, however within the curiosity of constructing this submit self-contained, right here’s a fast evaluate.
Least Squares Attenuation Bias
Suppose that we need to study the slope coefficient from a inhabitants linear regression of (Y) on (X^*):
[
beta equiv frac{text{Cov}(Y,X^*)}{text{Var}(X^*)}.
]
Sadly we observe not (X^*) however a loud measure (X = X^* + W_X) the place (W_X) is uncorrelated with each (X^*) and (Y).
Then
[
begin{aligned}
text{Cov}(Y, X) &= text{Cov}(Y, X^* + W_X) = text{Cov}(Y, X^*)
text{Var}(X) &= text{Var}(X^* + W_X) = text{Var}(X^*) + text{Var}(W_X).
end{aligned}
]
Now, outline the reliability ratio (lambda) as follows:
[
lambda equiv frac{text{Var}(X^*)}{text{Var}(X^*) + text{Var}(W_X)}.
]
Measurement error implies that (textual content{Var}(W_X)) is constructive.
Since variances can’t be destructive, this means (0 < lambda < 1).
Combining our definition of (lambda) with the expressions for (textual content{Cov}(Y,X)) and (textual content{Var}(X)) from above,
[
begin{aligned}
frac{text{Cov}(Y,X)}{text{Var}(X)} &= frac{text{Cov}(Y, X^*)}{text{Var}(X^*) + text{Var}(W_X)}
&=frac{text{Var}(X^*)}{text{Var}(X^*) + text{Var}(W_X)}cdot frac{text{Cov}(Y, X^*)}{text{Var}(X^*)}
&= lambda beta
end{aligned}
]
so we see that regressing (Y) on (X) provides (lambda beta) reasonably than (beta).
Since (0 < lambda < 1), this phenomenon is known as least squares attenuation bias: (lambda beta) has the identical signal as (beta) however is smaller in magnitude.
The higher the extent of measurement error, the bigger the variance of (W_X) and the smaller that (|lambda beta|) turns into.
Instrumental variables to the rescue
Suppose that (Y = alpha + beta X^* + U) the place (X = X^* + W_X) as above.
Now suppose that we are able to discover a variable (Z) that’s correlated with (X^*) however uncorrelated with (U) and (W_X).
Then
[
begin{aligned}
text{Cov}(Y,Z) &= text{Cov}(alpha + beta X^* + U, Z) = betatext{Cov}(X^*,Z)
text{Cov}(X,Z) &= text{Cov}(X^* + W_X, Z) = text{Cov}(X^*,Z)
end{aligned}
]
in order that (beta = textual content{Cov}(Y, Z) / textual content{Cov}(X,Z)).
If (X^*) is measured with classical measurement error, a easy instrumental variables regression solves the issue of attenuation bias.
Discover that we haven’t stated something about (U) in relation to (X^*).
If (beta) is the inhabitants linear regression slope, then (U) is uncorrelated with (X^*) by definition.
However this derivation nonetheless goes via if (Y = alpha + beta X^* + U) is a causal mannequin during which (X^*) is correlated with (U), e.g. if (Y) is wage and (X^*) is years of education, during which case (U) is likely to be “unobserved capability.”
On this manner, a single legitimate instrument can serve “double-duty,” eliminating each attenuation bias and choice bias.
Measurement error within the consequence
Now suppose that (X^*) is noticed however the true consequence (Y^*) will not be: we solely observe a loud measure (Y = Y^* + W_Y).
If (W_Y) is uncorrelated with (X^*),
[
frac{text{Cov}(Y,X^*)}{text{Var}(X^*)} = frac{text{Cov}(Y^* + W_Y, X^*)}{text{Var}(X^*)} = frac{text{Cov}(Y^*,X^*)}{text{Var}(X^*)}
]
so we’ll get hold of the identical slope from a regression of (Y) on (X^*) as we’d from a regression of (Y^*) on (X^*).
Classical measurement error within the consequence variable doesn’t introduce a bias.
Now that we’ve refreshed our reminiscences about classical measurement error, let’s a check out my pop quiz query from above:
If (D^*) and (D) are binary random variables and (D) is a loud measure of (D^*), is it attainable for the measurement error (W equiv D – D^*) to be classical? Clarify why or why not.
If (W) is a classical measurement error then, amongst different issues, it have to be uncorrelated with (D^*).
However that is unimaginable if each (D^*) and (D) are binary.
By the definition of (W), (D = D^* + W).
If (D^* = 1) then (D = 1 + W).
To make sure that (D) takes on a price in ({0, 1}), because of this (W) have to be both (0) or (-1).
If as a substitute (D^* = 0), then (D = W), so (W) have to be both (0) or (1).
Therefore, until (W) all the time equals zero, during which case there’s no measurement error, (W) should all the time be negatively correlated with (D^*).
In different phrases, measurement error in a a binary variable can by no means be classical.
The identical fundamental logic applies every time (X) and (X^*) are bounded: to make sure that (X) stays inside its bounds, any measurement error have to be correlated with (X^*).
Classical measurement error, as we’ve seen, is a really particular case.
Or to place it one other manner, non-classical measurement error isn’t as unique because it sounds.
As a result of discrete random variables can’t be topic to classical measurement error, non-classical measurement error needs to be on any utilized economist’s radar.
My subsequent few posts will present an summary of the only case: non-differential measurement error in a binary variable.
This assumption permits (D^*) to be correlated with (W), however assumes that conditioning on (D^*) is adequate to break the dependence between (W) and every little thing else.
Even on this comparatively easy case, every little thing we’ve discovered about classical measurement error goes out the window:
- Non-differential measurement error does not essentially trigger attenuation.
- The IV estimator doesn’t right for non-differential measurement error, and a single instrument can not serve “double-duty.”
- Non-classical measurement error within the consequence variable usually does introduce bias.
The excellent news is that there are strategies to deal with non-differential measurement error.
In my subsequent submit, I’ll begin by contemplating the case of a mis-measured binary consequence.
