Measurement Error

Question 1

2 ways we can have measurement errors

Use simple regression
Yi =β₀ +β₁ Xi +εi

Answer 1

Measure independent variable Xi with error
X*i = Xi + ui
With ui~N(0,σ²u)

Measure dependent variable with error
Y*i = Yi + wi
With wi~N(O,σ²w)

So the errors are normally distributed (mean 0, constant variance)

Question 2

Origins of measurement errors (4)

Answer 2

Misreporting figures
Age heaping
High frequency data - often has errors
Nefarious activities e.g corruption - incentive to lie

Question 3

We measure dependent variable with error as:
Y*i = Yi + wi with wi~N(0,σ²w)

What assumptions do we make? (4)
What is it known as?

Answer 3

E(wi)=0

Var(wi)=σ²w

Cov(wi,wj)=0 (errors are not correlated with each other)

i≠j

Known as classical measurement error!

Question 4

Estimate β₁ with the measurement error in Yi

B) what happens to our estimation of β₁ (β^₁)?

Answer 4

Yi* = Yi + wi
Rearrange to make Yi subject (Yi = Y*i - wi) , then sub into original model!

Y*i - wi = β₀ + β₁Xi + εi

Then rearrange this model
Y*i = β₀ + β₁Xi + wi + εi
Where wi + εi = vi (so combining measurment error wi and normal error εi)

B) due to our prev assumptions, β^₁ is unbias and consistent!

However variance is higher.

Question 5

So our estimation of β^₁ is unbias and consistent given those assumptions, but variance is higher.

Variance expression

Answer 5

Var(νi) = Var(wi+εi)= σ²w +σ²ε which is > σ²ε

Question 6

Now measurement error in independent variable

X*i = Xi + ui with ui~N(0,σ²u)

What assumptions do we make

Answer 6

The same as we do for dependent variables

E(ui) = 0
Var (ui) = 0
Cov(ui,uj) = 0
i≠j

Question 7

Estimate β^₁ with measurement error in Xi
(Learn first bit i have done, and final equation)

Answer 7

Rearrange to make Xi subject (Xi =X*i - ui), and sub into original model.

Yi = β₀ + β₁(X*i − ui) + εi

Expand to get
Yi = β₀ + β₁X*i + εi − β₁ui
Where εi − β₁ui = μi (measurement error and normal error)

Final equation for β^₁
β₁-β₁(σ²u/var(X*i))

Question 8

Equation for β^₁ estimation

Answer 8

β^₁ = β₁-β₁(σ²u/var(X*i)) ≠ β₁

Bias! Estimate is less than true parameter

Question 9

So our estimate for β₁ is bias; less than true parameter:

What is this called?

Answer 9

Attenuation bias

Question 10

Effect of σ²u (measurement error) on attenuation bias

Answer 10

As σ²u (measurement error) increases, attenuation bias increases

(Shown by the formula)

Question 11

What does it look like graphically

Answer 11

Regression line becomes flatter

Question 12

Summary of estimates as a result of dependent vs independent variable bias

Answer 12

Dependent error - unbias and consistent, but variance higher.

Independent - bias (attenuation)