Asymptotics and consistency

class: center, middle, inverse, title-slide

.title[
# Asymptotics and consistency
]
.subtitle[
## EC 421, Set 6
]
.author[
### Edward Rubin
]

---

class: inverse, middle

# Prologue

---
name: schedule

# Schedule

## Last Time

Living with heteroskedasticity

## Today

Asymptotics and consistency

## Next

- Midterm!
- Time series

---
# .mono[R] showcase

Need speed? .mono[R] allows essentially infinite parallelization.

Three popular packages:

- [`future`](https://github.com/HenrikBengtsson/future) and [`furrr`](https://github.com/DavisVaughan/furrr)

- `parallel`

- `foreach`

And here's a nice [tutorial](https://nceas.github.io/oss-lessons/parallel-computing-in-r/parallel-computing-in-r.html).

---
layout: true
# Consistency

---
class: inverse, middle

---
layout: true
# Consistency
## Welcome to asymptopia

---

*Previously:* We examined estimators (_e.g._, `$\hat{\beta}_j$`) and their properties using

1. The **mean**  of the estimator's distribution: `$\mathop{\boldsymbol{E}}\left[ \hat{\beta}_j \right] = ?$`

2. The **variance**  of the estimator's distribution: `$\mathop{\text{Var}} \left( \hat{\beta}_j \right) = ?$`

which tell us about the .hi[tendency of the estimator] if we took .hi[∞ samples], each with .hi[sample size] `$\color{#e64173}{n}$`.

This approach misses something.

---

.hi[New question:] How does our estimator behave as our sample gets larger (as `$n\rightarrow\infty$`)?

This *new question* forms a new way to think about the properties of estimators: .hi[asymptotic properties] (or large-sample properties).

A "good" estimator will become indistinguishable from the parameter it estimates when `$n$` is very large (close to `$\infty$`).

---
layout: true
# Consistency
## Probability limits

---

Just as the *expected value* helped us characterize **the finite-sample distribution of an estimator** with sample size `$n$`,

the .pink[*probability limit*] helps us analyze .hi[the asymptotic distribution of an estimator] (the distribution of the estimator as `$n$` gets "big".pink[†]).

.footnote[
.pink[†] Here, "big" `$n$` means `$n\rightarrow\infty$`. That's *really* big data.
]

---

Let `$B_n$` be our estimator with sample size `$n$`.

Then the .hi[probability limit] of `$B$` is `$\alpha$` if

$$ \lim_{n\rightarrow\infty} \mathop{P}\left( \middle| B_n - \alpha \middle| > \epsilon \right) = 0 \tag{1} $$

for any `$\varepsilon > 0$`.

The definition in `$(1)$` *essentially* says that as the .pink[sample size] approaches infinity, the probability that `$B_n$` differs from `$\alpha$` by more than a very small number `$(\epsilon)$` is zero.

*Practically:* `$B$`'s distribution collapses to a spike at `$\alpha$` as `$n$` approaches `$\infty$`.

---

Equivalent statements:

- The probability limit of `$B_n$` is `$\alpha$`.

- `$\text{plim}\: B = \alpha$`

- `$B$` converges in probability to `$\alpha$`.
---

Probability limits have some nice/important properties:

- `$\mathop{\text{plim}}\left( X \times Y \right) = \mathop{\text{plim}}\left( X \right) \times \mathop{\text{plim}}\left( Y \right)$`

- `$\mathop{\text{plim}}\left( X + Y \right) = \mathop{\text{plim}}\left( X \right) + \mathop{\text{plim}}\left( Y \right)$`

- `$\mathop{\text{plim}}\left( c \right) = c$`, where `$c$` is a constant

- `$\mathop{\text{plim}}\left( \dfrac{X}{Y} \right) = \dfrac{\mathop{\text{plim}}\left( X \right)}{ \mathop{\text{plim}}\left( Y \right)}$`

- `$\mathop{\text{plim}}\!\big( f(X) \big) = \mathop{f}\!\big(\mathop{\text{plim}}\left( X \right)\big)$`

---
layout: true
# Consistency
## Consistent estimators

---

We say that .hi[an estimator is consistent] if

1. The estimator .hi[has a prob. limit] (its distribution collapses to a spike).

2. This spike is .hi[located at the parameter] the estimator predicts.

*In other words...*

An estimator is consistent if its asymptotic distribution collapses to a spike located at the estimated parameter.

*In math:* The estimator `$B$` is consistent for `$\alpha$` if `$\mathop{\text{plim}} B = \alpha$`.

The estimator is *inconsistent* if `$\mathop{\text{plim}} B \neq \alpha$`.

---

*Example:* We want to estimate the population mean `$\mu_x$` (where `$X$`∼Normal).

Let's compare the asymptotic distributions of two competing estimators:

1. The first observation: `$X_{1}$`
2. The sample mean: `$\overline{X} = \dfrac{1}{n} \sum_{i=1}^n x_i$`
3. Some other estimator: `$\widetilde{X} = \dfrac{1}{n+1} \sum_{i=1}^n x_i$`

Note that (1) and (2) are unbiased, but (3) is biased.

---

To see which are unbiased/biased:

`$\mathop{\boldsymbol{E}}\left[ X_1 \right] = \mu_x$`

`$\mathop{\boldsymbol{E}}\left[ \overline{X} \right]$`--
 `$= \mathop{\boldsymbol{E}}\left[ \dfrac{1}{n} \sum_{i=1}^n x_i \right]$`--
 `$= \dfrac{1}{n} \sum_{i=1}^n \mathop{\boldsymbol{E}}\left[ x_i \right]$`--
 `$= \dfrac{1}{n} \sum_{i=1}^n \mu_x$`--
 `$= \mu_x$`

`$\mathop{\boldsymbol{E}}\left[ \widetilde{X} \right]$`--
 `$= \mathop{\boldsymbol{E}}\left[ \dfrac{1}{n+1} \sum_{i=1}^n x_i \right]$`--
 `$= \dfrac{1}{n+1} \sum_{i=1}^n \mathop{\boldsymbol{E}}\left[ x_i \right]$`--
 `$= \dfrac{n}{n+1}\mu_x$`

---
layout: true
# Consistency

Distributions of `$\color{#FFA500}{X_1}$`, `$\color{#e64173}{\overline{X}}$`, and `$\color{#314f4f}{\widetilde{X}}$`

---
 
`$n = 2$`

---
 
`$n = 5$`

---
 
`$n = 10$`

---
 
`$n = 30$`

---
 
`$n = 50$`

<img src="slides_files/figure-html/ex50-1.svg" style="display: block; margin: auto;" />
---
 
`$n = 100$`

---
 
`$n = 500$`

---
 
`$n = 1000$`

---
layout: true
# Consistency

---

The distributions of `$\color{#314f4f}{\widetilde{X}}$`
 
For `$n$` in `$\{\color{#FCCE25}{2},\, \color{#F89441}{5},\, \color{#E16462}{10},\, \color{#BF3984}{50},\, \color{#900DA4}{100},\, \color{#5601A4}{500}, \color{#0D0887}{1000}\}$`
<img src="slides_files/figure-html/ex biased consistent-1.svg" style="display: block; margin: auto;" />

---

## The takeaway?

- An estimator can be unbiased without being consistent
--
 (e.g., `$\color{#FFA500}{X_1}$`).

- An estimator can be unbiased and consistent
--
 (e.g., `$\color{#e64173}{\overline{X}}$`).

- An estimator can be biased but consistent
--
  (e.g., `$\color{#314f4f}{\widetilde{X}}$`).

- An estimator can be biased and inconsistent
--
  (e.g., `$\overline{X} - 50$`).

**Best-case scenario:** The estimator is unbiased and consistent.
---

## Why consistency (asymptotics)?

1. We cannot always find an unbiased estimator. In these situations, we generally (at least) want consistency.

2. Expected values can be hard/undefined. Probability limits are less constrained, _e.g._,
$$ \mathop{\boldsymbol{E}}\left[ g(X)h(Y) \right] \text{ vs. } \mathop{\text{plim}}\left( g(X)h(Y) \right) $$

3. Asymptotics help us move away from assuming the distribution of `$u_i$`.

.hi[Caution:] As we saw, consistent estimators can be biased in small samples.

---
layout: true
# OLS in asymptopia

---
class: inverse, middle

---

OLS has two very nice asymptotic properties:

1. Consistency

2. Asymptotic Normality

Let's prove \#1 for OLS with simple, linear regression, _i.e._,

$$ y_i = \beta_0 + \beta_1 x_i + u_i $$

---
layout: true
# OLS in asymptopia
## Proof of consistency

---

First, recall our previous derivation of of `$\hat{\beta}_1$`,

$$ \hat{\beta}_1 = \beta_1 + \dfrac{\sum_i \left( x_i - \overline{x} \right) u_i}{\sum_i \left( x_i - \overline{x} \right)^2} $$

Now divide the numerator and denominator by `$1/n$`

$$ \hat{\beta}_1 = \beta_1 + \dfrac{\frac{1}{n} \sum_i \left( x_i - \overline{x} \right) u_i}{\frac{1}{n}\sum_i \left( x_i - \overline{x} \right)^2} $$

---

We actually want to know the probability limit of `$\hat{\beta}_1$`, so

$$ \mathop{\text{plim}} \hat{\beta}_1 = \mathop{\text{plim}}\left(\beta_1 + \dfrac{\frac{1}{n} \sum_i \left( x_i - \overline{x} \right) u_i}{\frac{1}{n}\sum_i \left( x_i - \overline{x} \right)^2} \right) $$

which, by the properties of probability limits, gives us

$$ = \beta_1 + \dfrac{\mathop{\text{plim}}\left(\frac{1}{n} \sum_i \left( x_i - \overline{x} \right) u_i \right)}{\mathop{\text{plim}}\left(\frac{1}{n}\sum_i \left( x_i - \overline{x} \right)^2 \right)} $$

The numerator and denominator are, in fact, population quantities

$$ = \beta_1 + \dfrac{\mathop{\text{Cov}} \left( x,\, u \right)}{\mathop{\text{Var}} \left( x \right)} $$

---

So we have

$$ \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \dfrac{\mathop{\text{Cov}} \left( x,\, u \right)}{\mathop{\text{Var}} \left( x \right)} $$

By our assumption of exogeneity (plus the law of total expectation)

$$ \mathop{\text{Cov}} \left( x,\,u \right) = 0 $$

Combining these two equations yields

$$ \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \dfrac{0}{\mathop{\text{Var}} \left( x \right)} = \beta_1 \quad\text{🤓} $$

so long as `$\mathop{\text{Var}} \left( x \right) \neq 0$` (which we've assumed).

---
layout: true
# OLS in asymptopia
## Asymptotic normality

---

Up to this point, we made a very specific assumption about the distribution of `$u_i$`—the `$u_i$` came from a normal distribution.

We can relax this assumption—allowing the `$u_i$` to come from any distribution (still assume exogeneity, independence, and homoskedasticity).

We will focus on the .hi[asymptotic distribution] of our estimators (how they are distributed as `$n$` gets large), rather than their finite-sample distribution.

As `$n$` approaches `$\infty$`, the distribution of the OLS estimator converges to a normal distribution.

---
layout: false
# OLS in asymptopia
## Recap

With a more limited set of assumptions, OLS is .hi[consistent] and is .hi[asymptotically normally distributed].

### Current assumptions

1. Our data were **randomly sampled** from the population.
1. `$y_i$` is a **linear function** of its parameters and disturbance.
1. There is **no perfect collinearity** in our data.
1. The `$u_i$` have conditional mean of zero (**exogeneity**), `$\mathop{\boldsymbol{E}}\left[ u_i \middle| X_i \right] = 0$`.
1. The `$u_i$` are **homoskedastic** with **zero correlation** between `$u_i$` and `$u_j$`.

---
layout: false
class: inverse, middle
# Omitted-variable bias, redux

---
layout: true
# Omitted-variable bias, redux
## Inconsistency?
Imagine we have a population whose true model is

$$
`\begin{align}
 y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + u_i \tag{2}
\end{align}`
$$
---

*Recall.sub[1]:* .hi[Omitted-variable bias] occurs when we omit a variable in our linear regression model (_e.g._, leavining out `$x_2$`) such that

1. `$x_{2}$` affects `$y$`, _i.e._, `$\beta_2 \neq 0$`.

2. Correlates with an included explanatory variable, _i.e._, `$\mathop{\text{Cov}} \left( x_1,\, x_2 \right) \neq 0$`.
---

*Recall.sub[2]:* We defined the .hi[bias] of an estimator `$W$` for parameter `$\theta$`

$$ \mathop{\text{Bias}}\_\theta \left( W \right) = \mathop{\boldsymbol{E}}\left[ W \right] - \theta $$
---

We know that omitted-variable bias causes .pink[biased estimates].

*Question:* Do *omitted variables* also cause .pink[inconsistent estimates]?

*Answer:* Find `$\mathop{\text{plim}} \hat{\beta}_1$` in a regression that omits `$x_2$`.
---

but we instead specify the model as

$$
`\begin{align}
 y_i = \beta_0 + \beta_1 x_{1i} + w_i \tag{3}
\end{align}`
$$

where `$w_i = \beta_2 x_{2i} + u_i$`.
--
 We estimate `$(3)$` via OLS

$$
`\begin{align}
 y_i = \hat{\beta}_0 + \hat{\beta}_1 x_{1i} + \hat{w}_i \tag{4}
\end{align}`
$$

*Our question:* Is `$\hat{\beta}_1$` consistent for `$\beta_1$` when we omit `$x_2$`?
--

$$ \mathop{\text{plim}}\left( \hat{\beta}_1 \right) \overset{?}{=} \beta_1 $$
---
layout: true
# Omitted-variable bias, redux
## Inconsistency?
---

.pull-left[
.hi[Truth:] `$y_i = \beta_0 + \beta_1 x_{1i} + \beta_2 x_{2i} + u_i$`
]
.pull-right[
.hi-purple[Specified:] `$y_i = \beta_0 + \beta_1 x_{1i} + w_i$`
]

We already showed `$\mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \dfrac{\mathop{\text{Cov}} \left( x_1,\, w \right)}{\mathop{\text{Var}} \left( x_1 \right)}$`

where `$w$` is the disturbance.
--
 Here, we know `$w = \beta_2 x_2 + u$`. Thus,
--
$$ \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \dfrac{\mathop{\text{Cov}} \left( x_1,\, \beta_2 x_2 + u \right)}{\mathop{\text{Var}} \left( x_1 \right)} $$
--
Now, we make use of `$\mathop{\text{Cov}} \left( X,\, Y + Z \right) = \mathop{\text{Cov}} \left( X,\, Y \right) + \mathop{\text{Cov}} \left( X,\, Z \right)$`
--
$$ \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \dfrac{\mathop{\text{Cov}} \left( x_1,\, \beta_2 x_2 \right) + \mathop{\text{Cov}} \left( x_1,\, u \right)}{\mathop{\text{Var}} \left( x_1 \right)} $$
---

$$ \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \dfrac{\mathop{\text{Cov}} \left( x_1,\, \beta_2 x_2 \right) + \mathop{\text{Cov}} \left( x_1,\, u \right)}{\mathop{\text{Var}} \left( x_1 \right)} $$

Now we use the fact that `$\mathop{\text{Cov}} \left( X,\, cY \right) = c\mathop{\text{Cov}} \left( X,\,Y \right)$` for a constant `$c$`.
--
$$ \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \dfrac{\beta_2 \mathop{\text{Cov}} \left( x_1,\, x_2 \right) + \mathop{\text{Cov}} \left( x_1,\, u \right)}{\mathop{\text{Var}} \left( x_1 \right)} $$
--
As before, our exogeneity (conditional mean zero) assumption implies `$\mathop{\text{Cov}} \left( x_1,\, u \right) = 0$`, which gives us
--
$$ \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \dfrac{\beta_2 \mathop{\text{Cov}} \left( x_1,\, x_2 \right)}{\mathop{\text{Var}} \left( x_1 \right)} $$

---

Thus, we find that

$$ \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \beta_2 \dfrac{ \mathop{\text{Cov}} \left( x_1,\, x_2 \right)}{\mathop{\text{Var}} \left( x_1 \right)} $$

In other words, an .pink[omitted variable will cause OLS to be inconsistent if **both** of the following statements are true]:

1. The omitted variable .hi[affects our outcome], _i.e._, `$\beta_2 \neq 0$`.

2. The omitted variable correlates with included explanatory variables, _i.e._, `$\mathop{\text{Cov}} \left( x_1,\,x_2 \right) \neq 0$`.

If both of these statements are true, then the OLS estimate `$\hat{\beta}_1$` will not converge to `$\beta_1$`, even as `$n$` approaches `$\infty$`.

---
layout: true
# Omitted-variable bias, redux
## Signing the bias

---

Sometimes we're stuck with omitted variable bias..pink[†]

$$ \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \beta_2 \dfrac{ \mathop{\text{Cov}} \left( x_1,\, x_2 \right)}{\mathop{\text{Var}} \left( x_1 \right)} $$

.footnote[.pink[†] You will often hear the term "omitted-variable bias" when we're actually talking about inconsistency (rather than bias).]

When this happens, we can often at least know the direction of the inconsistency.

---

Begin with

$$ \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \beta_2 \dfrac{ \mathop{\text{Cov}} \left( x_1,\, x_2 \right)}{\mathop{\text{Var}} \left( x_1 \right)} $$

We know `$\color{#20B2AA}{\mathop{\text{Var}} \left( x_1 \right) > 0}$`. Suppose `$\color{#e64173}{\beta_2 > 0}$` and `$\color{#FFA500}{\mathop{\text{Cov}} \left( x_1,\,x_2 \right) > 0}$`. Then
--
$$
`\begin{align}
 \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \color{#e64173}{(+)} \dfrac{\color{#FFA500}{(+)}}{\color{#20B2AA}{(+)}} \implies \mathop{\text{plim}} \hat{\beta}_1 > \beta_1
\end{align}`
$$
∴ In this case, OLS is **biased upward** (estimates are too large).
--
$$
`\begin{matrix}
 \enspace & \color{#FFA500}{\text{Cov}(x_1,\,x_2)> 0} & \color{#FFA500}{\text{Cov}(x_1,\,x_2)< 0} \\
 \color{#e64173}{\beta_2 > 0} & \text{Upward} & \\
 \color{#e64173}{\beta_2 < 0} & &
\end{matrix}`
$$

---

Begin with

$$ \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \beta_2 \dfrac{ \mathop{\text{Cov}} \left( x_1,\, x_2 \right)}{\mathop{\text{Var}} \left( x_1 \right)} $$

We know `$\color{#20B2AA}{\mathop{\text{Var}} \left( x_1 \right) > 0}$`. Suppose `$\color{#e64173}{\beta_2 < 0}$` and `$\color{#FFA500}{\mathop{\text{Cov}} \left( x_1,\,x_2 \right) > 0}$`. Then
--
$$
`\begin{align}
 \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \color{#e64173}{(-)} \dfrac{\color{#FFA500}{(+)}}{\color{#20B2AA}{(+)}} \implies \mathop{\text{plim}} \hat{\beta}_1 < \beta_1
\end{align}`
$$
∴ In this case, OLS is **biased downward** (estimates are too small).

$$
`\begin{matrix}
 \enspace & \color{#FFA500}{\text{Cov}(x_1,\,x_2)> 0} & \color{#FFA500}{\text{Cov}(x_1,\,x_2)< 0} \\
 \color{#e64173}{\beta_2 > 0} & \text{Upward} & \\
 \color{#e64173}{\beta_2 < 0} & \text{Downward} &
\end{matrix}`
$$

---

Begin with

$$ \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \beta_2 \dfrac{ \mathop{\text{Cov}} \left( x_1,\, x_2 \right)}{\mathop{\text{Var}} \left( x_1 \right)} $$

We know `$\color{#20B2AA}{\mathop{\text{Var}} \left( x_1 \right) > 0}$`. Suppose `$\color{#e64173}{\beta_2 > 0}$` and `$\color{#FFA500}{\mathop{\text{Cov}} \left( x_1,\,x_2 \right) < 0}$`. Then

$$
`\begin{align}
 \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \color{#e64173}{(+)} \dfrac{\color{#FFA500}{(-)}}{\color{#20B2AA}{(+)}} \implies \mathop{\text{plim}} \hat{\beta}_1 < \beta_1
\end{align}`
$$
∴ In this case, OLS is **biased downward** (estimates are too small).

$$
`\begin{matrix}
 \enspace & \color{#FFA500}{\text{Cov}(x_1,\,x_2)> 0} & \color{#FFA500}{\text{Cov}(x_1,\,x_2)< 0} \\
 \color{#e64173}{\beta_2 > 0} & \text{Upward} & \text{Downward} \\
 \color{#e64173}{\beta_2 < 0} & \text{Downward} &
\end{matrix}`
$$

---

Begin with

$$ \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \beta_2 \dfrac{ \mathop{\text{Cov}} \left( x_1,\, x_2 \right)}{\mathop{\text{Var}} \left( x_1 \right)} $$

We know `$\color{#20B2AA}{\mathop{\text{Var}} \left( x_1 \right) > 0}$`. Suppose `$\color{#e64173}{\beta_2 < 0}$` and `$\color{#FFA500}{\mathop{\text{Cov}} \left( x_1,\,x_2 \right) < 0}$`. Then

$$
`\begin{align}
 \mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \color{#e64173}{(-)} \dfrac{\color{#FFA500}{(-)}}{\color{#20B2AA}{(+)}} \implies \mathop{\text{plim}} \hat{\beta}_1 > \beta_1
\end{align}`
$$
∴ In this case, OLS is **biased upward** (estimates are too large).

$$
`\begin{matrix}
 \enspace & \color{#FFA500}{\text{Cov}(x_1,\,x_2)> 0} & \color{#FFA500}{\text{Cov}(x_1,\,x_2)< 0} \\
 \color{#e64173}{\beta_2 > 0} & \text{Upward} & \text{Downward} \\
 \color{#e64173}{\beta_2 < 0} & \text{Downward} & \text{Upward}
\end{matrix}`
$$

---

Thus, in cases where we have a sense of

1. the sign of `$\mathop{\text{Cov}} \left( x_1,\,x_2 \right)$`

2. the sign of `$\beta_2$`

we know in which direction inconsistency pushes our estimates.

.center[
**Direction of bias**
]

$$
`\begin{matrix}
 \enspace & \color{#FFA500}{\text{Cov}(x_1,\,x_2)> 0} & \color{#FFA500}{\text{Cov}(x_1,\,x_2)< 0} \\
 \color{#e64173}{\beta_2 > 0} & \text{Upward} & \text{Downward} \\
 \color{#e64173}{\beta_2 < 0} & \text{Downward} & \text{Upward}
\end{matrix}`
$$

---
layout: true
# Measurement error

---
name: measurement_error

.hi[Measurement error] in our explanatory variables presents another case in which OLS is inconsistent.

Consider the population model: `$y_i = \beta_0 + \beta_1 z_i + u_i$`

- We want to observe `$z_i$` but cannot.

- Instead, we *measure* the variable `$x_i$`, which is `$z_i$` plus some error (noise):
$$ x_i = z_i + \omega_i $$

- Assume `$\mathop{\boldsymbol{E}}\left[ \omega_i \right] = 0$`, `$\mathop{\text{Var}} \left( \omega_i \right) = \sigma^2_\omega$`, and `$\omega$` is independent of `$z$` and `$u$`.

OLS regression of `$y$` and `$x$` will produce inconsistent estimates for `$\beta_1$`.

---
layout: true
# Measurement error
## Proof

---

`$y_i = \beta_0 + \beta_1 z_i + u_i$`
--
 `$\quad= \beta_0 + \beta_1 \left( x_i - \omega_i \right) + u_i$`
--
 `$\quad= \beta_0 + \beta_1 x_i + \left( u_i - \beta_1 \omega_i \right)$`
--
 `$\quad= \beta_0 + \beta_1 x_i + \varepsilon_i$`

where `$\varepsilon_i = u_i - \beta_1 \omega_i$`

What happens when we estimate `$y_i = \hat{\beta}_0 + \hat{\beta}_1 x_i + e_i$`?

`$\mathop{\text{plim}} \hat{\beta}_1 = \beta_1 + \dfrac{\mathop{\text{Cov}} \left( x,\,\varepsilon \right)}{\mathop{\text{Var}} \left( x \right)}$`

We will derive the numerator and denominator separately...

---

The covariance of our noisy variable `$x$` and the disturbance `$\varepsilon$`.

`$\mathop{\text{Cov}} \left( x,\, \varepsilon \right)$`
--
 `$= \mathop{\text{Cov}} \left( \left[ z + \omega \right],\, \left[ u - \beta_1 \omega \right] \right)$`
--
 `$\quad\quad\quad\quad\enspace= \mathop{\text{Cov}} \left( z,\,u \right) -\beta_1 \mathop{\text{Cov}} \left( z,\,\omega \right) + \mathop{\text{Cov}} \left( \omega,\, u \right) - \beta_1 \mathop{\text{Var}} \left( \omega \right)$`
--
 `$\quad\quad\quad\quad\enspace= 0 + 0 + 0 - \beta_1 \sigma_\omega^2$`
--
 `$\quad\quad\quad\quad\enspace= - \beta_1 \sigma_\omega^2$`

---

Now for the denominator, `$\mathop{\text{Var}} \left( x \right)$`.

`$\mathop{\text{Var}} \left( x \right)$`
--
`$= \mathop{\text{Var}} \left( z + \omega \right)$`
--
 `$\quad\quad\quad= \mathop{\text{Var}} \left( z \right) + \mathop{\text{Var}} \left( \omega \right) + 2\mathop{\text{Cov}} \left( z,\,\omega \right)$`
--
 `$\quad\quad\quad= \sigma_z^2 + \sigma_\omega^2$`

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Putting the numerator and denominator back together,

$$
`\begin{align}
 \mathop{\text{plim}} \hat{\beta}_1
 &= \beta_1 + \dfrac{\mathop{\text{Cov}} \left( x,\,\varepsilon \right)}{\mathop{\text{Var}} \left( x \right)} \\
 &= \beta_1 + \dfrac{-\beta_1 \sigma_\omega^2}{\sigma_z^2 + \sigma_\omega^2} \\
 &= \beta_1 - \beta_1 \dfrac{\sigma_\omega^2}{\sigma_z^2 + \sigma_\omega^2} \\
 &= \beta_1 \dfrac{\sigma_z^2 + \sigma_\omega^2}{\sigma_z^2 + \sigma_\omega^2} - \beta_1 \dfrac{\sigma_\omega^2}{\sigma_z^2 + \sigma_\omega^2} \\
 &= \beta_1 \dfrac{\sigma_z^2}{\sigma_z^2 + \sigma_\omega^2}
\end{align}`
$$

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# Measurement error
## Summary

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∴ `$\mathop{\text{plim}} \hat{\beta}_1 = \beta_1 \dfrac{\sigma_z^2}{\sigma_z^2 + \sigma_\omega^2}$`.

What does this equation tell us?

.hi[Measurement error in our explanatory variables] biases the coefficient estimates toward zero.

- This type of bias/inconsistency is often called .hi[attenuation bias].

- If .hi[the measurement error correlates with the explanatory variables], we have bigger problems with inconsistency/bias.

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What about **measurement in the outcome variable**?

It doesn't really matter—it just increases our standard errors.

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# Measurement error
## It's everywhere

**General cases**

1. We cannot perfectly observe a variable.
1. We use one variable as a *proxy* for another.

**Specific examples**

- GDP
- Population
- Crime/police statistics
- Air quality
- Health data
- Proxy *ability* with test scores

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exclude: true