Instrumental Variables Estimation

Paul Schrimpf

2024-12-14

Consistency

$\begin{align*} \hat{\beta}^{IV}_W - \beta_0 = & (X'Z W Z'W)^{-1}(X'Z W Z'u) \\ = & \left[ \left(\frac{1}{n}\sum_{i=1}^n X_i Z_i'\right) W \left(\frac{1}{n}\sum_{i=1}^n Z_i X_i'\right) \right]^{-1} \left(\frac{1}{n}\sum_{i=1}^n X_i Z_i'\right) W \left(\frac{1}{n}\sum_{i=1}^n Z_i u_i\right) \end{align*}$

Consistent if LLN applies to $\frac{1}{n}\sum_{i=1}^n Z_i X_i'$ and $\frac{1}{n}\sum_{i=1}^n Z_i u_i$
- E.g. if i.i.d. with $\Er[\norm{X_i}^4]$ and $\Er[\norm{Z_i}^4]$ finite and $\Er[u_i^2|Z_i=z] = \sigma^2$ ¹

Asymptotic Normality

$\sqrt{n}(\hat{\beta}^{IV} - \beta_0) \indist N(0, V)$ if LLN applies to $\frac{1}{n}\sum_{i=1}^n Z_i X_i'$ and CLT to $\frac{1}{\sqrt{n}}\sum_{i=1}^n Z_i u_i$
- E.g. if i.i.d. with $\Er[\norm{X_i}^4]$ and $\Er[\norm{Z_i}^4]$ finite and $\Er[u_i^2|Z_i=z] = \sigma^2$
- then $\frac{1}{\sqrt{n}} \sum Z_i u_i \indist N(0, \sigma^2 \Er[Z_iZ_i'])$
- $V = \sigma^2 (\Er[Z_iX_i']' W \Er[Z_iX_i'])^{-1} (\Er[Z_iX_i']' W \Er[Z_i Z_i'] W \Er[Z_i X_i']) (\Er[Z_iX_i']' W \Er[Z_iX_i'])^{-1}$

Over-identifying Test

Only has power when instruments have different covariances with $u$

Code

using Distributions, LinearAlgebra
import PlotlyLight
function sim(n; d=3, EZu = zeros(d), Exu = 0.5, beta = 1, gamma = ones(d))
  zu = randn(n,d)
  Z = randn(n,d) + mapslices(x->x.*EZu, zu, dims=2)
  xu = randn(n)
  X = Z*gamma + xu*Exu
  u = vec(sum(zu,dims=2) + xu + randn(n))
  y = X*beta + u
  return(y,X,Z)
end

biv(y,X,Z) = (X'*Z*inv(Z'*Z)*Z'*X) \ (X'*Z*inv(Z'*Z)*Z'*y)

function J(y,X,Z)
  n = length(y)
  bhat = biv(y,X,Z)
  uhat = y - X*bhat
  C = inv(1/n*sum(z*z'*u^2 for (z,u) in zip(eachrow(Z),uhat)))
  Zu = Z'*uhat/n
  J = n*Zu'*C*Zu
end

S = 1_000
n = 100
j0s = [J(sim(n)...) for _ in 1:S]
j1s = [J(sim(n,EZu=[0.,0., 3.])...) for _ in 1:S]
j2s = [J(sim(n,EZu=[1.,1., 1.])...) for _ in 1:S]

plt = PlotlyLight.Plot()
plt(x=j0s, type="histogram", name="E[Zu] = 0")
plt(x=j1s, type="histogram", name="E[Zu] = [0,0,3]")
fig=plt(x=j2s, type="histogram", name="E[Zu] = [1,1,1]")

fig

┌ Warning: `Plot(; kw...)` is deprecated. Use `plot(; kw...)` instead.
│   caller = top-level scope at iv.qmd:246
└ @ Core ~/626/site/iv/iv.qmd:246

Simulated Distribution of $\hat{\beta}^{IV}$

Code

function tiv(y,X,Z; b0 = ones(size(X,2)))
  b = biv(y,X,Z)
  u = y - X*b
  V = var(u)*inv(X'*Z*inv(Z'*Z)*Z'*X)
  (b - b0)./sqrt.(diag(V))
end
n = 100
S = 10_000
plt = PlotlyLight.Plot()
for g in [1, 0.2, 0.1]
  b = [tiv(sim(n,d=1,EZu=0,gamma=g)...)[1] for _ in 1:S]
  # crop outliers so figure looks okay
  b .= max.(b, -4)
  b .= min.(b, 4)
  plt(x=b, type="histogram",name="γ=$g")
end
fig=plt(x=randn(S), type="histogram", name="Normal")

fig

┌ Warning: `Plot(; kw...)` is deprecated. Use `plot(; kw...)` instead.
│   caller = top-level scope at iv.qmd:280
└ @ Core ~/626/site/iv/iv.qmd:280

Irrelevant Instrument Asymptotics

Suppose $\Er[Z_i X_i'] = 0$
CLT $\frac{1}{\sqrt{n}} \begin{pmatrix} vec(Z'X) \\ Z'u \end{pmatrix} \indist \begin{pmatrix} \zeta_1 \\ \zeta_2 \end{pmatrix} \sim N(0, \Sigma)$
Then $\begin{align*} \hat{\beta}^{IV} - \beta_0 = & \left((Z'X)'(Z'Z)^{-1}(Z'X)\right)^{-1} (Z'X)'(Z'Z)^{-1}(Z'u) \\ \indist & \left(H' \Er[Z_i Z_i]^{-1} H\right)^{-1} \left(H \Er[Z_i Z_i']^{-1} \zeta_2\right) \end{align*}$ where $vec(H) = \zeta_1$

Identification Robust Inference

Opinion: always do this, testing for relevance not needed
Test $H_0: \beta = \beta_0$ vs $\beta \neq \beta_0$ with Anderson-Rubin test $AR(\beta) = n\left(\frac{1}{n} Z'(y-X\beta) \right)' \Sigma(\beta)^{-1} \left(\frac{1}{n} Z'(y - X\beta)\right)$ where $\Sigma(\beta) = \frac{1}{n} \sum_{i=1}^n Z_iZ_i' (y_i - X_i'\beta)^2$
$AR(\beta) \indist \chi^2_d$ (under either weak instrument or usual asymptotics)
See my other notes for simulations and references

Identification Robust Inference

Two downsides of AR test:
1. AR statistic is similar to over-identifying test ( $AR(\hat{\beta}^{IV}) = J$ )
- Small (even empty) confidence region if model is misspecified
1. Only gives confidence region for all of $\beta$ , not confidence intervals for single co-ordinates
Kleibergen’s LM and Moreira CLR tests address 1, see my other notes for simulations and references

LATE

Wald estimator $\frac{\Er[Y_i | Z_i=1] - \Er[Y_i|Z_i=0]}{\Er[D_i|Z_i=1] - \Er[D_i|Z_i=0]} = \frac{\Er[Y_i(D_i(1))] - \Er[Y_i(D_i(0))]}{\Er[D_i(1)] - \Er[D_i(0)]} \text{ (exogeneity)$

$= \frac{\Er[Y_i(D_i(1)) - Y_i(D_i(0)) | D_i(1) \neq D_i(0)] P(D_i(1) \neq D_i(0))} { P(D_i(1) > D_i(0)) - P(D_i(1) < D_i(0))}$

Assume monotonicity $P(D_i(1)<D_i(0)) = 0$ , then $\frac{\Er[Y_i | Z_i=1] - \Er[Y_i|Z_i=0]}{\Er[D_i|Z_i=1] - \Er[D_i|Z_i=0]} = \Er[Y_i(1) - Y_i(0) | D_i(1)=1, D_i(0) = 0 ]$
local average treatment effect

IV=LATE

With single binary $Z$ and $D$ , $\begin{align*} \hat{\beta}^{IV} & = \frac{\sum Y_i(Z_i - \bar{Z})} {\sum D_i(Z_i - \bar{Z})} \\ & \inprob \Er[Y_i(1) - Y_i(0) | D_i(1)=1, D_i(0) = 0 ] \end{align*}$

How general is this interpretation?
- Multi-valued $D$ ?
- Multi-values or multiple $Z$ ?
- Exogenous controls $X$ ?
Can salvage some LATE like intrepretation with multiple treatments or instruments, but monotonocity assumption needs to be stronger
- Mogstad and Torgovitsky (2024) for comprehensive review

2SLS with Controls

$\begin{align*} \hat{\beta}^{IV} = & \frac{\sum y_i \tilde{Z}_i}{\sum D_i \tilde{Z}_i} \\ \inprob & \frac{\Er[Y_i \tilde{Z}_i]}{\Er[D_i \tilde{Z}_i]} \\ = & \frac{\Er[\cov(Y_i \tilde{Z}_i|X_i)] + \Er[\Er[Y_i|X_i]\Er[\tilde{Z}_i|X_i]]}{\Er[D_i \tilde{Z}_i]} \end{align*}$ - If $\Er[\Er[Y_i|X_i]\Er[\tilde{Z}_i|X_i]] = 0$ , we get average of $X$ specific LATEs - But unless $\Er[Z_i|X_i]$ is linear, $\Er[\Er[Y_i|X_i]\Er[\tilde{Z}_i|X_i]] \neq 0$

2SLS with Controls is not LATE

Blandhol et al. (2022) show $\begin{align*} \beta^{IV} \inprob & \Er\left[\omega(cp,X)\Er[Y(1) - Y(0) |D(1)>D(0), X] \right] + \\ & + \Er\left[\omega(at,X)\Er[Y(1) - Y(0) |D(1)=D(0)=1, X] \right] \end{align*}$ with $\begin{align*} \omega(cp,X) = & \Er[Z|X](1 - L[Z|X])P(D(1)>D(0)|X)\Er[\tilde{Z}D]^{-1} \\ \omega(at,X) = & \Er[\tilde{Z}|X] P(D(1)=D(0)=1|X)\Er[\tilde{Z}D]^{-1} \end{align*}$
- $\Er[\tilde{Z}] = 0$ , so unless $\Er[\tilde{Z}|X] = 0$ , it will sometimes be negative

References

Andrews, Isaiah, James H. Stock, and Liyang Sun. 2019. “Weak Instruments in Instrumental Variables Regression: Theory and Practice.” Annual Review of Economics 11 (1): 727–53. https://doi.org/10.1146/annurev-economics-080218-025643.

Blandhol, Christine, John Bonney, Magne Mogstad, and Alexander Torgovitsky. 2022. “When Is TSLS Actually Late?”

Chernozhukov, V., C. Hansen, N. Kallus, M. Spindler, and V. Syrgkanis. 2024. Applied Causal Inference Powered by ML and AI. https://causalml-book.org/.

Guggenberger, Patrik, Frank Kleibergen, and Sophocles Mavroeidis. 2024. “A POWERFUL SUBVECTOR ANDERSON–RUBIN TEST IN LINEAR INSTRUMENTAL VARIABLES REGRESSION WITH CONDITIONAL HETEROSKEDASTICITY.” Econometric Theory 40 (5): 957–1002. https://doi.org/10.1017/S0266466622000627.

Keane, Michael, and Timothy Neal. 2023. “Instrument Strength in IV Estimation and Inference: A Guide to Theory and Practice.” Journal of Econometrics 235 (2): 1625–53. https://doi.org/https://doi.org/10.1016/j.jeconom.2022.12.009.

Lee, David S, Justin McCrary, Marcelo J. Moreira, and Jack Porter. 2022. “Valid t-Ratio Inference for IV.” American Economic Review 112 (10): 3260–90. https://doi.org/10.1257/aer.20211063.

Lee, David S, Justin McCrary, Marcelo J Moreira, Jack R Porter, and Luther Yap. 2023. “What to Do When You Can’t Use ’1.96’ Confidence Intervals for IV.” Working Paper 31893. Working Paper Series. National Bureau of Economic Research. https://doi.org/10.3386/w31893.

Londschien, Malte, and Peter Bühlmann. 2024. “Weak-Instrument-Robust Subvector Inference in Instrumental Variables Regression: A Subvector Lagrange Multiplier Test and Properties of Subvector Anderson-Rubin Confidence Sets.” https://arxiv.org/abs/2407.15256.

Mogstad, Magne, and Alexander Torgovitsky. 2024. “Instrumental Variables with Unobserved Heterogeneity in Treatment Effects.” Working Paper 32927. Working Paper Series. National Bureau of Economic Research. https://doi.org/10.3386/w32927.

Song, Kyunchul. 2021. “Introduction to Econometrics.”

Stock, James H, Jonathan H Wright, and Motohiro Yogo. 2002. “A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments.” Journal of Business & Economic Statistics 20 (4): 518–29. https://doi.org/10.1198/073500102288618658.

Stock, James H, and Motohiro Yogo. 2002. “Testing for Weak Instruments in Linear IV Regression.” Working Paper 284. Technical Working Paper Series. National Bureau of Economic Research. https://doi.org/10.3386/t0284.

Tuvaandorj, Purevdorj. 2024. “Robust Permutation Tests in Linear Instrumental Variables Regression.” Journal of the American Statistical Association 0 (ja): 1–24. https://doi.org/10.1080/01621459.2024.2412363.

Instrumental Variables Estimation Paul Schrimpf 2024-12-14

Instrumental Variables Estimation
Reading
Instrumental Variables
Model
Identification
Estimation
Method of Moments Estimation
Method of Moments Estimation
Method of Moments Estimation
Asymptotic Properties
Consistency
Asymptotic Normality
Optimal $W$
Two Stage Least Squares
Testing Overidentifying Restrictions
Testing Overidentifying Restrictions
Over-identifying Test
Weak Instruments
Simulated Distribution of $\hat{\beta}^{IV}$
Simulated Distribution of $\hat{\beta}^{IV}$
Weak Instruments
Irrelevant Instrument Asymptotics
Weak Instrument Asymptotics
Testing for Relevance
Testing for Relevance
Testing for Relevance
Identification Robust Inference
Identification Robust Inference
Identification Robust Inference
Further Reading
IV with Treatment Effect Heterogeneity
Model
LATE
IV=LATE
Controls
2SLS with Controls
2SLS with Controls is not LATE
Simulation: Low Bias
Simulation: Low Bias
Simulation: Low Bias
Simulation: High Bias
Observations
What to do?
Further Reading
References