Instrumental Variables Estimation

Paul Schrimpf

2024-10-21

Reading

• Required: Song (2021) chapter 12

\[ \def\Er{{\mathrm{E}}} \def\En{{\mathbb{E}_n}} \def\cov{{\mathrm{Cov}}} \def\var{{\mathrm{Var}}} \def\R{{\mathbb{R}}} \newcommand\norm[1]{\left\lVert#1\right\rVert} \def\rank{{\mathrm{rank}}} \newcommand{\inpr}{ \overset{p^*_{\scriptscriptstyle n}}{\longrightarrow}} \def\inprob{{\,{\buildrel p \over \rightarrow}\,}} \def\indist{\,{\buildrel d \over \rightarrow}\,} \DeclareMathOperator*{\plim}{plim} \]

Instrumental Variables

Model

\[ Y_i = \underbrace{X_i}_{\in \R^k}' \beta_0 + u_i \]

• \(\Er[u_i] = 0\), but \(\Er[X_i u_i] \neq 0\)

• Instrument \(Z_i \in \R^d\) s.t.

  1. Relevant \(rank(\Er[Z_i X_i']) = k\)

  2. Exogenous \(\Er[Z_i u_i] = 0\)

Identification

• Exogeneity implies \[ \Er[Z_i Y_i] = \Er[Z_i X_i']\beta_0 \]

• If \(d=k\) (exactly identified), then relevance implies \(\Er[Z_i X_i']\) invertible, so \[ \beta_0 = \Er[Z_i X_i']^{-1} \Er[Z_i Y_i] \]

• For \(d>k\), relevance implies \(\Er[Z_iX_i']'\Er[Z_iX_i']\) invertible, so \[ \beta_0 = (\Er[Z_i X_i]' \Er[Z_i X_i'])^{-1} \Er[Z_i X_i']' \Er[Z_i Y_i] \]

Estimation

Method of Moments Estimation

• We assume \(\Er[Z_i u_i] = 0\), so \[ \Er[Z_i(Y_i - X_i'\beta_0)] = 0 \]
• Estimate by replacing \(\Er\) with \(\frac{1}{n}\sum_{i=1}n\)
• \(d\) equations, \(k \geq d\) unknowns, so find \[ \frac{1}{n} \sum_{i=1}^n Z_i(Y_i - X_i'\hat{\beta}^{IV}) \approx 0 \] by solving \[ \begin{align*} \hat{\beta}^{IV} & = \mathrm{arg}\min_\beta \norm{ \frac{1}{n} \sum_{i=1}^n Z_i(Y_i - X_i'\beta) }_{W}^2 \\ & = \mathrm{arg}\min_\beta \left( \frac{1}{n} \sum_{i=1}^n Z_i(Y_i - X_i'\beta\right)' W \left( \frac{1}{n} \sum_{i=1}^n Z_i(Y_i - X_i'\beta\right) \end{align*} \]

Method of Moments Estimation

\[ \hat{\beta}^{IV} = \mathrm{arg}\min_\beta \left( \frac{1}{n} \sum_{i=1}^n Z_i(Y_i - X_i'\beta\right)' W \left( \frac{1}{n} \sum_{i=1}^n Z_i(Y_i - X_i'\beta\right) \]

• \(\hat{\beta}^{IV}_W = (X'Z W Z'W)^{-1}(X'Z W Z'y)\)

Asymptotic Properties

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

1. these are stronger than needed, for LLN, but needed for CLT

Asymptotic Normality

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

• \(\sqrt{n}(\hat{\beta}^{IV} - \beta_0) \indist N(0, V)\) 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 \(\frac{1}{\sqrt{n}} \sum Z_i u_i \indist N(0, \sigma^2 \Er[Z_iZ_i'])\)
  • \(V = \sigma^2 (\Er[Z_iX_i']' W \Er[Z_iX_i'])^{-1} (\Er[Z_iX_i']' W \Er[Z_i Z_i'] W \Er[Z_i X_i']) (\Er[Z_iX_i']' W \Er[Z_iX_i'])^{-1}\)

Optimal \(W\)

Theorem 2.1

\(W^* = \Er[Z_iZ_i']^{-1}\) minimizes the asymptotic variance of \(\hat{\beta}^{IV}_W\)

• Estimate \(\hat{W}^* = \left(\frac{1}{n} Z'Z\right)^{-1}\) \[ \hat{\beta}^{IV} = (X'Z (Z'Z)^{-1} Z' X)^{-1} (X'Z(Z'Z)^{-1}Z'y) \]

Two Stage Least Squares

\[ \begin{align*} \hat{\beta}^{IV} & = (X'Z (Z'Z)^{-1} Z' X)^{-1} (X'Z(Z'Z)^{-1}Z'y) \\ & = (X'P_Z X)^{-1} (X' P_Z y) \\ & = ((P_Z X)'(P_Z X))^{-1} ((P_Z X)'y) \end{align*} \]

1. Regress \(X\) on \(Z\), let \(\hat{X} = P_Z X\)
2. Regress \(y\) on \(\hat{X}\)

Testing Overidentifying Restrictions

• \(H_0: \Er[Z_i(Y_i - X_i'\beta_0)] = 0\)
• \(k=d\), have \(\En[Z_i(Y_i - X_i'\hat{\beta}^{IV})] = 0\) exactly, and \(H_0\) is untestable
• \(k>d\), can test
• Test statistic \[ J = n \left(\frac{1}{n} Z'(y-X\hat{\beta}^{IV}) \right)' \hat{C} \left(\frac{1}{n} Z'(y-X\hat{\beta}^{IV}) \right) \]

Testing Overidentifying Restrictions

Theorem 2.3

Let \(\hat{C} = \left(\frac{1}{n} \sum_{i=1}^n Z_iZ_i' \hat{u}_i^2\right)^{-1}\). Assume:

1. \(\Er[ \norm{X_i}^4] + \Er[\norm{Z_i}^4] < \infty\)

2. \(\Er[u|Z] = \sigma^2\)

3. \(\Er[Z_i Z_i']\) is positive definite

Then, \[ J \indist \chi^2_{d-k} \]

Over-identifying Test

• Only has power when instruments have different covariances with \(u\)

using PlotlyLight, Distributions, LinearAlgebra
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 = 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

Weak Instruments

Simulated Distribution of \(\hat{\beta}^{IV}\)

• First stage \(X = Z\gamma + e\), simulation with \(\Er[Z_i Z_i] = I\) and \(e \sigma N(0,0.25)\), so first stage \(t \approx \sqrt{n}\gamma/0.5\)

• Distribution of \(\hat{\beta}^IV\) with \(\gamma = 1\), \(\gamma=0.2\), and \(\gamma=0.1\)

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 = 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

Weak Instruments

• Lessons from simulation:
  • When \(\Er[Z_i X_i']\) is small, usual asymptotic distribution is a poor approximation for the finite sample distribution of \(\hat{\beta}^{IV}\)
  • The approximation can be poor even when \(H_0: \gamma = 0\) in \(X = Z\gamma + e\) would be rejected
• Can we find a better approximation to the finite sample distribution when \(\Er[Z_i X_i']\) is small?

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 \(vec(H) = \zeta_1\)

Weak Instrument Asymptotics

• Let \(\Er[Z_i X_i'] = \frac{1}{\sqrt{n}} \Gamma\)
• Then \(\frac{1}{\sqrt{n}} Z' X = \Gamma + H\)
• and \[ \hat{\beta}^{IV} - \beta_0 \indist \left((\Gamma + H)' \Er[Z_i Z_i]^{-1} H\right)^{-1} \left((\Gamma + H) \Er[Z_i Z_i']^{-1} \zeta_2\right) \]
• \(\Gamma\) cannot be estimated, but we can try to develop estimators and inference methods for \(\beta\) that work for any \(\Gamma\)

Testing for Relevance

• Model , assume \(\Er[W_i u_i] = 0\) and \(\Er[Z_i u_i] = 0\) \[ Y_i = X_i'\beta + W_i'\beta_W + u_i \]
• First stage \[ X_i = Z_i' \pi_z + W_i' \pi_W + \nu_i \]
• Can test \(H_0 : \pi_z = 0\) vs \(H_1 : \pi_z \neq 0\) using F-test
  • With one instrument, \(F = t^2\)
  • Rejecting \(H_0\) at usual significance level is not enough for \(\hat{\beta}^{IV}\) to be well aproximated by its asymptotic normal distribution

Testing for Relevance

• Stock and Yogo (2002) (table from Stock, Wright, and Yogo (2002)): first stage F > threshold \(\approx 10\) implies \(Bias(\hat{\beta}^{IV}) < 10\% Bias(\hat{\beta}^{OLS})\) and size of 5% test < 15%

swy-tab1.png

Testing for Relevance

• David S. Lee et al. (2022) : F\(>>10\) is needed in practice¹

1. The argument here is that practitioners misuse the \(F>10\) threshold, not that Stock and Yogo (2002) is wrong.

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
  2. 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

• Various approaches to 2 see Andrews, Stock, and Sun (2019) for a review

• If you want something close to the usual t-test and have 1 endogenous regression and 1 instrument, the tF test from David S. Lee et al. (2022), or better yet, recently improved VtF test in David S. Lee et al. (2023)

Further Reading

• Recent reviews:
  • Andrews, Stock, and Sun (2019)
  • Keane and Neal (2023)

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.

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.

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.