Instrumental Variables Assumptions, Tests, and Interpretations

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# Instrumental Variables Assumptions, Tests, and Interpretations
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### Ian McCarthy | Emory University
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### Econ 771, Fall 2022
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# Instrumental Variables Assumptions

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# Key IV assumptions
1. *Exclusion:* Instrument is uncorrelated with the error term<br>

2. *Validity:* Instrument is correlated with the endogenous variable<br>

3. *Monotonicity:* Treatment more (less) likely for those with higher (lower) values of the instrument<br>

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Assumptions 1 and 2 sometimes grouped into an *only through* condition.

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# Exclusion

Kippersluis and Rietveld (2018), "Beyond Plausibly Exogenous"
- "zero-first-stage" test
- Focus on subsample for which your instrument is not correlated with the endogenous variable of interest
    1. Regress the outcome on all covariates and the instruments among this subsample
    2. Coefficient on the instruments captures any potential direct effect of the instruments on the outcome (since the correlation with the endogenous variable is 0 by assumption).

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# Exclusion

Beckert (2019), "A Note on Specification Testing in Some Structural Regression Models"
- With at least `$n$` valid instruments, test if all instruments are valid against the alternative that up to `$m - n$` instruments are valid
    1. Estimate the first-stage regressions and save residuals, denoted `$\hat{u}$`. 
    2. Estimate the "artificial" regression `$$y=\beta x + \delta \tilde{z} + \gamma \hat{u} + \varepsilon$$` where `$\tilde{z}$` denotes a subset of `$m-n$` instruments from the full instrument matrix `$z$`. 
    3. Test the null that `$\delta=0$` using a standard F-test

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# Solving and Exclusion Problem

Conley et al (2010) and "plausible exogeneity", union of confidence intervals approach

- Suppose extent of violation is known in `$y_{i} = \beta x_{i} + \gamma z_{i} + \varepsilon_{i}$`, so that `$\gamma = \gamma_{0}$`
- IV/TSLS applied to `$y_{i} - \gamma_{0}z_{i} = \beta x_{i} + \varepsilon_{i}$` works
- With `$\gamma_{0}$` unknown...do this a bunch of times!
    - Pick `$\gamma=\gamma^{b}$` for `$b=1,...,B$`
    - Obtain `$(1-\alpha)$` % confidence interval for `$\beta$`, denoted `$CI^{b}(1-\alpha)$`
    - Compute final CI as the union of all `$CI^{b}$`

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# Solving an Exclusion Problem

Nevo and Rosen (2012) ReStat:

`$$y_{i} = \beta x_{i} + \delta D_{i} + \varepsilon_{i}$$`

- Allow instrument, `$z$`, to be correlated with `$\varepsilon$`, but `$|\rho_{x, \varepsilon}| \geq |\rho_{z, \varepsilon}|$`
- i.e., IV is better than just using the endogenous variable
- Assume `$\rho_{x, \varepsilon} \times \rho_{z, \varepsilon} >0$` (same sign of correlation in the error)
- Denote `$\lambda = \frac{\rho_{z, \varepsilon}}{\rho_{x, \varepsilon}}$`, then valid `$IV$` would be `$V(z) = \sigma_{x} z - \lambda \sigma_{z} x$`
- Can bound `$\beta$` using range of `$\lambda$`

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# Validity and Weak Instruments

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# Validity

Just says that your instrument is correlated with the endogenous variable, but what about the **strength** of the correlation?

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# Why we care about instrument strength

Recall our schooling and wages equation, `$$y = \beta S + \epsilon.$$` Bias in IV can be represented as:

`$$Bias_{IV} \approx \frac{Cov(S, \epsilon)}{V(S)} \frac{1}{F+1} = Bias_{OLS} \frac{1}{F+1}$$`

- Bias in IV may be close to OLS, depending on instrument strength
- **Bigger problem:** Bias could be bigger than OLS if exclusion restriction not *fully* satisfied