4
$\begingroup$

I have a question related to measurement error bias and the use of instrumental variables. Assume I have a continuous outcome variable $Y$ which is observed with no error and an endogenous (continuous) variable $X$ with measurement error. Let $Z$ be a (continuous) instrument of $X$ which satisfies the relevance and exogeneity conditions but $Z$ is also observed with measurement error. My data is panel and I use fixed effects.

Can I use $Z$ as an instrument on a two-stage least squares (2SLS) IV model and under what conditions?

$\endgroup$
2
  • $\begingroup$ Is the measurement error of $X$ related to its endogeneity? Is the cause for the measurement error in $Z$ the same as in $X$? $\endgroup$
    – Durden
    Commented May 4 at 18:48
  • $\begingroup$ X is endogenous not only because of its measurement error but also because of omitted variable. The measurement error in Z is different from that of X. $\endgroup$
    – Panagiotis
    Commented May 5 at 6:22

1 Answer 1

2
$\begingroup$

With classical (i.e., independent and additive) measurement error, the attenuation bias in the first stage and reduced form will cancel each other out.

\begin{align*} Y &= \beta X + u &\quad &(\text{Structural equation}) \\ X &= \theta Z + v &\quad &(\text{First stage})\\ Y &= \phi Z + \varepsilon &\quad &(\text{Reduced form}) \end{align*}

where $\varepsilon = \beta v + u $ and $\phi = \beta\theta$.

\begin{align*} \theta &\neq 0 &\quad &(\text{Relevance}) \\ Cov(Z, u) &= 0 &\quad &(\text{Validity}) \end{align*}

Denote observed $X$ by $\tilde{X}$ and observed $Z$ by $\tilde{Z}$

$$ \tilde{X} = X + m_x $$ $$ \tilde{Z} = Z + m_z $$

where $m_x$ and $m_z$ are measurement errors that are assumed to be independent of each other, $Z$, $u$, and $v$. Regressing $Y$ and $\tilde{X}$ on $\tilde{Z}$ by OLS

$$ \hat{\phi}_{OLS} \overset{p}{\to} \frac{Cov(\tilde{Z},Y)}{Var(\tilde{Z})} = \frac{Cov(Z + m_z, \beta\theta Z + \beta v + u)}{Var(Z + m_z)} = \frac{\beta\theta Var(Z) + \beta Cov(Z,v)}{Var(Z) + Var(m_z)} $$

$$ \hat{\theta}_{OLS} \overset{p}{\to} \frac{Cov(\tilde{Z},\tilde{X})}{Var(\tilde{Z})} = \frac{Cov(Z + m_z, \theta Z + v + m_x)}{Var(Z + m_z)} = \frac{\theta Var(Z) + Cov(Z,v)}{Var(Z) + Var(m_z)} $$

Even though neither the first stage nor the reduced form are estimated consistently, the IV estimator for $\beta$ is still consistent becasue both the first stage and reduced form are biased in the same way

$$\hat{\beta}_{IV} = \frac{\hat{\phi}_{OLS}}{\hat{\theta}_{OLS}} \overset{p}{\to} \frac{\beta\left(\theta Var(Z) + Cov(Z,v)\right)/\left(Var(Z) + Var(m_z)\right)}{\left(\theta Var(Z) + Cov(Z,v)\right)/\left(Var(Z) + Var(m_z)\right)} = \beta $$

Assuming $Cov(Z, v) = 0$ to focus only on the role of the measurement error, we have

$$ \hat{\phi}_{OLS} \overset{p}{\to} \beta\theta\lambda$$ $$ \hat{\theta}_{OLS} \overset{p}{\to} \theta\lambda$$ $$ \lambda = \frac{Var(Z)}{Var(Z) + Var(m_z)}$$

The first stage and reduced form will be biased towards zero by the same degree, becasue $0 \leq \lambda \leq 1$.

To investigate how the panel-dimension of your data and your use of fixed effects will influence this result, you'll have to add subscripts (unit $i$, period $t$; $N$ units, $T$ periods) and within transform the variables. For example, within-transformed $Y_{it}$ (with unit fixed effects) is $Y_{it} - \bar{Y}_i$, where $\bar{Y}_i = \frac{1}{T} \sum_1^T Y_{it}$. Then write the probability limit of the reduced form fixed-effects estimator as

$$ \hat{\phi}_{FE} \overset{p}{\to} \frac{Cov(\tilde{Z}_{it} -\bar{\tilde{Z}}_{i},Y_{it} - \bar{Y}_{i})}{Var(\tilde{Z}_{it} -\bar{\tilde{Z}}_{i})} = \frac{Cov(Z_{it} + m_{z,it} -\bar{Z}_{i} - \bar{m}_{z,i},\beta\theta Z_{it} + \beta v_{it} + u_{it} - \beta\theta \bar{Z}_{i} - \beta\bar{v}_i - \bar{u}_i)}{Var(Z_{it} + m_{z,it} -\bar{Z}_{i} - \bar{m}_{z,i})} $$

If you exapand, the numerator will turn into a straigtforward but very long sum of covariances. Some of these might be cancelled by corresponding terms in the first-stage estimator, but others will have to be zero by assumption. Compared to the cross-section case, it will be necessary to assume that $Z_{it}$ is uncorrelated with $u_{is}$ for all $s = 1, ..., T$ (strict exogeneity).

If you're not assuming classical measurement errors, you'll have to determine what kind of dependence is alright and what is not. If I'm not mistaken, autocorrelation in $m_{z,it}$ should not matter, becasue $m_{z,it}$ only shows up to the left in the covariance expression. My intuition is that both $Z_{it}$ and $m_{z,it}$ have to be strictly exogenous with respect to $u_{it}$ and $m_{x,it}$.

REFERENCES

  • Ashenfelter, O., & Krueger, A. (1994). Estimates of the economic return to schooling from a new sample of twins. The American Economic Review, 1157-1173. https://www.jstor.org/stable/2117766
  • Angrist, J. D., & Pischke, J. S. (2014). Mastering 'Metrics: The path from cause to effect. Princeton university press. Appendix, Chapter 6.
$\endgroup$
3
  • $\begingroup$ Thanks so much for the analytic answer. This is really helpful $\endgroup$
    – Panagiotis
    Commented May 10 at 7:06
  • $\begingroup$ May I ask a follow up question? What would it mean if measurement errors m_z were autocorrelated (recall my data is panel)? $\endgroup$
    – Panagiotis
    Commented May 10 at 17:38
  • $\begingroup$ No problem. I've tried to address the panel dimension now. $\endgroup$
    – Jonathan
    Commented May 13 at 8:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.