# Instrumental variable with measurement error

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?

• Is the measurement error of $X$ related to its endogeneity? Is the cause for the measurement error in $Z$ the same as in $X$? Commented May 4 at 18:48
• 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. Commented May 5 at 6:22

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.
• Thanks so much for the analytic answer. This is really helpful Commented May 10 at 7:06
• May I ask a follow up question? What would it mean if measurement errors m_z were autocorrelated (recall my data is panel)? Commented May 10 at 17:38
• No problem. I've tried to address the panel dimension now. Commented May 13 at 8:22