Measurement error in independent variable May I know how do we prove that measurement error in the independent variable will lead to biased estimators? What assumptions do we need to have?
 A: The question concerns how to prove this result.  The necessary assumptions will become clear over the course of the analysis.
Let's start with some intuition afforded by the simplest (non-trivial) case of regression, where the true values of $x$ take on just two possible values $\xi_0$ and $\xi_1$ and corresponding to those are the values $\eta_0$ and $\eta_1,$ respectively.  The observed values of the response $y$ depart randomly from the $\eta_i.$  This situation is depicted by the points in the left hand scatterplot:

The scatterplot depicts observations made with no error in $x$ (so all points line up above $x=\xi_0$ and $x=\xi_1$) and all the variation is the vertical ($y$) direction.  This is the usual linear regression model.  The line shows the ordinary least squares (OLS) fit.  It is unbiased: this means that on average the OLS line passes through the points $(\xi_0,\eta_0)$ and $(\xi_1,\eta_1)$ (shown as large solid dots).
Measurement errors in $x$ displace those points randomly and horizontally, as shown in the middle picture.  That results in the scatterplot at right.
Randomly moving the $x$ values has changed the least squares fit.  Why?  Mainly because the slope is most strongly influenced by the points at the upper right (coming from positive errors in measuring $\xi_1$) and the points at the lower left (coming from negative errors in measuring $\xi_0$).  The other points pretty much cancel each other out, since they are scattered randomly through the middle of the scatterplot.  On average, the ordinary least squares estimate will have a slope that is smaller in magnitude: it is therefore is biased.
A similar explanation applies more generally when there are more than two values underlying the $x_i.$

Proving this result is now straightforward: we ought to be able to see the same phenomenon in the algebra.  Rather than going through this formally, let me sketch the underlying idea.  Recall that the slope estimate is the average product of the standardized variables.  When you include measurement errors in the first variable, that

*

*Increases the variance of the $x_i.$  Thus, on the whole, expect the standardized values of the $x_i$ to become smaller in size (because standardization divides all values by the square root of the variance).


*Does not change the variance of the $y_i$ (obviously).


*Does not introduce any more or less correlation, because the errors in the $x_i$ are assumed to be independent of those in the $y_i.$  (Apart from the usual linear regression assumptions, this is the only additional assumption.)
Consequently, the average product of the standardized variables has a tendency to decrease (due to #1), QED.
A: The proof for the classical errors-in-variables model is demonstrated, for example, in this educational reference Lecture Notes on Measurement Error.
Simply, in words to quote, where $\hat{x}$ is a random variable representing the measured value of the independent variable (which can randomly differ from its unobserved true underlying value of the independent variable ${x}$):

The measurement error in $\hat{x}$ becomes part of the error term in the regression equation thus creating an endogeneity bias. Since $\hat{x}$ and u are positively correlated (from (2)) we can see that OLS estimation will lead to a negative bias $\hat{\beta}$ if the true ${\beta}$ is positive and a positive bias if β is negative.

Relatedly, comments on endogeneity per Wikipedia, to quote:

In econometrics, endogeneity broadly refers to situations in which an explanatory variable is correlated with the error term.[1] The distinction between endogenous and exogenous variables originated in simultaneous equations models, where one separates variables whose values are determined by the model from variables which are predetermined;[2][3] ignoring simultaneity in the estimation leads to biased estimates as it violates the exogeneity assumption of the Gauss–Markov theorem. The problem of endogeneity is unfortunately, oftentimes ignored by researchers conducting non-experimental research and doing so precludes making policy recommendations.[4]

See also discussion in Wikipedia on errors-in-variables models (also referred to as measurement error models) where, to quote:

In the case when some regressors have been measured with errors, estimation based on the standard assumption leads to inconsistent estimates, meaning that the parameter estimates do not tend to the true values even in very large samples.

Remember in the simple classical regression model, the x-variable is fixed and known (not a random variable subject to any form of error that is rolled into the model's error term).
