Consider a linear model with unobserved beterogeneity (q) and measurement error in an explanatory variable:

y = B0 + B1x1 + ... + Bkx*k + q + v

where ek = xk - x*k is the measurement error and we set the coefficient on q equal to one without loss of generality. The variable q might be correlated with any of the explanatory variables, but an indicator, q1= 8o + 81q + a1, is available. The measurement error ek might be correlated with the observed measure, xk. In addition to q1, you also have variables z1,z2; ...zM, M > or equal to 2, that are uncorrelated with v, a1, and ek.

A.) Suggest an IV procedure for consistently estimating the Bj. Why is M > or equal to 2 required?

To consistently estimate the coefficients Bj in the presence of measurement error and unobserved heterogeneity, you can use an Instrumental Variables (IV) procedure. The IV approach relies on finding variables that are correlated with the endogenous explanatory variable (x) but are not directly associated with the error term (v). These instruments are used to identify the causal relationship between x and y.

In this case, the variable q1 serves as a potential instrument for the endogenous variable q. This instrument can help address the potential correlation between q and other explanatory variables. By using q1 as an instrument, you can obtain consistent estimates of the coefficients Bj in the linear model.

Now, let's discuss why M ≥ 2 is required. In IV estimation, it is crucial to have at least as many instruments as the number of endogenous regressors. In this model, the endogenous regressor is q, and you need at least one instrument to consistently estimate its coefficient. However, if you want to estimate multiple coefficients Bk for each xk, you require more than one instrument. Hence, M ≥ 2 ensures that you have enough instruments to consistently estimate all the coefficients in the linear model.

In summary, the suggested IV procedure involves using q1 as an instrument for q to consistently estimate the coefficients Bj. The requirement M ≥ 2 ensures that there are enough instruments available to estimate each relevant coefficient in the model.