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Here is some rough code in R implementing the non-linear two stage least squares estimator under the assumption that $\mathbb E[\epsilon\lvert x] = 0$. Just to see if it works ... it seems to be working up to the point of standard over-underflow issues there might be. But the non-linear two stage least squares estimator should be implemented in most ...

6

No. $R^2$ in instrumental variables regression is not useful. Since one of the explanatory variables $x$ is correlated with the error $\epsilon$ we can't decompose the variance of the outcome $y$ into $\beta^2 Var(x) + Var(\epsilon)$, so the obtained $R^2$ has neither a natural interpretation, nor can it be used for computation of F statistics. ...

3

The issue here is that equation $(1)$ does not fully specify the regression model, so yes, that equation is an incomplete statement of the model. In order to fully specify the regression model, you need to specify the distribution of $e$ conditional on $x$. For a homoscedastic Gaussian linear regression model, the defining equations are: $$y = a + b x + e ... 3 For time dependent regressors, it is pretty straightforward. Many classes of time series models can handle them, including from the ARIMA family (ex: ARIMAX and regression with ARIMA errors), BSTS, Facebook Prophet, and others. The tricky part is time independent regressors: Most people don't realize that time independent regressors are of no use ... 2 Start by defining parameters \beta_0 and \beta_1. Then select a distribution F for the error term \epsilon_t, for example a normal distribution in which case you have to specify mean 0 and then select some value for the variance \sigma^2. Then choose some value T number of timeperiods to simulate and some initial value for the process y_0 and ... 1 U is an orthonormal matrix, with columns having norm 1 and orthogonal to each other. The solution is not unique, so coming up with a solution should suffice for you I guess, if there are no other restrictions. You can create a random matrix, and then orthonormalize its columns using gram-schmidt process. The following R script does it for you: library('... 1 You could do that, or you could use a method that weights the composites (on both sides) based on the relationships between predictor and outcome variables.Back in the day, Jacob Cohen (1982) described what he called "set correlation," a generalization of regression that allowed the LHS to include a set of variables. However, there are a range of techniques ... 1 Note that omitted variable bias is only occuring when the regressors itself are correlated with eachother and the dependent variable. To answer your question: The coefficient is larger because the change in variation is unrightfully and indirectly attributed to the single regressor. I use the term indirectly, because the regression coefficient is inflated ... 1 First of all, it is SSR, and using regression coefficient estimates, we can calculate estimated target values: \hat y_i = \hat \beta x_i+\hat \beta_0, then substitute into the formulation of SSR:$$\begin{align}\text{SSR}&=\sum_{i=1}^n(\hat y_i-\bar y)^2=\sum_{i=1}^n \hat y_i^2 -2 \bar y\sum_{i=1}^n \hat y_i+\underbrace{\sum_{i=1}^n \bar y^2}_{\text{...

1

Exogeneity won't hold if the correlation between z and x is nonzero and $b\not=0$ because then the error will be correlated to x in model 2 with the omitted z (which is now part of the error). If the true value of b is zero then you would also still have exogeneity. Deriving the estimator of a in both cases might help you see this.

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The R package ivmodel implements the LIML estimator (link here).

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