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I have the following regression $$y_i=a_1+b_1x_i+c_1z_i+d_1w_i$$ $$x_i=a_2+b_2y_i+c_2z_i+d_2w_i$$

a, b, c, d are coefficients. x is endogenous.z and w are exogenous to errors. In this simultaneous regressions, I determined the variable z as an instrument for y in the regression x and the variable w as an instrument for x in the regression y. And then I randomly generate first, errors and second z and w variables. And then by using them and reduced form, I obtain x and y. However, now I need to change instruments' strength. I need to make instruments weak. My question:

how can I generate a weak instrument z? (valid but weak) in R.

I don't want any code.

Just give me a hint.

what I guess is that

For that, I need to do this calculation $y=\delta/\sqrt{n}z$.But in this way, I need to change (increase) its sample size(n) such that instrumental variable become weak. I don't want to change n in fact. Please suggest different ways. Thank you

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You can make the instrument irrelevant/weak by shrinking its coefficient to near zero so that is has little effect on the endogenous variable.

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  • $\begingroup$ that's, $\delta$ become close to zero, professor ? $\endgroup$
    – Example
    Commented Jan 31, 2018 at 23:44
  • $\begingroup$ I don't know what $\delta$ is here, so I can't help. $\endgroup$
    – dimitriy
    Commented Jan 31, 2018 at 23:46
  • $\begingroup$ okay I understand professor. Thanks a lot again! :) $\endgroup$
    – Example
    Commented Jan 31, 2018 at 23:50

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