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I want to generate data in R to solve the following problem:

Consider the following Data Generating Process (DGP) $Y_i = β_0 + β_1 · X_i + β_2 · Z_i + u_i$ , where $β_0 = 0.75$, $β_1 = 0.50$ and $β_2 = 0.25$. Imagine that somebody wishes to obtain a proper (unbiased and consistent) estimate for $β_1$ and decides to estimate $Y_i = b_0 + b_1 · X_i + v_i$ , instead of the true DGP.

Write the R code to show that:

  1. when Xi and Zi are uncorrelated, the OLS estimator for b1 is unbiased and consistent;

  2. when Xi and Zi are positively correlated, the OLS estimator for b1 is upward biased and inconsistent;

  3. when Xi and Zi are negatively correlated, the OLS estimator for b1 is downward biased and inconsistent.

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    $\begingroup$ Is this a question from a course or textbook? If so, please add the [self-study] tag & read its wiki. $\endgroup$ – deemel Dec 7 '19 at 15:55
  • $\begingroup$ It is not from a text book. Is an exercise for us to learn how to generate data with similar characteristics. I am just interested in the data generating process given a specific model as a general indication for R. I will add the tag! thanks for the repply $\endgroup$ – Daniel Ortiz Dec 7 '19 at 19:21
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I would use DeclareDesign for the simulation and bookkeeping, and lm to estimate the regression parameters

require(DeclareDesign)

N = 20
r = .2
b0 = 0.75
b1 = 0.50
b2 = 0.25
s = 1

design <- declare_population(N=N, X = rnorm(N), 
                                  Z = rnorm(N, r*X, sd=sqrt(1-r*r)), 
                                  Y = b0 + b1*X + b2 * Z +rnorm(N, sd=s)) +
          declare_estimands(b1=b1) +
          declare_estimators(Y~X, model=lm, term='X', estimand='b1')

diagnostics <- diagnose_design(redesign(design, r=-9:9/10), sims = 300, bootstrap_sims=0)

The actual amount of bias observed will of course also depend on the sample size, correlation, and variance of the error term.

For N=20 and s=1, we can plot the bias over r. Also note that without a large number of simulations, the results can be rather noisy.

bias over r

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