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I am concerned with simulating data for a linear regression model. I need to control the means, variances, and correlations (covariances) between the predictors and the criterion variable. In addition, I need to be able to vary the explained variances ($$R^2$$). It is obvious to me that the latter must be a function of the earlier, so at least one correlation (covariance) in $$\Sigma$$ is perhaps dependent on the choice of $$R^2$$, where $$\Sigma=(Y,X)(Y,X)^T$$$$\Sigma=E((Y,X)(Y,X)^T)$$, for centered $$X$$ and $$Y$$, is the variance-covariance matrix of all variables.

My plan is thus as follows:

1. Specify $$\Sigma$$, means, and $$R^2$$
2. Simulate data with these sufficient statistics, e.g. by sampling from the multivariate normal
3. Check estimated $$\beta$$ (regression coefficient) vector against population (theoretical) coefficients and use the model for unrelated tests/science.

Hence, my approach does not suggest specifying $$\beta$$ but letting the coefficients be a function of population $$\Sigma$$, means, and $$R^2$$. The reason I need to do this is to attribute some realisitc scale to $$X$$ and $$Y$$ (e.g., let $$Y$$ assume an 'income' scale and give $$X$$ a realistic scale for years of eduction). Therefore, I specify the sufficient statistics instead of regression coefficients. But maybe there is a better way.

Moreover, I have two specific questions:

1. Given the population variance-covariance matrix $$\Sigma$$ of one criterion variable $$Y$$ and a series of predictors (covariates) $$X$$ , I would like to calculate the vector of true population regression coefficients. Of course, I could simulate data $$X$$ and $$Y$$ and use the OLS estimator, but there should be a direct way to use $$\Sigma$$ in the estimation of population $$\beta$$?

2. Which options are there to specify covariances (correlations) in $$\Sigma$$ given I need a fixed $$R^2$$ of a linear regression of $$Y$$ on $$X$$? This, to systematically vary the explanatory power of the regression model.

I am concerned with simulating data for a linear regression model. I need to control the means, variances, and correlations (covariances) between the predictors and the criterion variable. In addition, I need to be able to vary the explained variances ($$R^2$$). It is obvious to me that the latter must be a function of the earlier, so at least one correlation (covariance) in $$\Sigma$$ is perhaps dependent on the choice of $$R^2$$, where $$\Sigma=(Y,X)(Y,X)^T$$, for centered $$X$$ and $$Y$$, is the variance-covariance matrix of all variables.

My plan is thus as follows:

1. Specify $$\Sigma$$, means, and $$R^2$$
2. Simulate data with these sufficient statistics, e.g. by sampling from the multivariate normal
3. Check estimated $$\beta$$ (regression coefficient) vector against population (theoretical) coefficients and use the model for unrelated tests/science.

Hence, my approach does not suggest specifying $$\beta$$ but letting the coefficients be a function of population $$\Sigma$$, means, and $$R^2$$. The reason I need to do this is to attribute some realisitc scale to $$X$$ and $$Y$$ (e.g., let $$Y$$ assume an 'income' scale and give $$X$$ a realistic scale for years of eduction). Therefore, I specify the sufficient statistics instead of regression coefficients. But maybe there is a better way.

Moreover, I have two specific questions:

1. Given the population variance-covariance matrix $$\Sigma$$ of one criterion variable $$Y$$ and a series of predictors (covariates) $$X$$ , I would like to calculate the vector of true population regression coefficients. Of course, I could simulate data $$X$$ and $$Y$$ and use the OLS estimator, but there should be a direct way to use $$\Sigma$$ in the estimation of population $$\beta$$?

2. Which options are there to specify covariances (correlations) in $$\Sigma$$ given I need a fixed $$R^2$$ of a linear regression of $$Y$$ on $$X$$? This, to systematically vary the explanatory power of the regression model.

I am concerned with simulating data for a linear regression model. I need to control the means, variances, and correlations (covariances) between the predictors and the criterion variable. In addition, I need to be able to vary the explained variances ($$R^2$$). It is obvious to me that the latter must be a function of the earlier, so at least one correlation (covariance) in $$\Sigma$$ is perhaps dependent on the choice of $$R^2$$, where $$\Sigma=E((Y,X)(Y,X)^T)$$, for centered $$X$$ and $$Y$$, is the variance-covariance matrix of all variables.

My plan is thus as follows:

1. Specify $$\Sigma$$, means, and $$R^2$$
2. Simulate data with these sufficient statistics, e.g. by sampling from the multivariate normal
3. Check estimated $$\beta$$ (regression coefficient) vector against population (theoretical) coefficients and use the model for unrelated tests/science.

Hence, my approach does not suggest specifying $$\beta$$ but letting the coefficients be a function of population $$\Sigma$$, means, and $$R^2$$. The reason I need to do this is to attribute some realisitc scale to $$X$$ and $$Y$$ (e.g., let $$Y$$ assume an 'income' scale and give $$X$$ a realistic scale for years of eduction). Therefore, I specify the sufficient statistics instead of regression coefficients. But maybe there is a better way.

Moreover, I have two specific questions:

1. Given the population variance-covariance matrix $$\Sigma$$ of one criterion variable $$Y$$ and a series of predictors (covariates) $$X$$ , I would like to calculate the vector of true population regression coefficients. Of course, I could simulate data $$X$$ and $$Y$$ and use the OLS estimator, but there should be a direct way to use $$\Sigma$$ in the estimation of population $$\beta$$?

2. Which options are there to specify covariances (correlations) in $$\Sigma$$ given I need a fixed $$R^2$$ of a linear regression of $$Y$$ on $$X$$? This, to systematically vary the explanatory power of the regression model.

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What is the best way to simulate data for a linear regression model?

I am concerned with simulating data for a linear regression model. I need to control the means, variances, and correlations (covariances) between the predictors and the criterion variable. In addition, I need to be able to vary the explained variances ($$R^2$$). It is obvious to me that the latter must be a function of the earlier, so at least one correlation (covariance) in $$\Sigma$$ is perhaps dependent on the choice of $$R^2$$, where $$\Sigma=(Y,X)(Y,X)^T$$, for centered $$X$$ and $$Y$$, is the variance-covariance matrix of all variables.

My plan is thus as follows:

1. Specify $$\Sigma$$, means, and $$R^2$$
2. Simulate data with these sufficient statistics, e.g. by sampling from the multivariate normal
3. Check estimated $$\beta$$ (regression coefficient) vector against population (theoretical) coefficients and use the model for unrelated tests/science.

Hence, my approach does not suggest specifying $$\beta$$ but letting the coefficients be a function of population $$\Sigma$$, means, and $$R^2$$. The reason I need to do this is to attribute some realisitc scale to $$X$$ and $$Y$$ (e.g., let $$Y$$ assume an 'income' scale and give $$X$$ a realistic scale for years of eduction). Therefore, I specify the sufficient statistics instead of regression coefficients. But maybe there is a better way.

Moreover, I have two specific questions:

1. Given the population variance-covariance matrix $$\Sigma$$ of one criterion variable $$Y$$ and a series of predictors (covariates) $$X$$ , I would like to calculate the vector of true population regression coefficients. Of course, I could simulate data $$X$$ and $$Y$$ and use the OLS estimator, but there should be a direct way to use $$\Sigma$$ in the estimation of population $$\beta$$?

2. Which options are there to specify covariances (correlations) in $$\Sigma$$ given I need a fixed $$R^2$$ of a linear regression of $$Y$$ on $$X$$? This, to systematically vary the explanatory power of the regression model.