# AR.power: Power of the Anderson-Rubin (1949) Test In ivmodel: Statistical Inference and Sensitivity Analysis for Instrumental Variables Model

## Description

`AR.power` computes the power of Anderson-Rubin (1949) test based on the given values of parameters.

## Usage

 ```1 2``` ```AR.power(n, k, l, beta, gamma, Zadj_sq, sigmau, sigmav, rho, alpha = 0.05) ```

## Arguments

 `n` Sample size. `k` Number of exogenous variables. `l` Number of instrumental variables. `beta` True causal effect minus null hypothesis causal effect. `gamma` Regression coefficient for effect of instruments on treatment. `Zadj_sq` Variance of instruments after regressed on the observed variables. `sigmau` Standard deviation of potential outcome under control. (structural error for y) `sigmav` Standard deviation of error from regressing treatment on instruments. `rho` Correlation between u (potential outcome under control) and v (error from regressing treatment on instrument). `alpha` Significance level.

## Value

Power of the Anderson-Rubin test based on the given values of parameters.

## Author(s)

Yang Jiang, Hyunseung Kang, and Dylan Small

## References

Anderson, T.W. and Rubin, H. (1949). Estimation of the parameters of a single equation in a complete system of stochastic equations. Annals of Mathematical Statistics, 20, 46-63.

See also `ivmodel` for details on the instrumental variables model.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```# Assume we calculate the power of AR test in a study with one IV (l=1) # and the only one exogenous variable is the intercept (k=1). # Suppose the difference between the null hypothesis and true causal # effect is 1 (beta=1). # The sample size is 250 (n=250), the IV variance is .25 (Zadj_sq =.25). # The standard deviation of potential outcome is 1(sigmau= 1). # The coefficient of regressing IV upon exposure is .5 (gamma= .5). # The correlation between u and v is assumed to be .5 (rho=.5). # The standard deviation of first stage error is .4 (sigmav=.4). # The significance level for the study is alpha = .05. # power of Anderson-Rubin test: AR.power(n=250, k=1, l=1, beta=1, gamma=.5, Zadj_sq=.25, sigmau=1, sigmav=.4, rho=.5, alpha = 0.05) ```