First, it has to be highly correlated with the instrumented, endogenous variable, that is, the HRSI. Based on both Staiger and Stock (1997) and Stock and Yogo (2005) criteria, we rejected the hypothesis of weak instrument.
(2011a), which also measured stringency using the HRSI.
The independent variables included the predicted HRSI and all the control variables.
Standard errors for the second-stage equations were estimated using bootstrapping that took into account the uncertainty in the predicted values of the instrumented variable as well as sampling error, as follows: (Step 1) We drew 500 random vectors of the predicted HRSIs (each consisting of 100 HRSI values for all 50 states in each of 2 years) from a normal distribution with means equal to the predicted HRSIs and variance-covariance matrix of the predicted HRSIs in the first-stage equation.
We used the Durbin-Wu-Hausman test statistic to test the hypothesis that the HRSI is endogenous with the quality measures.
To make the estimated coefficients for the HRSI more meaningful, we used the estimated models to predict the incremental change in quality (the dependent variables) for a one standard deviation increase in the HRSI for the average nursing home at the average level of quality.
We estimated models assuming that the staffing standards are also endogenous, including them in the HRSI as well, and using the same IV.
To estimate the cost-effectiveness of the regulation of quality, we estimated the costs and impact associated with an increase of 1 standard deviation in the HRSI for the average facility of 100 beds.
The HRSI and the IV were at the state level, with an N = 100 (50 states in each of 2 years).
The HRSI significantly increases only quality related to CNA staffing and risk-adjusted pressure sores.
The first-stage equation, shown in column 2, predicts the HRSI based on the IV.