Oxford Handbook of Human Resource Management

27. 
    
    
    
    4. 3 Simultaneity
    
    
    
    

In a recursive model, causation runs in one direction. In a non-recursive model,
causation is reciprocal (Duncan 1975 ). In other words, there is simultaneity. To
illustrate, consider the following model and example adapted from Duncan ( 1975 :
ch. 5 ), which has two exogenous variables,x 1 andx 2 , and perf and hr as endogenous
variables (variables in standard score form to simplify things):

perf¼bperf 1 x 1 þbperfhrhrþv

hr¼bhr 2 x 2 þbhrperfperfþu

Thus, in this model, hr!perf and perf!hr.
Ordinary least squares (OLS) estimates of the regression coeYcients in this
model are biased because the disturbances/residuals are no longer independent
of the right-hand-side variables in the model (see Duncan 1975 : 77 ). For example,
given that perf!hr, thenv, the disturbance term in the perf equation, must also be
related to hr. Thus, cov(hr,v)6¼ 0 , which will lead to bias in estimatingbperfhr.
SpeciWcally, Duncan shows that (as applied here) rather thanbperfhr, the OLS
estimate will equal:

bperfhrþrhrv=( 1 
r^2 hr 1 ):

The second part of this equation is referred to as simultaneity bias.

27. 
    
    
    
    4. 
       
       
       
       3. 1 Two-Stage Least Squares ( 2 SLS) and Instrumental Variables (IV)
       
       
       
       
    
    
    
    

One approach to dealing with simultaneity (or endogeneity in a purportedly
exogenous variable) is instrumental variables (IV), often estimated using two-
stage least squares ( 2 SLS). (As previously noted, IV/ 2 SLS can also be used to address
omitted variable and measurement error problems.) Consider again the equations:

perf¼bperf1x 1 þbperfhrhrþv

hr¼bhr2x 2 þbhrperfperfþu

The basic idea of instrumental variables is to replace a right-hand-side variable,
hr, suspected of being correlated with the error term with a predicted value, hr

hat, which is not correlated with the error term. Using 2 SLS, hr
hat is obtained by
regressing hr on the full set of exogenous variables, which must include at least one
instrument. This is theWrst stage of 2 SLS. The second stage is re-estimating the
perf-equation by replacing hr with hr
hat.
An instrument is a variable that is included in the equation for hr, but is
excluded from (orWxed to zero in) the equation for perf and thus inXuences
perf only through hr. This zero/exclusion restriction is necessary to identify the

modeling hrm and performance linkages 567