Two Stage Least Squares Regression

Boldina and Beninger (2016) state that a violation of any of the assumptions of OLS regression means that a more robust or better-fitting model should be employed.

Therefore, due to some of the violations of the OLS assumptions found in the regression diagnostics above, this study employs an alternative econometric method to address these issues. One of the biggest econometric issues faced in the data set is the problem of endogeneity and many previous corporate governance researchers have faced this problem. This study addresses the issue of endogeneity, the violation of the normality assumption, the issue of heteroskedasticity and the issue of autocorrelation

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in several ways. First, appropriate and relevant control variables are used in order to account for other potential influences on firm performance and board characteristics as recognised in the literature. Second, robust standard errors are used in the econometric model to deal with the presence of heteroskedasticity in the data. Third, a Two-Stage Least Square regression (2SLS) model is performed using an instrumental variables approach to deal with endogeneity. In addition, the 2SLS is also estimated using the other dependent variables as the Durbin Watson tests results indicated no autocorrelation for ROE and ROA. Lastly, an alternative regression model is used to deal with the presence of autocorrelation.

The 2SLS is an extension of OLS regression and is commonly used in applied econometrics to address the potential problem of endogeneity and reverse causality (Wooldridge, 2016). Specifically, this method deals with the possibility that the dependent variable has a correlation with the cause of the explanatory variable.

Previous researchers have suggested that this is an issue of concern in corporate governance research due to the possibility that companies may change their governance and board structures, only after a period of poor performance (Adams &

Ferreira, 2009; Carter et al., 2010). 2SLS also deals with the problem where an explanatory variable has a value that is determined by other variables in the system and therefore correlates with the random error in the regression model. In order to make a causal claim, an exogenous variable that is uncorrelated with the error term is needed (Verbeek, 2008).¹⁴ This study focuses on testing hypotheses formulated in Chapter 3 by considering how a range of independent variables individually affect a dependent variable. This relationship can normally be assessed through a basic econometric model that uses linear regressions estimated by OLS. This is the starting point for the analysis and the equation is represented below:

𝛾

_𝑖𝑡

= 𝜒

_𝑖𝑡−2

𝛽 + 𝜀

_𝑖𝑡

(4.1)

14 An endogenous variable is different to an exogenous one in that, it is an inherent function of the other variables in the study (Verbeek, 2008).

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Where Yit is the dependent variable, 𝜺_𝒊𝒕 is the random error term, Xit denotes the vector of all independent variables and their associated parameters, 𝜷

.

The FTSE 350 companies in the study’s sample are represented by subscript i and the years by subscript t. In equation 4.1 t-2 represents a two-year lag that was incorporated in the data set because the effects of board structure on firm performance are not likely to be immediate. However, as previously displayed in the diagnostic checks, the regression model needs to address the presence of unobserved heterogeneity. Two error terms are included in the model; the first term is 𝝁_𝒊 to cover unobserved heterogeneity at firm level; whilst the second is 𝜺_𝒊𝒕 which is an idiosyncratic error term. This gives the following equation:

𝛾

_𝑖𝑡

= 𝜒

_𝑖𝑡−2

𝛽 + 𝜇

_𝑖𝑡

+ 𝜀

_𝑖𝑡

(4.2)

In order to deal with endogeneity, this study uses a multi-equation model with joint estimation using two-stage least squares. In this equation (4.3), a new dependent variable is denoted by 𝒛_𝒊𝒕, whilst α, 𝜹 and 𝜽 denote the new parameters that will be estimated in the model. Lastly, in these equations, 𝝊_𝒊𝒕 represents the idiosyncratic error term. The 2SLS equations are as follows:

𝛾

_𝑖𝑡

= 𝑧

_𝑖𝑡

𝛼 + 𝜒

_𝑖𝑡−2

𝛽 + 𝜀

_𝑖𝑡

(4.3)

𝑧

_𝑖𝑡

= 𝛾

_𝑖𝑡

𝛿 + 𝜒

_𝑖𝑡−2

𝜃 + 𝜐

_𝑖𝑡

(4.4)

Further to this, in order for this study to measure a causal relationship, a structural equation is included in the second stage of the 2SLS model. These equations include 𝝌_{𝟏𝒊𝒕−𝟐} as an endogenous regressor that is a subset of 𝝌_{𝒊𝒕−𝟐} and in equation 4.6, 𝝌_𝟐 denotes a different subset of regressors with a causal effect on 𝝌_{𝟏𝒊𝒕−𝟐}.

𝛾

_𝑖𝑡

= 𝜒

_𝑖𝑡−2

𝛽 + 𝜀

_𝑖𝑡

(4.5)

𝜒

_{1𝑖𝑡−2}

= 𝜒

_{2𝑖𝑡−2}

𝜃 + 𝜐

_𝑖𝑡

(4.6)

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Although both the literature and the diagnostic checks have identified the 2SLS model as an appropriate method of analysis for this research, there are some limitations of using this model. Wooldridge (2016) states that one of the biggest limitations of using 2SLS, or any other instrumental variables approach, is the difficulty in obtaining instrumental variables that are adequately uncorrelated with the error term and adequately correlated with the explanatory variables. Previous research suggests that an ideal instrument (z) should satisfy two main conditions (Black et al., 2006b;

Castineira & Nunes, 1999; Wooldridge, 2016). The first condition is that a good instrumental variable must not be correlated to the disturbance in the equation, the error terms, and the second condition is that good instrumental variables should be correlated with the endogenous variables. Black et al. (2006b) argue that the majority of previous governance studies lacked plausible instruments. Similarly, Wintoki et al.

(2012) stated that although instrumental variables techniques can mitigate endogeneity, in corporate governance studies it is difficult to find instrumentals that are not affected by any firm characteristics.

The natural choice for corporate governance researchers has been to use lagged values of the endogenous regressors as instruments (Terjesen et al., 2016). However, the data set employed in this research has already incorporated a time lag in the data to capture the stance that the effects of a change in the governance structure on firm performance are not likely to be immediate. Therefore, the researcher uses a different set of instrumental variables following the work of Jermias and Gani (2014). The instrumental variables used are total equity, total sales, capital intensity, operating margin,¹⁵ length of operating cycle¹⁶ and sales growth. These instruments are selected in line with previous research that suggests these variables are all credible instruments for corporate governance because there is no theoretical reason for these variables to be endogenous to firm performance or corporate governance (Black et al., 2006b;

Ittner & Larcker, 2001; Jermias & Gani, 2014). The validity of the instrumental variables is tested using Basmann’s (1957) and Sargan’s (1958) test for over identified

15 Operating margin is calculated as sales minus cost of goods sold divided by sales (Jermias & Gani, 2014).

16 Length of operating cycle is calculated as average receivables divided by sales plus average inventory divided by cost of goods sold (Jermias & Gani, 2014).

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restrictions. These tests both examine the correlation between the instruments and the models’ residuals in order to check whether the instrumental variables are uncorrelated with the error term (Arora & Sharma, 2016). The results indicate that the null hypothesis of no correlation between the instrumental variables and the error term cannot be rejected (chi2(3) = 2.64474 p = 0.4497 for Sargan’s test and chi2(3) = 2.60742 p = 0.4562 for Basmann’s test). In addition, STATA has its own test for testing overidentifying restrictions and the score for this was chi2(3) = 4.05304 (p = 0.2558). This means thatthe null hypothesis that the instruments are valid is accepted.

For this and theoretical reasons, the instrumental variables selected are considered to be plausible and valid instruments.

In document A Quantitative Investigation into the Business Case for Diversity in the Boardroom: a Multi-Theory Framework (Page 151-155)