Mediation Analysis

In order to examine the influence of earnings quality on the relationship between corporate governance and firm value, this study employed mediating regression model. Earnings quality variables (accruals quality, predictability and conservatism) are considered as the mediators of the relationship.

In general, mediating variables are used to explain how or why two variables are related. A modern application of mediating variables is in treatment and prevention research,

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where interventions are designed to change mediating variables, such as norms, which are hypothesised to be causally related to a dependent variable.

Mediation analysis is chosen to be applied in this thesis as it can identify fundamental processes underlying one particular issue that is relevant across contexts. Once a true mediating process is identified, then more efficient and powerful interventions can be developed because these interventions can focus on variables in the mediating process (MacKinnon & Fairchild 2009).

Generally, in every mediation regression model, an antecedent variable affects a mediator variable and the mediator variable affects a dependent variable, thus forming a chain of relations among three variables (Baron & Kenny 1986). The chain of relations among the variables is called an indirect or mediated effect of the antecedent variable on the dependent variable. An effect that is not mediated this way is called a direct effect. Following MacKinnon (2008), a single-mediator model can be illustrated by the following three equations:

𝑌 = 𝑖₁+ 𝑐𝑋 + 𝑒₁ (4.8) 𝑌 = 𝑖₂+ 𝑐́𝑋 + 𝑏𝑀 + 𝑒₂ (4.9) 𝑀 = 𝑖3+ 𝑎𝑋 + 𝑒3 (4.10)

Where Y is the dependent variable, X is the independent variable and M is the mediating variable. The coefficient c represents how strongly X predicts Y; ć represents the strength of prediction of Y from X, with the strength of the M-to-Y relation removed; b is the coefficient for the strength of the relation between M and Y with the strength of the X-to-

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between X and M. The intercepts in each equation, i.e. 𝑖₁, 𝑖₂and 𝑖₃, represent the average score of each variable; and 𝑒_1,, 𝑒₂ and 𝑒₃ represent the error.

Several different effects are represented in the model. Firstly, a direct effect relating X to

Y with the strength of mediator relation removed, quantified by ć; secondly, a mediated or indirect effect of X to Y transmitted through the mediating variable, quantified by ab

(the numerical values of the mediated effect may be computed in one of two ways; as either the difference in coefficients c minus c’ or as the product of coefficients ab); and finally, a total effect of X on Y that is computed by the addition of these two parts (MacKinnon, David P & Dawyer 1993).

For this thesis, the relationships among variables will be represented by simultaneous mediation design, a mediation model which involves multiple mediators. The following additional set of equations is applicable for a multiple-mediator regression model:

𝑌 = 𝑖2+ 𝑐′𝑋 + 𝑏1𝑀1+ 𝑏2𝑀2+ 𝑒2 (4.11) 𝑀1 = 𝑖3+ 𝑎1𝑋 + 𝑒3 (4.12) 𝑀₂ = 𝑖₄+ 𝑎₂𝑋 + 𝑒₄ (4.13)

Equation (4.10) illustrates a multiple mediation model with two mediators, i.e. 𝑀₁ and

𝑀₂. The equation represents both the direct effect of X on Y (coefficient ć) and the indirect effects of X on Y via both mediators. The specific indirect effect of X on Y via a particular mediatoris defined as the product of the two unstandardized path linking X to Y via that particular mediator (for example, the specific indirect effect of X on Y through 𝑀1 is

quantified as 𝑎₁𝑏₁). Equation (4.11) and Equation (4.12) represent direct relationship between independent variable X and each of the mediators (𝑀₁and 𝑀₂). The total indirect effect of X on Y is the sum of the specific indirect effects presented as follow:

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(4.14)

Where product of coefficient a and coefficient b, (𝑎_𝑗𝑏_𝑗) in Equation (4.13) is the indirect effect of j mediator for k number of mediators. The total effect of X on Y is the sum of the direct effect and all k of the specific indirect effects, represented in Equation (4.14) below:

Total effects = c = 𝑐 +́ ∑ (𝑎_{𝑗 𝑗}𝑏_𝑗), 𝑗 = 1 to k.

(4.15)

The multiple mediation design is chosen as it premises on several advantages. Firstly, testing the total indirect effect of X on Y is similar to conducting a regression analysis with several predictors, with the aim of determining whether an overall effect exists. If the mediation effect is found, it can be concluded that the set of k variables mediates the effect of X on Y. Secondly, it is possible to determine as to what extent specific M

variables mediate the effect of X on Y, conditional on the presence of other mediators in the model. Thirdly, when multiple presumed mediators are diverted in a multiple mediation model, the likelihood of parameter bias due to omitted variables is reduced. As highlighted by Judd and Kenny (1981), contrary to multiple mediation, when several simple mediation hypotheses are each tested with simple mediator model, these separate models may suffer from the omitted problem, which can lead to biased parameter estimates. Finally, including several mediators in one model allows the researcher to determine the relative magnitudes of the specific indirect effects associated with all mediators, to dig deeply into competing theories against one another within a single model.

Assessing multiple mediations involves not only deciding whether or not an indirect effect exists, but also deciding on how to tear apart individual mediating effects often attributed to several potential mediators that may overlap in content (West & Aiken 1997).

Several methods have been implemented in this study to assess whether the mediated effect of earnings quality is large enough to be considered important in the relationships between the mechanisms of corporate governance and firm value. As suggested in

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statistical literature, there are four major groups of tests for mediation, i.e. causal steps approach, difference in coefficients test, product of coefficients test and bootstrapped confidence interval. Based on the accuracy of Type I error and statistical power, MacKinnon et al. (2002), found that among 14 widely used methods in mediation research, the most balance method (reasonably accurate Type I error and high statistical power) is the joint significance test.

Causal-Steps Approach

Although there are many methods available for testing hypotheses pertaining to mediation effects, the most widely used method is the causal-steps approach popularised by Baron and Kenny (1986) and Judd and Kenny (1981). MacKinnon et al. (2002), and Fritz and MacKinnon (2007), among others, discovered that this method is most frequently being employed in mediation studies due to its convenience, despite its low statistical power. This approach requires the researcher to estimate each of the paths in the model and then ascertain whether a variable functions as a mediator meets certain statistical criteria. The method entails separate significant tests of the strength of the overall relationship between X and Y, the strength of the relationship between X and M, the strength of the relationship between M and Y adjusted for X and visual inspection of whether coefficient c is greater than the coefficient c’.

The requirement of a significant overall relationship between X and Y is the central difference between the causal steps approach and other methods for testing mediation (Hayes 2009). Some researchers have treated this test of overall relationship between X

and Y as a perfect test of the relationship, failing to recognise that it is an inadequate statistical test that is subject to error and arguing that if there is no significant overall effect then mediation should not be examined. Even though the requirement that X is significantly related to Y is an important test in any research study, however, in most social science research, partial mediation is more realistic by which mediation can exist even in the absence of such a significant relationship (Baron & Kenny 1986).

Several scenarios had illustrated that although significant mediation exists but the overall effect of X on Y is not significant. Consider a case of mediation in which there are subgroups of individuals for whom the mediated effect is of opposite sign (i.e. positive versus negative), such that a test of the X-to-Y relationship for the pooled data would be

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zero even though mediation exists in the data. Considering another case in which the sign of mediated effects (ab) differs from the sign of the direct effect (ć), causing the overall relationship of X and Y (c) to be zero (such cases are known as inconsistent-mediation analysis).

Despite it being widely used in substantive research, there are several limitations to the causal-steps approach. Hayes (2009) and MacKinnon, David P and Fairchild (2009) highlight the critiques on the approach. Firstly, they found that simulation studies have shown that among the methods for testing intervening variables effects, the causal steps is among the lowest in statistical power. Secondly, they argued that the method is not based on a quantification of the strength of the mediated effect. Finally, they are concerned with the fact that the test requires that there be a significant overall relationship between X and Y for mediation to exist.

Difference in Coefficients Tests

As described by MacKinnon et al. (2002), this approach does not call for a significant of total effect of X on Y, coefficient c. The variation of Baron and Kenny’s (1986) causal steps approach ignores the significant of c but requires a and b coefficients to be significant for the mediation to exist and to be further analysed. It considers the variability of difference in coefficients c minus c’ and for the product of coefficients ab (MacKinnon et al. 2002), basically the difference between the regression coefficients before and after adjusted for the mediating variable.

Sobel First Order (Product of Coefficients) Test

This approach has its origin in sociology and is based on the product of coefficients involving paths in a path model (i.e. the indirect effect) (Sobel 1982). This approach is to test the significance of the mediating variables effect by dividing the estimate of the mediating variable by its standard error and comparing this value to a standard normal distribution.

Bootstrapped Confidence Intervals Method

In addition to testing the mediated effect, ab (or c minus c’), for significance, limits for the true value of ab can also be constructed. The confidence limits for the mediated effect provide information on the reliability or accuracy of the estimate of the mediated effect. Recent research has shown that confidence limits and significance testing for the

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mediated effect based on the normal distribution are often inaccurate and are likely to find a real mediated effect in a sample of data (MacKinnon et al. 2002). Asymmetric confidence limits based on the distribution of the product, ab, and methods based on repeatedly sampling the original data are more accurate. These tests capture the non- normal shape of the mediated-effect sampling distribution (which occurs because the strength of the mediated effect is the product of two coefficients and does not always have a normal distribution), thus improving statistical power (MacKinnon, David P & Fairchild 2009). Resampling methods are among options to handle the non-normality in the distribution of the mediated effect (Shrout & Bolger 2002).

Bootstrapping is one such resampling method that has been widely applied to cases in which classical methods do not perform well. Bootstrapping involves drawing a larger number of samples with replacement from the original sample. Sampling with replacement means the bootstrap samples, although all are of the same size as the original sample, but can exclude some cases from the original sample and include duplicates of others. The model of interest is estimated in each bootstrap sample as in the original data. The distribution of sample statistics estimated in each bootstrap sample can be used to perform significance tests or to form confidence intervals (Taylor, MacKinnon & Tein 2008).

For this thesis, regression models are first estimated for the original data to find the coefficients. A large number of bootstrap samples are drawn, the same models are estimated for each bootstrap sample, and the estimates (coefficients) from each bootstrap sample are used to form the bootstrap distribution. The limits of a percentile bootstrap confidence interval are simply the values of the coefficients (β1, β2 and β3) at the α/2 and

1- α/2 percentiles of the bootstrap distribution, where α is the nominal Type I error rate. As an example, for the typical α = 0.05, the limits are the 2.5th and 97.5th percentiles of the distribution. The bias corrected confidence interval limits are also taken from the bootstrap distribution, but they are adjusted if the bootstrap distribution fails to centre at the sample estimate of the mediated effect. Results and discussions on bootstrap procedures implemented for this thesis is presented in Chapter 5, and the procedures are run using bootstrap syntax of STATA 12.

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