Decomposition of Wage Differentials

An often used methodology to study labour market outcomes by groups is to decompose mean differences in log of wages known as the Blinder-Oaxaca decomposition (Jann, 2008). This Blinder-Oaxaca decomposition is a standard technique used to divide the wage differential between two groups into a part that is explained by differences in observable characteristics and a residual that cannot be explained by differences in characteristics (Jann, 2008; Pearlman and Tsao, 2008).

The explained part represents the part of the wage gap that is attributable to differences in group characteristics, that is the differences in wages that exists between groups if all groups had the same characteristics (Jann, 2008). The unexplained part is often used as a measure for discrimination, but it also subsumes the effects of group differences in unobserved predictors (Jann, 2008).

Hamilton and Hamilton (1997) in using the Oaxaca decomposition in their study, state that the unconditional earnings differential, measures the difference in earnings between two workers who have observable characteristics identical to the average person of each drinker type. The earnings differential is unconditional in that the predicted earnings are calculated independently of the workers actual choice of drinking status and hence the earnings differences are independent of selection effects. Hamilton and Hamilton (1997) state that the unexplained term is a pure wage differential and shows whether the returns to a representative set of observed traits vary by drinking status. It measures the differences in household income if observable characteristics are constant (Hamilton and Hamilton, 1997).

The Oaxaca method set out by Barrett (2002) in his study into the effects of alcohol consumption on wages as follows:

  ^{     }

Where: lnY_ij log of household income differences in characteristics across drinking categories.

 equals differences in productivity in status j

versus status k drinkers attributable to the differences in characteristics across drinking categories and is the explained part of the differential (Barrett, 2002).

The second term on the right hand side represents the component of the wage gap due to differences in coefficients and is the unexplained part of the differential (Barrett, 2002). This part tests whether the returns to a representative set of observed traits differ by drinking status and captures the effect of alcohol consumption on household income if observable characteristics are held constant (Hamilton and Hamilton, 1997).

2.1.9: Testing the relevance of instruments and post estimation tests

The significance of each of the instruments can be tested using a Wald test, which calculates a Z Statistic, which is then squared, yielding a Wald Statistic with a chi-squared distribution and will correspond to a two tailed P Value (Agresti, 1996). The Likelihood Ratio Test is another test which can be used to test the significance of coefficients (Gujarati, 2004). The likelihood-ratio test uses the ratio of the maximised value of the likelihood function for the full model over the maximised value of the likelihood function for the simpler model, the full model being that with an additional one or more parameters. The log transformation of the likelihood function yields a chi-squared statistic (Gujarati, 2004). The t and z statistics test whether a given coefficient is significantly different from zero (Gujarati, 2004).

Heteroskedasticity causes standard errors to be biased. OLS assumes that errors are both independently and identically distributed and that the variance of the error term is constant. If heteroskedasticity is present it would lead to bias in test statistics and confidence intervals (Berry and Feldman, 1985). The presence of heteroskedasticity can be tested using the Breusch Pagan test which tests the null hypothesis that the error variances are all equal (Berry and Feldman, 1985).

Whites’ general test for heteroskedasticity, which is a special case of the Breusch-Pagan test can also be used (Greene, 2000). This tests the error distribution by regressing the squared residuals on all distinct regressors, cross-products, and squares of regressors (Greene, 2000). A possible solution would be to use robust standard errors when heteroskedasticity is present as these relax the assumptions that the errors are both independent and identically distributed, hence robust standard errors tend to be more trustworthy (Berry and Feldman, 1985).

Multicollinearity arises when two or more predictor variables in a model are highly correlated and could cause coefficient estimates of particular variables to be to be incorrect.

2.1.10: Conclusion

This section reviews the literature on the effect of alcohol on income, and in looking at this relationship the issue of endogeneity and selection bias are reviewed in detail and the possible methods of estimation that can be used to account for these. Endogeneity arises when an explanatory variable such as alcohol is determined within the context of the model. Selection bias arises when an individual selects into different categories of drinking resulting in the sample not being random. The multinomial logit OLS two step estimate as proposed by Lee (1982) estimates alcohol consumption as a multinomial logit, from which the Inverse Mills Ratio is derived and included in the income equation. Separate income equations are estimated for each category of drinker. This method of estimation treats the sector choice as endogenous and accounts for selection bias.

The different factors that affect both alcohol consumption and income are also assessed. Similar to previous studies alcohol is assumed to be unordered (Hamilton and Hamilton, 1997; Barrett, 2002) and hence is estimated using a multinomial logit model. Alcohol consumption could however be viewed as ordered data and hence should be estimated as such (Harris et al, 2006).

2.2: Alternative Methods of Estimation

Previous studies into the effect of alcohol consumption on an individuals financial welfare such as Hamilton and Hamilton (1997) and Barrett (2002) among others, have assumed that alcohol status is unordered and hence have estimated the alcohol status equation using the multinomial logit model. Alcohol consumption could however be viewed as ordered data (Harris et al, 2006). If ordinality is ignored then this may lead to a loss of efficiency and an increased risk of getting insignificant results (Harris et al, 2006).

2.2.1: Definition and Estimation of Ordered Data

Ordered data is where the variable of interest follows a strict ordering based on the value of the latent variable (Hilmer, 2001). Some polychotomous dependent variables are in a natural order and are expressed in terms of categories (Kennedy, 2003). Measurement through the use of ordered categories is a common practice in marketing and behavioural sciences (Kennedy, 2003).

Ordered data avoids a false sense of precision that continuous scales convey (Sprinivasan and Basu, 1989).

Failure to account for the ordinal nature of the dependent variable can result in incorrect results (Greene, 2002). If a dependent variable is ordered, but the ordinality is ignored then this may lead to a loss of efficiency and an increased risk of getting insignificant results (Harris et al, 2006). If data is ordered, estimating the data by a multinomial logit or probit model would not be efficient because no account would be taken of the extra information of the ordinal nature of the dependent variable, nor would OLS be appropriate because the coding of the dependent variable reflects only a ranking, the difference between 1 and 2 cannot be treated as equivalent to the difference between a 2 and a 3 (Kennedy, 2003).

An ordered probit model is an econometric model that can be used to deal with ordered categorical variables and is designed to model a discrete dependent variable that has ordered multinomial outcomes (Jones 2005). An ordered probit

model can be expressed in terms of an underlying latent variable y* (Jones 2005). The ordered probit assumes that the variable of interest follows a strict ordering based on the value of the latent variable (Hilmer 2001). The ordered probit and logit models have come into fairly wide use as a framework for analysing such responses (Zavoina and McElvey, 1975). Hilmer (2001) states that the estimated thresholds in the ordered probit model should always be significant and if not, then one could conclude that the assumed natural ordering and consequently the ordered probit is an inappropriate specification. A primary difference between the multinomial logit and ordered probit is that due to the assumed natural ordering the latter does not require the Independence of Irrelevant Alternatives (IIA) property, however for the model to be appropriate, the assumed natural ordering must be realistic (Hilmer, 2001).

Wooldridge (2009) says that the ordered probit and logit models have come into fairly wide use as a framework for analysing such responses. The model is built around a latent regression in the same manner as the binomial probit model.



 

 x

y^* (2.2.1)

Where: y dependent variable x independent variable

 coefficient

 error term

y is unobserved but what is observed is *

0

y if y^* 0

1

y if 0 y^* ₁

2

y if ₁  y^* ₂ .

. .

= J if _J1  y^*

Where: y dependent variable

J known cutoffs

In this equation wherey^* 0, these respondents are in category 0. Where y is ^* greater than 0 but less than ₁ category 1 is observed and where y is greater ^*

1but less than ₂ category 2 is observed.

In order to address the issue of selection bias when data is of an ordered nature, various extensions of the Heckman two step model have been adopted (Greene and Hensher, 2010). A variety of extensions to the Heckman model (1979) have been developed for ordered choice models, one being to use an ordered probit extension of the Heckman correction (Vella, 1998; Greene and Hensher, 2010).

This is where the selection equation is estimated using an ordered probit model, from which an estimate of lambda is computed for each individual in the selected sample. This is then included as an additional regressor in the outcome equation (Vella, 1998).

Many studies have adopted this approach whereby the selection equation is estimated as an ordered probit which allows the inverse mills ratio to be derived (Garen, 1984; Butler et al, 1994, 1998; Frazis, 1993; Jimenez and Kugler, 1987;

Harmon and Walker,1995; Chiburis and Lokshin, 2007). Langpap and Kerkvliet (2002) in their study into whether the endangered species act in the US has been successful in promoting species recovery, estimated an probit in the first step, from which the inverse mills ratio is derived and the second step is then estimated as an ordered probit.

Chiburis and Lokshin (2007) in their study into the estimation of wages for public, private and informal sectors for male workers in India use an ordered probit selection model. The categorical variable describing the sector individuals work in, is estimated as an ordered probit on the basis of an ordered probit selection rule. They set out the model specification as follows;

Step One – Estimation of the Selection Equation

i i

i s

c^*   (2.2.2)

1

ci if c_i ₁

2

ci if ₁ c_i ₂

3

ci if ₂ c_i 

Where: c sector category

 unknown vector of parameters, s independent variables

 standard normal shock

J cutoffs

i indexes individuals

The category an individual is in depends on a range of independent variables s (Chiburis and Lokshin, 2007). It is assumed that the independent variables s_i and the categorical variables c are observed. It is important that the ordered probit _i selection model contains a variable that is not an independent variable in the income equation (Chiburis and Lokshin, 2007). There must be at least one instrument in the selection variable s that has no effect on y except through its effect on c . If all the variables in the selection equation are also in the wages equation, then the identification of the coefficient _j would be weak (Chiburis and Lokshin, 2007). This is due to the fact that additional variables in the first step selection equation are important for identification of the second step estimates which would inflate second step standard errors and unreliable estimates of coefficients (Vella, 1998).

In the first step the selection equation is estimated by an ordered probit of c on s, yielding the consistent estimatesˆ,ˆ₁,ˆ₂,...ˆ_J (Chiburis and Lokshin, 2007).

The probability of observing c=1,2,3 is defined as follows:

) term, _i can be obtained from the ordered probit equations (Chiburis and

Lokshin, 2007; Jimenez and Kugler, 1987; Hamilton and Nickerson, 2001).

ˆ )

iis included as an omitted variable in the OLS equation estimated in step 2.