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Do male and female employees have different wage structures?IntroductionThe wage equation is a model presenting the relationship between wages and exogenous variables that explains the variation in wages, such as an individual’s age, or experience. Rational employees aim to maximise this equation gi...

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Do male and female employees have different wage structures? Introduction The wage equation is a model presenting the relationship between wages and exogenous variables that explains the variation in wages, such as an individual’s age, or experience. Rational employees aim to maximise this equation given their position, however a higher wage is not the only objective. Wage differences between genders could be down to; differences in human capital, managerial discrimination, the cost of childcare, the need for flexible working hours, limited opportunities after maternity leave, or different occupational choices. There is debate over whether the wage structure is distinct for male or females. Description of the Model The data used is cross-sectional data taken from the Quarterly Labour Force survey (QLFS), January – March 2017 dataset. For a full explanation of the explanatory variables used in the model and the reason for omitting particular variables, see Appendix 1. I will model the data using a semi-log functional form: lnhourW= β 1 + β 2 potexp+ β 3 potexp2+ β 4 female+ β 5 PT+ β 6 married+ β 7 manager+ β 8 GCSE+ β 9 Alevel+ β 10 Degree+ β 11 SouthE+ ui I used this log-linear model after conducting a MacKinnon White Davidson (MWD) test. This will be presented in upcoming project sections. Ideally, I would have included additional variables, such as dummy variables for occupational sectors, ethnicity and more specific geographical location variables. However, this would reduce parsimony in the model, as all these variables will reduce simplicity. Data Issues, Limitations, and Concerns Measurement error bias is a type of model misspecification e.g. when collecting data on wages, individuals may round up their wage or slightly exaggerate, or if employees are paid annually, they may also not know what their hourly wage is. Missing data would make our sample unrepresentative despite being such a large sample (8470). This could come about if we only had wage data on people paying taxes, for example. Cross-sectional data is affected by multicollinearity, heteroscedasticity, spatial autocorrelation, and misspecification. I will go on to test for multicollinearity, misspecification and heteroscedasticity within the model. Hypotheses to be Tested 1.) Standard individual tests of significance H 0 : β 2 , β 3 ,…, β 11 =0 H 1 : β 2 , β 3 ,…, β 11≠ 0 I am uncertain about whether the t-vales will be positive or negative, so will use a two-tailed significance test.

2.) Overall significance test H 0 : β 2 = β 3 = β 4 = β 5 = β 6 = β 7 = β 8 = β 9 = β 10 = β 11 =0 H 1 : H 0 is false 3.) The Chow Test for structural equivalence H 0 : The model is structurally equivalent across males and females α 1 = λ1 , α 2 = λ2 ,…, α 11 = λ11 H 1 : H 0 is false α 1 ≠ λ1 , α 2 ≠ λ2 ,…, α 11 ≠ λ 11 4.) The Dummy Variable Approach for comparing 2 regressions H 0 : β 4 = β 12 = β 13 = β 14 =0 H 0 : H 0 is false Where β 4 is the differential intercept coefficient for the female dummy variable, and β 12 , β 13 , and β 14 are the regression coefficients for (female*GCSE), (female*Alevel), and (female*Degree) respectively. Presentation of the Model lnhourW = ^ β1 + ^ β 2 potexp+ ^ β 3 potexp2+ ^ β 4 female+ ^ β 5 PT+ ^ β 6 married+ ^ ^ ^ ^ ^ β 7 manager+ β 8 GCSE + β 9 Alevel+ β 10 Degree+ β 11 SouthE+ e i Where ^ β 1 is the estimated intercept coefficient and ^ β 2 ,…, ^ β 11 are the respective partial regression/differential intercept coefficient estimates, and where e i is the estimator of ui . Standard individual tests of significance 

Using the t-statistic t=

^β i−β i tn −k , I will individually test each partial regression sⅇ ( ^β ) i

/differential intercept coefficient β 2 , β 3 ,…, β 11 , to see if they are significantly different from 0. Overall Significance test 

R2 /k−1 F k−1 ,n −k , I will test if all the partial (1−R 2)/ n−k regression/differential intercept coefficients are jointly and simultaneously equal to zero. Using the F-statistic F=

The Chow Test for structural equivalence 

Estimate our original model lnhourW= β 1 + β 2 potexp+ β 3 potexp2+ β 4 female+ β 5 PT+ β 6 married+ β 7 manager+ β 8 GCSE+ β 9 Alevel+ β 10 Degr ee+ β 11 SouthE+ ui (1)

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Where n=8470, k=11, and obtain RSS(1). Estimate this model over the observations 1-4447 to obtain the regression model for female employees lnhourW= α 1 + α 2 potexp+ α 3 potexp2+ α 4 female+ α 5 PT+ α 6 married+ α 7 manager + α 8 GCSE+ α 9 Alevel+ α 10 Degree+ α 11 SouthE+ uia (2a) Where N a =4447, and obtain RSS(2a) Estimate the same model over the observations 4448-8470 to obtain the regression model for male employees lnhourW= λ1 + λ2 potexp+ λ3 potexp2+ λ 4 female+ λ5 PT+ λ6 married+ λ7 manager + λ8 GCSE+ λ9 Alevel+ λ10 Degree+ λ11 SouthE+ uib (2b) Where N b =4023, and obtain RSS(2b) RSS RSS N 2 a+RSS a+N ¿ ¿ (¿ )/(¿ 2b b −2 k ) F k , N + N −2 k Construct the F statistic using F= ¿ [¿ ¿ 1−( RSS2 a + RSS2 b )]/ k ¿ ¿ If our statistic is greater than the critical value for the given significance value we can reject our null hypothesis of the model being structurally equivalent across males and females. a

b

The Dummy Variable Approach for comparing 2 regressions 

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We can add our interaction variables into our unrestricted model lnhourW= β 1 + β 2 potexp+ β 3 potexp2+ β 4 female+ β 5 PT+ β 6 married+ β 7 manager+ β 8 GCSE+ β 9 Alevel+ β 10 Degree+ β 11 SouthE+ β 12 (female*GCSE)+ β 13 (female*Alevel) + β 14 (female*Degree)+ ui . Under the null hypothesis we have that β 4 = β 12 = β 13 = β 14 =0, subbing this in our unrestricted model becomes restricted: lnhourW= β 1 + β 2 potexp+ β 3 potexp2+ β 5 PT+ β 6 married+ β 7 manager + β 8 GCSE+ β 9 Alevel+ β 10 Degree+ β 11 SouthE+ ui Following this we estimate our restricted and unrestricted models to obtains RUR2 , and RR

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2

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2 2 (RUR −R R )/ m F m,n −k . If this is significant then it (1−RUR 2)/ n−k will show that the relationship between hourly wage and education levels is different for males and females.

Construct an F-test, using F=

MacKinnon White Davidson (MWD) test

H0 :

H1

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Linear Model hourW= β 1 + β 2 potexp+ β 3 potexp2+ β 4 female+ β 5 PT+ β 6 married+ β 7 manager+ β 8 GCSE + β 9 Alevel+ β 10 Degree+ β 11 SouthE+ ui : Semi-long Model lnhourW= β 1 + β 2 potexp+ β 3 potexp2+ β 4 female+ β 5 PT+ β 6 married+ β 7 manager+ β 8 GCSE+ β 9 Alevel+ β 10 Degree+ β 11 SouthE+ ui If we estimate each model from the null and alternative hypotheses we obtain estimates for each dependant variable ( ^ hourW , and ^ lnhourW ). ^ We must construct a new variable Z i =ln( hourW )- ^ lnhourW , and then Z regress hourW on our explanatory variables and i . This gives us: hourW= β 1 + β 2 potexp+ β 3 potexp2+ β 4 female+ β 5 PT+ β 6 married+ β 7 manager+ β 8 GCSE+ β 9 Alevel+ β 10 Degree+ β 11 SouthE+ β 12 Z i + ui If the coefficient of Z i , is statistically significant, we can reject our null hypothesis in favour of a log-linear model.

The Ramsey RESET Test H0 : Model is correctly specified H 1 : H 0 is false  Estimate our original model: lnhourW= β 1 + β 2 potexp+ β 3 potexp2+ β 4 female+ β 5 PT+ β 6 married+ β 7 manager+ β 8 GCSE+ β 9 Alevel+ β 10 Degree+ β 11 SouthE+ ui , and obtain the estimated lnhourW ( ^ lnhourW ). 2 3 ^ ^  Add lnhourW , and lnhourW into out model: lnhourW= β 1 + β 2 potexp+ β 3 potexp2+ β 4 female+ β 5 PT+ β 6 married+ β 7 manager+ β 8 GCSE+ β 9 Alevel+ β 10 Degree+ β 11 SouthE+ 2 3 β 12^ lnhourW + β 13^ lnhourW + v i , estimate this model.  Let the R2 from the original model be Rold 2 , and the R2 from the new 2 model be Rnew .  We can then use the F-test to see whether the increase R2 from the old model to the new model is statistically significant. R ¿ 2 (¿ new¿¿ 2−R old )/ Number of new regressors F= F m ,n −k (1−R new2 )/(n−Number of paramters∈the new model) ¿ ¿  If F is significant the we can reject the null hypothesis that the model is correctly specified. However this test does not tell us what the correct specification is.

Test for Multicollinearity

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If we make the variable ‘None’ the dependant variable and regress the explanatory variables GCSE, Alevel, and Degree, on ‘None’ we can test to see if there is a linear relationship between these explanatory variables. None= β 1 + β 2 GCSE+ β 3 Alevel+ β 4 Degree+ ui After estimating this model if R2 =0, then there is no multicollinearity, if 0< R2...