Main Feratures and Examples of Economic Mathematics - Math Problem Example

A. Simple Linear Regression Model Summary statistics of the given data is presented in Table below and the calculations are shown in the appendix. Table 1: Summary Statistics of the given data Column1 Mean 56935.71429 Standard Error 925.8561314 Median 56000 Mode 56000 Standard Deviation 3464.236433 Sample Variance 12000934.07 Kurtosis 0.732061491 Skewness 0.851323059 Range 12500 Minimum 52500 Maximum 65000 Sum 797100 Count 14 Q2) (i) The scatter diagram of annual salary versus years of experience is presented in figure 1, in the appendix. From the diagram the underlying relationship between these variables i.e. annual salary and the years of experience is positive correlation i.e. annual salary is increasing with increasing years of experience. (ii) The scatter diagram of annual salary versus years of post secondary school education is presented in figure 2, in the appendix. From the diagram the underlying relationship between these variables i.e. annual salary and the years of post secondary education is positive correlation i.e. annual salary is increasing with increasing years of post secondary school education. (iii) The scatter diagram of annual salary versus gender is presented in figure 3, in the appendix. From the diagram there is no explicit relationship between these two variables. Q3) Regression analysis was carried out to explore the relationship between Annual salary and years of experience. The regression equation as obtained by Ordinary Least Square (OLS) method is presented below: (1) Yi = 52498 + 748.58 X1i + i The regression line imposed on the scatter diagram is shown in figure 4, in appendix. The Summary output of the regression analysis is presented below: SUMMARY OUTPUT Regression Statistics Multiple R 0.636882196 R Square 0.405618932 Adjusted R Square 0.356087176 Standard Error 2779.848079 Observations 14 ANOVA   df SS MS F Significance F Regression 1 63281478.72 63281478.72 8.189068327 0.014309806 Residual 12 92730664.14 7727555.345 Total 13 156012142.9         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 52497.72296 1719.621294 30.52865368 9.57085E-13 48750.99009 56244.45583 X Variable 1 748.5768501 261.5887996 2.861654823 0.014309806 178.6238272 1318.529873 Hypothesis testing for the slope 1 Null Hypothesis Ho: 1 = 0 (There is no linear relationship) Alternate Hypothesis H1: 1≠ 0 (There is a linear relationship) Decision rule: Null hypothesis will be rejected if ttest> t0.025, 12 ttest = 2.86 (from summary output of regression analysis) tcritical = t0.025, 12 = 2.1788 As ttest> t0.025, 12; therefore, Null hypothesis is rejected and the Alternate hypothesis is accepted. This means there is linear relationship between annual salary and years of experience. Q4) Regression analysis was carried out to explore the relationship between Annual salary and years of post secondary school education. The regression equation as obtained by Ordinary Least Square (OLS) method is presented below: (2) Yi = 51517 + 1330.9 X2i + i The regression line imposed on the scatter diagram is shown in figure 5, in appendix. The Summary output of the regression analysis is presented below: SUMMARY OUTPUT Regression Statistics Multiple R 0.431873138 R Square 0.186514408 Adjusted R Square 0.118723942 Standard Error 3252.097149 Observations 14 ANOVA   df SS MS F Significance F Regression 1 29098512.42 29098512.42 2.751336857 0.123061856 Residual 12 126913630.4 10576135.87 Total 13 156012142.9         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 51517.17391 3380.358718 15.24015 3.24049E-09 44152.0051 58882.34273 X Variable 1 1330.869565 802.3495448 1.658715424 0.123061856 -417.299887 3079.039017 Hypothesis testing for the slope 2 Null Hypothesis Ho: 2 = 0 (There is no linear relationship) Alternate Hypothesis H1: 2≠ 0 (There is a linear relationship) Decision rule: Null hypothesis will be rejected if ttest> t0.025, 12 ttest = 1.66 (from summary output of regression analysis) tcritical = t0.025, 12 = 2.1788 As ttest< t0.025, 12; therefore, Null hypothesis is accepted. This means there is no linear relationship between annual salary and years of post secondary school education. Q5) Sample regression equation 1 is Yi = 52498 + 748.58 X1i For i = 12 X1i = X1, 12 = 6 Therefore, Y12 = 56989 For i = 13 X1i = X1, 13 = 2.5 Therefore, Y13 = 54369 For i = 14 X1i = X1, 14 = 1.5 Therefore, Y12 = 53620 There is difference between the actual salary and that predicted by this linear regression equation. This error is associated with the linear regression model and has been caused due to the assumption that annual salary depends on years of experience only, while in reality there are many other independent variables affecting the annual salary and effect of those variables has not been accounted for in this regression analysis. Q6) Calculation of R2 = 1 – (ESS/TSS) For equation (1); R2 = 1 – (92730664.14/156012142.9) = 0.4056 For equation (2); R2 = 1 – (126913630.4/156012142.9) = 0.1865 R2 or coefficient of determination tells how well the regression equation fits into the sample data. Its value lies between 0 and 1. Values close to 1 imply very good fit, while those close to 0 imply a poor fit. Higher the value of R2 better is the fit of the regression equation. Therefore, based on the values of equation (1) is better fit or better in determining the values of the dependent variable based on the values of independent variable. B. Multiple Linear Regression Model Q1) The multiple regression equation is presented below: (3) Yi = 45495 + 801.6 X1i + 1595.7 X2i + 382.6 X3i + i The coefficients of the multiple regression 3, 4 and 5 are partial coefficients. This means they give the change in the value of the dependent variable (Yi) per unit change in the respective independent variable provided the other independent variables remain constant. For example 3 gives by how much the annual salary will change with change in the experience of work by one year provided years of post secondary school education and gender remains same. Similar is the explanation for the other slope coefficients. The excel regression output report is presented below: SUMMARY OUTPUT Regression Statistics Multiple R 0.821590487 R Square 0.675010928 Adjusted R Square 0.577514206 Standard Error 2251.715826 Observations 14 ANOVA   df SS MS F Significance F Regression 3 105309901.3 35103300.42 6.923421787 0.008376182 Residual 10 50702241.6 5070224.16 Total 13 156012142.9         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 45495.32938 2809.771399 16.19182592 1.67346E-08 39234.76747 51755.89128 X Variable 1 801.5711025 228.4684282 3.508454576 0.005646207 292.511633 1310.630572 X Variable 2 1595.736538 560.6439792 2.846256443 0.017361158 346.5436898 2844.929387 X Variable 3 382.5720808 1287.410877 0.297163934 0.772422528 -2485.958608 3251.10277 Q2) The coefficients for years of experience X1 In simple linear regression equation (1) the coefficient is 1 = 748.58 In multiple regression equation (3) the coefficient is 3 = 801.57 In equation (3) this coefficient is accounting for the effect of only one variable (years of experience), while in equation (1) this coefficient is accounting for the composite effect of all the independent variables affecting annual salary of an employee. This is the reason why the two values are different in magnitude. Q3) Coefficient of multiple determination or R2 for the multiple regression model is computed below: R2 = SSR/SST = 105309901.3/156012142.9 = 0.675 SSR is sum of square of regression values and SST is sum of squares of all the values. This means 67.5% variations in the dependent variable can be explained by this multiple regression model. However, this method of computation for R2 does not take into account the sample size and also the number of independent variables. By accommodating the sample size and number of independent variables used to predict the value of the dependent variable, the value of R2 is modified and what one gets is adjusted R2. The computation of R2adj is shown below: R2adj = 1- [(1-R2)(n-1)/(n-k-1)] = 0.5775 Where n = 14 is the sample size and k = 3 is the number of independent variables This means the R2adj can explain 57.75% variation in the annual salary of employees. Q4) Hypothesis testing for the slope coefficients Hypothesis testing for 3 = 0 Significance level 5% and degrees of freedom = 10 Null Hypothesis Ho: 3 = 0 (There is no linear relationship) Alternate Hypothesis H1: 3≠ 0 (There is a linear relationship) Decision rule: Null hypothesis will be rejected if ttest> t0.025, 10 ttest = 3.5084 (from summary output of regression analysis) tcritical = t0.025, 10 = 2.2281 As ttest> t0.025, 10; therefore, Null hypothesis is rejected. This means there is linear relationship between annual salary and years of experience. Hypothesis testing for 4 = 0 Significance level 5% and degrees of freedom = 10 Null Hypothesis Ho: 4 = 0 (There is no linear relationship) Alternate Hypothesis H1: 4≠ 0 (There is a linear relationship) Decision rule: Null hypothesis will be rejected if ttest> t0.025, 10 ttest = 2.8462 (from summary output of regression analysis) tcritical = t0.025, 10 = 2.2281 As ttest> t0.025, 10; therefore, Null hypothesis is rejected. This means there is linear relationship between annual salary and years of education post secondary school examination. Hypothesis testing for 5 = 0 Significance level 5% and degrees of freedom = 10 Null Hypothesis Ho: 5 = 0 (There is no linear relationship) Alternate Hypothesis H1: 5≠ 0 (There is a linear relationship) Decision rule: Null hypothesis will be rejected if ttest> t0.025, 10 ttest = 0.2971 (from summary output of regression analysis) tcritical = t0.025, 10 = 2.2281 As ttest< t0.025, 10; therefore, Null hypothesis is accepted. This means there is no significant relationship between annual salary and gender of employees. Q5) Testing the joint hypothesis 3 = 4 = 5 = 0 Null Hypothesis H0: 3 = 4 = 5 = 0 Alternate hypothesis H1 either of 3, 4 and 5≠ 0 The hypothesis will be tested using F test which is described below: F statistics is equal to regression mean square (MSR) divided by error mean square (MSE) i.e. F = MSR / MSE Where F = test statistics from an F distribution with k and n-k-1 degrees of freedom, n = 14 is the sample size and k = 3 is the number of independent variables. MSR = SSR/k = 105309901.3/3 = 35103300.42 (SSR = Regression sum of squares) MSE = SSE / (n-k-1) = 50702241.6/10 = 5070224.16 (SSE = Error sum of squares) F = MSR / MSE = 6.9234 Decision Rule: Reject H0 at the  (=0.025) level of significance if F > Fk, n-k-1 Else accept H0 At 5% significance level ( = 0.025) F3, 10 = 3.71 As F > Fk, n-k-1; therefore, Null Hypothesis is rejected and alternate hypothesis is accepted that is at least one of multiple regression coefficients represents a significant linear relationship between the independent variable(s) and the dependent variable. Q6) Based on the simple and multiple regression analysis following conclusions can be drawn: (i) There is significant positive linear relationship between the annual salary and years of experience. (ii) There is significant positive linear relationship between annual salary and years of education post secondary school examination. (iii) There is no significant relationship between annual salary and gender of the technicians. Based on these conclusions it can be said that education increases human capital and therefore, has a positive impact on earnings as in this study the salary was found to be positively related to years of education post secondary school examination and years of experience and not at all related to the gender of the technicians. Thus findings of this study support the Human Capital Theory. Q7) Simple regression analysis and multiple regression analysis was carried out to explore the relationship between the dependent variable annual salary of technicians and the independent variables like years of experience, years of education post secondary school examination and gender. Predictive regression models (equations) were developed and the slope coefficients were examined for their statistical significance using t test and F test. From the analysis it came out that annual salary of the technicians is positively related to their years of experience and years of education post secondary school examination as the coefficients related to these independent variables are statistically significant and positive. On the other hand gender has no statistically significant bearing on the annual salary of the technicians. These findings thus can be taken as an affirmation of the Human Capital Theory. Appendix: Q1. Calculations for Mean, Median, Mode, Variance and Standard Deviation Mean Median = Mean of 7th and 8th term when salary are arranged in ascending/descending order = 56000 Mode = 56000 (term with maximum frequency) Variance Standard Deviation S = Q2) (i) Q2) (ii) Q2) (iii) 3) Q4) Read More