1)
a) Statement of the problem is to identify whether there is a difference in the intercepts or slopes or both by the use additive and multiplicative dummies. In this case, an additive dummy will be used to identify whether intercepts in the different sets differ while a multiplicative dummy will be used to find any difference in slopes.
b) Literatures reviewed include the testing of the intercepts and differential slopes in order to identify whether they are statistically significant or insignificant. Inorder to test for statistical significance or insignificance of the differential slopes, the intercept for the entire data had to be computed and in our case it is -1.0821 together with the intercept of the other two sets which are -0.2663 and -1.7502
c) The model Si=a0+aiD+a2Yi+a3(DYi),
= -0.2663 -1.4839D + 0.0470Yi + 0.1034(DYi)
d) All data in the model is obtained from the calculations and estimations shown in the excel template: a0 is the intercept of the first set, period between 1946- 1953; a2 is the intercept of the second set, the period between 1954- 1963. Since the intercept of the first set a1 is statistically insignificant, then the estimate of the common intercept of both sets is obtained by the difference between the intercept of the second set and the first set. Since the slope of the first set a3 is statistically insignificant, the common slope of both sets is obtained by the difference between the slope of the second set and the slope of the first set.
e) The method of estimation applied in testing for equality between the sets of coefficients in the two sets of linear regression is the dummy variable approach constituting the inclusion of additive and multiplicative dummies as described above where the additive dummies have been used to arrive at the common intercepts value and the multiplicative to arrive at the common slopes value.
f) The hypothesis testing is done by allowing a degree of freedom of n-x where n is the sample size and x represent the variable available in our case the variable are two. The period between 1946 -1963 have a size of 18 and thus degree of freedom is 18-2 =16. Period between 1946-1954, sample size is 9 and thus degree of freedom is seven and likewise for the period between 1955 -1963. The null hypothesis assumes there is no significant difference between the sets but by applying the additive and multiplicative dummies approaches, it’s clear that there is a significant difference between the intercepts and slopes of the two sets respectively: as such, the null hypothesis is rejected.
g) For the period between 1946 – 1963, the model relating savings and income is S= -1.0822+0.1178Y. The model shows a strong degree of correlation as indicated by the coefficient of determination R2=0.9134, which means that for every level of income there is a 90% level of confidence that it will give an actual estimate of the level of savings. For the first set, period between 1946-1953, it shows a weak degree of correlation as indicated by the coefficient of determination of 0.2105 while for the second set there is a strong degree of correlation as indicated by the coefficient of determination of 0.9006. This is has been revealed by the application of the dummy variable approach which indicates a significant difference in the intercepts and slopes of the two sets.
h) The linear equations derived are only estimators of savings: one cannot certainly explain every value of income that influences savings. The level of savings does not solely depend on the level of income.
There is also a limited scope of estimation: you cannot predict values outside those observed.
4) By using the dummy approach, the intercept of the first set was statistically insignificant and therefore there was need to compute the common intercept value for the sets. Due to the statistical insignificance of the slope of the first set, a common slope value for the sets has to be calculated as well. Due to these differences in intercepts and slopes, the null hypothesis which assumed that the two sets are the same is rejected.
Note: The calculations and analysis are shown on the separate excel file uploaded.
References
A.F George,(1977).Linear regression analysis. Canada: John Wiley & sons,Inc. W.Sanford,(2005) Third edition, Applied linear regression:Hoboken,New jersey: John Wiley & sons,Inc.H.Kutner Michael. (2005).Applied linear statistical Models: McGraw-Hill Irwin. Richard N.D&Hilliary S,(1981).Applied Regression Analysis, Second edition:: John Wiley & sons,Inc. J. Fox(1997).Applied Regression analysis, Linear models and related methods: London, Sage publications Ltd, London
