Regression Analysis Updating the Study by Barro and Sala I Martin for Income Convergence across States - Statistics Project Example

A regression analysis updating the study by Barro and sala I martin for income convergence across s Introduction According to Barro and Sala I martin, state income per capita were converging to constant value and will at some time in the future be equal across all the states. There was great shortcoming in their study since they considered only nominal income per capita, never adjusted for inflation, and never factored the cost of living differences across states. The study uses data on the deflated real income. In this study the economic time series that measure the relative cost of living differences between states. Barro and Sala I martin (1991, 1992a) found out that two concepts of convergence exist. According to Barro and Sala I martin (1991. 1992a), the convergence of income exist when the poor economy country is growing very fast than the rich country economy. This makes the poor country trend to catch up with the rich one in terms of the level of per capital of product. According to Easterlin (1960) and Borts ans stein (1964), converging arise if dispersion- measured, using the standard deviation of the logarithm of per capita income or product across a group of countries or regions declines over time. The convergence of the first kind arises when the poor countries tend to grow faster than rich ones. The convergence of the second kind arises when there is reduced dispersion per capita income or product but this situation can be eliminated when the process is offset by new disturbances that tend to increase the population. In this study, a regression analysis of the real income per capita will be used to obtain the regression analysis from the year 1990 to 2012. The growth of nominal personal income per capita shows a significant relationship with the time passage. The growth of real personal income per capital indicates a relationship with time. There are a great argument about the growth of nominal personal income per capital, growth of real personal income per capital and the time. However, it cannot be denied that the time passage increases. According to Norton, Olds, Olive and Craig (1996), the growth of nominal personal income per capital and the growth of real personal income per capital are highly significant as the time changes. Many researchers have devoted on the topic about the relationship between the growths of nominal personal income per capita with the time. The researchers have also indicated that there is a lot of work between the growths of real personal income per capital. Payne and Isaacs (2005) advocated that lack of growth of nominal personal income has a significant relationship with the time. As a result, it is believed that time is a significant element in the growth of the nominal income per capita and the growth of real personal income per capita. Besides, time is a factor in the economic element and they are values that influence the other factors of the income per capita. Dintiman and Ward (2003) the growth of nominal personal income per capital and the growth of real personal income per capita have a significant 2. Data Survey and theoretical argument The major neoclassical growth model property is that it is able to predict the conditional convergence. The data on per capita income for the states of USA were used to analyze the growth of nominal personal income per capita, growth of real personal income per capita and sigma convergence. In case for example, there are observations at only two points in time which is T and 0(zero).Basing on the below equation, this means that the mean growth rate for a given interval that is from 0 to T is as: Where the error terms average is represented by the intercept is and between 0 and T. The initial income expression is and it is considered to be the expression which decreases with interval length for a specific β. In this case, if the linear relationship is estimated between the initial income logs and the income growth rate, then the predicted coefficient will tend to be smaller as the time span becomes longer over which there is averaging of the growth rate. This is because there is a decline in the growth rate as the income increases. Thus, if the growth rate is computed over a time span that is longer, then the smaller future growth rates are combined with initial larger growth rates. Thus, the initial position effect on the growth rate will decline as the interval increases. In this case, the coefficient is seen to approach zero as the T approaches infinity where, where it tends to beta as T approaches zero. The beta estimate is got from the nonlinear form of the equation above and the T value is taken into account where, it is applied in each case. Such method is required to produce similar beta estimate regardless of the averaging interval length for the given data. Basing on the above literature, the equations used in the current study to meet the objectives were designed as follows: -----(1); growth=[(1/T)(GDP2013/GDP1990)] Where represents the intercept; represents the initial income coefficient; represents the initial income; represents the percentage growth rate. -----------(2) where every year from 1990 to 2013, t(year) uses dummy variable such like 1990=1, 1991=2…2013=24 The regression analysis is defined as; Regression model The model is a simple linear regression model. The parameter are called the regression coefficient. Such model describes the hyper plane in the 2-dimensional space of the regression variable. The parameters represent the expected change in the response variable y per a change in, while all the repressor remains constants. The regression equation is The model assumption 1 Or equivalently 2. or equivalently 3. Analysis The data used in the study was obtained from the BEA web site. The data is the annual income per capital for different states in USA from 1990 to 2012. 3.1 Growth of Nominal Personal Income per Capita 1990 personal income and 1990-2012 income growth Basing on the above scatter plot with fitted line, it is observed that the growth is negatively related to the Nominal Personal Income per Capita in 1990.This implies that beta convergence do exist for the states of USA. The large R squared (0.997) as seen in the table below can be appreciated from the above scatter plot with fitted line. In this case, the average growth rate per capita between 1990 and 2012 that is against the personal income per capita in 1990 is provided by the scatter plot. Test of assumption of the regression model Source | SS df MS Number of obs 52 =----------- -- +------------- ----------------- F( 22, 29 ) = 786.17 Model | 2.89E+09 22 131230250 Prob > F 0 Residual | 4840758 29 166922.689 R-squared 0.9983 ------------ -+ -------------- ---------------- Adj R-squared 0.9971 Total | 2.89E+09 51 56704044.2 Root MSE 408.56 From the regression analysis, it can be observed that the value of the F (22, 29) is equal to 786.17 with a p- value of 0.000 which is significantly less than the 0.05 level of confidence. This implies that there is significant in the convergence of the income per capita at 95% confidence for different states in USA from 1990 to 2012. The value of the coefficient of determination is equal to 0.9983. Which implies that the regression model account for 99.83% of all the errors in the regression model. The root mean square error of the regression model on the income per capita is equal to 408.56. It can be observed that the value of coefficient in the year 1993 is equal to 0.3526 which has a t- value of 1.48 and a p- value of 0.149 which is greater than the 0.05 level of confidence. This implies that the income per capita is significant in the year 1993 showed any convergence. It can be observed that the value of coefficient in the year 1994 is equal to -0.5564 which has a t- value of -1.89 and a p- value of 0.069 which is greater than the 0.05 level of confidence. This implies that the income per capita is significant in the year 19945 showed any convergence. 3.2.Growth of Real Personal Income per Capita According to the scatter plot with fitted line above, it is observed that the growth is negatively related to the real personal income per capita in 1990.This means that beta convergence do exist for the states of USA. The large R squared (0.9991) as seen in the table below can be appreciated from the above scatter plot with fitted line where, the average growth rate per capita between 1990 and 2012 that is against the real personal income per capita in 1990 is provided by the scatter plot. Note: F omitted because of collinearity Source | SS df MS Number of obs = 52 -------------+------------------------------ F( 22, 29) = 1504.45 Model | 2.2824e+09 22 103744117 Prob > F = 0.0000 Residual | 1999787.75 29 68958.1983 R-squared = 0.9991 -------------+------------------------------ Adj R-squared = 0.9985 Total | 2.2844e+09 51 44791575.7 Root MSE = 262.6 The analysis of the real personal income per capital indicate that the value of the F (22, 29) = 1504.45 with a significant value of 0.0000 which is less than the 0.05 level of confidence. This implies that there is sufficient evidence to conclude that the data contain sufficient information to conclude that the regression model on real personal income per capital can be predicted or forecasted using the regression model. From the coefficient of determination, it can be observed that the value of coefficient of determination is equal to 0.9991. This implies that the regression model of the real personal income per capital can account for 99.91% of all the errors in the regression model of real personal income per capital. 3.3. Sigma Convergence There are two different approaches that are used to quantify the level to which development process is leading to convergence or divergence in regional performance over time. The oldest and the ancient method is the sigma convergence and the new method is the neo- classical method known as the beta approach. The sigma convergence is used to measure the dispersion of the real per capita income. In this study, the standard deviation indicates a falling trend over time. In this case, we can conclude that there is a difference of per capita income between regions which decreases as time changes. We used the standard deviation divided by the mean to obtain the coefficient of determination. From this analysis, we can observe that there is falling trend in the coefficient of determination. The figure below represents the scatter plot between the growth of real personal income per capita and growth of nominal personal income per capita. From the scatter plot, it can be observed the there is significant relationship between the growth of real personal income per capita and the growth of nominal income per capital. The scatter plot indicates that the increase in the growth of real personal income per capital also increases the growth of nominal personal income per capita. The table below indicates the nonlinear least square estimates in the form of the equation; for the USA states for various periods of time. Year Coef. t- P>|t| ------------ -------------- ------------------ ------- 1 0.670131 2.45 0.02 2 0.356638 0.89 0.379 3 0.174468 0.61 0.548 4 0.331065 1.94 0.063 5 -0.40331 -1.16 0.254 6 -0.14758 -0.42 0.679 7 0.358391 1.48 0.149 8 -0.55648 -1.89 0.069 9 0.021856 0.07 0.948 10 0.291369 0.54 0.593 11 -0.5081 -1.29 0.206 12 -0.4581 -0.95 0.349 13 0.643315 1.86 0.073 14 1.603438 2.60 0.014 15 -2.05763 -3.64 0.001 16 0.018719 0.03 0.973 17 0.91914 0.89 0.38 18 -0.17019 -0.25 0.801 19 -0.13355 -0.23 0.82 20 0.307386 0.53 0.6 21 -0.79625 -1.72 0.097 22 0.503758 1.51 0.141 The regression coefficient indicates that there are several variables that are significant to the regression model while other variables are not significant. The variables that are independent in the following regression model are the income per capital for 1990, Income per capital for 2003, and income per capital for 2004. We also indicate that the variables that are not significant the regression variables are; the income per capital of the year 1991 to the income per capital for the year 2002, the income per capital for 2005 to the income per capital for year 2011. Conclusion As mentioned, the major neoclassical growth model property is that it is able to predict the conditional convergence. The data on per capita income for the states of USA were used in the study to analyze the growth of nominal personal income per capita, growth of real personal income per capita and sigma convergence. From the analysis, we can conclude that there is significant evidence to conclude that the classical and Keynesian indicate that there is convergence in the income per capita. As the year progresses, the convergence of the income per capita is observed. From the analysis, it can be observed that there is a significant relationship between the growths of nominal personal income per capital the time as it progresses. It can also be observed that there is a significant relationship between the growth of real personal income per capital and the time. From the analysis, the sigma convergence for the real personal income data and the growth of nominal income per capital is significant and decline as time progresses. References ATP Tour (1995). ATP 1995 player guide. Ponte Vedra Beach, Florida: ATP Tour. Borts ans stein (1964). Economic growth in the free market . The Economic Journal Vol. 75, No. 300 (Dec., 1965), pp. 822-824 Dintiman, G., & Ward, R. (2003). Sport speed. (3rd ed.). Champaign, IL: Human Kinetics. Easterlin (1960), Will raising the incomes of all increase the happiness of all?; Journal of Economic Behavior and Organization Vol. 27 (1995) 35-47 Barro and Sala-i-Martin (1990). "Economic Growth and Convergence across the United States." Working Paper 3419. Cambridge, Mass.: National Bureau of Economic Research (August) Read More