Quantitative and Analytical Techniques for Managers - Essay Example

work in Quantitative and Analytical Techniques for Managers Section 0 Exercise on Descriptive Statistics 1. Summary of data on the Gross Regional Product in yuan using measures of central tendency and dispersion (e.g., mean, median, variance and standard deviation) and discussion on the strengths and weaknesses of these measures from the perspective of a manager seeking to understand the regional distribution of gross regional product in China. The measures of central tendency and dispersion for the 2007 gross regional product measured in terms of 100 million yuan are as follows: Mean 8920.319355 Median 6091.1 Mode #N/A Standard Deviation 7596.482057 Sample Variance 57706539.64 Range 30742.2 Minimum 342.2 Maximum 31084.4 Sum 276529.9 The data indicate that the 30 provinces of the People’s Republic of China are highly variable with regard to the gross regional product. Based on the data above, the 2007 gross regional product is as low as 34,220 million yuan to as high as 3,108,440 million yuan. This indicates that some of the provinces may be very poor while others may be very rich although we still have to validate this when the data are transformed into gross regional product per capita. A standard deviation of 759,648.2 million yuan which is very close to the mean value of 892,031.93 million yuan. This indicates a very high variability. The high variability is also indicated by the high variance of 5,770,653,964 million yuan. From the perspective of the manager, the mean as measure of central tendency is very useful. However, the mean can mask a situation in which some of the provinces or cases have actually very high or very low variable values. The mode can be almost useless for ratio data but is very useful for nominal data or variables. The median is extremely useful to identify at what value the population is divided into 2 equal parts: half below the median while the other half is above the median. For instance, in the data above, the median is 609,110 million yuan versus the mean value of 892,031.93 million yuan or that the median is lower than the mean. This indicates that a few provinces with high values of the gross regional product are raising the mean to be above the median. Given a poverty figure, for example, we can determine through the median whether at least half of the population are below or above the poverty figure. Another option is to use a measure of living standard. A median above the living standard would indicate that at least half of the population are above the living standard. For ratio data, it is the belief of this writer that the using both the median and the mean simultaneously would be useful. However, for nominal variables, the identification of central tendency through the mode will the one useful. 1.2. The Pearson correlation coefficient between the gross regional product and gross capital is positive 0.97408077 versus the perfect correlation of positive 1. This indicates that the two variable as positively correlated, one increases as the other one increases, and the correlation magnitude is high. Intuitively we can say that the correlation between the two variables is highly significant because it is very close to perfect positive correlation. The correlation nevertheless is not surprising because output is a function of capital formation and, intuitively and from the standpoint of theory, output indeed is a function of capital or capital formation. 1.3. Given a mean price of 450 and standard deviation of 2870, the following insights can be made: a) The Z-score would indicate the position of a variable value in relation to the mean computed in terms of its distance from the mean in terms of standard deviation unit. A positive Z-score indicates that a variable value is higher than the mean while a negative Z-score indicates a variable value that is lower than the mean. For example, if Z=+1, this indicates that the variable value is higher than the mean by one standard deviation. b) The probability associated with the a variable value is indicated by a normal curve. For example, if price = 500, then the Z-score is 0.017421603. The value is very close to zero and for practical purposes we can say that the probability that the share price might exceed 500 is close to 50%: the probability is less than 50% but very close to 50%. This probability value is equal to 49.305% c) Similarly, based on the Z-score of 100 which is (100-450)/2870 = -0.12195122. Thus, based on the Z-score, the probability that the price might be less than 100 is less than 45.15%. Section 2.0 Miscellaneous 2.1. In Table 1, the relationship between X and Y may be specified as based on Walpole et al. 2007:397 and McClave et al. 2000:478-479. Based on the references, 2 can be computed as or 2 = [ (31*6,505,311.9)-(387*681,176)]/[(31*5,983.74)-(387*387)] = -1,734 while 1 can be computed as 1 = or 1 = 21,973.42-(-1,734*12.48) = 43,614. Thus, the regression function ix Y = 43,614 – 1,734 X. 2.2. 1 = 43,614 implies that our regression function begins from 43,614 base. This need not have an economic interpretation. 2.3. The null hypothesis for 2 is that it is equal to zero while the alternative hypotheses is that it less than zero. 2.4. 2 = -1,734 implies that gross regional product is negatively correlated with the percentage share of the primary sector in total output. The magnitude and sign implies that a percentage increase in the share of the primary sector in total output decreases the gross regional product per capita by 1,734 yuan. 2 is significant up to the 0.000 level and is therefore highly significant. Thus, the null hypothesis that 2 is zero can be rejected to accept the alternative hypothesis that 2 is less than zero. 2.5. Based on the adjusted R-squared of the model at 0.576291, the model can explain 57.62% of the variation in the dependent variable or the gross regional product per capita. Section 3.0: Work on data set “Chinese Statistical Yearbook 2007” Regional growth can be modelled in several ways. For instance, we can follow the standard income function which has been known, identified, or labeled as “Keynes” even if John Maynard Keynes himself did not actually use the mathematics in his work, The General Theory of Employment, Interest, and Money: Y = C + I + G + (X – M). Another option is to use an aggregate supply and demand model and there are several examples of modelling output in this manner in the literature. Beck et al. (2005) use however an innovative model which used of GDP per capita, an information variable, and a country index. On the other hand, the theoretical foundation as well as the empirical implementation of the models of Beck and Wobmann (2006) appears more complicated. In the models formulated by Beck and Wobmann, the gross domestic product is seen as a function of employment shares, total employment, equilibrium, wage differential, migration factors, and several other variables (p. 192-196). Further, the model of Beck and Wobmann (2006) employed the concept of Solow residual. Table 3.1. Table of coefficients, production function model for Gross Regional Product Nevertheless, a regression or an econometric model need not be overcomplicated. For instance, for for our regression or econometric model, we utilize a simple production function Y = Y (L, K) in which output or gross regional product is considered a function of labour and capital. The regression function is conventional, fundamental, often used in economics for regression and econometrics (Gujarati 2004, p. 11 & 288). Using data from the Chinese Statistical Yearbook of 2007 and using the gross regional product as the dependent variable and population and gross capital formation as the independent variables, the SPSS estimate on the empirical implementation of the model for the People’s Republic of China’s 31 provinces in the 2007 Chinese Statistical Yearbook is given next page. Table 3.2. Anova table of the econometric model reflected in Table 3.1. In our production model of the gross regional product of China’s 31 provinces as reflected in the Chinese Statistical Yearbook of 2007, the regional gross product is the dependent variable representing output. The independent variables are labour and capital. Labor is proxied by the population while capital represented by the gross capital formation. Economics is both careful with both overspecification/overfitting and underspecification/underfitting of economic models because both of them can distort the values of the coefficients thereby making the estimates of the coefficients invalid (Gujarati 2004, p. 510). With the specification of simple production function, however, we avoid both underspecification and overspecification because the production function is a standard function in economics. At the same time, as a simple production function, our model need not have a perfect tracking capability because the production is only one of the variables that can affect the market decisions of economic agents. The fundamental gain achieved in using a conventional and widely-accepted production function, however, is that the coefficient estimates derived therein would be considered valid. As shown in Table 3.3., the econometric model based on a simple production function produced signs that are consistent with theory. Both population (proxy for or representing labour) and gross capital formation (representing capital as a factor) have positive signs in the regression. The interpretation, of course, is that labour and capital are positively correlated with the gross regional product, consistent with the expected signs in a simple production function model. Table 3.3. Summary of the model described in Table 3.1. The regression data on Table 3.1. of the earlier page suggest that each increase in the population by 10,000 people will contribute around 22,500 million to the gross regional product while each 100 million yuan increase in capital formation will contribute around 214.6 million yuan to the gross regional product. Table 3.1. indicates that the t-statistics of coefficient is significant for the role of gross capital formation on the gross regional product. Unfortunately, Table 3.1. does not indicate a significant role for labour which is represented by its proxy, population. In particular, the null hypothesis that the coefficient of labor (proxied by population) is not equal to zero cannot be rejected based on the significance of the statistics at all traditional alphas considered significant or at alphas equal to 0.1, 0.05, and 0.01. In contrast, based on Table 3.1., it is possible for us to reject the null hypothesis that the coefficient of the regression associated for gross capital formation is equal to zero and accept the alternate hypothesis that the coefficient is greater than zero. The significance of 0.000 suggests that we can reject the null hypotheses at all traditional critical alphas 0.1, 0.05, and 0.01. As desired, we can even use critical alpha 0.001 to reject the null hypothesis and accept the alternative hypothesis that the coefficient of capital formation is greater than zero and that the variable significantly affect the gross regional product of China’s 31 provinces. The F-statistics of the regression model is significant up to the 0.000 level, indicating that we can reject the null hypothesis that all of the coefficients of the regressor variables are simultaneously equal to zero to accept the alternate hypothesis that at least two of the coefficients are not equal to zero. The F-statistics suggest that the R-square of the regression is positive and, in one interpretation, is considered as a test of the significance of the R-squared statistics. In other words, based on the F-statistics, it is possible to say that we can reject the null hypothesis that R-square is zero to accept the alternative hypothesis that R-square is positive. The R-square of the model is 0.96 and indicates that the model can explain around 96% of the variation in dependent variable. Annex 1. Provinces of China and gross regional product and gross capital formation Table 1. Regional Gross Product Formation in Chinese Provinces 2007 Province 2007 Gross Regional Product in 100 million yuan (Output) 2007 Gross Capital Formation in 100 million yuan (Investment) Beijing 9353.3 4558 Tianjin 5050.4 2922.5 Hebie 13709.5 6761.3 Shanxi 5731 3204.4 Inner Mongolia 6091.1 4494.4 Liaoning 11023.5 6336.9 Jilan 5601.1 3880 Heilongjiang 7065 3036.9 Shanghai 12188.9 5568.5 Jiangsu 25741.2 12371.2 Zhejiang 18780.4 8512 Anhui 7364.2 3419.7 Fujian 9339.5 4704.6 Jiangxi 5500.3 2767 Shandong 25965.9 12607.5 Henan 15012.5 8366.4 Hubie 9550 4450.3 Hunan 9200 4034.8 Guangdong 31084.4 11148.9 Guangxi 5955.7 3034.5 Hainan 1223.3 558.4 Chongqig 4303.8 2679.2 Sichuan 10505.3 5185.5 Guizhou 2741.9 1421 Yunnan 4741.3 2666.7 Tibet 342.2 272.5 Shaanxi 5465.8 3329.8 Gansu 2702.4 1322.5 Qinghai 783.6 496.7 Ningxia 889.2 654.3 Xinjiang 3523.2 2089.8 Source: Table 2.17 Chinese Central Statistical Office (2008) China Statistical Yearbook 2008, China Press Statistics Press, Beijing Annex 2: GDP per capita and % share of primary sector in total output Table 2. Regional Gross Product Per Capita and the Size of the Primary Sector in Chinese Provinces 2007 Province 2007 GDP per capita in Yuan (Y) % share of primary sector in total output (X) Beijing 58204 1.1 Tianjin 46122 2.2 Hebie 19877 13.2 Shanxi 16945 4.7 Inner Mongolia 25393 12.5 Liaoning 25729 10.3 Jilan 19383 14.8 Heilongjiang 18478 13 Shanghai 66367 0.8 Jiangsu 33928 7.1 Zhejiang 37411 5.3 Anhui 12045 16.3 Fujian 25908 10.8 Jiangxi 12633 16.5 Shandong 27807 9.7 Henan 16012 14.8 Hubie 16206 14.9 Hunan 14492 17.7 Guangdong 33151 5.5 Guangxi 12555 20.8 Hainan 14555 29.5 Chongqig 14660 11.7 Sichuan 12893 19.3 Guizhou 6915 16.3 Yunnan 10540 17.7 Tibet 12109 16 Shaanxi 14607 10.8 Gansu 10346 14.3 Qinghai 14257 10.6 Ningxia 14649 11 Xinjiang 16999 17.8 Source: Table 2.15 Chinese Central Statistical Office (2008) China Statistical Yearbook 2008, China Statistics Press, Beijing. Bibliography Beck, T., Kunt, A., & Levine, R., 2005. SMEs, growth, and poverty: Cross-country evidence. Journal of economic growth, 10 (3), 199-229. Gujarati, D., 2004. Basic econometrics. 4th Ed. New York & London: McGraw Hill Companies. Keynes, J., 1936. The general theory of employment, interest, and money. McClave, J., Benson, G., & Sinich, T., 2000. A first course in business statistics. Upper Saddle: Prentice Hall. Moore, D. & McCabe, G., 1999. Introduction to the practice of statistics. 3rd ed. New York: W.H. Freeman and Company. Temple, J. & Wobmann, L. 2006. Dualism and cross-country growth regressions. Journal of economic growth, 11 (3), 187-228. Walpole, R., 1983. Introduction to statistics. 3rd ed. London: Collier Macmillan Publisher. Walpole, R., Myers, R., Myers, S., & Ye, K., 2007. Probability and statistics for engineers and scientists. 8th ed. London: Pearson Education Ltd. Read More