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# Standard & Poor’s 500 in Multiple Regression Model Report (Assessment)

## Multiple Regression Model Analysis

In the current report, two statistical models are developed to analyze changes observed in the value of a stock market benchmark i.e. Standard & Poor’s (S&P) 500. For this purpose, the data was collected from 1980 to 2011. Each model is discussed in the following under separate headings to understand variables included and their impact on changes in the value of the S&P 500 over a period of 31 years.

The multiple-regression model is used for testing the relationship between the S&P 500 (%) and selected independent variables (Chatterjee and Hadi 4). The following null and alternative hypotheses are set for testing the relationship between S&P 500 (%) and each independent variable.

Null: H0: There is no significant relationship between changes in S&P 500 (%) and the independent variable.

Alternative: H1: There is a significant relationship between changes in S&P 500 (%) and the independent variable.

The level of significance is determined by comparing the p-value of a coefficient with the confidence level of 95% or 0.05. The criterion is to accept the null hypothesis if the value of p is greater than 0.05 and reject it otherwise (Seber and Lee 97).

### Model 1

The first model considers percentage changes in the value of the S&P 500 on a year-over-year basis (YoY) as the dependent variable (S&P 500 %). The independent variables included in this model are Annual CPI, Annual Average PPI, Annual, Average HPI, Annual Average Interest Rate, Percentage Change GDP for the US, Percentage Change GDP for Spain, and Percentage Change GDP for Germany. The data of these independent variables were also collected for the same period from 1980 to 2011.

The results obtained from testing are provided in the following along with their discussion.

Table 1

Model 1 Summary

 Regression Statistics R 0.4137 R-square 0.17114 Adjusted R-square -0.08112 S 17.42485 N 31

Table 1 provides the model summary of regression statistics. The value of R-square is 0.17114 that implies that the model could only explain 17.114% of the total variations observed in 31 data entries. The low value of R-square indicates the model implemented is not sufficient to explain the relationship between the selected variables.

Table 2

ANOVA Model 1

 ANOVA d.f. SS MS F p-level Regression 7 1,441.94 205.9915 0.67844 0.68857 Residual 23 6,983.38 303.62538 Total 30 8,425.32

Table 2 also supports that the findings of the regression model are not significant as the p-value is greater than 0.05. Only 1,441.94 of the total variations are explained by the model, which is very less.

Table 3

Coefficients Model 1

 Coefficient Standard Error LCL UCL t Stat p-level Intercept 49.44335 75.41912 -106.57298 205.45968 0.65558 0.5186 Annual CPI -0.00492 0.60432 -1.25507 1.24522 -0.00815 0.99357 Annual Average PPI -0.0747 0.58458 -1.28399 1.13459 -0.12778 0.89943 Annual Average HPI -0.0824 0.16388 -0.42142 0.25661 -0.50281 0.61988 Annual Average Interest rate -3.23241 3.83065 -11.15672 4.69189 -0.84383 0.40746 Percentage change GDP for the US 2.23877 1.74595 -1.373 5.85055 1.28227 0.21252 Percentage change GDP for Spain -0.48194 2.89487 -6.47044 5.50655 -0.16648 0.86923 Percentage change of GDP for Germany -0.52996 2.4306 -5.55804 4.49812 -0.21804 0.82932 T (5%) 2.06866

The regression equation obtained from the coefficients in Table 3 is provided in the following.

S&P 500 = 49.44335 – 0.00492 * Annual CPI – 0.0747 * Annual Average PPI – 0.0824 * Annual Average HPI – 3.23241 * Annual Average Interest rate + 2.23877 * Percentage change GDP for US – 0.48194 * Percentage change GDP for Spain – 0.52996 * Percentage change of GDP for Germany

The coefficients of slope obtained indicated that the is a negative relationship between S&P 500 (%) and Annual CPI, Annual Average PPI, Annual Average Interest rate, Percentage change GDP for Spain, and Percentage change of GDP for Germany. The negative relationship implies that the value of the S&P 500 increased in the past with a decrease in the value of these variables. There is a positive relationship between the S&P 500 (%) and Percentage change GDP for the US as values of both changes in the same direction. Table 3 also indicates that the null hypothesis is accepted for all relationships as the p-value is greater than 0.05.

### Model 2

The first model considers the value of the S&P 500 on a year-over-year basis (YoY) as the dependent variable (S&P 500 value). The independent variables included in this model are Annual Average CPI, Annual Average HPI, Annual Average Interest rate, Average annual Unemployment rate, GDP of US (trillions), GDP for Germany (trillions), and GDP for China (trillions). The results obtained from testing are provided in the following.

Table 4

Model 2 Summary

 Regression Statistics R 0.97984 R-square 0.96008 Adjusted R-square 0.94844 S 108.60518 N 32

Table 4 provides the model summary of regression statistics. The value of R-square is 0.96008 that implies that the model could only explain 96.008% of the total variations observed in 32 data entries. The high value of R-square indicates the model implemented is sufficient to explain the relationship between the selected variables.

Table 5

ANOVA Model 2

 ANOVA d.f. SS MS F p-level Regression 7 6,808,210.63 972,601.52 82.4582 0 Residual 24 283,082.06 11,795.09 Total 31 7,091,292.69

Table 2 also supports that the findings of the regression model are significant as the p-value is less than 0.05.

Table 6

Coefficients Model 2

 Coefficient Standard Error LCL UCL t Stat p-level Intercept 1,103.29 587.74259 -109.75232 2,316.33 1.87716 0.0727 Annual Average CPI -24.23912 6.5276 -37.71142 -10.76682 -3.71333 0.00108 Annual Average HPI -10.07565 1.61286 -13.40444 -6.74686 -6.24706 1.86E-06 Annual Average Interest rate 32.51144 21.62016 -12.11037 77.13325 1.50376 0.14569 Average annual Unemployment rate -29.19836 20.14888 -70.7836 12.38688 -1.44913 0.16024 GDP of US (trillions) 652.31822 90.32463 465.89734 838.73909 7.22193 1.84E-07 GDP for Germany (trillions) 59.92468 85.16407 -115.84533 235.69469 0.70364 0.48843 GDP for China (trillions) -210.26035 42.32493 -297.61472 -122.90599 -4.96777 0.00005 T (5%) 2.0639

The regression equation obtained from the results is provided in the following.

S&P 500 = 1,103.28877 – 24.23912 * Annual Average CPI – 10.07565 * Annual Average HPI + 32.51144 * Annual Average Interest rate – 29.19836 * Average annual Unemployment rate + 652.31822 * GDP of US (trillions) + 59.92468 * GDP for Germany (trillions) – 210.26035 * GDP for China (trillions)

The coefficients of slope obtained indicated that the is a negative relationship between S&P 500 (value) and Annual Average CPI, Annual Average HPI, Average annual Unemployment rate, and GDP for China. The negative relationship implies that the value of the S&P 500 increased in the past with a decrease in the value of these variables. There is a positive relationship between the S&P 500 (value) and the Annual Average Interest rate, GDP of the US, and GDP for Germany as values of both moves in the same direction. Table 3 also indicates that the null hypothesis is accepted for the Annual Average Interest rate, Average annual Unemployment rate, and GDP for Germany (trillions) as the p-value is greater than 0.05. On the other hand, the null hypothesis is rejected for Annual Average CPI, Annual Average HPI, GDP of US (trillions), and GDP for China as the p-value is less than 0.05.

## Conclusion

It could be concluded that Model 2 is superior to Model 1 as it explains the variations in S&P 500 in a better way.

## Works Cited

FRED, 2017. Web.

Chatterjee, ‎Samprit and Ali S. Hadi. Regression Analysis by Example. John Wiley & Sons, 2013.

US Inflation Calculator, 2017. Web.

Bureau of Labor Statistics, 2017. Web.

OECD, 2017. Web.

Seber, ‎George A. F. and Alan J. Lee. Linear Regression Analysis. John Wiley & Sons, 2012.

Multpl, 2017. Web.

## Appendix

Model-1 Data

 Year S&P 500 Annual CPI Annual Average PPI Annual Average HPI Annual Average Interest rate Percentage change GDP for US Percentage change GDP for Spain Percentage change of GDP for Germany 1980 82.40 89.80 102.69 11.43 9.62 2.21 1.41 1981 -9.73 90.90 98.00 107.78 13.92 9.69 -0.13 0.53 1982 14.76 96.50 100.00 111.47 13.01 3.79 1.25 -0.39 1983 17.27 99.60 101.30 115.78 11.10 11.39 1.77 1.57 1984 1.40 103.90 103.70 120.79 12.46 9.26 1.78 2.82 1985 26.33 107.60 103.20 126.96 10.62 7.37 2.32 2.33 1986 14.62 109.60 100.20 135.58 7.67 4.86 3.25 2.29 1987 2.03 113.60 102.80 144.30 8.39 7.57 5.55 1.4 1988 12.40 118.30 106.90 152.01 8.85 7.76 5.09 3.71 1989 27.25 124.00 112.20 160.01 8.49 6.48 4.83 3.9 1990 -6.56 130.70 116.30 165.00 8.55 4.51 3.78 5.26 1991 26.31 136.20 116.50 168.14 7.86 4.25 2.55 5.11 1992 4.46 140.30 117.20 172.83 7.01 6.66 0.93 1.92 1993 7.06 144.50 118.90 176.85 5.87 5.00 -1.03 -0.96 1994 -1.54 148.20 120.40 181.38 7.09 6.31 2.38 2.45 1995 34.11 152.40 124.70 186.91 6.57 4.32 2.76 1.7 1996 20.26 156.90 127.70 193.40 6.44 6.25 2.67 0.78 1997 31.01 160.50 127.60 199.93 6.35 6.05 3.69 1.82 1998 26.67 163.00 124.40 210.17 5.26 6.11 4.31 1.97 1999 19.53 166.60 125.50 220.53 5.65 6.44 4.48 1.99 2000 -10.14 172.20 132.70 234.64 6.03 5.50 5.29 2.98 2001 -13.04 177.10 134.20 252.17 5.02 2.19 4 1.7 2002 -23.37 179.90 131.10 268.14 4.61 3.76 2.88 0.01 2003 26.38 184.00 138.10 284.82 4.01 6.42 3.19 -0.72 2004 8.99 188.90 146.70 311.43 4.27 6.31 3.17 1.18 2005 3.00 195.30 157.40 346.72 4.29 6.52 3.72 0.71 2006 13.62 201.60 164.70 371.55 4.80 5.12 4.17 3.71 2007 3.53 207.30 172.60 375.70 4.63 4.40 3.77 3.27 2008 -38.49 215.30 189.60 356.74 3.66 -0.92 1.12 1.05 2009 23.45 214.54 172.90 336.96 3.26 0.11 -3.57 -5.64 2010 12.78 218.06 184.70 323.20 3.22 4.56 0.01 4.09 2011 -0.0032 224.94 201.00 310.93 2.78 3.64 -0.62 3.59

Model-2 Data

 Year S&P 500 Annual Average CPI Annual Average HPI Annual Average Interest rate Average annual Unemployment rate GDP of US (trillions) GDP for Germany (trillions) GDP for China (trillions) 1980 135.76 82.40 102.69 11.43 7.60 2.99 0.95 0.19 1981 122.55 90.90 107.78 13.92 7.50 3.28 0.80 0.19 1982 140.64 96.50 111.47 13.01 8.60 3.41 0.77 0.20 1983 164.93 99.60 115.78 11.10 10.40 3.80 0.77 0.23 1984 167.24 103.90 120.79 12.46 8.00 4.15 0.72 0.26 1985 211.28 107.60 126.96 10.62 7.30 4.45 0.73 0.31 1986 242.17 109.60 135.58 7.67 6.70 4.67 1.04 0.30 1987 247.08 113.60 144.30 8.39 6.60 5.02 1.29 0.27 1988 277.72 118.30 152.01 8.85 5.70 5.41 1.40 0.31 1989 353.40 124.00 160.01 8.49 5.40 5.76 1.39 0.35 1990 330.22 130.70 165.00 8.55 5.40 6.02 1.76 0.36 1991 417.09 136.20 168.14 7.86 6.40 6.28 1.86 0.38 1992 435.71 140.30 172.83 7.01 7.30 6.70 2.12 0.42 1993 466.45 144.50 176.85 5.87 7.30 7.03 2.07 0.44 1994 459.27 148.20 181.38 7.09 6.60 7.48 2.21 0.56 1995 615.93 152.40 186.91 6.57 5.60 7.80 2.59 0.73 1996 740.74 156.90 193.40 6.44 5.60 8.29 2.50 0.86 1997 970.43 160.50 199.93 6.35 5.30 8.79 2.22 0.96 1998 1229.23 163.00 210.17 5.26 4.60 9.33 2.24 1.03 1999 1469.25 166.60 220.53 5.65 4.30 9.93 2.20 1.09 2000 1320.28 172.20 234.64 6.03 4.00 10.47 1.95 1.21 2001 1148.08 177.10 252.17 5.02 4.20 10.70 1.95 1.33 2002 879.82 179.90 268.14 4.61 5.70 11.10 2.08 1.46 2003 1111.92 184.00 284.82 4.01 5.80 11.82 2.50 1.65 2004 1211.92 188.90 311.43 4.27 5.70 12.56 2.82 1.94 2005 1248.29 195.30 346.72 4.29 5.30 13.38 2.86 2.27 2006 1418.30 201.60 371.55 4.80 4.70 14.07 3.00 2.73 2007 1468.36 207.30 375.70 4.63 4.60 14.69 3.44 3.52 2008 903.25 215.30 356.74 3.66 5.00 14.55 3.75 4.56 2009 1115.10 214.54 336.96 3.26 7.80 14.57 3.42 5.06 2010 1257.64 218.06 323.20 3.22 9.80 15.23 3.42 6.04 2011 1257.60 224.94 310.93 2.78 9.20 15.79 3.76 7.49

Sources: (“Gross domestic product (GDP)”; “US GDP Growth Rate by Year”; “Databases, Tables & Calculators by Subject”; “Annual changes of the Producer Price Index (PPI) for commodities in the United States from 1990 to 2015”; “Consumer Price Index Data from 1913 to 2017”; “10-Year Treasury Constant Maturity Rate (DGS10)”)

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