## 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: H_{0}: There is no significant relationship between changes in S&P 500 (%) and the independent variable.

Alternative: H_{1}: 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

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

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

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

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

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

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

“10-Year Treasury Constant Maturity Rate (DGS10).” *FRED, *2017. Web.

“Annual changes of the Producer Price Index (PPI) for commodities in the United States from 1990 to 2015.” *Statista, *2017. Web.

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

“Consumer Price Index Data from 1913 to 2017.” *US Inflation Calculator, *2017. Web.

“Databases, Tables & Calculators by Subject.” *Bureau of Labor Statistics, *2017. Web.

“Gross domestic product (GDP).” *OECD,* 2017. Web.

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

“US GDP Growth Rate by Year.” *Multpl, *2017. Web.

## Appendix

Model-1 Data

Model-2 Data

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)”)