Question

Regression Statistics
Multiple R	0.451216205
R Square	0.203596063
Adjusted R Square	0.190097692
Standard Error	0.051791629
Observations	61
ANOVA
	df	SS	MS	F	Significance F
Regression	1	0.040458253	0.040458253	15.083009	0.000262577
Residual	59	0.158259997	0.002682373
Total	60	0.19871825
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	0.00987396	0.006785133	1.455234544	0.150904641	-0.00370306	0.023450979	-0.00370306	0.023450979
S&P	0.752212332	0.193685208	3.883684976	0.000262577	0.364649126	1.139775537	0.364649126	1.139775537
Current estimate given to us in the directions
	1.07
RESIDUAL OUTPUT
Observation	Predicted Y	Residuals	Standard Residuals
1	0.038198737	-0.01978845	-0.385302506
2	-0.00179574	0.144257104	2.808841664

1. How does your estimate of beta compare with the beta estimate provided (1.07)? Why might your estimate differ from estimated beta of 1.07?

2. How much of the variability of your security’s return is “explained” by the variability of returns in the “market”? (Note: In your case, the market is represented by the S&P 500 Index.) Do you think that a different market index might be a better representation of the market for your particular security? Why/Why not?

3. What is the correlation of returns for your security with the market for the selected time period? Might this relationship change over time, and if so, how and why?

4. Does the relationship between your security and the market appear to be statistically significantly different than zero? What evidence from the regression supports your conclusion?

5. Review the standardized residuals and comment about the importance of individual data points (if any) that may have influenced your estimation of beta. (observation 2 is the only skewed one)

Answer 1

1. I think the estimate of beta provided here is instructed, maybe it is a regression outcome from a different regression model. The current model has an output of beta which is in the same direction but as it is a linear regression model, so it will have intercept and slope. So, beta will determine the slope of the straight line. As the beta obtained from the outcome is smaller than the beta mentioned here in the question, hence we can say that the straight line will be slightly less steep than the mentioned regression line because the slope is smaller than the mentioned one.

2. The regression Sum of Square is reported to 0.040458253, which means that only 4% variability can be explained by the model.

I think maybe different variable can be more predictive power than the S&P500 index. Because as for the regression perspective the value of the explained variability is very much less. May be multiple linear regression (Regression with more than one explanatory variables) will give better results.

3. This question might need the data; if I am not wrong. In this case, feel free to write down the query in the comment section, and I shall be happy to answer it. For now, the correlation (rather, relation) with the security and the S&P500 index is slightly positive as per the regression outcome. one unit of increase in the S&P500 index will have 75% increment in the dependent variable, which is security.

4. Yes, as per the regression model, the relationship between the dependent and independent variable is statistically significant. Because, the p-value is 0.000262577, which is lesser than the model level of significance 0.05.

5. The standardized residual is a measure of the strength of the difference between observed and expected values. It’s a measure of how significant your cells are to the chi-square value. When you compare the cells, the standardized residual makes it easy to see which cells are contributing the most to the value, and which are contributing the least. If your sample is large enough, the standardized residual can be roughly compared to a z-score. Standardization can work even if your variables are not normally distributed.

A general rule of thumb for figuring out what the standardized residual means, is:

• If the residual is less than -2, the cell’s observed frequency is less than the expected frequency.
• Greater than 2 and the observed frequency is greater than the expected frequency.

Here, in this case, the standardized residual exceeds +2 and tends to +3 which indicates that the predicted value is not in the allows bound of the regression model. Hence you should check the model once again. It may happen that the mentioned point is a rare one (probably one) in the entire data model, but if the number of these kinds of points is more, then you should check the linearity of the model once again.

Hope this answer has helped you.

Thanks !!