Excel

SUMMARY OUTPUT- SIMPLE LINEAR REGRESSION X=Independent variable e is error term (residual)

Formula for line > Y = a + bX + e Regression Statistics a and b are estimates a is intercept (constant) b is relationship (slope coefficient)

Multiple R (square root) 0.65346695 < This is the absolute value of the correlation coefficient If our beta was negative we'd interpret this as negative.

R Square 0.4270191 < Tells us the strength of our relationship. In this case 42% of Y is explained by X. Adjusted R Square 0.42220409 R Squred is also the percentage of the stock's risk that is systematic Standard Error 0.01218616 of residuals (data points not explained by the line) - this is tracking error for ETFs Observations 121 data points

ANOVA is Analysis of Variance Sum of Squares Model SS F-stat tells us if there is any relationship at all between X & Y df is "degrees of freedom" df SS MS F Significance F If sig F this is <.5 then we are 95% conf.

Regression 1 0.0131701 0.0131701 88.68579 4.4945E-16 that we have found a relationship. Residual 119 0.0176718 0.0001485 Total 120 0.0308418

We want p-values to be low; less than 0.05 is 'stat. significant' Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%

Intercept (estimate of a) -0.00031701 0.0011187 -0.28338 0.7773767 -0.00253 0.00190 -0.00253 0.00190 X Variable 1 Beta (Slope) 0.641077 0.0680743 9.41731 4.494E-16 0.50628 0.77587 0.50628 0.77587 ^ ^Actual relationship and magnitude

^ A multiple regression would have multiple independent X variables listed here.

Standard deviation and standard error are not the same thing. Standard deviation is referring to the raw data. Standard error is referring to your parameter estimates.

Interpretation of the F-test for model signficance If the F-test is 0, don't consider anything else on the page. If the F-test is signficantly positive, the "Significance F" will be less than 0.05 If F-stat is significant (large) but R Square is very small, then there is a relationship but it is not explained by a straight line. F-stat doesn't tell us if we have the "right" model, only if there is a strong relationship. We need to make sure we have enough variables.

confidence intervals for the parameter estimates

Y=Dependent