Variance of X

The intercept (a) of the regression can be read in a number of ways. One interpretation is that it is the value that Y will have when X is zero. Another is more straightforward, and is based upon how it is calculated. It is the difference between the average value of Y, and the slope adjusted value of X.

Intercept of the Regression = a = - b* (^X) Regression parameters are always estimated with some error or statistical noise, partly because the relationship between the variables is not perfect and partly because we estimate them from samples of data. This noise is captured in a couple of statistics. One is the R-squared of the regression, which measures the proportion of the variability in the independent variable (Y) that is explained by the dependent variable (X). It is a direct function of the correlation between the variables -

R - squared of the Regression = Correlation^x = PyX

An R-squared value closer to one indicates a strong relationship between the two variables, !though the relationship may be either positive or negative. Another measure of noise in a regression is the standard error, which measures the "spread' around each of the two parameters estimated- the intercept and the slope. Each parameter has an associated standard error, which is calculated from the data -

Standard Error of Intercept = SEa =

Standard Error of Slope = SEb

If we make the additional assumption that the intercept and slope estimates are normally distributed, the parameter estimate and the standard error can be combined to get a "t statistic" that measures whether the relationship is statistically significant.

T statistic for intercept = a/SEa T statistic from slope = b/SEb For samples with more than 120 observations, a t statistic greater than 1.66 indicates that the variable is significantly different from zero with 95% certainty, while a statistic greater than 2.36 indicates the same with 99% certainty. For smaller samples, the t statistic has to be larger to have statistical significance.1

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