R-squared and the standard error are used to assess the accuracy of calculated statistics
R squared is a measure of how good a fit we get with the regression line
If every data point lies exactly on the line, R squared is 100%
R squared is the square of the correlation coefficient between the security returns and the market returns
It measures the portion of a security’s variability that is due to the market variability
The standard error is the standard deviation divided by the square root of the number of observations:
Standard Errors (cont’d)
The standard error enables us to determine the likelihood that the coefficient is statistically different from zero
About 68% of the elements of the distribution lie within one standard error of the mean
About 95% lie within 1.96 standard errors
About 99% lie within 3.00 standard errors
A runs test allows the statistical testing of whether a series of price movements occurred by chance.
A run is defined as an uninterrupted sequence of the same observation. Ex : if the stock price increases 10 times in a row, then decreases 3 times, and then increases 4 times, we then say that we have three runs.
R = number of runs (3 in this example)
n 1 = number of observations in the first category. For instance, here we have a total of 14 “ups”, so n 1 =14.
n 2 = number of observations in the second category. For instance, here we have a total of 3 “downs”, so n 2 =3.
Note that n 1 and n 2 could be the number of “Heads” and “Tails” in the case of a coin toss.
Let the number of runs R=23
Let the number of ups n 1 =20
Let the number of downs n 2 =30
About 2.5% of the area under the normal curve is below a z score of -1.96.
Since our z-score is not in the lower tail (nor is it in the upper tail), the runs we have witnessed are purely the product of chance.
If, on the other hand, we had obtained a z-score in the upper (2.5%) or lower (2.5%) tail, we would then be 95% certain that this specific occurrence of runs didn’t happen by chance. (Or that we just witnessed an extremely rare event)