Large
amounts of data are often compressed into more easily assimilated summaries,
which provide the user with a sense of the content, without overwhelming him or
her with too many numbers. There a number of ways in which data can be
presented. One approach breaks the numbers down into individual values (or
ranges of values) and provides probabilities for each range. This is called a
"distribution". Another approach is to estimate "summary
statistics" for the data. For a data series, X_{1}, X_{2},
X_{3}, ....X_{n}, where n is the number of observations in the
series, the most widely used summary statistics are as follows –

• the
__mean__ (m), which is the average
of all of the observations in the data series

• the
__median__, which is the mid-point of the series; half the data in the
series is higher than the median and half is lower

• the
__variance__, which is a measure of the spread in the distribution around
the mean, and is calculated by first summing up the squared deviations from the
mean, and then dividing by either the number of observations (if the data
represents the entire population) or by this number, reduced by one (if the
data represents a sample)

When
there are two series of data, there are a number of statistical measures that
can be used to capture how the two series move together over time. The two most
widely used are the __correlation__ and the __covariance__. For two data
series, X (X_{1}, X_{2},.) and Y(Y,Y... ), the covariance provides a non-standardized
measure of the degree to which they move together, and is estimated by taking
the product of the deviations from the mean for each variable in each period.

The** **sign on the covariance indicates the type of
relationship that the two variables have. A positive sign indicates that they
move together and a negative that they move in opposite directions. While the
covariance increases with the strength of the relationship, it is still
relatively difficult to draw judgements on the strength of the relationship
between two variables by looking at the covariance, since it is not
standardized.

The correlation is the standardized measure of the relationship between two variables. It can be computed from the covariance –

The correlation can never be greater than 1 or less than minus 1. A correlation close to zero indicates that the two variables are unrelated. A positive correlation indicates that the two variables move together, and the relationship is stronger the closer the correlation gets to one. A negative correlation indicates the two variables move in opposite directions, and that relationship also gets stronger the closer the correlation gets to minus 1. Two variables that are perfectly positively correlated (r=1) essentially move in perfect proportion in the same direction, while two assets which are perfectly negatively correlated move in perfect proportion in opposite directions.

A
simple regression is an extension of the correlation/covariance concept. It
attempts to explain one variable, which is called the __dependent variable__,
using the other variable, called the __independent variable__. Keeping with
statistical tradition, let Y be the dependent variable and X be the independent
variable. If the two variables are plotted against each other on a scatter
plot, with Y on the vertical axis and X on the horizontal axis, the regression
attempts to fit a straight line through the points in such a way as the
minimize the sum of the squared deviations of the points from the line.
Consequently, it is called ordinary least squares (OLS) regression. When such a
line is fit, two parameters emerge – one is the point at which the line
cuts through the Y axis, called the intercept of the regression, and the other
is the slope of the regression line.

OLS Regression: Y = a + b X

The slope (b) of the regression measures both the direction and the magnitude of the relation. When the two variables are positively correlated, the slope will also be positive, whereas when the two variables are negatively correlated, the slope will be negative. The magnitude of the slope of the regression can be read as follows - for every unit increase in the dependent variable (X), the independent variable will change by b (slope). The close linkage between the slope of the regression and the correlation/covariance should not be surprising since the slope is estimated using the covariance –

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.

Regression parameters are always estimated with some noise, partly because the data is measured with error 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 Y that is explained by X. It is a direct function of the correlation between the variables –

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 = SE_{a} =

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/SE_{a}

T statistic from slope = b/SE_{b}

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]

The regression that measures the relationship between two variables becomes a multiple regression when it is extended to include more than one independent variables (X1,X2,X3,X4..) in trying to explain the dependent variable Y. While the graphical presentation becomes more difficult, the multiple regression yields a form that is an extension of the simple regression.

Y = a + b X1 + c X2 + dX3 + eX4

The R-squared still measures the strength of the relationship, but an additional R-squared statistic called the adjusted R squared is computed to counter the bias that will induce the R-squared to keep increasing as more independent variables are added to the regression. If there are k independent variables in the regression, the adjusted R squared is computed as follows –

In theory, the independent variables in a regression need to be uncorrelated with each other. In practice, they often are, and this correlation across independent variables is called multi-collinearity. When there is multi-collinearity,

· The coefficients on each of the independent variables becomes much more difficult to read in isolation, since variables start proxying for each other.

· The reported t-statistics tend to overstate the significance of the relationship. There are statistical approaches available for dealing with multi-collinearity.

· The regression still has predictive power

Both the simple and the multiple regression are based upon a linear relationship between the dependent variable and the independent variable. When the relationship is non-linear, using a linear regression will yield incorrect predictions. In such cases, the independent variables will need to be transformed to make the relationship more linear.

[1] The actual values that t statistics need to take on can be found in a table for the t distribution, which is reproduced at the end of this book as an appendix.