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Regression analysis & Confidence Interval
Regression analysis
The standard mathematical way to present the relationship when Y
is a linear function of X
is:
In this relationship, Yis the dependent variable andXis the
independent variable while a
represents the value at which the line intersects the Y axis on
a graph and b is the slope
coefficient of the line.
Regression analysis is a statistical method to evaluate the
relationship and causality between
different variables by fitting a function to a given dataset. It
is a powerful and flexible tool
that allows to disintegrate the effect of many factors on a
specific indicator.
For example, the GDP growth in a country depends, to different
extent, on the
unemployment rate, terms of trade, wage levels, consumption etc.
Regressions are often
used to forecast or estimate values that are unobservable.
In the graph below, you can see a scatterplot where each blue
dot represents an
observation iwith the given values for X (horizontal axis) and Y
(vertical axis).
The red line is fitted to the dataset using the Ordinary Least
Squares (OLS) method. The OLS
method estimates the parameters of the relationship (the
intercept a and the slope
coefficient b) so that the sum of the squared deviations of each
observation from the fitted
line is minimized for the dataset.
Given that all of the data points do not lie on the line, an
error term must be added to the
relationship equation to reflect the uncertainty in the
estimation of unknown values. This
error term is assumed to be an independent and identically
distributed (i.i.d.) random
variable, meaning that the error terms across the sample are
uncorrelated and have the
same statistical distribution with an expected mean equal to
zero.
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A common regression model in finance is the market model (the
formula below) which
states that the return on a financial asset depends on the
return on the market and the
companys riskiness. Thus, the return on the asset is the
dependent variable and the return
on the market portfolio is the independent variable. The
intercept is noted with alpha, the
slope coefficient with beta and error term with epsilon. Beta
measures the responsiveness
of the stock to the market portfolio and is also considered to
be a measure of risk. If the
company is listed and its stock returns are observable, then the
riskiness of the companyrelative to the market can be estimated by
running a regression. Usually, the market return
is proxied by the return on the market index.
Confidence interval
The confidence interval indicates the reliability of an
estimate. The confidence interval is the
set of values reasonably consistent with the observed results.
For example, the 99%
confidence interval is the range of values that has a 99% chance
of containing the true value
of the estimated variable.
With less rigor, it is possible to say that the confidence
interval represents the range of
values within which we are 99% certain to find the true value we
are looking for.
Alternatively, we coud say that if an estimated variable is
within the 99% confidence interval,
then we are 99% sure that the true value of the estimated
variable is in the interval. We can
thus accept the estimated value as it has only 1% chance of
being outside the interval.
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The confidence interval gives an indication of the uncertainty
of the estimate. Within theconfidence interval, the true valueis
not statistically significantly different from the observed
result. The upper and lower bounds are by definition the limit
values. If the observed result
is above the upper limit or below the lower limit then the true
value is statistically
significantly different from the estimate. Consequently, the
estimate is not good enough.
The confidence level (or confidence coefficient) gives the
frequency at which the observed
interval contains the sought after parameter, the 95% level is
used most commonly.
