hi it is time for the actual
introduction of regressions let's start
with some dry theory a linear regression
is a linear approximation of a causal
relationship between two or more
variables regressions models are highly
valuable as they are one of the most
common ways to make inferences and
predictions the process goes like this
you get sample data come up with a model
that explains the data and then make
predictions for the whole population
based on the model you have developed
there is a dependent variable labeled Y
being predicted and an independent
variable labeled x1 x2 and so forth
these are the predictors Y is a function
of the X variables and the regression
model is a linear approximation of this
function the easiest regression model is
the simple linear regression y is equal
to beta 0 plus beta 1 times X plus
Epsilon let's see what these values mean
Y is the variable we are trying to
predict and is called the dependent
variable X is the independent variable
when using regression analysis we want
to predict the value of y provided we
have the value of x but to have a
regression Y must depend on X in some
causal way whenever there is a change in
X such change must translate into a
change in Y think about the following
equation the income a person receives
depends on the number of years of
education that person has received the
dependent variable is income while the
independent variable is the years of
education there is a causal relationship
between the two the more education you
get the higher income you are likely to
receive this relationship is so trivial
that it is probably the reason you are
watching this course right now you want
to get a higher income so you are
increasing your education now let's
pause for a second and think about the
reverse relationship
what if education depends on income this
would mean the higher your income the
more years you spend educating yourself
putting high tuition fees aside
wealthier individuals don't spend more
years
in school and high school and college
take the same number of years no matter
your tax bracket
therefore a causal relationship like
this one is faulty if not plain wrong
hence it is unfit for regression
analysis let's go back to the original
example income is a function of
education the more years you study the
higher income you will receive this
sounds about right
all right what we haven't mentioned so
far is that in our model there are
coefficients beta 1 is the coefficient
that stands before the independent
variable it quantifies the effect of
Education on income if beta 1 is 50 then
for each additional year of education
your income would grow by $50 in the USA
the number is much bigger somewhere
around three to five thousand dollars so
for each additional year you spend on
education your yearly income is expected
to rise by three to five thousand
dollars and that's not considering
higher education or tailored courses
like this one the other two components
are the constant beta 0 and the error
epsilon in this example you can think of
the constant beta 0 as the minimum wage
no matter your education if you have a
job you will get the minimum wage this
is a guaranteed amount so if you never
went to school and plug in an education
value of 0 years in the formula the
regression will predict that your income
will be the minimum wage makes sense
right
the last term is Epsilon this represents
the error of estimation the error is the
actual difference between the observed
income and B income the regression
predicted on average across all
observations the error is 0 if you earn
more than what the regression has
predicted then someone earns less than
what the regression has predicted
everything evens out alright the
original formula was written with Greek
letters what does this tell us
it was the population formula but we
know statistics is all about sample data
in practice
we use the linear regression equation it
is simply y hat equals b0 plus b1 times
X you heard it right the Y here is
referred to as Y hat whenever we have a
hat symbol it is an estimated or a
predicted value b0 is the estimate of
the regression constant beta 0 while b1
is the estimate of beta 1 and X is the
sample data for the independent variable
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