Implementing
OLS regression
2.
Implementing
goodness of fit –chi square
Concept:
In statistics, ordinary least squares (OLS) or linear
least squares is a method for
estimating the unknown parameters in a linear regression model. This
method minimizes the sum of squared vertical distances between the observed
responses in the dataset and
the responses predicted by the linear approximation. The resulting estimator can be expressed by a simple formula,
especially in the case of a single regressor on the right-hand side.
OLS is used in economics (econometrics),
political science and electrical engineering (control
theory and signal
processing), among many areas of application.
A chi-squared
test, also referred to as chi-square
test or 
test, is any statistical hypothesis
test in which
the sampling distribution of the test statistic is a chi-squared distribution when the null
hypothesis is
true. Also considered a chi-squared test is a test in which this is asymptotically true, meaning that the sampling
distribution (if the null hypothesis is true) can be made to approximate a
chi-squared distribution as closely as desired by making the sample size large
enough.
test, is any statistical hypothesis
test in which
the sampling distribution of the test statistic is a chi-squared distribution when the null
hypothesis is
true. Also considered a chi-squared test is a test in which this is asymptotically true, meaning that the sampling
distribution (if the null hypothesis is true) can be made to approximate a
chi-squared distribution as closely as desired by making the sample size large
enough.
Pearson's chi-squared test is used to assess two types of comparison:
tests of goodness of fit and tests of independence.
·
A test
of goodness of fit establishes whether or not an observed frequency distribution differs from a
theoretical distribution.
·
A test
of independence assesses whether paired observations on two variables,
expressed in a contingency table, are independent of each
other (e.g. polling responses from people of different nationalities to see if
one's nationality is related to the response).
The procedure of the test includes the
following steps:
1.
Calculate
the chi-squared test statistic, , which resembles a normalized sum of squared deviations
between observed and theoretical frequencies
, which resembles a normalized sum of squared deviations
between observed and theoretical frequencies
2.
Determine
the degrees of freedom, df,
of that statistic, which is essentially the number of frequencies reduced by
the number of parameters of the fitted distribution.
3.
Compare to the critical value from the chi-squared distribution with d degrees
of freedom, which in many cases gives a good approximation of the distribution
of .
to the critical value from the chi-squared distribution with d degrees
of freedom, which in many cases gives a good approximation of the distribution
of .
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