Consider a k-variables linear regression model, i.e., Y = X 1β1 + X 2 β2 + ε, Where, X1 is (N . k1 ) , X 2 is (N . k2 ) and k = k1 + k2 . As you may recall, adding columns to the X matrix (including additional regressors in the model) gives a positive definite increase in R2. The adjusted R2 attempts to avoid this phenomenon of ever-increasing R2. Show that the additional k2 number of variables (regressors) in this model increases R2 if the calculated F-statistic in testing the joint statistical significance of coefficients of these additional regressors (β2 ) are larger than one.
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
Consider a k-variables linear regression model, i.e.,
Y = X 1β1 + X 2 β2 + ε,
Where, X1 is (N . k1 ) , X 2 is (N . k2 ) and k = k1 + k2 . As you may recall, adding columns to
the X matrix (including additional regressors in the model) gives a positive definite increase in R2.
The adjusted R2 attempts to avoid this phenomenon of ever-increasing R2. Show that the
additional k2 number of variables (regressors) in this model increases R2 if the calculated F-statistic in testing the joint statistical significance of coefficients of these additional
regressors (β2 ) are larger than one.
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