The linear regression model

Key Takeaways

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 + beta 1 * x + 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 $50 in the USA the number is much bigger somewhere around $3 to $5,000 so for each additional year you spend on education your yearly income is expected to rise by $3 to $5,000 and that's not considering higher education or tailored courses like this one the other two components are the constant beta0 and the error Epsilon in this example you can think of the constant beta zero 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 zero 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 the income the regression predicted on average across all observations the error is zero if you earn more than what the regression has predicted then someone earns less than what the regression has predicted everything evens out all right 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 equation it is simply y Hat = B 0 + B1 * 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 B 0 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 for more videos like this one please subscribe