Simple Linear Model are models in which the behavior of a variable, Y, can be explained by a variable X: Y = f(X). If we consider that the relation f(), which connects Y with X, is linear, then it can be written as: y = β 0 + β 1x + ε. Since relations of the above type are seldom exact, but rather are approximations in which many variables of secondary importance have been omitted, we must include a random perturbation term, which reflects all factors - other than X - that influence the endogenous variable, but none of them is individually relevant. The parameters β 0 and β 1 determine the intercept and the slope of the line respectively. The intercept β 0 represents the predicted value of y when x = 0 . The slope β 1 represents the predicted increase in Y resulting from a one unit increase in X. The above expression reflects a linear relationship, and only one single explanatory variable, receiving the name of simple linear relation. H...
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The regression model y = Xβ + ε , divides the response into two components: one systematic, Xβ and one random, the error ( ε ). What we want is to choose β so that the systematic components explain as much of the response as possible. We have to find the unknown parameters β that make X β is as close to Y as possible. What is the analytical relationship that best fits our data? The least squares method is a general procedure that allows us to answer this question. Ordinary least squares (OLS) is a method for estimating the unknown parameters β in a linear regression model, with the goal of minimizing the sum of the squares of the differences between the observed response in the given dataset. We have to take into account that It is common to assume the hypothesis that the ordinary least squares method must be used to minimize the residuals (difference between the values in the data set and the adjusted line). Under this ...