Month: February 2023

Testing the significance of a slope coefficient is a first diagnostic check of the suitability of a regression model. However, per se, this test does not tell us much about the predictive power of the model as a whole. The reader may better appreciate the point by thinking about a multiple regression model involving several…

Now we are armed with the necessary knowledge to draw statistical inferences about the regression parameters. Mirroring what we did with the estimation of expected value, we should The technicalities involved here are essentially the same as those involved in dealing with estimation of the expected value, and we avoid repeating the reasoning. In the…

Equations (10.16) and (10.18) help us in assessing the uncertainty about the estimate of unknown regression parameters. A missing piece in this puzzle, however, is the standard deviation  of the random errors, which are not directly observable. The only viable approach we have is to rely on the residuals  as a proxy for the errors . The intuition is…

In this section we want to tackle a few statistical issues concerning the estimation of the unknown parameters of the data-generating model, featuring a nonstochastic regressor and homoskedastic errors: As we said, the values xi are numbers and the errors  are independent and identically distributed random variables satisfying the following assumptions: Our task mirrors what we did when…

So far, we have regarded linear regression as a numerical problem, which is fairly easy to solve by least squares. However, this is a rather limited view. To begin with, it would be useful to gain a deeper insight into the meaning of the regression coefficients, in particular the slope b. The careful reader might have…

If we label a model like y = a + bx as linear, no eyebrow should be raised. Now, consider a regression model involving a squared explanatory variable: Is this linear? Actually it is, in terms of the factor that matters most: fitting model coefficients. True, the model is nonlinear in terms of the explanatory variable x, but the actual unknowns when…

In the least-squares method, we square residuals and solve the corresponding optimization problem analytically. We should wonder what is so special with squared residuals. We might just as well take the absolute values of the residuals and solve Another noteworthy point is that in so doing we are essentially considering average values of squared or…

Consider the data tabulated and depicted in Fig. 10.1. These joint observations are displayed as circles, and a look at the plot suggests the possibility of finding a linear relationship between x and y. A linear law relating the two variables, such as1 can be exploited to understand what drives a social or physical phenomenon, and is one way of exploiting…

We take advantage of all the probabilistic and statistical knowledge we have built in the to get into the realm of empirical model building. Models come in many forms, but what we want to do here is finding a relationship between two variables, say, x and y, based on a set of n joint observations (xi, yi), i = 1,…,n.…