Assumption SLR.1 Linear in Parameters In the population model, the dependent var

Assumption SLR.1
Linear in Parameters
In the population model, the dependent variable, y, is related to the independent variable, x, and the error (or disturbance), u, as
[2.47]
where and are the population intercept and slope parameters, respectively.
To be realistic, y, x, and u are all viewed as random variables in stating the population model. We discussed the interpretation of this model at some length in Section 2-1 and gave several examples. In the previous section, we learned that equation (2.47) is not as restrictive as it initially seems; by choosingy and x appropriately, we can obtain interesting nonlinear relationships (such as constant elasticity models).
We are interested in using data on y and x to estimate the parameters and, especially, . We assume that our data were obtained as a random sample. (See Math Refresher C for a review of random sampling.)
Assumption SLR.2
Random Sampling
We have a random sample of size n, , following the population model in equation (2.47).
We will have to address failure of the random sampling assumption in later chapters that deal with time series analysis and sample selection problems. Not all cross-sectional samples can be viewed as outcomes of random samples, but many can be.
We can write (2.47) in terms of the random sample as
[2.48]
where is the error or disturbance for observation i (for example, person i, firm i, city i, and so on). Thus, contains the unobservables for observation i that affect . The should not be confused with the residuals, , that we defined in Section 2-3. Later on, we will explore the relationship between the errors and the residuals. For interpreting and in a particular application, (2.47) is most informative, but (2.48) is also needed for some of the statistical derivations.