tailieunhanh - Econometric theory and methods, Russell Davidson - Chapter 3

Chapter 3 The Statistical Properties of Ordinary Least Squares Introduction In the previous chapter, we studied the numerical properties of ordinary least squares estimation, properties that hold no matter how the data may have been generated. | Chapter 3 The Statistical Properties of Ordinary Least Squares Introduction In the previous chapter we studied the numerical properties of ordinary least squares estimation properties that hold no matter how the data may have been generated. In this chapter we turn our attention to the statistical properties of OLS ones that depend on how the data were actually generated. These properties can never be shown to hold numerically for any actual data set but they can be proven to hold if we are willing to make certain assumptions. Most of the properties that we will focus on concern the first two moments of the least squares estimator. In Section we introduced the concept of a data-generating process or DGP. For any data set that we are trying to analyze the DGP is simply the mechanism that actually generated the data. Most real DGPs for economic data are probably very complicated and economists do not pretend to understand every detail of them. However for the purpose of studying the statistical properties of estimators it is almost always necessary to assume that the DGP is quite simple. For instance when we are studying the multiple linear regression model yt Xtß Ut Ut IID 0 a2 we may wish to assume that the data were actually generated by the DGP yt Xtßo Ut Ut NID 0 a2 . The symbol in and means is distributed as. We introduced the abbreviation IID which means independently and identically distributed in Section . In the model the notation IID 0 a2 means that the ut are statistically independent and all follow the same distribution with mean 0 and variance a2. Similarly in the DGP the notation NID 0 a2 means that the ut are normally independently and identically distributed with mean 0 and variance a2. In both cases it is implicitly being assumed that the distribution of ut is in no way dependent on Xt. Copyright 1999 Russell Davidson and James G. MacKinnon 87 88 The Statistical Properties of Ordinary Least Squares The .