8In advanced treatment of econometrics, one can relax the assumption that the explanatory variables are nonstochastic (see introduction to Part II).
CHAPTER ONE: THE NATURE OF REGRESSION ANALYSIS 25
one explanatory variable, as in the crop-yield, rainfall, temperature, sunshine, and fertilizer examples, it is known as multiple regression analysis. In other words, in two-variable regression there is only one explanatory variable, whereas in multiple regression there is more than one explanatory variable.
The term random is a synonym for the term stochastic. As noted earlier, a random or stochastic variable is a variable that can take on any set of values, positive or negative, with a given probability.9
Unless stated otherwise, the letter Y will denote the dependent variable and the X's (X1, X2,..., Xk) will denote the explanatory variables, Xk being the kth explanatory variable. The subscript i or t will denote the ith or the tth observation or value. Xki (or Xkt) will denote the ith (or tth) observation on variable Xk. N (or T) will denote the total number of observations or values in the population, and n (or t) the total number of observations in a sample. As a matter of convention, the observation subscript i will be used for cross-sectional data (i.e., data collected at one point in time) and the subscript t will be used for time series data (i.e., data collected over a period of time). The nature of cross-sectional and time series data, as well as the important topic of the nature and sources of data for empirical analysis, is discussed in the following section.
1.7 THE NATURE AND SOURCES OF DATA FOR ECONOMIC ANALYSIS10
The success of any econometric analysis ultimately depends on the availability of the appropriate data. It is therefore essential that we spend some time discussing the nature, sources, and limitations of the data that one may encounter in empirical analysis.
Three types of data may be available for empirical analysis: time series, cross-section, and pooled (i.e., combination of time series and cross-section) data.
Time Series Data The data shown in Table I.1 of the Introduction are an example of time series data. A time series is a set of observations on the values that a variable takes at different times. Such data may be collected at regular time intervals, such as daily (e.g., stock prices, weather reports), weekly (e.g., money supply figures), monthly [e.g., the unemployment rate, the Consumer Price Index (CPI)], quarterly (e.g., GDP), annually (e.g.,
9See App. A for formal definition and further details.
10For an informative account, see Michael D. Intriligator, Econometric Models, Techniques, and Applications, Prentice Hall, Englewood Cliffs, N.J., 1978, chap. 3.
26 PART ONE: SINGLE-EQUATION REGRESSION MODELS
government budgets), quinquennially, that is, every 5 years (e.g., the census of manufactures), or decennially (e.g., the census of population). Sometime data are available both quarterly as well as annually, as in the case of the data on GDP and consumer expenditure. With the advent of high-speed computers, data can now be collected over an extremely short interval of time, such as the data on stock prices, which can be obtained literally continuously (the so-called real-time quote).
Although time series data are used heavily in econometric studies, they present special problems for econometricians. As we will show in chapters on time series econometrics later on, most empirical work based on time series data assumes that the underlying time series is stationary. Although it is too early to introduce the precise technical meaning of stationarity at this juncture, loosely speaking a time series is stationary if its mean and variance do not vary systematically over time. To see what this means, consider Figure 1.5, which depicts the behavior of the Ml money supply in the United States from January 1, 1959, to July 31, 1999. (The actual data are given in exercise 1.4.) As you can see from this figure, the M1 money supply shows a steady upward trend as well as variability over the years, suggesting that the M1 time series is not stationary.11 We will explore this topic fully in Chapter 21.
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