1
Elementary Data Analysis

1.1 Variables and Observations

Where to begin? Data analysis is the business of summarizing a large volume of information into a smaller compass, in a form that a human investigator can appreciate, assess, and draw conclusions from. The idea is to smooth out incidental variations so as to bring the 'big picture' into focus, and the fundamental concept is averaging, extracting a representative value or central tendency from a collection of cases. The correct interpretation of these averages, and functions of them, on the basis of a model of the environment in which the observed data are generated,1 is the main concern of statistical theory. However, before tackling these often difficult questions, gaining familiarity with the methods of summarizing sample information and doing the associated calculations is an essential preliminary.

Information must be recorded in some numerical form. Data may consist of measured magnitudes, which in econometrics are typically monetary values, prices, indices, or rates of exchange. However, another important data type is the binary indicator of membership of some class or category, expressed numerically by ones and zeros. A thing or entity of which different instances are observed at different times or places is commonly called a variable. The instances themselves, of which collections are to be made and then analyzed, are the observations. The basic activity to be studied in this first part of the book is the application of mathematical formulae to the observations on one or more variables.

These formulae are, to a large extent, human-friendly versions of coded computer routines. In practice, econometric calculations are always done on computers, sometimes with spreadsheet programs such as Microsoft Excel but more often using specialized econometric software packages. Simple cases are traditionally given to students to carry out by hand, not because they ever need to be done this way but hopefully to cultivate insight into what it is that computers do. Making the connection between formulae on the page and the results of running estimation programs on a laptop is a fundamental step on the path to econometric expertise.

The most basic manipulation is to add up a column of numbers, where the word "column" is chosen deliberately to evoke the layout of a spreadsheet but could equally refer to the page of an accounting led!ger in the ink-and-paper technology of a now-vanished age. Nearly all of the important concepts can be explained in the context of a pair of variables. To give them names, call them and . Going from two variables up to three and more introduces no fundamental new ideas. In linear regression analysis, variables are always treated in pairs, no matter how many are involved in the calculation as a whole.

Thus, let denote the pair of variables chosen for analysis. The enclosure of the symbols in parentheses, separated by a comma, is a simple way of indicating that these items are to be taken together, but note that is not to be regarded as just another way of writing . The order in which the variables appear is often significant.

Let , a positive whole number, denote the number of observations or in other words the number of rows in the spreadsheet. Such a collection of observations, whose order may or may not be significant, is often called a series. The convention for denoting which row the observation belongs to is to append a subscript. Sometimes the letters , , or are used as row labels but there are typically other uses for these, and in this book we generally adopt the symbol for this purpose. Thus, the contents of a pair of spreadsheet columns may be denoted symbolically as

We variously refer to the and as the elements or the coordinates of their respective series.

This brings us inevitably to the question of the context in which observations are made. Very frequently, macroeconomic or financial variables (prices, interest rates, demand flows, asset stocks) are recorded at successive dates, at intervals of days, months, quarters, or years, and then is simply a date, standardized with respect to the time interval and the first observation. Such data sets are called time series. Economic data may also be observations of individual economic units. These can be workers or consumers, households, firms, industries, and sometimes regions, states, and countries. The observations can represent quantities such as incomes, rates of expenditure on consumption or investment, and also individual characteristics, such as family size, numbers of employees, population, and so forth. If these observations relate to a common date, the data set is called a cross-section. The ordering of the rows typically has no special significance in this case.

Increasingly commonly studied in economics are data sets with both a time and a cross-sectional dimension, known as panel data, representing a succession of observations on the same cross section of entities. In this case two subscripts are called for, say and . However, the analysis of panel data is an advanced topic not covered in this book, and for observations we can stick to single subscripts henceforth.

1.2 Summary Statistics

As remarked at the beginning, the basic statistical operation of averaging is a way of measuring the central tendency of a set of data. Take a column of numbers, add them up, and divide by . This operation defines the sample mean of the series, usually written as the symbol for the designated variable with a bar over the top. Thus,

where the second equality defines the 'sigma' representation of the sum. The Greek letter , decorated with upper and lower limits, is a neat way to express the adding-up operation, noting the vital role of the subscript in showing which items are to be added together. The formula for is constructed in just the same way.

The idea of the series mean extends from raw observations to various constructed series. The mean deviations are the series

Naturally enough this 'centred' series has zero mean, identically:

Not such an interesting fact, perhaps, but the statistic obtained as the mean of the squared mean deviations is very interesting indeed. This is the sample variance,

which contains information about how the series varies about its central tendency. The same information, but with units of measurement matching the original data, is conveyed by the square root , called the standard deviation of the series. If is a measure of location, then is a measure of dispersion.

One of the mysteries of the variance formula is the division by , not as for the mean itself. There are important technical reasons for this,2 but to convey the intuition involved here, it may be helpful to think about the case where , a single observation. Clearly, the mean formula still makes sense, because it gives . This is the best that can be done to measure location. There is clearly no possibility of computing a measure of dispersion, and the fact that the formula would involve dividing by zero gives warning that it is not meaningful to try. In other words, to measure the dispersion as , which is what (1.3) would produce with division by instead of , would be misleading. Rather, it is correct to say that no measure of dispersion exists.

Another property of the variance formula worth remarking is found by multiplying out the squared terms and summing them separately, thus:

In the first equality, note that "adding up" instances of (which does not depend on ) is the same thing as just multiplying by . The second equality then follows by cancellation, given the definition (1.1). This result shows that to compute the variance, there is no need to perform subtractions. Simply add up the squares of the coordinates, and subtract times the squared mean. Clearly, this second formula is more convenient for hand calculations than the first one.

The information contained in the standard deviation is nicely captured by a famous result in statistics called Chebyshev's rule, after the noted Russian mathematician who discovered it.3 Consider, for some chosen positive number , whether a series coordinate falls 'far from' the central tendency of the data set in the sense that either or . In other words, does lie beyond a distance from the mean, either above or below? This condition can be expressed as

Letting denote the number of cases that satisfy inequality (1.5), the inequality

is true by definition, where the 'sigma' notation variant expresses compactly the sum of the terms satisfying the stated condition. However, it is also the case that

since, remembering the definition of from (1.3), the sum cannot exceed , even with . Putting together the inequalities in (1.6) and (1.7) and also dividing through by and by yields the result

In words, the proportion of series coordinates falling beyond a distance from the mean is at...