The study of mathematical statistics by economists typically falls under a broad sub-discipline called econometrics. Econometrics is typically defined as the use of statistics and mathematics along with economic theory to describe economic relationships (see the boxes titled Tinbergen on Econometrics and Klein on Econometrics). The real issue is what do we mean by de- scribe? There are two dominant ideas in econometrics. The first involves the scientific concept of using statistical techniques (or more precisely, statistical inference) to test implications of economic theory. Hence, in a traditional sci- entific paradigm, we expose what we think we know to experience (see the box
titledPopper on Scientific Discovery). The second use of econometrics in- volves the estimation of parameters to be used in policy analysis. For example, economists working with a state legislature may be interested in estimating the effect of a sales tax holiday for school supplies on the government’s sales tax revenue. As a result, they may be more interested in imposing economi- cally justified restrictions that add additional information to their data rather than testing these hypotheses. The two uses of econometrics could then be summarized as scientific uses versus the uses of planners.
Tinbergen on Econometrics
Econometrics is the name for a field of science in which mathematical-economic and mathematical-statistical research are applied in combination. Econometrics, therefore, forms a border- land between two branches of science, with the advantages and disadvantages thereof; advantages, because new combinations are introduced which often open up new perspectives; disadvantages, because the work in this field requires skill in two domains, which either takes up too much time or leads to insufficient training of its students in one of the two respects [51, p. 3].
Klein on Econometrics
The purely theoretical approach to econometrics may be envisioned as the development of that body of knowledge which tells us how to go about measuring economic relationships. This theory is often developed on a fairly abstract or general basis, so that the results may be applied to any one of a variety of concrete problems that may arise. The empirical work in econometrics deals with actual data and sets out to make numerical estimates of economic rela- tionships. The empirical procedures are direct applications of the methods of theoretical econometrics [24, p. 1].
Popper on Scientific Discovery
A scientist, whether theorist or experimenter, puts forward state- ments, or systems of statements, and tests them step by step. In the field of the empirical sciences, more particularly, he constructs hy- potheses, or systems of theories, and tests them against experience by observation and experiment.
I suggest that it is the task of the logic of scientific discovery, or logic of knowledge, to give a logical analysis of this procedure; that is to analyse the method of empirical sciences [38, p. 3].
1.1.1 Econometrics and Scientific Discovery
The most prominent supporters of the traditional scientific paradigm to econo- metrics are Theil, Kmenta, and Spanos. According to Theil,
Econometrics is concerned with the empirical determination of eco- nomic laws. The word “empirical” indicates that the data used for this determination have been obtained from observation, which may be either controlled experimentation designed by the econometrician interested, or “passive” observation. The latter type is as prevalent among economists as it is among meterologists [49, p.1].
Kamenta [26] divides statistical applications in economics into descriptive statistics and statistical inference. Kmenta contends that most statistical ap- plications in economics involve applications of statistical inference, that is, the use of statistical data to draw conclusions or test hypotheses about economic behavior. Spanos states that “econometrics is concerned with the systematic study of economic phenomena using observed data” [45, p. 3].
How it all began – Haavelmo
The status of general economics was more or less as follows. There were lots of deep thoughts, but a lack of quantitative results. Even in simple cases where it can be said that some economic magnitude is influenced by only one causal factor, the question of how strong is the influence still remains. It is usually not of very great practical or even scientific interest to know whether the influence is positive or negative, if one does not know anything about the strength. But much worse is the situation when an economic magnitude to be studied is determined by many different factors at the same time, some factors working in one direction, others in the opposite di- rections. One could write long papers about so-called tendencies explaining how this factor might work, how that factor might work and so on. But what is the answer to the question of the total net effect of all the factors? This question cannot be answered without measures of the strength with which the various factors work in their directions. The fathers of modern econometrics, led by the giant brains of Ragnar Frisch and Jan Tinbergen, had the vision that it would be possible to get out of this situation for the science of economics. Their program was to use available statistical mate- rial in order to extract information about how an economy works.
Only in this way could one get beyond the state of affairs where talk of tendencies was about all one could have as a result from even the greatest brains in the science of economics [15].
Nature of Econometrics – Judge et al.
If the goal is to select the best decision from the economic choice set, it is usually not enough just to know that certain economic variables are related. To be really useful we must usually also know the direction of the relation and in many cases the magnitudes involved. Toward this end, econometrics, using economic theory, mathematical economics, and statistical inference as an analytical foundation and economic data as the information base, provides an inferential basis for:
(1) Modifying, refining, or possibly refuting conclusions contained in economic theory and/or what represents current knowledge about economic processes and institutions.
(2) Attaching signs, numbers, and reliability statements to the co- efficient of variables in economic relationships so that this informa- tion can be used as a basis for decision making and choice [23, p.
1].
A quick survey of a couple of important economics journals provides a look at how econometrics is used in the development of economic theory. Ashraf and Galor [2] examine the effect of genetic diversity on economic growth. Specifi- cally, they hypothesize that increased genetic diversity initially increases eco- nomic growth as individuals from diverse cultures allow the economy to quickly adopt a wide array of technological innovations. However, this rate of increase starts to decline such that the effect of diversity reaches a maximum as the increased diversity starts to impose higher transaction costs on the economy.
Thus, Ashraf and Galor hypothesize that the effect of diversity on population growth is “hump shaped.” To test this hypothesis, they estimate two empirical relationships. The first relationship examines the effect of genetic diversity on each country’s population density.
ln (Pi) =β0+β1Gi+β2G2i+β3ln (Ti)+β4ln (X1i)+β5ln (X2i)+β6ln (X3i)+i
(1.1) where ln (Pi) is the natural logarithm of the population density for country i, Gi is a measure of genetic diversity in country i, Ti is the time in years since the establishment of agriculture in country i, X1i is the percentage of arable land in country i, X2i is the absolute latitude of country i, X3i is a variable capturing the suitability of land in countryifor agriculture, andi is the residual. The second equation then estimates the effect of the same factors on each country’s income per capita.
ln (yi) =γ0+γ1Gˆi+γ2Gˆ2i+γ3ln (Ti)+γ4ln (X1i)+γ5ln (X2i)+γ6ln (X3i)+νi
(1.2) whereyi represents the income per capita and ˆGi is the estimated level of ge- netic diversity. Ashraf and Galor use the estimated genetic diversity to adjust for the relationship between genetic diversity and the path of development
TABLE 1.1
Estimated Effect of Genetic Diversity on Economic Development
Population Income
Variable Density per Capita
Genetic Diversity (Gi) 225.440∗∗∗ 203.443∗∗
(73.781)a (83.368) Genetic Diversity Squared (G2i) -3161.158∗∗ -142.663∗∗
(56.155) (59.037) Emergence of Agriculture (ln(Ti)) 1.214∗∗∗ -0.151
(0.373) (0.197)
Percent of Arable Land (ln(X1i)) 0.516∗∗∗ -0.112
(0.165) (0.103)
Absolute Latitude (ln(X2i)) -0.162 0.163
(0.130) (0.117)
Land Suitability (ln(X3i)) 0.571∗ -0.192∗∗
(0.294) (0.096)
R2 0.89 0.57
a Numbers in parenthesis denote standard errors.∗∗∗ denotes statistical significance at the 0.01 level of confidence,∗∗ denotes statistical
significance at the 0.05 level of confidence, and∗ denotes statistical significance at the 0.10 level of confidence.
Source: Ashraf and Galor [2]
from Africa to other regions of the world (i.e., the “Out of Africa” hypothe- sis). The statistical results of these estimations presented in Table 1.1 support the theoretical arguments of Ashraf and Galor.
In the same journal, Naidu and Yuchtman [35] examine whether the “Mas- ter and Servant Act” used to enforce labor contracts in Britain in the nine- teenth century affected wages. At the beginning of the twenty-first century a variety of labor contracts exist in the United States. Most hourly employees have an implicit or continuing consent contract which is not formally bind- ing on either the employer or the employee. By contrast, university faculty typically sign annual employment contracts for the upcoming academic year.
Technically, this contract binds the employer to continue to pay the faculty member the contracted amount throughout the academic year unless the fac- ulty member violates the terms of this contract. However, while the faculty member is bound by the contract, sufficient latitude is typically provided for the employee to be released from the contract before the end of the academic year without penalty (or by forfeiting the remaining payments under the con- tract). Naidu and Yuchtman note that labor laws in Britain (the Master and Servant Act of 1823) increased the enforcement of these labor contracts by providing both civil and criminal penalties for employee breach of contract.
Under this act employees who attempted to leave a job for a better opportu- nity could be forced back into the original job under the terms of the contract.
TABLE 1.2
Estimates of the Effect of Master and Servant Prosecutions on Wages
Variable Parameter
Fraction of Textiles×ln(Cotton Price) 159.3∗∗∗
(42.02)a
Iron County×ln(Iron Price) 51.98∗∗
(19.48)
Coal County×ln(Coal Price) 41.25∗∗∗
(10.11)
ln(Population) 79.13∗∗
(35.09)
a Numbers in parenthesis denote standard errors.∗∗∗ denotes statistical significance at the 0.01 level of confidence, and∗∗ denotes statistical significance at the 0.05 level of confidence.
Source: Naidu and Yuchtman [35]
Naidu and Yuchtman develop an economic model which indicates that the en- forcement of this law will reduce the average wage rate. Hence, they start their analysis by examining factors that determine the number of prosecutions un- der the Master and Servant laws for counties in Britain before 1875.
Zit=α0+α1Si×X1,t+α2I2,i×ln (X2,t) +α3I3,iln (X3,t)
+α4ln (pi,t) +it (1.3)
whereZitis the number of prosecutions under the Master and Servant Act in countyi in year t, Si is the share of textile production in county i in 1851, X1,t is the cotton price at time t, I2,i is a dummy variable that is 1 if the county produces iron and 0 otherwise,X2,t is the iron price at timet,I3,i is a dummy variable that is 1 if the county produces coal and 0 otherwise,X3,t
is the price of coal,pi,t is the population of county iat timet, and it is the residual. The results for this formulation are presented in Table 1.2. Next, Naidu and Yuchtman estimate the effect of these prosecutions on the wage rate.
wit=β0+β1I4,t×ln ¯Zi
+β2X5,it+β3X6,it+β4ln (X7,it)
+β5ln (pit) +β6X8,it+νit (1.4) wherewitis the average wage rate in countyiat timet,Itis a dummy variable that is 1 ift >1875 (or after the repeal of the Master and Servant Act) and 0 otherwise,X5,it is the population density of countyiat time t,X6,it is the proportion of the population living in urban areas in countyiat timet,X7,itis the average income in countyiat timet,X8,itis the level of union membership in countyiat timet, andνitis the residual. The results presented in Table 1.3 provide weak support (i.e., at the 0.10 level of significance) that prosecutions under the Master and Servant Act reduced wages. Specifically, the positive
TABLE 1.3
Effect of Master and Servant Prosecutions on the Wage Rate
Variable Parameter
Post-1875×ln(Average Prosecutions) 0.0122∗ (0.0061)
Population Density -0.0570
(0.0583)
Proportion Urban -0.0488
(0.0461)
ln(Income) 0.0291
(0.0312)
ln(Population) 0.0944∗∗
(0.0389)
Union Membership 0.0881
(0.0955)
a Numbers in parenthesis denote standard errors.∗∗ denotes statistical significance at the 0.05 level of confidence and∗ denotes statistical significance at the 0.10 level of confidence.
Source: Naidu and Yuchtman [35]
coefficient on the post-1875 variable indicates that wages were 0.0122 shillings per hour higher after the Master and Servant Act was repealed in 1875.
As a final example, consider the research of Kling et al. [25], who examine the role of information in the purchase of Medicare drug plans. In the Medicare Part D prescription drug insurance program consumers choose from a menu of drug plans. These different plans offer a variety of terms, including the price of the coverage, the level of deductability (i.e., the lower limit required for the insurance to start paying benefits), and the amount of co-payment (e.g., the share of the price of the drug that must be paid by the senior). Ultimately con- sumers make a variety of choices. These differences may be driven in part by differences between household circumstances. For example, some seniors may be in better health than others. Alternatively, some households may be in better financial condition. Finally, the households probably have different at- titudes toward risk. Under typical assumptions regarding consumer behavior, the ability to choose maximizes the benefits from Medicare Part D to seniors.
However, the conjecture that consumer choice maximizes the benefit from the Medicare drug plans depends on the consumer’s ability to understand the benefits provided by each plan. This concept is particularly important given the complexity of most insurance packages. Kling et al. analyze the possibil- ity of comparison friction. Comparison friction is a bias from switching to a possibly better product because the two products are difficult to compare. To analyze the significance of comparison friction Kling et al. construct a sample of seniors who purchase Medicare Part D coverage. Splitting this sample into a control group and an intervention (or treatment) group, the intervention group was then provided personalized information about how each alternative
would affect the household. The control group was then given access to a web- page which could be used to construct the same information. The researchers then observed which households switched their coverage. The sample was then used to estimate
Di=α0+α1Zi+α2X1i+α3X2i+α4X3i+α5X4i+α6X5i
α7X6i+α8X7i+α9X8i+α10X9i+α11X10i+i (1.5) where Di is one if the household switches its plan and zero otherwise, Zi is the intervention variable equal to one if the household was provided individual information,X1i is a dummy variable which is one if the head of household is female,X2i is one if the head of household is married,X3i is one if the indi- vidual finished high school,X4iis one if the participant finished college,X5iis one if the individual completed post-graduate studies,X6iis one if the partic- ipant is over 70 years old,X7iis one if the participant is over 75 years old,X8i is one if the individual has over four medications,X9iis one if the participant has over seven medications, andX10i is one if the household is poor.
Table 1.4 presents the empirical results of this model. In general these results confirm a comparison friction since seniors who are given more in- formation about alternatives are more likely to switch (i.e., the estimated intervention parameter is statistically significant at the 0.10 level). However, the empirical results indicate that other factors matter. For example, mar- ried couples are more likely to switch. In addition, individuals who take over seven medications are more likely to switch. Interestingly, individual levels of education (i.e., the high school graduate, college graduate, and post-college graduate variables) are not individually significant. However, further testing would be required to determine whether education was statistically informa- tive. Specifically, we would have to design a statistical test that simultaneously restricted all three parameters to be zero at the same time. As constructed, we can only compare each individual effect with the dropped category (probably that the participant did not complete high school).
In each of these examples, data is used to test a hypothesis about individual behavior. In the first study (Ashraf and Galor [2]), the implications of indi- vidual actions on the aggregate economy (i.e., nations) are examined. Specif- ically, does greater diversity lead to economic growth? In the second study, Naidu and Yuchtman [35] reduced the level of analysis to the region, asking whether the Master and Servant Act affected wages at the parish (or county) level. In both scenarios the formulation does not model the actions themselves (i.e., whether genetic diversity improves the ability to carry out a variety of activities through a more diverse skill set or whether the presence of labor re- strictions limited factor mobility) but the consequences of those actions. The last example (Kling et al. [25]) focuses more directly on individual behavior.
However, in all three cases an economic theory is faced with observations.
On a somewhat related matter, econometrics positions economics as a pos- itive science. Econometrics is interested in what happens as opposed to what should happen (i.e., a positive instead of a normative science; see box The
TABLE 1.4
Effect of Information on Comparison Friction
Variable Parameter
Intervention 0.098∗
(0.041)
Female −0.023
(0.045)
Married 0.107∗
(0.045)
High School Graduate −0.044
(0.093)
College Graduate 0.048
(0.048)
Post-college Graduate −0.084
(0.062)
Age 70+ −0.039
(0.060)
Age 75+ 0.079
(0.048)
4+ Medications −0.054
(0.050)
7+ Mediations 0.116∗
(0.052)
Poor 0.097∗
(0.045)
a Numbers in parenthesis denote standard errors.∗ denotes statistical significance at the 0.10 level of confidence.
Source: Kling et al. [25]
Methodology of Positive Economics – Friedman). In the forgoing dis- cussion we were not interested in whether increased diversity should improve economic growth, but rather whether it could be empirically established that increased diversity was associated with higher economic growth.
The Methodology of Positive Economics – Friedman ... the problem how to decide whether a suggested hypothesis or theory should be be tentatively accepted as part of the “body of systematized knowledge concerning what is.” But the confusion [John Neville] Keynes laments is still so rife and so much a hin- drance of the recognition that economics can be, and in part is, a positive science that it seems to preface the main body of the paper with a few remarks about the relation between positive and normative economics.
... Self-proclaimed “experts” speak with many voices and can hardly all be regarded as disinterested; in any event, on questions that matter so much, “expert” opinion could hardly be accepted soley on faith even if the “experts” were nearly unanimous and clearly disinterested The conclusions of positive economics seem to be, and are, immediately relevant to important normative prob- lems, to questions of what ought to be done and how any given goal can be attained. Laymen and experts alike are inevitably tempted to shape positive conclusions to fit strongly held normative pre- conceptions and to reject positive conclusions if their normative implications – or what are said to be their normative implications – are unpalatable.
Positive economics is in principle independent of any partic- ular ethical position or normative judgments. As Keynes says, it deals with “what is,” not with “what ought to be.” Its task is to provide a system of generalizations that can be used to make cor- rect predictions about the consequences of any change in circum- stances. Its performance is to be judged by the precision, scope, and conformity with experience of the predictions it yields [13, pp.
3–5].
1.1.2 Econometrics and Planning
While the interaction between governments and their economies is a subject beyond the scope of the current book, certain features of this interaction are important when considering the development of econometrics and the role of mathematical statistics within that development. For modern students of eco- nomics, the history of economics starts with the classical economics of Adam Smith [44]. At the risk of oversimplication, Smith’s insight was that markets allowed individuals to make choices that maximized their well-being. Aggre- gated over all individuals, these decisions acted like an invisible hand that allocated resources toward the production of goods that maximized the over- all well-being of the economy. This result must be viewed within the context of the economic thought that the classical model replaced – mercantilism [43].
Historically the mercantile system grew out of the cities. Each city limited the trade in raw materials and finished goods in its region to provide economic benefits to the city’s craftsmen and merchants. For example, by prohibiting the export of wool (or by imposing significant taxes on those exports) the re- sulting lower price would benefit local weavers. Smith’s treatise demonstrated that these limitations reduced society’s well-being.
Thelaissez-faireof classical economics provided little role for econometrics as a policy tool. However, the onset of the Great Depression provided a sig- nificantly greater potential role for econometrics (see boxGovernment and