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Dummy variables have proven to be an analytically convenient and empirically productive means of handling econometric problems relating to seasonality, structural change and time series and cross section data pooling. Despite the advantageous of the approach there are few results regarding its use in simultaneous equation estimation. In this paper the problem of discrete parameter variations in simultaneous equation estimation is systematically investigated. The estimation procedures proposed are based on extensions of existing methods for seemingly unrelated regressions and two and three-strage least squares. In addition to obtaining the desired estimators and their properties a number of convenient advantages resulting from the use of dummy variables to treat parameter variations across subsets of simultaneous equations are indicated.

Nerlovian supply functions are fitted to post-World War II coffee data for São Paulo, Brazil. Supply shifts are permitted through the use of dummy variables, and recent specifications of irreversible functions are tested. The modified Nerlovian functions perform best from both the economic and statistical point of view.

Temporal patterns in fluctuating physical and biological phenomena are of great interest in several fields of biology, primarily because of their importance as evolutionary constraints. To clarify and simplify the wide variety of terms used to describe aspects of temporal pattern, simple measures of predictability, constancy, and contingency are proposed. These are sufficient to describe the general characteristics of periodic phenomena. The measures are based on the mathematics of information theory. Methods of testing the statistical significance of these measures are given. Predictability, constancy, and contingency can be determined for either qualitative (categorical) or quantitative (discrete or continuous) variables measured over a period of time. Alternative patterns of seasonal flowering and fruiting of tropical trees are given as an example of a qualitative variable; precipitation data are analyzed as an example of a quantitative variable.

This paper is a study of the relative importance of four factors-ability, social class, sex, and type of school-on the performance of children in the educational system of England and Wales. It uses a dummy variable linear regression analysis of data collected by the National Survey of Health and Development, tracing the progress of all children born in one week of 1946. The conclusions suggest among other things that reorganisation of educational institutions can have considerable effect on children's performance, irrespective of their class and intelligence; and also that sex plays an increasingly important determining role as pupils progress through the system.

Increasing interest has been expressed, particularly by social scientists, in the analysis of unbalanced data occurring in models not of full rank, i.e., linear models where the numbers of observations in the subclasses are unequal. The various discussions of this problem that have appeared in the psychological and educational literature differ depending upon the intent of the author(s). For example, some writers (Gourlay, 1955; Steinhorst & Miller, 1969; Tsao, 1942, 1946; Williams, 1972) have examined and at times compared different types of solutions, including approximate solutions, for the unbalanced case. Other writers (Overall & Spiegel, 1969; Rawlings, 1972) have focused upon the difficulty which arises in interpreting the results of unbalanced data analyses because the estimable functions involved in tests of hypotheses are not orthogonal. Though not specifically addressed to the unbalanced case but important in this regard is the work of Bottenberg and Ward (1960), Cohen (1968), Hurst (1970), and Jennings (1967) that demonstrates the equivalence of linear regression and the fixed-effects analysis of variance by the use of regression on dummy variables. This diversity in purpose, combined with the relative narrowness of the individual efforts, has resulted in a fragmented treatment of the problem of unbalanced data and in some cases confusion and controversy regarding methodology, e.g., the role of constraints in obtaining a solution for models not of full rank or the calculation of reductions in sums of squares (Overall & Spiegel, 1973; Rawlings, 1973; Smith, 1973). In response, at least one writer (Joe, 1971) has been prompted to call for a greater understanding and more detailed presentation of the models employed. Some of the more important issues encountered in the analysis of unbalanced data will be examined here within a single framework. These issues are broadly defined as (1) solutions to the normal equations for linear models where the incidence or design matrix is not of full column rank, (2) estimable functions of the parameters and tests of hypotheses, and (3) reductions in sums of squares and interpretation of results. Because of the increased availability of computer hardware and sophisticated least squares routines, the various approximate solutions for analyzing unbalanced data are not considered here.

OVER A DECADE has passed since Gurley and Shaw [5] alerted the economics profession to the importance of considering the implications of the growth of non-bank financial liabilities for monetary theory and policy. The empirical question at the heart of the issue concerning the effects of non-bank financial intermediaries focused on the degree of substitutability and the stability of the substitution relationship between demand deposits and other liquid assets. A large number of empirical studies were subsequently undertaken to examine the substitution relationships and the purpose of this paper is to inquire whether some consensus emerges from the empirical literature which examines demand functions for demand deposits utilizing pooled temporal cross-section data. Section I reviews and evaluates the existing cross-section evidence and suggests that while a fairly broad consensus emerges on the substitution relationships between demand deposits and other liquid assets, the differential use of dummy variables in the model specifications appears to be an important factor in explaining substantive variations between studies. Section II considers a covariance model of the demand for demand deposits which permit testing of alternative dummy specifications. Section III reports results based on a random coefficient model which is more restrictive than the covariance model with respect to temporal dummy variables but allows a richer specification of variation in behavior across states.

Two types of multivariate methods are distinguished from one another: structure-seeking methods and causal modeling. The former is particularly susceptible to coding artifacts based on improper use of dummy variables or of ranking procedures. Solutions for both problems are suggested.

This paper is concerned with the description of a specialized form of linear regression analysis commonly known as "dummy variable" regression analysis. Efforts are made to show the relationship between "dummy variable" regression analysis and other multivariate analysis techniques with a specific emphasis on the kinds of problems and data that sociologists deal with where "dummy variable" regression analysis is particularly appropriate and useful. Efforts are also made (1) to give illustrations and examples of problems to which this type of multiple-regression analysis might be applied productively; (2) to show how "dummy variable" regression analysis is both similar to and different from other multivariate techniques in terms of the analytical procedures and the kinds of interpretations that are made possible.

Aggregate time-series data are used to investigate the demand for new investment in farm buildings in eight of the ten production regions of the United States. The explanatory variables which were found highly significant included lagged gross farm income deflated by prices received by farmers, the lagged ratio of the value of farm real estate to the amount of farm mortgages outstanding used as an equity ratio, the lagged deflated rate of interest charged on farm mortgages, and the time trend representing change in technology. Projections were made for investment in each of the eight regions for 1980. All projections indicate lower expenditures than those currently being made. This may be explained, despite the expected increased demand for building services, if the technology of design and use of buildings continues to progress in the future. Least-squares regression with dummy variables to include all regions and several time periods in one equation was used.

The paper exposes some limitations of the abstract mode model based on the econometric theory as applied to transportation, and suggests modifications for improving its descriptive function as well as predictive capabilities for more accurate travel demand estimations. The modifications included model stratification by trip purpose, suggestions of significant forms of the explanatory variables entering the model, derivation of a time series model for travel forecasts, and the introduction of the dummy variable techniques for modal-split analysis. Habitual travel behavior which provides the theoretical background for the empirical analysis of any travel demand estimation, served as a basis for explaining the modifications. In order to illustrate the aforementioned proposals, versions of the abstract mode model were formulated by using a sample of city-pair data on the intercity passenger travel in West Virginia. A detailed discussion of model formulation procedures is presented. Thus, the aim of the proposed modifications to assist the accuracy of the abstract mode model is substantiated.

This paper considers social and economic correlates of age-specific 1950-1960 net migration of Negro males from a sample of 150 southern counties. A model is developed with five components: (1) economic activity and urbanization, (2) white traditionalism, (3) demographic and ecological pressure, (4) nonwhite poverty, and (5) nonwhite home ownership. The dominant migration forces, as evidenced by correlations with component indicator variables, are the “pull” factor of change in nonprimary industrial employment, the “push” factor of population pressure in the nonwhite rural-farm sector, and the “push” of white traditionalism. However, the significance of model components varied when analyzed along age and industrial development continua. In the younger age groupings, industrial employment growth, population pressure, and white traditionalism were dominant migratory forces while in the older age groupings, industrial employment growth and non-home ownership were most significant. For Negro males in agricultural counties, the major migration propellents appeared to be the “push” of population pressure in the rural farm sector and non-ownership of homes. On the other hand the statistical explanation for Negro migration in more industrialized southern counties rests primarily with the “pull” of increased employment in non-primary industries along with population pressure. The importance of the findings for migration theory is discussed.

is paper presents the results of a statis1tical test of Friedman's hypothesis (Friedman 1961) that the Federal Reserve's decision to count vault cash as reserves would increase the demand for free reserves. In addition, the results are interpreted in the light of the two main theories of portfolio management: the liquidity theory of Edgeworth (1888) and the risk theory of Markowitz (1952) and Tobin (1958). To test Friedman's hypothesis, member banks were divided into three categories: New York City banks, other reserve city banks, and country banks. Demand functions for free reserves were estimated for each bank class, with dummy variables being used to reflect the hypothesized shift in demand. The tests show that Friedman's hypothesis appears true for both classes of reserve city banks but is definitely false for country banks. These results reconcile both of the previously published empirical studies. Georgio Basevi (1963) "confirmed" Friedman's hypothesis by estimating the demand function for free reserves of all member banks together; and J. C. Rothwell (1962), using survey techniques, found little support for the hypothesis among country banks. It is argued that the different marginal responses of reserve city and country banks may be explained by assuming that country banks are motivated by risk considerations, while reserve city banks are motivated by liquidity considerations.

The logical implications of certain simple consistency requirements for appraising alternative procedures for seasonal adjustment constitute the first problem considered in this paper. It is shown that any sum preserving technique of seasonal adjustment that satisfies the quite reasonable requirements of orthogonality and idempotency can be executed on the electronic computer by standard least squares regression procedures. Problems involved in utilizing seasonally adjusted data when estimating the parameters of econometric models are examined. Extending the fundamental Frisch-Waugh theorem concerning trend and regression analysis to encompass problems of seasonality facilitates the comparison of the implications of running regressions on data subjected to prior seasonal adjustment with the effects of including dummy variables with unadjusted data. Complicated types of moving seasonal patterns that may be handled by the dummy variable procedure are considered. Although efficient estimates of the ...

The use of dummy variables requires the imposition of additional constraints on the parameters of regression equations if determinate estimates are to be obtained. Among the possible constraints the most useful are (a) to set the constant term of the equation to zero, or (b) to omit one of the dummy variables from the equation. In working with a single system of classes either constraint can be used, and results from the application of one are readily derived from those obtained from the other. If several systems of classes are involved the best procedure is to delete one dummy variable from each system.

An integrated intraseasonal and interseasonal stochastic dynamic programing model is developed to determine an optimal decision rule with respect to the following three classes of crop irrigation decisions: (1) intertemporal water application rates, (2) whether or not some acreage should be ‘abandoned’ from further irrigations for the remainder of the season, and (3) the optimal acreage to plant for potential irrigation at the beginning of the season. Solutions of the problem are shown to be a basis for optimizing the levels of three design variables, namely, developed irrigation acreage, reservoir capacity, and distribution system capacity. A method is presented for incorporating variance, as well as expected value, of net benefits into the decision criterion for optimal developed acreage. An application is made to a simplified real situation in which optimal acreage to develop is the only design variable. State variable transition probabilities are calculated by a simulation model. A significant trade off was found between expected net benefits and their variability in determination of the optimal developed acreage for irrigation.

In a recent article (this Review, 63 [June, 1969]), Bruce Russett attempts to answer the title question by using two-variable regression analysis to estimate the “trade-off” relationships between the defense spending proportion of Gross National Product (GNP) and the other nondefense expenditure proportions of GNP for the 1939–68 period. This paper contends that Russett's method of analysis is misleading because it does not permit the detection of possible subperiod shifts in the trade-off relationships. The results of a multiple regression, dummy variable analysis indicate that significant shifts in both the sign and the numerical values of the trade-off relationships occurred between the three war periods and the peacetime period. Two attempts to identify the reasons for the shifts produced mixed results, but our tentative conclusion is that the shifts are related to subperiod differences in the unemployment rate and the methods of financing defense expenditures.

SUMMARY The effects of industrialization on exports of primary-producing countries. In the period from pre-war to 1955, the volume of exports from the "semi-industrial" countries (i. e. the industrializing group of primary-producing countries) failed to rise, whereas exports from the "non-industrial" countries (i. e. all other primaryproducing countries outside Western Europe, North America and the Soviet bloc) rose by about 60% in volume. There are two schools of thought about the main causes of this lag in exports from the semi-industrial countries. The first school places emphasis on the defciency of demand by the industrial countries for the products of the semi-industrial countries; the second emphasises a deficiency of supply of exports resulting from the process of industrialization in the producing areas. Recent annual reports of GATT have presented statistical comparisons of the exports of the semi-industrial and the non-industrial countries which appear to support the supply-deficiency argument. The GATT analysis, however, suffers from two deficiences. First, it makes inadequate allowance for variations in world demand for particular commodities; second, it makes no allowance for the influence of relative price changes on the exports of particular countries. The present article attempts to build on the GATT analysis by extending it to include these two additional factors. A special index of world demand is constructed for the exports of each of the semi-industrialized countries, based on the movement in world trade of individual commodities and using the commodity-pattern of exports from each country as weights. Further, an index of export unit values of world trade in the commodity "bundle" of exports from each country is computed and related to the actual export unit value index of each country. Regressions of export volume on the indices of world demand and relative prices show a close correspondence between exports from a given country and the relevant world demand changes (as represented by world trade). The mean co-efficient for the relative price variable is -1.7, but the standard error is high (±0.9). The inclusion of a dummy variable to indicate the presence or absence of sheltered market yielded no significant result, but this was probably because the variable used is a very imperfect indicator of the degree of shelter enjoyed. The inclusion of industrial growth in the regressions indicated that if there is any significant relation between industrialization and exports it is a positive, and not a negative, one. A more detailed commodity analysis shows that the export experience of the non-industrial countries was in total much the same as that of the semi-industrial countries, if the comparison is limited to directly competitive commodities. Finally, it is shown that there is a positive association between industrial growth and the “capacity to import”. The implications of the analysis are (I) that industrialization per se in the period covered did not have a retarding effect on the exports of the semi-industrial countries as a whole; (2) the retardation in their exports was due essentially to the slower rate of growth in demand for their "bundle" of commodity exports than for those of the non-industrial countries; and (3) that the pace of industrialization has itself been heavily dependent on the progress of exports. These generalizations appear to fit most closely the views of the "demand-deficiency" school mentioned above.

In this study the mid-1960's new car market is analyzed by using a new, dummy variable, multivariate technique known as Multiple Classification Analysis (MCA). This approach helps provide a better understanding of this complex market and gives some interesting hints as to its possible future direction.

The method of canonical correlation between sets of dummy variables is used to assign weights to the categories of occupational mobility tables, in order to test two assumptions: 1. The socioeconomic distance between occupations determines the correlation across individuals in a mobility table. 2. The relative status of occupations has remained constant over time. The weights which yield the maximum correlation are found to correlate very highly with measures of socioeconomic status, thus validating the first assumption. Since the maximum correlation between fathers' and sons' occupations is obtained by assigning similar weights to both sets, the second assumption is also validated.

During the analysis of a large process system, the values of certain design variables must be preassigned to allow the solution of the design equations. A careless selection of these distinguished variables can lead to trouble, magnifying computational problems above what need be. A method is proposed for determining a set of design variables which could minimize the computational labor associated with process analysis. This work is part of a larger project on the computer aided design of process systems.

IN A RECENT STUDY,1 Charles T. Taylor found a significant relationship between the loan mix and the average loan interest rate at individual banks. The regression analysis also showed a negative relationship between bank size (total deposits) and loan interest rates. In addition, an analysis of variance of the regression residuals revealed significant differences between average loan rates at the banks grouped by type of charter, geographic area, and par status. Other categorical variables often used as measures of competition were found to be insignificant. These variables are the number of banks in the city, metropolitan character, and bank ownership type. However, improper specification of Taylor's regression casts doubt on the reliability of the results. This note has three primary purposes. First, Taylor's model is respecified to avoid the problems of multicollinearity. The respecification and empirical estimation of the model facilitate evaluation of the correctness of Taylor's regression results. Second, the model is also specified to facilitate estimation of the yields on the various types of loans across all Federal Reserve districts. Third, the model is modified to estimate the effects of regional location and bank size on the yields on the various loan types. This modification was accomplished through the use of dummy variables. It permits analysis of the effects and significance of these variables on the overall yield on bank loans while retaining direct estimation of the yields on the individual loan classes. In addition to respecifying the model, this study also differs from Taylor's as to such factors as geographic coverage, year, definition of loan classes, and organization of the data.

Given a linear time-invariant differential equation, there are many ways to simulate it in an analogue computer. In this paper comparisons of these various simulations are given. It is shown that the dummy variable method is the best in terms of easiness to set up and number of components used. However, if the sensitivity is considered, the Jordan canonical form simulation is the best.

The use in single-stage medical diagnosis of a recent, nearly assumption-free method of statistical discrimination is explored. The method involves the application of Bayes' theorem to linearly smoothed, joint-symptom-probability estimates. The linear smoothing filter is optimized by an empirical performance-scoring technique that employs a single sample for both development and verification. Advantages of the method: acceptability of a wide variety of nonnormal symptom variables, ranging from dichotomous to seemingly continuous; lack of necessity for independence of symptom variables; simultaneous use of a large number of variables, on the order of magnitude of the number of patients observed; capacity to handle missing observations by creating for each incompletely observed variable a dichotomous dummy variable that indicates the missing values. Disadvantage: lack of automatic identification of important variables, as is obtained when the standard discriminant-function method applies.

In a recent study of the socio-economic effects of the disease bilharzia on the population of the Caribbean island of St. Lucia, Weisbrod et al. (1973) make abundant use of regression methods to analyze the various relationships of interest. A common independent variable in their work is a dummy variable that indicates the presence or absence of the disease in the person sampled (the observation unit). The nature of diagnosis for this particular disease is such that if a person is diagnosed as having the disease, he does indeed have it. However, if the diagnosis is negative there is a non-zero probability, q, that he has been diagnosed incorrectly and has the disease. It is the intent of this note to consider the effects of such an independent variable - a binary variable subject td ‘errors of classification’ - in least squares regression. The results of both an analysis of bias in least squares parameter estimates and of the availability of alternative estimators is parallel to the classical case where both the variable and its measurement error are continuous random variables. Many details will not be repeated here in deference to the reader’s familiarity with that subject. The important practical difference between the two cases is that the information needed to obtain consistent (or nearly so) parameter estimates may be more readily available in the discrete case. In the study cited, for instance, extraneous information about q is available from patient medical histories and examination data. The following section of the present article contains a brief exposition of the nature of a discrete ‘classification error’. Previous authors who have treated this material include Neyman (1950), Bross (1954) and Lord and Novick (1968). Sect. 3 then takes up an analysis of the effects of including an independent variable subject to such measure

a The ratio correlation method of estimating county population may be improved with the use of dummy variables and stratification to represent county “type.” Unsatisfactory results were derived when the ratio correlation technique was applied to West Virginia counties using only such conventional symptomatic indicators as births, deaths, non-agricultural employment, and automobile registrations, probably due to the relatively unique absolute population decrease 1950–1960 that occurred in many counties. To improve the predictive power of the technique: counties were stratified along the lines recently suggested by Rosenberg; dummy variables were introduced to represent such county characteristics as region or rate of development; and stratification and dummy variables were combined in a single test. The average 1960 estimating error for the 55 counties was reduced from 4.6 to 2.3 percent when dummy variables and stratification were used to supplement the conventional indicators.

The interpretation of regression coefficients when the dependent variable is qualitative is not impaired by the scale variance which accrues from order preserving transformations of the dependent variable. The article compares the scale variance of dependent qualitative variables to the more familiar scale variance of independent or dummy variables. Further analysis is conducted about the properties of trichotomous dependent variables as applied to a specific example in the estimation of revealed preference and public choice.

Although the use of dummy variables in regression analysis is quite common, the implications of alternative models for incorporating dummy variables are not generally understood. References dealing with the use of dummy variables are numerous but scattered in the literature. The purpose of this article is to demonstrate, using numerical examples, the implications and interrelationships among various models which incorporate dummy variables. Five separate models are considered.

Federal milk marketing orders contain formulas that determine prices received by farmers for milk delivered to bottlers. Different orders contain various combinations of provisions. Cross-section data were used to test the hypothesis that order prices are related to the provisions contained in orders. The hypothesis was accepted. Some 78 percent of the variance in prices is explained by two economic variables: distance of order market from Eau Claire, Wisconsin, and class I utilization ratio. Inclusion of dummy variables to allow for effects of order provisions increased the proportion of explained variance to 94 percent. Covariance analysis found no significant changes between years in the effects of order provisions on prices. Many possible combinations of provisions are not used in any order. Many of these combinations would provide unacceptable prices if they were used. IN RECENT years the problem of maintaining proper milk price relationships among federal order fluid milk markets has become increasingly serious. This is mainly due to three factors: (a) increase in the size and number of orders, (b) improvements in the highway system and developments in longdistance hauling of bulk and of packaged milk, and (c) growth in size of grocery chain stores and bottling plants. The Federal Milk Order Study Committee was concerned with problems of intermarket relationships [2, pp. II-2-1 to 11-2-15, II-4-1 to II-4-56, III-19 to III-923. In 1955 the Agricultural Marketing Service published a study which dealt, in part, with intermarket price relationships [81. Later, Babb [1] analyzed intermarket price relationships. The objective of my study was to test hypotheses concerning effects of federal order provisions on relationships among prices in federal order markets. Dummy variables were used to study effects of order provisions. Covariance analysis was used to test for variation in the effects of these provisions. The hypotheses tested will be presented after a brief discussion of federal order provisions; more complete descriptions can be found in publications

In studies of the labor supply function, alternative measures have been used for the quantity of labor supplied. Data are available now on weekly hours in a specific week, weekly hours in a "normal week," weeks worked during the year, and labor-force participation (measured generally on a weekly reference period). Some of the more important studies of labor supply of the last few years have used labor-force participation (LFP) as the measure of labor supply. This can be done either with grouped data for cities or years where the variable to be explained is the labor-force participation rate (LFPR) or in individual data where participation is indicated by a dummy variable. The independent variables in such studies are the wage rate and "other income" (plus demographic and geographic control variables). The estimates of wage elasticity obtained from equations like (1) have often been decomposed into a substitution and an income effect, using the Slutzky equation (2).

While path analysis has great advantages for bridging the gap between sociological theory and statistical analysis, a major obstacle is the requirement that interval scales be assumed for the data. This problem is attacked, first by investigating what theoretically can go wrong when interval scales are assumed, second by presenting a technique (the use of dummy variables) for checking on the results of assuming equal intervals, and finally by applying this technique to one set of empirical data. The conclusion is that the empirical dangers of assuming equal intervals are not great. In addition to checking on this, the dummy-variable technique can otentially lead to improvements in sociological measurement.

En los últimos años ha crecido el número de personas inmigrantes sin hogar. El artículo muestra las relaciones entre la situación documental y el estrés aculturativo con variables sociodemográficas, apoyo social, abusos de sustancias, sucesos vitales estresantes y malestar emocional en una muestra de 107 personas migrantes sin hogar. Para ello se han realizado entrevistas en profundidad y autoinformes analizando las relaciones entre las variables mediante diversas técnicas de análisis de datos y análisis de contenido. Los resultados manifiestan la importancia de lo cualitativo para la variable «situación documental» que apenas tiene correlaciones con el resto de las variables. También destaca la importancia de los recursos sociales en el estrés aculturativo por encima de otras variables como el choque cultural o la nostalgia por el país de origen.

The most frequently used practical regression model is that in which it is supposed that the dependent variate is normal for given values of the independent variate. In this paper it is pointed out that another regression model of wide applicability may be furnished by the contrary assumption—that the independent variate is normal for given values of the dependent variate. The particular case where the dependent variate is a vector of dummy variables is explored in detail.

In the beginning of this presentation, the use of dummy variables was suggested to overcome problems of non-linearity and category type variables. Where these problems do not exist, it is not recommended that this particular technique be used. The dummy variable technique is simply a tool to aid the analyst in model building. This technique, like most others, has both advantages and disadvantages. It does seem to aid in the interpretation of more complex non-linear and category type variables. In many cases a more clear and concise analysis can be made using the dummy variable technique than in the transgeneration of the independent variables. The particular approach to the use of dummy variables taken in this presentation represents only one of several approaches. For the reader who would like to explore this technique in more depth, it is suggested that he review the references listed in this presentation. The approach taken will be influenced by the objective of the particular study.

variables 1 and 0 can be used as aii alternative to the standard analysis of variance and covariance methods and the Chow test. The subsequent discussion proceeds as follows: Section II is devoted to a brief discussion of the Chow test. There is no need to discuss the analysis of variance aad covariance techniques since they are discussed in any standard text on statistics. The dummy variable method is presented in Section III. A numerical example is solved in Section IV, illustrating the Chow test and the dummy variable method. Some of the advantages of the dummy variable approach over the Chow test are discussed in Section V.

IN A RECENT ARTICLE [4], Professor Ronald Wippern presented the results of several multiple regressions involving dummy variables. This note contains two criticisms of Wippern's published interpretation of these results. These criticisms detract little from Wippern's otherwise excellent article. Nonetheless, they seem worthwhile noting, since the use of dummy variables in multiple regressions is becoming widespread in the analysis of financial and economic data; therefore, the correct interpretation of such regression results is increasingly important. Wippern presents the following model as his equation (1)

The Jorgenson and Lovell axiomatizations of the process of seasonal adjustment are contrasted in this paper. It is argued that dummy variables rather than data subject to prior seasonal adjustment should be employed when time series are utilized in estimating the parameters of a linear economic model.

This paper proposes a method for transforming conventional dummy variable coefficients for the purpose of simplifying their interpretation. This method is especially useful in problems involving several dummy variable systems. The transformation was first presented by James A. M\lorgan, who illustrated the transformation for a hypothetical problem consisting of a single dummy variable system with three classes. The authors have extended Morgan's method to equations with several dummy variable systems as well as conventionally scaled independent variables. The transformation is illustrated for an empirical example involving several continuously scaled variables and five dummy variable systems, each with two or more classes.

Although I support the contention that concurrent use of tree analysis and regression analysis may offer advantages to the researcher, I would like to raise a few questions about the November 1970 JMR article by Armstrong and Andress, "Exploratory Analysis of Marketing Data: Trees vs. Regression." When conducted with understanding, regression can have great utility which, I believe, is not fairly represented in that comparison. First, the authors claim that dummy coding in regression analysis, which allows numerical analysis of attribute information, is "very cumbersome" (p. 487). Dummy variables need not be any more cumbersome than any other variable. I find dummy coding indispensable. It is the basis for the analysis of variance, a technique many researchers rely upon. Second, the authors do not sufficiently stress the superior ability of regression analysis to "measure relationships rather than cell means" (p. 487). A complete (as contrasted to "exploratory") investigation by regression could be used directly for cost analysis or for predicting the effect of changing the characteristics of stations. An equation of the tree, if constructed, could, of course, be similarly used. In measuring these relationships, questions of interpretation arise. For instance, what are the standard errors of the estimates shown in the tree of Figure 3? Standard errors are routinely obtained in regression (see Table 2). Further, the rating variable, a five-category polytomy, is represented by four values in the tree analysis but has only one coefficient in the regression model. Does the regression coefficient represent the difference in response between one category and the other four, or has this subjective description been numerically scaled? Why is the "typeof-area" variable missing from the tree while its inclusion in the regression model seems important and is statistically significant? Third, the guiding rule (p. 487) for a minimum sample size of "10 observations per variable" may be adequate for exploratory regression, but where the model has been established and precise estimates of the coefficients are wanted, the proper sample size may differ in either direction. This depends on: (1) residual error variance, (2) degree of intercorrelation in the data set, and (3) the specified degree of precision. Fourth, the restrictiveness of assumptions in regression analysis is seriously misrepresented (p. 488). In considering this question, it is important to distinguish between properties of models and properties of data. The properties discussed are model properties but are referred to as data problems. The first two "problems," interaction and nonlinear effects, are problems only when one is not sure about their existence. One is then limited in using associated techniques (e.g., establishing confidence intervals) which depend on the assumption that the correct model is being used. Exploratory investigations are conducted to check for such interactions, curvilinearities, or discontinuities which need to be structured into the model. Holding out a validation subsample helps to avoid "over fitting" with the use of such structures.

I am in complete agreement with the substantive content of Mr. Beck’s article. He did misunderstand however the purpose of my analysis. I shall try to clarify this misunderstanding, even though in the end he may disagree even more. I. In his second paragraph he compares my algebraic formulations with conventional statistical notations where dichotomies are treated as “dummy variables” taking the values, O-l. I am, of course, very aware of this “Lazarsfeld-translation” possibility. However for a long time I have proposed treating proportions and dichotomies in their own right without deriving them as special cases of conventional analyses of quantitative variables. I have been guided in my work by several considerations.

IN his rejoinder' to my analysis of the hypothesis posed in his earlier paper,2 C. J. Maule suggests that the tests I applied to the hypothesis are inadequate and that a more complete reporting of my results demonstrates this to be so. I do not propose to debate this issue further, but I should like to make the following points in clarification: 1. Leaving my tests aside altogether, Maule's hypothesis does not pass his own test. As noted in my analysis, using Maule's own data (or alternatively, the survey data on acquired firms) one finds a sharp increase in No*/M* (or No/M) "immediately after 1959 which it would be very difficult to associate with changes in United States merger law and law enforcement."3 2. Maule's regression estimates (Table 1),4 employing D1 or D2 [equations (2) and (5) ], correspond to those reported in my analysis and lead to exactly the same conclusion: the coefficients for both dummy variables, intended to identify the break in the legal environment that underlies his hypothesis, are highly insignificant. Nor is the multicollinearity between D1, which relates to the break identified by Maule, and the other explanatory variables included in the relationship so high as to cast much doubt on this result. The multicollinearity between D2, (relating to an alternative possibility that I suggested) and the other variables is higher, but it is by no means hopeless by the conventions of regression analysis.5 The insignificant coefficients for F and L are probably explained by the high multicollinearity between these two variables (r(F, L) = .97).

Entropy of a binary variable X can be estimated from frequencies calculated on the basis of N observations. The paper considers a reliability criterion of the form Prob(¦H−HN¦

Structural design problems are concerned with the transmission of forces and moments in space. The engineer usually selects his design from a subset of the total class of possible solutions according to the value of some objective junctional defined on the subset (for instance mass or, more generally, cost).

Empirical evidence on the differences in earnings between single women, married women and men is examined. First a mathematical function is developed which approximates to the "length of service-earnings" profile, the logarithmic transformation of salary being used to generate a distribution which does not grossly violate the restrictions of multiple regression analysis. Dummy variables are introduced into the basic multiple regression equation to represent different categories of teacher: various interaction effects between these variables are examined. The results show that although most of the difference between the earnings of married women and men can be accounted for through length of service standardization, there is still a significant difference remaining which cannot easily be explained. Single women were found to earn consistently more than their married counterparts but less than men, especially in primary schools. A slight deterioration over time in the position of married women is detected, when results for 1963 and 1971 are compared.

IN 1961, Arrow, Chenery, Minhas, and Solow (ACMS) (1961) introduced their now familiar production function V y[8 K-P + (1-8)L-P]-1/P (1.) where V is value added per man-year, K is capital, L is man-years of labor, and y, 8, and p are the efficiency, distribution, and substitution parameters, respectively. It is well known that the elasticity of substitution, 1/(1 + p), can be estimated by estimating b in the profit maximizing conditi'on log (V/L) _log a + b log w + u (2) where w is the annual wage rate of production workers. In fact, ACMS obtained very good results by using international data from 19 countries for various census years between 1949 and 1955. These data represented up to 24 ISIC industries at the three-digit level. In 1963, C. E. Ferguson (1963) used U.S. Census of Manfactures data to fit the regression equation (2). Whereas ACMS obtained good results, Ferguson was disappointed in his: "In the entire list of 129 items, R2 is significant at P < .05 in only 50%o of the cases. The bcoefficient is significant 70% of the time . . . But in more than half of these, b was not found to be significantly different from one" (1963, p. 306). Ferguson recognized a possible reason for such results. The model requires different relative factor prices for different observations. With only a little variation in the wage rate, the regression coefficients will have large standard errors. Unfortunately, when Ferguson's paper appeared there was no way to correct or improve the sample. Now, it is possible to pool time-series and cross-sectional data and to recognize the possibility of cross-sectional heteroscedasticity and time-wise autoregression of the disturbance terms. Jan Kmenta has termed this a "cross-sectionally heteroscedastic and time-wise autoregressive model" (1971, p. 509). We shall use this model to estimate the elasticity of substitution, b, in a modification of regression equation (2). We expect that the increased variability of the independent variable will improve the results. This system of production functions for various indtustries provides a classic example of a case where the method of seemingly unrelated regressions may be applied. Thus, we also obtain two-stage Aitken estimates of the elasticity of substitution' using the pooled data. Estimation of the elasticity of substitution by pooling time-series and cross-section data requires a modification of regression equation (2). As the regression model stands, there is an implicit assumption of no technological progress over time. This assumption is removed by specifying the model as log (V/L)_ log a + b logw + c2T2 + C3T3 + C4T4 + u (3) where T2, T3, and T4 are dummy variables representing the years 1958, 1963, and 1967, respectively.' The introduction of the dummy variables into equation (3) allows for the possibility of technological progress in each of the cross-section years of 1958, 1963, and 1967. Although this results in a loss of three degrees of freedom, it does allow our model to capture the influence of technological progress. The Received for publication August 8, 1973. Revision received for publication October 12, 1973. * We have benefited greatly from the comments offered by our colleagues,. David Denslow and Frank Sloan. The encouragement and helpful suggestions of the late C. E. Ferguson, Jan Kmenta, John Moroney, and an anonymous referee are gratefully acknowledged. Jerry R. Jackson provided invaluable assistance in the computations. Of course, we must exonerate everyone but ourselves of all blame for what follows. An earlier version of this paper was presented at the annual meetings of the Econometric Society in December 197 1. 1 We are indebted to an anonymous referee for suggesting this means of accounting for technological progress.

This paper presents estimates of the effects of co-payments upon the use of physicians' services by poor families. The study is based upon pooled cross-section random samples of the population of the Province of Saskatchewan. Data for the period 1963 to 1968 are analyzed. A dummy variable is used to capture the effects of introducing co-payment in 1968. Estimates are presented for various types of service in an attempt to observe differences between patient-elective and physician-elective services. The results indicate that co-payment is associated with an 18 percent reduction in utilization by the poor in 1968.