*08/11/98*

**Estimating the Demand for Higher Education in the United States,
1965-1995**

Introduction

According to the U.S. Department of Education’s *Projections of Education
Statistics to 2006*, enrollment in institutions of higher education is expected to rise
from 14.1 million in 1994 to 16.4 millions by the year 2006, an increase of 2.3 million or
16 percent increase. Since college enrollment is expected to grow substantially in the
coming decade as a result of expected surge in college-age population, a phenomenon known
as "tidal wave II", this large increase in expected enrollment may not be
surprising. An important policy issue is over the extent of such large expected growth in
college enrollment. Given the importance of projected college enrollment in the
formulation of policy for physical and human resource planning, our ability to forecast
the projected enrollment accurately is obviously an important concern to college and
university administrators and legislators, because enrollment is the principal determinant
of total revenues available to colleges and universities. In addition, this would be also
an important concern to economists and policy makers.

Reviewing enrollment growth path in the last three decades (See Figure 1), we note that
total college enrollment experienced a phenomenal growth during the early period from 1965
to 1975, nearly doubling its size from 5.9 million in 1965 to 11.5 million in 1975. During
the next ten year period from 1976 through 1985, however, the college enrollment showed
only a modest growth of one million to 12.2 million in 1985. Since then, college
enrollment started to show a steady growth path through 1992 with a peak enrollment of
14.5. In the most recent three years of 1993-1995, the total enrollment essentially
leveled off at around 14.3 million. What might explain these large variations in the
growth path of college enrollment? It is generally agreed that the
noted changes in college enrollment essentially reflect demographic shifts and economic
fluctuations. Since economists have long worked on modeling and estimating the demand for
higher* *education, it is reasonable to turn to their work and ask to what extent
economic models can provide help in evaluating the accuracy of the projected
college enrollment.

In order for economic model to be of help in the evaluation of the official enrollment projection in the next decade, it is necessary to first construct and estimate a generally acceptable model of the demand for higher education. In addition, it is highly desirable to have estimated parameters of the model to be robust and remain stable over time. Reviewing prior empirical studies, it was relatively easy to identify a easily agreeable and conventional model. But we note that there exists a wide variety of estimated parameters for such basic determinants as tuition and income elasticities. The existence of a disparate empirical evidence in the previous studies, which will be reviewed briefly later, is largely due to difference in the specification, type of data, level of aggregation, method of estimation, and the time period. Unfortunately, the existence of a variety of estimated parameters only raises serious doubts as to the use of conventional model of the demand for higher education toward the goal of evaluating various projected enrollment growth in the decade.

In the face of the unsatisfactory current status of empirical research on the demand for higher education, it is reasonable to examine the extent to which historical experience, as incorporated in estimated equations of the conventional model, can be used as a reliable guide in evaluating the future trends in college enrollment. In doing this, however, there are a number of unresolved issues in empirical studies of enrollment growth which need to be resolved. They include a wide ranging issues like the specification of the model, functional form, parameter stability, and time series properties of aggregate time series data. What we attempt to accomplish in the present paper is to focus on the first two issues, namely the specification of the model and its functional form. The remaining issues will be the topic of companion paper, which is in progress.

The primary goal of the paper is to conduct a rigorous analysis of the demand for higher education in the United Sates with special emphasis on the specification and functional form of the conventional model during the period from 1965-1995. Following the tradition in the existing literature, we will examine the demand for higher education at public and private institutions, respectively. What we attempt to present is not to obtain the best results in individual regressions for public or private institutions from theoretical and statistical viewpoints, but to examine how and why parameters of the conventional single-equation regression model, estimated from aggregate time series data, may change over different specifications and functional forms. A brief review of prior empirical studies follows to provide a perspective on major issues in the estimation of the conventional model for the demand for education.

*Organization Preview of empirical results??*

2. Review of Related Literature

While earlier works of Schultz(1961) and Becker(1964) laid theoretical foundation, it was Campbell and Siegel(1967) who first seriously attempted to measure the impact of tuition and disposable income on the demand for higher education. In their estimation, they used enrollment ratio against ‘eligible’ college population as the dependent variable. While there are many empirical studies examining different aspects of the demand for higher education, it is reasonable to limit our review only to those studies which focused on the aggregate enrollment behavior based on time series data, for they will have direct bearings for our ultimate goal of evaluating forecasting accuracy. We chose only studies by Campbell and Siegel(1964) and ensuing similar works by Hights(1970, 1975), Shim(1990), and Hsing and Chang(1996).

Looking for common features in these studies, we note that all of these is first based on single equation regression model with either linear or more commonly log-linear functional form. Secondly, they all used in the specification two common explanatory variables, namely the cost of education as own price variable and household or per capita disposable income as a measure of both amounts and sources of funds to purchase enrollment, with both explanatory variables deflated by the consumer price index. After these two common features, different analysts added different explanatory variables. For instance, Hights(1975) used additionally the cost of education at alternative educational institution, and ratio between tuition and tuition at alternative institutions to capture cross price effects. In more recent years, other analysts introduced two additional economic variables to reflect labor market conditions into the conventional model. They are unemployment rate for the civilian labor force to measure the effect of the tightness of the labor market conditions on the enrollment and wage rate to capture the opportunity cost of attending college. In the most recent study, Hsing and Chang(1996), who also used wage rate in the retail sector, focused on choosing the appropriate functional form of the conventional model. Their results indicate that the appropriate functional form of the conventional model is not the usual log-linear but non-linear and general. From their estimated model with preferred non-linear and general functional form, they obtained and presented evidence of increasing sensitivity of enrollment at private institutions to tuition and income.

Examining the magnitude of estimated parameters for tuition, income, and additional explanatory variables in these studies, we find that they vary substantially from one another, as mentioned earlier. Table 1 presents tuition and income elasticities from these studies. Note first that Campbell and Siegel(1967) obtained tuition and income elasticities of –0.4404 and 1.2036 by estimating a conventional model relating enrollment ratio to only two economic variables of tuition and disposable income. Using Campbell and Siegel’s data and linear regression formulation, Hight(1970) was not able to obtain statistically significant results for all equations. His price elasticities (calculated about mean values) were -1.058 and -0.6414 for the public and private institutions, both of which were substantially higher than that of Campbell and Siegel(1967). His income elasticities were 0.977 and 1.701 for public and private institutions, respectively. While it is difficult to directly compare price and income elasticities in these two studies, it is interesting to note that Hight’s price elasticties are substantially higher than that of Campbell and Siegel; but Campbell and Siegel’s income elasticity of 1.2036 appears to be a weighted averages of Hight’s two income elasticities. In a later study, however, Hights(1975) re-estimated the demand for higher education for the public and private institutions from a model, which includes both tuition and tuition at alternative institutions, in addition to income, during selected years between1927 and 1955 and 1957-72. Interestingly, he reported tuition elasticities of -1.783 and -0.714 for public and private institutions, respective, which were even higher than those in his earlier study. An important question is why Hight(1975) might have obtained higher own price elasticities in his later study when he added the tuition at alternative institutions than in his earlier study with a model excluding the alternative tuition. Anticipating a fuller discussion later, it is sufficient to say here that the inclusion of alternative tuition holds the key to the question. In a more recent study with additional variables (unemployment rate and two dummy variables to account for the impact of the Vietnam War draft calls during 1965-69 and of recession during 1981-83), Shim(1990) obtained tuition elasticity of –0.2354, a smallest estimate thus far. His income elasticity turns out to be 1.1251, a similar figure to that of Campbell and Siegel. Most recently, Hsing and Chang(1996) used the Box-Cox(1964) model to determine the correct functional form for a conventional model of the demand for higher education at private institutions. They reported that a general and non-linear functional form is chosen over the log-linear form, and they derived varying tuition and income elasticities, as the level of tuition and income vary, from the estimated regression results with the general form. Their tuition elasticity ranges from -0.261 in 1964-65 to -0.557 in 1990-91; and their income elasticity ranges from 0.493 in 1965-66 to 1.093 in 1990-91.

Reviewing estimated tuition and income elasticities in Table 1, we first note that there exists a large difference in the estimated tuition elasticities, ranging from low figure of –0.235 to high figure of –1.783. In contrast, there is much smaller discrepancy in the estimated income elasticities. With the exception of one small figure of 0.301 by Hights(1975) and one large income elasticity of 1.701 in his earlier study, the estimated income elasticity is close to unity in most cases. An important and related question to be asked is why there exists a large variance in tuition elasticity but only a small variance in income elasticity in the existing studies? Anticipating again a fuller discussion of this point later in our empirical investigation, it is sufficient to point out now that the noted large discrepancy in estimated own price elasticity results again from the inclusion or exclusion of tuition at alternative institutions.

Another source of difference in the estimated tuition and income elasticities in Table
1 may also results from the fact that while reported regression results in earlier studies
are without autocorrelation correction, Hsing and Chang(1996) reported regression results
with serial correlation correction. It is unfortunate that investigators in earlier
studies did not even bother to report such basic regression statistics as Durbin-Watson
statistics and R^{2}. Therefore, it is not only difficult to compare the results
of these studies, but also impossible to evaluate the reliability of their regression
results with key regression statistics. At least in one of these earlier studies,
Hights(1975) presented regression results with necessary regression statistics. In
explaining why the income elasticity coefficient in his work is not significant, he argues
that multicollinearity among explanatory variables and possible specification error
resulting from omitted variables might be responsible. While it is quite possible that the
noted two problems are likely to make regression results susceptible, we should also note
that reported Durbin-Watson statistics in all of his regressions are substantially lower
than corresponding R^{2} in all of his five regressions. This is an indication of
"spurious" regression which Granger and Newbold(1974) pointed out. Therefore, it
is not just the wide range of elasticities of tuition and income elasticities, which is
the source of concern. More serious concern may be the precision of estimated parameters
due to nonstationarity problem in the aggregate time series data used in these prior
studies.

After reviewing common and additional features in earlier empirical studies of the demand for higher education, we now turn to the formulation of a conventional model for the purpose of our estimation. Ideally this conventional model incorporates these features discussed earlier and hopefully most find it reasonable and good for the ultimate purpose of using it in the evaluation of the projected enrollment. Since the main economic determinants of the demand for higher education are deemed to be essentially the same for both public and private institutions, it is reasonable to specify such a model on a common theoretical framework.

2. The Model

Assuming an absence of nonprice rationing, like admission criteria and others, at the aggregate level, it is reasonable to view the demand for higher education as equivalent to enrollment at higher educational institutions. In the conventional model, the demand for higher education is invariably measured by the ratio of the number of enrollment in institutions of higher education and various measures of ‘eligible’ population.. What then explains the demand for higher education at public and private institutions? In answering to this question, we need to remind that the decision to enroll at higher education institutions is viewed both as investment and consumption decision. The investment approach asserts that the demand for higher education depends on the present value of expected stream of benefits resulting from the education and the present cost of education. The cost would include direct outlays like tuition and fees, room and board, and cost of textbooks, as well as opportunity cost of foregone earnings.

The demand for higher education may thus be specified as a function of the cost of education as own price variable and disposable income as a measure of both amounts and sources of funds to purchase enrollment. In addition, wage rate in the retail sector and civilian unemployment rate may be incorporated to reflect labor market conditions. The wage rate is first added to capture the opportunity cost of attending college; and the unemployment rate for the civilian labor force is also added to measure the effect of the tightness of the labor conditions on the college enrollment.

Specifically, the demand for higher education at both the public and private institutions may be specified in a common framework as follows:

ENRU_{t} = f (TFU_{t}, Y_{t}, WAGE_{t}, UR_{t})
(1A)

- + - +

ENRV_{t} = g(TFV_{t}, Y_{t}, WAGE_{t}, UR_{t})
(1B)

- + - +

where ENRU_{t} and ENRV_{t} are the ratios of total enrollment in
public and private institutions of higher education, respectively, to various
‘eligible’ college-age population or total population; TFU_{t} and TFV_{t}
are average undergraduate tuition and fees, room and board paid by students in public and
private institutions of higher education, respectively; Y_{t}, WAGE_{t}
and UR_{t} are disposable income, average wage rate in retail trade, and
unemployment rate of civilian workers as percent of civilian workers, with all explanatory
variables deflated by the consumer price index and subscript t in each variable refers to
year t.

The conventional model of the demand for higher education in Equations (1A) and (1B) may be expanded by including, as Hights(1975) did, a cost of education at alternative institutions to capture the cross price effects. If the true model explaining the behavior of college enrollment calls for the inclusion of the tuition at the alternative institutions, the omission of the variable, as in Equations (1A) and (1B), will result in usual specification bias. [Gujarati(1995), pp. 204-7; Greene(1990), pp. 259-61]. In the absence of alternative tuition in the model, the parameter for the tuition as own price actually measures the gross effect of tuition on enrollment ratio, which is equal to the sum of both direct and indirect effects of tuition on enrollment. Thus an expanded model of the demand for higher education by including tuition at alternative institutions may be specified as follows:

ENRU_{t} = f(TFU_{t}, Y_{t}, WAGE_{t},
UR_{t}, TFV_{t}) (2A)

- + - + +

ENRV_{t} = g(TFV_{t}, Y_{t, }WAGE_{t},
UR_{t}, TFU_{t}) (2B)

- + - + +

The signs below the equations indicate the expected direction of change in the relationship between the dependent variable and each explanatory variables. The sign of the cost of education as own price is expected to be negative as higher cost of education will make higher education less affordable. The sign for the income variable is usually expected to be positive with disposable income as a measure of both amounts and sources of funds to purchase enrollment, particularly for the enrollment at private institutions. But as Hight(1975) points out, rising income may also cause a shift from attending lower-priced public institutions to attending higher-priced private institutions and it may also cause a shift on the part of some of those who would have been in the market in any case to the private institutions. Depending on the relative strength of these effects of a rising income, it may be difficult to determine a priori the net effect of a change in income on total college enrollment. With wage rate in the retail sector(WAGE) serving as a proxy for the opportunity cost of attending colleges, it is expected that higher the wage rate, lower the enrollment. Finally, when unemployment rate (UR) is lower, it is easier for potential students to find jobs and thus work, rather than attending colleges and vice versa. Therefore, the expected sign for the UR is positive. Finally, the expected sign of the tuition at alternative institutions is expected to be positive as is usual with the cross-price effect is positive. As the tuition at alternative institutions rises, this will make enrollment at the institutions under consideration more attractive.

As mentioned earlier, the dependent variable in the conventional model of the demand for higher education is defined as the ratio of total enrollment at public and private institutions, respectively, to various definition of ‘eligible’ college-age population or total population. Starting with Campbell and Siegel(1967), they used the seven year cumulative totals of high school graduates of 18-24 year age group who are not in the armed forces. In later studies, Hights(1975) and Shim(1990) used the cumulative sum of high school graduates in the current and three or four preceding academic years as their measure of ‘eligible’ college age population. In the most recent study, Hsing and Chang(1996) used total population in thousand as the denominator of the ratio of the enrollment ratio.

Which between the two measures, namely ‘eligibles’ and total population, would serve better as the denominator in the calculation of enrollment ratios for our purpose? Examination of historical growth paths of underlying data in the calculation of enrollment ratios convinces us that total population would serve better now than variously defined ‘eligibles’. To understand this, let us examine first the growth patterns of number of high school graduates and eligibles which is measured the cumulative sum of number of high school graduates. In the earlier period from 1965 to 1982, the number of high school graduates and thus the ‘eligible’ college age population continued to grow at a steady rate, as total population did. Thus enrollment ratios using either the eligibles or total population would have served well. In a sense, one may argue that the eligibles might have served better, for it measured more narrowly the total sum of those people who would most likely consider enrolling colleges. In the later period since 1983, however, the number of ‘eligible’ started to decline gradually, as the number of high school graduates started to decline two years earlier. (See Figure 1). During the same period, total enrollments at public and private institutions, respectively, grew only slowly. Consequently, enrollment ratios against ‘eligible’ college-age population rose sharply, not because total enrollment increased sharply, but because denominator reversed its trend and continued to declined substantially. In addition, as larger number of older people started to attend the college in recent years for various reasons, variously measured cumulative sum of high school graduates appear no longer to be a good measure for so-called ‘eligible’ college age population.

In light of these new developments, a broad measure like total population would appear to serve better as a denominator for the calculation of the enrollment ratio. One added advantage of using the total population lies in the simple fact that such population projection by the Census of Bureau is readily available. For the same reason, the U. S. Department of Education develops its projected enrollment ratios or ‘participation rate’ by different category, which are then multiplied by population projection developed by the Bureau of Census. Given the above considerations, it would seem quite reasonable to use the total population as the appropriate denominator. Enrollment ratios against total population I thousand at public and private institutions, respectively, are presented in Figure 3. It is worthwhile to note that enrollment ratios in Figure 3 appear to capture the general growth patterns of total enrollment at two institutions in Figure 1, mainly because growth path of total population is steady.

**3. Empirical Results**

A. __Data__

Yearly data on fall enrollment in public and private institutions of higher education,
and average undergraduate tuition and fees, and room and board, paid by students in public
and private educational institutions are obtained from *Digest of Education Statistics
1997 and prior years*; other macroeconomic data like disposable personal income per
capita, unemployment rate, and consumer price index are obtained from *1998 Economic
Report of the President;* average hourly earnings of production workers in retail trade
is downloaded from *National Employment, Hours, and Earnings* of BLS home page.

B. __Regression Results__

Since the specification of the model is in general terms with no discussion of function form for an empirical model, we now turn to the functional form for an empirical model. In prior studies, log-linear is commonly used for its use of interpretation. However, since there is no a prior knowledge to choose such a functional form, it is advisable to use the Box-Cox transformation, as Hsing and Chang(1996) did, in choosing an appropriate functional form. Table 2 presents regression results for enrollment ratios for Equations (1A) and (1B) at public and private institutions, respectively, under three functional forms: linear, log-linear, and general and non-linear. Regression results in Table 2 are estimated from the time series data for the period from 1965-1995 and they are after the autocorrelation correction. The autocorrelation correction was necessary, for autocorrelation was evident from the initial regression results in all cases.

Examining regression results, we note that virtually all the coefficients have the correct signs and highly significant at either 5% or 1% level. The only exception is the positive and insignificant parameter for wage rate for public institutions. Of particular interest are estimated tuition or own price and income elasticities. Using the double-log linear version for its ease of interpretation, we note that own price elasticity at public institutions (-0.797) is substantially higher than corresponding own price elasticity for private institutions (-0.154). Looking at income elasticities at two institutions, we note that income elasticity at public institutions (1.558) is also substantially higher than the corresponding income elasticity for private institutions (0.359). So it appears that tuition and income do affect enrollment decision at private institutions, their role is larger in the larger in the determination of enrollment ratios at public institutions than at private institutions. This may be true when a great majority of those who decide to enroll at private institutions do so regardless of tuition and income, while tuition and income are the two major concern to those who are making decision to enroll at public institutions.

The role of wage rate in the determination of enrollment is somewhat mixed, while the role of unemployment rate is unambiguously positive.

*Box-Cox tests of appropriate functional form* with regression results in Table 1.

Once we choose a correct functional form, now discuss how tuition elasticities change by adding tuition at alternative institutions by comparing regression results in Tables 2 and 3. Role of cross price elasticity.

*Box-Cox test again to the new specification* and determine the correct functional
form. Evaluate the stability of regression results with focus on the specification and
choice of functional form.