Estimating permanent and transitory income elasticities of education spending from panel data

Journal of Public Economics 92 (2008) 2132–2145

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Journal of Public Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o n b a s e

Estimating permanent and transitory income elasticities of education spending from panel data☆ Stephen J. Schmidt ⁎, Therese A. McCarty Department of Economics, Union College, Schenectady NY 12308, USA

a r t i c l e

i n f o

Article history: Received 1 December 2006 Received in revised form 12 February 2008 Accepted 13 March 2008 Available online 21 March 2008 JEL classifications: H72 I22 C23 Keywords: Education spending Panel data Income

a b s t r a c t We use a twenty-one year panel of data to examine the role of past income and aid, and expectations of future income, in regressions explaining state and local education spending. We show that simple estimates of the elasticity of spending with respect to financial resources are not robust to specification changes because the variables are non-stationary over time, causing inconsistent estimation of model parameters. Estimation in first differences (or equivalently, in growth rates) solves the time-series problems and produces robust estimates of the model's parameters. We then show that current spending by states responds to changes in expected future income. This explains why using fixed effects in simpler models reduces estimated income elasticities; fixed effects partially capture permanent income effects on spending. Estimates of lagged income are significant when used in models that do not explicitly model the expectations process, but present and past aid both have no effect on education spending. Models with structural assumptions about expected income produce estimates very similar to simpler models which include lagged information on income as a control variable. We conclude with recommendations for estimating models when only cross-section data or only short panels are available. © 2008 Elsevier B.V. All rights reserved.

1. Introduction The funding levels that state and local governments choose to allocate to public schools depend on the financial resources available to those governments, the costs of providing education, and willingness of citizens to pay those costs to receive the benefits that education brings. States with high incomes may choose to spend more on education, both because higher-income citizens are willing to spend more on education and because districts in areas with high costs of living may feel pressure to pay higher wages to teachers and other education staff. Empirical work on education spending has produced a wide range of estimates of the effects of income on education spending, with results that are often quite sensitive to model specification. In the last ten to fifteen years, econometricians have started using panel data, rather than single-year cross-section data, to estimate the elasticities of state and local education spending with respect to aid and income. Panel data offer many advantages for estimating these elasticities, but also present some difficulties. One advantage, widely recognized in the literature, is that panel data sets increase the size of the sample available; cross-section regressions on U.S. states are necessarily limited to 50 observations. Another advantage, less widely recognized, is that panel data permit econometricians to examine the difference

☆ Some of the work in this paper was done while Schmidt was at Rensselaer Polytechnic Institute and while McCarty was at the Center for State and Local Taxation at the University of California at Davis, whose support is gratefully acknowledged. The authors thank Alison Payne, Terri Sexton, two anonymous referees, participants at the Union and Vassar economics seminars, and at the 2006 American Education Finance Association meetings for their very helpful comments on an earlier draft of this paper. All remaining errors are the sole responsibility of the authors. ⁎ Corresponding author. Tel.: +1 518 388 6078; fax: +1 518 388 6988. E-mail addresses: [email protected] (S.J. Schmidt), [email protected] (T.A. McCarty). 0047-2727/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jpubeco.2008.03.005

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between permanent and transitory changes in income. A state or school district facing a permanent increase in income or aid is likely to increase school spending by more than one facing a transitory increase that is not expected to continue into the future. Making this distinction requires the use of panel data, so that permanent changes in income and aid can be distinguished from temporary changes. Cross-section regressions will estimate a relationship between income, aid, and spending in the year of the cross-section, but cannot identify whether the relationship is the response of states and school districts to permanent shocks or transitory shocks if those responses are different, as theory suggests they should be. The difficulties of using panel data arise because panel data include time variance of the variables, and the non-stationarity of these variables over time causes bias in OLS estimates. The long-run relationship of income, aid, and spending may be different from the ceterus paribus response of spending to changes in the first two variables, and there may be spurious correlations between unrelated variables due to their time-series properties. It is not possible to draw valid conclusions about the relationship of education spending, income, and aid from panel data without considering the time-series properties of that data, to be sure that the elasticities produced are estimated consistently. In this paper we use a twenty-one year panel data set to estimate elasticities of education spending with respect to financial resources by state and local governments in the United States, dealing appropriately both with changes in resources over time and with the time-series properties of the data. Recent literature in state-level public finance using panel data strongly suggests that results are critically dependent on how issues in using panel data, such as lagged independent variables, fixed effects, and period effects, are handled. Misspecifying the error structure of the data can create bias in the parameter estimates; in particular, the literature shows that including fixed effects in a regression often causes substantial changes in the parameter estimates and their interpretation. Failing to include dynamic effects in the model can lead to omitted variable bias as well. Explicitly dynamic equations in education spending models are not common; where dynamic effects such as permanent income are not included, fixed effects may be correlated with them, but may not fully or correctly control for them. We begin by estimating a simple expenditure equation, and demonstrate the sensitivity of the results to different specifications of fixed and period effects. We show that the variables in the equation are not stationary, which may cause the problem of spurious regression and thus contribute to the instability of the parameter estimates. We consider several different methods of addressing the problem of non-stationary variables, and find that estimating regressions in first differences (or equivalently, in growth rates) is the most satisfactory way to handle the problem. We then turn to conceptual explanations of why the variables have these properties. We propose a model in which states and school districts use expectations of future income, as well as present income, to set levels of education spending, using lagged income to predict future income. We show that current spending does react to changes in expected future income, but there is little or no effect from current income changes when expected future income is included in the model. This explains why fixed effects are often significant in models without measures of permanent income – they capture the income history of the state, which changes only slowly over time – and why lagged income is statistically significant when added to simpler models that do not explicitly model expectations. It also explains why including fixed effects in models without lagged income changes elasticity estimates so dramatically. The fixed effects are correlated with the omitted permanent income, and when permanent income is included, current income has a much reduced effect, or no effect, on spending. We also show that, in regressions where time issues are addressed, neither present aid nor future aid has any significant effect on spending. We conclude with some implications about alternative ways to estimate income elasticities when fixed effects are not feasible, or to properly control for income differences across states when estimating other parameters of education spending models. The rest of this paper is organized as follows. Section 2 reviews the literature using panel data and fixed effects in state-level public finance. Section 3 presents a simple reduced-form model to be used as a reference point in the subsequent analysis, and shows that different assumptions about the error structure for the estimates produce greatly varying estimates of the parameters. Section 4 shows that including lagged values of the variables does not eliminate the variation of the estimates, and that the variables in the analysis are not stationary, which explains the instability of the estimates. Section 5 shows that first differencing, or equivalently, estimating models in terms of growth rates, is the most reliable econometric method of resolving the problems. Section 6 presents a two-period model of spending that separates transitory and permanent income effects on spending, and implies that lagged terms should be significant. Estimating the model shows that spending reacts strongly to permanent changes in income, but little or not at all to transitory changes, consistent with theory. Section 7 concludes. 2. Literature review: panel data in empirical public finance Many studies of state and local government expenditure estimate resource elasticities using single-year cross-section data, often because of difficulty obtaining data for municipalities or school districts on an annual basis. However, a significant number use panel data, either with annual observations or with observations separated by 5 or 10 year intervals. Most papers with panel data include fixed and period effects, and most of those show results of both OLS regressions and regressions that include fixed and period effects. We have identified seven papers, discussed below, that report both types of regressions, allowing us to observe the consequences of including or excluding fixed and period effects. Almost all of these papers report substantially different coefficient estimates when fixed and period effects are included. This pattern is not surprising, as we would expect variation across states and variation within states to be quite different, especially if the variation within states depends on only a few observations for each state, and particularly if variation in financial resources is correlated with unobserved determinants of spending. Holtz-Eakin (1986) estimates a spending equation using an eight-year panel of municipal governments, and argues that models without fixed effects are misspecified as they exclude unobservable characteristics of municipalities that fixed effects

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can capture. He finds significantly different estimated parameters when fixed and period effects are included. Aronsson et al. (2000) also find significantly different parameter estimates with fixed effects in a short panel, in this case seven years for data on Swedish counties and municipalities. They note that, “Earlier studies of local public expenditure determination using Swedish cross-sectional data have found stronger tax price effects than what we find in the present analysis. These differences are not explained by the neglect of county expenditures in previous studies. Instead, they are related to the use of fixed effects in the present study.” (p. 196). Goldhaber (1999) uses fixed effects in a model of public school expenditure. He finds that the sign on the coefficient of primary interest in the paper, the one on private school enrollment rate, changes when district fixed effects are included. The results without fixed effects are consistent with the author's theoretical predictions while the results with fixed effects are not. Shadbegian (1999a) and Jin and Zou (2002) use longer panels and find less dramatic changes when fixed and period effects are added. Shadbegian (1999b) uses state fixed effects in a model in which counties are the unit of observation, rather than countylevel fixed effects, but gets similar results. One notable change in Jin and Zou (2002) when fixed effects are added is the change in the coefficient for real GDP, which becomes negative and statistically insignificant. Painter and Bae (2001) use a 28-year panel and find that some coefficients are “improved” by the introduction of fixed effects. In Shadbegian (1999a), income coefficients become more negative when fixed effects are added. Jin and Zou (2002) note the disadvantage of having to drop variables that are constant over time and that reflect important economic and institutional aspects of the model. A small number of studies have used more sophisticated panel data techniques. Harris et al. (2001) estimate a spending equation on school districts using both district-level fixed effects and period effects which are allowed to vary by state. They show that a larger elderly population decreases state and local education spending but not district spending; they include fixed effects in all regressions they report. Bradbury et al. (2001) estimate a spending model using first differences of spending to remove fixed effects in a short (four-year) panel. They find that percentage changes in school spending are significantly lower in towns with higher incomes, but the effect is quite small. Papers that include lagged spending, income, or aid in panel data estimation of spending equations are less common, and confront econometric complications raised by the lagged variables. Most of them respond to these complications by using vector autoregression (VAR) methods, or related methods. McCarty and Schmidt (1997, 2001) estimate VAR models including education spending, income, and aid, but do not give structural interpretations of the parameters. Bensi et al. (2004) also estimate VAR models of education and income, finding that causality runs from education expenditures to income rather than vice-versa. More work has been done with lags in studying other types of public spending. Connolly (1999) estimates a VAR for welfare spending and determines the effect of various shocks on spending over time. Buettner and Wildasin (2006) estimate a vector error correction (VEC) model for a sample of 1270 cities from 1972 to 1997. They find that lagged values of spending, revenue, and vertical grants are significant in the model, and trace the dynamic relationships between these variables over time. They find that responses are systematically different between large and small municipalities, with large municipalities relying much more on grants to smooth expenditures over time. 3. Simple reduced-form analysis Our goal is to use panel data to estimate correctly the response of state and local education spending to changes in financial resources. In this section, we begin by estimating some simple models of state and local education spending, to establish a baseline against which later results can be compared, and to establish comparability with other results in the literature.1 We show that the estimates change substantially when different error structures are used, and when lagged terms are introduced to include dynamics, suggesting that more complex models are needed to produce reliable or robust estimates of elasticities. In this baseline model, we estimate an equation of the form log ðSpendit Þ ¼ b0 þ b1 4 log ðIncomeit þ b2 4EdAidit þ b3 4GenAidit Þ þ b4 4StudentFracit þ b5 4Reformit þ b6 4ItemizeFracit þ b7 4Xit þ eit

(1)

ð1Þ where Spendit is state and local spending per capita in state i in year t, EdAidit is federal aid that is restricted to education spending, 2 GenAidit is general (unrestricted) federal aid, Incomeit is per capita income , StudentFracit is the fraction of the state's population that is school-aged (6 to 17 inclusive), Reformit is a dummy variable that is 1 if the state has experienced a court-ordered reform of its education finance system and 0 if it has not, ItemizeFracit is the portion of tax returns filed in the state that itemized deductions (thus exporting a portion of the taxpayer's school taxes to the Federal government)3, and X is a vector of demographic variables 1 The amount of spending a state will choose is the product of the quantity of education services demanded by the residents of the state and the unit cost of providing such services. Thus, the dependent variable in the equation is total expenditure, rather than quantity demanded. The equation should not be interpreted as a demand curve and the elasticities estimated are expenditure elasticities, not demand elasticities. 2 We have also estimated the model using median household income in the equation. There are several reasons why we prefer per capita income. The main one is that per capita income is available annually, while median household income is available only in census years. It can be interpolated into an annual series, but the resulting series is subject to multicollinearity problems; the series and two lags of itself are perfectly multicollinear. In particular, no more than two lags can be used in unit root tests. Results with median household income are not very different from those reported here, but are more difficult to interpret, probably because of the problems caused by the interpolation. 3 We have also interacted this variable with a dummy variable indicating whether the state has a sales tax and no income tax (since sales taxes are not deductible) but the interaction is not significant.

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Table 1 Means and standard deviations of variables Mean Observed variables Aid Income Revenue Ed Aid Ed Exp Enrollment Reform StudentFrac ItemizeFrac High school College Poverty Over 64 Urban Asian White Derived variables Log(Aid) Log(Income) Log(Revenue) Log(Ed Aid) Log(Ed Exp)

656.60 21,233.19 2517.88 113.15 843.51 874,912 0.20 17.40 30.73 76.06 19.57 12.86 12.40 68.26 1.50 84.60

6.433 9.946 7.796 4.689 6.714

Std. Dev.

Maximum

Minimum

227.75 3967.34 678.35 33.71 184.92 918,100 0.40 1.86 7.31 7.88 4.66 3.78 1.82 15.00 1.54 9.59

1754.13 37,118.20 5020.27 376.03 1473.11 6,142,348 1.00 25.82 50.29 91.80 34.60 25.50 18.31 94.00 10.91 99.11

249.81 12,894.17 1252.73 30.95 463.74 89,940 0.00 13.41 13.43 53.10 10.20 6.18 7.50 31.76 0.20 59.40

0.327 0.183 0.265 0.281 0.216

7.470 10.522 8.521 5.930 7.295

5.521 9.465 7.133 3.432 6.139

including the fraction of the state's population that is high-school educated, college educated, below the poverty line, elderly, living in urbanized areas, ethnically Asian, or ethnically Caucasian. The fraction of educated populations captures increased demand for education by highly educated parents, and the college-educated variable may also captured the availability of qualified teachers, affecting costs. The fractions of the population that are elderly or in poverty reflect competing needs for government funds and, in the case of elderly, voters without children. The fractions of population that are Asian and white have been shown to reflect demand for education spending. The fraction that are urban and the fraction that are in poverty capture differences in the perstudent cost of providing education. We use a functional form which is generally log-linear, but in which the financial resources variables are related additively to one another, because they may be fungible. The parameter β1 is the elasticity of spending with respect to total financial resources; the parameters β2 and β3 allow for the possibility that a dollar of aid may have greater effects on spending than a dollar of state income, either because there are flypaper effects or because there are deadweight losses associated with taxing private income.4 In the estimation, we use a panel of data from 1980 to 2000 on 48 states (Alaska and Hawaii are excluded). Data on state income, spending and aid are from the Bureau of the Census's annual publication, State Government Finances.5 School finance reform data are from Yinger (2002) and Murray et al. (1998). Data on school-age population are from the Digest of Education Statistics. Demographic data are taken from the Census Bureau, and where not available on an annual basis, are interpolated to annual observations using the 1980, 1990, and 2000 censuses with a quadratic fit. All financial data are in real 1996 dollars and in per capita terms, except data on spending and education-specific aid, which are in per capita terms. Means and standard deviations of all variables used in the analysis are found in Table 1. We first estimate the model assuming independent and identically distributed errors across states and across time; no fixed effects, no trend or period effects, and no dynamic structure. Results of the estimation are found in Table 2, column 1. We then make a variety of assumptions about the error structure and re-estimate the model. In Table 2, columns 2, 4, and 6 include fixed effects to control for unobserved variation across states. Columns 3 and 4 include a time trend to allow for a general increase (or decrease) in spending over time; columns 5 and 6 instead include period effects to allow for variation over time in the estimation that is not linear, such as business cycle variation. The results change considerably depending on what assumptions are made about the error structure. The resources elasticity of spending, for example, ranges from a high of 1.29 to a low of 0.19. The variation in these elasticities implies a corresponding range in the policy implications of the model. There is also a large variation in the coefficients on education-specific aid and general aid, ranging from − 2.88 to − 19.85 for education-specific aid and 2.13 to 10.03 for general aid. Each type of aid has larger effects on

4 We thank an anonymous referee for suggesting this functional form. We have also estimated the model in functional forms where aid and income enter loglinearly as separate variables (a more traditional functional form; see, for example, Baqir 2002) and in linear functional forms. Results are very similar to the results presented here, and are available on request from the authors. 5 Spending includes only public school funds, and student counts include only public school students.

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Table 2 Estimation results, Eq. (1) Baseline model, no dynamic terms Variable

Regression 1

Intercept log(Resources) EdAid GenAid StudentFrac Reform Itemize fraction High school College Poverty Over 64 Urban Asian White

−5.2631 0.6059 1.1023 0.0592 −7.6337 3.5624 10.0295 1.0271 0.0119 0.0033 0.0308 0.0092 0.5156 0.0625 0.0046 0.0009 −0.0054 0.0014 0.0046 0.0019 0.0033 0.0027 −0.1094 0.0324 −0.0108 0.0030 0.0015 0.0006

2

3

4

−1.1030 0.5849 0.5786 0.0541 −13.8038 4.0937 2.1379 0.9211 0.0215 0.0034 0.0113 0.0086 −0.0173 0.0516 0.0142 0.0011 0.0003 0.0015 0.0140 0.0032 −0.0007 0.0054 −0.2446 0.1130 0.0304 0.0073 0.0067 0.0023

−7.3653 0.7260 1.2966 0.0697 −3.5850 2.9181 8.3800 0.8059 0.0158 0.0034 0.0289 0.0091 0.4774 0.0621 0.0072 0.0010 −0.0056 0.0014 0.0083 0.0020 0.0080 0.0028 −0.1776 0.0346 −0.0124 0.0030 0.0005 0.0006 −0.0072 0.0014 No No 0.8037 1008

− 0.3365 0.6234 0.4927 0.0594 −17.5090 4.9333 2.2083 1.0574 0.0214 0.0034 0.0061 0.0087 0.0062 0.0517 0.0110 0.0014 0.0002 0.0015 0.0124 0.0033 − 0.0088 0.0059 − 0.2915 0.1131 0.0334 0.0074 0.0116 0.0027 0.0068 0.0020 Yes No 0.9453 1008

Trend Fixed effects? Period effects? R2 Observations

No No

Yes No

0.7985 1008

0.9446 1008

5 −7.3941 0.7188 1.2880 0.0684 −2.8821 2.9118 9.2989 0.8628 0.0159 0.0033 0.0425 0.0087 0.8693 0.0725 0.0065 0.0010 −0.0069 0.0013 0.0079 0.0020 0.0089 0.0027 −0.1885 0.0329 −0.0175 0.0028 0.0002 0.0006

6 3.5257 0.5960 0.1931 0.0558 −19.8543 11.6732 3.6396 2.8389 0.0102 0.0031 0.0173 0.0072 1.2830 0.0769 0.0095 0.0012 0.0001 0.0012 0.0081 0.0028 −0.0254 0.0051 −0.1759 0.0950 −0.0032 0.0069 0.0036 0.0026

No Yes

Yes Yes

0.8313 1008

0.9646 1008

Standard errors in italics. Parameters in bold are statistically significantly different from zero at the 5% level.

spending than income does, although the coefficient on education aid is implausibly negative; it is significant in only three of the six cases.6 Some of the other results are more consistent than these across different assumptions about the error structure. For example, student fraction always increases spending per capita by 1% to 2% per 1% increase in student fraction. But other results are not so consistent; for example, the estimates on many of the demographic parameters change significance or sign or both in different specifications. Furthermore, the variation in the income and aid parameters is large enough to cause considerable difficulty in using the model to guide policy. Appropriate F-tests show that both fixed effects and period effects are statistically significant and that the linear time trend is not an acceptable simplification of the model with period effects. Still, the variation in parameter estimates is troubling, especially since data limitations may make it impossible to include fixed and period effects in the model if a shorter panel (or worse, a cross-section) is used. Thus, it is necessary to understand why this variation in parameter estimates occurs, and how to specify the model to eliminate or reduce it. 4. Adding simple dynamic elements Dynamic effects may help explain why changes in the assumptions about the error structure change the results. For example, past resources may matter in determining spending as well as current resources. If so, then lagged values should be included in the equation; if they are not, the regression suffers from omitted variable bias. Including fixed effects may reduce this bias and thus change the estimates of the parameters of the model. The ranking of states by income changes relatively slowly over time, so fixed effects will capture some of the distinction between a historically high-income state (e.g., Connecticut) and a historically low-income state (e.g., Mississippi). The situation may be as shown in Fig. 1. Holding all other

6 The significant cases are spurious, because the time-series properties of the data result in a substantial downward bias in the estimates of the standard errors of the parameters. We return to this point below.

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Fig. 1. Hypothetical data showing possible results of including fixed effects.

factors constant, state 3 has higher levels of income in all years than states 1 and 2, and thus for any given level of current income, spends more on education. If so, then including fixed effects would then reduce the estimated effect of income on spending. In fact, that is exactly what including fixed effects does; in all three columns using fixed effects in Table 2 (columns

Table 3 Estimation results, Eq. ( 2) including dynamic terms Variable

Regression 1

Intercept log(Resources) EdAid GenAid log(Resources− -1) StudentFrac Reform Itemize fraction High school College Poverty Over 64 Urban Asian White

−5.9724 0.6260 0.6767 0.1188 −7.3841 3.3868 9.0789 0.9209 0.4952 0.1205 0.0139 0.0033 0.0280 0.0091 0.5356 0.0622 0.0042 0.0009 −0.0058 0.0014 0.0061 0.0019 0.0039 0.0027 −0.1285 0.0324 −0.0112 0.0030 0.0017 0.0006

2 −2.3407 0.6074 0.0607 0.0681 −9.9311 3.2829 −0.4390 0.6707 0.6394 0.0709 0.0259 0.0033 0.0146 0.0083 −0.0240 0.0494 0.0135 0.0011 −0.0004 0.0015 0.0174 0.0031 0.0037 0.0052 −0.2816 0.1092 0.0226 0.0071 0.0068 0.0022

Trend Fixed effects? Period effects? R2 Observations

No No

Yes No

0.8018 1008

0.9491 1008

3

4

−8.2905 0.7491 0.8204 0.1258 −3.0849 2.7605 7.5344 0.7191 0.5663 0.1243 0.0181 0.0033 0.0259 0.0090 0.4966 0.0616 0.0069 0.0010 −0.0060 0.0014 0.0101 0.0020 0.0089 0.0028 −0.2025 0.0346 −0.0129 0.0030 0.0007 0.0006 −0.0076 0.0014 No No 0.8077 1008

−1.5750 0.6349 0.0015 0.0668 −11.5550 3.6148 −0.7360 0.7240 0.6134 0.0703 0.0256 0.0033 0.0096 0.0084 0.0015 0.0495 0.0104 0.0014 −0.0004 0.0015 0.0157 0.0032 −0.0045 0.0057 −0.3240 0.1092 0.0256 0.0071 0.0115 0.0026 0.0066 0.0019 Yes No 0.9497 1008

Standard errors in italics. Parameters in bold are statistically significantly different from zero at the 5% level.

5 −8.1792 0.7331 0.5203 0.1464 −0.9711 2.7999 8.7344 0.7932 0.8482 0.1488 0.0181 0.0033 0.0410 0.0085 0.8565 0.0713 0.0058 0.0010 −0.0069 0.0013 0.0080 0.0020 0.0100 0.0026 −0.2021 0.0325 −0.0186 0.0028 0.0002 0.0006

6 3.0098 0.6441 0.0757 0.0751 − 15.2861 10.3731 2.9879 2.3923 0.1708 0.0811 0.0116 0.0032 0.0177 0.0072 1.2388 0.0800 0.0092 0.0012 −0.0001 0.0013 0.0087 0.0028 −0.0247 0.0051 −0.1781 0.0949 −0.0043 0.0069 0.0034 0.0026

No Yes

Yes Yes

0.8369 1008

0.9648 1008

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Table 4 Summary of unit root tests

Unit root tests in levels Levin, Lin, and Chu ADF-Fisher

Unit root tests in first differences Levin, Lin, and Chu ADF-Fisher

Log(EdExp)

Log(EdAid)

Log(Aid)

Log(Income)

11.666 (1.000) 7.226 (1.000)

13.61 (1.000) 5.937 (1.000)

11.358 (1.000) 5.003 (1.000)

15.597 (1.000) 2.083 (1.000)

− 7.721 (0.000) 137.206 (0.004)

− 7.492 (0.000) 160.331 (0.000)

−9.985 (0.000) 188.722 (0.000)

−8.137 (0.000) 149.265 (0.000)

P-values in parenthesis. Critical value for LLC test is 1.645; 5% critical value for ADF-Fisher test is 65.17. All tests use 4 lags of the dependent variable.

2, 4, and 6), estimated resource elasticities are much lower than in the corresponding column without fixed effects (columns 1, 3, and 5 respectively). If so, however, the model is still misspecified, because state income patterns do change over long periods of time. In a long panel the fixed effects would not perfectly control for dynamic effects, and so the parameter estimates would still be biased. Also, the estimated elasticity would be neither the response to a change in temporary income nor to a change in permanent income; it would overestimate the former and underestimate the latter. A better solution is to include lagged income and aid variables directly in the model, in hopes of explaining the variation across states (thus eliminating the need for fixed effects). Also, a model with both current and lagged income allows for explicit consideration of the differences between temporary and permanent changes in state income. To test the possibility that dynamic effects may be important, we include lagged financial resources in the equation:
 log ðSpendit Þ ¼ b0 þ b1 4 log ðIncomeit þ b2 4EdAidit þ b3 4GenAidit Þ þ b4 4 log Incomeiðt1Þ þ b2 4EdAidiðt1Þ þ b3 4GenAidiðt1Þ þb5 4StudentFracit þ b6 4Reformit þ b7 4ItemizeFracit þ b8 4Xit þ eit

ð2Þ

Results of estimating Eq. (2) under the same six assumptions about the error structure are shown in Table 3. The results show that dynamic effects do matter; the lag of resources is significant in all six specifications, and its coefficient is generally large (an elasticity between 0.49 and 0.85, except for one case of 0.17). The inclusion of lagged resources has altered the estimated parameters on the contemporaneous financial resources; the estimates are smaller than before, and are not significant in the specifications including fixed effects. F-tests show that both the fixed effects and period effects are significant, and the linear trend is not an acceptable simplification of the period effects. However, there is still considerable variation in the parameter estimates across the different error specifications. Lagged resources do not explain why estimated resource elasticities are sensitive to changes in the error specification. There may be more complex dynamic effects in the equation. In particular, it may be that the variables in the equation have unit roots, and thus are not stationary. If so, then OLS estimation of the equation is subject to the problem of spurious regression, which could explain why the parameter estimates of both Tables 2 and 3 are not robust across specifications.7 There are a variety of methods for testing for non-stationary variables using panel data, most of which are variations of the basic augmented Dickey– Fuller test. They work with equations of the form Dyit ¼ b0 þ b1 4yitðt1Þ þ b2 4Dyitðt1Þ þ N þ bSþ1 4DyiðtsÞ þ eit

ð3Þ

where yit is the variable which may not be stationary, and the null hypothesis is β1 = 0 (implying that y is non-stationary) against the alternative hypothesis β1 b 0 (implying that y is stationary). In particular, some tests impose that requirement that β1 takes the same value for all cross-section units, while other tests allow for the possibility that β1 takes different values for different crosssection units. Table 4 contains a summary of unit root test results for the four financial variables. The Levin, Lin, and Chu (LLC) tests assume a common value of β1 across states, while the ADF-Fisher tests allow differing values for each state. The LLC test statistic has an asymptotically normal distribution, since it tests for a single value of β1, while the ADF-Fisher test statistic, which tests one value for each state, has an asymptotically chi-square distribution with (in this case) 48 degrees of freedom. Table 4 shows test statistics and p-values for all four variables, both in levels and in first differences. The results show clearly that the variables are not stationary. All four variables fail to reject the null hypothesis of a unit root in their levels. Because of this, regressions estimated using panel data that do not account for the non-stationarity of the data are likely to produce biased and inconsistent parameter estimates, and may not be reliable guides to use in making policy for education

7

The problem of spurious regression was identified by Granger and Newbold (1974).

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finance. In contrast, taking first differences removes the unit roots; we reject unit roots for the first differences of all four variables with both tests. Therefore the original variables are integrated of order 1, and regressions using the first differences should not suffer from the problem of spurious regression. Redoing the tests to include fixed effects or even state-specific trends (results not shown) improves the test statistics somewhat but in general does not permit us to reject the null hypothesis of a unit root in the levels of the variables. 5. Addressing the problem of non-stationary variables There are several possible approaches to dealing with non-stationary variables. The one we pursue here is to estimate the equation in first differences.8 Since the first differences of the variables are stationary, equations containing them can be consistently estimated by least squares. This method has not been widely used in the literature, but occasionally has been, as in Bradbury et al. (2001), although it is usually used to eliminate fixed effects and not to deal with stationarity of the variables. Taking first differences of Eq. (1) produces: D log ðSpendit Þ ¼ b0 þ b1 4DlogðIncomeit þ b2 4EdAidit þ b3 4GenAidit Þ þ b4 4DStudentFracit þ b5 4DReformit þ b6 4DItemizeFracit þ b7 4DXit þ tit

ð4Þ

where υit = Δɛit. The dependent variable, Δlog(Spendit), is approximately equal to the percentage growth rate of spending. Thus, this model is essentially equivalent to regressing the growth rate of spending on the growth rate of financial resources.9 It is important to note that estimation in first differences does not change the interpretation of the regression coefficients. In particular, β1 is still an estimate of the elasticity of per capita spending with respect to financial resources. This is true because taking first differences is merely a mathematical transformation of the original equation, not a reparameterization of it. However, taking first differences does change the way in which the estimate is inferred from the data. Because the dependent variable in the first-differenced model is the growth rate of spending, not the level of spending, the model no longer takes advantage of the differences between high-spending states and low-spending states in estimating the relationship between spending and resources. This has the effect of reducing the fit of the model. However, there are many differences between high-and lowspending states other than differences in aid and income. Regressions of Eqs. (1) and (2) implicitly attribute those differences to the effects of changes in aid and income, which may be incorrect, and can be a source of spurious correlation between financial resources and spending when the model is estimated in its non-stationary form. We estimate this equation using the same six specifications used in estimating Eqs. (1) and (2).10 Results are shown in Table 5. The most important point to note is that the parameter estimates are now far more robust to changes in the specification than previously. Most variables are either significant in all specifications in which they are used, or insignificant in all of them. Estimated parameter values for the resources parameter range from 0.187 to 0.238; most other parameter estimates are similarly robust. The primary exception is that StudentFrac becomes insignificant when period effects are included, while ItemizeFrac becomes significant. The second point to note is that in these regressions, the fixed effects are not statistically significant; appropriate F-tests comparing column 1 to column 2, column 3 to column 4, and column 5 to column 6 all show the fixed effects may be excluded from the model. This implies that variation in spending growth rates across states is constant across time and not a function of differences in trends across states. The period effects remain statistically significant, and the null hypothesis that they can be represented by a linear trend is again rejected. Also, the demographic variables have become, with two exceptions, insignificant; first differencing appears to largely control for these sorts of variations in spending. Using percentage growth rates of financial resources and spending, instead of their levels, produces a specification for the regression that is defensible on econometric grounds, and also much more robust to different assumptions about the error structure of the model. The model estimated in first differences is also the appropriate model for the counterfactuals implied by most policy analysis, because it eliminates all relevant variables about a state that are not changing over time, hence would not change in the counterfactual case. Because fixed effects are not significant in the estimates of Eq. (4), the first differencing has

8 Other options are to specify a functional form for the equation using variables whose values are bounded (bounded variables cannot be non-stationary) or to test the equation for cointegration. In work not reported here, we have pursued both of these approaches. Alternative functional forms giving expenditure and aid as shares of state income do not produce robust parameter estimates and are difficult to interpret in economic terms. Tests for cointegration of the variables give results which are very sensitive to assumptions about the structure of the model, but do not clearly support cointegration. Lags and leads estimators based on the assumption of cointegration (see Stock and Watson, 1993) also do not produce robust parameter estimates and in general do not appear plausible. Results are available from the authors upon request. 9 An alternative strategy is to include the demographic variables in levels, not first differences, since their levels are stationary. The results reported here assume that demographic variables affect the level of spending (i.e., states with high percentages of college graduates spend more) while the alternative specification assumes demographic variables affect the growth rate of spending (i.e., states with high percentages of college graduates increase spending at a higher rate). Estimates of equations with the demographic variables in levels are substantially very similar to the ones reported here. Results are available from the authors upon request. 10 Note that the inclusion of fixed effects in the first-differenced model has a different implication than including fixed effects in the levels model. Including fixed effects in the differenced regressions implies that the constant term in the equation for the change in spending differs between states. This implies that there are state-specific time trends in the levels model; spending grows faster in some states than others. Fixed effects in the levels model only imply that the amount of spending differs from state to state. If the fixed effects in the differenced regressions were statistically significant, then we would need to re-estimate Eq. (1) using state-specific time trends. However, as we will show, the fixed effects in the differenced equations are insignificant, and therefore state-specific trends in the levels model are not necessary.

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Table 5 Estimation results, Eq. ( 4) regression in first differences Variable

Regression 1

2

3

4

5

6

Intercept

0.0105 0.0062 0.2379 0.0516 18.0751 11.3210 3.3532 2.6077 0.0249 0.0047 0.0120 0.0100 −0.0648 0.0588 0.0093 0.0048 0.0022 0.0022 0.0081 0.0069 0.0006 0.0165 −0.3401 0.3360 0.0038 0.0200 0.0068 0.0072

0.0179 0.0123 0.2250 0.0531 19.4318 12.6971 3.5938 2.8909 0.0250 0.0049 0.0123 0.0104 −0.0795 0.0600 0.0017 0.0103 0.0019 0.0024 0.0041 0.0078 0.0072 0.0234 −0.5232 0.4814 −0.0120 0.0418 0.0045 0.0120

0.0191 0.0113 0.1873 0.0670 12.4886 12.4185 1.5361 3.0178 −0.0013 0.0060 0.0149 0.0097 0.3831 0.1378 −0.0036 0.0095 0.0000 0.0023 0.0096 0.0074 −0.0233 0.0259 −0.0768 0.4618 −0.0271 0.0405 −0.0113 0.0125

Yes No 0.1073 960

0.0343 0.0138 0.2189 0.0529 23.4315 14.0717 3.6493 3.0295 0.0263 0.0049 0.0140 0.0103 −0.0809 0.0598 0.0003 0.0102 0.0020 0.0024 0.0000 0.0079 −0.0332 0.0280 −0.1455 0.5012 −0.0469 0.0438 −0.0121 0.0135 −0.0015 0.0006 Yes No 0.1140 960

0.0097 0.0057 0.2066 0.0644 10.9000 10.6446 1.6416 2.6703 −0.0004 0.0057 0.0150 0.0094 0.4468 0.1308 0.0086 0.0045 0.0005 0.0021 0.0145 0.0066 −0.0218 0.0173 −0.0564 0.3255 −0.0182 0.0221 −0.0016 0.0081

No No 0.0867 960

0.0197 0.0075 0.2336 0.0515 21.1114 12.2929 3.5685 2.7323 0.0257 0.0047 0.0137 0.0100 −0.0665 0.0587 0.0097 0.0048 0.0022 0.0022 0.0048 0.0071 −0.0189 0.0188 −0.0997 0.3534 −0.0249 0.0240 −0.0040 0.0088 −0.0010 0.0004 No No 0.0912 960

No Yes 0.2448 960

Yes Yes 0.2636 960

log(Resources) EdAid GenAid StudentFrac Reform Itemize fraction High school College Poverty Over 64 Urban Asian White Trend Fixed effects? Period effects? R2 Observations

Standard errors in italics. Parameters in bold are statistically significantly different from zero at the 5% level.

removed the effect of unobserved attributes that vary between states, giving us a better estimate of the true effect of changing aid or income within a given state, which is generally what interests policymakers. Correcting for the non-stationary variables recalls the question of the role of lagged aid and income in spending. It may be that the significance of the lagged variables in the undifferenced Eq. (2) was an artifact of the spurious regression, and the lagged variables do not actually affect spending. We therefore also estimate first differences of Eq. (2):
 D log ðSpendit Þ ¼ b0 þ b1 4D log ðIncomeit þ b2 4EdAidit þ b3 4GenAidit Þ þ b4 4D log Incomeiðt1Þ þ b2 4EdAidiðt1Þ þ b3 4GenAidiðt1Þ þb5 4DStudentFracit þ b6 4DReformit þ b7 4DItemizeFracit þ b8 4DXit þ tit

ð5Þ

Results of estimating this equation are shown in Table 6. Both current and lagged resources remain statistically significant in all six specifications. General aid and education-specific aid now have much lower estimated parameter values and are never statistically significantly different from either 0 (implying aid has no effect on spending) or from 1 (implying that aid has the same effect as income on spending, that is, no flypaper effects). Inclusion of the lags has only slightly altered the estimated values of the coefficients of sameperiod resources; this occurs because the levels of income and aid are highly correlated within each state over time (due to the unit root structure of those variables) but their first differences are not highly correlated. The parameter estimates also remain fairly consistent across specifications; in particular, the coefficient on current resources ranges from 0.192 to 0.223, and the coefficient on lagged resources ranges from 0.249 to 0.381. In this model the fixed effects are again statistically insignificant, and the period effects remain significant and significantly different from the linear trend. Demographic variables are almost entirely insignificant. 6. A simple economic model including dynamic effects The results from the first difference estimates are statistically satisfying, but they do not explain why state and local expenditure should depend on lagged values of income, as they appear to do. In this section we develop a simple model of expenditure in which expectations of future income affects the level of education spending. In that model, governments and/or voters use past information on income to forecast future levels of income, so past information which

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Table 6 Estimation results, Eq. ( 5) regression in first differences with lags Variable

Regression 1

2

3

4

5

6

Intercept

0.0062 0.0060 0.2220 0.0491 1.3014 4.3107 −0.6881 1.0033 0.3795 0.0488 0.0254 0.0046 0.0088 0.0097 −0.0749 0.0574 0.0081 0.0047 0.0006 0.0022 0.0130 0.0068 0.0081 0.0160 −0.3197 0.3266 −0.0047 0.0195 0.0063 0.0070

0.0089 0.0120 0.2161 0.0505 0.7946 4.4871 − 0.7490 1.0368 0.3765 0.0502 0.0254 0.0047 0.0085 0.0101 − 0.0860 0.0586 0.0020 0.0100 0.0003 0.0024 0.0083 0.0076 0.0203 0.0229 − 0.4249 0.4686 0.0021 0.0407 − 0.0003 0.0117

0.0136 0.0113 0.1927 0.0662 4.0422 6.9120 0.4523 1.7860 0.2493 0.0649 0.0024 0.0060 0.0130 0.0096 0.4067 0.1371 −0.0034 0.0094 −0.0005 0.0023 0.0119 0.0074 −0.0135 0.0258 −0.0477 0.4583 −0.0201 0.0403 −0.0120 0.0124

Yes No 0.1562 960

0.0233 0.0135 0.2146 0.0506 2.1885 4.6048 −0.7750 1.0474 0.3748 0.0502 0.0266 0.0048 0.0100 0.0101 −0.0862 0.0585 0.0007 0.0100 0.0004 0.0024 0.0049 0.0077 −0.0154 0.0274 −0.0940 0.4879 −0.0290 0.0427 −0.0148 0.0132 −0.001321 0.00056 Yes No 0.1614 960

0.0068 0.0057 0.2039 0.0637 4.4869 6.4762 0.7234 1.6933 0.2569 0.0628 0.0035 0.0057 0.0133 0.0093 0.4505 0.1300 0.0076 0.0044 −0.0001 0.0021 0.0168 0.0066 −0.0156 0.0172 −0.0616 0.3229 −0.0228 0.0220 −0.0012 0.0080

No No 0.1380 960

0.0144 0.0073 0.2233 0.0493 2.5152 4.4046 −0.6080 1.0091 0.3808 0.0489 0.0261 0.0046 0.0102 0.0097 −0.0752 0.0573 0.0085 0.0047 0.0005 0.0022 0.0102 0.0069 −0.0093 0.0183 −0.1044 0.3436 −0.0305 0.0234 −0.0033 0.0085 −0.000869 0.000438 No No 0.1416 960

No Yes 0.2581 960

Yes Yes 0.2755 960

log(Resources) EdAid GenAid log(Resources− 1) StudentFrac Reform Itemize fraction High school College Poverty Over 64 Urban Asian White Trend Fixed effects? Period effects? R2 Observations

Standard errors in italics. Parameters in bold are statistically significantly different from zero at the 5% level.

increases (or decreases) expected future income will affect present spending. This model will help us give economic interpretations of the estimates of the earlier equations, or of other equations estimated with shorter panels or with crosssection data. Let there be a median voter who has preferences for present and future private consumption of a private good and for present and future education spending. This voter will have an expenditure function for present education spending which will generally depend on both current and future income. Higher income may increase demand for education spending, or may increase the perunit cost of education spending (perhaps by driving up wages for teachers). If the voter's preferences are not time-separable, so that the marginal utility of education spending today depends on the amount that will be spent in the future, then future income will affect today's desired level of spending. Alternatively, if some education spending contains a commitment to future spending (for example, construction of facilities which will have future operating expenses, or hiring teachers to multiple-year contracts, or making any other kind of sunk investment) then today's desired spending will depend on expectations about desired future spending, therefore on future income. Since future income is not known, the voter must form an expectation of it based on information available in the current period. For a simple model, we assume that expectations of future income are by taking the weighted average of past income. This method of forming expectations can be justified in a number of ways. For example, if expectations are rational (a common assumption in the literature) and income follows a unit root process but is measured with error, then a weighted average of past income is the rational expectation of future income. This includes a measure of permanent income in the model, which permits the model to distinguish between a historically high-income state with an unusually low present income, and a historically low-income state with an unusually high present income, as suggested by Fig. 1. Specifically, let expectations of real per capita future income EIF be given by

EIFit ¼ k

l X s¼0

ð1  kÞs IiðtsÞ

ð6Þ

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Table 7 Estimation results, Eq. ( 8) simple expectations of future income Variable

Regression 1

2

3

4

5

6

Intercept

−11.2331 1.8745 0.0648 0.0090 0.1143 0.0543 1.7432 0.2263 0.0267 0.0045 0.0082 0.0097 −0.0953 0.0578 −0.0055 0.0044 0.0000 0.0020 0.0090 0.0057 0.0019 0.0102 −0.7830 0.1726 −0.0142 0.0115 0.0033 0.0020

−6.9869 1.1453 0.2850 0.0199 −0.1376 0.0733 1.3040 0.1494 0.0220 0.0047 0.0056 0.0092 −0.0431 0.0576 0.0042 0.0023 0.0000 0.0018 0.0070 0.0057 0.0259 0.0110 −0.3659 0.1849 0.0537 0.0181 0.0144 0.0036

−1.9332 1.2834 0.2966 0.0225 −0.0800 0.0888 0.8934 0.1753 0.0070 0.0050 0.0145 0.0087 0.4496 0.1276 0.0087 0.0026 0.0005 0.0017 0.0035 0.0052 −0.0132 0.0112 −0.0455 0.1704 0.0174 0.0170 −0.0027 0.0049

Yes No 0.9803 960

−7.4820 1.2667 0.2814 0.0200 −0.1373 0.0729 1.3663 0.1633 0.0226 0.0047 0.0065 0.0093 −0.0585 0.0596 0.0059 0.0030 −0.0001 0.0018 0.0065 0.0057 0.0302 0.0119 −0.3353 0.1888 0.0497 0.0187 0.0109 0.0051 −0.0012 0.0012 Yes No 0.9803 960

−10.6522 2.4238 0.0528 0.0087 0.0933 0.0656 1.7051 0.2807 0.0019 0.0054 0.0131 0.0092 0.2694 0.1280 0.0008 0.0054 −0.0010 0.0019 0.0166 0.0058 −0.0150 0.0113 −0.7446 0.1867 −0.0211 0.0129 0.0028 0.0024

No No 0.9758 960

− 12.9901 2.1431 0.0650 0.0093 0.1081 0.0544 1.9066 0.2471 0.0276 0.0045 0.0098 0.0097 −0.1084 0.0581 0.0000 0.0053 −0.0003 0.0020 0.0112 0.0058 −0.0009 0.0104 −0.8126 0.1711 −0.0210 0.0121 0.0016 0.0022 −0.0007 0.0004 No No 0.9759 960

No Yes 0.9798 960

Yes Yes 0.9836 960

Lag term (λ) log Income Future income StudentFrac Reform Itemize fraction High school College Poverty Over 64 Urban Asian White Trend Fixed effects? Period effects? R2 Observations

Standard errors in italics. Parameters in bold are statistically significantly different from zero at the 5% level.

where the λ term outside the summation implies that if income has taken a constant value I in the past, then the expectation of future income will take the same value. Under these assumptions, spending at time t would be given by the equation log ðSpendit Þ ¼ b0 þ b1 4 log ðIncomeit þ b2 4EdAidit þ b3 4GenAidit Þ þ b4 4StudentFracit þ b5 4Reformit l X ð1  kÞs log ðIncomeÞiðtsÞ þeit þ b6 4ItemizeFracit þ b7 4Xit þ b8 4k4

ð7Þ

s¼0

which is the same as Eq. (1) with the addition of the future income term. This equation is not estimable in this form, because of the infinite lag structure of expectations. However, if we lag the equation by one time period, multiply the result by (1–λ), and subtract, we cancel out all but one of the terms in the summation, producing: log ðSpendit Þ ¼ k4b0 þ b1 4 log ðIncomeit þ b2 4EdAidit þ b3 4GenAidit Þ þ b4 4StudentFracit þ b5 4Reformit þ b6 4ItemizeFracit þ b7 4Xit  
 þð1  kÞ log Spendiðt1Þ þ ðk  1Þ4b1 4 log Incomeiðt1Þ þ b2 4EdAidiðt1Þ þ b3 4GenAidiðt1Þ þðk  1Þ4b4 4StudentFraciðt1Þ þ ðk  1Þ4b5 4Reformiðt1Þ þ ðk  1Þ4b6 4ItemizedFraciðt1Þ þ ðk  1Þb7 4Xiðt1Þ

ð8Þ

þb8 4k4 log ðIncomeit Þ þ vit

which can be estimated using nonlinear least squares.11 Initial estimations of Eq. 8, not shown here, showed that the effects of aid were not statistically significantly different from zero, as they were not in the differenced estimates in Tables (5) and (6). Estimates of Eq. (8) with the restriction β2 and β3 equal to zero are 11 Eq. (8) can also be estimated by ordinary least squares. However, imposing the nonlinear restrictions on λ and those β coefficients which appear twice in Eq. (8) improves the efficiency of the estimates. Also, because β0 and β8 appear in Eq. (8) only when multiplied by λ, it is difficult to calculate OLS standard error estimates for them. NLLS allows us to easily estimate their standard errors.

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shown in Table 7, under the same six specifications as in previous tables. Results are generally consistent with the earlier regressions and fairly stable with regard to the error specification, although several of the demographic estimates change when we add the fixed effects, and the fixed effects also substantially increase the estimate of λ, from about 0.06 to about 0.29. The model fits the data extremely well even without period effects or fixed effects (R2 = 0.9758) and the period and fixed effects improve the fit only slightly, though both are statistically significant. Unit root tests of the residuals show that the residuals are stationary. The model is estimated using the levels of the variables, not their first differences, but the transformation of the equation to remove the lag structure of expectations solves the problem of non-stationarity in a similar way, and we need not worry about the results of this estimation being spurious. The estimated parameters of current and expected future resources offer an interesting economic interpretation. The elasticity of spending with respect to expected future resources is statistically significant and quite large (between 0.89 and 1.74) in all specifications, suggesting that expectations of future resources, based on past values of income, do play an important role in determining state and local expenditure on education, as the model predicts.12 In contrast, the coefficient on current resources is not significant in four of the six specifications, and is quite small (0.11) when it is significant. This is consistent with the idea that states do not increase education spending, or do so only slightly, in response to transitory changes in resources. Most education expenditures are difficult to start and stop in response to resource fluctuations, so a state with a temporary increase in resources is not likely to want to increase education spending. Education expenditures should change only when there is a change in the permanent level of resources, and that is what the results of this model indicate. What do these regression results tell us about the actual elasticity of education spending with respect to financial resources? Because we have found that lagged resources matter, and the variables are not stationary, we should draw conclusions from one of the models that includes lags and deals appropriately with the problem of non-stationarity. We should also use one with an error structure that is not rejected by the data. Two of the regressions we have reported meet these criteria. The first is the regression in first differences, including lagged resources and period effects (but not fixed effects, which are statistically insignificant in that model) from column 5 of Table 6. The second is the regression explicitly including expected future income, as well as period and fixed effects, from Table 7. The difference between them is that the model including expected future income makes greater use of the information about the relationship between income and spending over time, but requires an assumption about how expectations are formed, which may be incorrect. Fortunately, both regressions give very similar estimated elasticities. The regression in first differences produces direct estimates of the elasticity of education spending with respect to current resources and the previous year's resources, while those elasticities can be easily calculated for the model that includes expectations of future income. The model including expected future income shows that the elasticity of education spending with respect to future resources is 0.893, and the elasticity with respect to current resources (holding expected future income constant) is not statistically significantly different from zero, with a point estimate of − 0.080. These estimates are economically reasonable. The former implies that states which are 20% richer, in the long run, spend about 18% more on schools. The latter implies no response to a temporary increase in resources, which is not surprising given the difficulty of adjusting education spending on a year-to-year basis. The models are also consistent in showing that neither present nor lagged aid affect spending at all. This implies that states and school districts reduce their own spending dollar for dollar in response to Federal aid. This is what we would expect them to do if they have a desired level of spending, and the income effects caused by the aid are too small to significantly affect that desired level of spending. In the model with expected future income, the elasticity of spending with respect to current resources can be found by differentiating Eq. (7) with respect to log(Income)it: eE;Yt ¼ b1 þ b8 4k

ð9Þ

The β1 term represents the direct effect of transitory resource changes to education spending, and the β8 ⁎ λ term represents the effect of an increase in current income on expected future income. The estimates from Table 7, column 6, give a value of − 0.080 for the direct effect, and 0.893 ⁎ 0.297=0.265 for the indirect effect, for a total elasticity of 0.185. This is quite close to the directly estimated elasticity from Table 6, column 5, which is 0.204.13 Both models predict a small increase in education spending in response to a rise in current resources — however, the expectations model implies that this rise is entirely due to the effect of rising current income on expectations of future income. The elasticity with respect to last year's resources in the expectations model is given by differentiating Eq. (7) with respect to log(Income)i(t1): eE;Y ðt1Þ ¼ b8 4k4ð1  kÞ

ð10Þ

The estimates from Table 7, column 6, give an elasticity of 0.893 ⁎ 0.297 ⁎ 0.703 = 0.186, which is quite close to the directly estimated value from Table 6 of 0.257.14 Thus, for both current and past resources, the two models imply very similar elasticities. 12 With β2 = β3 = 0, income is the only variable contributing to financial resources. In the discussion that follows, we will continue to refer to the elasticity of spending with respect to resources, to facilitate comparison of these results with earlier results. Another way to compare these results to the earlier results would be to re-estimate the model in Table 6, imposing the restriction that β2 = β3 = 0 there as well. We have done so; the results are omitted here to conserve space, but are available from the authors. We will indicate how those results compare to the ones from the structural model below. 13 Imposing the restriction β2 = β3 = 0 on Table 6, column 5, produces an income elasticity which, like that of Table 7, excludes aid from the model. The estimate income elasticity from that restricted model is 0.183, which is even closer to 0.185 than is the result from the model of Table 6, column 5 without β2 = β3 = 0. 14 Imposing the restriction β2 = β3 = 0 on Table 6, column 5, produces an elasticity of spending with respect to past income of 0.253, which is also close to the structural model’s estimate. The regressions from Table 6 omit lags of income longer than one period, which the structural model includes. Thus, in the Table 6 model, the effect of longer lags is partly captured by the one-year lag, increasing the estimated elasticity.

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The primary difference is that the expectations model implies that longer lags of income matter too, and should be included in the model in first differences. Their omission creates an omitted variable bias; however, because changes in log income are not very correlated over time, the implied omitted variable bias is small. 7. Implications and Conclusions In this paper, we have shown that lagged income has a statistically significant and economically substantial effect on state and local education spending. An economic model in which expectations of future income affect current spending, and states and school districts use past income to form expectations of future income, can explain this result. When lagged income is omitted from regressions of expenditures and income, fixed effects will capture much of its effect, although not perfectly because patterns of income do change slowly over time across states. When lagged income is included, current income has a much smaller effect on spending than when it is excluded, because spending does not respond as much to a transitory increase in income as it does to a permanent one. We anticipate that a similar effect would occur in analysis of any type of state or local government expenditure where spending levels are related from year to year, either because of long-term spending commitments or an expectation that expenditure, once begun, will continue. The inclusion of fixed effects in the regression alters the estimated effect of current income in a way similar to the inclusion of lagged income, because expectations of future income change only slowly over time. This explains the finding in the literature that including fixed effects in panel data estimation of income elasticities changes the estimates substantially. Time effects are also significant, even in models with lags and first differences, and linear time trends are not an acceptable method of correcting for their effects. Present and lagged values of Federal aid, both general and education-specific, are not significant in our preferred specifications. When long panels of data are available, so that there are both measurements of income over time and enough observations to use those measurements, it is desirable to use them. But what should be done when there is only cross-section data, or a panel too short to allow the use of fixed effects? We have shown that simple estimates that do not deal with the issue of income over time can be extremely sensitive to changes in the regression specification, which is undesirable for policy analysis. Our results suggest two procedures for including the effects of income over time when a long panel of data is not available. The first is to recognize that when only current income is included in a regression, it serves as a measure of both current income and of expectations of future income. Thus its parameter needs to be interpreted carefully. A transitory increase in income will produce a smaller change in education spending than the regression suggests, because a transitory increase in current income will not greatly change expectations of future income. In contrast, a permanent increase in income will produce a greater change in education spending than the regression suggests. In a short panel, including fixed effects will help control for the effects of expected future income (which will not change greatly over a shorter period of time) and so income elasticities estimated in models with fixed effects will be more similar to the effects of transitory increases in income. However, such elasticities cannot be interpreted as measuring the ceterus paribus effects of changes in current income. If a model must be estimated in cross-section because lagged data on some variables are not available, including lagged income can improve the estimates of the model substantially. The second procedure is to estimate models either in first differences or using growth rates of education spending and income rather than their levels, if the data permit doing so. This produces estimates that are consistent even when the levels of the variables are not stationary, whereas OLS estimates of levels are not consistent in that case. Economically, estimating by these methods eliminates all state-specific, time-invariant factors affecting the level of spending, and produces estimated elasticities which are much more robust to variation in the error specification than simple estimation in the levels of the variables. The resulting estimates are better indicators of the ceterus paribus effect of a change in income, holding other factors about the state constant. 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