Conceptual and econometric model

The causal link between per capita county public health expenditures and county-level self-rated health is placed in the context of an aggregate demand for health model. We define an aggregate demand for health model is an aggregate version of the individual demand for health model where health status is a function of the price of medical care, income, age, education and variables affecting preferences for health, such as race/ethnicity (Zweifel and Breyer, Reference Zweifel and Breyer1997). As decision making regarding county department of public health expenditures occurs at the county level, the county is used as the unit of analysis.

As noted above, the main hypothesis is that the key variable of interest, per capita county public health expenditures, will positively affect the health status of the population, both contemporaneously and in the future. While it is possible to reformulate the hypothesis to whether per capita county public health expenditures will positively affect the health status of the average individual in the population (as opposed to the health status of the overall population) contemporaneously and in the future, the necessary econometric model, which uses a lagged dependent variable, would require individual-level panel data. Individual-level panel data are not available, therefore the hypothesis is formulated at the population level and available county-level panel data are used.

A standard Koyck distributed lag model is estimated, which assumes that the effect of county public health expenditures is cumulative (Wooldridge, Reference Wooldridge2003). The cumulative impact is the sum of a contemporaneous impact, and any subsequent future impacts, which are assumed to decline geometrically over time as it may take longer periods of time for some programs to create measureable impacts on individuals with higher levels of health status and the impacts of such programs on individuals with higher levels of health status are likely to be smaller (Bopp et al., Reference Bopp, Braun, Gutzmiller and Faeh2012). In other words, the cumulative impact of public health expenditures becomes larger over time, but at a declining rate.

The Koyck model can be described as follows:

$${\rm y}_{t} ={\rm \alpha }{\plus}\beta _{0} x_{t} {\plus}\beta _{1} \lambda x_{{t{\minus}1}} {\plus}\beta _{2} \lambda ^{2} x_{{t{\minus}2}} {\plus}\cdots{\plus}u_{t} ,$$

where y is the percentage of individuals who rate their health as good, very good or excellent on a five-level health status item (poor, fair, good, very good, excellent), x is county public health expenditures per capita, and λ, where 0<λ<1, is the estimated rate of change of the distributed lag. The number of lags included in the model is theoretically infinite, but practically, the number of relevant lags depends on the size of λ. Larger values of λ yield lag structures where relatively large coefficients are present further and further back in time (in other words, the impact of public health expenditures is substantial for a longer period of time in the future). Additional parameters to be estimated are represented by α and β. Finally, u _t is the error term. To simplify, we lag equation (1) by one period, multiply it by λ, subtract this lagged equation from equation (1), and then rearrange:

$$y_{t} =\alpha _{0} {\plus}\beta _{0} x_{t} {\plus}{\rm \lambda }y_{{t{\minus}1}} {\plus}v_{t} ,$$

where v _t=(u _t−λu _t−1) and α ₀=α(1−λ). Other public-health researchers have also taken a dynamic approach in recent research (Macinko et al., Reference Macinko, de Oliveira, Turci, Guanais, Bonolo and Lima-Costa2011). The cumulative or long-run impact can be determined by substituting long-run values of both county-level general health status and per capita county public health expenditures into the above equation. We define y* as the long-run value of y and x* as the long-run value of x. Substituting these values into equation (2) yields the following equation:

$$y^{\asterisk}=\alpha _{0} {\plus}\lambda y^{\asterisk}{\plus}\beta x^{\asterisk}.$$

We then solve for y*:

$$y^{\asterisk}=\alpha _{0} /(1{\minus}\lambda ){\plus}[(\beta )/(1{\minus}\lambda )]x^{\asterisk}.$$

To determine the long-run cumulative impact of increasing (or decreasing) per capita county public health expenditures, we take the first derivative with respect to x*:

$${{\partial y^{\asterisk}} \over {\partial x^{\asterisk}}}=\beta /(1{\minus}\lambda ).$$

Many readers will recognize this as the standard result for the sum of an infinite geometric series where 0<λ<1. This can also be referred to as the long-run propensity (LRP; Wooldridge, Reference Wooldridge2003). The short-run propensity (SRP), or initial impact, can be defined as β alone.

In order to be empirically estimated, equation (2) must be adjusted in order to include standard factors relevant to the aggregate demand for health. As noted above, these include the price of medical care, income, age, education and factors that influence population preferences for health, such as race/ethnicity. Prices of medical care can be proxied using the proportions of the population covered by health insurance including Medicaid and related programs, Medicare, and private health insurance. Income is measured as per capita income. Age is measured by the age structure of the population (proportion of the population in the following ranges: 0–14, 15–24, 25–34, 35–44, 45–54, 55–64, 65–74, 75–84, 85 and older). Education is measured by the extent to which the population has a college education, which is measured by the proportion of the population who have a bachelor’s degree or higher. The racial/ethnic structure of each county’s population is measured by the proportion of individuals in each racial/ethnic group (White, Black, Hispanic, Asian/Pacific Islander, Other).

None of the variables used in the analysis are age-adjusted. We follow the approach of Rosenbaum and Rubin (Reference Rosenbaum and Rubin1984), who note that when using an age-adjusted population rate as a dependent variable in regression analysis, all independent variables must also be age-adjusted in order to avoid biased parameter estimates. When age-adjusted variables are not available for every variable in the model, they show that using crude population rates for both the dependent and independent variables, while adjusting for age by including variables indicating the proportion of the population in each age category, will yield consistent estimates.

We include per capita public health expenditures to complete the description of the institutional context. County health expenditures in California are formally categorized into four reporting categories (public health, medical care, drug and alcohol abuse and mental health). Note that these categories all represent health expenditures that flow through each county government. Because each of these is endogenous in our model (at least subject to omitted variable bias and measurement error), we only include public health expenditures following Angrist and Pischke (Reference Angrist and Pischke2009) who recommend omitting uncorrected endogenous variables in instrumental variables equations. All control variables (health insurance, income, age, education, race/ethnicity) in the model are lagged in order to render all control variables strictly exogenous or weakly exogenous (predetermined).

Two variables in the model are endogenous to the percentage of the population reporting good, very good or excellent health: the lagged dependent variable and per capita county public health expenditures. The main aspects of endogeneity with regard to per capita county public health expenditures are omitted variable bias and measurement error. For example, the equation does not contain information on varying health policy ordinances, varying environmental conditions, varying population health behaviors, or other factors that may correlate with both self-rated health and county public health expenditures. See the ‘Discussion’ section below regarding per capita county public health expenditures being measured with error. These two endogenous variables, the lagged dependent variable and per capita county public health expenditures, can be handled in at least three ways. One is through standard instrumental variables. However, standard instrumental variables are often unavailable or too weak. In these cases, at least two alternative approaches are available. One is the generalized method of moments (GMM) system estimator, which is designed for panels with a small number of time periods and a large number of observations within each time period where it is not assumed that adequate instruments are available outside of the core data set (Roodman, Reference Roodman2009). However, if the available valid instruments (lagged levels and lagged differences) used by the GMM system estimator (Blundell and Bond, Reference Blundell and Bond1998) are weak, the result can be significant parameter bias, similar to that found in traditional instrumental variable models with weak instruments (Bun and Windmeijer, Reference Bun and Windmeijer2010). Only ad-hoc methods of testing for weak instruments are currently available when using the GMM system estimator (Bazzi and Clemens, Reference Bazzi and Clemens2013).

An alternative approach that allows traditional weak instrument testing and also does not require outside instruments is that of Lewbel (Reference Lewbel2012). Lewbel’s approach was disseminated through working papers before publication resulting in a large number of studies being published that applied Lewbel’s methodology both before and after the formal publication of Lewbel’s methodology (Stifel and Alderman, Reference Stifel and Alderman2006; Block, Reference Block2007; Sabia, Reference Sabia2007a, Reference Sabia2007b, Reference Sabia2007c; Haung et al., Reference Huang, Lin and Yeh2009; Kelly and Markowitz, Reference Kelly and Markowitz2009; Drichoutis et al., Reference Drichoutis, Nayga and Lazaridis2012; Emran and Shilpi, Reference Emran and Shilpi2012; Kelly et al., Reference Kelly, Dave, Sindelar and Gallo2012; Lewbel, Reference Lewbel2012; Denny and Oppedisano, Reference Denny and Oppedisano2013; Huang and Xie, Reference Huang and Xie2013; Schroeter et al., Reference Schroeter, Anders and Carlson2013).

Lewbel (Reference Lewbel2012) uses any heteroscedasticity present in the data to generate sets of valid instruments. Lewbel’s approach does not identify the endogenous variables in the second-stage equation based on traditional exclusion restrictions. Rather, he achieves identification using higher moments. Sufficient conditions for Lewbel’s method to be applied are as follows: $$Cov(Z,{\varepsilon}_{1} {\varepsilon}_{2} )=0$$ , $$E(X{\varepsilon}_{1} )=0$$ , and $$E(X{\varepsilon}_{2} )=0$$ , where Z is a subset of X, the vector of exogenous variables in the second stage (and Z may include all of X) along with some heteroscedasticity in $${\varepsilon}_{1} $$ and $${\varepsilon}_{2} $$ , the errors from both the first and second stages of a standard IV regression. As noted by Lewbel (Reference Lewbel2012), a standard overidentication test can be used to evaluate the validity of these assumptions.

Potentially valid instruments are constructed by multiplying the heteroscedastic residuals from the first-stage regressions with a subset (or all) of the mean-centered exogenous variables $$(Z{\minus}\bar{Z})\hat{{\varepsilon}}_{1} $$ , where $$\hat{{\varepsilon}}_{1} $$ is the vector heteroscedastic residuals from the first-stage regressions and $$\bar{Z}$$ is a vector of the means of Z.

These generated instruments may be weak or strong, depending on the degree of scale-related heteroscedasticity in the data. Heteroscedasticity can be detected using a standard Breusch–Pagan type test (Lewbel, Reference Lewbel2012). The strength of the generated instruments can be evaluated using standard weak instrument tests (Stock and Yogo, Reference Stock and Yogo2005). As this approach may produce a mix of strong and weak instruments, it is valuable to remove weak instruments from the set of instruments used. The approach taken here is to remove instruments as needed if the absolute value of the instrument’s t-statistic is <1.96 in both of the first-stage regressions.

A final point is that when using panel data, two independent panels can be spuriously associated if they both have unit roots (Granger and Newbold, Reference Granger and Newbold1974; Wooldridge, Reference Wooldridge2003). We therefore test for the existence of unit roots using a test that assumes that the number of units, N, can be constant, as counties are unlikely to be added to California. In particular, we perform Phillips–Perron (Reference Phillips and Perron1988) tests, which use Newey-West (Reference Newey and West1987) standard errors to account for any serial correlation. The null hypothesis is that the data follow a random walk with or without drift. We use the z-statistic as our test statistic, which is recommended by Choi (Reference Choi2001) as offering the best trade-off between size and power. Following the recommendation of Levin et al. (Reference Levin, Lin and Chu.2002), all series are first demeaned by subtracting the cross-sectional means to mitigate the influence of cross-sectional dependence.

All analyses are performed using Stata 11 and ivreg2 using two-step feasible GMM estimation and ordinary least squares (OLS) for comparison purposes (Baum et al., Reference Baum, Schaffer and Stillman2010; Lewbel, Reference Lewbel2012). After generating IVs from non-clustered, heteroscedastic regressions, we cluster our data by county to correct for any downward bias in the standard errors from the presence of any serial correlation following the suggestion of Angrist and Pischke (Reference Angrist and Pischke2009). Petersen’s (Reference Petersen2009) work on panel data also suggests clustering by county to avoid biasing the standard errors when seeking to take into account both the correlation of observations within clusters and the correlation of observations within time periods, with the latter being taken into account parametrically (by including time dummies). The cumulative impact of county-level per capita public health expenditures is then estimated by evaluating the overall non-linear combination of [ $$(\beta _{0} )/(1{\minus}\lambda )$$ ] from equation (5).

Data

Self-rated health status at the county-level comes from the California Health Interview Survey (CHIS), which is fielded biennially. Health status is measured from high to low in five categories as excellent, very good, good, fair and poor. Categories are collapsed in accordance with the common practice of dichotomising the variable into two categories: poor/fair and good/very-good/excellent. Thus, the county-level measure is the percentage of individuals in the county with good, very good or excellent health status.

Information on the per cent of the population with Medicaid or other public insurance, the per cent of the population with Medicare, per cent of population with private insurance and the per cent of the population with a bachelor’s degree or higher also come from the CHIS. The CHIS is designed to yield statistically valid county-level measures for 41 of California’s 58 counties. The survey incorporates large statistically valid samples in each county from all age groups including children, teens and adults. However, subsetting self-rated health status by race/ethnicity was not feasible as sample sizes then became small and statistically unstable in many counties. For this reason subgroup analysis by race/ethnicity was not performed.

Data were obtained on county-level expenditures for public health from the 2001, 2003, 2005, 2007 and 2009 versions of the Counties Annual Report issued by the California State Controller’s Office. The Counties Annual Report details county expenditures, including public health expenditures (California State Controller’s Office, 2011). The report separates county health expenditures into four categories: public health, medical care, mental health and drug and alcohol abuse. It is not possible to disaggregate county health expenditures further using this report. The determination of how services are categorized across these four classes is a county decision and not all counties categorize services in the same way. Thus, county public health expenditures are subject to some degree of measurement error.

San Francisco, the only county in California that is also a city, is included in the Counties Annual Report, but is accorded a separate appendix due to its combined city-county functions. Because of its unique organizational structure, its expenditures are difficult to compare with the expenditures of other counties. It was therefore omitted from the analysis. The remaining 40 out of 58 counties represent ∼96% of the population of California.

The proportions of the population in nine age groups (0–14, 15–24, 25–34, 35–44, 45–54, 55–64, 65–74, 75–84, 85 and older) and in five racial/ethnic categories (White, Black, Hispanic, Asian/Pacific Islander and other race) come from the US Census (RAND, 2012). Per capita income comes from the US Bureau of Economic Analysis. All dollar figures are adjusted for inflation and reported in constant 2009 dollars. The analysis includes a total of 199 observations, 40 county-level observations for each year (with the exception of one county for which some data elements were unavailable for one year). The years included are 2001, 2003, 2005, 2007 and 2009.