2. Data

The control variable of interest is the Stansel (Reference Stansel2013) economic freedom index. An index value is available for each of the 384 US metropolitan areas for the year 2002. The index is designed to provide a comprehensive measure of the extent to which private property is protected from expropriation and individuals are allowed to engage in voluntary transactions.

The index is constructed on a scale from 0 to 10, with 10 indicating most free. The comprehensive index is an average of the scores that metropolitan areas receive for three areas of economic freedom separately (each also on a scale of 0 to 10). Each area of economic freedom is itself scored based on a number of related components:

• • Area 1: Size of Government: (a) government consumption expenditures by government; (b) transfers and subsidies; (c) social security payments.

• • Area 2: Takings and Discriminatory Taxation: (a) total tax revenue; (b) individual income tax revenue; (c) indirect tax revenue; (d) sales taxes collected.

• • Area 3: Labor Market Freedom: (a) minimum wage annual income; (b) state and local government employment; (c) union density.Footnote ⁷

Most of the components listed above are in dollar terms and are taken as a percentage of metropolitan area personal incomes. Government employment is taken as a percentage of total employment, as are union employees to arrive at a measure of union density.

The 384 metropolitan areas include 355 metropolitan statistical areas (MSAs) and 29 metropolitan divisions (MDs). These MSAs and MDs are defined according to the 2009 Office of Budget and Management definitions. The 29 MDs are portions of 11 large MSAs that Stansel breaks down so that all of the metropolitan areas are of comparable sizes. Since we are pursuing a spatial econometric analysis, we limit ourselves to the 381 metropolitan areas of the contiguous US states (i.e., Alaska and Hawaii areas are excluded). As can be seen in Figure 2, there is considerable variation in the economic freedom scores across US metropolitan areas. Figure 2 also seems to suggest that levels of economic freedom are similar across neighboring metropolitan areas. This would be consistent with yardstick competition playing an important role in determining institutional quality and resulting in spatial correlation in economic freedom scores. Casual inspection suggests that the spatial correlation is not entirely accounted for by intra-state similarities. All else equal, then, economic freedom scores may be converging across metropolitan areas over time. However, as our data only contain metropolitan area freedom index values for one year, 2002, we cannot provide more systematic evidence for or against such convergence.Footnote ⁸ Even so, the idea of yardstick competition is important to keep in mind while interpreting the results.

Figure 2.

Economic freedom scores for US metropolitan areas.

Source: Stansel (2013) metropolitan area economic freedom index for the year 2002. Index is on a scale of 0–10 with 10 indicating most free.

Our dependent variable will be metropolitan area real (log) personal per capita income, or its average annual rate of growth. We use the US Bureau of Economic Analysis's (BEA’s) measure of personal income. The BEA's measure of personal income includes transfer payments. Since we are only interested in the income individuals receive from production, we net out the BEA's estimate of transfer payments for each metropolitan area.

Institutions and their effects on incomes are persistent. In lieu of a well-defined time frame for those effects, we consider income levels in 2005 and also in 2011 (or the rate of growth over 2002–2005 and also over 2002–2011). We will often refer to results in terms of those ‘over the 3-year horizon’ or ‘over the 9-year horizon’ and, when we do so, we are referring to those where the dependent variable is based on, respectively, the 2005 or 2011 income level. In growth rate regressions, we also include 2002 personal income per capita as an initial income level control (to account for convergence effects). Personal income is always converted into 2005 constant dollars.Footnote ⁹ As in the case of economic freedom scores, per capita income levels appear to be spatially correlated across metropolitan areas (Figure 3; 2011 income levels); as do also their growth rates (Figure 4; 2002–2011 income growth rates).

Figure 3.

Personal income per capita by metropolitan area in 2011.

Source: Bureau of Economic Analysis (BEA). Personal income per capita excludes transfer payments and is expressed in 2005 dollars.

Figure 4.

Personal income per capita average growth rate from 2002–2011 by metropolitan area

Source: Bureau of Economic Analysis (BEA). Personal income per capita excludes transfer payments and is expressed in 2005 dollars.

We also employed 25 control variables that are motivated by Higgins et al. (Reference Higgins, Levy and Young2006) and Young et al. (Reference Young, Higgins and Levy2010, Reference Young and Levy2013) county-level studies of income growth. These data are all from either the US Census Bureau or the BEA.Footnote ¹⁰ The control variables include (i) a number of industry employment shares; (ii) three different educational attainment population shares (high school diploma, some college, and bachelor's degree or higher); and (iii) a number of demographic population shares (e.g., age and racial demographics).Footnote ¹¹ We also control for a metropolitan area's land and water areas per capita. All estimations include state-level fixed effects to account for unobserved characteristics at that level.Footnote ¹² The data for (i), (ii), and (iii) are all based on either the year 2000 or the year 2002.

Definitions, years, and sources for all variables are reported in Table 1, along with summary statistics.

Table 1.

Variable definitions, sources, and summary statistics

3. Empirical methodology and model choice

Our dependent variable is either a metropolitan area's (log) real per capita income level or its average annual growth rate. We relate this to, among other controls, a metropolitan area's economic freedom score from Stansel (Reference Stansel2013). We employ Elhorst's (Reference Elhorst2010) testing procedure to determine whether there is spatial dependence in our data and, if there is, what is the appropriate spatial econometric model for our analysis.

In addition to (1) OLS, the Elhorst testing procedure evaluates four spatial models: (2) the spatial autoregressive model (SAR), (3) the spatial error model (SEM), (4) the spatial lag of X model (SLX), and (5) the spatial Durbin model (SDM). A general form that nests each of these alternative models is

(3.1)\begin{equation} y = \alpha + \rho Wy + X\beta + WX\theta + \varepsilon ;\end{equation}

(3.2)\begin{equation} \varepsilon = \lambda W\varepsilon + \upsilon ,\end{equation}

where y is an n × 1 vector of cross-sectional observations on the dependent variable; X is an n × m matrix of control variables weighted by the m × 1 vector of coefficients, β; and ɛ is an n × 1 vector of errors.Footnote ¹³ In equations (3.1) and (3.2), W is an n × n weight matrix that defines neighbor relations, ρ and λ are scalar parameters, and θ is a k × 1 vector of parameters (where k is the number of X variables) defining spatial effects across neighbors. In the simplest case, if ρ, θ, and λ are all zero then we are left with a standard OLS model. Alternatively, spatial dependence can exist in the dependent variable (y), the independent variables (X), or the error term (ɛ). Each of the specific models (1) through (5) can be arrived at through parameter restrictions on the general model (3.1) and (3.2):

• • OLS: ρ = θ = λ = 0;

• • SAR: θ = λ = 0;

• • SEM: ρ = θ = 0;

• • SLX: ρ = λ = 0;

• • SDM: λ = 0.

As LeSage and Pace (Reference LeSage and Pace2009) explain, given that the true data generating process is uncertain, in the presence of spatial dependence the SDM model, (5), is the most likely model to produce unbiased estimates since it is actually a generalization of models (1) through (4).Footnote ¹⁴ Consequently, SDM is likely to be the appropriate model unless testing provides statistically significant evidence to the contrary. The Elhorst (Reference Elhorst2010) testing procedure systematically searches for such evidence.

To test for spatial dependence we first need to specify the weight matrix, W. We define neighbor relations according to a ‘k-nearest-neighbors’ criterion. Since US metropolitan areas have on average six contiguous neighboring areas, a reasonable assumption is k = 6. We therefore define a metropolitan area's neighbors as the six other metropolitan areas that are closest to the center of its geographic mass.Footnote ¹⁵ Based on the six closest neighbors, entries in the W matrix are assigned 1s to indicate metropolitan area neighbor pairs. All of the other elements of W are assigned 0s. Following convention we ‘row normalize’ the entries such that the rows of the W matrix all sum to one.Footnote ¹⁶ While k = 6 is reasonable, it is also to some extent arbitrary. However, for spatial models that include a ρ parameter, as long as the estimates are interpreted properly they will not be sensitive to this particular choice of k (LeSage and Pace, Reference LeSage and Pace2014).

With W specified as described above, we begin the Elhorst (Reference Elhorst2010) testing procedure by conducting Lagrange multiplier (LM) tests to determine if there is any spatial dependence in our data. Specifically, we estimate the OLS model and test the hypothesis of no spatial dependence in the dependent variable (SAR) using an LM lag test; then we test the hypothesis of no spatial dependence in the error term (SEM) using the LM error test. The appeal of these tests is that they only require the residuals of the OLS estimation and W. A drawback of both of the LM tests is that the specific sources of spatial dependence can sometimes be hard to distinguish from one another (i.e., each of the standard LM tests has power over the alternative form of spatial dependence). We employ robust versions of these LM tests that take this into account.

Tables 2 and 3 report the results of these LM tests for per capita income levels (INCOME_05 and INCOME_11) and average growth rates (GROWTH_05 and GROWTH_11), respectively. The levels are for 2005 or 2011; the average growth rates are for 2002–2005 or 2002–2011. The results are uniform. The LM tests always reject the null hypothesis of no spatial dependence in the OLS model, suggesting that we have spatial dependence in the dependent variable, the error term, or both.

Table 2.

Model specification tests: per capita income levels

Notes: Table gives LM value for the LM lag and LM error tests, p value for likelihood ratio tests and WALD tests, and estimated coefficient for RHO in the SDM. *** indicates statistically significant at 1% level, ** indicates statistically significant at 5% level, * indicates statistically significant at 10% level.

Table 3.

Model specification tests: average per capita income growth rates

Notes: Table gives LM value for the LM lag and LM error tests, p value for Likelihood Ratio tests and WALD tests, and estimated coefficient for RHO in the SDM. *** indicates statistically significant at 1% level, ** indicates statistically significant at 5% level, * indicates statistically significant at 10% level.

Having rejected the null hypothesis of no spatial dependence for both income levels and growth rates, the next step of the procedure is to estimate the SDM model to determine if the spatial dependence is more general than either the SAR or the SEM models allow for. We use the estimate of the SDM model to conduct a likelihood ratio (LR) test to determine if SDM is more appropriate than either SAR or SEM. If SDM is found to be more appropriate than the SAR and SEM models, we then also need to test if the ρ parameter in the SDM model is statistically significant once we allow for spatial dependence in the explanatory variables. Specifically, we first test the following two hypotheses: H₀: θ = 0 and H₀: θ + ρβ = 0, where θ, β, and ρ are as defined in equation (3.1). The first hypothesis is that the SDM model can be collapsed into the SAR model, while the second hypothesis is that SDM model can be collapsed into the SEM model. If both hypotheses are rejected, then we accept that the SDM is more appropriate than both the SAR and SEM models. If H₀: θ = 0 is not rejected then we employ the SAR model as long as the robust LM tests point to SAR. Similarly, if H₀: θ + ρβ = 0 is not rejected then we employ the SEM model as long as the robust LM tests point to SEM. If either of these conditions is not satisfied then the SDM model is employed by default.

Furthermore, if we deem SDM more appropriate than either SAR or SEM, or both, we need to test if ρ is statistically different from zero in the SDM model as we are now allowing for spatial dependence in the explanatory variables. If we cannot reject the null hypothesis that ρ is statistically different from zero, we then need to re-estimate OLS including spatially lagged explanatory variables, with no spatially lagged dependent variable, i.e., the SLX model. This SLX estimate would be used to test the hypothesis H₀: θ = 0; i.e., if the spatially lagged explanatory variables are collectively statistically different from zero. If this hypothesis cannot be rejected, we still use the SDM model since we found the spatial dependence in the data to be more general than the SAR or SEM models allow.

The results from this testing procedure are reported in Tables 2 and 3 for, respectively, income levels and growth rates. In all cases, the hypotheses that SDM can be collapsed into either the SAR or SEM models area rejected. Furthermore, the estimate of ρ is statistically different from zero in all cases. Notice, that according to the Wald test, θ is not statistically different from zero in any case for the income levels (Table 2) and only for one case for the growth rates (Table 3). However, even with this finding, according to the testing procedure SDM is the most appropriate model for all specifications. This is because we have already found that the spatial dependence is more general than a simple SAR model would allow. SDM results are estimated using maximum likelihood. Maximum likelihood avoids the simultaneity bias that would arise in an OLS estimation that includes a spatially lagged dependent variable.