The robust relationship between taxes and U.S. state income growth.

An alternative is to estimate a reduced form version of column 1, omitting the terms DLNK, DLNL, and DLNN. The last column of Table 3 reports the results of this exercise. As expected, the combined effect of taxes is estimated to be substantially larger. A one percentage point increase in tax burden is associated with a contemporaneous decrease of 2.59 percent in real PCPI growth. In addition, future five-year growth rates are estimated to be lower by 1.56 percent.

ROBUSTNESS CHECKS

Robustness with Respect to Alternative Specifications

One concern with the previous set of results is that the estimated tax effects may suffer from omitted variable bias. Thus, it is important to control for the influence of other variables that may affect state income growth. The subsequent analysis takes the specification in column 1 of Table 1 as its starting point, and appends this with theoretically appropriate control variables.

It is clear from equation [5] that any number of variables could be included as proxies for the unobserved term, [C.sub.t]. For example, Garcia-Mila and McGuire (1993) argue that a state's industrial's composition will matter if agglomeration economies or knowledge spillovers differ by industry. Ciccone and Hall (1996) hypothesize that the density of economic activity influences productivity due to externalities associated with physical proximity, among other reasons. Education and other demographic variables can proxy for productivity growth from human capital.

Reed (2008) identifies 32 variables that have been used or suggested by previous studies. Eliminating the public sector variables (such as categories of public spending or taxes)--since including these would change the nature of the tax variables--leaves 13 non--tax variables. These are identified in Table 2. Each of these can be argued to be included in differenced or level (initial value) form. If one also allows the initial value of income to be included as a regressor, (15) and recalls that the differenced form of the population variable (DLNN) is already included in the core specification, one obtains a total of 26 possible control variables. (16)

While it is likely that many of these variables do not really belong in the regression equation, it is not apparent a priori which ones should be excluded. Choosing one or a few sets of control variables is potentially a problem, since previous literature (e.g., Leamer, 1985; Levine and Renelt, 1992; Crain and Lee, 1999; Sala-i-Martin et al., 2004) has shown that estimated coefficients are often fragile, sensitive to the particular composition of conditioning variables.

The problem is complicated by the fact that there are [2.sup.26] [congruent to] 67 million ways to combine 26 variables, each one a possible regression specification. I address the issue of variable specification in the following way. First, I estimate a complete specification that includes all 26 variables. Next, I identify and estimate the "best" specifications as determined by two different model selection criteria, the Schwarz Information Criterion (SIC) and the corrected Akaike Information Criterion (AICc). (17) This produces three sets of regression results, each of which is reported in Table 3.

Of greatest interest are the first two rows of Table 3. These report the estimated coefficients of TaxBurden(D) and TaxBurden(L) after including alternative sets of control variables. Both tax coefficients are smaller in absolute value compared to column 1 of Table 1, where the estimated values are -1.37 and -0.90, respectively. Nevertheless, they remain negative across all the expanded specifications of Table 3. Further, they continue to be highly significant. In the "All Variables" specification, TaxBurden(D) and TaxBurden(L) have t-statistics (p-values) of, respectively, -2.58(0.011) and -2.87(0.004). The corresponding t-statistics are even higher in the "Best SIC" and "Best AICc" specifications. And while these latter two specifications are the product of sequential search, the t-statistics /p-values for the two tax variables can still be interpreted in the classical manner because the search procedure includes these two variables in every specification.

Turning to the other variables, I find that the estimated coefficients are generally consistent with the predictions of growth theory, or at least not inconsistent. Focusing on the coefficients from column 2 of Table 3--the "Best SIC Specification"--we observe the following results (ignoring the distinction between initial levels and contemporaneous changes): higher educational attainment, a greater percentage of the population who are of working age, a greater percentage of the population that is nonwhite, a larger population, a greater reliance on agriculture, and a more unionized workforce are associated with higher income growth. A larger female population, a larger mining sector, and greater industrial diversity are associated with lower income growth. Lastly, ceteris paribus, states with a greater initial value of real PCPI grow slower than other states.

In conclusion, I find that the significant, negative tax effects first reported in column I of Table I are robust to the inclusion of a wide variety of control variables. (18) The next section investigates the robustness of the relationship between tax burden and state income growth when alternative estimation procedures are employed.

Robustness with Respect to Alternative Estimation Procedures

The subsequent analysis employs the variable specification of column 2 of Table 3 for additional robustness checks. This OLS equation displays good properties. It has a high [R.sup.2], the key explanatory variables all have large t-statistics, the Durbin-Watson statistic is close to two, and a test of error normality fails to be rejected at the five percent level. (19)

However, there are at least two concerns. First, panel data are often characterized by complex error structures. Using the residuals from this specification, I tested for (1) first-order serial correlation, (2) groupwise heteroscedasticity, and (3) cross-sectional correlation. I found no evidence of significant serial correlation (the estimated value of the AR(1) parameter was -0.02). However, I reject the hypothesis of no groupwise heteroscedasticity (20) and find substantial evidence of cross-sectional correlation. (21) This raises worries about the inefficiency of the coefficient estimates and biasedness in the estimates of the standard errors. (22)

Unfortunately, while one can estimate an error variance-covariance matrix that allows for cross-sectional correlation, one cannot invert that matrix, since N = 48 > T = 6. This precludes the use of Parks-type, feasible FGLS. However, there are several alternatives. One approach is to continue to use OLS, but adjust the standard errors for cross-sectional correlation; either by using Beck and Katz's "panel--corrected standard error" procedure (Beck and Katz, 1995), or by using a more robust estimator of the error variance-covariance matrix. Another is to follow--up a suggestion by Greene (2003, cf 333f) and use FGLS, weighting on groupwise heteroscedasticity while adjusting the standard errors for cross-sectional correlation. Accordingly, I check for robustness of the estimated tax effects across the following alternative estimation procedures.

1. OLS with panel-corrected standard errors.

2. OLS with heteroscedasticity and cross-sectional correlation robust standard errors.

3. FGLS (weighted on groupwise heteroscedasticity) with heteroscedasticity robust standard errors.

4. FGLS (weighted on groupwise heteroscedasticity) with panel-corrected standard errors.

5. FGLS (weighted on groupwise heteroscedasticity) with heteroscedasticity and cross-sectional correlation robust standard errors.

There is an additional concern. The explanatory variables include both fixed effects and a lagged form of the dependent variable as explanatory variables. This generates correlation between the error term and the lagged form of the dependent variable, causing biased coefficient estimates (Nickell, 1981). To address this concern, I use two DPD estimators: the Arellano-Bond (difference) one-step and two--step procedures. (23)

Table 4 reports the estimates from these alternative estimation procedures. For comparison's sake, the first row duplicates the tax coefficient estimates from column (2), Table 3. There are two main findings from this analysis: Both FGLS and DPD confirm earlier results in that they produce negative coefficient estimates for each of the tax variables. The FGLS estimates are similar in size to the OLS estimates, while the DPD estimates are generally larger (in absolute value). In addition, the statistical significance of the tax effects is confirmed across all alternative estimation procedures. Of the 16 t-statistics reported in Table 4, 14 imply significance at the one percent level, with the remaining two significant at the five and ten percent levels. Accordingly, I conclude that my main findings of negative, statistically significant tax effects are robust across alternative estimation procedures.

Robustness across Alternative Cuts of the Data

The preceding analyses divide the 30 years of data from 1970-1999 into six periods of five-years each: 1970-1974, 19751979 ..., 1995-1999. This section looks at two alternative ways of dividing the data.