5.2.3. Inequality Regressions
Before running any regressions, one needs to think about the best method of estimation to chose, given the panel data at our disposal.
Generalized Method of Moments (GMM) as the Method of Estimation
Panel data lend themselves to dynamic models. The chosen method of estimation is a dynamic panel data model where the general approach relies on instrumental variables estimators and on a generalized method of moments estimator. Several reasons justify this choice. First, performing a Hausman test on the growth equation leads to reject the assumption that the regressors are uncorrelated with the error terms. In other words, ordinary least squares estimates cannot be consistent. An alternative method to least squares estimation is the instrumental variable technique. In other words, one needs to introduce instrumental variables to the dynamic panel data specification which is given by Equation (5.3)
(5.3) y it = α i + βx it + δy i,t -1 + ε it .
Second, the time-span 1913-2003 is long enough to use the differences (y i,t -2 – y i,t -3) as instrumental variables for (y i,t -1 – y i,t -2). The first differences transformation applied to the dynamic model corrects for the cross-section effects and produces an equation of the form 31 :
(5.4)y it – y i,t -1 = β (x it – x i,t -1) + δ (y i,t -1 – y i,t -2) + (ε it – ε i,t -1).
Third, the dynamic model adopted here corrects for time-period fixed effects with dummy variables and corrects for heteroscedasticity with a White diagonal instrument weighting matrix.
Inequality and Initial Income Level: Significant Positive Correlation
The table below shows a strong positive correlation between inter-state inequality (AI99-100i / ybar US) and the state average income at the beginning of each period.
| Dependent Variable: INTER99100 | |||||||||
| Sample | Variable | Coefficient | Prob. | ||||||
| 1913-1928 |
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| 1929-1939 |
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| 1944-1979 |
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| 1980-1989 |
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| 1990-1999 |
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| 1999-2003 |
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Coefficients do not vary much when AI99.99-100 is used instead. Do the results change after the addition of another regressor in the GMM regressions? Reflective of the industrial composition of income in each state, this variable added to the X i matrix is borrowed from Barro and Sala-i-Martin (1991, p. 117). It breaks down “state i’s personal income into nine standard sectors: agriculture; mining, construction; manufacturing; transportation; wholesale and retail trade; finance, insurance, real estate; services, and government.” The data relate to the personal income accruing to each sector in 1930, 1940, 1950, 1960, 1970, and 1980 at the national level. The two authors first calculated the national growth rate of each sector from one decade to the other, and then weighted the national growth rates by the share of each sector in state i. The variable they end up with is a single number per state summing up all sectoral (weighted) growth rates together. The authors also computed the agricultural share of personal income for 1920-1930, therefore included in the vector of the sectoral regressor here.
| Sample | Variable | Coefficient | Prob. | |||||||||
| 1920-1980 |
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With some disappointment, the sectoral variables added to the basic equation fail to improve the results. They also affect the number of observations dropping from 4,539 to 288. This is due to the decennial nature of the sectoral variable, measuring growth rates of a sector in a state income from one decade to the next (and not from one year to the next) during the time-period 1920-1988.
The correlation between the growth rate of the average income and inequality across states is the focus of the next section.
Growth and Inter-State Inequality: Significant Negative Correlation
In this section, the main variable is no longer the initial income average, but the initial income in the top decile of the distribution. To which extent does the initial income of the wealthiest tax filers affect the growth rate of the state average income? The main equation is written as
(5.5)(1/T) . log [y i,(t0+T)/y i,t0] = α + β log [ytop i,t0] + γX i+ εit,
as opposed to
(5.6)(1/T) . log [y i,(t0+T)/y i,t0] = α + β log [y i,t0] + γX i+ εit,
where the new items are:
ytop i,t0 is the initial per-tax-unit income in the top decile,
log [ytop i,t0] is a one-percent increase in initial income in the top decile. 32
The variables of the table below show the correlation coefficients (and their corresponding probability of being insignificant) between G, the annual growth, and the inter-state inequality indicator, labeled INTER99100, named after the 51 ratios dividing each year the top 1 percent fractile on the national average income.
What appears from the table above is unequivocal: whenever the inequality coefficient is negative, it is statistically significant. In other words, a decrease in the growth rate occurs along with a wider inequality gap, and vice-versa, but the two variables hardly move in the same direction at the same time. These results need to be taken with caution, as correlation does not mean causality. Granger tests of causality show that growth does not Granger cause inequality, and inequality does not Granger cause growth.
With the dependent variable being the annual growth rate of personal income in state i, the results of the GMM regressions including the industrial composition of income are summarized in the table below.
| Variable | Coefficient | t-Statistic | Prob. |
| G(-1) | -0.5204 | -1.2333 | 0.2185 |
| INTER99100 | -0.9112 | -1.4141 | 0.1584 |
| SECTOR | 36.5199 | 1.1143 | 0.2661 |
| 1920-1930 | 11.4669 | 1.6569 | 0.0987 |
| 1930-1940 | 34.9368 | 5.4154 | 0.0000 |
| 1940-1950 | -15.0182 | -8.3573 | 0.0000 |
| 1950-1960 | -8.7370 | -1.9994 | 0.0465 |
| 1960-1970 | -0.3046 | -0.6538 | 0.5138 |
| 1970-1980 | -5.4314 | -1.6213 | 0.1061 |
The R² jumps from 0.59 to 0.77, and the inter-state inequality indicator keeps its negative sign but loses its statistical significance. Note again that the sample size shrank drastically from 4,539 to 288, as the sectoral variable is available for 6 years and 48 states only.