The purpose of the CDR cross-country model ( Ridley, 2020 and Appendix B below) is to understand gross domestic product from levels of capitalism (C), degrees of democracy (D) and degrees of rule of law (R). These are policy variables that government can pursue, and citizens can understand to develop an income maximizing mindset. Together with natural variables natural resources and geographic latitude, the model explains 90% of per capita real gross domestic product (GDP) adjusted for purchasing power parity (GDPppp).

In this paper we will explore the concept in terms of a cross-state CDR model. The CDR model will be used to explain per capita real gross state product (GSP) adjusted for purchasing power parity (GSPppp), in terms of state capital (C), degree of state democracy (D) and degree of state rule of law (R). In the cross-country model, C is total market capitalization. It measures the degree of organization of capital and is the total value of all outstanding stocks on the stock markets. It measures the amount of capital available for investment. It also measures the sum of all future value of stocks, discounted to their present value. In the state model, C is not derived from a stock market, so it is not a measurement of capitalism and is not a policy variable. It is not the same value proposition. However, like in the cross-country model, rule of law attracts capital and protects democracy, which deploys capital optimally to maximize GSPppp. That is, maximize standard of living.

There is no such measure as total market capitalization for US states. Therefore, a proxy variable must be substituted. The proxy must relate to human capital imagination and creativity. The combination of imagination, creativity and innovation is the source of wealth. In this paper, we choose the product of average state scholastic aptitude test (SAT) score and population. So, SAT is the human capital potential, and population is the vessel that contains the volume of that potential. The state with the most capital, has the most people with the highest aptitude. In low CDR states, said aptitude is not promoted and may even be suppressed. In high CDR states, exogenous aptitude is extracted in schools and universities and converted into a measure of academic achievement such as grade point average (GPA), and endogenous stock of knowledge, high skill, machines, programming, recording, teaching material, etc. Said capital stock is deployed optimally to produce goods and services and maximize standard of living. The traditional economic model might choose years of education as the relevant input. However, the quality of education varies from state to state whereas SAT is standardized. Also, in the CDR model, the capital that is of interest is exogenous aptitude and potential to create knowledge that becomes the subject of endogenous stock of knowledge education.

Prior to the industrial revolution, apart from a few sporadic increases in living standards during the Roman Empire and in China during the Song dynasty, economic growth was negligible. Some 19^th century occurrences that set some countries on a path of sustained growth were explained by unified growth theory ( Galor, 2011 ). Pre 1850, increases in population size followed increases in living standards. However, Malthus (1798) attributed that to the standard of living which then fell, after the population increase. But, after 1850 the standard of living rose in England, Western Europe and the USA with similar population growth. The 20^th century brought growth in the standard of living that was faster than population growth. This demographic transition contradicted Malthus. There are any number of factors that may have led to the industrial revolution (see Senna, 2013 ). Whatever the reasons are, the standard of living in the industrial world has risen continuously ever since. It is inescapable that England’s invention of Magna Carta and all the countries that adopted it, have better economies than those that have not ( Ridley & Nelson, 2022a ). This is consistent with the idea of the importance of institutions such as rule of law and democracy ( North, 1991 ; North & Weingart, 1989 ). Rupasingha, Goetz & Freshwater (2002) also said that low income developing countries have the potential to grow faster than high income developed countries. But these principles are conditional on good governance and good institutions. They also claim that ethnic diversity is associated with faster rates of economic growth.

Solow (1956) advised that economies tend to converge to a balanced growth path. As the economy moves farther away from the balanced growth path, the marginal return on capital increases. The poorer the economy is, the faster it grows as it moves back towards the growth path, then converges more slowly to a limiting value as capital is accumulated and the marginal return on capital falls. This can take decades. This explanation is appealing, but it is only apparent for countries that were developed and experienced setbacks such as war. For example, Germany after WWII. Other countries seem to have stubborn zero or negative growth. Is their growth path zero or negative? Solow does not explain how developed countries got to be rich. For the reader who is curious about a log linear model Solow approach, the result of fitting a cross-country GDPppp = ${\text{β}}_0{C}^{{\text{β}}_C}{D}^{{\text{β}}_D}{R}^{{\text{β}}_R}\text{є}$, where $\text{є}$ is random error, and the 𝛽′𝑠 are output elasticities, is the very low value of $R_{adj}^2$ = 0.36. Exogenous catalysts D and R augment C but are not complementary. On the other hand, this paper proposes GDPppp = ${\text{β}}_{C}C+{\text{β}}_{D}D+{\text{β}}_{R}R+{\text{β}}_{CDR}C\cdot D\cdot R+\text{ε},$ where $\text{ε}$ is random error, includes an interaction term, and yields very high correlations, irrespective of government spending, natural resources, country size, location, culture, and other commonly held beliefs.

Jones (1995a, 1995b) observed that the share of GDP going to and the share of workers doing research and development increased during the 20^th century. Still the long-run growth rate remains the same. It appears that what determines growth rate is the speed of technological progress. And technological progress is correlated with population. Colacito et al. (2019) provides empirical evidence that temperature has significant effects on economic growth in the USA, both at the aggregate level and across a wide cross section of economic sectors.

Yu (2010) , Hall, Lacombe & Shaughnessy (2019) and Akai & Sakata (2002) discuss economic growth variations across states within the USA. Yu (2010) obtained high $R^2$ values, but their data are longitudinal and contain lagged variables. Income in any one year will always be similar to that in the previous year. Hall, Lacombe & Shaughnessy (2019) found an effect on GSP from economic freedoms. It appears reasonable that economic freedoms are effective in the presence of USA rule of law, but not in the absence of rule of law in many other countries elsewhere. Gwartney, Holcombe and Lawson (2006) used the EFW in a cross-country model to obtain $R_{adj}^2$ = 52.5%, considerably lower than the 90% obtained from the CDR cross-country model. Whetting the economic freedoms appetite of the population in corrupt countries only causes them to flee to Western Europe and the USA. Akai & Sakata (2002) found that fiscal decentralization contributes to economic growth in the USA. The opposite result was found for developing countries. This appears reasonable since decentralization is a positive activity of capitalism and opposite to the centralization activity of communism. Still the $R^2$ = 40% result is low compared with the CDR model. The negative result for developing countries may be due to the absence of rule of law where decentralization exacerbates corruption and theft.

Mankiw, Romer & Weil (1992) gives an augmented Solow model that includes accumulation of human as well as physical capital that better describes cross-country data. But the elasticities would all have to be equal for it to be a legitimate aggregate model. Furthermore, the log linear Solow type model explains only about 36% of GDPppp. Islam (1995) added a panel analysis to growth modeling in which the independent variables were lagged dependent variables. Income from year to year is almost the same everywhere so such a model will have an artificially high $R^2$. Ridley (2020) also estimated a CDR panel regression but there were no lagged variables. The model coefficients remained practically unchanged and $R^2$ = 90% persisted, reinforcing the global time invariance of the CDR model. Caselli, Esquivel & Lefort (1996) apply a generalized method of moments estimator to eliminate problems of correlated individual effects and endogenous explanatory variables in cross-country models. They did not report the impact on model $R^2$ to compare. Still, they suggested future research that applies new and better methods to cross-regional data. This paper applies the CDR model to US cross-state data. Sala-i-Martin, Doppelhofer & Miller (2004) introduced Bayesian averaging of classical estimates (BACE), to determine the “importance” of variables in cross country growth regressions. They also did not report the impact on model $R^2$ to facilitate comparison.

There is no history of cross-state US CDR models that can be reviewed. The Solow (1956, 1957) aggregate (national) growth model was proved mathematically by Ridley & Ngnepieba (2018) to be a fallacy of composition ( Cohen & Harcourt, 2003 ) to think that we can simply jump from Cobb & Douglas (1928) microeconomic production function conceptions to an understanding of aggregate production by society as a whole. Cobb-Douglas applies to a single machine. The only way that the Solow model could work as an aggregate model is if the elasticities were equal for every Cobb-Douglas machine function in the national sum. Furthermore, the Solow model explains only 36% of GDP. Acemoglu, Johnson & Robinson (2005) , Acemoglu & Robinson (2012) and Acemoglu, Naidu, Restrepro & Robinson (2014) expounded on the importance of institutions and both cross-country and cross-state CDR models are comprised of variables that represent institutions.

$H_0$: The cross-state US CDR model is not a good predictor of GSPppp.

$H_1$: The cross-state US CDR model is a good predictor of GSPppp.

The CDR model is a statistical model in which GSP is regressed on C, D, R and L. Ordinarily, the regression coefficients in a g = f(C,D,R,L) economic model would simply be weights and g would be a weighted average of C, D, R and L. However, the weights would be impossible to interpret in any meaningful way. The model would not provide a reference from which G can be estimated outside of the data sample. That is, a model that is estimated in any one year would not apply to subsequent years. This problem is overcome by creating a constant global index from the combination of the model parameters and weights C, D, R and L. That is, a CDR index. To do this, we standardize the variables such that they are always on or between 0 and 1, as defined below, and the model parameters become scale factors. The standardized variables are obtained from the following reversible transformations.

The terms capital, democracy and rule of law are often confused. In this research we are interested in capital, democracy and rule of law based on epistemological, metaphysical, and axiological insights ( Randrup, Druckemiller, & Briggs, 2016 ), so for clarity of purpose we begin with the following definitions. The data are almost never available for the same year, but they change so slowly that selecting from different years, as available, is satisfactory. The data for this study are given in Table A1 and Table A2 in Appendix A.

The results of the regression analysis are given in Table 1 . The cross-state CDR model is consistent with the cross-country model. In both models, the coefficients for C, D and R are all positive and the coefficient for C∙D∙R is negative. Natural resources data are not available, so no comparison is possible. The coefficient for Latitude is positive. Longitude was not statistically significant and was omitted from the model.

The cross-state model $R_{}^2$ = 0.8 and $R_{adj}^2$ = 0.778, or approximately 0.8. The $R_{adj}^2$ for the cross-country model is 0.9. The lower $R_{adj}^2$ for the cross-state model could be for any number of reasons. We assume that the primary reasons are the absence of natural resources, and the approximation of total capitalization. The coefficients of R, $\text{C}\cdot \text{D}\cdot \text{R}$ and L are significant at the 5% level. This typically is what economists consider statistically significant. The coefficients of C and D are not statistically highly significant but are important for general understanding of the model. The model retains the general characteristics of the CDR model and for the first time, places the cross-state macroeconomics on a sound scientific footing.

In the cross-country model capitalism contributes 60% to GDPppp. Capital included both exogenous and endogenous capital. In the cross-state model capital contributes 1.9%. We are unable to capture the effect of organizing this capital into capitalism. It is exogenous and does not include endogenous capital stock of raw materials, machines, knowledge, recordings, computers, etc. The SAT measures academic participation of children, not the wisdom of adults. This accounts for the lower percentage of contribution. Nevertheless, human capital of imagination and creativity is the source of wealth and is critical to economic development. The product of SAT and population appears to capture this capital. As an aside, state budget and state spending were tried as the surrogates for capital and $R_{adj}^2$ was approximately 0.4 and 0.5 respectively, much lower than 0.8 for SATxPopulation. The greatest single (non-interactive) partial contribution to GSP is democracy (20%). Rule of law contributes 2.4%. It is important to understand that rule of law protects democracy and attracts capital. So, the functionality of rule of law is paramount. We now know from Ridley and Nelson (2022a) that while cooperation is an obstacle to rule of law, collaboration is required to permit rule of law. So, the point of beginning is collaboration. The absence of rule of law, especially the presence of corruption, repels capital and minimizes economic growth. Geographic latitude contributes 2.24%. Geography is obviously important. It can be absolutely restrictive to certain activities and crops. And it might have contributed more in former times. But with the advent of modern communications and transportation, obstacles related to geography are easily overcome simply by division of production, and trade, not unlike that suggested by Smith (1776) . All of these intangible variables are responsible for the production of gross state product of tangible goods and services.

The regression model tells us how much variation in GSP is explained by the variations in C, D, R and L. Although all variations in these variables are important to characterize the cross-state model, we are unable to determine the variation in total capitalization (since those data are not available). But there is a sizable effect of variation in democracy across states. This could be due to variation in collaboration skills in different communities. Ridley & Nelson (2022b) explain how a negative epigenetic transgenerational psycho sequela can be inherited by members of some US subpopulations as a result of historical environmental stresses such as forced labor, excessive discrimination and exposure to dangerous chemicals. Even when the stresses are removed, formerly oppressed communities can continue to suffer the same genetic maladies. Collaboration skills may have been lost. The ability to participate fully in democracy may have been affected similarly. ( Ridley, 2022, 2023 ) show how collaboration trumps IQ for the prediction of GDPppp. Rosier, Llaugel & Ridley (2024) explain how US corporations use job design to train employees to collaborate, thereby greatly increasing corporate profit. Ridley, Lee & Nelson (2023) and Lee & Ridley (2024) explain how Singapore uses mandatory school sports and music education to raise national collaboration skill, rule of law and ultimately GDPppp to a level 50% higher than the US. The US should consider a similar strategy as part of reparations for its formerly oppressed communities. See Ridley & Korovyakovskaya (2025) for a full discussion.

As part of the exploration effort to find a model that fits the data well, a logarithmic model was considered. Logarithms were taken of all variables and the model re-estimated (see Appendix D). The logarithmic transformation yields elasticities for parameters so comparisons can be made with some traditional economic models. However, the ability to explain the variation in GSPppp and ultimately the predictive ability of the logarithmic model is significantly reduced. The logarithmic model $R_{adj}^2$ = 0.44, very much lower than (approximately 0.8) that for the proposed CDR model. Therefore, logarithms were not considered further. Rupasingha, Goetz & Freshwater (2002) investigated a variety of social and institutional factors as determinants of economic growth. The highest $R_{adj}^2$ obtained there was 27.7%. The CDR model explains 80%.

In passing we mention a phenomenon that is never considered in explaining the political choices that Americans make. One of the obstacles to communications along potential pathways is legitimate logical and binary thinking, and group thinking. Music can connect that which people have in common. Music and sports can facilitate political and other problem solving by cutting through polarization and binary logic. People who might otherwise not think to associate might come together through music and sports. Consider also, the discovery by Hibbing, Smith and Alford (2014) that there are genes that make people predisposed to liberalism and other genes that make people predisposed to conservatism. In the advanced democracy of the USA, it is quite astonishing to find that these characteristics appear nearly equally bifurcated in the population. Due to the remarkably peaceful political party and presidential transitions, it is reasonable to assume that democracy is responsible for the meaningful deployment of liberal and conservative traits. And that both traits are required for the economic growth that we observe. In contrast, the absence of democracy could permit one of these two traits to dominate the economy. Such an outcome could be a reduction in diversity and the lack of economic growth that persists in poor countries.

Also varying substantially across states is the effect of geographic latitude. As latitude varies so does climate and altitude. The US terrain varies substantially from the lowlands of Florida to the highlands of Alaska. However, latitude is a natural variable that is fixed outside the control of government policy. An alternative to latitude might well be temperature. Colacito, Hoffmann, & Phan (2019) provides empirical evidence that temperature affects economic growth negatively in the United States. Since temperature and latitude are negatively correlated, as expected, this current paper provides evidence that latitude has a positive effect on GSP. In the interest of exploration for the best variable, latitude was replaced by temperature in the model (see Appendix D). The result was $R^2$ = 0.78. This is the same as that for the CDR model that uses latitude. However, one major difference between latitude and temperature is that whereas latitude is a fixed time invariant variable of geography, temperature could vary albeit over centuries in time. Latitude was chosen for compatibility and comparison with the published cross-country CDR model.

Another variable that has been reported in the published literature is ethnolinguistic fractionalization. This variable was obtained from Lu (2024) and included in the model. However, it had no impact on $R_{adj}^2$ (see Appendix D). The t statistic was near zero. This implies that race and ethnicity are irrelevant to GSP. Ethnolinguistic fractionalization is also a natural variable that is fixed outside the control of government policy. Although some ethnic communities in the US have suffered historical deprivation of education and earn less than is fair from a human justice point of view, the good fit of the CDR cross-state economic model to the data, suggests that they earn what is equatable for their average level of education. That is, they are economically placed in jobs that they are currently suited for. The economic model does not know who is who, only their contribution to GSP. Human justice will be fulfilled when they acquire education that is comparable to their fellow citizens of equal aptitude. Olson (1982) warns against artificial attempts at attaining equity when equality of outcomes is best.

The fitted OLS model is

$\begin{array}{l}\hat{\text{g}}=0.5586+0.1305\text{C}+0.0489\text{D}+0.1174\text{R}-0.6732\text{C}\cdot \text{D}\cdot \text{R}+0.2862\text{L},R_{adj}^2=0.8\\ \left|\text{t}\right|=\left(1.38\right)\left(0.75\right)\left(2.09\right)\left(9.13\right)\left(2.24\right)\\ \text{Partial correlation}=1.9\%20.0\%2.4\%51.5\%2.0\%\end{array}$

State standard of living can be estimated from the CDR index for any combination of C, D, R, L and inverse transformation as follows:

$\widehat{\text{GSPppp}}$ = g∙(highest GSPppp − lowest GSPppp) + lowest GSPppp.

From Table A1 highest GSPppp = 107450.43 and lowest GSPppp = 34890.25.

$\begin{array}{l}\widehat{\text{GSPppp}}=\left(0.5586+0.1305\text{C}+0.0489\text{D}+0.1174\text{R}-0.6732\text{C}\cdot \text{D}\cdot \text{R}+0.2862\text{L}\right)\\ \times \left(107450.43-34890.25\right)+34890.25\end{array}$.

The cross-state US CDR model fitted F statistic = ${\text{F}}_{5\text{,50-5-1}}$ = ${\text{F}}_{5\text{,44}}$ = 35 > the theoretical ${\text{F}}_{0.05,v1\text{,v2}}$ = 2.4. Therefore, at a level of significance of 5%, we reject $H_0$ and accept the hypothesis $H_1$, and concluded that the cross-state US CDR model is a good predictor of GSPppp.

The fitted values $\widehat{\text{GSPppp}}$ are plotted in Figure 1 . The values are best described as lying on a straight line. Alaska (AK) is far right because of the influence of extreme latitude. A histogram of the fitted values is plotted in Figure 2 . The residuals from the CDR model are plotted in Figure 3 . They appear to be randomly distributed with no patterns. A histogram of the residuals from the CDR model is plotted in Figure 4 . They appear to be symmetrical and normally distributed. The normal probability plot in Figure 5 is an approximately straight line suggesting a normal distribution. A series of statistical tests to determine the aptness of the model is given in Appendix C.

From the regression analysis, the greatest single partial contribution to GSP is democracy at 20.0%. This suggests that the effect of democracy matters greatly. The US enjoys a high world democracy ranking and high GDPppp. And there are countries that also have high democracy rankings and high GDPppp. This implies that states with relatively low democracy ranking should have their level of democracy improved. The US has a well-defined system of democracy nationally and there is fairly uniform application of the rule of law. But there may be pockets of corruption such that democracy may not benefit from the full protection of the rule of law (contribution 2.4%), or at least not as well as in other states. The contribution from the CDR interaction is 51.5%. A negative CDR coefficient represents distortions. These distortions occur when there is too much democracy and too much rule of law. Excessive democracy is the result of unnecessary discussions that delay investments and opportunities. US corporations and institutions should limit advice and discussion inputs to the various classes of experts to save time in decision making. Excess rule of law is the result of over regulation that obscures opportunities. US corporations and institutions should limit regulations and laws to those which are necessary and eliminate law making for the sake of making laws. Laws should be enforced rather than making new laws for political gamesmanship. The contribution from latitude is 2.0%. Latitude is a natural variable. It is not a policy variable.

The cross-state CDR economic growth model fits state data well and explains GSP. It is comparable to and generally consistent with the cross-country CDR economic growth model. The model coefficients are somewhat different as expected, but remarkably, the coefficient signs are the same. The different sizes of the coefficients are due to the use of a surrogate for state capital in the cross-state model and the absence of data for natural resources. The inability to obtain a measure of capitalism and natural resources is an inherent limitation of the cross-state model. The cross-state CDR can be used to accurately estimate state income. It is the first economic model to place macro growth economics on sound scientific ground. GSPppp is a function of C, D, R, and L. But it is called the CDR model because C, D and R are policy variables that can be determined by government policy, and L is fixed, not determined by government. In summary, collaboration permits rule of law, rule of law attracts capital and protects democracy, democracy deploys capital optimally to create GSP of products and services. That is, collaboration is the point of beginning. Singapore used mandatory school sports and music education to raise the collaboration skill of its children to number one in the world. Therefore, recommendations for future research include mandatory sports and music education, and gene therapy reparations to improve collaboration skills that may have been lost due to historic environmental stresses such as forced labor, excessive discrimination and exposure to dangerous chemicals. Recovery of collaboration skill may bode well for GSP.

All data used in this study are given in the paper.

The author declares that no animals were used in this research.

Source: Democracy rank https://archive.fairvote.org/?page=2117 List of Latitudes and Longitudes for Every State (inkplant.com). Temperature. World population review (2024): Average Temperatures by State 2024.

Source: Real GSP 2023. BEA 2017 Chain: chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://www.bea.gov/sites/default/files/2024-06/stgdppi1q24.pdf. Consumer price index SAT https://blog.prepscholar.com/average-sat-scores-by-state-most-recent Population (year 2020) https://simple.wikipedia.org/wiki/List_of_U.S._states_by_population Rule of Law index (2016) https://ncaj.org/state-rankings/justice-index .

$\begin{array}{l}\hat{\text{g}}=1.11\text{C}+0.14\text{D}+0.20\text{R}-0.91\text{C}\cdot \text{D}\cdot \text{R}+0.32N+0.26{}^{\hat{\text{g}}}\\ \left|t\right|=\left(1.78\right)\left(1.63\right)\left(2.11\right)\left(1.83\right)\left(2.98\right)\left(0.72\right)R_{}^2=\mathrm{0.84844.}\end{array}$

The number of parameters estimated p = 6.

$t_{\text{α},n-p-1}=t_{0.05,50-6-1}=t_{0.05,43}=2.0$

The coefficient of ${}^{\hat{\text{g}}}$ is 0.26 with t = 0.72.

t = 0.72 < $t_{0.05,43}=2.0$ implies that at a 5% level of significance, we fail to reject Ho and accept that the linear specification is appropriate for the CDR model.

Jarque and Bera (1980, 1987) test for normality of residuals.

Consider the following hypotheses.

$H_0$: Residuals from the CDR model are normally distributed

$H_1$: Residuals from the CDR model are not normally distributed

The number of parameters estimated p = 2.

The number of observations n = 50.

Skewness S = 0.302

Kurtosis K = 3.194

$\text{JB}=\frac{n}{6}\left(S^2+\frac{1}{4}{\left(K-3\right)}^2\right)=\frac{50}{6}\left({0.302}^2+\frac{1}{4}{\left(3.194-3\right)}^2\right)=0.84$

${\chi }_{\text{α},p}^2={\chi }_{0.01,2}^2=9.21$

JB $=$ 0.84 < ${\chi }_{0.01,48}^2$ = 9.21 implies that at 1% level of significance, we fail to reject Ho and accept that the residuals from the CDR model are normally distributed.

See also the histogram in Figure 4 .

Breusch and Pagan (1979) test for homoscedasticity of the residuals.

Consider the following hypotheses

$H_0$: The residuals from the CDR model are homoscedastic

$H_1$: The residuals from the CDR model are heteroscedastic

Regressing the variance of residuals on the independent variables,

$\begin{array}{l}{}^{\widehat{\text{ε}}}=0.0-0.0013\text{C}-0.0005\text{D}-0.0013\text{R}+0.00599\text{C}\cdot \text{D}\cdot \text{R}+0.0\text{L}\\ \left|t\right|=\left(1.57\right)\left(0.78\right)\left(0.61\right)\left(1.64\right)\left(0.95\right)\left(0.85\right)R_{}^2=\mathrm{0.093.}\end{array}$

The number of parameters estimated p = 5.

The number of observations n = 50.

F test:

$\text{F}=\frac{R^2/p}{\left(1-R^2\right)/\left(n-p-1\right)}=\frac{0.093/5}{\left(1-0.093\right)/\left(50-5-1\right)}=\frac{0.0186}{0.206}=0.09$

$F_{\text{α},p,n-p-1}=F_{0.01,5,50-5-1}=F_{0.01,5,44}=3.29$

F = 0.09 < $F_{0.01,5,744}$ = 3.29 implies that at 1% level of significance, we fail to reject Ho and accept that the residuals of the CDR model are homoscedastic.

Chi Square Lagrange Multiplier (LM) test:

LM = n $R^2$ = 50 $\cdot 0.09$ = 4.5

${\chi }_{\text{α},p}^2={\chi }_{0.01,6}^2=16.81$

LM $=4.5<{\chi }_{0.01,6}^2$ $=16.81$ implies that at 1 % level of significance, we fail to reject Ho and accept that the residuals of the CDR model are homoscedastic.

White (1980) test for homoscedasticity of the residuals.

Consider the following hypotheses

$H_0$: The residuals from the CDR model are homoscedastic

$H_1$: The residuals from the CDR model are heteroscedastic

Regressing the variance of residuals on the independent variables,

$\begin{array}{l}{}^{\widehat{\text{ε}}}=0.0013-0.0006\hat{\text{g}}+0.0129{}^{\hat{\text{g}}}\\ \left|t\right|=\left(0.601\right)\left(0.056\right)\left(1.111\right)R_{}^2=\mathrm{0.228.}\end{array}$

The number of parameters estimated p = 2.

The number of observations n = 50.

F test:

$\text{F}=\frac{R^2/p}{\left(1-R^2\right)/\left(n-p-1\right)}=\frac{0.228/2}{\left(1-0.228\right)/\left(50-5-1\right)}=\frac{0.114}{\left(0.175\right)}=0.65$

$F_{\text{α},p,n-p-1}=F_{0.01,1,50-2-1}=F_{0.01,1,47}=7.3$

F = 0.65< $F_{0.01,1,47}$ = 7.3 implies that at 1% level of significance, we fail to reject Ho and accept that the residuals of the CDR model are homoscedastic.

Chi Square Lagrange Multiplier (LM) test:

LM = n $R^2$ = 50∙0.09 = 4.5

${\chi }_{\text{α},p}^2={\chi }_{0.01,6}^2=16.81$

$\text{LM}=4.5<{\chi }_{0.01,2}^2=9.21$ implies that at 0.01% level of significance, we fail to reject Ho and accept that the residuals of the CDR model are homoscedastic.

See also the plot of residuals vs. fitted values of g in Figure 3 .

The fitted OLS model with temperature (T) substituted for latitude (L) is

$\begin{array}{l}\hat{\text{g}}=0.8432+0.1634\text{C}+0.0294\text{D}+0.1152\text{R}-0.6592\text{C}\cdot \text{D}\cdot \text{R}-0.2730T，R_{adj}^2=0.8\\ \left|t\right|=\left(1.74\right)\left(0.46\right)\left(2.10\right)\left(9.04\right)\left(2.67\right)\end{array}$

The fitted OLS model with ethnolinguistic fractionalization (EF) included is

$\begin{array}{l}\hat{\text{g}}=0.5635+0.1319\text{C}+0.0439\text{D}+0.1192\text{R}-0.6728\text{C}\cdot \text{D}\cdot \text{R}+0.2872L-0096EF，R_{adj}^2=0.8\\ \left|t\right|=\left(1.36\right)\left(0.64\right)\left(1.93\right)\left(8.98\right)\text{}\left(2.13\right)\left(0.06\right)\end{array}$

The fitted OLS model with all variables replaced by logarithms is

$\begin{array}{l}\hat{\text{g}}=2.5586+0.0893\text{C}+0.1830\text{D}+0.1591\text{R}-0.0188\text{C}\cdot \text{D}\cdot \text{R}+0.7775L,R_{adj}^2=0.44\\ \left|t\right|=\left(2.38\right)\left(1.36\right)\left(1.03\right)\left(1.66\right)\left(3.93\right)\end{array}$