Scott 1 DAVIDSON COLLEGE

Patent Applications and School Enrollment in the Variation of GDP Matthew Scott Statistics (Economics 105) 08/4/2008

Scott 2 Executive Summary My multiple regression analysis estimates a model to account for some variation in the GDP per capita of nations. The independent variables I have chosen to explain that variation are National Patent Applications and Gross School Enrollment. The theoretical base contributing to the development of the model suggested positive marginal effects on GDP for both independent variables. Drawing from the theoretical base I estimated the initial model in the following form:

GDPi   0   P Patentsi   E Enrollmenti   i The initial multiple regression analysis based on my sample of 172 countries, as expected, showed positive marginal effects for Patent Applications and Enrollment on GDP. Past literature suggested several factors at work in GDP that are not included in this model. Omitted variables introduced some heteroskedasticity into the model error term. An analysis of the 172-nation sample showed that the marginal effect of Enrollment on GDP increased as Enrollment increased. I devoted a section of the paper to gauging the implications of a variable marginal effect on GDP for Enrollment:

GDPi   0   P Patentsi   E Enrollmenti   E2 Enrollmenti2   i The new model with a squared term for Enrollment resulted in a better fit than the initial estimation, reducing the variation in GDP per capita left unexplained.

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Introduction In this study I will estimate a model to explain variation in a country’s GDP per capita. The model is based on the marginal effects of total Patent Applications to the US Patent and Trademark Office (from 1964 to 2006), and the gross enrollment of school age children (primary, secondary, and tertiary levels). I will look at that relationship through a cross section of 172 nations. I am interested in patent applications as a proxy for technological advancement. Advancement of technology should represent the force of any creative ability behind the production in a nation. Differences in creative ability would represent a major difference between the GDP of nations. Creative ability has to start somewhere though. Universally and traditionally, creative ability is fostered in schools. For that reason, I am also interested in the School Enrollment of a country as a proxy for education. Education and technological advancement both have a well-grounded economic relationship to production. My proposal is a multiple regression analysis to estimate the dynamics of their effect on GDP. Survey of the Literature Studies relating general production, education, and the advancement of technology and ideas have been conducted in the past. In some of these analyses technological or scientific advancement served as the binding feature. Zvi Griliches (1990) conducted a useful overview of using patent statistics as economic indicators. He made the assertion that economic research often requires a measurement of creativity: “…invention as shifting outward the production possibilities frontier (PPF) for some generalized aggregate of potential human wants (Griliches, 1990, 1669).” In building his argument, Griliches hypothesized that researchers look to patent statistics as the measure of that outward PPF shift, or the output of technological change. Griliches brought his previous studies together with the work of Jacob Schmookler (1966) to argue that patent statistics are far more useful as an input index of inventive activity than as an output index (Griliches, 1990, 1670). The input Griliches gave to the usage of patent statistics as economic indicator is that patent statistics would serve well as an input measure when analyzing production. In contrast to the overview of Griliches (1990), Schofer, Ramirez, and Meyer (2000) conducted tests of technological advancement as an indicator of economic growth. They used a model with many applications to my own. Schofer et al. discussed the direct and important link between scientific activity and patent applications (Schofer, Ramirez, and Meyer, 2000, 874). Instead of using patent statistics they used measures of each nation’s scientific labor force and research to measure scientific activity (Schofer, Ramirez, and Meyer, 2000, 872-874). At the same time, they noted that Patent Applications would serve as an effective measure of scientific activity in place of the measures they used. The measure used by Schofer et al. to control for an educated labor force was Secondary School Enrollment as a proportion of the relevant population age group (Schofer, Ramirez, and Meyer, 2000, 876). They also used a United Nations Educational, Scientific, and Cultural Organization (UNESCO) ratio for enrollment like the one I selected for this model. While there are many similarities between the model of Schofer, Ramirez, and Meyer (2000) and this one, there are differences as well. The difference between the Gross Enrollment of this model and the Secondary Enrollment of Schofer et al. is the primary and tertiary levels included in the gross calculation.

Scott 4 A major difference between the models is the dependent variable chosen. Schofer, Ramirez, and Meyer (2000) used economic growth, the growth of real GDP per capita from 1970 to 1990. The regression models estimated by Schofer et al. tested growth by using a log ratio of GDP from period “t” and GDP from period “t – 1” (Schofer, Ramirez, and Meyer, 2000, 875). They also tested their findings by replacing GDP as a measure of production with the National Development Index. The National Development Index is an index of industrial development that incorporates GDP per capita, energy consumption in kilowatt hours, and the percentage of the labor force outside of the agricultural sector (Schofer, Ramirez, and Meyer, 2000, 880). While I can expect the same factors to influence GDP in this model, the difference in the dependent variable makes interpretation different. The use of the logged GDP per capita ratio by Schofer et al. (2000) translates to a percentage marginal effect on GDP growth rather than the marginal effect on GDP. Although the economic growth of GDP is different from GDP, Schofer et al. (2000) remains very relevant to the development of this model. In all of the fifteen estimated models, Schofer et al. found positive marginal effects of their variables associated with technological advancement and enrollment, on those associated with production or economic growth. Model Development I will estimate the marginal effects of the two independent variables Patent Applications and Enrollment, on GDP, using an initial model of the following form:

GDPi   0   P Patentsi   E Enrollmenti   i

Equation (1)

Patent Applications Utilizing the work of Griliches (1990) suggesting the use of patent statistics as an input measure, the first independent variable is the number of Patent Applications to the US from each nation “i”, from 1964 to 2006. The work of Schofer, Ramirez, and Meyer (2000) indicates a positive marginal effect of scientific activity on economic growth. Schofer et al. specify a linear marginal effect of scientific activity on economic growth. They do not indicate any other functional forms, so I expect the marginal effect of Patent Applications on GDP to be positive and linear in this model, holding Enrollment constant. For the magnitude of the marginal effect of Patent Applications on GDP, there is no theoretical grounding to support an exact estimate. To expect a small magnitude ($0 < αP <$1) would support the notion that a change of one Patent Application would not influence the GDP per capita for a nation substantially. An increase of one Patent Application could possibly mean a slightly bigger difference in GDP per capita for a smaller nation but it still should not have a large effect on the GDP of an entire country. Schofer et al. (2000) estimated a separate effect of “technically relevant” scientific activity on economic growth from a proposed negative effect of “socially relevant” scientific activity on economic growth (Schofer, Ramirez, and Meyer, 2000, 869-872). In connecting the inversely related measures of scientific activity to patent statistics, Zvi Griliches (1990) grappled with a similar issue. Griliches (1990) expressed a parallel concern with regard to the variance in the technical and economic significance of the average patent invention (Griliches, 1990, 1666). In doing so he foreshadowed problems with aggregate measures of patent statistics like total Patent Applications. The indication of two separate and opposing marginal effects on economic growth introduces a possible downward bias into the estimate of the coefficient for Patent Applications.

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I expect a positive marginal effect of Patent Applications on GDP holding Enrollment constant. There is also enough information to reasonably speculate that the marginal effect of an increase of one application on GDP per capita might fall between zero dollars and one dollar. I will be testing this against an alternative one-tailed hypothesis that the marginal effect is less than or equal to zero to determine the effect from aggregating two separate effects in the variable Patent Applications. Gross Enrollment Ratio The second independent variable is gross enrollment of school age children at all levels (primary, secondary, and tertiary) in each country “i”. The measure is represented by that nations’ UNESCO Gross Enrollment Ratio. The ratio is an average ratio for each country calculated over the decade from 1996 until 2006. Schofer, Ramirez, and Meyer (2000) used Secondary School Enrollment in their model to control for an educated labor force, which spurs economic growth according to their literature review (Schofer, Ramirez, and Meyer, 2000, 876). The models they estimated reflected their expectation showing positive effects for all of their coefficient estimates of the marginal effect of secondary school enrollment. Their use of Secondary School Enrollment was based upon a linear form in all of their models. Schofer et al. added tertiary school enrollment to several models in a linear form, as a separate variable. The positive marginal effect of Secondary School Enrollment on Economic Growth deduced in Schofer, Ramirez, and Meyer (2000) is a linear and positive effect in all models. That leads me to expect a positive and linear marginal effect on GDP in this model, holding Patent Applications constant. Regarding an estimate of the magnitude of the marginal effect of the Gross Enrollment Ratio, Schofer et al. (2000) provides some insight. With most of the values for the Gross Enrollment Ratio under 100, an increase of one would make a large difference in GDP. Schofer, Ramirez, and Meyer (2000) used a similar ratio multiplied by (10-3) for secondary education, making their estimates of αE comparable. The average marginal effect of Enrollment in the five models with the largest sample sizes was a $399 increase in GDP for an increase of 1 in the Enrollment Ratio (Schofer, Ramirez, and Meyer, 2000, 878-881). In forming theoretical expectations for the model then, I expect a positive marginal effect of Enrollment on GDP with an estimated magnitude close to 400, holding Patent Applications constant. There is no theoretical evidence to suggest whether the addition of primary and tertiary enrollment together will have a positive or negative effect on the coefficient for Enrollment. Therefore, I will test a double-sided hypothesis. GDP per Capita Gross Domestic Product per person for each country “i”, or GDP per capita, serves as the measure of the production of each nation in this multi-variable regression. The difference between the dependent variable used by Schofer et al. and GDP may have some implications for the magnitude estimate of the marginal effect of Enrollment on GDP. Because Enrollment is itself a ratio though, the implications of the difference in dependent variables may be offset. I can still assume that the marginal effect on GDP will be great for a change in Enrollment because Enrollment is a ratio, so a $399 marginal effect on GDP is still a reasonable estimate. There would not be any implications outside of the difference in interpretations and the estimate for the marginal effect of Enrollment. Because I am still looking at the differentiation in GDP we can expect a positive and constant marginal effect for both variables on GDP, even with the difference in functional forms between models.

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The Model Intercept Term The intercept in the multiple regression estimate modeled after Equation (1) is associated with the value of GDP per capita for a country with no Patent Applications, and with a Gross Enrollment Ratio of zero. Given the period of 42 years for Patent Applications, and the decade average for School Enrollment, it is unlikely that there would be a nation with no history of Patent Applications and no School Enrollment that would also have a publicly available GDP per capita. Such an observation would fall outside of the range of interpretation this model is specified to achieve. Past studies by neither Griliches (1990), nor Schofer, Ramirez and Meyer (2000) grant a theoretical basis to derive meaning from the intercept term in this model. Since I cannot speculate on an expected value or meaning of the intercept term there is no theoretical grounding to support an expectation of the sign or magnitude of α0. The intercept term in this model, then, is an extrapolation. The Additive Error Term Schofer, Ramirez, and Meyer (2000), in using logged GDP growth as their dependent variable (and the National Development Index), controlled for several variables that will not be included in this model. Other independent variables used to control for separate effects on economic growth included the Investment Share of GDP in period 1, a Political Democracy Index, and Trade as a Share of GDP (Schofer, Ramirez, and Meyer, 2000, 875-877). As discussed in the variable development for Patent Applications, Schofer et al. (2000) also estimated the separate effect of “technically relevant” scientific activity on economic growth from the proposed negative effect of “socially relevant” scientific activity on economic growth. The supportive claim of two separate effects in patent statistics raised by Zvi Griliches (1990), is an additional indicator of the two separate effects in one variable (Griliches, 1990, 1666). The estimate of two separate and opposing marginal effects on economic growth introduces the further potential of an omitted effect by the use of only Patent Applications here. Total Patent Applications are not a uniform technological advancement then, and treating them as such is an omission of what may even be an inverse effect on GDP from economically irrelevant patents. There are, then, at least a few factors with a theoretical relationship to GDP per capita that remain unaccounted for in this model. Excluded relevant variables will mean variation in GDP that would be explained by the omitted variables will be attributed to the error term, Patent Applications, and School Enrollment. Therefore, a priori, I expect to have some problems resulting from misspecification of the model such as autocorrelation or heteroskedasticity. Because of the omitted variables, I would also expect a poor fit for this model in explaining the full variation in GDP per capita.

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The Multiple Regression Equation Based on the theoretical structure outlined above, the model estimation for this multiple regression analysis will take the form of Equation (1). Theoretically, I expect a constant and positive marginal effect of both independent variables on GDP per capita in the multiple regression analysis. Because I expect a positive linear relationship between both independent variables and GDP, and given the magnitude estimates, I will be testing the significance of the relationship between each independent variable and GDP using hypothesis tests of the following specifications:

H0 : P  0 H1 :  P  0

H0 : E  0 H1 :  E  0

H0 : 0   P  1 H1 :  P  0

H 0 :  E  $399 H1 :  E  $399 Discussion of the Data Data Description

In this section I will discuss the characteristics of my sample to estimate the multivariable regression model established in the Model Development section. To estimate the multivariable regression after the model indicated in the Model Development section, I have collected data for a sample of 172 countries. The CIA World Factbook estimates that there are 266 nations, dependent areas, and other entities, making this sample size almost 65% of the population sample size (Central Intelligence Agency, 2008). The data for GDP per capita are the latest estimate obtained by the CIA for each of the 172 countries. Though I am unable to locate the calculation method used, these data from the CIA World Factbook are likely sufficient to test hypotheses of the marginal effects of Patents and School Enrollment on GDP. Data from the US Patent and Trademark Office are the combined Patent Applications to the US Patent and Trademark Office between 1965 and 2006 for each of the 172 nations. The country is designated by the home country listed by the patent applicant. The Gross Enrollment Ratio (GER) for all levels is a statistic compiled by the United Nations Educational, Scientific, and Cultural Organization (UNESCO). The Enrollment Ratio is the average gross enrollment for primary, secondary, and tertiary levels of schooling combined, over the decade from 1996 until 2006. It is important to note that I am not able to locate the exact calculation procedure for the gross enrollment ratio. A similar ratio was used by Schofer, Ramirez, and Meyer (2000) as discussed in the Model Development section. The ratio is helpful in relating the study by Schofer, Ramirez, and Meyer to this one, and is likely sufficient to test hypotheses of the marginal effect of Gross Enrollment on GDP.

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Summary Statistics In this section I will discuss the sample distribution of both independent variables and the dependent variable GDP using scatter plots that show the relationships of each independent variable to GDP per capita.

A scatter plot displaying the relationship between GDP and Patent Applications is shown in Graph 1. Many of the countries in this sample of 172 had a single digit number of Patent Applications. At the same time, many countries such as European countries, Japan, China, and others heavily involved in the US and other global markets, had a great deal of applications. The first quartile of 6.75 and the median of 52.5 for Patent Applications look like one line against the y-axis in Graph 1. The third quartile is the separate dotted line ahead of the other two in Graph 1, at 573.25 patents. Graph 1 has a modified view to bring the mean of 17,686 patents into the range of relative comparison. Over 75 percent of the observations fall well below the mean number of Patent Applications, indicating a heavy skewness in the distribution of Patent Applications. The standard deviation of 99,024.79 patents, far higher than the mean number of patents, gives a clear signal of the percentage of the distribution that even lies beyond the range of Graph 1. The minimum number of Patent Applications is one, shared by fifteen countries including Belize, Chad, Malawi, Sierra Leone, and Mongolia. The maximum number of patent applications is 1,154,384 patent applications from Japan. The sampling distribution of Patent Applications, as the sample shows, is a long tailed distribution highly skewed to the right. For GDP per capita, most countries fall around the median of $8,650, or between the median and the first Quartile of $2,888. The mean GDP per capita for the sample is $14,844. The world GDP per capita was estimated at $10,000 by the CIA in 2007, which would fall between the mean and median for

Scott 9 this sample of 172 countries (Central Intelligence Agency, 2008). The standard deviation for GDP is $15,862. The maximum GDP per capita in the sample is Qatar with $80,900, and the minimum GDP in the sample is Zimbabwe, with a GDP per capita of $200. There is a great deal of variation in GDP as well, which should be expected with the general economic trend of variation between the wealth and production of nations. Using the indicator that the mean is higher than the median, I can conclude that the sampling distribution of GDP per capita is skewed to the right.

Graph 2 shows a scatter plot of the Gross Enrollment Ratio as it relates to GDP per capita. The first quartile is 60.86, and the mean Enrollment Ratio is 70.74. The median of 73.43 is higher than the mean, and the third quartile is an Enrollment Ratio of 84. Most of the observations are in the first, third, and fourth quadrants of the junction between the mean enrollment rate and the mean GDP per capita, and are centered evenly about the mean. The number of observations below the mean and median Enrollment Ratio appears to be almost the same as the number of observations above the mean and median Enrollment Ratio. The standard deviation for the Enrollment Ratio is 17.23, making Gross Enrollment the only of the three variables in this data set with a standard deviation less than its mean. Although there is a visual balance in the distribution of the sample of Enrollment Ratio values the mean is less than the median. The sampling distribution for the Enrollment Ratio is therefore slightly skewed to the left. The minimum is an Enrollment Ratio of 20.24 in the nation of Niger, and the maximum is a ratio of 102.59 in Sweden. The minimum of 20 falls well above an Enrollment Ratio of zero, enforcing the conclusion reached in the Model Development section that the intercept term in this model is an extrapolation.

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Analyses The following section will concern the analyses associated with the multivariable regression to estimate the effects of National Patent Applications and the Enrollment Ratio on GDP per capita. Most of the hypothesis tests for the model estimates in this study are contained in this section. In the tests I will conduct in this study, most of the alternative hypotheses can be classified as the more speculative of the two hypotheses. I have selected the use of a .05 level of significance because I want a low probability of rejecting a true null hypothesis in my tests. I am more concerned about rejecting a true null hypothesis in my hypothesis testing, or the commission of a Type I error, than I am about accepting a false null hypothesis. I am willing to sacrifice some power in the tests of my model estimates to avoid a relatively higher possibility of rejecting a true null. For these reasons, in all of the subsequent hypothesis tests I will be using the .05 level of significance (α = .05). Graphical Analyses A scatter plot of the sample relationship between GDP and Patent Applications is shown in Graph 1 of the Discussion of the Data section. The estimate of the correlation for that relationship is .21205, which I interpret to mean that there is a weak but positive relationship between GDP and Patent Applications in the sample. In the theory I used to develop the overall model in Equation (1), I estimated a positive and constant marginal effect of Patent Applications on GDP. I also estimated a small magnitude falling somewhere between zero and one. The gently rising trend line indicates the validity of the magnitude prediction. As nearly as I can tell from a graphical analysis there is no indication that the relationship is not linear (with a constant slope). Some observations are close to the trend line and others are far from it, indicating a varying size of the squared errors. The varying errors reflect the predicted differences in GDP and Patent Applications between nations of varying sizes and resources. Based on Graph 1 and the correlation coefficient between the two variables, all of the model expectations developed for the relationship between GDP and Patent Applications appear to be satisfied in this sample. A scatter plot of the sample relationship between GDP and the Gross Enrollment Ratio is shown in Graph 2. The correlation estimate between GDP and Enrollment from this sample is .54456. That correlation translates to a positive relationship between GDP and the Gross Enrollment Ratio in this sample, which confirms the theoretical expectation of a positive relationship. The slope of the trend line for Enrollment is much higher than the slope for the trend line of Patents and GDP. The trend line slope seems visually to approach the estimated magnitude of $399 for the marginal effect of Enrollment on GDP. The most noticeable aspect of the relationship between GDP and the Enrollment Ratio in the sample is that the relationship appears to change as Enrollment increases. There is a definite non-linear drift in the pattern of the data, and the variance from the linear trend line increases as Gross Enrollment rises. The trend indicates that higher enrollment allows more distributed values of GDP. This pattern is different from the theoretical expectation of a linear relationship, or a constant marginal effect of Enrollment on GDP. No indication of a non-linear relationship was given with the similar Enrollment ratio used by Schofer, Ramirez, and Meyer (2000), making the obvious non-linear relationship here an unexpected mystery in this model.

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Linear Regression Analysis Table #1 Comparison of Regression Results Equation (1) GDP = α0 + αPP + αEE + ε

Equation (2) GDP = β0 + βPP + ε

Estimate of α0, β0, or γ0

-19,867.1

13,958.855

Estimate of αP or βP

$0.01924

$0.034

Estimate of αE or γE

$481.864

Coefficient of Determination Adjusted R

2

Equation (3) GDP = γ0 + γEE + ε -20,900.39 $501.282

0.31053

0.04496

0.29654

0.30237

0.03935

0.29240

Table #1 displays the regression coefficients for three models estimated using different combinations of the independent variables Patent Applications and Gross Enrollment, including the multivariable regression model for Equation (1). Equation (1) Analysis The Equation (1) regression coefficient for Patent Applications can be translated to mean that for every increase of one National Patent Application there is an average increase of $.01924 in GDP, holding Gross Enrollment constant. This makes sense because, as outlined in the Model Development, the magnitude of an increase in GDP resulting from an increase of one in Patent Applications should be small and the sign should be positive. The value of the coefficient actually falls within the predicted interval of a magnitude between zero and one. The Equation (1) regression coefficient for Patent Applications can be translated to mean that for every increase of one in the Enrollment Ration I can expect an average increase of $481.86 in GDP per capita, holding the number of Patent Applications constant. This value also makes sense because it indicates a positive marginal effect of Gross Enrollment on GDP, and because it falls fairly close to the estimated marginal effect of $399. Furthermore, the fact that this estimate of the Enrollment coefficient is greater than $399 accounts for the use of Gross Enrollment rather than just secondary enrollment, allowing primary and tertiary levels to increase the marginal effect of Enrollment on GDP. The coefficient for the intercept term, if we were to interpret it, would translate to an average GDP per capita of -$19,867.1 for countries with no Patent Applications and a Gross Enrollment Ratio of Zero. This is a meaningless value for GDP per capita, and it re-enforces the specification of the intercept term as an extrapolation. The F-Test for the overall model significance of Equation (1) yields an F-statistic of 38.0573 and a p-value of 2.263E-14, which is sufficient to reject the null hypothesis of model insignificance at the .05 level of significance. Using the Coefficient of Determination we can see that about 31% of the variation in GDP per capita is accounted for by the multivariable regression estimated in Equation (1). The hypothesis that the R2 for the overall model in Equation (1) is zero implies that the model is insignificant in explaining the variation in GDP. Since we used the F-test for overall model significance to reject the null hypothesis in favor of a significant model, we can reject the hypothesis that R2 equals zero.

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The null hypothesis that the intercept term is equal to the population average of GDP per capita is the same as the null hypothesis in the whole model F-test. The two tests are the same because if the independent variable coefficients were zero, the estimated expected value of GDP would be equal to the intercept term. I have already shown that the intercept term is an extrapolation, and rejected the insignificance of the whole model in Equation (1) using the F-test. We can also use the F-test results for overall model significance to reject the null hypothesis. The average population GDP per capita is not equal to the intercept term (μy ≠ α0). Equation (2) and Equation (3) Comparative Analysis Comparing the coefficient for Patent Applications in Equation (1) to the coefficient in Equation (2), there is an increase in the estimate for the marginal effect of patent applications on GDP when Enrollment is dropped. The adjusted R2 decreases substantially from Equation (1) to Equation (2). The decrease in the adjusted R2 combined with the increase in the coefficient for Patent Applications indicates that the portion of the variation in GDP explained by Enrollment is a significant portion of the variation explained by the overall model. When Enrollment is dropped the amount of variation unexplained by the model increases as the model loses explanatory power. The increase in the coefficient αP indicates that some of the explanatory power of Enrollment is falsely attributed to Patent Applications. There is an increase in the estimated marginal effect of School Enrollment on GDP when the variable Patent Applications is retained from the model estimation equation. The adjusted R2 decreases in the shift from Equation (1) to Equation (3), but the change is slight. The increase in the coefficient for School Enrollment, combined with a decrease in the adjusted R2, indicates that some explanatory power becomes unexplained variation in GDP when the variable Patent Applications is dropped from the model between Equation (1) and Equation (3). Some of the variation in GDP explained by Patent Applications is falsely attributed to Gross Enrollment as well, resulting in the increase of αE. Relying on the analysis of past studies and theoretical speculation in the Model Development section, I developed hypotheses for the sign and magnitude of the alpha coefficients estimated using Equation (1). Those tests, which I will conduct in this section, are specified in the following form:

H0 : P  0 H1 :  P  0

H0 : E  0 H1 :  E  0

H0 : 0   P  1 H1 :  P  0

H 0 :  E  $399 H1 :  E  $399 Patent Applications Coefficient Tests

A one sided hypothesis test on the coefficient of Patent Applications yields a t-statistic of 1.85 and a p-value of .033. I will use those results to reject the null hypothesis in favor of the alternative, that the marginal effect of Patent Applications on Enrollment is positive. This aligns with the expectations of a positive relationship discussed in the Model Development section. I will use a 90% confidence interval to test the magnitude of αP for agreement with the hypothesis that it falls between the values zero and one. The coefficient point estimate is .01924 and the Margin of Error is .0172. The upper value of the confidence interval is .0364 and the lower value is .00205. The

Scott 13 estimated value for the coefficient of Patent Applications is contained within the range of zero to one. At the .05 level of significance I reject the null hypothesis for the alternative that the marginal effect falls between zero and one, because 90% of the intervals estimated in this way would contain the true population mean which would subsequently fall in the interval between zero and one. While the sign and magnitude of the coefficient for Patent Applications align with theoretical predictions, the difference between accepting the null and rejecting the null is close here. At the .01 level of significance I would have failed to reject the null hypothesis of a non-zero or negative relationship. This is reflective of the great difference in economic significance between patents, and the problems that arise from an aggregate specification of patent statistics or scientific activity. This problem with the connected possibility of introducing a downward bias in the marginal effect point estimate was discussed in the Model Development section. Gross Enrollment Coefficient Tests Testing the positive significance of the marginal effect of Gross Enrollment using Equation (1) yields a t-statistic of 8.068 and a p-value of 6.32E-14. At this value, and using the .05 level of significance I reject the null hypothesis in favor of a positive marginal effect of Gross Enrollment on GDP. To test the prediction for the magnitude of the marginal effect of Secondary Enrollment on GDP per capita, I removed a degree of freedom and used the sample mean Enrollment. This two-tailed test calculation yields a t-statistic of 5.5 and a p-value of 1.42E-07. That is sufficient to reject the null hypothesis that the marginal effect of Enrollment on GDP is $399. The sample estimated effect is significantly greater than the estimate of Schofer, Ramirez, and Meyer (2000). The magnitudes are fairly close though. This indicates that the inclusion of the primary and tertiary education levels in the Gross Enrollment results in an increase in the marginal effect of Enrollment on GDP from that estimated by Schofer, Ramirez, and Meyer (2000). Multicollinearity An examination of the Pearson correlation coefficient between Patent Applications and School Enrollment reveals an rE,P of .175609. There is a small positive relationship between the independent variables in this model then, but it is not substantial enough to interfere with our analysis of the relationship between either of the independent variables and GDP per capita.

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Residuals The unrepresented factors of GDP discussed in the Model Development section suggest the expectation that the error term in the model estimate of Equation (1) would fail to uphold some assumptions of the error term for a normal multivariable regression model. Those assumptions about the error term are necessary for obtaining unbiased results in our hypothesis testing. It would be prudent, then, to analyze the residuals for patterns and indications of model specification problems.

Graph 3 is a graph of the deviation of the deviation of the residuals relative to the model estimated GDP per capita. The measures can be interpreted as the number of standard deviations each observation deviates from an error of zero as the model estimated GDP per capita rises. One assumption about the error term which we can visually test is the assumption that the mean value (and therefore the mean z-value) of the error term is zero. The z-values of the observation errors appear to be evenly distributed about a z-value of zero, and the mean z-score is -1.2E-15, so that assumption holds. The initial indication of the graph is a pattern though, and a pattern is not indicative of the random residuals assumed for an unbiased regression model. The residual variation appears to increase as the estimate for GDP increases. This pattern indicates a non-constant relationship between the estimate of GDP and the residuals, also referred to as a heteroskedastic error term. The pattern in the residuals is a violation of the assumption that the variance of the error term is the same for all values of the dependent variable, and the violation suggests that we cannot be sure that our hypothesis tests based on this multivariable regression model are accurate. There are three outliers with z-scores greater than 3. Those outliers are the GDP per capita estimate residuals for Bermuda, Luxembourg, and Qatar.

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Dropping the outliers Bermuda, Luxembourg, and Qatar does change the dynamic of the residual variation plot slightly, introducing a relatively more uniform variation with 169 observations (see Graph 4). The change introduces three new outliers with z-scores higher than 3 though – the British Virgin Islands, Cuba, and Malaysia. Even when dropping the outliers the pattern of a heteroskedastic error term persits in the model. Because this variation violates the assumption of the error term for a regression model, we cannot be sure that the hypotheses testing in the analysis of these data has yielded accurate results. Other Useful Analyses Exploring the Marginal Effect of Enrollment I have added this section to explore this possibility that the Gross Enrollment ratio has a variable marginal effect on GDP rather than a constant marginal effect on GDP. In the Graphical Analysis portion of the Analyses section I noted an interesting trend. The analysis of the scatter plot relationship between GDP per capita and Gross Enrollment, shown in Graph 2, indicated a change in GDP as Gross Enrollment rises. The pattern suggests that that the marginal affect of Gross Enrollment on GDP is variable, not constant. The curvature of the relationship between GDP and Gross Enrollment suggests an exponential, concave-up pattern. For the purposes of analysis, this would suggest a modification to the model estimate of the following form:

GDPi   0   P Patentsi   E Enrollmenti   E2 Enrollmenti2   i Adding a squared term for Enrollment will permit the analysis of a non-constant marginal effect of Gross Enrollment on GDP per capita. If the marginal effect is properly variable, this newly specified functional form should reduce some of the unexplained error in the previously estimated model.

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Table #2 Squared Functional Form Regression Analysis Equation (1) Y = α0 + αPP + αEE + ε

Equation (4) Y = δ0 + δPP1 + δEE2 + δE2E22 + ε

Estimate of α0 or δ0

-19,867.08

3,667.45

Estimate of αP or δP

0.01924

0.017

Estimate of αE or δE

481.864

-297.612

Estimate of δE2

5.9717

Coefficient of Determination Adjusted R

2

0.31053

0.32889

0.30237

0.31690

The coefficients for the model estimated with a squared term for Gross Enrollment are displayed above in Table #2 alongside the results from the multivariable regression model estimated earlier in this study. The change between Equation (1) and the newly estimated Equation (4) for the coefficient of Patent Applications is a decrease. The most notable change though is the now negative coefficient for Enrollment, coupled with the positive coefficient of Enrollment Squared. The difference between the adjusted R2 for the two models is a slightly higher adjusted R2 in Equation (4), indicating a better fit for that model. Testing the model in Equation (4) for overall significance yields an F-statistic of 27.44 and a p-value of 1.7E-14. I use those results to reject the null in favor of the alternative of model significance. The implications and interpretation of a variable marginal effect of Gross Enrollment on GDP should be considered closely. As an example, we would like to know why the marginal effect of Gross Enrollment on GDP is now estimated at a negative value. With a squared term in the model, the effect of a change in Gross Enrollment on GDP per capita is no longer just the accompanying coefficient:

GDPi   0   P Pi   E Ei   E 2 Ei2   i GDP   E  2 E 2 Ei E In Equation (4) the marginal effect of Enrollment on GDP is a function of the variable element in the marginal effect, Enrollmenti, and an autonomous factor, represented by the newly estimated and negative δE. Using the estimates obtained for the constants obtained from the regression of Equation (4) using the dataset of 172 countries, we can estimate the marginal effect as follows:

GDP   E  2 E 2 Ei  297.612  (2)(5.9717) Ei E GDP  11.94333Ei  297.612 E This indicates a concave up curve. The autonomous negative value for the new estimate of δE suggests that for the lowest values of Enrollment GDP actually decreases as Enrollment increases. The positive slope suggests that the marginal effect of Enrollment on GDP is increasing as Enrollment rises.

Scott 17 The bottom of that U-shaped relationship between GDP and Gross Enrollment, where this estimate would minimize the marginal effect of Enrollment on GDP, is as follows:

0  11.9433Ei  297.612 297.612  11.9433Ei 297.612 Ei   24.92 11.9433 A glance at Graph 2 shows that there is at least one nation with a Gross Enrollment Ratio lower than 24.92 (Niger). This estimate means that there is a theoretical Gross Enrollment Ratio (the sample point estimate of it is 24.92) after which the GDP per capita of a nation will rise as the Gross Enrollment rises, and before which the GDP per capita of a nation will rise as the Gross Enrollment Ratio falls. The model in Equation (4) suggests the possibility of an increasing and positively sloped marginal of Enrollment on GDP. This also suggests that the former tests for the marginal effect of Enrollment on GDP using Equation (1) were based on a mis-specified model, and are inaccurate. To prove this theory would require me to estimate and test the significance of this variable marginal effect. The necessary tools to test such a relationship for its significance are, unfortunately, outside of the scope of this course. If the relationship proved significant though, the model of best fit for a multivariable model using Patent Applications and Gross Enrollment would be modeled after Equation (4). Conclusion To conclude, both independent variables (Patents and the Gross Enrollment Ratio) show a positive and significant relationship with GDP. The initial multivariable regression model for this study was estimated in the following form:

GDPi   0   P Patentsi   E Enrollmenti   i

Equation (1)

Schofer, Ramirez, and Meyer (2000) was an integral study to building upon the past theory of the relation of Enrollment and Patent Applications to GDP per capita, and Griliches (1990) was useful in establishing the specifications of the relationship between GDP per capita and Patent Applications. The model of Equation (1), while significant, lacks several important factors that contribute to the variation of GDP. Those absent factors contributed to systematic variation in the error term of this model, raising the concern of heteroskedasticity. There is not as much cause for concern multicollinearity in this model – there is a weak positive relationship between Patent Applications and the Gross Enrollment Ratio indicated in the using the sample estimates. I also explored the variable relationship between GDP per capita and the Gross Enrollment Ratio. I used a new functional form for Enrollment Ratio in the estimation of Equation (4). The new model accounted for the increasing effect of Enrollment on GDP as Gross Enrollment increased as follows:

GDPi   0   P Patentsi   E Enrollmenti   E 2 Enrollmenti2   i

Equation (4)

The model in Equation (4) provided the estimate of best fit in explaining the variation in GDP per capita. A future study along these lines might expand on this proposed marginal effect of Gross

Scott 18 Enrollment on GDP by testing the significance and magnitude of the relationship with a new sample of nations.

Bibliography and References Central Intelligence Agency, U.S Government. "GDP Per Capita by Country." CIA World Factbook. Last Accessed June 15, 2008. Griliches, Zvi. “Patent Statistics as Economic Indicators: A Survey.” Journal of Economic Literature, Vol. 28, No. 4, (December 1990), pp. 1661-1707. Schmookler, Jacob. Invention and Economic Growth. Harvard University Press. Cambridge, Massachusetts, 1966. Schofer, Evan, Francisco O. Ramirez, John W. Meyer. “The Effects of Science on National Economic Development, 1970 to 1990. American Sociological Review, Vol. 65, No. 6, (December 2000), pp. 866-887. United Nations Educational, Scientific, and Cultural Organization. Gross Enrollment Ratio, All Levels Combined (except pre-primary), 1975 to 2007. Institute for Statistics. Last accessed July 15, 2008. US Patent Technology Monitoring Team (2007). Number of Utility Patent Applications filed in the United States by Country of Origin, Calendar Years 1965 to Present. U.S. Patent and Trademark Office. Last Accessed June 11, 2008.