Spatial Econometrics

It's Over. 

Developmental Asset Profile: A Spatial Model

Joshua Bates

Dr. Murdoch

Econ 6316

9 May 2012

Introduction

            “It’s difficult to convey the almost unbearable sweetness of this kind of American childhood to anybody who didn’t live it,” wrote blogger Jim Manzi, referring to his childhood.  “The safety and freedom that Krugman describe are rare now even for the wealthiest Americans – by age 9, I would typically leave the house on a Saturday morning on my bike, tell my parents I was “going out to play,” and not return until dinner; at age 10, would go down to the ocean to swim with friends without supervision all day; and at age 11 would play flashlight tag across dozens of yards for hours after dark” (2011).  Paul Krugman and Jim Manzi appeared to have the kind of perfect childhoods, rendered impossible by modern life and the risks children face today.  In fact, Carol Tilley wrote that modern childhood has been “Pathologized” (2011). 

Fortunately there have been recent attempts to strengthen children against the dangers of society, “In the past few decades, researchers and policymakers from fields including education, social work, and public health have proposed an alternative approach to understanding childhood and adolescence: positive youth development (PYD)” (Tilley 2011, 42).  Developmental assets are the evolution of positive youth development; the idea is that there are areas of development that allow children to resist societal ills, and that these areas, these assets, can be built if they do not occur naturally.  The Developmental Assets Profile is a tool for measuring these assets; perhaps their development in peers can lead to better academic outcomes for the student body.

Literature

            The literature for developmental assets is scarce outside of Search Institute, the company that devised the framework and tests (Stevens et al. 2010).  However, much of it speaks to the value of the assets.  Search Institute’s website states that, “Search Institute research has found that these assets are powerful influences on adolescent behavior—both protecting young people from many different risky behaviors, and promoting positive attitudes and actions” (Search 2011).  Scales et al. conclude that, “Specifically, youth with higher levels of developmental assets were considerably more likely than other youth to report being successful at school, overcoming adversity, and maintaining physical health, and delaying gratification” (2000, 44).  Furthermore one implementation of a project to develop assets has shown success as illustrated by St. Louis Park’s 9^th Grade Program (Roehlkepartain et al. 2003).  In contrast, Stevens et al. conclude this to be an outlier, noting that, “While the Developmental Assets can serve as a potentially productive programming component for the [American School Counselor Association] domains of personal/ social and academic development, the assets fell short in the area of career development. In addition, explicit implementation examples and outcome research are lacking” (2010).  Thus, assets appear to be an indicator of social progress, but their usefulness in predicting material success may be lacking.  Perhaps the latest edition of their developmental assets profile, spanning 53 items, will provide a better predictor of scholastic outcomes.

Data

            The data consists of 4030 surveyed ninth graders from Dallas Independent School District; the survey was administered in the spring semester of the 2009/2010 school year.  This was the first year of the survey.  Included in the model is GPA, the student’s second semester grade from 55 to 100, with over 90 being an A, between 80 and 90 being a B, between 70 and 80 being a C and below 70 being an F.    Spring Att is a ratio of days attended over days enrolled; gender and racial binaries are also included.    Limited Eng and Exited LEP are those enrolled in the limited English proficiency program and those who have exited the program respectively; since the district is largely Hispanic, this racial binary is omitted and these categories attempt to recapture the effect of being Hispanic.  Reading, Math, Science and Social Study are the scale score for the TAKS tests from the 8^th grade of each subject.  These scores are scaled such that a 2400 is a passing grade.  Does Hmwk, Avoids Alc, and Motivated are the answers from three of the Developmental Assets Profile questions coded from 0 to 3 with 3 being the “correct” answer.  The first asks whether a student regularly does homework, the second asks if the student avoids alcohol, tobacco and other drugs and the third asks if the student is motivated to do well in school.  From my previous studies, I found that these questions were the most correlated with student performance.

Variable	Obs	Mean	Std. Dev.	Min	Max
GPA	4030	81.72539	6.26722	55.8333	97.8125
Spring Att	4030	0.951753	0.063843	0.247191	1
Female	4030	0.50273	0.500055	0	1
Hispanic	4030	0.705459	0.455893	0	1
Native Am	4030	0.001985	0.044516	0	1
Asian	4030	0.0134	0.114992	0	1
Black	4030	0.233251	0.422953	0	1
White	4030	0.045906	0.209307	0	1
LEP	4030	0.135732	0.342546	0	1
Exited	4030	0.459057	0.498383	0	1
Reading	4030	2345.932	200.4315	1163	2734
Math	4030	2236.348	197.3286	1239	2796
Social Study	4030	2310.81	227.3386	1335	2805
Science	4030	2172.417	233.289	835	3020
Does Hmwk	4030	1.758809	0.88387	0	3
Avoids Alc	4030	2.35335	0.972548	0	3
Motivated	4030	2.23995	0.838537	0	3

            The sample attended just over 95% of classes enrolled, and is slightly more than half female.  Notably, Hispanics make up 70% of the population of this sample with blacks being the other sizable proportion.  80.6% of the district is of low socioeconomic status.  Nearly half the district was in a limited English proficiency program at one time; this could help explain why the mean student does not pass any of the TAKS tests.  This mean student does a good job avoiding substances and is motivated, but not quite motivated to do their homework regularly.  This adds up to a pedestrian GPA of 81.7.

Female Variable	Obs	Mean	Std. Dev.	Min	Max
GPA	2026	82.97088	6.077696	55.8333	97.8125
Spring Att	2026	0.951712	0.063191	0.247191	1
Hispanic	2026	0.692991	0.461367	0	1
Native Am	2026	0.001481	0.038462	0	1
Asian	2026	0.01382	0.116774	0	1
Black	2026	0.244817	0.430085	0	1
White	2026	0.04689	0.211456	0	1
LEP	2026	0.124383	0.3301	0	1
Exited	2026	0.455578	0.498146	0	1
Reading	2026	2362.658	199.7468	1163	2734
Math	2026	2235.4	196.0376	1239	2796
Social Study	2026	2305.097	217.2718	1335	2805
Science	2026	2162.556	221.9311	835	3020
Does Hmwk	2026	1.863771	0.864213	0	3
Avoids Alc	2026	2.446199	0.906121	0	3
Motivated	2026	2.291708	0.822102	0	3

Male

Variable	Obs	Mean	Std. Dev.	Min	Max
GPA	2004	80.46622	6.205453	59	97.3889
Spring Att	2004	0.951793	0.064511	0.359551	1
Hispanic	2004	0.718064	0.450055	0	1
Native Am	2004	0.002495	0.0499	0	1
Asian	2004	0.012974	0.113191	0	1
Black	2004	0.221557	0.415398	0	1
White	2004	0.04491	0.207159	0	1
LEP	2004	0.147206	0.354399	0	1
Exited	2004	0.462575	0.498722	0	1
Reading	2004	2329.022	199.7518	1163	2734
Math	2004	2237.306	198.6696	1239	2796
Social Study	2004	2316.586	236.9962	1335	2805
Science	2004	2182.386	243.8858	835	3020
Does Hmwk	2004	1.652695	0.891037	0	3
Avoids Alc	2004	2.259481	1.027109	0	3
Motivated	2004	2.187625	0.851844	0	3

Females have lower mean grades on the TAKS test across the board, but a lower variation; the opposite is true for GPA.  Females also average more of the three assets analyzed here with a similar variation.  Demographics and attendance are both quite similar as well.

Results

            The original incarnation of this model was an ordinary least-squares estimate with GPA as a dependent variable, and the others as independent variables.

Ordinary Least-squares Estimates
R-squared      =    0.5037
Rbar-squared   =    0.5019
sigma^2        =   19.5657
Durbin-Watson  =    2.0052
Nobs, Nvars    =   4030,    16

***************************************************************

Variable         Coefficient      t-statistic    t-probability 
Constant          15.608814        11.229595        0.000000
Spring Att        33.011007        28.816670        0.000000
Female              2.126967        14.941337         0.000000
Native Am        0.156757         0.099524          0.920727
Asian                2.201661         3.588635           0.000336
Black               -0.856488        -3.541753          0.000402
White                1.314982         3.477649          0.000511
Limited Eng      0.030157         0.106476         0.915210
Exited LEP        0.099791         0.461501         0.644464
Reading             0.002253         4.631903          0.000004
Math                  0.005589        11.089776         0.000000
Science              0.002795         5.849540          0.000000
Social Study      0.002294         5.113665          0.000000
Does Hmwk      1.361978        15.041336         0.000000
Avoids Alc        0.354113         4.557107         0.000005
Motivated          0.580438         6.194243         0.000000

            In this model, full attendance goes a long way toward improving grades.  The coefficients on the TAKS scores look quite small, but actual TAKS scores are large numbers, so passing the Reading TAKS results in a 5.4072 improvement in GPA compared to not taking the TAKS.  The Does Hmwk, Avoids Alc and Motivated coefficients are all positive and significant, though small; this confirms the results Scales et al. found (2000).  My previous research has indicated that there are large marginal improvements to be had in graduation rate among students in the low 70’s range; perhaps improving these assets a little could go a long way toward improving graduation rates among these types of students.

            One thing these models leave out it the social component.  Perhaps a student’s friends have a significant effect on his grade.   In this case, the student’s ten nearest geographical neighbors substitute for friends.

LM error tests for spatial correlation in residuals

LM value                  12.01322186
Marginal Probability       0.00052824
chi(1) .01 value          17.61100000

LM lag tests for spatial autocorrelation in the dependent variable                  

LM value                   5.5815
Marginal Probability        0.0182
chi(1) .01 value          17.61100000

LR tests for spatial correlation in residuals                 

LR value                  11.07305971
Marginal Probability       0.00087591
chi-squared(1) value       6.63500000

The LM error test is a Lagrangian Multiplier test; if it returns a significant value, a model with a spatial error correction is required.  Similarly, LM lag tests for the necessity of spatial lags.  Finally, the LR test is a log likelihood ratio test between a spatial error model and ordinary least-squares; a significant value from this test represents the need for a model that combines errors and lags.  LeSage and Pace term this model the spatial Durbin model (SDM), noting that  it “subsumes the spatial error model (SEM) and the spatial autoregressive model (SAR)” (2009, 46).  However, the spatial Durbin model did not yield useful results; the spatial lag term was highly insignificant and it biased the rest of the model.  A spatial error model yielded both better results and a higher log likelihood.

In this model, X is the same matrix of independent variables used in the least-squares model,  is a matrix of coefficients on the independent variables,  is the spatial coefficient, W is the spatial weights matrix formed by row standardizing a matrix of the inverse distance of the ten nearest neighbors and  is the error term.

Spatial error Model Estimates
R-squared       =    0.5055  
Rbar-squared    =    0.5037  
sigma^2         =   19.4163  
log-likelihood  =       -10300.746 
Nobs, Nvars     =   4030,    16

***************************************************************
Variable         Coefficient  Asymptot t-stat    z-probability
Constant          15.687526        24.354044         0.000000
Spring Att        32.827916        81.225720         0.000000
Female               2.122448        15.000892         0.000000
Native Am         0.037254         0.023766         0.981040
Asian                  2.203241         3.584278         0.000338
Black                -0.831360        -3.371972         0.000746
White                 1.265046         3.321480         0.000895
Limited Eng      -0.030236        -0.106922         0.914851
Exited LEP         0.085797         0.408859         0.682643
Reading              0.002154         4.573428         0.000005
Math                  0.005730        11.451815         0.000000
Science              0.002814         5.992110         0.000000
Social Study       0.002280         5.104536         0.000000
Does Hmwk       1.360496        15.252849         0.000000
Avoids Alc         0.362467         4.761261         0.000002
Motivated           0.582799         6.324687         0.000000
lambda               0.107000        12.395513         0.000000

The first result seems to be not much of a result at all.   does not seem to be much higher, and the coefficients are not changed much; if the spatial correlation is in the error term, limited change in coefficients is expected.  The t statistics have changed, most notably Spring Att, but the significance of the coefficients has not changed.  Given the conflicting reports from the preliminary tests, perhaps there is more here than these models can parse out on their own.

            The next few models were obtained by splitting the dataset by gender, starting with females.

LM error tests for spatial correlation in residuals

LM value                        2.52433590
Marginal Probability       0.11210197
chi(1) .01 value               6.63500000

LM lag tests for spatial autocorrelation                    

LM value                          5.5143
Marginal Probability        0.0189
chi(1) .01 value               17.61100000

LR tests for spatial correlation in residuals                     

LR value                          2.52691063
Marginal Probability       0.11191915
chi-squared(1) value       6.63500000

These tests are much less contradictory than the tests with combined genders; Wald’s tests and Moran’s I both confirm the results presented here.  Girls clearly require a model with a spatial lag.

The difference between this spatial autoregressive model and the earlier spatial error model is that this one lacks a spatial term in the error, replaced by , a spatial lag term.

Spatial autoregressive Model Estimates
R-squared          =    0.5193
Rbar-squared       =    0.5160
sigma^2            =   17.7196
Nobs, Nvars        =   2026,    15
log-likelihood     =       -5085.2068
# of iterations    =     13  
min and max rho    =   -1.0000,   1.0000
total time in secs =    1.0340
time for lndet     =    0.1490
time for t-stats   =    0.0600
time for x-impacts =    0.7890
# draws  x-impacts =      1000  
Pace and Barry, 1999 MC lndet approximation used
order for MC appr  =     50 
iter  for MC appr  =     30 

***************************************************************

Variable	Coefficient		Asymptot t-stat		z-probability
constant	8.729415		4.545145		0.000005
Spring Att	34.751276		22.194922		0
Native Am	2.862102		1.169645		0.242144
Asian	2.795347		3.424698		0.000615
Black	-1.039033		-3.241972		0.001187
White	1.189606		2.349351		0.018806
LEP	0.580998		1.495408		0.134808
Exited	-0.105703		-0.366219		0.714201
Reading	0.002676		4.033385		0.000055
Math	0.005498		8.045126		0
Science	0.002128		3.103902		0.00191
Social Study	0.003052		4.80879		0.000002
Does Hmwk	1.36035		10.846303		0
Avoids Alc	0.371607		3.311854		0.000927
Motivated	0.642385		4.98659		0.000001
rho	0.073974		57.746766		0
Direct	Coefficient	t-stat		t-prob		lower 01	upper 99
Spring Att	34.72403	22.27542		0		30.54179	38.71908
Native Am	2.861258	1.154805		0.248306		-4.58458	9.908883
Asian	2.785587	3.489691		0.000494		0.662066	4.650815
Black	-1.028367	-3.3248		0.000901		-1.88678	-0.27646
White	1.194016	2.323855		0.020232		-0.19186	2.634852
Limited Eng	0.582436	1.521414		0.128312		-0.43645	1.537326
Exited	-0.093548	-0.33132		0.740436		-0.80506	0.617553
Reading	0.002694	3.952995		0.00008		0.000736	0.004338
Math	0.005487	7.582869		0		0.003432	0.007138
Science	0.00213	3.122681		0.001817		0.000297	0.003947
Social Study	0.003049	4.895223		0.000001		0.001425	0.004531
Does Hmwk	1.358165	10.83116		0		1.041243	1.693066
Avoids Alc	0.368106	3.081119		0.00209		0.041072	0.657877
Motivated	0.641236	5.090491		0		0.295233	0.93549
Indirect	Coefficient	t-stat		t-prob		lower 01	upper 99
Spring Att	2.833569	1.639035		0.101361		-1.00872	7.770335
Native Am	0.235155	0.842495		0.399611		-0.40765	1.246565
Asian	0.228718	1.389939		0.1647		-0.08654	0.797703
Black	-0.081068	-1.50239		0.133152		-0.26759	0.029558
White	0.094726	1.246272		0.212808		-0.04857	0.368586
Limited Eng	0.047238	0.98591		0.324295		-0.05365	0.211492
Exited	-0.00685	-0.2435		0.807645		-0.09158	0.087753
Reading	0.000221	1.46769		0.142344		-8.7E-05	0.000711
Math	0.000449	1.593311		0.111247		-0.00015	0.001272
Science	0.000174	1.387211		0.16553		-7.4E-05	0.000571
Social Study	0.000247	1.548003		0.121778		-9.9E-05	0.000739
Does Hmwk	0.110734	1.617528		0.10592		-0.03945	0.304966
Avoids Alc	0.029519	1.428026		0.153439		-0.01392	0.096282
Motivated	0.052658	1.50034		0.133682		-0.01695	0.173835
Total	Coefficient	t-stat		t-prob		lower 01	upper 99
Spring Att	37.5576	15.5429		0		31.59877	44.03484
Native Am	3.096412	1.150815		0.249944		-4.85865	10.48798
Asian	3.014305	3.41973		0.000639		0.706157	5.112976
Black	-1.109435	-3.36324		0.000785		-2.07971	-0.31887
White	1.288743	2.321587		0.020354		-0.21912	2.821958
Limited Eng	0.629673	1.510178		0.131154		-0.49214	1.746009
Exited	-0.100397	-0.32746		0.743354		-0.84197	0.677741
Reading	0.002915	3.866509		0.000114		0.00081	0.004672
Math	0.005936	7.127401		0		0.003541	0.007825
Science	0.002304	3.082612		0.00208		0.00032	0.004242
Social Study	0.003295	4.823438		0.000002		0.001563	0.00493
Does Hmwk	1.468899	9.733525		0		1.105742	1.847141
Avoids Alc	0.397625	3.082265		0.002082		0.04418	0.712712
Motivated	0.693893	4.89533		0.000001		0.324606	1.064411

Since this model has a spatial lag, it incorporates both the effect of the subject’s characteristics on the dependent variable, but also the characteristics of the subject’s neighbors.  Therefore, the direct effect of attendance can be interpreted as the effect attendance has on one’s grade, and the indirect effect is the effect of one’s classmates’ attendance on one’s grade.  Although the rho is significant, none of the indirect effects are, and their potential effects seem to be miniscule anyway.  In any case, the coefficients on the DAP items are a bit larger than in the combined model.  The model for boys is next.

            Whether inherent or conditioned, there are striking differences in modeling boys.

LM error tests for spatial correlation in residuals

LM value                        4.6380
Marginal Probability       0.0313
chi(1) .01 value               17.6110

LM lag tests for spatial autocorrelation                    

LM value                          0.1085
Marginal Probability        0.7418
chi(1) .01 value               17.6110

LR tests for spatial correlation in residuals                     

LR value                          4.09937645
Marginal Probability       0.04289903
chi-squared(1) value       6.63500000

Quite the opposite of the girls, the boys seems to point directly toward a spatial error model; once again, these results are backed up by further tests not printed here. 

Spatial error Model Estimates

R-squared       =    0.4556  
Rbar-squared    =    0.4518  
sigma^2         =   20.9518  
log-likelihood  =       -5198.1853 
Nobs, Nvars     =   2004,    15

***************************************************************

Variable	Coefficient	Asymptot t-stat	z-probability
Constant	18.23194	6.171766	0
Spring Att	31.313473	11.694191	0
Native Am	-1.674218	-0.810512	0.417646
Asian	1.653483	1.791058	0.073284
Black	-0.471267	-1.265921	0.205541
White	1.385997	2.420828	0.015485
LEP	-0.430574	-1.03465	0.300832
Exited	0.363712	1.041399	0.29769
Reading	0.001759	2.378652	0.017376
Math	0.005835	7.844373	0
Science	0.003387	4.972903	0.000001
Social Study	0.001591	2.506893	0.01218
Does Hmwk	1.338494	10.210851	0
Avoids Alc	0.35403	3.139855	0.00169
Motivated	0.506962	3.580966	0.000342
Lambda	0.091	4.786	0.000002

As expected, the  is small, but significant.   has decreased by .0499; perhaps this is evidence that these boys try not to care what their peers do.  Furthermore, nearly all of the demographic binaries have been rendered insignificant by this model, when they were all significant except Native Am in the combined model.  Although they remained significant, the DAP variables have smaller coefficients than the combined model; either girls get more mileage out of their assets or boys are not answering the question in the same manner as girls.  This is particularly perplexing in light of the fact that girls have lower variance in responses.  The spatial model seems to have much less to say about boys all around than girls.

Conclusion

            The explanatory power of the Developmental Assets Profile remains small, but significant; the explanatory power of the spatial models is complicated by the apparent disconnect between boys and girls.  To further elaborate on this, the results of a spatial autocorrelation model indicated a significant spatial lag term and an insignificant spatial error term for girls, but both were significant for boys, although with much less effect.  There does seem to be evidence in the literature of girls being more reliant on their peers than boys.  Frydenberg and Lewis find that, “The study supports the assertion that boys turn to sport and physical relaxation whilst girls turn to others and make more of connectedness and relationships in coping” (1993, 264).  In any case, it seems certain that any further study of the scholarly endeavors of adolescents should separate boys and girls, if nothing else, to see the effects.

           

Works Cited

Frydenberg, Erica and Ramon Lewis. 1993. “Boys play sport and girls turn to others: age, gender and ethnicity as determinants of coping.” Journal of Adolescence 16, no 3: 253-66.

LeSage, James and R. Kelly Pace. 2009. Introduction to Spatial Econometrics. Boca Raton: Chapman & Hall/CRC.

Manzi, Jim. 2011. “A Moment of Communion with Paul Krugman.” The American Scene, April 29. http://theamericanscene.com/2011/04/29/a-moment-of-communion-with-paul-krugman.

Roehlkepartain, Eugene C., Peter L. Benson and Arturo Sesma. 2003. Signs of Progress in Putting Children First. Minneapolis: Search Institute.

Scales, Peter C, Peter L. Benson, Nancy Leffert and Dale A. Blyth. 2000. “Contribution of Developmental Assets to the Prediction of Thriving Among Adolescents.” Applied Developmental Science 4, no 1:27-46.

Search Institute. 2011. “Developmental Assets.” http://www.search-institute.org/developmental-assets.

Stevens, Holly, and Kevin Wilkerson. 2010. “The Developmental Assets and ASCA’s National Standards: A Crosswalk Review.” Professional School Counseling 13, no. 4: 227-233. Academic Search Complete, EBSCOhost (accessed July 26, 2011).

Tilley, Carol L. 2011. “Developmental Assets.” School Library Monthly 27, no. 7: 41-44.