Sugar Cane Production in Puerto Rico, 1958/59-1973/74: A Comparison of Four Model Specifications for Describing Small Heterogeneous Space-Time Datasets. by Daniel A. Griffith Ashbel Smith Professor of Geospatial Information Sciences. ABSTRACT. - PowerPoint PPT Presentation
Production in Production in Puerto Rico, Puerto Rico, 1958/59-1973/74: A 1958/59-1973/74: A Comparison of Four Comparison of Four Model Model Specifications for Specifications for Describing Small Describing Small Heterogeneous Heterogeneous Space-Time Space-Time Datasets Datasets by by Daniel A. Griffith Daniel A. Griffith Ashbel Smith Ashbel Smith Professor of Professor of Geospatial Geospatial Information Sciences Information Sciences
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Sugar Cane Production Sugar Cane Production in Puerto Rico, 1958/59-in Puerto Rico, 1958/59-1973/74: A Comparison 1973/74: A Comparison of Four Model of Four Model Specifications for Specifications for Describing Small Describing Small Heterogeneous Space-Heterogeneous Space-Time DatasetsTime Datasets
bybyDaniel A. GriffithDaniel A. Griffith
Ashbel Smith Professor Ashbel Smith Professor of Geospatial Information of Geospatial Information
ABSTRACTABSTRACTResearchers increasingly are accounting for heterogeneity in their empirical analyses. When data form a short time series—too short to utilize an ARIMA model—a random effect term can be employed to account for serial correlation. When data also are georeferenced, forming a space-time dataset, a random effect term can be included that is spatially structured in order to account for spatial autocorrelation, too. But space-time heterogeneity can be accounted for in various ways, including specifications involving recently developed spatial filtering methodology. This paper summarizes comparisons of four model specifications—simple pooled space-time; sequential, comparative statics; temporally varying coefficients with a spatially unstructured random effect; and, temporally varying coefficients with a spatially structured random effect—illustrating implementations with annual sugar cane production data for the 73 municipalities of Puerto Rico during 1958/59-1973/74. Covariates whose importance is assessed include elevation and distance from the primate city.
Panel data versus space-time dataPanel data are a form of longitudinal data, and can
be a cross-section (i.e., the spatial dimension) of individuals (e.g., farms) that are surveyed periodically over a given time horizon.
With repeated observations of the same individuals, panel data permit a researcher to study the dynamics of change with short time series.
A main advantage of panel data: controlling for unobserved heterogeneity (the fundamental complication of non-experimental data collection)
BUT longitudinal data need not involve the same individuals: if a sample is not the same, observed changes also may result from sampling error
Spatial filtering
A given random variable can be decomposed into a spatial component and an aspatial component: impulse-response function approach (based upon the autoregressive model), Getis approach (based on the K function), eigenfunction spatial filtering approach.
The spatial component relates to spatial autocorrelation
High Peak district biomass index:ratio of remotely sensed data spectral
bands B3 and B4
Spatially autocorrelated Geographically random
Defining spatial autocorrelation
Auto: self
Correlation: degree of relative correspondence
Positive: similar values cluster together on a map
Negative: dissimilar valuesCluster together on a map
Spatial auto-correlation
n
)x(x
n
)y(y
)/nx)(xy1(y
n
1i
2i
n
1i
2i
n
1iii
n
)y(y
n
)y(y
c/)y)(yy(yc
n
1i
2i
n
1i
2i
n
1i
n
1i
n
1jij
n
1jjiij
from r to MC
Constructing eigenfunctions for filtering spatial autocorrelation out of georeferenced variables:
Moran Coefficient = (n/1T C1)x
YT(I – 11T/n)C (I – 11T/n)Y/ YT(I – 11T/n)Y
the eigenfunctions come from
(I – 11T/n)C (I – 11T/n)
Eigenvectors for spatial filter construction
The first eigenvector, say E1, is the set of real number numerical values that has the largest MC achievable by any set for the spatial arrangement defined by the geographic connectivity matrix C. The second eigenvector is the set of values that has the largest achievable MC by any set that is uncorrelated with E1. The third eigenvector is the third such set of values. And so on. This sequential construction of eigenvectors continues through En, the set of values that has the largest negative MC achievable by any set that is uncorrelated with the preceding (n-1) eigenvectors.
Useful citation
Random effects model
is a random observation effect (differences among individual observational units)
is a time-varying residual error (links to change over time)
The composite error term is the sum of the two.
) , f( εξXβY ξ
ε
Random effects model: normally distributed intercept term
• ~ N(0, ) and uncorrelated with covariates
• supports inference beyond the nonrandom sample analyzed
• simplest is where intercept is allowed to vary across areal units (repeated observations are individual time series)
• The random effect variable is integrated out (with numerical methods) of the likelihood fcn
• accounts for missing variables & within unit correlation (commonality across time periods)
2σξ
Sugar cane production in Puerto Rico• Began in the 1530s• Experienced a sharp decline during 1580-1650• Introduction of slave labor resulted in considerable
expansion during 1765-1823• By 1828, sugar exports were sizeable• Spanish monarchy discouraging expansion
throughout much of the 1800s• United States took possession of the island in
1899, fully developing the long-demanded railroad on the island and channeling considerable investment into sugar cane production, achieving maximum expansion in the 1920
• Production peaked around 1950
Island-wide time series
, )I0.81524(I1.68I7.231000
tons0.15060LN
1.68I7.231000
tons0.84940LN0.009811.68I7.23
1000
tonsN̂L
1tt2-t2t
1-t1t
tt
US intervention
1924 sugar cane railroad
Finally started by the Spanish Crown, but aggressively completed by US investors
Covariates of sugar cane production
elevationelevation distance from San Juandistance from San Juan
Spatial filters for space-time spatially structured random effects
1958/591958/59MC = 0.77, GR = 0.30MC = 0.77, GR = 0.30
1963/641963/64MC = 0.93, GR = 0.18MC = 0.93, GR = 0.18
1968/691968/69MC = 0.86, GR = 0.18MC = 0.86, GR = 0.18
1973/741973/74MC = 0.94, GR = 0.22MC = 0.94, GR = 0.22
(normally distributed) random intercept: areal unit specific across all years
feature Spatially unstructured Added to spatial structure
Sample mean -0.00864 -0.00665
Sample variance 1.63044 1.63797
Moran Coefficient (MC) 0.08672 0.08778
Geary Ratio (GR) 1.10196 1.09907
P(Shapiro-Wilk) < 0.0001 (4 lower tail outliers)
< 0.0001 (4 lower tail outliers)
Correlations with covariates
(-0.17873, 0.32086) (-0.17833, 0.32095)
Time series plots: intercept &
covariate binomial regression coefficients
interceptintercept
● simple pooled model■ comparative static model
♦ model with a spatially unstructured random effect ▲mixed model with spatially structured random effect
mean elevationmean elevation distancedistance
Time series plots: covariate
binomial regression coefficient
standard errors mean elevationmean elevation
distancedistance
● simple pooled model■ comparative static model♦ model with a spatially
unstructured random effect ▲ mixed model with spatially
structured random effect
Residual serial correlation
The random effects estimator approximates the degree of serial correlation (or its importance in the model), and hence allows the computation of corrected estimates.
The 73 residual Durbin-Watson statistics have a range of (0.140, 2.513), with a mean of 0.836 and a standard deviation of 0.546.
Determining significance here is complicated because of small T, inclusion of a random effects term, and variable SF eigenvecvtor #s
Graphical portrayal of DWs
GLM residuals (heuristic using 4 dfs lost)
0 – 0.74 1.93 – 2.08 3.26 – 4
0.74 – 1.93 2.07 – 3.26
undecided
positive serial correlation
Summary of results
STAR-binomial specification
ti,
73
1j 1tj,ijs
1ti,T
ielev
idisttti,
εarea
scwρ
area
scρ
elevβ
distβμarea
scLN
time
space
space-time
Pseud- & quasi-likelihood estimation
885.0R-pseudo
3.38ρ̂
7.75ρ̂
0.19β̂
0.05β̂
0.03T1.40μ̂
2
s
T
elev
dist
T
Extra binomial variation remains
1958/59 1565 473 378
1959/60 1503 403 321
1960/61 1561 368 326
1961/62 1543 303 279
1962/63 1490 261 252
1963/64 1544 281 271
1964/65 1467 263 254
1965/66 1523 266 254
1966/67 1610 302 273
1967/68 1601 329 290
1968/69 1259 320 218
1969/70 1273 310 250
1970/71 1149 339 181
1971/72 1164 384 207
1972/73 1146 427 158
1973/74 899 347 167
● pineapple production■ milk production♦ sugar cane production ▲ tobacco production
implications
1. spatial autocorrelation appears to be a source of part of the overdispersion
2. random effects (e.g., missing covariates) appear to be a source of part of the overdispersion
3. land use competition may be a source of part of the overdispersion
4. spatial filters for mean elevation and distance have six eigenvectors in common; of these, one is shared with most of the annual comparative static spatial filters, and two with most of the spatially structured random effect term spatial filters
5. the components of spatial autocorrelation in sugar cane production vary over time
6. a spatially unstructured random effect term that seeks to account for serial correlation in multiple short time series can better highlight latent spatial autocorrelation
7. a spatial filter can effectively structure a random effect term
8. failure to include a spatially structured random effect term can result in biased parameter estimates (largely because of the nonlinear nature of the model specification)
9. spatial and temporal autocorrelation interact in a complex way
