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9    <meta name="author" content="Felipe Santos-Marquez  Master’s student (M2)  Graduate School of International Development Nagoya University, JAPAN" />
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16class: center, middle, inverse, title-slide
17
18# Spatial Beta-Convergence Forecasting Models:
19## Evidence from Municipal Homicide Rates in Colombia
20### Felipe Santos-Marquez <br />Master’s student (M2) <br />Graduate School of International Development<br />Nagoya University, JAPAN
21### Prepared for the 57th Annual Conference of JSLA <br /><br />[ slides available at: <a href="https://felipe-santos.rbind.io/" class="uri">https://felipe-santos.rbind.io/</a> <br /> &amp; <a href="https://jsla-2020.netlify.app/" class="uri">https://jsla-2020.netlify.app/</a>]
22
23---
24
25
26
27
28
29## Motivation:
30
31- Policy development and management should consider not only current challenges but also **the potential challenges in the near future**.
32
33- Persistent income differences, differences in health indicators and in "general" regional inequality in Colombia.
34
35- Scarce academic literature on the **forecasting of crime** at the municipal level in Colombia. 
36
37
38## Research Objective:
39  
40- **To build a forecasting model that can predict homicide rates 4 years into the future**
41
42- Analyze the spatial distribution of the forecasted homicide rates in 2022
43
44## Methods:
45
46- Exponential smoothing
47- ARIMA models
48- STAR models
49- **Classical convergence framework (Barro and Sala-i-Martin 1992)**
50
51---
52
53class: middle
54
55# Main Results:
56
571. **Beta and Spatial-beta forecasting models** are more accurate than other conventional forecasting models.
58
592. **768 out of 1120** Colombian municipalities are expected to reach the 2022 national SDG target. (**23 extra municipalities when compared to 2018**)
60
613. **Increasing spatial polarization of the data**. higher crime areas are more likely to be surrounded by high crime areas and conversely for low-low cluster formation. 
62
634. **Crime spillovers are forecasted** for municipalities in the pacific region and positive spillovers are forecasted in the north and north-eastern parts of the country. 
64
65---
66
67class: middle
68
69# Outline of this presentation
70
711. **Data description** administrative levels in Colombia and homicide rates.
72
733. **Forecasting models** 
74
75  - Exponential smoothing, ARIMA and STAR models
76
772. **Beta convergence models**
78
79  - Beta convergence and forecasting
80
814. **Main results and prediction:**
82
83  - Best models and forecasted data
84  
855. **Policy discussion**
86
87  - The Colombian National Development Plan
88 2018-22  
89  
905. **Concluding Remarks**
91  
92---
93# Colombian administrative levels and the spatial distribution of crime 
94
95.pull-left[ ## States and Municipalities
96
97![](figs/col_mun.png)
98]
99
100.pull-right[
101## Crime
102![](figs/01_eb_mr_2018.png)
103**(In Japan in 2018 about 0.02 per 10.000 people)**]
104
105---
106
107# Data:
108
109- Total number of **homicides** and in Colombia per year from 2003 until 2018 (**data taken from Municipal data-set Universidad de los Andes, Bogota, Colombia**).
110
111- Data is aggregated at the municipal and departmental levels.
112
113- Population census and estimates for states and municipalities (data from the same dataset).
114
115- Raw rates computed `$$raw\space rates = crimes / population$$`
116
117- Non crime rates  computed　
118   `$$NCR= 10000- raw\ rate * 10000$$`
119- **Survival rates** are chosen because positively defined variables are a **standard** in the convergence literature.
120
121- The variables are transformed from raw rates to Empirical Bayes rates in order to control for variance instability.
122
123---
124class: center
125# Global trend of crime
126
127&lt;img src="figs/ebnmr.png" style="width: 100%" /&gt;
128---
129class: middle 
130

131# Exponential smoothing methods
132
133In general terms, these methods use a **weighted
134average of past observations** in order to draw a forecast of future observations
135
136`$$\hat{y}_{T+1 \mid T}=\alpha y_{T}+\alpha(1-\alpha) y_{T-1}+\alpha(1-\alpha)^{2} y_{T-2}+\cdots$$`
137
138# ARIMA models
139
140An autoregressive model of order p
141
142`$$y_{t}=c+\phi_{1} y_{t-1}+\phi_{2} y_{t-2}+\cdots+\phi_{p} y_{t-p}+\varepsilon_{t}$$`
143A moving average model considers past forecast
144errors in a linear model:
145
146`$$y_{t}=c+\varepsilon_{t}+\theta_{1} \varepsilon_{t-1}+\theta_{2} \varepsilon_{t-2}+\cdots+\theta_{q} \varepsilon_{t-q}$$`
147An autoregressive model, a moving average model and the differentiating of the data can be combined into a non-seasonal ARIMA model:
148
149`$$y_{t}^{\prime}=c+\phi_{1} y_{t-1}^{\prime}+\cdots+\phi_{p} y_{t-p}^{\prime}+\theta_{1} \varepsilon_{t-1}+\cdots \theta_{q} \varepsilon_{t-q}+\varepsilon_{t}$$`
150---
151class: middle
152# Space time Autoregressive models (STAR)
153
154`$$y_{i t}=\sum_{k=1}^{p} \sum_{l=0}^{\lambda_{k}} \phi_{k l} \sum_{j=1}^{N} w_{i j}^{(l)} y_{j t-k}+a_{i t}$$`
155
156- `\(y_it\)` is the variable of interest at location `\(i\)` during time `\(t\)`
157- `\(a_{it}\)` is the random normal error associated with that location and time. 
158- `\(y_{jt-k}\)` are the
159time lagged variables at location `\(j\)` 
160- `\(w_{i j}^{(l)}\)` is the distance weight matrix of
161order (l).
162
163
164These models have been **successfully used to forecast crime data in the United States**. 
165See Shoesmith GL (2013) Space-time autoregressive models and forecasting
166national, regional and state crime rates.
167
168---
169class: center
170
171# **Beta convergence and forecasting** (catch-up process)  
172
173<img src="figs/betaexpl.png" style="width: 35%" />
174
175When studying regional data sets, Barro and Sala-i
176Martin (1995) proposes the following equation:
177
178      
179`$$\log \frac{y_{i T}}{y_{i 0}}=\alpha-\left[1-e^{-\beta T}\right] \cdot \log \left(y_{i 0}\right)+w_{i, 0 T}$$`
180
181Solving the equation for `\(y_{i T}\)`: 
182`$$\log \left(y_{i T}\right)=\alpha+e^{-\beta T} \cdot \log \left(y_{i 0}\right)+w_{i, 0 T}$$`
183Therefore, a plausible 4-year ahead forecast is:
184
185`$$\log \left(\hat{y}_{i(t+4) \mid t}\right)=\hat{\alpha}_{t+4 \mid t}+\hat{\beta}_{t+4 \mid t} \cdot \log \left(y_{i t}\right)+w_{i, t}$$`
186
187---
188class: middle, justify
189
190#**Spatial Beta convergence model**
191
192- Spatial beta convergence models have been used to model the dynamics of crime **(Santos-Marquez and Mendez, 2020)**
193- In order to simplify the computations instead of a SAR
194(spatial autoregressive model) or a SEM (spatial error model), **an SLX model
195(Spatially Lagged X) is proposed to forecast homicide**.
196- the spatial equivalent
197of the equation in the previous slide is:
198
199`$$\log \frac{y_{i T}}{y_{i 0}}=\alpha-\left[1-e^{-\beta T}\right] \cdot \log \left(y_{i 0}\right)+\theta W \cdot \log \left(y_{i 0}\right)+\epsilon_{i, 0 T}$$`
200Therefore, a plausible 4-year ahead forecast is:
201
202`$$\log \left(\hat{y}_{i(t+4) \mid t}\right)=\hat{\alpha}_{t+4 \mid t}+\hat{\beta}_{t+4 \mid t} \cdot \log \left(y_{i t}\right)+\hat{\theta}_{t+4 \mid t} W \cdot \log \left(y_{i t}\right)+\epsilon_{i, t}$$`
203Where three coefficients need to be computed: `\(\alpha\)`, `\(\beta\)` and `\(\theta\)`.
204
205---
206class: middle, center
207
208# Time series forecast accuracy and cross-validation
209
210#How can the best forecasting model be chosen?
211
212`$$\begin{aligned}
213&\text { Mean absolute error: } \operatorname{MAE}=\operatorname{mean}\left(\left|e_{t}\right|\right)\\
214&\text { Root mean squared error: RMSE }=\sqrt{\operatorname{mean}\left(e_{t}^{2}\right)}
215\end{aligned}$$`
216# Cross-validation of forecasts
217<img src="figs/traintest.png" style="width: 100%" />
218<img src="figs/traintest2.png" style="width: 100%" />
219
220---
221
222class: center, middle
223
224# The prediction of the coefficients of the beta models
225
226<img src="figs/beta_forecast.png" style="width: 70%" />
227
228Linear regressions are used to compute the coefficients in:
229
230`$$\log \left(\hat{y}_{i(t+4) \mid t}\right)=\hat{\alpha}_{t+4 \mid t}+\hat{\beta}_{t+4 \mid t} \cdot \log \left(y_{i t}\right)+\hat{\theta}_{t+4 \mid t} W \cdot \log \left(y_{i t}\right)+\epsilon_{i, t}$$`
231`$$\log \left(\hat{y}_{i(t+4) \mid t}\right)=\hat{\alpha}_{t+4 \mid t}+\hat{\beta}_{t+4 \mid t} \cdot \log \left(y_{i t}\right)+w_{i, t}$$`
232---
233
234class: center, middle
235
236# Cross-validation results (Beta and spatial Beta)
237
238<img src="figs/cross.png" style="width: 100%" />
239
240---
241class: justify
242# Forecasting homicide rates for 2022
243
244- In the Colombian National Development Plan 2018-2022 DNP (2018) there are national targets for a variety of SDG related variables for the year 2022.
245- In terms of homicides the baseline for 2018 is 2.58 (per 10.000 people) and
246the **2022 target is 2.32 per 10.000 people (nmr= 9997.68)**
247
248
249<img src="figs/trend.png" style="width: 90%" />
250
251
252---
253class: center
254# Forecasting homicide rates for 2022
255# (2022 Target = 9997.68)
256<img src="figs/forecast.png" style="width: 100%" />
257
258**A noticeable error in this forecasting model is the fact that a total of 68 municipalities have a predicted non crime rate slightly above 10000 (WHICH IS PRACTICALLY IMPOSSIBLE)**
259---
260class: center
261## The spatial distribution of the forecasted crime
262**municipalities in dark green are lagging behind**
263
264<img src="figs/belo1822.png" style="width: 100%" />
265
266---
267class: center
268## The spatial distribution of the forecasted crime
269**municipalities in dark green are lagging behind**
270
271<img src="figs/belo1822cc.png" style="width: 100%" />
272
273---
274class: center
275# Measuring the polarization of regional crime 
276## Local join count statistic (Anselin (1995), Anselin and Li (2019))
277<img src="figs/ljcs1822.png" style="width: 100%" />
278
279---

280# Policy discussion 
281
282.pull-left[ 
283- Vertical and horizontal policy coordination, spillovers and borders.
284
285- Spatial spillovers from neighbors can have **both positive and negative effects**.
286
287- It could be more appropriate for the formulation of development plans to have targets at the **state level or regional level**.
288
289- For example, there is a clear negative spill-over in the municipalities in **the pacific region (in purple)**.
290
291]
292
293.pull-right[
294## Colombian natural regions
295![](figs/col_regions.png)
296]
297
298
299---
300
301# Concluding Remarks
302
303## Uplifting results "on average" :
304
305- The dispersion of non-crime (crime) rates at the has decreased level **has decreased**.
306
307- It was found that the forecasted mean in 2022
308is still below the 2022 target. 
309
310- Nevertheless, major improvements are foreseen
311as **23 extra municipalities are expected to reach the SDG homicide target by 2022**. This will bring the total of municipalities that have accomplished the
312target to 768 out of 1120 in 2022.
313
314## The Role of Space
315
316- There are **positive and negative spillovers**.
317
318- **Policy should be localized in order to reverse the creation of crime clusters**.
319
320## Further Research 
321
322- **Other determinants** may help to improve the forecasting accuracy of the beta models.
323- A **damped trend method** may help to correct for municipalities that are forecasted to have 0.0 crime rates (over 10.000 non crime rates)
324---
325class: center, middle
326
327# Thank you very much for your attention
328
329 You can find more about my research outputs on my website https://felipe-santos.rbind.io
330
331If you are interested in our research please check the  QuaRCS lab website <br />
332
333https://quarcs-lab.org/
334
335 <img src="figs/logo2.png" style="width: 30%" />
336
337**Quantitative Regional and Computational Science Lab**
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