Spatial Beta-Convergence Forecasting Models:

# Spatial Beta-Convergence Forecasting Models:
## Evidence from Municipal Homicide Rates in Colombia
### Felipe Santos-Marquez Master’s student (M2) Graduate School of International Development Nagoya University, JAPAN
### Prepared for the 57th Annual Conference of JSLA [ slides available at: <a href="https://felipe-santos.rbind.io/" class="uri">https://felipe-santos.rbind.io/</a> & <a href="https://jsla-2020.netlify.app/" class="uri">https://jsla-2020.netlify.app/</a>]

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## Motivation:

- Policy development and management should consider not only current challenges but also **the potential challenges in the near future**.

- Persistent income differences, differences in health indicators and in "general" regional inequality in Colombia.

- Scarce academic literature on the **forecasting of crime** at the municipal level in Colombia.

## Research Objective:
  
- **To build a forecasting model that can predict homicide rates 4 years into the future**

- Analyze the spatial distribution of the forecasted homicide rates in 2022

## Methods:

- Exponential smoothing
- ARIMA models
- STAR models
- **Classical convergence framework (Barro and Sala-i-Martin 1992)**

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# Main Results:

1. **Beta and Spatial-beta forecasting models** are more accurate than other conventional forecasting models.

2. **768 out of 1120** Colombian municipalities are expected to reach the 2022 national SDG target. (**23 extra municipalities when compared to 2018**)

3. **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.

4. **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.

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# Outline of this presentation

1. **Data description** administrative levels in Colombia and homicide rates.

3. **Forecasting models**

- Exponential smoothing, ARIMA and STAR models

2. **Beta convergence models**

- Beta convergence and forecasting

4. **Main results and prediction:**

- Best models and forecasted data
  
5. **Policy discussion**

- The Colombian National Development Plan
 2018-22  
  
5. **Concluding Remarks**
  
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# Colombian administrative levels and the spatial distribution of crime

![](figs/col_mun.png)
]

.pull-right[
## Crime
![](figs/01_eb_mr_2018.png)
**(In Japan in 2018 about 0.02 per 10.000 people)**]

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# Data:

- 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**).

- Data is aggregated at the municipal and departmental levels.

- Population census and estimates for states and municipalities (data from the same dataset).

- Raw rates computed `$$raw\space rates = crimes / population$$`

- Non crime rates  computed　
   `$$NCR= 10000- raw\ rate * 10000$$`
- **Survival rates** are chosen because positively defined variables are a **standard** in the convergence literature.

- The variables are transformed from raw rates to Empirical Bayes rates in order to control for variance instability.

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# Global trend of crime

<img src="figs/ebnmr.png" style="width: 100%" />
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# Exponential smoothing methods

In general terms, these methods use a **weighted
average of past observations** in order to draw a forecast of future observations

`$$\hat{y}_{T+1 \mid T}=\alpha y_{T}+\alpha(1-\alpha) y_{T-1}+\alpha(1-\alpha)^{2} y_{T-2}+\cdots$$`

# ARIMA models

An autoregressive model of order p

`$$y_{t}=c+\phi_{1} y_{t-1}+\phi_{2} y_{t-2}+\cdots+\phi_{p} y_{t-p}+\varepsilon_{t}$$`
A moving average model considers past forecast
errors in a linear model:

`$$y_{t}=c+\varepsilon_{t}+\theta_{1} \varepsilon_{t-1}+\theta_{2} \varepsilon_{t-2}+\cdots+\theta_{q} \varepsilon_{t-q}$$`
An autoregressive model, a moving average model and the differentiating of the data can be combined into a non-seasonal ARIMA model:

`$$y_{t}^{\prime}=c+\phi_{1} y_{t-1}^{\prime}+\cdots+\phi_{p} y_{t-p}^{\prime}+\theta_{1} \varepsilon_{t-1}+\cdots \theta_{q} \varepsilon_{t-q}+\varepsilon_{t}$$`
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# Space time Autoregressive models (STAR)

`$$y_{i t}=\sum_{k=1}^{p} \sum_{l=0}^{\lambda_{k}} \phi_{k l} \sum_{j=1}^{N} w_{i j}^{(l)} y_{j t-k}+a_{i t}$$`

- `$y_it$` is the variable of interest at location `$i$` during time `$t$`
- `$a_{it}$` is the random normal error associated with that location and time. 
- `$y_{jt-k}$` are the
time lagged variables at location `$j$` 
- `$w_{i j}^{(l)}$` is the distance weight matrix of
order (l).

These models have been **successfully used to forecast crime data in the United States**. 
See Shoesmith GL (2013) Space-time autoregressive models and forecasting
national, regional and state crime rates.

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# **Beta convergence and forecasting** (catch-up process)

When studying regional data sets, Barro and Sala-i
Martin (1995) proposes the following equation:

`$$\log \frac{y_{i T}}{y_{i 0}}=\alpha-\left[1-e^{-\beta T}\right] \cdot \log \left(y_{i 0}\right)+w_{i, 0 T}$$`

Solving the equation for `$y_{i T}$`: 
`$$\log \left(y_{i T}\right)=\alpha+e^{-\beta T} \cdot \log \left(y_{i 0}\right)+w_{i, 0 T}$$`
Therefore, a plausible 4-year ahead forecast is:

`$$\log \left(\hat{y}_{i(t+4) \mid t}\right)=\hat{\alpha}_{t+4 \mid t}+\hat{\beta}_{t+4 \mid t} \cdot \log \left(y_{i t}\right)+w_{i, t}$$`

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#**Spatial Beta convergence model**

- Spatial beta convergence models have been used to model the dynamics of crime **(Santos-Marquez and Mendez, 2020)**
- In order to simplify the computations instead of a SAR
(spatial autoregressive model) or a SEM (spatial error model), **an SLX model
(Spatially Lagged X) is proposed to forecast homicide**.
- the spatial equivalent
of the equation in the previous slide is:

`$$\log \frac{y_{i T}}{y_{i 0}}=\alpha-\left[1-e^{-\beta T}\right] \cdot \log \left(y_{i 0}\right)+\theta W \cdot \log \left(y_{i 0}\right)+\epsilon_{i, 0 T}$$`
Therefore, a plausible 4-year ahead forecast is:

`$$\log \left(\hat{y}_{i(t+4) \mid t}\right)=\hat{\alpha}_{t+4 \mid t}+\hat{\beta}_{t+4 \mid t} \cdot \log \left(y_{i t}\right)+\hat{\theta}_{t+4 \mid t} W \cdot \log \left(y_{i t}\right)+\epsilon_{i, t}$$`
Where three coefficients need to be computed: `$\alpha$`, `$\beta$` and `$\theta$`.

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# Time series forecast accuracy and cross-validation

#How can the best forecasting model be chosen?

`$$\begin{aligned}
&\text { Mean absolute error: } \operatorname{MAE}=\operatorname{mean}\left(\left|e_{t}\right|\right)\\
&\text { Root mean squared error: RMSE }=\sqrt{\operatorname{mean}\left(e_{t}^{2}\right)}
\end{aligned}$$`
# Cross-validation of forecasts
<img src="figs/traintest.png" style="width: 100%" />
<img src="figs/traintest2.png" style="width: 100%" />

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# The prediction of the coefficients of the beta models

Linear regressions are used to compute the coefficients in:

`$$\log \left(\hat{y}_{i(t+4) \mid t}\right)=\hat{\alpha}_{t+4 \mid t}+\hat{\beta}_{t+4 \mid t} \cdot \log \left(y_{i t}\right)+\hat{\theta}_{t+4 \mid t} W \cdot \log \left(y_{i t}\right)+\epsilon_{i, t}$$`
`$$\log \left(\hat{y}_{i(t+4) \mid t}\right)=\hat{\alpha}_{t+4 \mid t}+\hat{\beta}_{t+4 \mid t} \cdot \log \left(y_{i t}\right)+w_{i, t}$$`
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# Cross-validation results (Beta and spatial Beta)

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# Forecasting homicide rates for 2022

- 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.
- In terms of homicides the baseline for 2018 is 2.58 (per 10.000 people) and
the **2022 target is 2.32 per 10.000 people (nmr= 9997.68)**

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# Forecasting homicide rates for 2022
# (2022 Target = 9997.68)
<img src="figs/forecast.png" style="width: 100%" />

**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)**
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## The spatial distribution of the forecasted crime
**municipalities in dark green are lagging behind**

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## The spatial distribution of the forecasted crime
**municipalities in dark green are lagging behind**

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# Measuring the polarization of regional crime 
## Local join count statistic (Anselin (1995), Anselin and Li (2019))
<img src="figs/ljcs1822.png" style="width: 100%" />

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# Policy discussion

- Spatial spillovers from neighbors can have **both positive and negative effects**.

- It could be more appropriate for the formulation of development plans to have targets at the **state level or regional level**.

- For example, there is a clear negative spill-over in the municipalities in **the pacific region (in purple)**.

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# Concluding Remarks

## Uplifting results "on average" :

- The dispersion of non-crime (crime) rates at the has decreased level **has decreased**.

- It was found that the forecasted mean in 2022
is still below the 2022 target.

- Nevertheless, major improvements are foreseen
as **23 extra municipalities are expected to reach the SDG homicide target by 2022**. This will bring the total of municipalities that have accomplished the
target to 768 out of 1120 in 2022.

## The Role of Space

- There are **positive and negative spillovers**.

- **Policy should be localized in order to reverse the creation of crime clusters**.

## Further Research

- **Other determinants** may help to improve the forecasting accuracy of the beta models.
- 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)
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# Thank you very much for your attention

You can find more about my research outputs on my website https://felipe-santos.rbind.io

If you are interested in our research please check the QuaRCS lab website

https://quarcs-lab.org/

**Quantitative Regional and Computational Science Lab**