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.

Outline of this presentation

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

2. Forecasting models

   • Exponential smoothing, ARIMA and STAR models

3. 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

6. Concluding Remarks

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:

$$y_{t}^{\prime}=c+\phi_{1} y_{t-1}^{\prime}+\cdots+\phi_{p} y_{t-p}^{\prime}+\theta_{1} \varepsilon_{t-1}+\cdots \theta_{q} \varepsilon_{t-q}+\varepsilon_{t}$$

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.