This exercise aims to forecast the Colombian GDP. To achieve this, we will use two sources of information , 1. quarterly information of the exports supported by procolombia and quarterly information about colombia’s GDP. In order to achieve this, We will use models such as ETS, ARIMA and VAR in order to identify the best-fitting model to later perform forecasting exercises.

The data

Colombia’s GDP

The National Department of Statistics (DANE) reports two series of real Gross Domestic Product (GDP). The first refers to the value of the real product with a base period of 2015. The second is the real GDP corrected for seasonal factors and for calendar effects. For the analysis of the state of the economy and growth forecasts, we will use this last series (real GDP corrected for seasonal factors and business days)

Procolombia’s exports

The second source of information we use are the exports supported by ProColombia. But first, what is Procolombia?

Procolombia is an entity of the Colombian state that seeks to support exports of non-traditional goods or services, promote the image of the country and attract foreign investment. This company receives a budget from the state, which must be managed efficiently in order to promote the Colombian economy from the areas mentioned above. Specifically, Procolombia supports (1) exports in the sectors of agribusiness, industries 4.0, metalworking, fashion and chemical systems, and life sciences, (2) foreign investment according to the country of origin and the economic sector, and (3) tourism. from the areas of vacation tourism and corporate tourism. The company has commercial offices in more than 20 countries around the world, which support the objectives of exports, investment and tourism from abroad.

The data that we will use, give us information about monthly exports measured in us dollars. We have information from january 2009 to december 2021.

Lets get a graph of the exports

As expected, we see that exports tend to grow over time. We also see some peaks that show us seasonality. Following this idea we will start with a regression only controlling for this two variables.

The models

1. ETS model - Error, Trend and Seasonal component

The exponential smoothing method is a way to forecast a series in a given period. It estimates that the series will be equal to, for example, the average of historical values for a given period, giving greater weight to the closest values in time. In addition, it takes into account the current forecast error in subsequent forecasts.

## ETS(M,A,N) 
## 
## Call:
##  ets(y = GDP) 
## 
##   Smoothing parameters:
##     alpha = 0.8345 
##     beta  = 1e-04 
## 
##   Initial states:
##     l = 77626.3497 
##     b = 2761.1318 
## 
##   sigma:  0.0328
## 
##      AIC     AICc      BIC 
## 1464.601 1465.569 1475.699

The ETS model that give us the lowest AIC value is an ETS(M,A,N) which means that the error type is multiplicative and the trend type is additive. As expected, the season does not need decomposition since the data we are using takes that into account.

Following this results, we will use a model with multiplicative decomposition for the error and additive for the variance but no decomposition for seasonality to predict the future values of the series.

##                    ME    RMSE      MAE       MPE     MAPE      MASE      ACF1
## Training set 920.9457 8218.59 3503.934 0.2879688 1.676985 0.2277208 0.1059353

2. ARIMA model

An autoregressive integrated moving average (ARIMA) model is a statistical model that uses variances and regressions of statistical data to find patterns for a prediction into the future. In this model we are taking into account 3 aspects; AR, MA and the I component.

## Series: GDP 
## ARIMA(0,1,0) with drift 
## 
## Coefficients:
##          drift
##       3575.575
## s.e.  1000.291
## 
## sigma^2 = 68054710:  log likelihood = -698.77
## AIC=1401.53   AICc=1401.72   BIC=1405.94
## Warning in plot.Arima(ARIMA1): No roots to plot

As we can see, we got an ARIMA(0,1,0) This means that we only have to differentiate one time in order to get a stationary series.

## Series: GDP 
## ARIMA(0,1,0) 
## 
## sigma^2 = 79823713:  log likelihood = -704.61
## AIC=1411.22   AICc=1411.29   BIC=1413.43

##                    ME     RMSE      MAE      MPE    MAPE      MASE        ACF1
## Training set 3524.187 8868.474 5105.621 1.943172 2.67101 0.3318145 -0.02261219

3. Vector Autoregressive

A VAR is a simultaneous equations model formed by a system of equations reduced without restriction. We will use the vector autoregressive (VAR) model to characterize the simultaneous interactions between the exports of ProColombia and the Colombian GDP.

## AIC(n)  HQ(n)  SC(n) FPE(n) 
##      4      4      4      4
## 
## VAR Estimation Results:
## ======================= 
## 
## Estimated coefficients for equation GDP: 
## ======================================== 
## Call:
## GDP = GDP.l1 + EXPORT1.l1 + GDP.l2 + EXPORT1.l2 + GDP.l3 + EXPORT1.l3 + GDP.l4 + EXPORT1.l4 + const 
## 
##        GDP.l1    EXPORT1.l1        GDP.l2    EXPORT1.l2        GDP.l3 
##  9.184849e-01 -3.501472e-06  6.522598e-02  7.148131e-06 -1.388338e-01 
##    EXPORT1.l3        GDP.l4    EXPORT1.l4         const 
##  8.397840e-06  6.788083e-02  5.709185e-07  1.336762e+04 
## 
## 
## Estimated coefficients for equation EXPORT1: 
## ============================================ 
## Call:
## EXPORT1 = GDP.l1 + EXPORT1.l1 + GDP.l2 + EXPORT1.l2 + GDP.l3 + EXPORT1.l3 + GDP.l4 + EXPORT1.l4 + const 
## 
##        GDP.l1    EXPORT1.l1        GDP.l2    EXPORT1.l2        GDP.l3 
##  4.562427e+03  5.368368e-03 -3.731246e+03 -2.048899e-01  1.246175e+04 
##    EXPORT1.l3        GDP.l4    EXPORT1.l4         const 
##  9.741074e-02 -9.743765e+03  8.301648e-01 -4.652709e+08

In both graphs we see a significant lag in 4.

Looking at the 3 graphs of the residuals we see that the VAR model is the only one with a significant lag. In addition to this, if we choose a model guided by the RMSE, we would choose the ETS(M,A,N) model since it has a lower statistic.