Análisis interno: cadena de valor de la producción de fréjol (Michael Porter)

We explore the applicability of our algorithm to the EAGDP task by varying two dimensions. Firstly, we consider the calculation of our model fit (by recent MSE) on the monthly or quarterly series. This is such that we can predict the growth rate of the EAGDP after our model is fit, then choose to compare it directly to the disaggregated series or instead transform it back to a quarterly series and compare it against the actual values (though they are still under revision). Comparing on a monthly level allows for a direct comparison without a loss in information during transformation, however only the quarterly values are actual observations. Our second dimension is time, such that we restrict the data to end a certain number of months earlier. This allows us to monitor the change in the structure selected by our algorithm through time. We vary this by up to 3 and a half years, or 42 months.

Results are presented on two main areas. Firstly we review the models chosen by the algorithm in their primary endpoint, nowcasting efficiency. This is done by a comparison of estimated growth rates against competing models, the actual observed and the Eurostat t + 30 flash estimate produced 30 days after the quarter ends. Similarly a comparison of the squared errors is presented for clarity. Our

Data: Yt, Euro Area Gross Domestic Product Monthly for t = 1, . . . n ;

Data: Xt,j For j = 1, . . . , m indicator series over timepoints t = 1, . . . n ;

input : n∗, The amount of recent points to consider in MSE calculation ;

1 δ = ∞ − ,  > 0 The minimum MSE of the current stage ;ˆ

2 δ = ∞ The minimum MSE of fitted models in all previous stages ; 3 δ_i∗ = ∞ ∀i ∈ N The current MSE of fitted models ;

4 p = 1 The current amount of dynamic factors ;ˆ 5 while ˆδ ≤ δ and m > 0 do

6 for j in 1 : m do

7 Fit DFM using data {Y, X_j} with ˆp dynamic factors. ; 8 Calculate MSE δ_i∗ over n∗ recent points (Equation (6.6)) ;

9 end

10 Let ˆδ = min({δ_j∗ : j = 1, . . . , m}) ; 11 if ˆδ < δ then

12 Reset δ∗_j = ∞ ∀j ∈ N ; 13 Re-assign δ = ˆδ ;

14 Expand the forecasting dataset: {Y} = {Y, {X_j:δ∗ j=min(δ

∗₎}};

15 Remove that variable from the indicators dataset:

{X} = {X\{Xj:δ∗_j=min(δ∗)}};

16 else

17 if ˆp ≤ m then

18 Let ˆp = ˆp + 1 ;

19 for j in 1 : m do

20 Fit DFM using data {Y, X_j} with ˆp dynamic factors ; 21 Calculate MSE δ_j∗ over n∗ recent points (Equation (6.6)) ;

22 end

23 Let ˆδ = min({δ_j∗ : j = 1, . . . , m}) ; 24 if ˆδ < δ then

25 Re-assign δ = ˆδ ;

26 Reset δ∗_j = ∞ ∀j ∈ N ;

27 Expand the forecasting dataset: {Y} = {Y, Xj:δ∗ j=min(δ

∗₎};

28 Remove that variable from the indicators dataset:

{X} = {X\{Xj:δ∗ j=min(δ

∗₎}} ;

29 end

30 end

31 Recalculate the amount of explanatory series still unselected

m = |X|. ;

32 end

33 end

output: Best dataset {Y} and best number of dynamic factors p. ;

Algorithm 3: Proposed Model Selection: Variable Selection and Number of Dynamic Factors.

second area of interest is the chosen structure of the models from our algorithm. We explore the datasets chosen at each of the given time points and the amount of factors chosen.

Forecast Efficiency

To begin our comparison of results, we look at Quarterly GDP growth against our estimated growth rates and those given by competing models. As we estimate a monthly growth rate we have to consider how we transform this into a comparable quarterly rate given which month we are in. For comparison we present the Eu- rostat t + 30 estimate, which we extend two months previous to their generation for comparison. Thus when we are in the first month of a new quarter, we use the fitted monthly growth rates from the prior three months to create a quarterly growth rate of the previous quarter. However, if we are in the second month of a quarter, we then look to nowcast the current quarter we are in, thus we require a one step ahead forecast alongside the fitted values. For the third month, we use the current and previous two values of the fitted series to compute a nowcast for the current quarter’s GDP. Table 6.3 shows how the months and the predicted quarter line up more succinctly.

A comparison of these calculated quarterly growth rates is given in Figure 6.1a alongside competing models. Note that the graph separates out the estimations made by a previous model and those under consideration. These are the estimates from a newly proposed VAR model implemented within Matlab with the full data set (unknown for the previous model). Additionally we include the Eurostat Flash Estimate of GDP for that quarter. Note the discontinuous nature of the actual growth rate and the flash estimate is such that we are comparing the actual quarter at month T with that which would be predicted at that time.

It can be seen that the DFM with quarterly variable selection is not as smooth as the monthly variable selection to begin, but smooths out throughout the period. Overall it appears to follow closely to the truth and the flash estimate. Most

notably in comparison to the currently implemented model the proposed model predicts a higher level, following more closely with the flash estimates and truth. The fit generally follows consistently the growth rate over the period 2014-2017 except for the jump and correction in early 2015. Numerically, in comparison to the VAR model estimates given towards the end, the DFM reduces the MSE over the period by 67.45%.

When using a monthly MSE the predictions can be more erratic on a month by month basis. However this allows for quick adaptation to the movements within the true growth rate such as in 2016 and the placement between the truth and the flash estimate in late 2014. However, as in the quarterly case, the level of the series follows that of the truth without deviating too much and in the very recent past draws closest to the truth. Here also there is a reduction in the MSE in comparison to VAR by 82.23%.

Figure 6.1b depicts the point wise squared error (SE) of the predictions from Figure 6.1a. The large departures prior to 2015 can be seen to be an overestimate of the jump to come in the truth, before recalculating a much lower estimate in the following monthly as the Eurostat Flash Estimate had. Beyond 2015 the jumps are much less erratic for both comparisons and often below that of the previous contractor until an underprediction of a large jump, which the previous model consistently overestimated. However, in comparison to the newly proposed VAR model, we see large reductions in error for both the monthly and quarterly variable selection procedures.

Choice of Structure

As detailed in Algorithm 3, a selection process was put in place such that only the most favourable data is used and most predictive factor structure is selected in the prediction for time period which we restrict to. Here we present which variables were being used throughout the previous 42 month period and the amount of factors that were selected. The results in Figure 6.2a show variable selection

Variable 1 2 3 4 5 6 7 8 9

Quarterly 0 27.91 0 65.12 100 32.56 6.98 53.49 16.28

Monthly 46.51 34.88 11.63 25.58 41.86 48.84 37.21 58.14 20.93 Table 6.2: Inclusion rate (%) of variable within each comparison methodology.

Month 1 2 3 4 5 6 7 8 9 10 11 12

Actual Quarter 1 2 3 4

Predicted Quarter 4 1 2 3 4

Table 6.3: How each calendar month corresponds to the actual quarter and the quarter predicted at that time.

throughout the period, where as Figure 6.2c shows the amount of factors selected. The index on the y axis corresponds to the ID within Table 6.1, but also to the amount of variables selected in total. Further we show the rate of inclusion as a percentage of each variable in Table 6.2.

Reviewing the variable selection it can be seen that this is a highly variable process which selects a diferent number and selection at each time point. Most prevalent is the complete inclusion of Variable 5 for the quarterly predictions. Further, there are some continuous streaks of variable usage, such as variable 8 during 2014 and partly 2015. An interesting trend is the inclusion of variable 1 towards the end of 2015 within the monthly comparison, and variable 4 similarly for the quarterly. However the other variables are only chosen for short periods of time before being removed. There is evidence of indicative behaviour here, but it is not necessarily consistent between the two comparisons, potentially reflecting the quick changes that can be expected in a monthly comparison rather than a quarterly. Certainly there appears to be a change in behaviour around the end of 2015.

Figure 6.2c shows how the behaviour of the comparisons differ greatly. Prior to 2015 it would appear that the quarterly comparison was changing the amount of factors it was using to predict very frequently, before returning to 1 and holding consistent as 2015 began. However, the monthly comparison shows little change in the amount of factors, choosing only to depart from 1 sporadically in 2016.

Time Gro wth Rate 2014 2015 2016 2017 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Actual Flash t+30 Previous Model VAR Model Quarterly Comparison Monthly Comparison

(a) Predictions for each quarter with aligned prediction period.

Time SE 0.00 0.05 0.10 0.15 0.20 2014 2015 2016 2017 Previous Model VAR Model Quarterly Comparison Monthly Comparison

(b) Squared Error (SE) in prediction of quarterly growth rate.

Figure 6.1: Rolling forecasts and errors for quarterly Euro-area GDP. n∗ = 24.

Time V ar iab le 1 2 3 4 5 6 7 8 9 2014 2015 2016 2017 o oo o o o o o o o oo o o oo o o o o x x x x x x x x x x x x x x x x x x x xo o oo o o ooo o oo oo o o o o o o x x x x x x xo x ox x x x xo o o o o ox x xo ox x x x x x x x x x x x x x x x x x xo o x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x o o o o o o o oooo o o o o oooo o o o oo oo x x xo x x x x xo o o o o ox x x x x x x xo o o o o x xoo o oo o o o ooo xx x o o o o oo o o o oo oo o o o o ooo x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x o o o o o o o o o o oo oo o o o o o o oo o o o o o oo o o x x x x x x x o o o o o oo o o o oo o o oo x o Quarterly Comparison Monthly Comparison

(a) Variables selected over the 42 months. Time T otal Amount of V ar iab les 1 2 3 4 5 6 7 2014 2015 2016 2017 Quarterly Comparison Monthly Comparison

(b) Total variables se-

lected throughout period.

Time F actors 1 2 3 4 2014 2015 2016 2017 Quarterly Comparison Monthly Comparison (c) Amount of dynamic factors estimated.

Figure 6.2: Structure of dataset selected throughout the 42 months.