Abstract
This chapter reviews different methods to construct density forecasts and to aggregate forecasts from many sources. Density evaluation tools to measure the accuracy of density forecasts are reviewed and calibration methods for improving the accuracy of forecasts are presented. The manuscript provides some numerical simulation tools to approximate predictive densities with a focus on parallel computing on graphical process units. Some simple examples are proposed to illustrate the methods.
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Notes
- 1.
Davidson and MacKinnon (2006) suggest to rescale the residuals so that they have the correct variance by \(\check {e}_{t} \equiv \left (\frac {n}{n-k}\right )^{0.5} \widehat {e}_{t}\).
- 2.
- 3.
For longer horizons, test for independence is skipped.
- 4.
See for the complete list of functions http://www.mathworks.com/help/distcomp/using-gpuarray.html.
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Appendix
Appendix
There is little difference between a CPU and a GPU MATLAB code as Listings 15.1 and 15.2, for example, show. The pseudo code, reported in the listings, generates random variables Y and X and estimates the linear regression model Y = Xβ + 𝜖, on CPU and GPU, respectively.
The GPU code, Listing 15.2, uses the command gpuArray.randn to generate a matrix of normal random numbers. The build-in function is handled by the NVIDIA plug-in that generates the random number with an underline raw CUDA code. Once the variables vY and mX are created and saved in the GPU memory all the related calculations are automatically executed on the GPU, e.g., inv is executed directly on the GPU. This is completely transparent to the user.
If further calculations are needed on the CPU then the command gather transfers the data from GPU to the CPU, see line 5 of Listing 15.2. There exist already a lot of supported functions and this number continuously increases with new releases.Footnote 4
Listing 15.1 MATLAB CPU code that generate random numbers and estimate a linear regression model
1 iRows = 1000; iColumns = 5; % number of rows and columns
2 mX = randn (iRows , iColumns ) ; % generate random numbers
3 vY = randn (iRows , 1) ;
4 vBeta = inv ( mX ’ * mX) * mX’ * vY;
Listing 15.2 MATLAB GPU code that generate random numbers and estimate a linear regression model
1 iRows = 1000; iColumns = 5; % number of rows and columns
2 mX = gpuArray . randn (iRows , iColumns ) ; % generate random numbers
3 vY = gpuArray . randn (iRows , 1) ;
4 vBeta = inv ( mX ’ * mX) * mX’ * vY;
5 vBeta = gather ( vBeta ) ; % transfer data to CPU
As further examples in Listings 15.3 and 15.4 we show the GPU implementation of the accept/reject and the importance sampling algorithms presented in Sect. 15.5.
Listing 15.3 Accept/reject MATLAB GPU code
1 sampsize = 1000000; % sample s i z e to use for examples
2 sig = 1; % standard deviation of the instrumental density
3 samp = gpuArray . randn ( sampsize , 1) .* sig ; % step 1 in the A/R algorithm
4 ys = exp((− samp .ˆ2) /2) .* ( sin (6 * samp) .ˆ2 + 3 *(( cos (samp) .ˆ2) .*( sin (4*samp) .ˆ2) ) + 1) ;
5 wts = (1/ sqrt (2* pi ) ) .* exp (− samp .ˆ2/2) ;
6 samp2 = gpuArray . rand ( sampsize , 1) ;
7 dens = samp(samp2<=(ys ) ./ wts ) ; % step 2 in the A/R algorithm
8 target = gather ( dens ) ; % step 3 in the A/R algorithm
Listing 15.4 Importance sampling GPU code
1 nIS = 10000; nu = gpuArray (12) ; nustar = gpuArray (7) ; % number of simulations ; degree of freedom of the target density ; degree of freedom of the proposal
2 muIS = gpuArray . nan( nIS , 2) ;
3 wIS = gpuArray . nan( nIS , 2) ;
4 x1 = rant_GPU( nIS , nustar ) ; % Student t proposals
5 x2 = tan (( gpuArray . rand ( nIS , 1) − 0.5) * pi ) ; % Cauchy proposals
6 wIS ( : , 1) = w1_GPU(x1 , nu , nustar ) ; % Importance weights
7 wIS ( : , 2) = w3_GPU(x2 , nu) ; % Importance weights
8 muIS ( : , 1) = sqrt ( abs (x1 ./(1− x1 ) ) ) ;
9 muIS ( : , 2) = sqrt ( abs (x2 ./(1− x2 ) ) ) ;
10 muIScum ( : , 1 )= cumsum(muIS ( : , 1 ) .* wIS ( : , 1 ) ) . / ( 1 : nIS ) ’;
11 muIScum ( : , 2 )= cumsum(muIS ( : , 2 ) .* wIS ( : , 2 ) ) . / ( 1 : nIS ) ’;
12 %
13 % Additional functions
14 function w = w1_GPU(x , nu , nustar ) % Student ’ s t weights
15 w = tpdf_GPU(x , nu) ./ tpdf_GPU(x , nustar ) ;
16 end
17 function w = w3_GPU(x , nu) % Cauchy weights
18 w = tpdf_GPU(x , nu) ./ pdfcauchy_GPU(x , 0 , 1) ;
19 end
20 function f = tpdf_GPU(x , v) % Student ’ s t GPU pdf
21 k = find (v>0 & v <Inf ) ;
22 i f any(k)
23 term = exp (gammaln(( v(k) + 1) / 2) − gammaln ( v ( k ) /2) ) ;
24 f (k) = term ./ ( sqrt (v(k)* pi ) .* (1 + (x(k) .ˆ 2) ./ v(k) ) .ˆ (( v(k) + 1) /2) ) ;
25 end
26 end
27 function f = pdfcauchy_GPU(x , a , b) % Cauchy GPU pdf
28 f = 1./( pi .* b .* (1 + (( x − a ) ./ b ) .ˆ2) ) ;
29 end
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Bassetti, F., Casarin, R., Ravazzolo, F. (2020). Density Forecasting. In: Fuleky, P. (eds) Macroeconomic Forecasting in the Era of Big Data. Advanced Studies in Theoretical and Applied Econometrics, vol 52. Springer, Cham. https://doi.org/10.1007/978-3-030-31150-6_15
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