To check if financial big data can help in raising the financial cycles forecasting efficiency, we apply
(MSSA) of the form \citep{Ghil2002} on residential property prices, credit to private non-financial sector, credit share in the GDP:
\(\begin{array}{l}{\left(C_{l, l^{\prime}}\right)_{i, j}=\frac{1}{\bar{N}} \sum_{t=\min \left(1,1,1^{\prime}+i-j\right)}^{\max (N+i-j)} x_{l}(t) x_{l^{\prime}}(t+i-j)} \\ {\widetilde{N}=\min \{N, N+i-j\}-\max \{1,1+i-j\}+1}\end{array}\)(6)
where
cij = lag covariance matrix
cl,lj =lag cross-covariance matrix
N = number of data points in a time series
i,j = indices of time
t = continuous time t ∈ R
x(t) = L-channel vector time series.
As in the case of (SSA), using the R-forecasting (MSSA forecasting method) we now get the (MSSA) forecasting results (forecasting including financial big data) for the residential property prices, credit to private non-financial sector, credit share in the GDP over the period 2017Q2 - 2019Q1.
To test the improved accuracy (better forecasting results) we obtain when using financial big data we use the Diebold - Mariano test \citep{Diebold1995,Diebold2015} of the form:
\(D M_{12}=\frac{\bar{d}_{12}}{\hat{\sigma}_{\bar{d}_{12}}} \rightarrow N(0,1)\) (7)
where
\(\bar{d}_{12}=\frac{1}{T} \sum_{t=1}^{T} d_{12 t}\) = sample means loss differential
\(\hat{\sigma}_{\bar{d}_{12}}\)= consistent estimate of standard deviation.
Diebold - Mariano test results confirm our hypothesis that financial big data improve forecasting accuracy for financial cycles (see table 1 for the UK).