Time series

Benchmark

TSW

Gross Domestic Product (GDP)

ARIMA (0,1,1) (0,1,1)₄

ARIMA (1,1,0) (0,1,1)₄

$\nabla {\nabla }_4{Y}_t=\left(1+0.118{\rm B}\right)\left(1+0.425{{\rm B}}^4\right){\epsilon }_t$

$\left(1-0.160{\rm B}\right)\nabla {\nabla }_4\ln Y_t=\left(1+0.267{{\rm B}}^4\right){\epsilon }_t$

Industrial Production Index (IPI)

ARIMA (2,0,0) (0,1,1)₁₂

ARIMA (1,1,0) (0,1,1)₁₂

$\left(1+0.379{\rm B}+0.547{{\rm B}}^2\right){\nabla }_{12}{Y}_t=\left(1+0.950{{\rm B}}^{12}\right){\epsilon }_t$

$\left(1-0.554{\rm B}\right)\nabla {\nabla }_{12}\ln Y_t=(1+0.831{{\rm B}}^{12}){\epsilon }_t$

Consumer Price Index (CPI)

ARIMA (0,1,0) (0,1,1)₁₂

ARIMA (1,1,0) (0,1,0)₁₂

$\nabla {\nabla }_{12}{Y}_t=\left(1+0.260{{\rm B}}^{12}\right){\epsilon }_t$

$\left(1+0.146{\rm B}\right)\nabla {\nabla }_{12}\ln Y_t={\epsilon }_t$

Harmonised Index of Consumer Prices (HICP)

ARIMA (0,1,0) (0,1,1)₁₂

ARIMA (0,1,0) (0,1,1)₁₂

$\nabla {\nabla }_{12}{Y}_t=\left(1+0.347{{\rm B}}^{12}\right){\epsilon }_t$

$\nabla {\nabla }_{12}\ln Y_t=\left(1+0.326{{\rm B}}^{12}\right){\epsilon }_t$

Unemployment – thousands

ARIMA (3,2,1) (0,1,1)₁₂

ARIMA (3,2,1) (0,1,1)₁₂

$\begin{array}{l}\left(1-0.681{\rm B}-0.674{{\rm B}}^2+0.062{{\rm B}}^3\right){\nabla }^2{\nabla }_{12}{Y}_t\\ =\left(1+0.758{\rm B}\right)\left(1+0.938{{\rm B}}^{12}\right){\epsilon }_t\end{array}$

$\begin{array}{l}\left(1-1.153{\rm B}-1.123{{\rm B}}^2-0.340{{\rm B}}^3\right){\nabla }^2{\nabla }_{12}{Y}_t\\ =\left(1+0.614{\rm B}\right)\left(1+0.907{{\rm B}}^{12}\right){\epsilon }_t\end{array}$

Unemployment – percentage

ARIMA (2,2,1) (0,1,1)₁₂

ARIMA (2,2,1) (0,1,1)₁₂

$\left(1-0.726{\rm B}-0.715{{\rm B}}^2\right){\nabla }^2{\nabla }_{12}{Y}_t=\left(1+0.734{\rm B}\right)\left(1+0.816{{\rm B}}^{12}\right){\epsilon }_t$

$\begin{array}{l}\left(1-0.726{\rm B}-0.715{{\rm B}}^2\right){\nabla }^2{\nabla }_{12}{Y}_t\\ =\left(1+0.734{\rm B}\right)\left(1+0.816{{\rm B}}^{12}\right){\epsilon }_t\end{array}$

Retail sales

ARIMA (0,1,1) (0,1,1)₁₂

ARIMA (0,1,1) (0,1,1)₁₂

$\nabla {\nabla }_{12}{Y}_t=\left(1+0.364{\rm B}\right)\left(1+0.566{{\rm B}}^{12}\right){\epsilon }_t$

$\nabla {\nabla }_{12}\ln Y_t=\left(1+0.334{\rm B}\right)\left(1+0.631{{\rm B}}^{12}\right){\epsilon }_t$

M1

ARIMA (0,2,1) (0,1,1)₁₂

ARIMA (3,1,0) (0,1,1)₁₂

${\nabla }^2{\nabla }_{12}{Y}_t=\left(1+0.838{\rm B}\right)\left(1+0.682{{\rm B}}^{12}\right){\epsilon }_t$

$\begin{array}{l}\left(1+0.0001{\rm B}+0.178{{\rm B}}^2+0.422{{\rm B}}^3\right)\nabla {\nabla }_{12}\ln Y_t\\ =\left(1+0.685{{\rm B}}^{12}\right){\epsilon }_t\end{array}$

M2

ARIMA (3,1,0) (1,0,1)₁₂

ARIMA (1,1,1) (0,1,1)₁₂

$\begin{array}{l}\left(1+0.328{\rm B}+0.040{{\rm B}}^2+0.307{{\rm B}}^3\right)\\ \left(1+0.868{{\rm B}}^{12}\right)\nabla Y_t=\left(1+0.656{{\rm B}}^{12}\right){\epsilon }_t\end{array}$

$\begin{array}{l}\left(1+0.933{\rm B}\right)\nabla {\nabla }_{12}\ln Y_t\\ =\left(1+0.627{\rm B}\right)\left(1+0.800{{\rm B}}^{12}\right){\epsilon }_t\end{array}$

M3

ARIMA (0,2,1) (0,1,1)₁₂

ARIMA (0,2,1) (0,1,1)₁₂

${\nabla }^2{\nabla }_{12}{Y}_t=\left(1+0.695{\rm B}\right)\left(1+0.824{{\rm B}}^{12}\right){\epsilon }_t$

${\nabla }^2{\nabla }_{12}\ln Y_t=\left(1+0.660{\rm B}\right)\left(1+0.838{{\rm B}}^{12}\right){\epsilon }_t$

Balance of payments (BOP) – Transport – Payments

ARIMA (0,1,1) (0,1,1)₁₂

ARIMA (0,1,1) (0,1,1)₁₂

$\nabla {\nabla }_{12}{Y}_t=\left(1+0.188{\rm B}\right)\left(1+0.847{{\rm B}}^{12}\right){\epsilon }_t$

${\nabla }^2{\nabla }_{12}\ln Y_t=\left(1+0.440{\rm B}\right)\left(1+0.841{{\rm B}}^{12}\right){\epsilon }_{\text{t}}$

Balance of payments (BOP) – Transport – Receipts

ARIMA (3,1,1) (0,1,1)₁₂

ARIMA (3,1,0) (0,1,1)₁₂

$\begin{array}{l}\left(1-0.393{\rm B}-0.050{{\rm B}}^2+0.264{{\rm B}}^3\right)\nabla {\nabla }_{12}{Y}_t\\ =\left(1-0.288{\rm B}\right)\left(1+0.950{{\rm B}}^{12}\right){\epsilon }_t\end{array}$

$\left(1-0.208{\rm B}-0.110{{\rm B}}^2+0.229{{\rm B}}^3\right)\nabla {\nabla }_{12}\ln Y_t=\left(1+0.895{{\rm B}}^{12}\right){\epsilon }_t$

Balance of payments (BOP) – Travelling – Payments

ARIMA (1,0,0) (0,1,1)₁₂

ARIMA (1,0,0) (1,0,0)₁₂

$\left(1+0.339{\rm B}\right){\nabla }_{12}{Y}_t=\left(1+0.506{{\rm B}}^{12}\right){\epsilon }_t$

$\left(1+0.328{\rm B}\right)\left(1+0.608{{\rm B}}^{12}\right)\ln Y_t={\epsilon }_t$

Balance of payments (BOP) – Travelling – Receipts

ARIMA (1,0,0) (1,1,0)₁₂

ARIMA (1,0,0) (1,1,0)₁₂

$\left(1+0.731{\rm B}\right)\left(1-0.371{{\rm B}}^{12}\right){\nabla }_{12}{Y}_t={\epsilon }_t$

$\left(1+0.598{\rm B}\right)\left(1-0.422{{\rm B}}^{12}\right){\nabla }_{12}\ln Y_t={\epsilon }_t$

Balance of payments (BOP) – Sea transport – Payments

ARIMA (0,1,1) (0,0,0)₁₂

ARIMA (0,1,1) (0,1,1)₁₂

$\nabla Y_t=\left(1+0202{\rm B}\right){\epsilon }_t$

$\nabla {\nabla }_{12}\ln Y_t=\left(1+0.334{\rm B}\right)\left(1+0.824{{\rm B}}^{12}\right){\epsilon }_t$