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L34 Further Extensions and Use Cases

Example: Bivariate \(MA(1)\) process

\[ \begin{pmatrix} Y_{1t} \\ Y_{2t} \end{pmatrix} = \begin{pmatrix} e_{1t} \\ e_{2t} \end{pmatrix} - \begin{pmatrix} 0.2 & -0.4 \\ -0.2 & 0.6 \end{pmatrix}\begin{pmatrix} e_{1t-1} \\ e_{2t-2} \end{pmatrix};\quad \Sigma = \begin{pmatrix} 4 & 1 \\ 1 & 4 \end{pmatrix} \]

Compute:

  • \(\Gamma(0) = \Theta_{1} \Sigma \Theta_{1}'\)
  • \(\Gamma(1)= \Sigma \Theta_{1}'\)
  • \(\Gamma(2+)=0\)
  • \(\rho(0)\)
  • \(\rho(1)\)

Example: \(VARP(p)\) process

  • \(\Phi_{i}\) = k-dim square matrices
  • \(e_{t}\) = k-dim vector of residuals (purely random process)
  • \(\delta\) = vector of constants
\[ \Phi(B)Y_{t} = \delta + e_{t} \]
  • We can express \(VMA(1) \equiv VAR(\infty)\)
  • We can express \(VAR(p) \equiv VMA(\infty)\)

Further

  • We look at the covariance matrices
  • Correlation matrices
  • The expression and formulae remain the same, but we are having corresponding notation and forms for multivariate notation.

Application Areas

Macroeconomics

  1. Monetary policy analysis: interest rate changes, inflation and monetary policies → GDP, unemployment, exchange rages
  2. Economic forecasting: VARMA is used to forecast GDP growth, inflation employment, considering interactions.

Supply chain an operations

  1. Inventory Managements:
    • Improve inventory management
    • reduce stockouts, excess inventory
  2. Logistics and shipping:
    • forecast shipping demand and lead times
    • interrelated demand taken into account

Financial markets

  1. Asset Pricing
    • co-movements between assets
  2. Volatility Forecasting

Energy Markets

  1. Electricity Load Forecasting
    • Overloads
  2. Oil and Gas

Healthcare

  1. Epidemiology
  2. Hospital Resource planning

Climate Science

  1. Temperature & Weather Forecasting
  2. Air Quality Monitoring