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L31 Multivariate Time Series Analysis

Motivation

  • Multivariate/Vector processes
    • Several related TS processes observed simultaneously over time
  • Interested in inter relationships (e.g. \(Y_{t}\) and \(X_{t}\))
  • Interested in Cross relationships between the series (\(Y_{t}\) and \(X_{t-2}\) for example)

Objectives

  • Understand Dynamic Relationships over time
  • Utilize additional information from other time series (\(X_{t}\)) to improve forecasts for \(y_{t}\).

Example

Finance

  • Price movements in one market spread easily and instantly to another market.
    • E.g. Bitcoin and Ethereum prices
    • Consider them jointly to understand the dynamic structure of global market
  • For an investor holding multiple assets, making investment decisions

Economics

  • Simultaneous behaviors
    • Interrelationships between different variables

Environmental sciences and agriculture

  • Joint study Min max temperature, humidity, wind speed and direction → Total production of wheat
  • Air pollution level, number of asthma patients, registered cars etc.

L31__Multivariate Time Series Analysis-1769149238997.webp

  • We note that there are dependencies between the different indices

  • Another example is Nottem temperatures and "heating demand"

  • Another one "population", "employment" etc

Potentially Interesting Question

  • Is there a causal direction?
  • Are there feedback between the different series?
  • How do impulses (shocks) transfer between the series?
  • What are the common factors that cause disturbances across series?

Refresher on Matrix Algebra

  • \(A\) = \(n \times n\) matrix
  • \(f(\lambda) = \lvert A - \lambda I_{n} \rvert\) is a polynomial in \(\{ \lambda_{i} \}\) are the \(n\) roots of \(\lvert A - \lambda I_{n} \rvert =0\)

Thus,

\[ (A - \lambda_{i}I_{n})q_{i} = 0 \quad \text{or} \quad Aq_{i} = \lambda_{i} q_{i} \]
  • \(\lambda_{i}\) are the \(n\) eigenvalues of \(A\)
  • \(q_{i}\) are the \(n\) eigenvectors of \(A\). Let \(Q = (q_{1},q_{2},\dots,q_{n})\)

Then,

\[ AQ = (\lambda_{1}q_{1},\lambda_{2}q_{2},\dots,\lambda_{n}q_{n}) = Q\Lambda \]

Where \(\Lambda = diag(\lambda_{1},\lambda_{2},\dots,\lambda_{n})\)

\[ Q^{-1} A Q = \Lambda \]

Some properties:

  • \(\lvert A \rvert = \Pi \lambda_{i}\)
  • \(trace(A) = \sum\lambda_{i}\)
  • \(A^m = Q \Lambda^m Q^{-1}\)

Random Vector properties

Let \(X\) be a \(p\times 1\) random vector

  • Mean: \(E(X_{i})\)
  • Variance: \(E(X_{i}-\mu_{i})^{2}=\sigma_{ii}\)
  • Covariance: \(E(X_{i}-\mu_{i})(X_{j}-\mu_{j})=\sigma_{ij}\)