LRM with time series data – the static model

Summary

Setup

  • Time series data on a dependent variable \(y\) and \(k-1 \) explanatory variables \(x_2,…,x_k\) .
  • \(y_1,…,y_T\) and \(x_{1,j},…,x_{T,j}\) for \(j=2,…,k\) are stationary processes.
  • We formulate a linear regression model

\[y_t=β_1+β_2x_{t,2}+…+β_kx_{t,k}+ε_t , t=1,…,T\]

Exogeneity in time series models

  • We say that the \(x\) -variables are exogenous if

\[E\left( x \right)=0 , t=1,…,T\]

  • where \(E\left( x \right)\) is the conditional expectation of \(y_t\) given all data on the explanatory variables at all points in time.
  • If the \(x\) -variables are exogenous then

\[E\left( x \right)=β_1+β_2x_{t,2}+…+β_kx_{t,k} , t=1,…,T\]

  • and

\[y_t=E\left( x \right)+ε_t , t=1,…,T\]

  • or

\[ε_t=y_t-E\left( x \right) , t=1,…,T\]

Static and dynamic models

  • The model is static if only observations made at time \(t\) affect \(E\left( x \right)\) . If past values affect \(E\left( x \right)\) then the model is dynamic .

Homoscedasticity in time series models

  • We say that the error terms are homoscedastic if

\[Var\left( ε_t|x \right)=σ^2 , t=1,…,n\]

  • Otherwise it they are heteroscedastic.

Autocorrelation

  • We say that the error terms are autocorrelated if, conditionally on \(x\) ,

\[Cov\left( ε_t,ε_s \right)≠0 for any t≠s\]

  • We say that we have no autocorrelation if \(Cov\left( ε_t,ε_s \right)=0 for any t≠s\)

Gauss-Markov assumptions in time series models

  • All data is stationary
  • The explanatory variables are exogenous
  • The error terms are homoscedastic
  • There is no autocorrelation