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