Serially Correlated Regressors

Question

I am trying to find information (without success) regarding serially correlated regressors in linear time series regression setting.

The topics covered are either correlation between regressors, or serial correlation of the error, but not serially correlated regressors.

To give an example, in the following model

$y_t = \beta x_{t-1} + \epsilon_t$

where $y_t$ and $x_t$ are $I(0)$ - order of integration is 0.

If I replace $x_t$ with

$z_t = (x_t + x_{t-1} + ... + x_{t-n+1})/n$, some averaging scheme

$z_t$ is still $I(0)$ but it is autocorrelated up to lag $n$. Now the model is

$y_t = \gamma z_{t-1} + \nu_t$

What are the pitfalls in this situation?

Thanks.

Answer 1

I see it as a special case of Structural Vector AutoRegression (SVAR). A standard SVAR model is: $$ \textbf{A}(L)X_t = \mu +\textbf{e}_t $$

where $X_t$ is a vector of k endogenous variables, $\mu$ is a vector of k constant parameters $\textbf{e}_t = N(0,I) $ is a random vector and $A(L)$ is a matrix polynomial of order n. Where $\textbf{A}(L)X_t = A_0X_t+A_1X_{t-1}+\cdots +A_nX_{t-n}.$ In your particular case: $$ X_t = \begin{bmatrix} x_t \\ y_t \end{bmatrix} $$ $$ A_0= \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} $$ $$ A_t=\begin{bmatrix} \mbox 0 & 0 \\ \mbox - \gamma/n & 0 \end{bmatrix} $$ for $t = 1,\ldots,n.$ SVAR models were introduced in the 80´s and they have been widely studied mainly within the econometric realm. In this paper, Lutz Kilian reviews the econometric literature regarding the identification of causal effects in the data in the context of SVARs, but in the beggining he provides a nice introduction to the topic.

Answer 2

After the moving average, the regressor in your model becomes more persistent, i.e. having stronger serial correlations. The OLS estimate would still be asymptotically consistent, but the finite-sample bias likely becomes larger. You could run a simple simulation study to see the finite-sample bias.

In finance, there is a literature on forecasting stock return with financial/macroeconomic variables. These variables are often highly persistent, similar to the problem you are facing. It might be useful for you to have a look at the finance studies that investigate the impact of highly persistent regressors, e.g. "Spurious Regressions in Financial Economics?" by Ferson etc, Journal of Finance, August 2003. Hope it helps.

Answer 3

I would look into multivariate ARMA modeling for generic models of this form. (It's a sort of hacky way to approach regression, but it'll work.)