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Say that we are regressing a variable $Y_t$ on another variable $X_t$, and both series are non-stationary. Specifically, let's say that both are $I(1)$ and trend upwards over time.

Now, say we regress: $$Y_t = \alpha + \beta X_t + u_t \tag{1}$$

I have always been under the impression that the coefficient $\hat{\beta}$ from this regression is biased, due to the problem of spurious regression. I've also been told that another way of thinking about it is that there is an omitted variable $t$ in (1) which causes the coefficients to be biased.

Today, someone told me that actually the coefficients are unbiased, and it is simply the standard errors from estimating (1) which are wrong.

What is the truth?

EDIT: Assume that $X_t$ and $Y_t$ are NOT cointegrated.

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  • $\begingroup$ Is $X_t$ a variable for time? $\endgroup$ – Dayne Oct 18 '19 at 11:21
  • $\begingroup$ @Dayne Apologies - edited the typo. $\endgroup$ – Thev Oct 18 '19 at 11:23
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Assuming you are using OLS method for estimation and and $X_t$ is exogenous variable, there are three cases with regards to the estimated residuals $\hat{u}_t:

  1. $\{\hat{u}_t\}$ are uncorrelated. If so, the estimate $\hat{\beta}$ is not only unbiased but also efficient.

  2. $\{\hat{u}_t\}$ is stationary but not uncorrelated. If so, the estimate $\hat{\beta}$ is unbiased but the standard errors can be improved by using ECM.

The above two cases are that if cointegration (or that that both series have linear time trend).

  1. $\{\hat{u}_t\}$ is non-stationary. In this case also $\hat{\beta}$ is unbiased but standard errors reported are practically meaningless. This is the case of spurious regression. This, I think, means that the model is incorrect so there's no sense of estimating such a model (this last statement needs confirmation!).
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  • $\begingroup$ why are you making statistical assumption on the residuals? Should it not rather be on the errors $u_t$? $\endgroup$ – Jesper for President Oct 18 '19 at 13:45
  • $\begingroup$ Absolutely. Theoretically it will be $u_t$. It's just that I was writing from the practical perspective. As in you get $\hat{u}_t$ only from data. $\endgroup$ – Dayne Oct 18 '19 at 14:34
  • $\begingroup$ Please note that residuals $\hat u$ from an OLS regression are related to the errors $u$ via $\hat u=Mu$, where the "residual-maker matrix" is $M=I-X(X'X)^{-1}X'$. So $Var(\hat u)=MVar(u)M'$, which reduces to, assuming classical assumptions on the errors, $MM'$, which equals $M$ due to symmetry and idempotency of $M$. Now, $M$ is not a diagonal matrix even if $Var(u)$ is. $\endgroup$ – Christoph Hanck Oct 18 '19 at 16:25
  • $\begingroup$ @ChristophHanck: This mean that in any regression (not restricting to time series), we should not expect residuals to be uncorrelated, even if the true errors are? How do we check for independence of true errors then? $\endgroup$ – Dayne Oct 19 '19 at 3:36
  • $\begingroup$ It rather means that such tests will typically be based on asymptotic approximations. Consider the simple case of $X=\iota$, a vector of ones, i.e., a regression of a constant. In that case, you can easily check that the off-diagonal elements of $M$ equal $-1/n$, which evidently tends to zero as sample size $n$ grows. $\endgroup$ – Christoph Hanck Oct 19 '19 at 11:05

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