# Is it possible to simplify linear regression if one of the variables does not have errors and is equispaced?

Specifically, I have a time series with fixed $\Delta t$ of the following form: $x_0, x_1, x_2, ... x_n$ and $t_0, t_1, t_2, ... t_n$ where $t_{i+1}-t_i =\Delta t$.

I'm interested in the improved efficiency for large data sets, if it is possible to simplify linear regression in this context.

Edit: I editet the title since equispaced is an important property.

• It is likely won't do you a whole lot of good. All you gain are closed formulas for the moments of $t$: the mean is $n(\Delta t)/2$ and the variance is $((n+1)^2-1)(\Delta t)^2/12$. You still have to compute moments of the dependent variable $y$ as well as the covariance of $t$ and $y$. And for rolling regressions, which are updated quickly in constant time, even those formulas accomplish nothing. The only situation I have seen that benefits is where calculations are limited by software (in)capabilities or there are a huge number of dependent variables.
– whuber
Jan 20, 2015 at 16:03

In fact in linear regression it's already assumed that the predictor variable (IV) has no error.

If not for that, you'd have errors-in-variables regression (though it has various other names). OLS is biased in this case.

So there's no opportunity for efficiency when you impose a condition that's already assumed.

There is some opportunity for some efficiency gain because the $t$'s are equispaced.

However, if you need to refit these models each time a new data point arrives, then there are larger opportunities for gain.

A warning, too, about fitting regression to time series; it must be done with care.

• Thanks for clearing this up for me! As it happens the refit you've spoken of when a new data point arrives is of great interest - could you elaborate or point me to a relevant source?
– JHK
Jan 20, 2015 at 10:15
• Search our site for rolling regression.
– whuber
Jan 20, 2015 at 13:07