Recently I read an paper where in a time series data has been modelled according to the equation $$ Y_t=\beta_1 Y_{t−1}+\beta_2X+\varepsilon. $$ OLS was used here (with the lm() command in R) for obtaining the coefficient of $Y_{t-1}$. Is it statistically correct?

I understand when we deal with time series data, this actually means an ARX process and can be represented as $$ Y_t=\theta Y_{t-1}+\beta X + \varepsilon, $$ where $\theta$ comes from the Yule-Walker equations.

Will $\theta$ and $\beta_1$ yield the same result? Won't the OLS estimator suffer from autocorrelation problem as $E[x_t \varepsilon_t] \ne 0$ ? My statistics knowledge is beginner level. Please guide me on understanding this.

  • $\begingroup$ See Can I use OLS to estimate an ARX(1) model? $\endgroup$ – Firebug Oct 17 '17 at 10:34
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    $\begingroup$ In the absence of Moving Average (MA) terms, AR models (with or without exogenous variables) can be fit with OLS without big problems. Compared to MLE it will have small sample bias, but doesn't affect consistency. And we are not scared of bias boogeyman anymore. $\endgroup$ – Cagdas Ozgenc Oct 17 '17 at 11:06
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    $\begingroup$ @CagdasOzgenc, if I am not mistaken, MLE will also have bias, won't it? $\endgroup$ – Richard Hardy Oct 17 '17 at 12:07
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    $\begingroup$ I stand corrected. I always thought that bias disappeared when exact likelihood is used. It seems conditional likelihood and OLS have the same properties while exact likelihood has a smaller bias, yet still biased. All schemes are consistent, that's for sure. $\endgroup$ – Cagdas Ozgenc Oct 17 '17 at 14:48

Hi: Your model is also called a koyck distributed lag and it can be difficult to estimate with small samples. With larger samples, my experience is that there is not a problem with bias. ( I used simulation to check this ).

The link discusses the statistical properties of the estimates briefly on pages 12 and 13. Essentially the problems with it are similar to those of the estimates of an AR(1).


I would check out hamilton or the little koyck book (1954) for a more in depth discussion but hopefully the above helps some.


From what I have read, the Yule-Walker equations use least squares to estimate the AR-1 lag coefficient (what you call $\beta_1$ in display 1 and $\theta$ in display 2). The joint estimation of the lag coefficient and the $X$ coefficient is correctly done using a least squares model adjusting for the lag and the concurrent $X$ in the model you have written. If the model is misspecified due to either cross lags, omitted variables, or higher lag orders, and display 1 does not describe the data generation process, the coefficients can be massively biased.


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