# Can I modify the Chow Statistic for use with ARIMAx models?

I'm trying to come up with a test for parameter constancy that's general enough to use for both OLS models and ARIMAx models. In all cases, we have non-stochastic exogenous variables.

Here's the chow prediction interval statistic for OLS with $$k$$ parameters, equation (31) in Chow's paper from 1960, "Tests of Equality Between Sets of Coefficients in Two Linear Regressions": $$\frac{||X_1\beta_1-X_1\beta_0||^2+||Y_2-X_2\beta_0||^2}{||Y_1-X_1\beta_1||^2} \cdot\frac{n-k}{m} \sim F_{(m,n-k)}$$ where $$Y_0=\begin{bmatrix}Y_1\\Y_2\end{bmatrix}, X_0=\begin{bmatrix}X_1\\X_2\end{bmatrix}, \beta_j=(X_j'X_j)^{-1}X_j'Y_j$$ and $$X_1$$ is $$n$$ by $$k$$, and $$X_2$$ is $$m$$ by $$k$$

Here's the question: If I modify the this chow prediction interval test statistic by replacing all the $$X\beta$$ terms with the analogous one step ahead projections from an ARIMAx model, what conclusions can I still draw from the modified test statistic, if any? (Is this modified statistic still distributed $$F_{(m,n-k)}$$?)

i.e. $$\frac{||f_1(X_1)-f_0(X_1)||^2+||Y_2-f_0(X_2)||^2}{||Y_1-f_1(X_1)||^2} \cdot\frac{n-k}{m}$$ where $$f_j$$ is the ARIMAx model with parameters estimated using $$X_j$$ and $$Y_j$$

• I've so far taken a monte carlo approach to see how the statistic is distributed when there are ARMA terms, and it looks to have a distribution exactly characterized by the F distribution with $m$ and $n-k$ degrees of freedom. Does anyone have a pier reviewed paper, or even a blog post or wiki article, that shows analytically (or even asserts) that the statistic has this F distribution when ARMA terms are included to control for autocorrelation? – Elon Plotkin Mar 16 '17 at 15:14

To my way of thinking ... arma coeffficients are simply coefficients capable of eating up (using ) degrees of freedom. Since the CHOW test employs degrees of freedom ... you are good to go ....

• Thanks for taking the time to think about this. Would you know of any sources I could cite though, or maybe any leads? – Elon Plotkin Jan 2 at 1:36
• think of an ar 1 parameter being the functional equivalent of a lag of Y thus decreasing the degrees of freedom by 1 . .. an ar 12 parameter ... by 12 etc. – IrishStat Jan 2 at 18:44