Choosing ARMA Lag Length with BIC and Model Comparisons

Summary

The document asks whether Bayesian Information Criterion should select autoregressive lag length using a model containing only the dependent variable and its lags, or the full specification that also includes exogenous predictors. The responses advise comparing candidate lag orders within the model structure intended for use, so the candidates differ in lag count while retaining the other regressors. They also suggest inspecting autocorrelation and partial autocorrelation patterns to help identify plausible lag choices, then fitting alternatives and comparing their criteria.

The discussion mentions AIC and other selection criteria, but its criticism of BIC and preference for AIC are an individual respondent’s opinion rather than a settled conclusion supported in the text. Information criteria help compare specified models; they do not establish that the chosen lag structure is correctly specified or that omitted variables, residual behavior, and forecasting performance are acceptable. The advice is therefore a practical starting point, not a full model validation procedure.

Key ideas

• Compare lag candidates while holding the rest of the intended model specification constant.
• BIC can be minimized across candidate autoregressive or moving-average lag orders.
• ACF and PACF plots can help suggest plausible lag choices before fitting models.
• The response’s preference for AIC over BIC is an opinion and is not established by evidence in the document.

Tags

# Limit of conditional expectations (when limit linked to the conditionning)


# Limit of conditional expectations (when limit linked to the conditionning)












I am working with conditional expectations and am trying to derive a limit property.

Consider $(Y_n)_{n \in \mathbb{N}}$ a sequence of square integrable random variables, that converge in $L^2$ to a square integrable random variable $Y$. Additionally assume that $\mathbb{E}[Y_n|Y] = Y_n$ (for example, $Y_n$ is a sequence of discrete quantizers of $Y$).

Is there anyway at all of guaranteeing that for some other $X$ in $L^2$, and for some form of convergence ($L^2$, $\mathbb{P}$ etc.) :

$$ \lim_{n \rightarrow + \infty} \mathbb{E}[X|Y_n] = \mathbb{E}[X|Y]. $$

I am aware of the following similar question :

https://math.stackexchange.com/questions/3096326/conditional-expectation-of-asymptotically-independent-random-variables

but in that case $\mathbb{E}[Y_n|Y] = Y_n$ does not hold...

With a $L^2$-projection approach to conditional expectation, and with $Y_n$ converging in $L^2$ to $Y$, I keep thinking there must be some way of getting this to work... But maybe it just won't.

Thank you for any suggestions !

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