Ljung-Box finite sample adjustments

Question

What is the intuition behind the finite sample adjustments in the Ljung-Box test: $Q = n\left(n+2\right)\sum_{k=1}^h\frac{\hat{\rho}^2_k}{n-k}$?

Degrees of freedom adjustments usually involve subtraction, so where does the $(n+2)$ come from?

Also, why are farther lags weighted more in the $(n-k)^{-1}$ adjustment?

Answer 1

I will only answer your first question. I will show that the "finite sample adjustment" is not really an adjustment and that the Ljung-Box statistic is only natural (more so than the Box-Pierce statistic).

(For the second and third questions, you could consult Anderson (1942), which is unfortunately quite technical. Probably another user will offer a more intuitive answer.)

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Take an ARMA($p$,$q$) model

$$ \phi(B) w_t = \theta(B) a_t $$

where $B$ is the backshift (or lag) operator. Define the $k$-th order autocorrelation of model errors (not residuals) as

$$ r_k := \frac{ \sum_{t=k+1}^n a_t a_{t-k} }{ \sum_{t=1}^n a_t^2 } $$

and collect the first $m$ autocorrelations in one vector $r:=(r_1,\dotsc,r_m)$.

Box & Pierce (1970) claim on p. 1510 that for large $n$,

• $r$ has a multivariate normal distribution,
• $r_i$ and $r_j$ are uncorrelated for $i \neq j$ and
• the variance of $r_k$ is

$$ \text{Var}(r_k) = \frac{n-k}{n(n+2)}. $$

Then it follows that the sum

$$ \sum_{k=1}^m \frac{n(n+2)}{n-k} \text{Var}(r_k) = n(n+2) \sum_{k=1}^m \frac{1}{n-k} \text{Var}(r_k) $$

is distributed as $\chi_m^2$ for large $n$ (because you get $\chi^2_m$ distribution by summing up $m$ squares of independent standard normal random variables).

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Up to this point we have an expression of the Ljung-Box (rather than Box-Pierce) test statistic. So apparently there is no "finite sample correction".

What happens next is that Box & Pierce (1970) note that

$$ \text{Var}(r_k) \approx \frac{1}{n} $$

since $\frac{n+2}{n-k} \approx 1$ for large $n$, and then also

$$ n \sum_{k=1}^m \text{Var}(r_k) \sim \chi_m^2. $$

Here is where the Box-Pierce statistic (different from the exact statistic above) is introduced.

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This concerns the case where model errors are known, which is not what we encounter in practice. Therefore, Box & Pierce (1970) go on to examine the case with estimated residuals in place of the true model errors.

After some elaboration on the pure autoregressive AR($p$) case, they note on p. 1517 that when errors are unknown and are replaced by residuals, for large $n$ it is sufficient to replace $m$ with $m-p$ ($=m-p-q$ since $q=0$) in the asymptotic distribution and the result will still hold:

$$ n \sum_{k=1}^m \text{Var}(\hat r_k) \sim \chi_{m-p}^2 $$

where $\hat r_k$ is the sample counterpart of $r_k$.

Further they show that the case of ARMA($p$,$q$) in place of pure AR($p$) does not change the essence, and so for a general ARMA($p$,$q$) model one still has that

$$ n \sum_{k=1}^m \text{Var}(\hat r_k) \sim \chi_{m-p-q}^2. $$

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In these last few expressions, the approximation $\frac{n+2}{n-k} \approx 1$ is used. It does not hurt in large samples, but apparently it causes trouble in small samples, which Ljung & Box (1987) note (citing a few studies). Therefore, they suggest dropping the approximation and going back to the original statistic.

References:

• Anderson, Ronald L. "Distribution of the serial correlation coefficient." The Annals of Mathematical Statistics 13.1 (1942): 1-13.
• Box, George EP, and David A. Pierce. "Distribution of residual autocorrelations in autoregressive-integrated moving average time series models." Journal of the American statistical Association 65.332 (1970): 1509-1526.
• Ljung, Greta M., and George EP Box. "On a measure of lack of fit in time series models." Biometrika 65.2 (1978): 297-303.