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Simultaneous Inference Bands for Autocorrelations
\noindentKeywords: Autocorrelogram, confidence bands, joint significance, regression residuals.
JEL classification: C12 (hypothesis testing), C22 (time-series models).
The autocorrelation function (ACF) is a fundamental tool in time series analysis. A plot of the empirical autocorrelation function (sample autocorrelogram) is the common starting point to the statistical analysis of a time series or a series of regression residuals, giving a first overview of the dependence structure.
The autocorrelogram is usually accompanied by a pointwise non-rejection band for the null hypothesis of temporal independence or uncorrelatedness, i.\,e., white noise. Under this hypothesis, the empirical autocorrelation at a certain lag falls inside this band with a probability of $1 - \alpha$, for example 0.9; see the blue lines in Figure (ref) for an example for regression residuals from a static Phillips curve regression (for details see Section (ref)). Such bands, which represent regions of insignificance of a certain null, have recently been called significance bands by inoue2025 and consistency bands by dimitriadis2021 in other contexts. We adopt the former terminology. Those pointwise significance bands for autocorrelation functions are discussed in basically every textbook on time series analysis (e.\,g., BrockwellDavis91, Hamilton94, Fuller96, shumway2017) and are added to plots of empirical autocorrelation functions by default in most statistical software.
There are two shortcomings of those significance bands. First, they are pointwise bands, arising as the Cartesian product of significance intervals of a certain level $1 - \alpha$ for individual autocorrelations. As the null hypothesis refers to the whole autocorrelation function, we are rather interested in joint inference, for which the pointwise bands are not valid. Simultaneous (or joint) bands are appropriately scaled-up versions of pointwise bands and provide valid joint inference, that is, we can reject the white noise hypothesis on the significance level $\alpha$ if the empirical autocorrelation function (up to a certain maximum lag) leaves the simultaneous significance band of level $1-\alpha$ at any point. Amended with those bands, the correlogram provides an overview of the dependence structure and valid inference on the white noise hypothesis at a glance. Figure (ref) contains a simultaneous significance band, too, which is represented by the red lines.
The second issue concerns significance bands in general. They are tied to a certain null hypothesis, which might not always be of particular interest. For example for an inflation series no one would seriously doubt that there is serial dependence. What is always of interest is a quantification of sampling uncertainty as provided by confidence bands. While significance bands are much more common, sometimes pointwise confidence bands are used. Like the pointwise significance bands, they are only valid for one autocorrelation at a single lag, while one is usually interested in the whole path of autocorrelations and joint inference for it. Again, simultaneous confidence bands are a natural choice here, see Figure (ref), which shows the empirical autocorrelation function for monthly US inflation alongside a pointwise and two types of simultaneous (sup-t and Bonferroni) confidence bands (for details see again Section (ref)). Note that some authors use the term `uniform' instead of `simultaneous' when addressing confidence bands, see, e.\,g., FreybergerRai2018. The simultaneous confidence bands cover the whole autocorrelation function up to a chosen maximum lag with a specified (asymptotic) coverage probability of at least $1-\alpha$. At the same time they still directly provide the result of the same hypothesis test as the significance bands and additionally of other hypotheses on the autocorrelation function (by simply checking if the hypothesized path lies within the band). While simultaneous confidence bands are widespread for example in impulse response analysis or in nonparametric regression (see montielolea2019 and references therein), they surprisingly have not been used in the context of autocorrelations to the best of our knowledge. The same seems to be the case for simultaneous significance bands, which have recently been introduced for impulse responses by inoue2025 and for probability integral transform histograms by Demetrescu2025.
We subsume both significance and confidence bands under the term inference bands. In this paper we introduce simultaneous inference bands for autocorrelation functions of stationary time series as well as regression residuals. To construct the inference bands we rely on asymptotic normality of empirical autocorrelations. For the case of a stationary time series, asymptotic normality holds under mild assumptions and the asymptotic covariance matrix is given by Bartlett's formula Bartlett1946. Its estimation is required to set up asymptotic confidence bands. We estimate the covariance matrix with a nonparametric estimator as proposed by melard1987, which resembles long-run variance or heteroskedasticity and autocorrelation consistent (HAC) estimation and makes use of Bartlett's formula. For regression residuals, we distinguish between static and dynamic regressions and several assumptions regarding (non)stationarity and establish asymptotic normality of the empirical autocorrelation vector and the formula for the covariance matrix, partly building on results by CumbaHuizinga1992. For static regressions under stationarity and exogeneity or under nonstationarity and cointegration the classical Bartlett formula for the covariance matrix well-known from the case of stationary time series continues to hold (Proposition (ref)). For dynamic regressions with lagged endogenous regressors the formula needs to be modified (Proposition (ref)).
Simultaneous significance bands with exact asymptotic coverage are easy to construct and do not even require variance estimation in the case of time series and static regressions. For dynamic regressions significance bands are particularly important since the errors being white noise is required to ensure consistent estimation of the regression coefficients. Here, the above-mentioned modification of the asymptotic covariance matrix can easily be estimated to construct bands that achieve exact coverage. Interestingly, the bands from the observed time series and static regression case are also valid, but conservative (Proposition (ref)).
We present two ways of constructing confidence bands, sup-t and Bonferroni bands, and discuss their properties and relations. While the sup-t bands have exact asymptotic coverage, the Bonferroni bands are always (in every sample) wider and thus conservative. While the differences between the bands are small for weak temporal dependence, they become considerable for stronger dependence.
We analyse the finite-sample performance of our inference bands in a simulation study. Interestingly, when testing for temporal independence, our simultaneous significance bands - even though being primarily a nice graphical diagnostic tool - show a comparable or even better performance than the classical tests of BoxPierce70 and LjungBox78. The sup-t confidence bands are very close to nominal coverage, even under strong temporal dependence. This is remarkable because the variance estimator of melard1987 does not seem to suffer from the same problems as HAC-type estimators in the case of the sample mean, which are well-known to cause severe oversizing of tests and undercoverage of confidence intervals (see e.\,g., lazarus2018 and references therein). When testing for white noise in regression residuals from dynamic regressions, our modified bands perform very well, again comparable to or better than the classical Breusch-Godfrey test (Breusch78, Godfrey78).
As already foreshadowed in this introduction, we illustrate the use of our inference bands in an application to monthly US inflation and regression residuals from static and dynamic Phillips curve regressions.
The rest of the paper is structured as follows. In Section (ref) we discuss preliminaries, introducing simultaneous inference bands in general and the limiting distribution for empirical autocorrelations, in particular Bartlett's formula for the asymptotic covariance matrix. In Section (ref) we introduce significance bands and confidence bands for stationary time series. Section (ref) treats the asymptotic distribution of empirical autocorrelations of regression residuals. Section (ref) introduces the corresponding inference bands in the residual case. Section (ref) presents the simulation study, Section (ref) the empirical applications and Section (ref) concludes. The appendix contains proofs and additional material. We provide a software implementation of our bands in the R package ACFbands, which is available at \url{https://github.com/TanjaZahn/ACFbands}. Replication material is provided at \url{https://github.com/TanjaZahn/ACFbands_replication}.
As $T\rightarrow \infty$, let $\Rightarrow$ stand for weak convergence and $\stackrel{d}{\to}$ specifically for convergence in distribution; $\stackrel{p}{\to}$ is short for convergence in probability. Moreover, $\lfloor x \rfloor = \max\left\lbrace m \in \mathbb{Z} \, | \, m \leq x \right\rbrace $, $x \in \mathbb{R}$. Further, bold capital letters stand for matrices with the $H$-dimensional identity matrix $\mI_H$, and bold lower case letters denote vectors. We write $\mA >0$ to symbolize that the matrix $\mA$ is (strictly) positive definite. Finally, let $T$ stand for the sample size, i.\,e., for the length of the time series considered, and $z_\tau$ for the $\tau$-quantile of the standard normal distribution.
To allow for a unified treatment we adopt the econometric framework by montielolea2019. Assume an $H$-dimensional parameter vector of interest $\vtheta = (\theta_1,\ldots,\theta_H)^\prime$ with a limiting normal estimator $\widehat \vtheta = (\widehat{\theta}_1,\ldots, \widehat{\theta}_H)^\prime$:
Here, $\bm{\Sigma} = \left(\sigma_{gh}\right)_{g,h = 1, \ldots, H}$ is the covariance matrix of the Gaussian random vector $\mV$. Let $M$ denote the maximum of the absolute values of correlated normal variates upon standardization:
and $q_\tau (\mSigma)$ is short for the $\tau$-quantile of $M$. Note that $q_\tau (\mSigma)$ is sometimes called an equicoordinate quantile of the multivariate normal distribution with covariance matrix $\bs \Sigma$ as it fulfils $ P \left(\sigma_{11}^{-1/2} | V_1| \leq q_\tau (\mSigma), \ \dots \ , \sigma_{HH}^{-1/2} | V_H| \leq q_\tau (\mSigma) \right) = \tau \, $. Equicoordinate quantiles can be easily calculated (essentially requiring evaluations of the CDF of a multivariate normal distribution), e.\,g., by the R package mvtnorm genz2023, which we will use for the empirical applications and simulations.
Consider now a null hypothesis $H_0$ that implies a certain value $\vtheta_0 = (\theta_{1,0},\ldots,\theta_{H,0})^\prime$ for the parameter vector and possibly a certain restriction on the covariance matrix, which we then call $\bm{\Sigma}_0 = \left(\sigma_{gh,0} \right)_{g,h = 1, \ldots, H}$. In many cases, the hypothesis will not have implications for the variance or one may decide to use the unrestricted covariance matrix such that simply $\bm{\Sigma}_0 = \bm{\Sigma}$.
\sloppy A univariate non-rejection region when testing at significance level $\alpha$ is given by $\left[\theta_{h,0} \pm z_{1-\alpha/2} \cdot \sqrt{\frac{\sigma_{hh,0}}{T}}\right]$. A simultaneous rectangular region of non-rejection is given by the Cartesian product of scaled-up univariate non-rejection regions:
We call this non-rejection region simultaneous significance band (following inoue2025) as it can be visualized easily when plotting $\widehat \theta_h$ against $h$ and controls simultaneous or joint significance in that \[ \lim_{T \to \infty} P \left(\widehat \vtheta \notin {SB}_{\vtheta} (\mSigma_0) \right) = \alpha \quad \mbox{under } H_0 . \] Similarly, one may construct rectangular confidence regions of confidence level $1-\alpha$, which amount to confidence bands when plotting $\widehat \theta_h$ against $h$:
Just combining the confidence intervals for every $\theta_h$, that is choosing the constant $c$ as $z_{1-\alpha/2}$, would lead to serious undercoverage. Thus, as with the significance bands, these pointwise confidence bands need to be scaled up appropriately to achieve a coverage of at least $1 - \alpha$. We consider two choices for the constant $c$. First, the classical Bonferroni bands closely related to the Bonferroni correction in multiple testing:
such that \[ \lim_{T \to \infty} P \left(\vtheta \in {CB}_{\vtheta}^{bf} (\mSigma) \right) \geq 1 - \alpha \, . \] Second, we consider so-called sup-t bands. They build on equicoordinate quantiles and are (asymptotically) exact by construction:
with \[ \lim_{T \to \infty} P \left(\vtheta \in {CB}_{\vtheta}^{supt} (\mSigma) \right) = 1 - \alpha \, . \] Bonferroni bands are always at least as wide as sup-t bands due to $q_{1-\alpha} (\mSigma) \leq z_{1-\alpha/(2H)}$, see Lemma (ref) in Appendix (ref) and the discussion in montielolea2019. Under independence (i.\,e., under $\mSigma=\mI_H$), the Bonferroni band becomes virtually identical to the sup-t band proschan2011, but under dependence it can be considerably wider, see Subsection (ref) for further discussion and illustration. For an overview of further alternative ways to construct simultaneous confidence bands, their relations and their inferiority to the sup-t band in that they are wider and do not achieve exact asymptotic coverage we refer again to montielolea2019.
The covariance matrix $\mSigma$ or $\mSigma_0$, respectively, is typically unknown and has to be estimated consistently. In specific cases, e.\,g., for the significance bands for the hypothesis of white noise in Subsection (ref), the covariance matrix can be fully implied by the null hypothesis such that no estimation is necessary.
Throughout, we work under the assumption that the underlying stochastic process $\{y_t\}_{t \in \mathbb{Z}}$ is covariance stationary. Define the autocovariances and autocorrelations as
We often omit the index indicating the process and just write $\gamma(h)$ or $\rho(h)$. Denote the vector of all autocovariances and autocorrelations up to a maximum lag $H \in \mathbb{N}$ by
Given a time series $\{y_t\}_{t=1}^T$ of length $T$, we estimate the autocovariances and autocorrelations via
Denote the empirical autocovariance and autocorrelation vector up to order $H$ by
Throughout, we maintain the assumption of limiting normality,
where the asymptotic covariance matrix is positive definite. This parallels ((ref)) such that the considerations of the previous subsection apply. When it comes to the estimation of $\mB^*$, however, we will work under the less general Assumption (ref) below that restricts the limiting covariance matrix to Bartlett's formula, since it underlies the construction of the conventional and very simple pointwise significance bands as discussed and depicted in the introduction, which are ubiquitous in practice and the natural competitors to our simultaneous bands. Further, Assumption (ref) ensures the classical construction of pointwise confidence bands that we extend to simultaneous ones. However, our methods for the construction of simultaneous inference bands extend in a straightforward manner to other sets of assumptions as long as $\mB^*$ is estimated consistently.
The original formula for $\mB$ by Bartlett1946 has been expressed conveniently as given in ((ref)) by BrockwellDavis91. For $y_t = \varepsilon_t$ it reduces to $\mB = \mI_H$. Different technical conditions with respect to the sequence of linear prediction errors $\{\varepsilon_t \}$ imply limiting normality with $\mB$ as in Assumption (ref). The classical restrictions by AndersonWalker1964 rely on independent and identical distribution (iid) and $\sum_{j=0}^\infty j c_j^2 < \infty$. This result is also proved in familiar textbooks, see, e.\,g., Fuller96 or BrockwellDavis91. The restriction to iid innovations was relaxed by HannanHeyde1972 who assumed a conditionally homoskedastic martingale difference sequence (MDS) and $\sum_{j=0}^\infty \sqrt{j} c_j^2 < \infty$. FrancqZakoian2009 showed that the restriction to conditional homokedasticity can be weakened to allow for conditional heterokedasticity under a symmetry condition for the MDS; as long as the squares $\varepsilon_t^2$ are not correlated (ruling out (G)ARCH, of course), the limiting Bartlett covariance matrix from ((ref)) arises.
In general, however, conditional heteroskedasticity of $\{\varepsilon_t \}$ violates Assumption (ref). FrancqZakoian2009 established a generalized Bartlett formula for the covariance matrix $\mB^*$ in ((ref)) that can be estimated along the lines of FrancqZakoian2009. RomanoThombs1996 worked in a different framework, namely weak mixing conditions, under which the Bartlett formula also does not continue to hold. This set of assumptions has been particularly popular in the literature on testing for white noise, while allowing for higher-order dependence, in which case the covariance matrix is not equal to the identity matrix and not even necessarily diagonal, see, e.\,g., lobato2002 and for an overview of this literature escanciano2009. To robustify against certain temporal dependence under absence of autocorrelation, taylor1984 suggested a self-normalizing t-type statistic. DallaGiraitisPhillips2022 carried out a rigorous asymptotic treatment with further refinements added by giraitis2024, addressing the estimation of $\mB^*$ in ((ref)), too. A different procedure to set up robust confidence intervals was advocated in hwang2024. However, none of these papers addresses simultaneous significance or confidence bands.
Building on the previous section, we now introduce simultaneous significance and confidence bands for autocorrelations and discuss their properties.
We begin with the hypothesis of white noise, that is the absence of serial correlation:
It implies by Bartlett's formula (ref) that $\bs B = \bs I_H$. Under independence, the equicoordinate quantile from (ref) equals $q_\tau (\mI_H) = z_{(1+\tau^{1/H})/2}$, which also shows up in the \v{S}id{\'a}k-correction for multiple testing sidak1967 and in \v{S}id{\'a}k confidence bands montielolea2019. This leads to a simultaneous significance band (see (ref) with $\vtheta_0=\vzeros$ and $\mSigma_0=\mI_H$), which does not require any estimation:
It is instructive to compare ${SB}_{\bm{\rho}} (\mI_H) $ with the conventional pointwise band,
Both significance bands are equally simple to construct, but under $H_0$ ${SB}_{\bm{\rho}} (\mI_H)$ provides a valid test with $\lim_{T \to \infty} P \left( \widehat \vrho \notin {SB}_{\bm{\rho}} (\mI_H) \right) = \alpha$, while pointwise bands are all the more oversized the larger $H$ is:
The pointwise band is arguably a nice and simple graphical diagnostic tool to inspect serial independence by counting the number of empirical autocorrelations that fall out of the band and checking if it is close to the expected number under independence $\alpha H$. However, the simultaneous band is certainly an at least as nice and simple diagnostic tool, where one simply has to check if the autocorrelation function leaves the band, and at the same time a valid inferential procedure. On the other hand, compared to the classical formal tests of serial independence by BoxPierce70 and LjungBox78, the simultaneous significance band adds interpretability due to its graphical representation.
Replacing $\mSigma$ in (ref) by a consistent estimator $\widehat \mB$ of the Bartlett matrix yields conservative confidence bands at level $1- \alpha$ of the Bonferroni type:
Similarly, the sup-t principle from ((ref)) provides by simply replacing $z_{1-\alpha/(2H)}$ with the respective equicoordinate quantile an asymptotically exact confidence band
The pointwise band reads
We are hence only left with estimation of the Bartlett covariance matrix.
melard1987 proposed to estimate $\bs B$ by replacing each $\rho(h)$ in (ref) by $K(h/L) \widehat{\rho}(h)$, where $K (\cdot)$ denotes a kernel and $L$ denotes the bandwidth. This parallels estimation of the long-run variance, or HAC estimation. We maintain the classical assumptions that the kernel is continuous at the origin with $K(0)=1$, has a finite number of discontinuities and is square integrable. Further, the kernel and the bandwidth fulfil $K(x) \rightarrow 0$ as $x \rightarrow \infty$, $L \rightarrow \infty$ and $L/T \rightarrow 0$. The M\'elard-Roy estimator amounts to:
melard1987 show that the resulting estimator $\bm{\widehat{B}}$ is nonnegative definite and consistent, $ \bm{\widehat{B}} \stackrel{p}{\to} \bm{B}$. We will use the triangular Bartlett kernel popularized by newey1987 for HAC estimation. For the bandwidth we consider the proposal $L = m \sqrt{T}$ by melard1987 with $m=1,3,5$. An alternative textbook rule from stock2020 for HAC-estimation of the variance of the sample mean is $L = 0.75 T^{\frac 1 3}$. A more recent recommendation by lazarus2018 is $L = 1.3 T^{\frac 1 2}$. We recommend the bandwidth rule-of-thumb $L = \sqrt{T}$ based on our simulations, for details see Section (ref).
We now analyse and illustrate the properties (i.\,e., width and coverage) of our inference bands and the relations between the different types of confidence bands on the one hand and the confidence and significance bands on the other.
The width of the generic confidence band from (ref) at lag $h$ equals
where for the bands for $\bm{\rho}$ proposed above $\sigma_{hh} = \widehat b_{hh}$ and $c$ equals $z_{1-\alpha/(2H)}$ for the Bonferroni band (ref), $q_{1-\alpha} (\widehat \mB)$ for the sup-t band (ref) and $z_{1-\alpha/2}$ for the pointwise band (ref). The formula also holds for the significance bands from Subsection (ref) with $\sigma_{hh} = 1$ and $c$ equalling $z_{(1+(1-\alpha)^{1/H})/2}$ for the simultaneous band (ref) and $z_{1-\alpha/2}$ for the pointwise band (ref). Thus, the width of all the bands grows with the scaling factor $c$ and shrinks with the sample size $T$. The only influence of the lag $h$ is through the variance $\sigma_{hh}$. Consequently, the width of the significance bands for $\bm{\rho}$ is constant over the lags. The width of the confidence bands depends on the data only through the estimated covariance matrix $\widehat \mB$, where for the Bonferroni bands the influence is only through the variance $\widehat b_{hh}$ and for the sup-t bands additionally through the equicoordinate quantile $q_{1-\alpha} (\widehat \mB)$.
We now analyse the properties and relations of the inference bands for a specific data generating process (DGP), the autoregressive process of order 1 (AR(1) process),
with four different parameter values, $\phi=0, 0.25, 0.5, 0.75$, representing varying degrees of temporal dependence from white noise to rather strong persistence. The normality of the innovations does not play a role here, but has to be specified for later on when we use this as DGP in our simulations. This DGP is simple in that it allows to control the degree of temporal dependence in one parameter, but at the same time it is realistic and relevant in that it yields a good approximation to the behaviour of many economic time series.
We do not estimate the Bartlett covariance matrix $\mB$, but assume that the AR(1) parameter $\phi$ is known and thus the covariance matrix can be calculated by the formulas in Cavazos-Cadena1994; see Appendix (ref) for the $\mB$-matrices for $\phi=0,0.25, 0.5, 0.75$. Thus, the following width comparisons can be regarded as asymptotic, complementing the width and coverage comparisons in the simulation study. Also the coverages we plot below are asymptotic, making use of the central limit theorem from (ref).
In the left panel of Figure (ref) we plot the relative width of the $90\%$ sup-t and pointwise confidence band, which equals $q_{0.9} (\mB)/z_{0.95}$ and does not change with $h$, against different values for the length of the band $H$ and for the different values of $\phi$. Note that the confidence bands for $\phi=0$ also represent the respective significance bands. Of course, the width of the sup-t band rises with its length $H$, but grows slowly and at a decreasing rate. The stronger the temporal dependence is, the smaller the relative width (as equicoordinate quantiles naturally get smaller under stronger dependence). Under independence and $H=25$, the sup-t band is roughly 1.75 times as wide as the pointwise band. The right panel of Figure (ref) depicts the asymptotic coverage of the pointwise bands. As expected, they show strong undercoverage increasing with $H$ and decreasing with the degree of temporal dependence, while the sup-t bands have exact asymptotic coverage of 0.9.
The left panel of Figure (ref) depicts the relative width of the $90\%$ Bonferroni and sup-t band, $z_{1-1/(20H)}/q_{0.9} (\mB)$, the right panel depicts the asymptotic coverage of the Bonferroni band. As mentioned in Section (ref), the Bonferroni band is always at least as wide as the sup-t band. Its conservativeness in terms of width and coverage increases with the degree of persistence. For $\phi=0.75$ it is about $15\%$ wider than the sup-t band for most values of $H$ and has an asymptotic coverage of about $95\%$ instead of the desired $90\%$.
Now we analyze how the width of the sup-t bands changes with the lag $h$. Figure (ref) plots the standard deviation $\sqrt{b_{hh}}$ against $h$, which determines the behaviour of the width for a certain value of $\phi$. An interesting pattern emerges. While for the case $\phi=0$, the width of the band is constant over $h$, this is not the case for the other values of $\phi$. There, for $h=1$ the variance $b_{11}$ is smaller than 1 (the higher $\phi$, the smaller) and larger than 1 for all other lags (the higher $\phi$, the larger) and rising with $h$, quickly approaching a certain limit. That is, the width of the confidence band first grows with $h$ and from a certain small lag on stays virtually constant. When computing the average width over $h$, the bands are the wider the stronger the degree of temporal dependence is, but the width for the first lag (which shows the opposite behaviour) can be especially important in terms of power (see our simulations in Section (ref)).
Overall, the counteracting effects on the width, see (ref), of the equicoordinate quantile (which decreases with degree of temporal dependence) and standard deviation (which increases with degree of temporal dependence) is dominated by the standard deviation so that the average width of the confidence bands (averaged over h) increases with the degree of temporal dependence (see again also our simulation results in Section (ref)) as one would expect. In particular, the significance bands (with unchanged width over $h$) are narrower than the corresponding confidence bands (again averaged over $h$).
Consider the regression model
with ${\vx}_t^\prime = (x_{1,t}, \ldots, x_{K,t})$ being a vector process of dimension $K$. In ((ref)), the effective sample size is $T$ such that we assume additional starting values in case that the vector ${\vx}_t$ contains lagged values. In particular, lagged endogenous regressors are not ruled out. Throughout, we work under the assumption that the errors are stationary processes as in Assumption\,(ref) and have mean zero.
We are interested in simultaneous inference bands for the true autocorrelation structure $\vrho_{ e} $, where the vector $\vrho_{ e} $ contains the theoretical error autocorrelations. Obviously, ((ref)) holds for $\widehat \vrho_{ e} = (\widehat \rho_{e} (1), \ldots, \widehat \rho_{ e} (H))^\prime $, and we could proceed as in the previous section if the regression errors $\{e_t\}$ were observable. Since they are not, we investigate the least squares (LS) residual autocorrelations:
\[ \widehat e_t := y_t - \widehat a - \widehat \beta^\prime \vx_t \, = e_t - (\widehat a - a) - (\widehat \vbeta - \vbeta )^\prime \vx_t \, . \] They build on \[ \widehat \gamma_{\widehat e} (h) := \frac{1}{T} \sum_{t=h+1}^T \widehat e_t \widehat e_{t-h} \, , \quad \widehat \vgamma_{\widehat e} := (\widehat \gamma_{\widehat e} (1), \ldots, \widehat \gamma_{\widehat e} (H))^\prime . \] In this section, we establish the limiting distributions under several sets of assumptions, such that simultaneous inference bands are available asymptotically. Two cases have to be distinguished. First, we treat static regressions under different assumptions concerning (non)stationarity; second, we turn to dynamic regressions (lagged endogenous regressors).
Note that $\widehat e_t = e_t + O_p (T^{-0.5})$, which will be met in this section, does not imply that ((ref)) continues to hold when replacing $\widehat \vrho_{e}$ by $\widehat \vrho_{\widehat e}$. Famous (counter)examples are lagged endogenous regressors under stationarity and tests against residual autocorrelation (Durbin's $h$ instead of the usual Durbin-Watson statistic, see Durbinsh); and under cointegration Shin1994 showed that the validity of a residual test requires so-called efficient estimation beyond LS.
We first assume that $\vx_t$ from ((ref)) does not contain lagged endogenous regressors. We allow for different assumptions concerning the regressors, leading all to the identical limiting behaviour summarized in Proposition\,(ref). We begin with the stationary case where lagged exogenous regressors are not ruled out, e.\,g., $\vx_t^\prime =(z_t, z_{t-1}, \ldots, z_{t-K+1})$.
Next, let $\mB (r) = (B_e (r), \mB_x^\prime (r))^\prime$ denote a vector Brownian motion of dimension $K+1$ to cover the case of regressors that are integrated of order 1, $I(1)$, but not cointegrated. This turns ((ref)) into a cointegrating regression. In order not to burden the exposition with too many technicalities, we simply assume a functional central limit theorem to hold without discussing assumptions behind it; for details see, e.\,g., PhillipsDurlauf86.
On top of being integrated, the regressors are allowed to be driven by linear time trends (“integrated with drift”); not all components of $\vdelta$ have to be different from 0.
For all three cases we can prove one common result: The limit result ((ref)) is recovered notwithstanding the replacement of $\{ e_ t \}$ by the residual process $\{ \widehat e_ t \}$.
{\sc Proof} See Appendix (ref).
Proposition (ref) is not restricted to LS. Obviously, one could consider valid instrumental variable estimation and relax exogeneity in Assumption (ref) accordingly. Similarly, under Assumption (ref) or (ref), one may allow for so-called efficient cointegrating regression as defined by saikkonen1991, see also phillips1990 and park1992.
Proposition (ref) continues to hold when detrending the regressors from Assumption (ref) through Assumption (ref), see Corollary (ref) in Appendix (ref). Further, Assumption (ref) can be generalized to account for trend-stationary regressors, see Corollary (ref) in Appendix (ref).
Now we allow for lagged endogenous regressors in ${\vx}_t$, which covers autoregressions and the autoregressive distributed lag (ARDL) model. Here, we restrict the regression errors to be white noise as required to ensure LS consistency in the presence of lagged endogenous regressors:
The limiting theory relies on predetermined regressors relative to the errors defined in terms of the following information set, \[ \mathcal{I}_t := \sigma(\varepsilon_{t}, \varepsilon_{t-1}, \ldots , \vx_{t+1}, \vx_t, \vx_{t-1}, \ldots) \, , \] where $\sigma(...)$ denotes the sigma algebra generated by the respective random vectors. We maintain stationary processes meeting the following assumption.
By Assumption (ref) it follows that $\{\vx_t \varepsilon_t \}$ forms a vector MDS, and the regressors are predetermined in that \[ \E[\vx_t \varepsilon_{t+h}] = \vzeros \, , \quad h \geq 0 \, , \] although they may correlate with past errors:
By an MDS central limit theorem and the continuous mapping theorem it follows for the LS estimator under Assumption 6 that \[ \sqrt{T} (\widehat \vbeta - \vbeta) \ \stackrel{d}{\to} \ \mathcal{N}_K (\vzeros, \mSigma_x^{-1} \mSigma_{x \varepsilon} \mSigma_x^{-1} ) \, . \] For LS residuals $\widehat e_t = y_t - \widehat a - \vx_t^\prime \widehat{\vbeta}$ we have the following result.
{\sc Proof} Follows from CumbaHuizinga1992, see Appendix (ref) for details.
If there are no lagged endogenous regressors, $\mGamma = \mZeros_{HK}$, then standard normal inference arises, $\mSigma_\rho = \mI_H$. With endogenous regressors, the asymptotic covariance matrix $\mSigma_\rho$ differs from $\mI_H$ by a correction term, which is in contrast to the cases of observed time series and regression residuals from static regressions.
Remember the discussion following Assumption (ref), which does not rule out conditional heteroskedasticity. In such a framework, consistent estimation of $\mSigma_\rho$ in the tradition of Eicker-White is straightforward, using plug-in estimation based on (ref) and the following estimators: \[ \widehat \mSigma_{x\varepsilon} := \frac{1}{T}\sum_{t=1}^T \widehat e_t^2 ( \vx_t - \overline \vx ) (\vx_t - \overline \vx )^\prime\ \stackrel {p}{\to} \ \mSigma_{x\varepsilon}\, , \] \[ \widehat \mSigma_{x} := \frac{1}{T}\sum_{t=1}^T ( \vx_t - \overline \vx ) (\vx_t - \overline \vx )^\prime\ \stackrel {p}{\to} \ \mSigma_{x}\, , \]
and (with $\mGamma$ from ((ref))) \[ \widehat \vc_h := \frac{1}{T}\sum_{t=h+1}^T \vx_t \widehat e_{t-h} \ \stackrel {p}{\to} \ \vc_h \, ; \] see also CumbaHuizinga1992. Denote the resulting estimator for $\mSigma_\rho$ by $\widetilde \mSigma_\rho^{het}$. It is, however, not guaranteed to be positive definite. We will address this issue below after discussing the special case of conditionally homoskedastic errors.
Under Assumption (ref) it follows that $\mSigma_{x\varepsilon} = {\mSigma}_x \sigma^2_\varepsilon$ and\footnote{Under this assumption, Davidson00 derived $\mSigma_\rho$ for the univariate case of $H=1$.}
The matrix $\mSigma_\rho^{hom}$ is (weakly) smaller than $\mI_H$ in every entry since the matrix that is subtracted from $\mI_H$ is nonnegative in every entry due to the positive definiteness of $\mSigma_x$. This observation is crucial to establish a role for the significance bands from the case of observed time series or errors from static regressions -- they are valid, but conservative, see Proposition (ref). Consistent estimation of $\mSigma_\rho^{hom}$ is again straightforward, plugging in $\widehat \mSigma_{x}$, $\widehat{\sigma}^2_\varepsilon$ and $\widehat \vc_h$ as defined above for their theoretical counterparts in (ref). Denote the resulting estimator by $\widetilde \mSigma_\rho^{hom}$.
The consistent plug-in estimators $\widetilde \mSigma_\rho^{hom}$ and $\widetilde \mSigma_\rho^{het}$ described so far are not guaranteed to be positive definite. In fact, they tend to be negative definite quite often. For an explanation of this behaviour and a better understanding of our proposed solution note that usually current regressors will only correlate strongly with recent errors. Thus, $\vc_h$ from (ref) will be essentially zero except for small $h$, leading to the correction matrix that is subtracted from $\mI_H$ in (ref) and (ref) being close to the zero matrix except for its upper left part. Due to estimation uncertainty, the parts that are approximately zero can become considerably positive and lead to $\widetilde \mSigma_\rho^{hom}$ or $\widetilde \mSigma_\rho^{het}$ becoming negative definite. This is especially easy to see for $\widetilde \mSigma_\rho^{hom}$, where the matrix that is subtracted is nonnegative in all of its entries, that is, can only fluctuate in the positive direction. When we are concerned with any of the two plug-in estimators, we just write $\widetilde \mSigma_\rho$.
This behaviour suggests a simple solution. The subtracted matrix should be shrunk to zero except for its upper left part. We propose a simple shrinkage algorithm, which on the one hand reduces estimation uncertainty as it avoids estimation of many entries of a matrix that are essentially zero and on the other hand guarantees positive definiteness of the resulting estimator.
As a special case of lagged endogenous regressors we briefly discuss the error-correction model (ECM) that combines Assumption\,(ref) with Assumption\,(ref) in Appendix (ref).
The results from Subsection (ref) imply that for error autocorrelations $\bm{\rho}_{e}$ from a static regression model estimated by residual autocorrelations, see (ref), inference bands can be constructed as laid out in Section (ref) for observed time series. More precisely, the significance band ${SB}_{\bm{\rho}_{e}} (\mI_H)$ follows formula (ref); the Bonferroni and sup-t confidence bands ${CB}_{\bm{\rho}_e}^{bf}$ and ${CB}_{\bm{\rho}_e}^{supt}$ follow formulas (ref) and (ref), respectively, when replacing empirical autocorrelations of observed time series $\widehat \rho (h)$ in the formulas and in the estimator for the covariance matrix $\widehat \mB$ by empirical residual autocorrelations $\widehat \rho_{\widehat e} (h)$.
For residual autocorrelations from dynamic regressions, however, the results from Subsection (ref) have several implications for the construction of inference bands. Confidence bands do not really make sense in this setting as one needs the assumption of white noise for the consistency in a dynamic regression model. In contrast, significance bands are all the more important since testing the white noise hypothesis for the regression errors $\{\varepsilon_t\}$ amounts to a test of model misspecification. Proposition (ref) shows that the formula for the covariance matrix needs to be adapted and in contrast to the case of time series and static regressions is not equal to the identity matrix. More precisely, under Assumption (ref), which implies
we obtain a significance band with asymptotic coverage $1-\alpha$ combining ((ref)) with Proposition (ref):
where $\widehat{\mSigma}_\rho$ stands either for the Eicker-White-type estimator $\widehat{\mSigma}_\rho^{het}$ or the estimator assuming homoskedasticity $\widehat{\mSigma}_\rho^{hom}$ (see Algorithm (ref)) and $\widehat{\sigma}_{\rho,gh}$ denotes one of the elements of $\widehat{\mSigma}_\rho$. This exact simultaneous significance band is in contrast to the naive simultaneous significance band. The naive significance band is defined by simply ignoring the residual effect in the presence of lagged endogenous regressors. This amounts to the simultaneous band in the case of observed time series (see (ref)):
Interestingly, under conditional homoskedasticity of the errors, the naive significance band ${SB}_{\bm{\rho}} (\mI_H)$ remains valid, but is conservative.
{\sc Proof} See Appendix (ref).
The first result remarkably is a finite-sample result that states that as long as the exact bands are estimated using homoskedastic standard errors, the naive bands are always weakly wider (in every sample). The second result states that when the true covariance matrix is used, under homoskedasticity the naive bands are at least as wide as the exact bands. This in particular means that as long as a consistent estimator is used, this relation holds asymptotically, which also implies that the naive bands are conservative, that is, have asymptotic overcoverage. These results follow essentially from the facts that equicoordinate quantiles are larger under independence than under any other correlation structure (see Lemma (ref) in the Appendix, which essentially follows from results by sidak1967) and that the covariance matrix under homoskedasticity $\mSigma_{\rho}^{hom}$ and its estimator $\widehat \mSigma_\rho^{hom}$ are weakly smaller than the unity matrix (in every element).
We now analyse and illustrate the structure of the asymptotic variance $\mSigma_\rho$ and the naive and exact simultaneous significance band for a dynamic regression. We consider again an AR(1) process as in (ref). We then estimate an AR(1) model with intercept by LS,
compute the residuals, $\widehat e_t = y_t - \widehat a - \widehat \vbeta^\prime \vx_t = y_t - \widehat a - \widehat \phi y_{t-1}$, i.\,e., $\vx_t = y_{t-1}$, and obtain $\widehat{\vrho}_{\widehat e}$. We now calculate the covariance matrix of the residual autocorrelations from (ref) (since the error term is homoskedastic). It holds that $\sigma^2_\varepsilon = 1$ and that $\mSigma_x = \mathrm{Var}(y_{t-1}) = \mathrm{Var}(y_t) = \frac{1}{1-\phi^2}$. Further, it holds that
Thus, $$\mGamma := (\vc_1, \ldots, \vc_H)^\prime = (1, \phi, \phi^2, \ldots, \phi^{H-1}) \, .$$ Collecting terms yields
where $( a_{ij} )_{1 \leq i,j \leq H}$ denotes a matrix with entries $a_{ij}$.
As discussed before Algorithm (ref), the covariance matrix is substantially different from the identity matrix in its upper left corner, but approaches the identity matrix quickly when moving away from that corner. Consequently, the naive significance band ${SB}_{\bm{\rho}} (\mI_H)$ differs substantially in its width from the exact band ${SB}_{\bm{\rho}_{\varepsilon}} (\mSigma_\rho^{hom})$ for small $h$ due to the asymptotic variances being much smaller than 1 (the equicoordinate quantiles $q_{1-\alpha} (\mSigma_\rho^{hom})$ and $q_{1-\alpha} (\mI_H)$ do not differ substantially), but is virtually identical for moderate and large $h$. Figure (ref) depicts the two bands for $1-\alpha=0.9$, $\phi=0.5$ and $T = 200$.
This behaviour of the width carries over to the asymptotic coverage. Short naive bands, i.\,e., with small values of $H$, will have severe overcoverage, while the overcoverage will be mild for larger values of $H$. Figure (ref) plots the asymptotic coverage for different values of $\phi$.
We analyse the finite-sample behaviour of our inference bands in simulations. Firstly, we consider significance bands, then confidence bands for observed time series. Finally, we turn to significance bands for dynamic regression errors.
We simulate time series from an AR(1) process as in (ref) with $\phi=0,0.25,0.5,0.75$. We use the sample sizes $T=50,200,800$, the maximum lags $H=1,10,25$, a nominal coverage of $1- \alpha = 0.9$ and 1000 simulation runs. How to choose $H$ is not addressed in our paper. Different rules of thumb are employed in applied work. The R function acf of the stats package, e.\,g., uses $10 \, \log_{10} (T)$ by default. In our Monte Carlo study, we also include smaller values of $H$.
In Table (ref) we report the rejection frequencies of the null hypothesis of white noise (i.\,e., one minus the coverage) of the simultaneous and the pointwise significance bands ${SB}_{\bm{\rho}} (\mI_H)$ from (ref) and ${PSB}_{\bm{\rho}} (\mI_H)$ from (ref), respectively. The case $\phi=0$ corresponds to size and all other cases to power. We also consider the classical test for the null of white noise of BoxPierce70 and its modification by LjungBox78 (and the sup-t confidence bands ${CB}_{\bm{\rho}}^{supt} (\widehat \mB)$ from (ref), which we discuss in the next subsection). For the simultaneous significance bands the size is very close to 0.1 for $T=800$, while they are conservative for the smaller sample sizes, especially for large values of $H$. Compared to the Box-Pierce test, which behaves similarly in terms of size, they tend to have higher power. Even for the Ljung-Box test, which is oversized for $H=25$, they behave similarly in terms of power. Note, however, that the power of the Box-Pierce and Ljung-Box tests decreases with growing $H$. This is due to our specific alternative where the violation of $\rho(h)=0$ is particularly strong for small values of $h$, such that power is watered down with growing $H$. The latter is true only to a lesser extent for the simultaneous bands where $c = z_{(1+(1-\alpha)^{1/H})/2}$ and more generally the width does not vary strongly with large $H$. Thus, even though we view the significance bands rather as a very useful graphical diagnostic tool, which allows to visually assess how much variation in the empirical autocorrelations is still compatible with the null hypothesis of white noise, they seem to be a valid competitor in terms of size and power for classical tests of this hypothesis. Unsurprisingly, the pointwise significance bands are seriously oversized (with size around $1-0.9^H$ as expected, see (ref)) and thus not useful for statistical inference. In Table (ref) in the Appendix we also report the width of the significance bands. As expected, the simultaneous bands get wider with growing dimension of the autocorrelation vector $H$ and narrower with growing sample size $T$.
We again simulate time series from an AR(1) process as in (ref) with $\phi=0,0.25,0.5,0.75$, using sample sizes $T=50,200,800$, maximum lags $H=1,10,25$, a nominal coverage of $1- \alpha = 0.9$ and 1000 simulation runs. We report the coverage of sup-t, Bonferroni and pointwise confidence bands in Table (ref) (by checking the relative frequency with which the true autocorrelation function of the respective AR(1) process $\bm{\rho} := (\phi,\phi^2,...,\phi^H)^\prime$ is contained in the confidence band). We also report the corresponding average widths (averaged over the lags $h$ and simulation runs) of the bands in Table (ref) in the Appendix. The coverage of the sup-t bands is very satisfactory, that is, quite close to nominal coverage. This is remarkable since at least in the case of the sample mean the estimation of its variance under temporal dependence is notoriously difficult and confidence intervals employing HAC-type estimators are known to suffer from serious undercoverage (see, e.\,g., lazarus2018 and references therein). However, in the case of empirical correlations, the estimator of melard1987 does not seem to suffer from similar problems. As expected, the average width of the sup-t bands increases with the length of the bands $H$ and degree of persistence represented by $\phi$ and decreases with sample size $T$. In line with the theoretical discussion in Section (ref) (see in particular Figure (ref)), the Bonferroni bands are very close in coverage and width to the sup-t bands under independence and for weak temporal dependence, but show overcoverage and are markedly wider for $\phi=0.5$ and especially for $\phi=0.75$. Unsurprisingly, the pointwise bands show serious undercoverage. For the results discussed so far, estimation of $\mB$ relies on the Bartlett window with bandwidth choice $L=T^{1/2}$. We report simulation results comparing the different bandwidth choices discussed in Section (ref) in Table (ref) in the Appendix. All bandwidth choices work quite well and are comparable in terms of coverage. Thus, the performance of the bands is quite robust with respect to the bandwidth choice. The smaller bandwidths work a bit better under weaker temporal dependence, the larger bandwidths under stronger temporal dependence. Thus, the medium bandwidth $T^{1/2}$ seems to be a good all-purpose and default choice.
The confidence bands can also be used for testing for white noise (by checking whether $\bf{0}$ is covered by the band). In terms of size and power, see Table (ref), the confidence bands behave very similar to the significance bands. This is interesting since the two types of bands have quite different properties. The confidence bands are on average (averaged over simulation runs and lags $h$) wider, see Table (ref) in the Appendix, than the significance bands, see Table (ref) in the Appendix. They tend to get wider with the lag $h$, see the theoretical discussion in Section (ref) and in particular Figure (ref) and the empirical examples in Section (ref), and depend on an estimated variance, while the significance bands have constant width and do not depend on estimated quantities. For $h=1$, however, which is particularly important for power, the confidence bands are narrower than the significance bands, which is probably the reason why they perform as good in terms of power.
Finally, we analyse the finite-sample performance of our significance bands for the null hypothesis that the errors in a dynamic regression model are white noise. We simulate time series of length $T$ from an AR(2) process,
As the model we assume an AR(1) process as in (ref) and run the corresponding LS regression (ref). The true error in this AR(1) model is $e_t = \phi_2 y_{t-2} + \varepsilon_{t}$, that is, size corresponds to the case $\phi_2=0$. Table (ref) reports size and power for the exact simultaneous significance band ${SB}_{\bm{\rho}_{\varepsilon}} (\widehat \mSigma_\rho^{hom})$ from (ref) (we use the variance estimator $\widehat \mSigma_\rho^{hom}$ since the errors under the null ${\varepsilon_t}$ are homoskedastic), the naive simultaneous band ${SB}_{\bm{\rho}} (\mI_H)$ from ((ref)) and the naive pointwise significance band ${PSB}_{\bm{\rho}} (\mI_H)$ currently used in practice. Furthermore, we consider the Ljung-Box test, which is not tailored to residuals similarly to the naive simultaneous band, and the Breusch-Godfrey test Breusch78, Godfrey78, which is designed for this setting. The exact simultaneous band has nearly exact size for $T=800$ and is conservative for the smaller sample sizes as in the case of significance bands for observed time series. The naive simultaneous bands are in most cases slightly more conservative and for $H=1$ very conservative, which is explained by the way too large variance of the naive simultaneous band for $h=1$, see (ref) and figures (ref) and (ref). The naive pointwise band is heavily oversized. In most cases the exact simultaneous band is more powerful than the Portmanteau tests, which are in turn more powerful than the naive simultaneous bands. We also report the average widths of the confidence bands (averaged over $h$ and over simulation runs) in Table (ref) in the Appendix.
We construct inference bands for inflation as well as for regression residuals from various specifications of a Phillips curve regression. For our analysis, we use data from the FRED-MD database mccracken2016 from 1961:01 to 2024:06. Plots of the monthly inflation and unemployment series used below can be found in Figure (ref) in the Appendix.
Figure (ref) displays the autocorrelation function with simultaneous $90\%$ confidence bands for monthly US inflation (month-on-month, seasonally adjusted). We show sup-t, Bonferroni and pointwise bands with a bandwidth choice of $T^{1/2}$ for variance estimation. The width of the bands increases from the first lag to the second and then again to the third and then stays almost constant over the lags. This behaviour is due to the smaller asymptotic variance for the first lags as we also observe in the AR(1) example in Section (ref). Of course the pointwise band is the narrowest, but essentially useless due to its heavy undercoverage. The sup-t band is in turn narrower than the Bonferroni band as expected.
Now, we turn to regression residuals of the Phillips curve. The Phillips curve is a widespread econometric model used to study the relation between unemployment and inflation (e.\,g., Gordon2013, Coibion2015, Blanchard2015, Blanchard2016, Ball2019, Smith2023). Another strand of the literature employs the model for forecasting (e.\,g., Atkeson2001, Stock1999, Stock2007, Stock2008, Dotsey2018). Throughout we assume that the so-called non-accelerating inflation rate of unemployment (NAIRU) is constant. Additionally, we approximate expected inflation by past inflation. Coefficient estimates as well as plots of the residuals for the following regression models are reported in tables (ref) and (ref) as well as figures (ref) to (ref) in the Appendix. We begin by estimating a static Phillips curve by LS,
where $\pi_t$ is inflation, $\Delta \pi_t = \pi_t - \pi_{t-1}$, and $u_t$ is the unemployment rate. The residual autocorrelogram is shown in Figure (ref) together with the pointwise significance band ${PSB}_{\bm{\rho} } (\mI_H)$ from (ref) and the simultaneous band ${SB}_{\bm{\rho}} (\mI_H)$ according to ((ref)). From the empirical autocorrelation function together with the simultaneous band we can conclude that the errors from the static Phillips curve regression still contain significant (on the $10\%$ level) dynamics, which are not explained by the model. Thus, one should probably move to a dynamic Phillips curve specification.
Adding $p$ and $r$ lags of the dependent variable and unemployment yields:
We again estimate this model by LS. Figure (ref) displays the autocorrelation function with the estimated significance bands: the naive pointwise significance band ${PSB}_{\bm{\rho}} (\mI_H)$, the naive simultaneous band ${SB}_{\bm{\rho}} (\mI_H)$ from ((ref)) and the exact simultaneous significance band ${SB}_{\bm{\rho}} (\widehat \mSigma_\rho^{hom})$ from (ref). We use the estimator $\widehat \mSigma_\rho^{hom}$ as for macroeconomic time series we do not expect much heteroskedasticity in the errors, which is confirmed by residual plots discussed at the end of this Section. The left side of the figure shows that using only one lag of the dependent variable ($p=1$) and the contemporaneous unemployment rate ($r = 0$) as regressors still yields significant (on the $10\%$ level) autocorrelations of the error. Apart from the first couple of lags, both the simultaneous bands (naive and exact) are very similar. Adding a full year of lags of both variables on the right-hand side of the figure ($p=r=12$) visibly reduces the empirical autocorrelations. The white noise hypothesis for the regression errors is no longer rejected according to both simultaneous bands. In contrast, the invalid pointwise bands would lead to the opposite conclusion.
While the forecasting literature tends to formulate the Phillips curve in terms of differences, the traditional Phillips curve is usually studied with inflation in levels. Thus, Figure (ref) shows the autocorrelation function of the residuals with significance bands for the following model, again estimated by LS:
Again, for $p=1$ and $r=0$ on the left-hand side there are significant (on the significance level $10\%$) dynamics in the error, likely invalidating the LS estimates. As before, adding lags ($p=r=12$) reduces the magnitude of the autocorrelation function. However, it slightly exceeds the upper bound of both simultaneous significance bands at lag 15 leading to an overall rejection of the null hypotheses at $10\%$.
Finally, we perform some graphical checks of the assumptions underlying the construction of our inference bands for dynamic regressions (assumptions (ref) and (ref)) and a robustness check with respect to the sampling window. The right-hand side of Figure (ref) in the Appendix shows the time plot of the residuals for the case for which the null of white noise is not rejected, i.\,e., $p=r=12$ in equation (ref). The plot suggests that there is virtually no autoregressive conditional heteroskedasticity. We also plot the residuals from this regression against the most relevant (recent) regressors, i.\,e., against $\Delta \pi_{t-1}$ and $u_{t}$, in Figure (ref) in the Appendix. Again, there seems to be virtually no conditional heteroskedasticity. Thus, the assumption of conditional homoskedasticity seems to be a good approximation of reality.
As a robustness check with respect to the time period, we run the same dynamic regression models from 1985 onwards instead of 1961, see Figures (ref) and (ref) in the Appendix. The empirical autocorrelations and significance bands look very similar to the ones above. For both models the null of white noise is again clearly rejected in the case $p=1$ and $r=0$. Interestingly, for the model in differences we have a different test decision of the naive and the exact band for $p=r=12$ due to lag 4 on the right-hand side of Figure (ref), which falls a tiny bit out of the exact band. Lastly, tables (ref) and (ref) in the Appendix show that most lags of unemployment are insignificant at $10\%$ significance level. Discarding them in equations (ref) and (ref) yields very similar plots of the empirical autocorrelation functions and the respective inference bands, which we do not report here.
We propose and discuss simultaneous significance and confidence bands for autocorrelations, which should supersede the classical pointwise non-rejection bands that are added by default to a plot of the empirical autocorrelation function in most statistical software. If the null hypothesis of white noise is of primary interest, in particular in the case of regression residuals, our simultaneous significance bands are the right choice, being a graphical diagnostic tool as simple to construct as the pointwise bands and even simpler to interpret, but at the same time providing valid statistical inference. For observed time series and regression residuals from static regressions we show that those bands have exact asymptotic coverage, while they are conservative for dynamic regressions, where we provide a simple modification that leads again to exact coverage. If the white noise hypothesis is not of particular interest, simultaneous confidence bands of the sup-t type are our recommended choice, providing valid uncertainty quantification around the empirical autocorrelation function. Bonferroni bands are a valid, but unnecessarily conservative alternative.
Since we are the first to propose simultaneous inference bands for autocorrelations, there are several directions for extension. We work under fairly classical assumptions, that is, we exclude innovations with correlated squares in Assumptions\,(ref) or (ref) such that the limiting Bartlett covariance $\mB$ arises. However, as discussed around Assumption\,(ref), more general sets of assumptions still lead to limiting normality as in ((ref)). Equipped with limiting normality and a consistent estimator of the generalized covariance matrix, the construction of our proposed inference bands can proceed by simply replacing $\widehat \mB$ by an estimator of the generalized matrix.
Additional extensions of interest may address further time series correlations. First, asymptotic significance bands for cross-correlations are immediately available. Hypotheses of interest could be “no cross-correlation up to a certain lead and lag”, “no cross-correlations and one process is white noise”, “no cross-correlation and both processes are white noise”. Limiting normality like in (ref) arises, where the shape of the Bartlett covariance matrix depends on the specified null hypothesis; see Box1994 for details. Second, bands for partial autocorrelations up to some order $H$ are straightforward, see Box1994 for limiting normality and references. The asymptotic covariance matrix depends on the null hypothesis under test, e.\,g., AR($p$) for some specified $p$. Third, our approach can be carried to a multivariate framework. Lutkepohl2005 established limiting normality and provides the covariance matrix for a vector of residual autocorrelations from vector autoregressions under the null hypothesis of white noise. Since vector autoregressions contain lagged endogenous regressors, the covariance matrix displays a similar structure like in Proposition (ref) above.
A further possible extension is to replace Pearson correlation in the definition of the autocorrelation by other dependence measures. To measure e.\,g., autocorrelation in the tails and not around the means one could use the so-called quantilogram linton2007, which measures dependence around quantiles. This is also possible for arbitrary statistical functionals instead of quantiles fissler2023. Again, only limiting normality and the form of the asymptotic variance are required for the empirical versions of those dependence measures and then the construction of the inference bands can be carried out as laid out in this paper.
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