2.a Gaussian test statistic matrix

The fundamental tools of testing is a gausisan test statistic matrix whose entries marginally follow standard normal under null.

The test statistic is constructed as follows. Observation \(\mathbf{y}_t\) follows an autoregressive structure, \(\mathbf{y}_{t+1} = \mathbf{A}_* \mathbf{y}_{t} + \mathbf{e}_{t}\), with the error term \(\mathbf{e}_{t} = - \mathbf{A}_* \mathbf{\epsilon}_{t}+ \mathbf{\epsilon}_{t+1} + \mathbf{\eta}_{t}\). Then the lag-1 auto-covariance of the error \(\mathbf{e}_t\) is of the form, \[ \mathbf{\Sigma}_e = \mathrm{Cov}(\mathbf{e}_{t},\mathbf{e}_{t-1}) = -\sigma_{\epsilon,*}^2 \mathbf{A}_*. \] This suggests that we can apply the covariance testing methods on \(\mathbf{\Sigma}_e\) to infer transition matrix \(\mathbf{A}_*\). However, \(\mathbf{e}_t\) is not directly observed. Define generic estimates of \(\Theta_*\) by \(\left \{\hat{\mathbf{A}},\hat\sigma_{\epsilon}^2, \hat\sigma_{\eta}^2 \right\}\) (sparse EM estimates also work). We use them to reconstruct this error, and obtain the sample lag-1 auto-covariance estimator, \[\hat{\mathbf{\Sigma}}_e = \frac{1}{T-2} \sum_{t=2}^{T-1} \hat{\mathbf{e}}_{t}\hat{\mathbf{e}}_{t-1}^\top, \ \text{where} \ \ \hat{\mathbf{e}}_{t} = \mathbf{y}_{t+1} - \hat{\mathbf{A}} \mathbf{y}_{t} - \frac{1}{T-1}\sum_{t'=1}^{T-1} (\mathbf{y}_{t'+1} - \hat{\mathbf{A}}\mathbf{y}_{t'}).\]

This sample estimator \(\hat{\mathbf{\Sigma}}_e\), nevertheless, involves some bias due to the reconstruction of the error term, and also an inflated variance due to the temporal dependence of the time series data. Bias and variance correction lead to the Gaussian matrix test statistic \(\mathbf{H}\), whose \((i,j)\)th entry is,
\[ H_{ij} = \frac{ \sum_{t=2}^{T-1} \{ \hat{e}_{ t,i}\hat{e}_{ t-1,j} + \left( \hat{\sigma}_{\eta}^2 +\hat{\sigma}_{\epsilon}^2 \right) \hat{A}_{ij} - \hat{\sigma}_\eta^2 A_{0,ij} \} }{\sqrt{T-2} \; \hat{\sigma}_{ij}}, \quad i,j \in [p]. \] Lyu et al. (2020) proves that, under mild assumptions,
\[ \frac{ \sum_{t=2}^{T-1} \{\hat{e}_{ t,i}\hat{e}_{ t-1,j} + \left( \hat{\sigma}_{\eta}^2 +\hat{\sigma}_{\epsilon}^2 \right) \hat{A}_{ij} - \hat{\sigma}_\eta^2 A_{*,ij} \}}{\sqrt{T-2} \; \hat{\sigma}_{ij}}\rightarrow_{d} \mathrm{N}(0, 1) \] uniformly for \(i,j \in [p]\) as \(p, T \to \infty\).

2.b Global testing

The key insight of global testing is that the squared maximum entry of a zero mean normal vector converges to a Gumbel distribution. Specifically, the global test statistic is
\[ G_{\mathcal{S}} = \max_{(i,j) \in \mathcal{S}} H_{ij}^2. \] Lyu et al. (2020) justifies that the asymptotic null distribution of \(G_{\mathcal{S}}\) is Gumbel, \[ \lim_{|\mathcal{S}| \rightarrow \infty} \mathbb{P} \Big( G_\mathcal{S} -2 \log |\mathcal{S}| + \log \log |\mathcal{S}| \le x \Big) = \exp \left\{- \exp (-x/2) / \sqrt{\pi} \right\}. \] It leads to an asymptotic \(\alpha\)-level test, \[\begin{eqnarray*} \Psi_\alpha = \mathbb{1} \big[ G_\mathcal{S} > 2 \log |\mathcal{S}| - \log \log |\mathcal{S}| - \log \pi -2 \log\{-\log(1-\alpha)\} \big]. \end{eqnarray*}\] The global null is rejected if \(\Psi_\alpha=1\).

2.c Simultaneous testing

Let \(\mathcal{H}_0 = \{(i,j) : A_{*,ij}=A_{0,ij}, (i,j) \in \mathcal{S} \}\) denote the set of true null hypotheses, and \(\mathcal{H}_1 = \{ (i,j) : (i,j)\in \mathcal{S} , (i,j) \notin \mathcal{H}_0\}\) denote the set of true alternatives. The test statistic \(H_{ij}\) follows a standard normal distribution when \(H_{0;ij}\) holds, and as such, we reject \(H_{0;ij}\) if \(|H_{ij}| > t\) for some thresholding value \(t > 0\). Let \(R_{\mathcal{S}}(t) = \sum_{(i,j) \in \mathcal{S}} \mathbb{1} \{ |H_{ij}|> t\}\) denote the number of rejections at \(t\). Then the false discovery proportion (FDP) and the false discovery rate (FDR) in the simultaneous testing problem are, \[\begin{eqnarray*} \textrm{FDP}_{\mathcal{S}}(t)=\frac{\sum_{(i,j) \in \mathcal{H}_0} \mathbb{1} \{ |H_{ij}|> t\}}{R_{\mathcal{S}}(t)\vee 1}, \;\; \textrm{ and } \;\; \textrm{FDR}_{\mathcal{S}}(t) = \mathbb{E} \left\{ \textrm{FDP}_{\mathcal{S}}(t) \right\}. \end{eqnarray*}\] An ideal choice of the threshold \(t\) is to reject as many true positives as possible, while controlling the false discovery at the pre-specified level \(\beta\), that is \(\inf \{ t > 0 : \text{FDP}_{\mathcal{S}} (t) \le \beta \}\). However, \(\mathcal{H}_0\) in \(\text{FDP}_{\mathcal{S}} (t)\) is unknown. Observing that \(\mathbb{P} ( |H_{ij}|> t ) \approx 2\{ 1- \Phi (t) \}\), where \(\Phi (\cdot)\) is the cumulative distribution function of a standard normal distribution, the false rejections \(\sum_{(i,j) \in\mathcal{H}_0} \mathbb{1} \{ |H_{ij}|> t\}\) in \(\text{FDP}_{\mathcal{S}} (t)\) can be approximated by \(\{ 2- 2 \Phi(t) \} |\mathcal{S}|\). Moreover, the search of \(t\) is restricted to the range \(\left(0, \sqrt{2\log |\mathcal{S}|} \right]\), since \(\mathbb{P}\left( \hat{t} \text{ exists in } \left(0, \sqrt{2\log |\mathcal{S}|}\right] \right) \to 1\) as shown in the theoretical justification of Lyu et al. (2020). The simultaneous testing procedure is justified that consistently control FDR, \[ \lim_{|\mathcal{S}| \to \infty} \frac{\text{FDR}_{\mathcal{S}} (\, \hat{t} \; )}{\beta |\mathcal{H}_0|/|\mathcal{S}|} = 1, \quad \textrm{ and } \quad \frac{\text{FDP}_{\mathcal{S}} (\, \hat{t} \; )}{\beta | \mathcal{H}_0|/|\mathcal{S}|}\rightarrow_{p} 1 \;\; \textrm{ as } \; |\mathcal{S}| \to \infty. \]