General diagnostic tests for cross-sectional dependence in panels

Abstract

This paper proposes simple tests of error cross-sectional dependence which are applicable to a variety of panel data models, including stationary and unit root dynamic heterogeneous panels with short T and large N. The proposed tests are based on the average of pair-wise correlation coefficients of the OLS residuals from the individual regressions in the panel and can be used to test for cross-sectional dependence of any fixed order p, as well as the case where no a priori ordering of the cross-sectional units is assumed, referred to as $\hbox {CD}(p)$ and $\hbox {CD}$ tests, respectively. Asymptotic distribution of these tests is derived and their power function analyzed under different alternatives. It is shown that these tests are correctly centred for fixed N and T and are robust to single or multiple breaks in the slope coefficients and/or error variances. The small sample properties of the tests are investigated and compared to the Lagrange multiplier test of Breusch and Pagan using Monte Carlo experiments. It is shown that the tests have the correct size in very small samples and satisfactory power, and, as predicted by the theory, they are quite robust to the presence of unit roots and structural breaks. The use of the $\hbox {CD}$ test is illustrated by applying it to study the degree of dependence in per capita output innovations across countries within a given region and across countries in different regions. The results show significant evidence of cross-dependence in output innovations across many countries and regions in the World.

Appendices

Appendix A: Properties of residuals in regression models subject to structural breaks

Abstracting from the cross-sectional index i, the regression model (11) with a single break can be written as

$$\begin{aligned} y_{t}=\left\{ \begin{array}{l} \mu _{y}+\varvec{\beta }_{1}^{\prime }({\mathbf {x}}_{t}-\varvec{\mu } _{x})+\sigma _{1}\varepsilon _{t},\qquad t=1,2,\ldots ,T_{1}, \\ \mu _{y}+\varvec{\beta }_{2}^{\prime }({\mathbf {x}}_{t}-\varvec{\mu } _{x})+\sigma _{2}\varepsilon _{t},\qquad t=T_{1}+1,\ldots ,T \end{array} \right. \end{aligned}$$

where the $k\times 1$ slope coefficients, $\varvec{\beta }_{j}$, and the error variances, $\sigma _{j}^{2},$ for $j=1,2$, are subject to a single break at time $t=T_{1}$, and $\varepsilon _{t}\sim iid(0,1)$. The implied intercepts are given by

$$\begin{aligned} \alpha _{j}=\mu _{y}-\varvec{\beta }_{j}^{\prime }\varvec{\mu }_{x},j=1,2, \end{aligned}$$

The unconditional means of $y_{t}$ and ${\mathbf {x}}_{t}$, namely ${\mu }_{y}$ and $\varvec{\mu }_{x}$, are not subject to change. We also assume that ${\mathbf {x}}_{t}$ follows the covariance stationary process:

$$\begin{aligned} {\mathbf {x}}_{t}=\varvec{\mu }_{x}+\sum \limits _{s=0}^{\infty }\varvec{\Psi } _{s}\varvec{\nu }_{t-s},\qquad \varvec{\nu }_{t}\sim iid(0,{\mathbf {I}} _{k}),~\sum \limits _{s=0}^{\infty }\varvec{\parallel \Psi }_{s}\varvec{\parallel <\infty }\text {.} \end{aligned}$$

(32)

where $\varvec{\nu }_{t}$ and $\varvec{\varepsilon }_{t^{\prime }}$, are independently distributed for all t and $t^{\prime }$. We shall also assume that the innovations in the ${\mathbf {x}}_{t}$ process, $\varvec{\nu } _{t}$, are symmetrically distributed around zero.

Suppose now that the breaks are ignored and the residuals$,e_{t},$ are computed by running the ordinary least squares regression of $y_{t}$ on ${\mathbf {x}}_{t}$ over the whole sample, $t=1,2,\ldots ,T$. We have

$$\begin{aligned} e_{t}=y_{t}-{\hat{\alpha }}-{\hat{\varvec{\beta }}}^{\prime }{\mathbf {x}}_{t}, \end{aligned}$$

(33)

where

$$\begin{aligned} {\hat{\varvec{\beta }}}= & {} \left[ \sum \limits _{t=1}^{T}({\mathbf {x}}_{t}- \overline{{\mathbf {x}}})({\mathbf {x}}_{t}-\overline{{\mathbf {x}}})^{\prime }\right] ^{-1}\left[ \sum \limits _{t=1}^{T}({\mathbf {x}}_{t}-\overline{{\mathbf {x}}})( {\mathbf {y}}_{t}-{\overline{y}})\right] , \\ {\widehat{\alpha }}= & {} {\overline{y}}-\widehat{\varvec{\beta }}^{\prime } \overline{{\mathbf {x}}}, \qquad \overline{{\mathbf {x}}}=\frac{1}{T} \sum \limits _{t=1}^{T}{\mathbf {x}}_{t},\text {and} \ {\overline{y}}=\frac{1}{T} \sum \limits _{t=1}^{T}y_{t}. \end{aligned}$$

In what follows we establish that for all $t=1,2,\ldots ,T,$ the OLS residuals $e_{t}$ are odd functions of the disturbances, $\varepsilon _{t}$ and $E(e_{t})=0$.

We first note that

$$\begin{aligned} {\overline{y}}= & {} \mu _{y}+\lambda _{1}\varvec{\beta }_{1}^{\prime }(\overline{ {\mathbf {x}}}_{1}-\varvec{\mu }_{x})+(1-\lambda _{1})\varvec{\beta } _{2}^{\prime }(\overline{{\mathbf {x}}}_{2}-\varvec{\mu }_{x}) \\&+\,\sigma _{1}\lambda _{1}{\overline{\varepsilon }}_{1}+\sigma _{2}(1-\lambda _{1}){\overline{\varepsilon }}_{2}, \end{aligned}$$

where $\lambda _{1}=T_{1}/T,$ $\overline{{\mathbf {x}}}_{1}=\frac{1}{T_{1}} \sum \nolimits _{t=1}^{T_{1}}{\mathbf {x}}_{t},$ $\overline{{\mathbf {x}}}_{2}=\frac{ 1}{T-T_{1}}\sum \nolimits _{t=T_{1}+1}^{T}{\mathbf {x}}_{t},$ etc. Hence, for $t\le T_{1}$

$$\begin{aligned} y_{t}-{\overline{y}}=\mu _{y}-{\overline{y}}+\varvec{\beta }_{1}^{\prime }( {\mathbf {x}}_{t}-\varvec{\mu }_{x})+\sigma _{1}\varepsilon _{t}, \end{aligned}$$

and for $t>T_{1}$

$$\begin{aligned} y_{t}-{\overline{y}}=\mu _{y}-{\overline{y}}+\varvec{\beta }_{2}^{\prime }( {\mathbf {x}}_{t}-\varvec{\mu }_{x})+\sigma _{2}\varepsilon _{t}. \end{aligned}$$

Also since

$$\begin{aligned} E\left( {\mathbf {x}}_{t}\right) =\varvec{\mu }_{x}, \end{aligned}$$

then for all t,

$$\begin{aligned} E(y_{t}-{\overline{y}})=0,\quad \text {for}\quad \,t=1,2,\ldots ,T. \end{aligned}$$

Consider now the residuals defined by (33) and note that

$$\begin{aligned} e_{t}=(y_{t}-{\overline{y}})-\widehat{\varvec{\beta }}^{\prime }({\mathbf {x}} _{t}-\overline{{\mathbf {x}}}). \end{aligned}$$

(34)

Hence, it is sufficient to show that the second term has zero expectations for all t. Under the data generating mechanism

$$\begin{aligned} {\mathbf {Q}}~\widehat{\varvec{\beta }}= & {} {\mathbf {Q}}_{1}\varvec{\beta } _{1}+{\mathbf {Q}}_{2}\varvec{\beta }_{2}+\sigma _{1}\sum \limits _{t=1}^{T_{1}}( {\mathbf {x}}_{t}-\overline{{\mathbf {x}}})\varepsilon _{t} \\&+\,\sigma _{2}\sum \limits _{t=T_{1}+1}^{T}({\mathbf {x}}_{t}-\overline{{\mathbf {x}} })\varepsilon _{t}, \end{aligned}$$

where

$$\begin{aligned} {\mathbf {Q}}= & {} \sum \limits _{t=1}^{T}({\mathbf {x}}_{t}-\overline{ {\mathbf {x}}})({\mathbf {x}}_{t}-\overline{{\mathbf {x}}})^{\prime } \\ {\mathbf {Q}}_{1}= & {} \sum \limits _{t=1}^{T_{1}}({\mathbf {x}}_{t}-\overline{\mathbf { x}})({\mathbf {x}}_{t}-\overline{{\mathbf {x}}})^{\prime },\text {and}\,{\mathbf {Q}} _{2}={\mathbf {Q}}-{\mathbf {Q}}_{1}. \end{aligned}$$

Therefore,

$$\begin{aligned} \widehat{\varvec{\beta }}^{\prime }({\mathbf {x}}_{t}-\overline{{\mathbf {x}}})= & {} \varvec{\beta }_{1}^{\prime }{\mathbf {Q}}_{1}{\mathbf {Q}}^{-1}({\mathbf {x}}_{t}- \overline{{\mathbf {x}}})+\varvec{\beta }_{2}^{\prime }{\mathbf {Q}}_{2}{\mathbf {Q}} ^{-1}({\mathbf {x}}_{t}-\overline{{\mathbf {x}}}) \\&+\,\sigma _{1}\sum \limits _{t=1}^{T_{1}}\varepsilon _{t}({\mathbf {x}}_{t}- \overline{{\mathbf {x}}})^{\prime }{\mathbf {Q}}^{-1}({\mathbf {x}}_{t}-\overline{ {\mathbf {x}}}) \\&+\,\sigma _{2}\sum \limits _{t=T_{1}+1}^{T}\varepsilon _{t}({\mathbf {x}}_{t}- \overline{{\mathbf {x}}})^{\prime }{\mathbf {Q}}^{-1}({\mathbf {x}}_{t}-\overline{ {\mathbf {x}}}). \end{aligned}$$

Since ${\mathbf {x}}_{t}$ and $\varepsilon _{t^{\prime }}$ are independently distributed for all t and $t^{\prime }$, the last two terms have zero unconditional expectations. Also using (32), it is easily seen that

$$\begin{aligned} {\mathbf {x}}_{t}-\overline{{\mathbf {x}}}=\sum \limits _{s=0}^{\infty }\varvec{\Psi }_{s}\left( \varvec{\nu }_{t-s}-{\bar{\varvec{\nu }}}_{-s}\right) , {\bar{\varvec{\nu }}}_{-s}=\frac{1}{T}\sum \limits _{t=1}^{T}\varvec{\nu }_{t-s}, \end{aligned}$$

which establishes that ${\mathbf {x}}_{t}-\overline{{\mathbf {x}}}$ is an odd function of $\left\{ \varvec{\nu }_{t}\right\}$, the innovations in ${\mathbf {x}}_{t}$. Since ${\mathbf {Q}}$ and ${\mathbf {Q}}_{1}$ are even functions of $\left\{ \varvec{\nu }_{t}\right\} ,$ it follows that ${\mathbf {Q}}_{j} {\mathbf {Q}}^{-1}({\mathbf {x}}_{t}-\overline{{\mathbf {x}}})$, for $j=1,2$ are also odd functions of $\left\{ \varvec{\nu }_{t}\right\}$ and in view of the symmetry of $\left\{ \varvec{\nu }_{t}\right\}$ will have zero mean unconditionally. Thus, under our assumptions, $e_{t}$ and $\xi _{t}=\left( \sum _{\tau =1}^{T}e_{\tau }^{2}\right) ^{-1/2}e_{t}$ are odd functions of $\left\{ \varepsilon _{t},\varvec{\nu }_{t}\right\} ,$ and therefore have zero expectations for all t, despite the breaks in the slopes and the error variances. This result continues to hold under multiple breaks and/or even if $\varvec{\Psi }_{i}$ are subject to one or more breaks. The key assumptions are symmetry of the innovations, $\varepsilon _{t}$ and $\varvec{\nu }_{t}$, and the time-invariance of the unconditional means of $y_{t}$ and ${\mathbf {x}} _{t}$.

Appendix B: Residuals from AR(p) models subject to breaks

Consider the AR(p) model defined over the period $t=1,2,\ldots ,T;$ and assumed to have been subject to a single structural break at time $T_{1}:$

$$\begin{aligned} y_{t}=\left\{ \begin{array}{l} \alpha _{1}+\beta _{11}y_{t-1}+\beta _{12}y_{t-2}+\cdots +\beta _{1p}y_{t-p}+\sigma _{1}\varepsilon _{t},\quad \text {for}\,\quad t\le T_{1}, \\ \alpha _{2}+\beta _{21}y_{t-1}+\beta _{22}y_{t-2}+\cdots +\beta _{2p}y_{t-p}+\sigma _{2}\varepsilon _{t},\quad \text {for}\quad \,t>T_{1}, \end{array} \right. . \end{aligned}$$

(35)

where $\varepsilon _{t}\thicksim iid(0,1)$ for all t,

$$\begin{aligned} \alpha _{j}=\mu _{j}(1-\varvec{\tau }_{p}^{\prime }\varvec{\beta }_{j}), \quad j=1,2, \end{aligned}$$

(36)

$\varvec{\beta }_{j}=(\beta _{j1},\beta _{j2},\ldots ,\beta _{jp})^{\prime }$ and $\varvec{\tau }_{p}$ is a $p\times 1$ unit vector.

Suppose that the structural break is ignored and residuals are computed by estimating the AR(p) model in $y_{t}$ using the OLS regression $y_{t}$ on an intercept and ${\mathbf {x}}_{t}=(y_{t-1},y_{t-2},\ldots ,y_{t-p})^{\prime }$ making using of the available observations $\digamma _{T}=\left( y_{1-p},y_{2-p},\ldots ,y_{0},y_{1},\ldots ,y_{T}\right)$. In this case, the fitted residuals are given by

$$\begin{aligned} e_{t}=y_{t}-{\hat{\alpha }}-{\hat{\varvec{\beta }}}^{\prime }{\mathbf {x}}_{t},~t=1,2,\ldots ,T \end{aligned}$$

(37)

where ${\mathbf {x}}_{t}=(y_{t-1},y_{t-2},\ldots ,y_{t-p})^{\prime }$,$\,{\hat{\varvec{\beta }}}=({\hat{\beta }}_{1},{\hat{\beta }}_{2},\ldots ,{\hat{\beta }}_{p})^{\prime }\ \$

$$\begin{aligned} {\hat{\varvec{\beta }}}&=\left( {\mathbf {X}}^{\prime }\mathbf {MX}\right) ^{-1} {\mathbf {X}}^{\prime }\mathbf {My}, \end{aligned}$$

(38)

$$\begin{aligned} {\hat{\alpha }}&=\frac{\varvec{\tau }_{T}^{\prime }{\mathbf {y}}-\varvec{\tau } _{T}^{\prime }\mathbf {X}{\hat{\varvec{\beta }}}}{T}, \end{aligned}$$

(39)

$\mathbf {y{=}}\left( y_{1},y_{2},\ldots ,y_{T}\right)$, $\mathbf {X{=}}\left( {\mathbf {y}}_{0},{\mathbf {y}}_{-1},\ldots ,{\mathbf {y}}_{-p+1}\right)$, ${\mathbf {y}} _{-j+1}{=}\left( y_{-j+1},y_{-j+2},\ldots ,y_{T-j}\right) ^{\prime }$, $\varvec{\tau }_{T}$ is a $T\times 1$ vector of ones, and ${\mathbf {M}}= {\mathbf {I}}_{T}-\varvec{\tau }_{T}(\varvec{\tau }_{T}^{\prime }\varvec{\tau } _{T})^{-1}\varvec{\tau }_{T}^{\prime }$. In what follows, we shall establish that $E(e_{t})=0$ for $t=1,2,\ldots ,T$ so long as $\mu _{1}=\mu _{2},$ $\varepsilon _{t}$ is symmetrically distributed, and $E(e_{t})$ exists. We shall provide a proof for the stationary case with a single break, although the result holds much more generally both in the presence of multiple breaks and if there are unit roots in the pre- and/or post-break processes.

In the case where the pre-break regime is stationary, the distribution of the initial values, ${\mathbf {x}}_{p}=(y_{1-p},y_{2-p},\ldots ,y_{0})^{\prime },$ can be written as

$$\begin{aligned} {\mathbf {x}}_{p}-\mu _{1}\varvec{\tau }_{p}\thicksim ({\mathbf {0}},\sigma _{1}^{2}{\mathbf {V}}_{p}), \end{aligned}$$

(40)

where ${\mathbf {V}}_{p}$ is a positive definite matrix.

Using (40) and (35) for $t=1,2,\ldots ,T$, in matrix notations we have

$$\begin{aligned} {\mathbf {B}}~{\mathbf {y}}^{*}={\mathbf {d}}+{\mathbf {D}}~\varvec{\varepsilon }^{*}, \end{aligned}$$

(41)

where ${\mathbf {y}}^{*}=\left( {\mathbf {x}}_{p}^{\prime },{\mathbf {y}}^{\prime }\right) ^{\prime }$, $\varvec{\varepsilon }^{*}\mathbf {=(}\varepsilon _{1-p},\varepsilon _{2-p},\ldots \varepsilon _{0},\varepsilon _{1},\varepsilon _{2},\ldots ,\varepsilon _{T})^{\prime }$

$$\begin{aligned} {\mathbf {D}}&=\sigma _{1}\left( \begin{array}{ccc} \varvec{\psi }_{p} &{} {\mathbf {0}} &{} {\mathbf {0}} \\ {\mathbf {0}} &{} {\mathbf {I}}_{T_{1}} &{} {\mathbf {0}} \\ {\mathbf {0}} &{} {\mathbf {0}} &{} \left( \sigma _{2}/\sigma _{1}\right) {\mathbf {I}} _{T_{2}} \end{array} \right) ,~\mathbf {d} =\left( \begin{array}{c} \mu _{1}\varvec{\tau }_{p} \\ \mu _{1}(1-\beta _{1}^{*})\varvec{\tau }_{T_{1}} \\ \mu _{2}(1-\beta _{2}^{*})\varvec{\tau }_{T_{2}} \end{array} \right) , \end{aligned}$$

(42)

$$\begin{aligned} {\mathbf {B}}&=\left( \begin{array}{ccc} {\mathbf {I}}_{p} &{} {\mathbf {0}} &{} {\mathbf {0}} \\ {\mathbf {B}}_{21} &{} {\mathbf {B}}_{22} &{} {\mathbf {0}} \\ {\mathbf {0}} &{} {\mathbf {B}}_{32} &{} {\mathbf {B}}_{33} \end{array} \right) . \end{aligned}$$

(43)

The sub-matrices, ${\mathbf {B}}_{ij}$, depend only on the slope coefficients, $\varvec{\beta }_{1}$ and $\varvec{\beta }_{2}$ and are as defined in “Appendix B” of Pesaran and Timmermann (2005). ${\mathbf {I}}_{T_{1}}$ and ${\mathbf {I}}_{T_{2}}$ are identity matrices of order $T_{1}$ and $T_{2}$, respectively, $T_{2}=T-T_{1}$, $\varvec{\varepsilon }^{*}\sim ({\mathbf {0}} ,{\mathbf {I}}_{T+p})$, and $\varvec{\psi }_{p}$ is a lower triangular Cholesky factor of ${\mathbf {V}}_{p}$, namely ${\mathbf {V}}_{p}=\varvec{\psi }_{p}\varvec{\psi }_{p}^{\prime }$.

Using (41), it is easily seen that

$$\begin{aligned} {\mathbf {y}}_{-j+1}={\mathbf {G}}_{j}(\mathbf {c+H}\varvec{\varepsilon }^{*}\mathbf {),}\,\quad \text {for}\quad \,j=0,1,\ldots ,p, \end{aligned}$$

(44)

where ${\mathbf {G}}_{j}$ are $T\times (T+p)$ selection matrices defined by ${\mathbf {G}}_{j}=({\mathbf {0}}_{T\times p-j}\varvec{\vdots I}_{T}\varvec{\vdots 0 }_{T\times j})$, ${\mathbf {H}}={\mathbf {B}}^{-1}{\mathbf {D}}$, and $\mathbf {c=B} ^{-1}{\mathbf {d}}$. In particular,

$$\begin{aligned} {\mathbf {y}}={\mathbf {G}}_{0}(\mathbf {c+H}\varvec{\varepsilon }^{*}\mathbf {),} \end{aligned}$$

and

$$\begin{aligned} {\mathbf {X}}=\left[ {\mathbf {G}}_{1}(\mathbf {c+H}\varvec{\varepsilon }^{*}\mathbf {),G} _{2}(\mathbf {c+H}\varvec{\varepsilon }^{*}\mathbf {),\ldots ,G}_{p}(\mathbf { c+H}\varvec{\varepsilon }^{*}\mathbf {)}\right] . \end{aligned}$$

However, as shown in Pesaran and Timmermann (2005, “Appendix B”), under $\mu _{1}=\mu _{2}=\mu$, ${\mathbf {G}}_{j}\mathbf {c}={\mu {\varvec{\tau }}_{\varvec{T}}}$, and the (i, j) element of ${\mathbf {X}}^{\prime }\mathbf {MX}$ will be given by $\varvec{\varepsilon }^{*\prime }{\mathbf {H}}^{\prime }{\mathbf {G}} _{i}^{\prime }{\mathbf {M}}_{\tau }{\mathbf {G}}_{j}\mathbf {H}\varvec{\varepsilon }^{*}$ , for $i,j=1,2,\ldots ,p$, and the $j\mathrm{th}$ element of ${\mathbf {X}}^{\prime } \mathbf {My}$ by $\varvec{\varepsilon }^{*\prime }{\mathbf {H}}^{\prime } {\mathbf {G}}_{j}^{\prime }{\mathbf {M}}_{\tau }{\mathbf {G}}_{0}\mathbf {H}\varvec{\varepsilon }^{*}$, for $j=1,2,\ldots ,p$. Hence, under $\mu _{1}=\mu _{2}$, ${\hat{\varvec{\beta }}}$ will be an even function of $\varvec{\varepsilon }$. Similarly, using (39) and recalling that ${\mathbf {G}}_{j}\mathbf { c}={\mu {\varvec{\tau }}_{\varvec{T}}}$, we have

$$\begin{aligned} {\hat{\alpha }}=\mu \left( 1-\sum _{j=1}^{p}{\hat{\beta }}_{j}\right) +\left( \frac{ \varvec{\tau }_{T}^{\prime }{\mathbf {G}}_{0}\mathbf {H}\varvec{\varepsilon }^{*}}{T} \right) \mathbf {-}\sum _{j=1}^{p}\left( \frac{\varvec{\tau }_{T}^{\prime } {\mathbf {G}}_{j}\mathbf {H}\varvec{\varepsilon }^{*}}{T}\right) {\hat{\beta }}_{j}. \end{aligned}$$

(45)

Using this result in (37) and noting that for $t\le T_{1}$

$$\begin{aligned} y_{t}-\mu =\sum _{j=1}^{p}\beta _{1j}(y_{t-j}-\mu )+\sigma _{1}\varepsilon _{t}, \end{aligned}$$

we have

$$\begin{aligned} e_{t}=-\sum _{j=1}^{p}\left( {\hat{\beta }}_{j}-\beta _{1j}\right) (y_{t-j}-\mu )+\sigma _{1}\varepsilon _{t}-\left( \frac{\varvec{\tau }_{T}^{\prime } {\mathbf {G}}_{\mathbf {0}}\mathbf {H}\varvec{\varepsilon }^{*}}{T}\right) \mathbf {+} \sum _{j=1}^{p}\left( \frac{\varvec{\tau }_{T}^{\prime }{\mathbf {G}}_{j}\mathbf { H}\varvec{\varepsilon }^{*}}{T}\right) {\hat{\beta }}_{j}. \end{aligned}$$

Similarly, for $t>T_{1}$:

$$\begin{aligned} e_{t}=-\sum _{j=1}^{p}\left( {\hat{\beta }}_{j}-\beta _{2j}\right) (y_{t-j}-\mu )+\sigma _{2}\varepsilon _{t}-\left( \frac{\varvec{\tau }_{T}^{\prime } {\mathbf {G}}_{\mathbf {0}}\mathbf {H}\varvec{\varepsilon }^{*}}{T}\right) \mathbf {+} \sum _{j=1}^{p}\left( \frac{\varvec{\tau }_{T}^{\prime }{\mathbf {G}}_{j}\mathbf { H}\varvec{\varepsilon }^{*}}{T}\right) {\hat{\beta }}_{j}. \end{aligned}$$

It is now easily seen that in both regimes $e_{t}$ is an odd function of the standardized errors, $\varepsilon _{t}$, $t=-p+1,-p+2,\ldots ,T$, and under the distributional symmetry of the errors, we have

$$\begin{aligned} E\left( \xi _{t}\right) =E\left[ \left( \sum _{\tau =1}^{T}e_{\tau }^{2}\right) ^{-1/2}e_{t}\right] =0,\quad \text {for}\quad \,t=1,2,\ldots ,T. \end{aligned}$$

Note that for $\xi _{t}$ to be well defined we need $T>p+1$, and $E\left( \xi _{t}\right)$ exists for all $T>p+1$. In contrast, the condition for the existence of the moments of $e_{t}$ is much more complicated and demanding. For example, $E\left( e_{t}\right)$ exists if $E\left( {\hat{\beta }} _{i}\right)$ exists. A sufficient condition for the latter is known in the literature only for the simple case of $p=1$. In this case,

$$\begin{aligned} {\hat{\beta }}_{1}=\frac{\varvec{\varepsilon }^{*\prime }{\mathbf {H}}^{\prime }{\mathbf {G}}_{1}^{\prime }{\mathbf {M}}_{\tau }{\mathbf {G}}_{0}\mathbf { H}\varvec{\varepsilon }^{*}}{\varvec{\varepsilon }^{*\prime }{\mathbf {H}} ^{\prime }{\mathbf {G}}_{1}^{\prime }{\mathbf {M}}_{\tau }{\mathbf {G}}_{1}\mathbf { H}\varvec{\varepsilon }^{*}}, \end{aligned}$$

and $E\left( {\hat{\beta }}_{1}\right)$ exists if $T>3$ (see Pesaran and Timmermann 2005).