Estimating Nonlinear Network Data Models with Fixed Effects

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\makeatletter [\mathcal{R}^{s}]_{g} & =\big[(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}\big]_{g} \end{align*}

By definition we have that $\mathcal{H}=-\partial_{\phi\phi'}\mathcal{L}$. Differentiating gives \[ \mathcal{H}_{b}=W^{-1}\big(\partial_{\beta\phi\phi'}\mathcal{L}+\sum_{f}(\partial_{\phi\phi'\phi_{f}}\mathcal{L})\big[\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\big) \]

Differentiating a second time gives

The third derivatives are

Fourth derivative:

Expressions for $\mathcal{P}$ terms

First derivatives:

Second derivatives:

Third derivatives:

Expressions for $\mathcal{R}$ terms

First derivatives:

Second derivatives:

Third derivatives:

Expressions for $\mathcal{F}$ terms

Define $\mathcal{E}^{s}=\partial_{\beta^{s}\phi\phi'}\mathcal{L}$, $\mathcal{F}^{s,t}=(\partial_{\beta^{s}\phi'}\mathcal{L})\mathcal{H}^{-1}(\partial_{\beta^{t}\phi}\mathcal{L})$, and

Then, differentiating gives

(1)

(2)

Differentiating a second time gives

(1)

(2)

(3)

Third derivative:

Expressions for $\mathcal{G}$ terms

Differentiating $\mathcal{G}=\mathcal{H}^{-1}(\partial_{\beta\phi}\mathcal{L})(\partial_{\beta\phi'}\mathcal{L})\mathcal{H}^{-1}$ with respect to $b$ and $s$ gives

Differentiating a second time gives

Finally, the third derivatives are

Expressions for $W$ terms

We begin by expressing the first three derivatives of $W$ in terms of $\mathcal{F}$. To do this, first note that by the definition of $\mathcal{F}$

Differentiating with respect to $b$ and $s$ gives

Differentiating a second time gives

(1)

(2)

(3)

Third derivative:

Expansion for $\widehat{\beta}$

In Section (ref) we provide bounds for each of the above terms that by Assumptions 1 and 2 can be shown to hold uniformly over a neighborhood of the truth. Using these results, we will be able to show that

where $(\bar{b},\bar{s})$ are intermediate values between $(0,0)$ and $(\mathcal{S}_{\beta},\mathcal{S})$, which under Assumptions 1 and 2 must lie in $\mathcal{B}(r_\beta,\beta_0)\times\mathcal{B}_q(r_\phi,\phi_0)$ for sufficiently large $N$ (see Corollary B.3 in FW16).

Hence, we have the first-order expansion

Lemmas for bounding individual components

In order to provide bounds on the components in the asymptotic expansion above, we first demonstrate some basic results that are used frequently. We begin with a result from FW16 (Lemma S.6 in that paper). Through the proofs we apply the theorem with $q=16$ which is justified by Assumption 2.

Bounds for $\bar{\mathcal{H}}^{-1}$ and $\overline{W}_{N}^{-1}$

Here we provide important results related to $\bar{\mathcal{H}}^{-1}$ and $\overline{W}_{N}^{-1}$. The objective function for the estimator is given by \[ \mathcal{L}_{N}(\beta,\phi_{N})=\frac{1}{N-1}\sum_{i}\sum_{j\ne i}\ell(Z_{ij},\beta,\alpha_{i},\gamma_{j})-\frac{1}{N}b(v_{N}'\phi_{N})^{2}/2 \] As in FW16 and Dzemski 2019, we may write \[ \bar{\mathcal{H}}=

+\frac{1}{N}bv_{N}v_{N}' \] where $\bar{\mathcal{H}}_{\alpha\alpha}^{*}$ and $\bar{\mathcal{H}}_{\gamma\gamma}^{*}$ are the diagonal matrices with elements

and $\bar{\mathcal{H}}_{\alpha\gamma}^{*}=(\bar{\mathcal{H}}_{\gamma\alpha}^{*})'$ has off-diagonal entries $\big(\bar{\mathcal{H}}_{\alpha\gamma}^{*}\big)_{ij}=-\bar{E}[\partial_{\pi^{2}}\ell_{ij}]/(N-1)$ and zeroes in diagonal entries. Following Lemma A.1 in D19 and Lemma D.1 in FW16, we may prove the following approximation result.

The proof of this Lemma is identical to the versions in D19 and FW16. We next bound some deviations between the sample average values and conditional expectations of the Hessian matrix.

In the proofs below we will frequently replace $\mathcal{H}^{-1}$ and $W^{-1}$ by the following approximations.

Products of matrices

The next result demonstrates that the properties of $\mathcal{\bar{H}}^{-1}$ (a dominant diagonal and smaller off-diagonal terms) transfer to certain products of this matrix.

Further terms

The remaing parts of this section provide bounds on a series of terms that appear in the expansion. These bounds are used in later sections to bound the expansion terrms in Section (ref).

Bounds on individual components

This section provides bounds on the terms $\mathcal{P}$, $\mathcal{R}$, $\mathcal{F}$, $\mathcal{G}$, $W$, and $\mathcal{H}$, and their derivatives. These form the basis for bounding the terms in the asymptotic expansion. The bounds follow from the definition of the terms (see Section (ref)) as well as the bounds presented in the previous section.

Bounds on $\mathcal{F}, \mathcal{P}, \mathcal{R}$ terms

Bounds on $W$ terms

Bounds on $\mathcal{H}$ terms

Bounds on $\mathcal{G}$ terms

Bounds for expansion terms

Finally, we make use of the bounds in the previous section to give bounds on the asymptotic expansion terms, which justifies the expansion in (ref). All results follow through applcation of the expressions derived in Section (ref), along with the bounds from Section (ref), and application of the Cauchy-Schwarz/triangle inequalities.

Third derivative terms

Fourth derivative terms

Using the bounds provided in Section (ref) we find

Fifth derivative terms

Application of the bounds provided in Section (ref) gives

Sixth derivative terms

Jackknife expansion

Asymptotic expansion in leave-out samples

In order to derive results for the jackknife estimator for $\beta$, we first establish the asymptotic expansion for $\widehat{\beta}_{(k)}$, i.e. the leave-out sample parameter estimate. The expansions can be derived identically to the full sample estimators, replacing $\ell_{ij}$ in the objective function with $\frac{N-1}{N-2}\ell_{ij}1_{ij}^{k}$ where $1_{ij}^{k}$ is an indicator variable that is equal to one whenever the observation $(i,j)$ is included in the $k$-th leave-out sample, and is zero when that observation has been dropped. In comparing the asymptotic expansions of the full-sample and leave-out sample estimators, we will replace $\mathcal{H}^{-1}$ and $W_{N}^{-1}$ terms with their conditional expectations. For this purpose, we first state a new version of Lemma (ref) for the leave-out sample.

Using this result, the first-order expansion for $\widehat{\beta}_{(k)}$ can be shown to be given by

Since $\lVert W_{N,(k)}-\bar{W}_{N}\rVert=o_{p}(1)$ by Lemma (ref), the same expansion up to first order applies to $N\bar{W}_{N}(\widehat{\beta}_{(k)}-\beta)$.

Jackknife results for higher-order terms

In the main appendix, the first-order terms of the jackknife estimator $\widehat{\beta}_{J}$ are derived. Here we show that the remaining terms up to $O_{p}(N^{-1})$ in the expansion for $N(\widehat{\beta}_{J}-\beta)$ are in fact $o_{p}(1)$ and so do not affect the asymptotic distribution of the estimator. As can be seen in Section (ref), there is an extremely large number of terms in the asymptotic expansion that must be considered. However, inspection of the terms shows that they share a common structure, that is, they can be expressed as V-statistics of a certain order that depend on sums of the derivatives of $\ell_{ij}$ up to sixth order. Here we prove the result for an example term, and discuss how this same proof can be used to show that the remaining terms will also be $o_{p}(1)$.

Consider the expansion term $\mathcal{S}'(\partial_{bbss'}\mathcal{L}_{(1)}^{*})\mathcal{S}\mathcal{S}_{\beta}$, which contains the term \[ \mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta} \] Expanding out this terms gives

Take the first term in this expression. We can replace $W$ with $\bar{W}+(W-\bar{W})$, and similarly for $W_{b}$, $\mathcal{H}^{-1}$ and $\partial_{\phi\phi'\phi_{f}}\mathcal{L}$ to give \[ \bar{W}^{-2}\bar{W}_{b}\mathcal{S}'\bar{\mathcal{H}}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{\bar{H}}^{-1}\mathcal{S}\mathcal{S}_{\beta}+o_{p}(N^{-2}) \] This result holds identically for the leave-out samples, replacing $\mathcal{S}$ with $\mathcal{S}_{(k)}$ and similarly for $\partial_{\beta\phi}\mathcal{L}$ and $\mathcal{S}_{\beta}$. Ignoring the $\bar{W}^{-2}\bar{W}_{b}$ term for the moment, which will not be affected by the jackknifing, we can decompose the above sum further into the components of $\phi=(\alpha,\gamma)$, for example the first of these terms would be

Define

and note that, by Lemma (ref), we have $\max_{i\ne s\ne t}Q_{ist}=O_{p}(N^{-2})$, while the max such that two of $(i,s,t)$ are equal is $O_{p}(N^{-1})$ and $Q_{iii}=O_{p}(1)$. We can write the term now as

which is the sum of two V-statistic type terms, of fourth order. We now consider the effect of the jackknife operation on these terms. Consider the first of the terms (the result will clearly be identical for both terms) and note that $Q_{ist}^{\alpha}$ is fixed across leave-out samples, so that the $k$-th leave-out version of this term is \[ \frac{1}{(N-2)^{4}}\sum_{i,i'\ne i}\sum_{s,s'\ne s}\sum_{t,t'\ne t}\sum_{r,r'\ne r}Q_{ist}^{\alpha}(\partial_{\pi}\ell_{i,i'})(\partial_{\pi}\ell_{s,s'})(\partial_{\beta\pi}\ell_{t,t'})(\partial_{\beta}\ell_{r,r'})1_{i,i'}^{k}1_{s,s'}^{k}1_{t,t'}^{k}1_{r,r'}^{k} \]

The compute the average over the $N-1$ leave-out samples first define \[ n(i,i',s,s',t,t',r,r')=N-1-\sum_{k}1_{i,i'}^{k}1_{s,s'}^{k}1_{t,t'}^{k}1_{r,r'}^{k} \] which counts the number of leave-out samples in which all four observations appear -- this number depends on whether the four observations appear in the same set $\mathcal{I}_{k}$, or in two, three of four different sets. We can write the average as

The jackknifed term is then equal to

since $(\frac{1}{(N-1)^{3}}-\frac{N-1-n(i,i',s,s',t,t',r,r')}{(N-1)(N-2)^{3}})=O(N^{-3})$.

To bound the summation, we decompose the sum based on the order of $Q_{ist}^{\alpha}$, using the shorthand notation $1_{n=j}=1\{n(i,i',s,s',t,t',r,r')=j\}$. For some $j$ we have

The first term is

Since we have $\max_{i\ne s\ne t}Q_{ist}=O_{p}(N^{-2})$, and $\bar{E}[\partial_{\pi}\ell_{ii'}]=\bar{E}[\partial_{\beta}\ell_{ii'}]=0$, this term is $O_{p}(N^{6})$. Similarly, for the final term we have

and hence the final term is $O_{p}(N^{4})$. The remaining terms can be shown to also be $O_{p}(N^{4})$. This gives $\mathcal{J}=O_{p}(N)$. Nearly identical steps apply to the remaining components of the term $\mathcal{S}'\bar{\mathcal{H}}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{\bar{H}}^{-1}\mathcal{S}\mathcal{S}_{\beta}$ and adding back the terms $\bar{W}^{-2}\bar{W}_{b}$, we find that the jackknife operator applied to the full term satisfies \[ \mathcal{J}\Big[\bar{W}^{-2}\bar{W}_{b}\mathcal{S}'\bar{\mathcal{H}}^{-1}\sum_{f}(\partial_{\phi\phi'\phi_{f}}\bar{\mathcal{L}})\big[\bar{\mathcal{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})\big]_{f}\mathcal{\bar{H}}^{-1}\mathcal{S}\mathcal{S}_{\beta}\Big]=o_{p}(1). \] Each of the remaining terms in $\mathcal{S}'\mathcal{H}^{-1}\mathcal{H}_{bb}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}$ can similarly be shown to have the same V-statistic like structure, up to fifth-order. Inspection of the expressions in Section (ref) shows that an essentially identical proof can be used for any of these terms. That is, we first replace matrices with expansions around their conditional expectations, and the expand the terms to give sums of products of terms like the above. The jackknifed versions of these sums can then straightforwardly be shown to have the same asymptotic order as the original terms, and hence each of the jackknifed terms is $o_{p}(1)$ as required.

Results for weighted jackknife

In order to complete the proof of Theorem 1 in Section (ref), we must show:

(1) $(N-1)\lVert W_{J}-W_{N}\rVert=o_{p}(1)$

(2) $(N-1)(\widehat{W}_{J}-W_{J})(\widehat{\beta}-\beta)-(N-2)\frac{1}{N-1}\sum_{k}(\widehat{W}_{(k)}-W_{(k)})(\widehat{\beta}_{(k)}-\beta)=o_{p}(N^{-1})$

(3) $\lVert \widehat{W}_J - \overline{W}_N\rVert=o_p(1)$

(1) $(N-1)(W_{J}-W_{N})=o_{p}(1)$

Expanding out this term we get

where the remainder term is $o_p(1)$ since $\lVert\mathcal{H}^{-1}-\mathcal{\bar{H}}^{-1}+\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}-\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}\rVert=O_{p}(N^{-\frac{3}{2}+6\epsilon})$ by Lemma (ref), $\lVert\partial_{\beta\phi'}\mathcal{L}\rVert=O_{p}(N^{1/2})$, and similarly for the leaveout versions.

Further expanding out the first of these approximation terms gives

since $\Gamma_{isjt}$ is $O_p(1)$ only when either $i=s$ or $j=t$, and is $O_p(N^{-1})$ otherwise. Expanding out the next term in the approximation gives

where the third equality follows from the fact that $\big(\frac{(N-1)^{2}}{(N-2)^{3}}\sum_{k}1_{ii'}^{k}1_{jj'}^{k}1_{ss'}^{k}-1\big)$ is $O(N^{-1})$ whenever $\sum_{k}1_{ii'}^{k}1_{jj'}^{k}1_{ss'}^{k}$ is equal to $(N-2)$ or $(N-3)$, but is $O(N^{-2})$ when the three observations span three different $\mathcal{I}_k$, while the final equality is due to the fact that: (1) $\Gamma_{ii'jj'}\Gamma_{jj'ss'}$ is $O_{p}(1)$ only if one of $i=j,i'=j'$ holds and one of $j=s,j'=s'$ holds, and is $O_{p}(N^{-1})$ if only one of the two holds, and is $O_{p}(N^{-2})$ if neither hold, and (2) when $I^1_{(ii')(ss')(jj')}=1$, the indices $s'$ and $j'$ are fixed given $i'$, while for $I^2_{(ii')(ss')(jj')}=1$, choosing two of these fixes the third. Similar steps show that \[ \frac{1}{N-1}\sum_{k}(\partial_{\beta\phi'}\mathcal{L}_{(k)})\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}_{(k)}\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}_{(k)}\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L}_{(k)})-(\partial_{\beta\phi'}\mathcal{L})\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}\tilde{\mathcal{H}}\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi}\mathcal{L})=o_{p}(1) \] This gives the required result.

(2) $(N-1)(\widehat{W}_{J}-W_{J})(\widehat{\beta}-\beta)-(N-2)\frac{1}{N-1}\sum_{k}(\widehat{W}_{(k)}-W_{(k)})(\widehat{\beta}_{(k)}-\beta)=o_{p}(N^{-1})$

We first provide a first-order expansion for $\widehat{W}$. Recall that we can write $\widehat{W}=\big(\partial_{bb}\mathcal{L}^{*}(0,0)\big)^{-1}$, and so letting $\mathcal{W}^{*}(b,s)=\big(\partial_{bb}\mathcal{L}^{*}(b,s)\big)^{-1}$ and $\mathcal{W}^{*}=\big(\partial_{bb}\mathcal{L}^{*}(\mathcal{S}_{\beta},\mathcal{S})\big)^{-1}$ we can expand to give

Up to first order we find

and similarly for the leaveout versions $\widehat{W}_{(k)}$. We can then write

Using these expansions (and the equivalent leaveout versions) we can write

Following similar steps to that of the higher-order jackknife expansion, we can show that each of the terms in the product $N(\widehat{W}_{J}-W_{J})\times(\widehat{\beta}-\beta)$ is $o_{p}(1)$ when jackknifed. For example, take the first term in the expansion of both $N(\widehat{W}_{J}-W_{J})$ and $(\widehat{\beta}-\beta)$ and apply the jackknife to give

where the final line follows from application of Lemma (ref) in the main appendix. Similarly, each of the remaining terms can be shown to be jackknife versions of products of up to four random sums, all of which are $o_p(N^{-1})$, from which the result follows.

\paragraph*{(3) $\lVert \widehat{W}_J - \overline{W}_N\rVert=o_p(1)$}

For the final result, we can apply: (a) the first result $\lVert W_{J}-W_{N}\rVert=o_p (1)$, (b) $\lVert \widehat{W}_{J}-W_{J}\rVert=o_p (1)$ (which following from parameter consistency and the fact that $W$ is a smooth function of parameters), and (c) $\lVert W_{N}-\overline{W}_{N}\rVert=o_p (1)$ from Lemma (ref), along with the triangle inequality to give the result.

Asymptotic expansion for average effect

Expansion for $\widehat{\phi}$

In order to derive an expansion for the average effect parameter, we first give an expansion for the fixed effect parameters $\phi=(\alpha',\gamma')'$. Since we have $\partial_{s}\mathcal{L}^{*}(0,0)=-\widehat{\phi}$ and $\partial_{s}\mathcal{L}^{*}(\mathcal{S}_{\beta},\mathcal{S})=-\phi_{0}$, we take a Taylor expansion to give

where the remainder is

Bounds for derivation of expansion

In order to provide bounds on the terms in the asymptotic expansion for the average effect, we first state some additional bounds on terms that are used in the expansion for $\widehat{\phi}$. The first set of bounds shows that $N\partial_{\phi}\bar{\Delta}$ has $O_{p}(1)$ elements, while $N\partial_{\phi\phi'}\bar{\Delta}$ is a matrix that satisfies the conditions in Lemma (ref). In addition, we show that $N\partial_{\phi}\tilde{\Delta}$ satisfies the conditions in Lemma (ref), while $N\partial_{\phi\phi'}\tilde{\Delta}$ satisfies a bound like that in Lemma (ref).

The next lemmas are used to bound the jackknifed versions of the new forms of product terms appearing in the expansion for the average effects.

Asymptotic expansion

An expansion of the average effects estimator is

where the remainder term satisfies

Substitution of the expansion for $\widehat{\beta}$ up to order $N^{-2}$ and the expansion for $\widehat{\phi}$ shown above then gives an expansion with remainder $o_p(N^{-2})$, since for example

It follows that the expansion up to first-order is

The expansion for the leave-out estimator follows similarly, replacing $m_\lambda$ with $\frac{N-1}{N-r-1} m_\lambda 1^k_\lambda$. Note that the same bounds on $(N\partial_{\beta^{s}\alpha_{i}}\tilde{\Delta})$ derivied in Lemma (ref) also apply to $(N\partial_{\beta^{s}\alpha_{i}}\tilde{\Delta}_{(k)})$, so that we may replace the $\partial_{\beta}\Delta_{(k)}$ and $\partial_{\alpha}\Delta_{(k)}$ terms with $\partial_{\beta}\bar{\Delta}$ and $\partial_{\alpha}\bar{\Delta}$ in the first-order expansion. This gives the expansion for $N(\widehat{\Delta}_{(k)} - \Delta_{(k)})$ as shown in Lemma 7.

Jackknifing higher-order terms for average effect

Analogously to the jackknife of the common parameter $\widehat{\beta},$ we demonstrate the effect of the jackknife for just one higher-order term in the expansion for $\widehat{\Delta}$. Inspection of this expansion shows that the structure of each of the terms is similar, i.e. a V-statistic like term of some order, multiplied by $(\partial_{\phi'}\Delta)$ or $(\partial_{\phi\phi'}\Delta)$. The bound for the jackknife version of each of these many terms is therefore highly similar to the specific term we demonstrate here. Consider the expansion term $(\partial_{\phi'}\Delta)(\partial_{bss'}\mathcal{L}_{(1)}^{*})\mathcal{S}\mathcal{S}_{\beta}$, which contains the term \[ (\partial_{\phi'}\Delta)\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta} \] Expanding out this terms gives

Take the first term in this expression. Replacing $W$ with $\bar{W}+(W-\bar{W})$, and similarly for $\mathcal{H}^{-1}$ and $\partial_{\beta\phi\phi'}\mathcal{L}$ we get \[ \frac{1}{N}\bar{W}_{N}^{-1}(\partial_{\phi'}\Delta)\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}\mathcal{S}\mathcal{S}_{\beta}+o_{p}(N^{-2}) \] This result holds identically for the leave-out samples, replacing $\mathcal{S}$ with $\mathcal{S}_{(k)}$ and similarly for $\partial_{\phi'}\Delta$ and $\mathcal{S}_{\beta}$. Ignoring the $\frac{1}{N}\bar{W}_{N}^{-1}$ term for the moment, which will not be affected by the jackknifing, and letting $\bar{M}=\mathcal{\bar{H}}^{-1}(\partial_{\beta\phi\phi'}\bar{\mathcal{L}})\bar{\mathcal{H}}^{-1}$, we can decompose the above sum further into the components of $\phi=(\alpha,\gamma)$, for example the first of these terms would be

and note that, by Lemma (ref), we have $\max_{i\ne s}\bar{M}_{is}=O_{p}(N^{-1})$, and $\max_{i}\bar{M}_{ii}=O_{p}(1)$. This is again a V-statistic like term, and we will show that the jackknifed version of this term is $o_{p}(1)$. The $k$-th leave-out version of this term is

The jackknifed term is then

Note that, since $\lambda$ contains $r$ observations, the set $\{\lambda,(s,t),(j,l)\}$ spans at most $r+2$ different sets $\mathcal{I}_{k}$ and so $\sum_{k}1_{\lambda}^{k}1_{st}^{k}1_{jl}^{k}\in\{N-r-3,\dots,N-2\}$, depending on how many of the $\mathcal{I}_{k}$ the observations are contained in. Then, letting $1_{n}$ be an indicator variable that is equal to one when the set $\{\lambda,(s,t),(j,l)\}$ covers $n$ different $\mathcal{I}_{k},$ we can write

The second moment of the RHS is

Since $\bar{E}[\partial_{\pi}\ell_{ij}]=\bar{E}[\partial_{\beta}\ell_{ij}]=0$, while $\max_{i\ne s}\bar{M}_{is}=O_{p}(N^{-1})$, and $\max_{i}\bar{M}_{ii}=O_{p}(1)$, we can conclude that the above sum is $O_{p}(N^{2})$, and hence $\vert\mathcal{J}\vert=O_{p}(N^{-1})$. The same steps can be shown for the remaining elements of this term, from which we then conclude that the jackknifed version of $(\partial_{\phi'}\Delta)\mathcal{H}^{-1}\mathcal{H}_{b}\mathcal{H}^{-1}\mathcal{S}\mathcal{S}_{\beta}$ is $O_{p}(N^{-2})$, and hence does not show up in the asymptotic distribution of $\widehat{\Delta}_{J}$. Inspection of the expansion for $\widehat{\Delta}$ shows that the remaining terms of the expansion share this similar V-statistic like structure, and so similar proofs could be applied to each of the terms.

This then gives the result \[ (N-1)(\widehat{\Delta}-\Delta)-(N-2)\frac{1}{N-1}\sum_{k}(\widehat{\Delta}_{(k)}-\Delta_{(k)})=\frac{1}{N-1}\sum_{i}\sum_{j\ne i}h_{ij}+o_{p}(1). \]

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