this paper studies the following panel data models with unobservable interactive effects : where is the dependent variable ; is a row vector of explanatory variables ; is an intercept ; the term is unobservable and has a factor structure , is an vector of factor loadings , is a vector of factors and is the idiosyncratic error .the interactive effects ( ) generalize the usual additive individual and time effects ; for example , if , then .a key feature of the model is that the regressors are allowed to be correlated with .this situation is commonly encountered in economics and other social sciences , in which some of the regressors are decision variables that are influenced by the unobserved individual heterogeneities .the practical relevance of the model will be further discussed below .the objective of this paper is to obtain consistent and efficient estimation of in the presence of correlations between the regressors and the factor loadings and factors .the usual pooled least squares estimator or even the within - group estimator is inconsistent for .one method to obtain a consistent estimator is to treat as parameters and estimate them jointly with .the idea is `` controlling through estimating '' ( controlling the effects by estimating them ) .this is the approach used in and . while there are some advantages , an undesirable consequence of this approach is the incidental parameters problem .there are too many parameters being estimated , and the incidental parameters bias arises ; see . in and authors consider the generalized method of moments ( gmm ) method .the gmm method is based on a nonlinear transformation known as quasi - differencing that eliminates the factor errors .quasi - differencing increases the nonlinearity of the model especially with more than one factor .the gmm method works well with a small .when is large , the number of moment equations will be large , and the so called many - moment bias arises . in , the author considers an alternative method by augmenting the model with additional regressors and , which are the cross - sectional averages of and .these averages provide an estimate for .the estimator of becomes inconsistent when the factor loadings in the equation are correlated with those in the equation , as shown in .a further approach to controlling the correlation between the regressors and factor errors is to use the mundlak chamberlain projection ( and  ) .the latter method projects and onto the regressors such that , where ( are parameters to be estimated , and is the projection residual ( a similar projection is done for  ) .the projection residuals are uncorrelated with the regressors so that a variety of approaches can be used to estimate the model .this framework is designed for small and is studied by .  in this paperwe consider the pseudo - gaussian maximum likelihood method under large and large .the theory does not depend on normality . in view of the importance of the mle in the statistical literature ,it is of both practical and theoretical interest to examine the mle in this context .we develop a rigorous theory for the mle .we show that there is no incidental parameters bias for .we allow time - invariant regressors such as education , race and gender in the model .the corresponding regression coefficients are time - dependent .similarly , we allow common regressors , which do not vary across individuals , such as prices and policy variables .the corresponding regression coefficients are individual - dependent so that individuals respond differently to policy or price changes . in our view , this is a sensible way to incorporate time - invariant and common regressors . for example , wages associated with education and with gender are more likely to change over time rather than remain constant . in our analysis , time invariant regressors are treated as the components of that are observable , and common regressors as the components of that are observable .this view fits naturally into the factor framework in which part of the factor loadings and factors are observable , and the maximum likelihood method imposes the corresponding loadings and factors at their observed values .  while the theoretical analysis of mle is demanding , the limiting distributions of the mle are simple and have intuitive interpretations .the computation is also easy and can be implemented by adapting the ecm ( expectation and constrained maximization ) of .in addition , the maximum likelihood method allows restrictions to be imposed on or on to achieve more efficient estimation .these restrictions can take the form of known values , being either zeros , or other fixed values .part of the rigorous analysis includes setting up the constrained maximization as a lagrange multiplier problem .this approach provides insight into which kinds of restrictions provide efficiency gain and which kinds do not .panel data models with interactive effects have wide applicability in economics . in macroeconomics , for example , can be the output growth rate for country in year ; represents production inputs , and is a vector of common shocks ( technological progress , financial crises ) ; the common shocks have heterogenous impacts across countries through the different factor loadings ; represents the country - specific unmeasured growth rates . in microeconomics , and especially in earnings studies, is the wage rate for individual for period ( or for cohort ) , is a vector of observable characteristics such as marital status and experience ; is a vector of unobservable individual traits such as ability , perseverance , motivation and dedication ; the payoff to these individual traits is not constant over time , but time varying through ; and is idiosyncratic variations in the wage rates . in finance, is stock s return in period , is a vector of observable factors , is a vector of unobservable common factors ( systematic risks ) and is the exposure to the risks ; is the idiosyncratic returns .factor error structures are also used as a flexible trend modeling as in .most of panel data analysis assumes cross - sectional independence ; see , for example , and .the factor structure is also capable of capturing the cross - sectional dependence arising from the common shocks .further motivation can be found in .  throughout the paper, the norm of a vector or matrix is that of frobenius , that is , ^{1/2} ] for a sufficiently large constant .it is a well - known result in factor analysis that the factors and loadings can only be identified up to a rotation ; see , for example , .the models considered in this paper can be viewed as extensions of the factor models . as suchthey inherit the same identification problem .we show that identification conditions can be imposed on the factors and loadings without loss of generality . to see this , model ( [ comshkmaineq2 ] )can be rewritten as \bigl[r'm_{f f}^{-1/2}(f_t- \bar f)\bigr]+\varepsilon_t,\ ] ] where is an orthogonal matrix , which we choose to be the matrix consisting of the eigenvectors of associated with the eigenvalues arranged in descending order . treating as the new , as the new and as the new , we have with and being a diagonal matrix .thus we impose the following restrictions for model ( [ comshkmaineq2 ] ) , which we refer to as ( _ identification _ restrictions for _ basic _ models ) . ;   , where is a diagonal matrix with its diagonal elements distinct and arranged in descending order ;   .the objective function considered in this section is ,\ ] ] where and .the latter is the data matrix .the parameters are .the mle is defined as where the parameter space is defined to be a closed and bounded subset containing the true parameter as an interior point ; and are positive definite matrices , as in assumption [ assd ] .the boundedness of implies that the elements of and are bounded .this is for theoretical purpose and is usually assumed for nonconvex optimizations , as in and . in actual computation with the em algorithm , we do not find the need to impose an upper or lower bound for the parameter values .the likelihood function involves simple functions and are continuous on ( in fact differentiable ) , so the mle exists because a continuous function achieves its extreme value on a closed and bounded subset .note that the determinant of is 1 , so the jacobian term does not depend on .if and are independent and normally distributed , the likelihood function for the observed data has the form of ( [ comshkmaineq3 ] ) . here recall that are fixed constants , and are not necessarily normal ; ( [ comshkmaineq3 ] ) is a pseudo - likelihood function .  for further analysis , we partition the matrix and as where for any , and are both matrices .let and denote the mle .the first order condition for satisfies \biggr\}\dot x_{it}=0,\hspace*{-35pt}\ ] ] where .the first order condition for satisfies post - multiplying on both sides of ( [ comshkmaineq5 ] ) and then taking summation over , we have the first order condition for satisfies where is a matrix such that its upper - left and lower - right submatrices are both zero , but the remaining elements are undetermined .the undetermined elements correspond to the zero elements of .these first order conditions are needed for the asymptotic representation of the mle .theorem [ comshkthm2 ] states the convergence rates of the mle .the consistency is implied by the theorem .[ comshkthm2 ] let be the solution by maximizing ( [ comshkmaineq3 ] ) . under assumptions [ assa][assd ] and the identification conditions , we have   bai considers an iterated principal components estimator for model ( [ comshkmaineq1 ] ) .his derivation shows that , in the presence of heteroscedasticities over the cross section , the pc estimator for has a bias of order . as a comparison , theorem [ comshkthm2 ] shows that the mle is robust to the heteroscedasticities over the cross section .so if is fixed , the estimator in is inconsistent unless there is no heteroskedasticity , but the estimator here is still .let denote the project matrix onto the space orthogonal to , that is , .we have  [ comshkthm3 ] under the assumptions of theorem [ comshkthm2 ] , we have where is a matrix whose element with being the element of matrix .[ rmk25 ] in appendix a.3 of the supplement  , we show that the asymptotic expression of can be alternatively expressed as & \cdots & \operatorname{tr}\bigl[\ddot mx_1\mathcal{m}(\widebar{\mathbb f } ) x_k ' \bigr ] \vspace*{3pt}\cr \vdots & \vdots & \vdots \vspace*{3pt}\cr \operatorname{tr}\bigl[\ddot mx _k\mathcal { m}(\widebar{\mathbb f})x_1 ' \bigr ] & \cdots&\operatorname{tr}\bigl[\ddot mx _ k\mathcal { m}(\widebar{\mathbb f } ) x _ k ' \bigr]}^{-1 } \nonumber\\[20pt]\\[-30pt ] & & { } \times\pmatrix { \operatorname{tr}\bigl[\ddot mx_1\mathcal{m}(\widebar{\mathbb f } ) e ' \bigr ] \vspace*{3pt}\cr \vdots \vspace*{3pt}\cr \operatorname{tr}\bigl[\ddot mx _ k\mathcal{m}(\widebar{\mathbb f } ) e'\bigr]}\nonumber \\ & & { } + o_p \bigl(t^{-3/2}\bigr)+o_p\bigl(n^{-1}t^{-1/2}\bigr)+o_p\bigl(n^{-1/2}t^{-1}\bigr),\nonumber\end{aligned}\ ] ] where is ( the data matrix for the regressor , ) ; is ; with and ; ; where is a vector with all 1 s .  theorem [ comshkthm3 ] shows that the asymptotic expression of only involves variations in and .intuitively , this is due to the fact that the error terms of the equation share the same factors with the explanatory variables .the variations from the common factor part of ( i.e. , ) do not provide information for since this part of information is offset by the common factor part of the error terms ( i.e. , ) in the equation .[ comshkthm4 ] under the assumptions of theorem [ comshkthm3 ] , if , we have where , and is also the limit of & \cdots & \operatorname{tr}\bigl[\ddot mx_1\mathcal{m } ( \widebar{\mathbb f } ) x_k ' \bigr ] \vspace*{3pt}\cr \vdots & \vdots & \vdots \vspace*{3pt}\cr \operatorname{tr}\bigl[\ddot mx _ k\mathcal { m}(\widebar{\mathbb f})x_1 ' \bigr ] & \cdots&\operatorname{tr}\bigl[\ddot mx _ k\mathcal { m}(\widebar{\mathbb f } ) x _ k ' \bigr]}.\ ] ]  [ remark2 ] matrix can be consistently estimated by & \cdots & \operatorname{tr}\bigl[\widehat{\ddot m}x_1\mathcal{m}(\widehat{\widebar{\mathbb f } } ) x_k ' \bigr ] \vspace*{3pt}\cr \vdots & \vdots & \vdots \vspace*{3pt}\cr \operatorname{tr}\bigl[\widehat{\ddot m}x_k\mathcal{m}(\widehat{\widebar{\mathbb f}})x_1 ' \bigr ] & \cdots & \operatorname{tr}\bigl[\widehat{\ddot m}x_k\mathcal { m}(\widehat{\widebar{\mathbb f } } ) x _ k ' \bigr]},\ ] ] where is the data matrix for the regressor ,  with and here and are the maximum likelihood estimators .the basic model in section [ sec2 ] assumes that the explanatory variables share the same factors with .this section relaxes this assumption .we assume that the regressors are impacted by additional factors that do not affect the equation .an alternative view is that some factor loadings in the equation are restricted to be zero .consider the following model : \\[-8pt ] x_{itk}&=&\mu_{ik}+\gamma_{ik}^{g\prime}g_t+ \gamma_{ik}^{h\prime } h_t+v_{itk } \nonumber\end{aligned}\ ] ] for , where is an vector representing the shocks affecting both and , and is an vector representing the shocks affecting only .let , and , the above model can be written as which is the same as model ( [ comshkmaineq1 ] ) except that elements of are restricted to be zeros .for further analysis , we introduce some notation .we define we also define and similarly as , that is , , .this implies that .the presence of zero restrictions in ( [ zerores1 ] ) requires different identification conditions .zero loading restrictions alleviate rotational indeterminacy . instead of restrictions , we only need to impose restrictions .these restrictions are referred to as iz restrictions ( _ identification _ conditions with _ zero _ restrictions ) .they are :   ;   and , where and are both diagonal matrices with distinct diagonal elements in descending order ;   and .in addition , we need an additional assumption for our analysis .[ asse ] is of full column rank .identification conditions iz are less stringent than of the previous section .assumption [ asse ] says that the factors are pervasive for the equation . in appendixb of the supplement  , we explain why restrictions are sufficient .the likelihood function is now maximized under three sets of restrictions , that is , , and where denotes the zero factor loading matrix in the equation . the likelihood function with the lagrange multipliersis \\ & & { } + \operatorname{tr } \biggl[\upsilon_1 \biggl(\frac{1}{n } \gamma^{g\prime } \sigma_{\varepsilon \varepsilon}^{-1}\gamma^g -d_1 \biggr ) \biggr]+\operatorname{tr } \biggl[\upsilon_2 \biggl(\frac { 1}{n}\gamma^{h\prime}\sigma _ { \varepsilon\varepsilon}^{-1 } \gamma^h - d_2 \biggr ) \biggr ] \\ & & { } + \operatorname { tr}\bigl [ \upsilon_3'\phi\bigr],\end{aligned}\ ] ] where ; is and is , both are symmetric lagrange multipliers matrices with zero diagonal elements ; is a lagrange multiplier matrix of dimension .let \widehat\sigma_{z z}^{-1} ] .( this equation would be the first order condition for if it were unknown . )this equality can be simplified as \widehat\sigma_{\varepsilon\varepsilon}^{-1}\widehat\gamma=0\ ] ] because with .next , we partition the matrix and as follows : = { \left}[\matrix{\widehat g_{11 } & \widehat g_{12 } \vspace*{3pt}\cr \widehat g_{21 } & \widehat g_{22 } } { \right } ] , \qquad \widehat h= { \left } [ \matrix{\widehat h_1 \vspace*{3pt}\cr \widehat h_2 } { \right } ] = { \left}[\matrix { \widehat h_{11 } & \widehat h_{12 } \vspace*{3pt}\cr \widehat h_{21 } & \widehat h_{22 } } { \right}],\ ] ] where are , while are .  notice and .substitute these results into ( [ extenmaineq1 ] ) , and use ( [ extenmaineq2 ] ) .the first order condition for can be simplified as where is the first column of the identity matrix of dimension .similarly , the first order condition for is where is a matrix , obtained by deleting the first column of the identity matrix of dimension .the first order condition for is \\[-8pt ] & & \qquad { } -\sum_{i=1}^n \bigl(\widehat bm_{z z}^{ji}\widehat b{}'-\widehat\sigma_{z z}^{ji } \bigr)\widehat\sigma_{ii}^{-1}\widehat\gamma_i ' \widehat g\widehat\gamma_j=\mathbb { w } , \nonumber\end{aligned}\ ] ] where is defined following ( [ comshkmaineq7 ] ) .the first order condition for is \biggr\}\dot x_{it}=0,\hspace*{-35pt}\ ] ] which is the same as in section [ sec2 ] .we need an additional identity to study the properties of the mle .recall that , by the special structures of and , the three submatrices of can be directly derived to be zeros .the remaining submatrix is also zero , as shown earlier .however , this submatrix being zero yields the following equation ( the detailed derivation is delivered in appendix f ) : these identities are used to derive the asymptotic representations .the results on consistency and the rate of convergence are similar to those in the previous section , which are presented in appendixes b.1 and b.2 . for simplicity , we only state the asymptotic representation for the mle here .  [ extenthm1 ] assumptions and the identification restriction iz, we have where is a symmetric matrix with its element equal to ; ' ] ; and .  proposition [ extenthm1 ] is derived under the identification conditions iz . in appendixb.3 of the supplement  , we show that for any set of factors and factor loadings , it can always be transformed into a new set , which satisfies iz , and at the same time , leaving intact .given the asymptotic representation in proposition [ extenthm1 ] , together with the relationship between the two sets , we have the following theorem , which does not depend on iz .  [ extenthmnew ] under assumptions [ assa][asse ], we have where  is a symmetric matrix with its element equal to +\frac{1 } n\sum_{i=1}^n\sigma _ { iie}^{-1}\sigma_{iix}^{(p , q)},\ ] ] where ; , .[ remark - asy - beta ] in appendix b.3 , we show that the asymptotic expression of in theorem [ extenthmnew ] can be expressed alternatively as & \cdots & \operatorname{tr}\bigl[\ddot mx_1\mathcal{m}(\widebar{\mathbb g } ) x_k ' \bigr ] \vspace*{3pt}\cr \vdots & \vdots & \vdots \vspace*{3pt}\cr \operatorname{tr}\bigl[\ddot mx_k\mathcal { m}(\widebar{\mathbb g})x_1 ' \bigr ] & \cdots&\operatorname{tr}\bigl[\ddot mx_k\mathcal{m}(\widebar{\mathbb g } ) x_k ' \bigr]}^{-1 } \\[4pt ] & & { } \times\pmatrix{\operatorname{tr}\bigl[\ddot mx_1\mathcal{m}(\widebar{\mathbb g } ) e ' \bigr ] \vspace*{3pt}\cr \vdots \vspace*{3pt}\cr \operatorname{tr}\bigl[\ddot mx _k\mathcal{m}(\widebar{\mathbb g } ) e'\bigr ] } \\ & & { } + o_p \bigl(t^{-3/2}\bigr)+o_p\bigl(n^{-1}t^{-1/2 } \bigr)+o_p\bigl(n^{-1/2}t^{-1}\bigr),\end{aligned}\ ] ] where and are defined below ( [ alternative1 ] ) and .notice is defined as , which is equal to since in the present context . in appendixb.3 of the supplement  , we also provide an intuitive explanation for this alternative expression .given theorem [ extenthmnew ] and remark [ remark - asy - beta ] we have the following corollary .[ extenthm2 ] under assumptions [ assa][asse ] , if , we have where , and is also the probability limit of & \cdots & \operatorname{tr}\bigl[\ddot mx_1\mathcal{m}(\widebar{\mathbb g } ) x_k ' \bigr ] \vspace*{3pt}\cr \vdots & \vdots & \vdots \vspace*{3pt}\cr \operatorname{tr}\bigl[\ddot mx _ k\mathcal { m}(\widebar{\mathbb g})x_1 ' \bigr ] & \cdots&\operatorname{tr}\bigl[\ddot mx _ k\mathcal { m}(\widebar{\mathbb g } ) x_k ' \bigr]}.\ ] ]  [ remark1 ] compared with the model in section [ sec2 ] , is more efficient under the zero loading restrictions .the reason is intuitive . in the previous model ,only variations in provide information for .but in the present case , variations in of also provide information for .this can also be seen by comparing the limiting variances of corollaries [ comshkthm4 ] and [ extenthm2 ] .notice the projection matrix now only involves instead of ; and is a submatrix of .in addition , the covariance matrix can be estimated by the same method as in estimating ; see remark [ remark2 ] .in this section , we extend the basic model in section [ sec2 ] to include time - invariant regressors and common regressors .examples of time - invariant regressors include gender , race and education ; and examples for common regressors include price variables , unemployment rate , or macroeconomic policy variables .these types of regressors are important for empirical applications .we first consider the model with only time - invariant regressors , \\[-8pt ] x_{itk}&=&\mu_{ik}+\gamma_{ik}^{g\prime}g_t+ \gamma_{ik}^{h\prime } h_t+v_{itk } \nonumber\end{aligned}\ ] ] for , where is an -dimensional vector , and is an dimensional vector .let , an -dimensional vector .the key point of model ( [ obseffeq2 ] ) is that the s are known ( but not zeros ) .we treat as new added time - invariant regressors , whose coefficient is allowed to be time - varying .the parameter of interest is still .the inference for is provided in appendix c.4 of the supplement  .the model in the previous section can be viewed as , where .however , the earlier derivation is not applicable here because now is a general matrix with full column rank , which provides more information ( restrictions ) on the rotation matrix .thus the number of restrictions required to eliminate rotational indeterminacy is even fewer than in section [ sec3 ] .this point can be seen in the next subsection .we define the following notation for further analysis : then equation ( [ obseffeq2 ] ) has the same matrix expression as ( [ comshkmaineq2 ] ) .note that  ] is of full column rank .   for clarity , in this subsection, we use to denote the observed value for . recall that , where contains the factor loading coefficients ( including ) ; contains the sub - blocks , and ; contains the heteroskedasticity coefficients .the regression coefficient is contained in matrix .the maximization of the likelihood function is now subject to four sets of restrictions , , , and .the likelihood function augmented with the lagrange multipliers is +\operatorname { tr } [ \upsilon _ 1m_{gh } ] \\ & & { } + \operatorname{tr } \bigl[\upsilon_2(m_{g g}-i_{r_1 } ) \bigr]+\operatorname{tr } \biggl[\upsilon_3 \biggl(\frac { 1}{n } \gamma^{g\prime}\sigma_{\varepsilon\varepsilon}^{-1 } \gamma^g - d \biggr ) \biggr]+\operatorname{tr } \bigl[\upsilon_4\bigl(\phi-\phi ^ * \bigr ) \bigr],\end{aligned}\ ] ] where and are all lagrange multipliers matrices ; is an matrix ; is an symmetric matrix ; is an symmetric matrix with all diagonal elements zeros ; is an matrix ; and . using the same arguments in deriving in section [ sec3 ], we have . then the likelihood function is simplified as \nonumber \\[-8pt ] \\[-8pt ] & & { } + \operatorname{tr}[\upsilon_1m_{g h } ] + \operatorname{tr}\bigl[\upsilon_2(m_{g g}-i_{r_1 } ) \bigr]+\operatorname { tr}\bigl[\upsilon_4\bigl(\phi-\phi^*\bigr ) \bigr ] .\nonumber\end{aligned}\ ] ] the first order condition for gives \widehat\sigma_{z z}^{-1}=w',\ ] ] where is defined in ( [ extenmaineq1 ] ) .pre - multiplying and post - multiplying , and by the special structures of and , we have \widehat\sigma_{z z}^{-1}\widehat\gamma \\ & & \qquad = - { \left } [ \matrix{0_{r_1\times r_1 } & 0_{r_1\times r_2 } \vspace*{3pt}\cr \dfrac{1 } n \widehat m_{hh}^{-1}\upsilon_4 ' \widehat\psi & \dfrac{1 } n\widehat m_{hh}^{-1}\upsilon _ 4'\phi}{\right}].\end{aligned}\ ] ] but the first order condition for gives \widehat\sigma_{z z}^{-1}\widehat\gamma= { \left}[\matrix { \upsilon_2 & \upsilon_1 ' \vspace*{3pt}\cr \upsilon_1 & 0_{r_2\times r_2 } } { \right}].\ ] ] comparing the proceeding two results and noting that the left - hand side is a symmetric matrix , we have \widehat\sigma_{z z}^{-1}\widehat\gamma=0 ] ; ; ' ] .proposition [ obseffthm1 ] is derived under the identification conditions io . in appendixc.3 , we show that for any set of factors and factor loadings , we can always transform it to another set which satisfies io , and at the same time , still maintains the observability of ( i.e. , is untransformed ) .this is in agreement with the lagrange multiplier analysis , in which ( , but the multiplier for is nonzero . using the relationship between the two sets , we can generalize proposition [ obseffthm1 ] into the following theorem , which does not depend on io .[ obseffthmnew ] under assumptions [ assa][assd ] and [ assf ] , we have where  is a symmetric matrix with its element equal to +\frac{1 } n\sum _ { i=1}^n\sigma _ { iie}^{-1 } \sigma_{iix}^{(p , q)}\ ] ] and , and are defined in proposition [ obseffthm1 ] .[ remark3 ] in appendix c.3 we show that the asymptotic expression of in theorem [ obseffthmnew ] can be expressed alternatively as & \cdots & \operatorname{tr}\bigl[\ddot mx_1\mathcal{m}(\widebar{\mathbb g } ) x_k ' \bigr ] \vspace*{3pt}\cr \vdots & \vdots & \vdots \vspace*{3pt}\cr \operatorname{tr}\bigl[\ddot mx _k\mathcal { m}(\widebar{\mathbb g})x_1 ' \bigr ] & \cdots&\operatorname{tr}\bigl[\ddot mx _ k\mathcal { m}(\widebar{\mathbb g } ) x _ k ' \bigr]}^{-1 } \\[4pt ] & & { } \times\pmatrix{\operatorname{tr}\bigl[\ddot mx_1\mathcal{m}(\widebar{\mathbb g } ) e ' \bigr ] \vspace*{3pt}\cr \vdots \vspace*{3pt}\cr \operatorname{tr}\bigl[\ddot mx _k\mathcal{m}(\widebar{\mathbb g } ) e'\bigr]}+o_p \bigl(t^{-3/2}\bigr ) \\ & & { } + o_p\bigl(n^{-1}t^{-1/2 } \bigr)+o_p\bigl(n^{-1/2}t^{-1}\bigr),\end{aligned}\ ] ] where and are defined below ( [ alternative1 ] ) and .we also show in appendix c.3 that this alternative expression has an intuitive explanation .  from theorem [ obseffthmnew ], we obtain the following corollary .[ obseffcor ] under the conditions of theorem [ obseffthm1 ] , if , we have where , which has an alternative expression & \cdots & \operatorname{tr}\bigl[\ddot mx_1\mathcal{m}(\widebar{\mathbb g } ) x_k ' \bigr ] \vspace*{3pt}\cr \vdots & \vdots & \vdots \vspace*{3pt}\cr \operatorname{tr}\bigl[\ddot mx _k\mathcal { m}(\widebar{\mathbb g})x_1 ' \bigr ] & \cdots&\operatorname{tr}\bigl[\ddot mx_k\mathcal { m}(\widebar{\mathbb g } ) x _ k ' \bigr]}.\ ] ]  compared with the model in section [ sec2 ] , is more efficient with observable fixed effects ( time - invariant regressors ) . the reason is provided in remark [ remark1 ] .   in this subsection , we consider the joint presence of time - invariant regressors and common regressors .consider the following model : \\[-8pt ] x_{itk}&=&\gamma_{ik}^{g\prime}g_t+ \gamma_{ik}^{h\prime}h_t+\gamma _ { ik}^{d\prime}\,d_t+v_{itk } \nonumber\end{aligned}\ ] ] for , where , and are , and vectors , respectively .a key feature of model ( [ obseffeq7 ] ) is that and are observable for all and .we call the time - invariant regressors because they are invariant over time and the common regressors because they are the same for all the cross - sectional units . in this model ,the time - invariant regressors have time - varying coefficients , and the common regressors have heterogeneous ( individual - dependent ) coefficients . if , plays the role of in ( [ obseffeq2 ] ) .so the model here is more general .similar to the previous subsection , we make the following assumption :  [ assg ] the matrices and are both of full column rank , where and .let , and .the model can be written as {it}= \gamma_i'f_t + \delta_i'd_t+ \varepsilon_{it},\ ] ] where are defined in section [ sec2 ] ; let .then where the symbols are defiend in section [ sec2 ] .the likelihood function can be written as ' \sigma_{z z}^{-1}\bigl[(i_n\otimes b)z_t-\delta d_t\bigr].\ ] ] take and as given . maximizes the above function at substituting into the above likelihood function , we obtain the concentrated likelihood function ,\ ] ] where , and , a projection matrix .consider ( [ obseffeq8 ] ) , which is equivalent to where .post - multiplying on both sides , we have if we treat as the new observable data , as the new unobservable factors , the preceding equation can be viewed as a special case of ( [ obseffeq2 ] ) .invoking theorem [ obseffthmnew ] , which does not need io [ the factors may not satisfy io ] , we have the following theorem :  [ obseffthm3 ] under assumptions [ assa][assd ] and [ assg ] , the asymptotic representation of in the presence of time invariant and common regressors is where ^{-1 } \bigl(g_t-\mathbb g'\mathbb d\bigl(\mathbb d'\mathbb d\bigr)^{-1}\,d_t\bigr);\ ] ] is a symmetric matrix with its element equal to +\frac{1 } n\sum_{i=1}^n \sigma_{iie}^{-1}\sigma _ { iix}^{(p , q)},\ ] ] where and , a matrix of dimension ; ; .[ remark - invariant - common ] the asymptotic expression of can be alternatively expressed as & \cdots & \operatorname{tr}\bigl[\ddot mx_1\mathcal{m}(\mathbb b ) x_k ' \bigr ] \vspace*{3pt}\cr \vdots & \vdots & \vdots \vspace*{3pt}\cr \operatorname{tr}\bigl[\ddot mx_k\mathcal { m}(\mathbb b)x_1 ' \bigr ] & \cdots & \operatorname{tr}\bigl[\ddot mx_k\mathcal{m}(\mathbb b ) x_k ' \bigr]}^{-1 } \\[4pt ] & & { } \times\pmatrix{\operatorname{tr}\bigl[\ddot mx_1\mathcal{m}(\mathbb b ) e ' \bigr ] \vspace*{3pt}\cr \vdots \vspace*{3pt}\cr\operatorname{tr}\bigl[\ddot mx_k\mathcal{m}(\mathbb b ) e'\bigr ] } \\ & & { } + o_p \bigl(t^{-3/2}\bigr)+o_p\bigl(n^{-1}t^{-1/2 } \bigr)+o_p\bigl(n^{-1/2}t^{-1}\bigr).\end{aligned}\ ] ] if , the above asymptotic result reduces to the one in theorem [ obseffthmnew ] since .given theorem [ obseffthm3 ] and remark [ remark - invariant - common ] , we have the following corollary .  [ corsub44 ] under assumptions [ assa][assd ] and [ assg ] , if , then where , and can also be expressed as & \cdots & \operatorname{tr}\bigl[\ddot mx_1\mathcal{m}(\mathbb b ) x_k ' \bigr ] \vspace*{3pt}\cr \vdots & \vdots & \vdots \vspace*{3pt}\cr \operatorname{tr}\bigl[\ddot mx_k\mathcal { m}(\mathbb b)x_1 ' \bigr ] & \cdots&\operatorname{tr}\bigl[\ddot mx_k\mathcal{m}(\mathbb b ) x_k ' \bigr]}.\ ] ]to estimate the model by the maximum likelihood method , we adapt the ecm ( expectation and conditional maximization ) procedures of .more specifically , in the m - step we split the parameter into two blocks , and , and update to given and then update to given , where is the estimated value at the iteration . in this section , we only state the iterating formulas for basic models .the iterating formulas for the models in sections [ sec3 ] and [ sec4 ] can be found in appendix e of . in appendix e, we also show that the iterated em solutions satisfy the first order conditions .so the em estimators are at least locally optimal .  in the basic model , .so the parameters to be estimated reduce to .let be the estimated value at the iteration .we update according to \biggl[\frac{1 } t\sum _ { t=1}^te\bigl(f_tf_t'|z , \theta^{(k)}\bigr ) \biggr]^{-1},\ ] ] where & & \qquad = i_r-\gamma ^{(k)\prime } \bigl(\sigma_{z z}^{(k ) } \bigr)^{-1}\gamma^{(k)}\label{52 } \\ & & \quad\qquad { } + \gamma^{(k)\prime}\bigl(\sigma_{z z}^{(k ) } \bigr)^{-1}\bigl(i_n\otimes b^{(k ) } \bigr)m_{z z}\bigl(i_n\otimes b^{(k)\prime}\bigr ) \bigl ( \sigma_{z z}^{(k)}\bigr)^{-1}\gamma ^{(k ) } , \nonumber \\[-2pt ] \label{53 } & & \frac{1 } t\sum _ { t=1}^te\bigl(z_tf_t'|z , \theta ^{(k)}\bigr)=\bigl(i_n\otimes b^{(k ) } \bigr)m_{z z}\bigl(i_n\otimes b^{(k)\prime } \bigr ) \bigl ( \sigma _ { z z}^{(k)}\bigr)^{-1}\gamma^{(k)}\end{aligned}\ ] ] with .we update and according to \label{54 } \\[-9pt ] & & \hspace*{27pt } { } \times\bigl(i_n\otimes b^{(k)}\bigr)m_{z z } \bigl(i_n\otimes b^{(k)\prime}\bigr ) \bigr\ } , \nonumber \\ \beta^{(k+1)}&= & \biggl(\sum_{i=1}^n \sum_{t=1}^t\dot x_{it } ' \bigl(\sigma _ { iie}^{(k+1)}\bigr)^{-1}\dot x_{it } \biggr)^{-1 } \nonumber \\[-9pt]\label{55 } \\[-9pt ] & & { } \times \biggl(\sum_{i=1}^n\sum _ { t=1}^t\dot x_{it } ' \bigl(\sigma _ { iie}^{(k+1)}\bigr)^{-1}\bigl(\dot y_{it}-\lambda_i^{(k+1)\prime } f_t^{(k ) } \bigr ) \biggr ) , \nonumber\end{aligned}\ ] ] where is the transpose of the row of where with ; is the operator that sets the entries of its argument to zeros if the counterparts of are zeros .putting together , we obtain .the above iteration continues until is smaller than a preset error tolerance .the initial values use the iterated pc estimators of .in this section , we consider the finite sample properties of the mle .data are generated according to \\[-8pt ] x_{itk}&=&\mu_{ik}+\gamma_{ik}^{g\prime}g_t+ \gamma_{ik}^{h\prime } h_t+\gamma_{ik}^{d\prime}\,d_t+v_{itk } , \qquad k=1,2 . \nonumber\end{aligned}\ ] ] the dimensions of are each fixed to 1 .we set and .we consider four types of dgp ( data generating process ) , which correspond to the four models considered in the paper . and are fixed to zeros ; and are generated from and . and are fixed to zeros ; , and are generated from ; . and are fixed to zeros ; and are generated from ; and . here is observable . and are generated from ; , , and . here and are observable .  using the method of writing( [ comshkmaineq2 ] ) , we can rewrite ( [ datagen ] ) as where for dgp1 ; for dgp2 and dgp3 ; for dgp4 , and is the corresponding loadings matrix .let be the row of .we generate the cross - sectional heteroscedasticity , an vector , according to , where is drawn from $ ] with .a similar way of generating heteroscedasticity is also used in and .let be an block diagonal matrix , in which , with being a standard normal random matrix for each .once is generated , the error term  , which is defined as with , is calculated by , where is an vector with all its elements being i.i.d . , where denotes the chi - squared distribution with two freedom degrees , which is normalized to mean zero and variance one .additional simulation results for normal and student- errors are given in appendix d. once is obtained , we use to yield the observable data .  in the basic model ,the number of factors is determined by with where and are the respective estimators of and when the factor number is set to and . in the simulation , we set . for the model with zero restrictions , we consider a two - step method to determine  and  .first , we use ( [ eq0 ] ) to estimate the total number , denoted by , and obtain by the method of the basic model under .then we calculate the matrix with and use the information criterion proposed by to determine the factor number in , which we use to denote . in the second step ,the upper bound of the factor number is set to .then . for models in section [ sec4 ] , even though there are observable common regressors and time invariant regressors in the equation , we treat them as part of the unknown factor structure when estimating the total number of factors .once the total number of factors are obtained , the dimension of is obtained by subtracting the dimension of and that of because and are observable in section [ sec4 ] .this approach works very well .other methods may also be considered .we consider an unified way to estimate the model in section [ sec2 ] and the model in section [ sec3 ] ( with zero restrictions ) .more specifically , for a given data set , we calculate and .if , we turn to the basic model ; if , we turn to the model with zero restrictions .  tables [ table1][table2 ] report the simulation results based on 1000 repetitions .bias and root mean square error ( rmse ) are computed to measure the performance of the estimators .the percentage that the factor number is correctly estimated by the above procedure is given in the third column of each table . for comparison, we also report the performance of the within - group ( wg ) estimators and bai s iterated principal components estimators ( pc ) .simulations for the models in section [ sec4 ] are provided in the supplement  ..0d3.1ccccccccd2.4ccc@ & & & & & + & & & & & + & & & & & & & & + & & & & & & & & + & & & & & & & & & & & & & & + 50&75&99.9&0.1562&0.1616&0.1550&0.1600&0.0174&0.0405&0.0171&0.0411&-0.0001&0.0020&0.0000&0.0034 + 100&75&100.0&0.1539&0.1568&0.1558&0.1587&0.0061&0.0228&0.0062&0.0224&0.0000&0.0011&0.0000&0.0010 + 150&75&100.0&0.1534&0.1556&0.1540&0.1561&0.0029&0.0168&0.0028&0.0146&0.0000&0.0007&0.0000&0.0007 + 50&125&100.0&0.1559&0.1605&0.1588&0.1636&0.0182&0.0389&0.0184&0.0409&0.0000&0.0017&0.0000&0.0016 + 100&125&100.0&0.1561&0.1586&0.1554&0.1579&0.0050&0.0167&0.0052&0.0167&0.0000&0.0009&0.0000&0.0008 + 150&125&100.0&0.1546&0.1565&0.1551&0.1570&0.0025&0.0108&0.0025&0.0106&0.0000&0.0006&0.0000&0.0005 +  .0d3.1cccccccccccc@ & & & & & + & & & & & + & & & & & & & & + & & & & & & & & + & & & & & & & & & & & & & & + 50&75&99.7&0.1098&0.1137&0.1095&0.1135&0.0097&0.0245&0.0099&0.0246&0.0000&0.0012&0.0000&0.0011 + 100&75&100.0&0.1088&0.1111&0.1092&0.1114&0.0038&0.0140&0.0038&0.0140&0.0000&0.0006&0.0000&0.0006 + 150&75&100.0&0.1086&0.1102&0.1083&0.1099&0.0011&0.0075&0.0015&0.0076&0.0000&0.0004&0.0000&0.0004 + 50&125&99.7&0.1089&0.1121&0.1097&0.1130&0.0076&0.0199&0.0077&0.0196&0.0000&0.0009&0.0000&0.0009 + 100&125&100.0&0.1088&0.1107&0.1087&0.1106&0.0029&0.0104&0.0026&0.0100&0.0000&0.0005&0.0000&0.0004 + 150&125&100.0&0.1086&0.1099&0.1076&0.1090&0.0011&0.0055&0.0010&0.0054&0.0000&0.0003&0.0000&0.0003 +  from the tables , we can see that the factor number can be correctly estimated with very high probability .it is also seen from the simulations that the wg estimators are inconsistent .the bias of the wg estimators shows no signs of decreasing as the sample size grows .the iterated pc estimators are consistent , but biased .as the sample size becomes large , the bias decreases noticeably .however , when the sample size is moderate , the bias of the iterated pc estimators is still pronounced . in comparison ,the ml estimators are consistent and unbiased .for all the sample sizes , the biases of the ml estimators are very small and negligible . in addition , the rmses of the ml estimators are always the smallest among the three estimators , illustrating the efficiency of the ml method .the same patten is observed for all of the four models considered .this paper considers estimating panel data models with interactive effects , in which explanatory variables are correlated with the unobserved effects .standard panel data methods ( such as the within - group estimator ) are not suitable for this type of models .we study the maximum likelihood method and provide a rigorous analysis for the asymptotic theory .while the analysis is difficult , the limiting distributions of the mle are simple and have intuitive interpretations .the maximum likelihood method can incorporate parameter restrictions to gain efficiency , a useful feature in view of the large number of parameters under large and large .we analyze the restrictions via the lagrange multiplier approach , which is capable of revealing what kinds of restrictions lead to efficiency gain .we allow the model to include time invariant regressors and common regressors .the coefficients of the time invariant regressors are time dependent , and the coefficients of the common regressors are cross - section dependent .this is a sensible way for modeling the effects of such variables in panel data context and fits naturally into the framework of interactive effects .the likelihood method is easy to implement and performs very well , as demonstrated by the monte carlo simulations .the authors thank two anonymous referees , an associate editor and an editor for constructive comments .

Abstract: this paper considers the maximum likelihood estimation of panel data models with interactive effects . motivated by applications in economics and other social sciences , a notable feature of the model is that the explanatory variables are correlated with the unobserved effects . the usual within - group estimator is inconsistent . existing methods for consistent estimation are either designed for panel data with short time periods or are less efficient . the maximum likelihood estimator has desirable properties and is easy to implement , as illustrated by the monte carlo simulations . this paper develops the inferential theory for the maximum likelihood estimator , including consistency , rate of convergence and the limiting distributions . we further extend the model to include time - invariant regressors and common regressors ( cross - section invariant ) . the regression coefficients for the time - invariant regressors are time - varying , and the coefficients for the common regressors are cross - sectionally varying .


the rapid development of video techniques in recent years has resulted in the massive use of video cameras in meteor observations . the number of new meteor networks has increased , and the efficiency of those already existing has improved . in three years ,the japanese meteor network database , containing around 30 low - light level camera observations , grew to 65000 orbits ( ; ) .the recently established system cams ( cameras for allsky meteor surveillance ) in the united states obtained 47000 orbits of meteors just in the first year of its operation ( ) . in europe , between 2000 and 2013 , the imo video meteor network collected over 1.2 million single - station meteors ( ) .also , in europe , the continuous monitoring of meteors and fireballs is conducted by the 25 stations the spanish meteor and fireball network ( spmn ; ) , which has been working now for 5 years . while till mid-2011 , nasa s all - sky fireball network , established in 2008 , with its 6 video cameras , detected 1796 multi - station meteors ( ) .another good example of well developed regional networks are the canadian automated meteor observatory ( camo ; ) and the croatian meteor network ( cmn ; ) .thanks to the broad international cooperation of video meteor observers from several european countries , a multi - national network edmond ( european videometeor observation network ) was created . as a result of its work , the first version of the edmond database , containing data from the years 2009 , 2010 , 2011 and half of 2012 ,was presented at the imc conference in la palma , spain in 2012 ( ) . in the last year , observers affiliated to the international meteor organization video meteor network ( imo vmn ) have started to share their data , whereas the data of edmond and imo vmn have been merged . nowadays , the data is collected from observers from a substantial part of europe and , due to this international cooperation , meteor activity is monitored over almost the entire europe . in effect , the database has accumulated 1639358 records of single - station meteors between 2000 and 2013 ( edmond 447266 and imo vmn 1192092 ) .the computation of meteor orbits is performed by the ufoorbit software ( ) . as the single - station video data are obtained and reduced using two different tools , the metrec ( ) and ufoanalyzer tools ( ) , the ufo data can be used without any changes .however , the data obtained by the metrec software has to be first converted into the ufo format using the program inf2mcsv written by sonotaco .the present database contains about 72 % of metrec data .as the conversion is not fully compatible , the computation of orbits is performed in two steps .first , preliminary orbits are computed using ufoorbit with basic parameter settings and sec ( which means that all combinations of single - station meteors within 5 second intervals are computed ) , and with additional settings : beginning and terminal heights have to be km , the empirically derived quality parameter , and the largest difference in velocity among considered stations in the orbit computation is / s .  after that , to reject the less precise orbits and false orbits , another filter of parameters is applied : the angle of observed trajectory has to be , the duration of the meteor , the convergence angle , the difference between the two poles of ground trajectory , and the difference in velocity between unified velocity and velocity from one of the stations . in comparison to the previous versions of the database ,the most important modification is the restriction of the difference in velocities for stations used in meteor orbit computation .the definition of all parameters is in the ufo manual .more details can be found in ( ) .  at present ,the 4th version of the edmond database containing 83369 video meteor orbits , has been released .most of them ( % ) are double - stations orbits .about 48800 orbits belong to the sporadic background and 34500 are shower meteors ( 59% and 41% , respectively ) .the edmond database was examined in several tests and compared with other meteor orbits databases .the examination allowed us to demonstrate the characteristic features of the edmond , which are particularly important for future analyses based on the data .the derivation of orbital elements , which define the shape of the orbit , is highly dependent on the uncertainty of the determination of the meteor velocity .one of the parameters used in the data reduction is dv12% ( the difference in the velocities , given in percentage ) .the geocentric velocity is the decisive parameter in the calculation of the orbit .thus , the dv12% parameter is an important indicator of its accuracy .the smaller the difference between velocities from different stations , the more accurate the orbit determination is .therefore , in the distribution of dv12% , a decrease in the number of orbits with increasing values of dv12% should be the most rapid .the comparison of the distribution of dv12% parameter of the edmond ( ) and sonotaco catalogue showed similar decrease ; with a slightly slower one in the edmond .the distributions of orbital parameters within several meteoroid streams from edmond were also analysed . in  ,the dispersions of orbital elements of the lyrids from edmond were studied .comparing them with the sonotaco video orbits , the consistency of both datasets was demonstrated . moreover ,if we compare both sets of video data ( figure [ fig1 ] ) , the dispersions in of the meteor orbits within individual streams obtained from the edmond data are about 1.3 times larger than the values from the sonotaco catalogue .figure [ fig1 ] shows the observed differences in the semi - major axes within the meteor streams compared with the orbital deviations in the streams determined from different datasets .the median semimajor axes of video meteor orbits in both the edmond and sonotaco data are systematically biased , probably as a consequence of the method used to determine the orbits . in comparison with the iau mdc photographic database ( ) , they are shifted towards the short - period side ; the velocities determined in the video data are slightly underestimated .  :thin line - the interval between two limiting values of , which includes 50 percent of all orbits .bold line - the interval between two limiting values of the uncertainty of the resulting values of median .dotted vertical line - the parabolic limit . dashed vertical lines - parent comets ( the figure is taken from the paper ( ).,width=226 ]  [ fig1 ]an important indicator of the quality of data is the relative number of hyperbolic orbits , because the probability of registering real hyperbolic orbits is very small ( ) .the apparent hyperbolicity of the orbits is , generally , caused by a high spread in velocity determination , shifting a part of the data through the parabolic limit .this , however , does not explicitly mean large measurement errors . of the 83369 meteor orbits collected in the edmond , 5.7% are determined as hyperbolic .this percentage is roughly comparable to that in the sonotaco database .initially , the proportion of hyperbolic meteors in the latter was 11.58% , but after the selection of quality orbits ( ) , this was reduced to 3.28% .of the 4712 hyperbolic meteors in the edmond , 43% are shower meteors .shower meteors which have heliocentric velocities with excesses over the parabolic limit offer proof of the false hyperbolicity of their orbits .the hyperbolic orbits in our data were analysed separately in the paper  .a comparison of both the edmond and the sonotaco catalogue , in terms of orbital parameters , showed an equivalence of the data .meteor showers in the edmond database were identified using the iau meteor data center database ( iau mdc ; ) . at the end of august 2013 ,the iau mdc list of showers contained 461 showers , 95 of them established and 366 in the working list .  in the first search , the radiant position - geocentric velocity method was used ( we hereafter call it _ radiant- _meteors were selected according to the peaks activity of meteor showers (  deg ) given in the iau mdc list , and fulfilling the conditions for radiant position (  deg ) and geocentric velocity ( % ) .a shower was considered only if at least 5 orbits had been identified . in this way , 267 meteor showers were identified , where 200 of them are meteor showers from the working list and 67 are the established showers .we focused on _ pro tempore _showers from the iau mdc working list . of 65_ pro tempore _showers in the list , 61 were identified using the _ radiant- _ method . to determine their fundamental parameters more precisely ,the first part of the welch method ( ) with southworth - hawkins criterion (  ) was used . according to the equation ( 4 ) in paper    where is a density at a point in orbital elements space , is the number of meteors of a _ pro tempore _shower found in the first step , is the value obtained for the -th meteor in the _ pro tempore _shower by comparing its orbit with orbits of each member of the identified shower , and is the threshold value that determines the dynamical similarity among meteor orbits .we searched for the core of each _ pro tempore _ shower identified in the first step of the analysis ( i.e. by _radiant- _ method ) .the procedure creates a group of meteors around each meteor orbit from the examined shower , which fulfil the condition of the limiting value of southworth and hawkins criterion . on the basis of the equation ( [ eqn1 ] ) ,the value of the density ( ) is determined for each group .the higher the density value , the more important the group in the examined shower is .however , the highest value of does not always mean it is the core of the stream because the initial set could be contaminated by a nearby separate small shower ; or because the mdc data are not yet accurate enough .we therefore compared all the available parameters of each _ pro tempore _ shower at the iau mdc with the mean values of the same parameters of each found group .we compared as well the mean orbits , radiant positions and geocentric velocities with newly meteor showers found in the sonotaco ( 2007 - 2009 ) and cams ( 2010 - 2011 ) databases ( ) .the mean values of the orbital elements and other parameters of each group were obtained as a weighted arithmetic mean , where the weight was determined by ( ) .the results obtained from this procedure are given in table [ tab1 ] and table [ tab2 ] .table [ tab1 ] contains 22 showers for which the identification was certain , i.e. the parameters of which agree well with those from the iau mdc list .another 18 showers , for which the comparison showed quite considerable differences in some parameters , and thus making their identification questionable , are shown in table [ tab2 ] .for instance , the difference in solar longitude or right ascension of radiant position reaches 10 , while the difference in eccentricities and perihelion distances , probably due to their high geocentric velocity , is greater than 0.1 . in the edmond database ( as of august 2013 )we could not identify 25 _ pro tempore _ meteor showers .the reason is either the number of orbits in the particular showers was insufficient ( less than 5 ) or the differences between some of the compared parameters were too big ( larger than in table [ tab2 ] ) .a few low inclined meteor showers ( # 449 , # 467 , # 473 , # 475 , # 476 , # 478 ) seem to be represented as separate branches , where one ( or both ) of the branch includes from 1 to 3 members .however , as the amount of meteors in the northern and/or southern branch is small ( ) , and there is no evident splitting , in those cases we considered such shower as one meteor shower .therefore , we added ( or subtracted ) 180 degrees to the angular elements ( , ) of the smaller branch , and then the weighted mean of and of the shower was calculated ..mean values of the parameters : solar longitude ( ) , radiant position , geocentric velocity ( ) , orbital elements and ( ) southworth - hawkins criterion of reliably identified _ pro tempore _showers from the iau mdc in the database edmond .number of meteors . in the second line of each shower , there are standard deviations . [ cols= " > , > , > , > , > , > , > , > , > , > , > , > " , ]   [ tab2 ]in the work , the european video meteor network database ( edmond ) is introduced .its 4th version contains 83369 video meteor orbits .the database was created thanks to the broad international cooperation of several european national networks and the international meteor organization video meteor network , with the aim of connecting observers within a wide area .this has made it possible to combine those observations which otherwise would have stayed as single - station data .we expect to use this expanding database , particularly to study minor streams .the first results are here presented . using the _ radiant- _ method we identified 267 meteor showers of the iau mdc , where 67 of them are established showers and 200 are showers from the working list .  making a more detailed analysis of _ pro tempore _showers from the iau mdc working list , we determined their orbital elements , radiant positions , geocentric velocities and solar longitudes .the results were divided into two groups , based on a comparison of the mean values with those available in the iau mdc .table [ tab1 ] contains 22 showers of the 65 _ pro tempore _showers in the working list of the iau mdc ( august 2013 ) , identification of which was clear and reliable .identification of 18 meteor showers listed in table [ tab2 ] is questionable , as some their parameters differ considerably from those at the iau mdc .this work showed that the edmond database is able to provide relevant data convenient for the confirmation of meteor showers from the working list of the iau mdc , which can improve their orbital and geophysical parameters .andrei , . ,egon , d. , 2010 , in proceedings of the international meteor conference , achtika , slovakia , 1821 september 2008 , 16 brown , p. , weryk , r. j. , kohut , s. , edwards , w. n. , krzeminski , z. , 2010 , wgn , journal of the international meteor organization , 38 , 25 cooke , w. j. , moser , d. e. , 2012 , in proceedings of the international meteor conference , sibiu , romania , 1518 september 2011 , 9 hajdukov , m. jr . , 2013 , in proceedings of the international meteor conference , pozna , poland , 2225 august 2013 , in press hajdukov , m. jr . ,korno , l. , tth , j. 2014a , meteoritics planet ., 49 , 63 hajdukov , m. jr . ,korno , l. , tth , j. 2014b , in this proceedings , p ? ?jenniskens , p. , gural , p. s. , dynneson , l. , grigsby ,b. j. , newman , k. e. , borden , m. , koop , m. , holman , d. , 2011 , icarus , 216 , 40 jopek , t. j. , kauchov , z. , 2014 , in this proceedings , p ? ?korno , l. , koukal , j. , piffl , r. , tth , j. , 2013a , in proceedings of the international meteor conference , canary islands , spain , 2023 september 2012 , 21 korno , l. , koukal , j. , piffl , r. , tth , j. , 2013b , in proceedings of the international meteor conference , pozna , poland , 2225 august 2013 , in press lindblad , b. a. , nesluan , l. , poruban , v. , svore j.,2005 earth , moon , planets 93 , 249 molau , s. , 1999 , in proceedings of the international conference meteoroids 1998 , tatranska lomnica , slovakia , 1721 august 1998 , 131 molau , s. , 2014 , in this proceedings , p ? ?pujols , p. , ambls , m. , trigo - rodrguez , j. m. , dergham , j. , madiedo , j. m. , montany , j. , van der velde , o. , 2013 , lunar and planetary institute science conference abstracts , 44 , 2054 rudawska , r. , jenniskens , p. , 2014, in this proceedings , p ?sonotaco , 2009 , wgn , journal of the international meteor organization , 37 , 55 sonotaco , molau , s. , koschny , d. , 2010 , european planetary science congress 2010 , 798 southworth r. b. , hawkins g. s. , 1963 , smithsonian contrib .astrophys . 7 , 261 tth , j. , korno , l. , vere , p. , ilha , j. , kalmanok , d. , zigo , p. , vilgi , j. , 2011 , publ .japan 63 , 331 vere , p. and tth , j. , 2010 , wgn , journal of the international meteor organization , 38 , 54 welch , p. g. , 2001 , mon . not .r. astron .

Abstract: the european video meteor network database ( edmond ) is a database of video meteor orbits resulting from cooperation and data sharing among several european national networks and the international meteor organization video meteor network , imo vmn , ( ) . at present , the 4th version of the edmond database , which contains 83369 video meteor orbits , has been released .   the first results of the database analysis , in which we studied minor streams , are presented . using the radiant - geocentric velocity method we identified 267 meteor showers , among them 67 established showers and 200 from the working list of the iau mdc . making a more detailed examination , we clearly identified 22 showers of 65 _ pro tempore _ showers of the working list of the iau mdc ( updated in august 2013 ) . the identification of 18 meteor showers was questionable , while 25 showers were not found . for all the identified temporary meteor showers , we list the weighted mean orbital elements , the radiant position and the geocentric velocity .


the tcp protocol is one of the main data transmission protocols of the internet .it has been designed to adapt to the various traffic conditions of the actual network . for a connection ,the maximum number of packets that can be sent at each round is given by a variable , called the _ congestion window size_. if all the packets are successfully transmitted , then is increased by , otherwise it is multiplied by ( detection of a congestion ) . as shown in , a correct scaling of this process leads to a continuous time markov process , called general tcp window size process .this process has as state space and its infinitesimal generator is given , for any smooth function , by for some probability measure supported in .this window size increases linearly ( this is the part of ) until the reception of a congestion signal which forces the reduction of the window size by a multiplicative factor of law or equal to in the simplest case ( this is the jump part of ) .the sample paths of are deterministic between jumps , the jumps are multiplicative , and the whole randomness of the dynamics relies on the jump mechanism . of course, the randomness of may also come from a random initial value .the process appears as an additive increase multiplicative decrease process ( aimd ) , but also as a very special piecewise deterministic markov process ( pdmp ) initially introduced in . in this direction, gives a generalization of the scaling procedure to interpret various pdmps as the limit of discrete time markov chains . in the real world ( internet ) , the aimd mechanism allows a good compromise between the minimization of network congestion time and the maximization of mean throughput .our aim in this paper is to get quantitative estimates for the convergence to equilibrium of this general tcp window size process.this process is ergodic and admits a unique invariant law , as can be checked using a suitable lyapunov function ( for instance , see e.g. for the meyn - tweedie - foster - lyapunov technique ) .nevertheless , this process is irreversible since time reversed sample paths are not sample paths and it has infinite support .this makes meyn - tweedie - foster - lyapunov techniques inefficient for the derivation of quantitative exponential ergodicity .the _ embedded chain _ of the process is the sequence of its positions just after a jump .it is an homogeneous discrete time markov chain with state space . as already observed in , it is also the square root of a first order auto - regressive process with non - gaussian innovations and random coefficients .we obtain the following results concerning .we show first that it admits a unique invariant probability measure , and that it converges in law to given any ( random ) initial value .more precisely , using a coupling technique on trajectories , we prove an ergodic theorem of geometric convergence to equilibrium with respect to any wasserstein distance. then we provide non asymptotic concentration bounds , thanks to gross s logarithmic sobolev inequalities .similarly , the continuous time process admits a unique invariant probability measure , and converges in law to , for any ( random ) initial value .the reader may find explicit series for the moments of and in . nevertheless , quantitative convergence to equlibium have not yet been obtained .we will adress this question for a slight generalization of the tcp process given by its infinitesimal genrerator : where .we obtain a good answer if . in this casewe first show the existence of a coupling with exponential decay .we use this result to prove an exponential ergodic theorem in term of wasserstein distance .eventually , we provide a uniform bound over the starting law that implies strong ergodicity .this kind of uniform estimates , though classical for processes on a compact set , is rather unusual for real valued processes . nevertheless ,if , we are not able to derive exponential bounds .the remainder of the paper is organized as follows . in the next preliminary section ,we introduce some notations and give the statements of the main results . in section [ se : embedded ] , we focus on the embedded chain and establish its convergence to equilibrium .the last section is devoted to the study of the continuous time process and its generalization and contains the proof of the results announced in section [ se : results ] .let us first explain how the trajectories of the process may be constructed .the _ jump rate _ ( or _ jump intensity _ ) of is given by for every .if then the process will experience its first jump at a random time solution of where is an exponential random variable of unit mean .since the trajectories of are piecewise deterministic with slope 1 , this is nothing else but which leads to .then , the sample paths of the process generated by may be constructed recursively as follows .let be its non - negative random initial position , be a sequence of i.i.d .exponential random variables of unit mean , and be a sequence of i.i.d .random variables of law .assume that , and are independent .we define by induction the jump times and the positions just after the jumps as if we set , then for every and , we have and in particular , . for every ,one can also write the series representation the sequence is the _ embedded chain _ of . according to, this discrete time markov chain with state space satisfies the recursion thus , the embedded chain is the square root of a first order auto - regressive process with non - gaussian innovations and random coefficients , as already observed in .the embedded chain is homogeneous , and its transition kernel is given , for any and every bounded measurable , by the formula }}}\ ] ] where is an exponential random variable of unit mean and is a random variable of law independent of .we show in section [ se : embedded ] that the embedded markov chain admits a unique invariant probability measure , and converges in law to given any ( random ) initial value .similarly , the continuous time process admits a unique invariant probability measure , and converges in law to , for any ( random ) initial value .we recall that explicit series for the moments of and can be found in .  despite the apparent simplicity of the dynamics , the quantitative study of the long time behavior of is not easy , mainly because the jump rate depends on the position of the process .our strategy is to couple two trajectories starting at two different points in such a way that they get closer and closer .it seems difficult to stick the two trajectories in order to get total variation estimates since the sample paths are parallel between jump times .thus , we provide quantitative bounds in terms of the wasserstein coupling distance .recall that for every , the wasserstein distance between two laws and on with finite moment is defined by where the infimum runs over all coupling of and .in other words , runs over the convex set of laws on with marginals and , see e.g. .it is well known that for any , the convergence in wasserstein distance is equivalent to weak convergence together with convergence of all moments up to order .the jump part of ensures that the process will remain essentially in a compact set .the jumps act in a way like a confining potential . on the other hand ,the jump rate is small when the process is close to the origin .this prevents the decay of the wasserstein distance to be exponential for small times .  in section [ se : embedded ]we first establish the following geometric convergence to equilibrium of the embedded markov chain for any wasserstein distance .[ th : convk ] let and be two processes generated by .assume that and have finite moment for some real .let and be the embedded chains of and .then , for any , with a random variable , in particular , if is the invariant law of then   we also establish in section [ se : embedded ] non asymptotic concentration bounds in the ergodic theorem by using gross logarithmic sobolev inequalities :  [ thm : gdevemb ] let be the embedded chain associated to and starting from .assume that is the dirac mass at point .then for any and any 1-lipschitz function ,   the convergence to equilibrium of the continuous time process with generator is addressed in section [ se : coupling ] .the idea is to exhibit a _ coupling _ , _ i.e. _ a markov process on for which the marginal components are generated by , with prescribed initial laws .the infinitesimal generator of this coupling is defined for every smooth by if and if , where .this coupling is the only one such that the lower component never jump alone .let us give the pathwise interpretation of this coupling .all the heuristic statements below are made more precise hereafter .the positions of both `` components '' increase linearly with slope 1 .the jump rate is an increasing function of the position .thus , `` the higher a component is , the sooner it will jump '' .the dynamics of the couple of components is as follows :  1 .after an `` appropriate '' time which depends only on the initial position of the upper component , this one jumps .2 .  simultaneously, the other one `` tosses an appropriate coin '' whose probability of success depends on the positions on the two components to decide whether or not it jumps too .3 .  in the case of joint jumps ,both components use the same multiplicative factor .then , we repeat these three first steps again and again   this coupling provides the following quantitative exponential upper bounds .[ th : wun - gen ] assume that . for any processes and generated by with finite first moment at initial time , and for any , we have where . in particular ,when follows the invariant law of , we get for every   the following theorem , proved in section [ se : coupling ] , shows that the convergence to equilibrium is in fact uniform over the starting laws , as it could be for a process living in a compact set .  [ th : strong ] assume that . for two processes and generated by with arbitrary initial laws and and for every and such that , one has   theorem [ th : strong ] provides in particular a uniform bound in if and .this kind of uniform estimates are classical for processes on a compact set but rather unusual for real valued ones .[ th : realtcp ] assume that and that with . then the process driven by the infinitesimal generator defined in satisfies for any .in particular , for any and , we have    the inequality should provide a better bound than . as pointed out in lemma [ le : youpee ] , one can actually expect an exponential rate , but this remains an open problem .one may also ask for a version involving for any or even the total variation distance .  beyond the tcp window size dynamics , one may ask about the speed of convergence of ergodic pdmps , for which necessary and sufficient ergodicity criteria are already known , see e.g. .one may also study the long time behavior of interacting processes associated to or , for instance mac kean - vlasov mean field interactions as in .it is shown in ( * ? ? ? * proposition 8) , by laplace transform inversion , that if is a dirac mass at point , the invariant measure of the embedded chain has lebesgue density it is unimodal , of order when , and all its derivatives vanish at .if is not a dirac mass , the invariant measure of the embedded markov chain is no longer explicit .nevertheless , the recursion formula provides the following result , see , which establish the existence of an invariant measure with sub - gaussian tails .[ th : cvemb ] given any , the embedded markov chain associated to the dynamics converges in distribution to the law of the random variable which is a.s .finite , where and are independent sequences of i.i.d .random variables following respectively the exponential law of unit mean and the law which appear in .in particular , is the unique invariant law of and which is finite if and infinite if , where .let us now turn to our quantitative estimate for the convergence to equilibrium for the embedded chain .it is sufficient to provide a good coupling .let and be two non - negative real numbers , and let and be two independent sequences of i.i.d .random variables with respective laws the exponential law of unit mean and the law which appears in .let and be the discrete time markov chains on defined by by the analogue of for , the law of ( respectively ) is the law of the embedded chain of a process generated by and starting from ( respectively ) .now , for any , since is a 1-lipschitz function on for any , we get a straightforward recurrence leads to this gives the desired inequality when the initial laws are dirac masses .the general case follows by integrating this inequality with respect to couplings of the initial laws .let us now investigate some properties of the kernel defined by that will be used to provide concentration bounds for the ergodic theorem .the key point is that and satisfy a gross ( or logarithmic sobolev ) inequality .  a law on satisfies a gross ( or logarithmic sobolev ) inequality with constant when for any smooth compactly supported , we denote by  ] .now , for every and , also , since is negative , we obtain , by using jensen s inequality , as a consequence , where is the solution of the riccati differential equation denoting , one gets , for , and therefore on the other hand , we have also , and thus for every , consider now two processes and generated by with arbitrary initial laws . for any , andtherefore the upper bound above gives together with theorem [ th : wun - gen ] , this gives the following uniform estimate , for every ,     we start the proof as in theorem [ th : wun - gen ] : the first bound is better .nevertheless , if is the set , one has to notice that the process can not exit .then , thanks to markov property , we get the following bound : jensen s inequality ensures that and thus , for any ,   figure [ fi : comp ] suggests that the convergence rate given by theorem [ th : realtcp ] is far from being satisfactory .the non - optimality of the coupling is clear .however , even with such a coupling , one could expect an explicit exponential upper bound .let us denote where is defined in theorem [ th : realtcp ] .we think that is in fact of the order of ( instead of ) .indeed , with a rate of order a nonsimultaneous jump occurs at time and then is again of order one . in the following lemma , we introduce a simple markov chain which captures the essential feature the dynamics of ( division by 2 with probability ) and we show that the expected position at time goes to zeros exponentially fast as goes to infinity .additionally the recursive equation plays the role of .   and .this picture presents the three following functions of time : where is driven by with ( blue curve ) , where is driven by with ( red curve ) , the upper bound of theorem [ th : realtcp ] ( green curve ) .the first and second curves have been obtained by monte - carlo simulations . ][ le : youpee ] consider the homogeneous irreducible markov chain with state space such that , for any and , on the event  denote by the quantity . then , for any , and there exists a constant such that for any ,   the markov chain is transient ( and converges to 0 ) since since for any , we get . in particular , and is decreasing .similarly , as a consequence , for any , and follows since for any ,   * acknowledgments .* the authors are grateful to the anonymous referees for their quick reports and fine comments which helped to improve the paper .dc and fm would like to thank philippe robert and his team for kind hospitality during their visits to inria rocquencourt in december 2007 and june 2009 .    c. an , s. blachre , d. chafa , p. fougres , i. gentil , f. malrieu , c. roberto , and g. scheffer , _ sur les ingalits de sobolev logarithmiques _ , panoramas et synthses [ panoramas and syntheses ] , vol .10 , socit mathmatique de france , paris , 2000 , with a preface by d. bakry and m. ledoux .s. t. rachev , _ probability metrics and the stability of stochastic models _, wiley series in probability and mathematical statistics : applied probability and statistics , john wiley & sons ltd . , chichester , 1991 .

Abstract: the tcp window size process appears in the modeling of the famous transmission control protocol used for data transmission over the internet . this continuous time markov process takes its values in , is ergodic and irreversible . it belongs to the additive increase multiplicative decrease class of processes . the sample paths are piecewise linear deterministic and the whole randomness of the dynamics comes from the jump mechanism . several aspects of this process have already been investigated in the literature . in the present paper , we mainly get quantitative estimates for the convergence to equilibrium , in terms of the wasserstein coupling distance , for the process and also for its embedded chain .  * keywords . * network protocols ; queueing theory ; additive increase multiplicative decrease processes ( aimd ) ; piecewise deterministic markov processes ( pdmp ) ; exponential ergodicity ; coupling .  * ams - msc . * 68m12 ; 60k30 ; 60k25 ; 90b18