In my role, I manage complex strategy projects for clients within the Life Sciences, Healthcare, and Consumer Health industries. Projects range from launch strategy (i.e. which customers/physicians should we target), growth strategy (which new countries/regions should we enter), and transaction strategy (which products/companies should we acquire). In my role, I manage a team of Consultants and Analysts to create an independent point of view to help companies grow and deliver innovative products to market to improve the lives of patients and consumers worldwide.
During my undergrad experience at Wake Forest, I was a science major through and through. I was pre-med, an anatomy T.A. for Dr. Marsh, and worked two research-based summer internships in Winston Salem for Dr. King in Chemistry and Dr. Miller in HES. I love science and medicine, but realized I didn't want to practice or pursue a PhD. I realized I wanted to explore a career that blended business and science and allowed me to focus on problem-solving and analytics, and so I pursued opportunities in consulting. I took a job out of Wake Forest as an Analyst at a small life sciences consulting group in Boston, and have moved around the industry since then.
After an academic career in science, the transition to business was challenging. I had never analyzed P&Ls, built forecasting models, or analyzed company growth strategies. But Wake's small classroom experience and focus on teamwork led me to learn from my peers and managers to pick things up quickly.
Keep asking questions! When you come out of undergrad, you might think you know what you want to do, but you definitely don't know all of the potential options and outlets that exist. Continuing to develop your professional network and really understand all of the potential options is necessary. Also – it's incredibly important to focus on your health. Making time for exercising and eating right can be tough when you're working long hours at your first job, but it helps you to think more clearly and get into a disciplined routine.
When moving to Boston, my initial personal and professional network was very alumni driven! One colleague at my first company, Rick Sullivan ('09) graduated a year ahead of me and helped introduce me to a variety of folks in the Boston area. I also attended a variety of Wake Forest alumni networking events and happy hours to meet more people in the area, which is actually where I met my now wife (Alyson Cooper, '08).
Work hard and ask for help! Coming out of undergrad, I had some challenges asking for help at first, because I didn't want to be perceived as not knowing how to accomplish basic tasks. But it's a lot easier to ask a question early, than it is to try to do something new for 5 hours only to ask for help when you're stuck later on. It's also important to always be thinking about how what you're doing now is helping you achieve your goals – keeping focused on where you want to go and who you want to be will help you stay focused on your development.
I want to help build innovative, global companies in the Life Sciences and Consumer Health industries. There are such interesting technologies being developed across genome editing, machine learning, and data analytics that will forever change the way we think about healthcare and wellness. I want to work on the front-lines, meeting with scientists, entrepreneurs, and executives that are helping to bring these ideas to market, and working in Life Sciences strategy consulting with EY-Parthenon lets me do that every day.
Story published in January 2018. For current updates about Joe, visit his LinkedIn page.


Image © Mark Driedger at ATA Architects Inc.
As a child in the early 1980s, this article's author had a clubhouse he called the 'Millennium Falcon'—standing 2 m (6 ft) off the ground, it was a simple A-frame held in place by four old telephone posts, complete with a fireman's pole and pully system for escape when under attack by the Imperial Starfleet. However, it was also an experiment in solar energy storage.
In its former location, the 'Falcon' had been filled with stones from the surrounding fields, and its south façade was fully glazed, allowing the sun to access the stones. The theory was the sun's heat could be harvested from the stones long after it had dropped below the horizon.
The author's parents' plan was to build a house that was extremely energy efficient, and this was the prototype to provide the heating energy. In the early '80s, natural gas pipelines had yet to reach the farm, and heating with electricity or fuel oil were the only options. With rising energy costs, it became important to save money, and using glazing to access passive solar energy was heavily featured in the media.
With the input of an American architect who had experience with passive solar design, traditional south-facing windows were chosen for the house, and a well-sealed staggered-stud enclosure was employed rather than the more exotic thermal mass method. Unfortunately, the chosen system simply was not balanced to provide additional energy to the house—although some sunlight was harnessed, the reality was the low thermal performance of this particular glazing caused much more energy to be lost than gained.
The glazing was also not large enough to collect sufficient heat, but increasing its size would also increase conduction losses during the cold winter nights, as glass is inherently a terrible insulator. Increasing the thermal properties of the windows reduced these losses, but it also reduced the sunlight that was allowed through. The system may have worked in a more temperate climate, but in Kingsville, Ont., the structure was destined to become the Millennium Falcon.
Figure 1: Residential energy usage in Canada, as surveyed by National Resources.
Some 30 years later, as an architect, this author's interest in the problem remains. Why are architects not doing more to harness the sun? Yes, photovoltaics (PVs) have become more efficient, and large-scale battery installations are being considered, but battery storage still has a long way to go. Given a large portion of the energy used in buildings by Canadians is for heating (Figure 1), it would be very difficult and expensive to build a battery-storage system that would power a house through the winter. (This data was obtained from National Resources Canada (NRCan). For more information, click here.) It does not help that the cost of natural gas is still affordable at the moment, and it is cheaper for Canadians to run pipelines from Alberta than convert furnaces to electric heat and build solar panels with huge battery banks. Technology has changed rapidly over the last 30 years, but the equation still comes down to how energy from the sun can be efficiently stored. Research seems to be focused on the complex transformation of solar energy using batteries, rather than the simple transfer of energy into a thermal mass.


text:
Andrew Yang: Businessman, Meme, and Presidential Candidate
By Lucy Ritzmann / May 28, 2019, 4:08 p.m.
The Democrats are facing many challenges in 2020, but finding people to run is not one of them. As of April 22, twenty Democrats have tossed their hat in the ring. And the race is already contentious: Joe Biden has been smelling women's hair, Amy Klobuchar might not be a good boss and "Mayor Pete" is the first unexpected hit. Amongst this sea of candidates is Andrew Yang, a forty-four-year-old former tech executive, who joined the fray on a platform to protect Americans from the effects of automation.
Yang has not been shy about putting forward bold policies, but his candidacy in itself is distinct. Yang's parents immigrated to America from Taiwan in the 1960s. They met at graduate school and each went on to have successful careers in technology. Yang attended Brown University and then went on to Columbia Law School. On his campaign website, Yang states that he "grew up believing in the American Dream—it's why my parents came here." He and his family are uniquely optimistic in this experience of prosperity; only 23 percent of Baby Boomers say that the American Dream was alive for them and only 21 percent of all respondents say that the American Dream will be alive for the next generation. Yang, however, still has faith, and if he were to secure the nomination, he would be the first ever Asian American person to be nominated by a major party.
As a candidate, Yang also has a unique professional background. He recently stepped down as CEO of Venture for America, a company he founded, to run for President. Venture for America, a philanthropic organization, "places top college graduates in startups in low-cost US cities for two years, generating job growth and training the next generation of entrepreneurs." He has also published two books, the second of which, The War on Normal People, lays out his plan to utilize Universal Basic Income to help Americans, which is essentially the core of his platform. Yang is very much a twenty-first century candidate. He is extremely tech-literate and proposes policies for an automated future that he believes is imminent. Whether or not that makes him seem attractive or ludicrous to voters remains to be seen.
Yang has also shared an element of his personal life that makes him unique: one of his sons was diagnosed with autism. Yang has been outspoken about the need to provide better resources for people with autism and their families. On his website, he writes, "As a country, we should provide ample resources to parents to be able to intervene to support the development of children with autism or who are exceptional in other ways. Many of these children have something unique to offer." This is a far cry from President Donald Trump's campaign: in 2016, Rosie O'Donnell re-posted a video claiming that Barron Trump was autistic. First Lady Melania Trump threatened a lawsuit if the video was not removed and insisted that Barron was not autistic. Regardless of whether or not it was true, Melania Trump's reaction is not a kind one to the autism community. Voters may be excited to have a candidate like Yang who is addressing the concerns of Americans with special needs.
Yang as a candidate stands out from the crowd but his proposed legislation is what really makes him a distinct candidate. He is centering his campaign on a three-pronged policy strategy and he has no time for moderacy or gradualism: each of which would be a significant departure from the status quo. His first policy is arguably his least radical: he supports Medicare for All. Yang considered the Affordable Care Act a step in the right direction but draws on his own personal experience as a CEO to explain why this initiative needs to go further. On his campaign website, he cites a passage from his own book: "Technology that should decrease costs has been kept at the door because for most actors in the system, the goal is to increase revenue and profitability. The more services, tests, appointments, procedures and expensive gadgets you use, the better. The system rewards activity and output over health improvements and outcomes." Yang carves out a compelling space for himself here as a tech CEO and trained economist who wants to use his expertise to fight for Americans who cannot afford healthcare. This may not be enough to convince those Americans to support Medicare for All; however, a recent poll shows that while 56 percent of Americans initially support Medicare for All, that number drops to 37 percent when they are told taxes would be raised, and decreases even further, to 26 percent, when told wait times at doctors' offices would increase. Yang, armed with a BA in economics from Brown and a new model based of the highly effective Cleveland Clinic, has an uphill battle here but one that he is equipped to fight.
Yang's second policy initiative takes a step into the realm of economic philosophy. He is proposing a new type of capitalist theory and has coined the name "Human-Centered Capitalism." Yang argues that capitalism is still the correct form for the American market—a point that puts him squarely at odds with fellow candidate Bernie Sanders who is polling highly across the board—but that the current form of capitalism prioritizes the dollar, not the human. Yang, not unlike Smith or Marx, even provides three core tenets of Human-Centered Capitalism: "1) Humans are more important than money; 2) The unit of of a Human Capitalism economy is each person, not each dollar; 3) Markets exist to solve our common goals and values."
This policy has noticeably less data and even less text on Yang's campaign website. A quick Google search reveals that Yang really is the only person talking about this. As of now, there is no literature, data or theory that supports him—but Yang is certainly one to pick up a pen and write his own book in lieu of corroborating sources. He will have to fight against two centuries of history and a plethora of economic theorists who argue that capitalism, by its very nature, is about money. If anything, it seems as though Yang is proposing something that is certainly not capitalism but that, as a businessman candidate, he is afraid of losing voters who will only vote for a capitalist. The general impression of this policy is that it is purely rhetorical: it will sound good in a speech that Yang is a capitalist who cares, but it seems unlikely that Yang will be changing up the very premise of American capitalism by 2020.
Yang's third policy initiative is the keystone of his campaign: he is proposing a Universal Basic Income, or UBI. A UBI, as Yang envisions it, is that every single American citizen over the age of eighteen will receive $1,000 a month, every month. Yang's explanation of the UBI paints a grim picture for the future of the United States: "By 2015, automation had already destroyed four million manufacturing jobs, and the smartest people in the world now predict that a third of all working Americans will lose their job to automation in the next twelve years. Our current policies are not equipped to handle this crisis. Even our most forward-thinking politicians are unprepared." And Yang has a plan to get ahold of the billions of dollars that it would take for implement a UBI. He suggests that the United States pull the money from four places: 1) shifting money away from current welfare programs; 2) implementing a Value-Added Tax of 10 percent, which is a consumption tax where tax is added at each stage of production where the value of the product increases; 3) using the new revenue generated by UBI; and 4) eventually cutting funding from prisons, homeless shelters and health care centers because they will be less needed when everyone is able to financially care for themselves. Yang is clearly sold on the merits of UBI but understands that the American public may need convincing. On his campaign site, he lists the 461 research papers that have been written about UBI; the projects that he highlights include the "Mincome" experiment, an experiment completed forty years ago in which people of a rural Canadian town were all given a basic income, which resulted in less hospitalization; and the BIG pilot program, which gave a basic income to a Namibian village resulting in improved health and reduced crime. Yang doesn't simply want to ameliorate Americans' financial situations—he wants to save our minds. One of his favorite studies to cite is a Princeton survey that found that persistent financial insecurity lowers a person's IQ by thirteen points. If these sources aren't enough to convince voters, on his campaign site, Yang has anticipated twenty-plus questions that a voter may ask, ranging from "Isn't this Communism?" to "I don't see robots. Isn't this early?" and provided answers.
While Yang and his staff do a thorough job at explaining a future with UBI, there is much to be understood about the past of UBI, which is extensive and spans several countries and centuries. The idea first originated in England as the product of humanists Thomas More and Johannes Vives's collective musings when they realized that society would not provide a basic standard of living for everyone and that some people would be living in desperate poverty. Vives even developed a plan for a basic income that was implemented in the Flemish municipality of Ypres, but most of this work goes unremembered by history. Other advocates of UBI include Thomas Paine, Napoleon and Martin Luther King Jr., so Yang is in good company. The last politician to seriously take up the cause in the United States was Huey Long, a populist senator from Louisiana who rose to prominence during the Great Depression; he proposed a basic income of $2,000. Today, that amount would be about $37,000. While UBI has a significant amount of famous supporters, it has yet to ever come close to being implemented. Yang believes that at this critical juncture of technology advancement, when thirty-six million American jobs are at risk of automation, people may begin to seriously consider UBI as a viable and even critical policy.
This is because, while UBI certainly has its merits, it doesn't seem all that feasible. Simply from a logic perspective, any plan that requires its own success to access its benefits seems problematic: a significant part of the funding for the UBI comes from the revenue that will be supposedly generated by UBI, as well as from the programs that the United States will be able to cut because of UBI. If UBI isn't a fairly immediate success, it will not be able to continue. And while Yang is able to provide sources that point to the fact that UBI will be successful, other ongoing studies involving UBI have yet to find any promising results. In Finland, a research agency began paying workers and non-workers between 550–700 euros a month. The government, however, quickly grew critical of the program and cut funding to the people who were employed, defeating the purpose of the experiment and demonstrating that it may be difficult to remain committed to the program. Moreover, the National Bureau of Economic Research just published a paper stating that current plans for UBI are still too vague and underdeveloped to actually achieve any of the proposed benefits and that we are still a long way away from developing a successful UBI program. However, it is certainly too soon to draw any definitive conclusions from this research, and most economic scholars seem to think that while finding a working UBI would be a challenge, it would still be worth the endeavor.
Yang's campaign, with UBI as its keystone, has yet to garner any significant attention in the 2020 Presidential Race. He is a businessman-turned-politician, which is not something Democrats are feeling great about since Trump, another businessman-turned-politician, took the White House; Yang has had to take a defensive stance in attempting to separate his story from Trump's. Moreover, while Yang claims to not ascribe to the traditional capitalist/socialist dichotomy in the United States, a lot of people still see him as a socialist; to combat this, he has re-named his UBI plan as "the freedom dividend." Strangely enough, Richard Spencer, a prominent white nationalist, has tweeted his support for Yang; Yang has firmly separated himself from his fringe white-nationalist support, but Democrats are much less likely to find his campaign appealing if they know that white supremacists share their interests. The support from Spencer could certainly turn into a larger issue if Yang garners more attention, but currently, Yang's biggest issue is the latest polls that show that he does not have much popularity or name recognition. His greatest supporters are millennials and Asian and Hispanic voters; there is a path to victory for Yang but currently, he is quite the longshot.
Yang is an example of a candidate who may not have the poll numbers but who does have the potential. Crucially, he knows how to win over the Internet: he has amassed a significant following on Facebook and on Reddit, prompting the New York Times to dub Yang a "meme-worthy candidate." As America learned in 2016 with the Sanders campaign, if a candidate can win over the Internet, he or she will enjoy near limitless free promotion and enthusiasm from millennials, a notoriously un-interested group of voters. In addition to potential, Yang has also demonstrated his commitment to this race. He is headed to Iowa, where he will personally implement the freedom dividend for one family, giving them $1,000 a month for the next month to see how it may change their lives. He recently did the same for a family in New Hampshire. In Iowa, where Yang has been polling at a dismal 1 percent, becoming a Yang supporter is, if nothing else, essentially buying a lottery ticket for $12,000.
Yang is approaching the Presidential race with creativity, ingenuity and no intention to stay within the lines. That alone merits America's attention; while 2016 was filled with cries to "Drain the Swamp" that led to very little actual change in America's political landscape, Yang has put time and effort into thoughtful and effective ways to shake up the system. While it isn't likely that President Yang will step into the Oval Office in 2020, he is certainly a candidate to watch and the race for the Democratic Primary will be more interesting with Yang as a part of it.
The featured image is courtesy of Flickr user Collision Conf and licensed under Creative Commons 2.0. The original can be found here.
Lucy Ritzmann
Lucy Ritzmann is a first year prospective Political Science major interested in political media and law. Last summer, she interned at the Manhattan Borough President's Office. For winter quarter, she is a Fellow's Ambassador at the IOP. In her free time, she enjoys being with her friends and zumba.

redpajama_set_name: RedPajamaCommonCrawl


This is a field study/independent studies course for students in a study abroad program at a recognized Camosun College exchange/partner institution overseas for one semester or less. Curriculum and evaluation will be determined in consultation between the host institution, the appropriate Camosun College program/department, International Education, and the Registrar.
$177.75 plus additional fees. Effective until August 31, 2019.


As our readers know we've made a business out of taking a very functional thing and turning it into an art object (In fact, that's the basis of our slogan!) – so it's always neat to see what other everyday objects people can take and turn into something beautiful and well-crafted. Take staircases: We've compiled ten examples below where someone has been able to take the staircase — an everyday construction "designed to bridge a large vertical distance" — to another level.
Tell us in the comments section below if you have any favourites – or if we've missed any other examples that belong among these ten!


text:
\section{Introduction}

In an important paper, \citet{Lancaster02} studies, from a Bayesian
perspective, estimation of the structural parameters of a dynamic panel data
model with fixed effects and initial observations equal to zero. His method
involves reparameterizing the model so that the information matrix is block
diagonal, with the common parameters in one block and the incidental
parameters in the other. His estimator is then defined as one of the local
maxima of the integrated likelihood function, integrating with respect to
the Lebesgue measure on $\mathbb{R}^{N}$. However, Lancaster leaves
unanswered the question of how to uniquely determine the consistent root of
his proposed methodology. Some authors, including \citet{DhaeneJochmans16}
and \citet{Kruiniger14}, have proposed different ways to find the consistent
estimator in Lancaster's approach.

In this paper, we explain the shortcoming of Lancaster's estimator: it
ignores available information in the model. In particular, Lancaster's
posterior distribution uses only part of the maximal invariant statistic's
log-likelihood function; when the full likelihood function is used in the
estimation, a unique, consistent, and asymptotically normal efficient
estimator is obtained; see \citet{Moreira09b}. The estimator obtained using
the full likelihood is asymptotically more efficient than Lancaster's
estimator. Therefore, trying to correct the nonuniqueness issue of
Lancaster's estimator is unnecessary and leads to inefficient estimators.

\citet{Lancaster02} and \citet{Moreira09b} study consistent estimation of
dynamic panel models under the same set of assumptions and with initial
observations equal to zero. The zero initial condition is trivially
satisfied when the initial observations are subtracted from the
autoregressive variables. We then show that efficiency is improved by
conditioning on the initial observations instead of differencing out the
data. The conditional argument is essentially a fixed-effects approach in
which we make no further premises on unobserved data. This is in contrast
with commonly-used estimators in the literature which make further
assumptions; see \citet{Bai13,Bai13b} on correlated random effects or
\citet{BlundellBond98} on stationarity.

A potential advantage of conditioning on the first observation is
robustness. As \citet{BlundellSmith91} point out, asymptotic arguments are
usually based on the average temporal effect, calculated on the individual
dimension. Therefore, the importance of unobserved data does not disappear
asymptotically for relatively short panels. For example, take data on firms
or individual earnings. \citet{CabralMata03} show that the distribution of
firms is very skewed, with most of the mass being small firms, while
\citet{Evans87, Evans87b} and \citet{Hall87} show that Gilbrat's Law
(independent firm size and growth) is rejected for small firms. As only a
handful of large firms provide the bulk of the data, we should not expect
estimators based on assumptions about unobserved data to be approximately
normal. As the firms' entry states do not disappear asymptotically for
relatively short panels, conditioning on the first observation would be
preferable to assuming known processes for unobserved data.

The remainder of this paper is organized as follows. Section 2 introduces a
simple dynamic panel data model without covariates and a zero initial
condition. This section determines the maximal invariant statistic and
summarizes the asymptotic theory for the maximum invariant likelihood
estimator (MILE). Section 3 develops the asymptotic theory for Lancaster's
estimator. Section 4 shows that this estimator is less efficient than MILE\
because it ignores relevant data. Section 5 shows that it is less efficient
to difference out the first observation than to condition on it. Section 6
compares the conditional argument to standard random effects approaches for
the unobserved data. Section 7 concludes and discusses how to extend the
model to a more useful form.

\section{The Model and the Maximal Invariant Likelihood \label{Model Sec}}

We consider a simple homoskedastic dynamic panel model with fixed effects
and without covariates:
\begin{equation}
y_{i,t+1}=\rho y_{i,t}+\eta_{i}+\sigma u_{i,t},\;i=1,\cdots,N;\;t=1,\cdots,T
\label{eq:Model}
\end{equation}
where $N\geq T+1$, $y_{i,t}\in\mathbb{R}$ are observable variables and
u_{i,t}\overset{iid}{\sim}N\left(0,1\right)$ are unobservable errors;
\eta_{i}\in\mathbb{R}$ are incidental parameters and $\left(\rho,\sigma^{2
\right)\in\mathbb{\mathbb{R}\times R}$ are structural parameters. We denote
the true unknown parameters by $\left(\rho^{\ast},\sigma^{\ast2},\eta_{i}^
\ast}\right)$. We assume that the parameter space is a compact set and
\sigma^{\ast2}>0$. For now, we assume that the initial \emph{observed}
condition is $y_{i,1}=0$ as \citet{Lancaster02} does. We relax this
assumption in Sections \ref{Initial Condition Sec} and \ref{Random Effects
Sec}.

Solving model (\ref{eq:Model}) recursively and writing it in matrix form
yields
\begin{eqnarray}
Y_{T} & = & \eta1_{T}^{\prime}B_{T}^{\prime}+\sigma U_{T}B_{T}^{\prime}
\text{ where}  \label{eq:Model-Matrix-Form} \\
U_{T} & \sim & N\left(0_{N\times T},I_{N}\otimes I_{T}\right),  \notag
\end{eqnarray}
$\eta=\left(\eta_{1},\cdots,\eta_{N}\right)^{\prime}\in\mathbb{R}^{N\times1}
, $1_{T}=\left(1,\cdots,1\right)^{\prime}\in\mathbb{R}^{T\times1},$
\begin{equation*}
Y_{T}=\left
\begin{array}{ccc}
y_{1,2} & \cdots & y_{1,T+1} \\
\vdots & \ddots & \vdots \\
y_{N,2} & \cdots & y_{N,T+1
\end{array
\right],\;U_{T}=\left
\begin{array}{ccc}
u_{1,2} & \cdots & u_{1,T+1} \\
\vdots & \ddots & \vdots \\
u_{N,2} & \cdots & u_{N,T+1
\end{array
\right],\text{ and }B_{T}=\left
\begin{array}{ccc}
1 &  &  \\
\vdots & \ddots &  \\
\rho^{T-1} & \cdots &
\end{array
\right].
\end{equation*}
When there is no confusion, we will omit the subscript from the matrices;
e.g., $Y$ instead of $Y_{T}$, $B$ instead of $B_{T}$, etc.

The inverse of $B$ has a simple form,
\begin{equation*}
B^{-1}\equiv D=I_{T}-\rho J_{T},\text{ where }J_{T}=\left
\begin{array}{cc}
0_{T-1}^{\prime} & 0 \\
I_{T-1} & 0_{T-1
\end{array
\right]
\end{equation*}
and $0_{T-1}$ is a $\left(T-1\right)$-dimensional column vector with zero
entries.

If individuals $i$ are treated equally, the coordinate system used to
specify the vector $\left(y_{1,t},\cdots,y_{N,t}\right)$ should not affect
inference based on them. Therefore, it is reasonable to restrict attention
to coordinate-free functions of $\left(y_{1,t},\cdots,y_{N,t}\right)$.
\citet{ChamberlainMoreira09} and \citet{Moreira09b} show that, indeed,
orthogonal transformations preserve both the model (\re
{eq:Model-Matrix-Form}) and the structural parameters $\left(\rho,\sigma^{2
\right)$ and this yields a maximal invariant statistic, the $T\times T$
matrix $Y^{\prime}Y$. So, if the researcher finds that it is reasonable to
restrict attention to statistics that are invariant to orthogonal
transformations, the maximal invariant statistic plays a crucial role: a
statistic is invariant to orthogonal transformations if, and only if, it
depends on the data through the maximal invariant statistic $Y^{\prime}Y$.

The maximal invariant statistic $Y^{\prime}Y$ has a noncentral Wishart
distribution and depends only on $\rho$, $\sigma^{2}$, and
\omega_{\eta}^{2}\equiv\frac{\eta^{\prime}\eta}{\sigma^{2}N}$. The
noncentral Wishart distribution is the multivariate generalization of the
noncentral Chi-squared distribution and it depends on the modified Bessel
function of the first kind. We use uniform approximations of Bessel
functions (see \citet{AbramowitzStegun65}), which allows us to write the
density of the noncentral Wishart distribution in a more tractable form.
Specifically, the log-likelihood of $Y^{\prime}Y$ is, up to an
o_{p}\left(N^{-1}\right)$ term, proportional to
\begin{eqnarray}
Q_{N}^{M}\left(\theta\right) & = & -\frac{1}{2}\ln\left(\sigma^{2}\right)
\frac{1}{2\sigma^{2}}\frac{tr\left(DY^{\prime}YD^{\prime}\right)}{NT}-\frac
\omega_{\eta}^{2}}{2}  \notag \\
& & +\frac{\left(1+A_{N}^{2}\right)^{1/2}}{2T}-\frac{\ln\left(1
\left(1+A_{N}^{2}\right)^{1/2}\right)}{2T},
\label{eq:Objetive-Function-MILE1}
\end{eqnarray}
where $\theta=\left(\rho,\sigma^{2},\omega_{\eta}^{2}\right)$ and $A_{N}=
\sqrt{\omega_{\eta}^{2}\frac{1_{T}^{\prime}DY^{\prime}YD^{\prime}1_{T}}
\sigma^{2}N}}$.

The log-likelihood $Q_{N}^{M}\left( \theta \right) $ is free from the
incidental parameter problem since $Y^{\prime }Y$ is parametrized by the
fixed dimensional vector of parameters $\theta $. Although the dimension is
fixed, the parameter
\begin{equation}
\omega _{\eta }^{2}\equiv \frac{\eta ^{\prime }\eta }{\sigma ^{2}N}=\frac
\sum_{i=1}^{N}\eta _{i}^{2}}{\sigma ^{2}N}
\end{equation
depends on the sample size $N$. For simplicity, we omit the dependence on $N$
from the parameter $\omega _{\eta }^{2}$. However, the asymptotic properties
of the estimator will be derived under different sequences of $\omega _{\eta
}^{2}$. The asymptotic properties of the estimator obtained by maximizing
the objective function $Q_{N}^{M}\left( \theta \right) $ are studied by
\citet{Moreira09b} and we reproduce these results here for convenience. The
information matrix $\mathcal{I}_{T}\left( \theta ^{\ast }\right) $ in
\citet{Moreira09b} contains typographical errors which are corrected here.
Define the matrix
\begin{equation*}
F_{0}=\left[
\begin{array}{ccccc}
0 & 0 & \cdots & 0 & 0 \\
\rho & 0 & \cdots & 0 & 0 \\
\frac{1}{2}\rho ^{2} & \rho & \cdots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
\frac{1}{T-1}\rho ^{T-1} & \frac{1}{T-2}\rho ^{T-2} & \cdots & \rho &
\end{array
\right]
\end{equation*
and its derivatives:
\begin{equation*}
F_{j}=\frac{d^{j}}{d\rho ^{j}}F_{0}\text{ and }F_{j}^{\ast }=\left. \frac
d^{j}}{d\rho ^{j}}F_{0}\right\vert _{\rho =\rho ^{\ast }}.
\end{equation*}

\bigskip{}

{\large {}{}{}{}{}{}Theorem 1}: Let
\begin{equation}
\hat{\theta}=\underset{\theta\in\varTheta}{\arg\max}Q_{N}^{M}\left(\thet
\right).  \label{eq:MILE}
\end{equation}

(A.1) Under the assumption that $N\rightarrow\infty$ with $T$ fixed, (i) if
\omega_{\eta}^{2}$ is fixed at $\omega_{\eta}^{\ast2}$, then $\hat{\theta
_{M}\rightarrow_{p}\theta^{\ast}=\left(\rho^{\ast},\sigma^{\ast2},\omega_
\eta}^{2\ast}\right)$; (ii) if $\omega_{\eta}^{2}\rightarrow\omega_{\eta}^
\ast2}$, then $\hat{\theta}_{M}\rightarrow_{p}\theta^{\ast}=\left(\rho^
\ast},\sigma^{\ast2},\omega_{\eta}^{\ast2}\right)$; and (iii) if
\limsup\omega_{\eta}^{\ast2}<\infty$, then $\hat{\theta}_{M}=\theta^
\ast}+o_{p}(1)$, where $\theta^{\ast}=\left(\rho^{\ast},\sigma^{\ast2}
\omega_{\eta}^{\ast2}\right)$.

(A.2) Under the assumption that $T\rightarrow\infty$ and $\left\vert
\rho^{\ast}\right\vert <1$, (i) if $\omega_{\eta}^{2}$ is fixed at
\omega_{\eta}^{\ast2}$, then $\hat{\theta}_{M}\rightarrow_{p}\theta^{\ast}
\left(\rho^{\ast},\sigma^{\ast2},\omega_{\eta}^{\ast2}\right)$; (ii) if
\omega_{\eta}^{2}\rightarrow\omega_{\eta}^{\ast2}$, then $\hat{\theta
_{M}\rightarrow_{p}\theta^{\ast}=\left(\rho^{\ast},\sigma^{\ast2},\omega_
\eta}^{\ast2}\right)$; and (iii) if $\limsup\omega_{\eta}^{\ast2}<\infty$,
then $\hat{\theta}_{M}=\theta^{\ast}+o_{p}(1)$, where $\theta^{\ast}=\left
\rho^{\ast},\sigma^{\ast2},\omega_{\eta}^{\ast2}\right)$.

(B) Assume that $\omega_{\eta}^{\ast2}>0$ is fixed, and let the score
statistic and the Hessian matrix be
\begin{equation*}
S_{N}^{M}\left(\theta\right)=\frac{\partial Q_{N}^{M}\left(\theta\right)}
\partial\theta}\;\text{and }H_{N}^{M}\left(\theta\right)=\frac
\partial^{2}Q_{N}^{M}\left(\theta\right)}{\partial\theta\partial\theta^
\prime}},
\end{equation*}
respectively, and define the matrix
\begin{equation*}
\mathcal{I}_{T}^{M}\left(\theta^{\ast}\right)=\left
\begin{array}{ccc}
h_{T}^{M} & \frac{\omega_{\eta}^{\ast4}}{\sigma^{\ast2}}\frac
1_{T}^{\prime}F_{1}^{\ast}1_{T}}{1+2\omega_{\eta}^{\ast2}T} & \frac
1+\omega_{\eta}^{\ast2}T}{1+2\omega_{\eta}^{\ast2}T}\frac
1_{T}^{\prime}F_{1}^{\ast}1_{T}}{T} \\
\frac{\omega_{\eta}^{\ast4}}{\sigma^{\ast2}}\frac{1_{T}^{\prime}F_{1}^
\ast}1_{T}}{1+2\omega_{\eta}^{\ast2}T} & \frac{1}{2\left(\sigma^{\ast2
\right)^{2}}\left(1+\frac{\omega_{\eta}^{\ast4}T}{1+2\omega_{\eta}^{\ast2}T
\right) & \frac{1}{2\sigma^{\ast2}}\frac{1+\omega_{\eta}^{\ast2}T}
1+2\omega_{\eta}^{\ast2}T} \\
\frac{1+\omega_{\eta}^{\ast2}T}{1+2\omega_{\eta}^{\ast2}T}\frac
1_{T}^{\prime}F_{1}^{\ast}1_{T}}{T} & \frac{1}{2\sigma^{\ast2}}\frac
1+\omega_{\eta}^{\ast2}T}{1+2\omega_{\eta}^{\ast2}T} & \frac{T}
2\left(1+2\omega_{\eta}^{\ast2}T\right)
\end{array
\right],
\end{equation*}
where
\begin{equation*}
h_{T}^{M}=\frac{tr\left(F_{1}^{\ast}F_{1}^{^{\prime}\ast}\right)}{T}+\frac
\omega_{\eta}^{\ast4}T}{1+\omega_{\eta}^{\ast2}T}\left(\frac
1_{T}^{\prime}F_{1}^{\ast}F_{1}^{^{\prime}\ast}1_{T}}{T}+\frac{1}
1+2\omega_{\eta}^{\ast2}T}\left(\frac{1_{T}^{\prime}F_{1}^{\ast}1_{T}}{T
\right)^{2}\right).
\end{equation*}
As $N\rightarrow\infty$ with $T$ fixed, (i) $\sqrt{NT}S_{N}^{M}\left
\theta^{\ast}\right)\rightarrow_{d}N\left(0,\mathcal{I}_{T}^{M}\left
\theta^{\ast}\right)\right)$; (ii) $H_{N}^{M}\left(\theta^{\ast}\right
\rightarrow_{p}-\mathcal{I}_{T}^{M}\left(\theta^{\ast}\right)$; (iii) $\sqrt
NT}\left(\hat{\theta}_{M}-\theta^{\ast}\right)\rightarrow_{d}N\left(0
\mathcal{I}_{T}^{M}\left(\theta^{\ast}\right)^{-1}\right)$; and (iv) the
log-likelihood ratio is
\begin{eqnarray*}
\Lambda_{N}\left(\theta^{\ast}+h\cdot\left(NT\right)^{-1/2},\theta^{\ast
\right) & = &
NT\left(Q_{N}^{M}\left(\theta^{\ast}+h\cdot\left(NT\right)^{-1/2
\right)-Q_{N}^{M}\left(\theta^{\ast}\right)\right) \\
& = & h^{\prime}\sqrt{NT}S_{N}^{M}\left(\theta^{\ast}\right)-\frac{1}{2
h^{\prime}\mathcal{I}_{T}\left(\theta^{\ast}\right)h+o_{Q_{N}^{M}\left
\theta^{\ast}\right)}(1),
\end{eqnarray*}
$\sqrt{NT}S_{N}^{M}\left(\theta^{\ast}\right)\rightarrow_{d}N\left(0
\mathcal{I}_{T}^{M}\left(\theta^{\ast}\right)\right)$ under
Q_{N}^{M}\left(\theta^{\ast}\right)$. Furthermore, $\hat{\theta}_{M}$ is
asymptotically efficient within the class of regular invariant estimators
for the differenced model (\ref{eq:Dif-Model}) under large $N$, fixed $T$
asymptotics.

\bigskip{}

Part (A) of the above theorem implies that $\hat{\rho}_{M}\rightarrow_{p
\rho^{\ast}$ and $\hat{\sigma}_{M}^{2}\rightarrow_{p}\sigma^{\ast2}$
regardless of the growth rate of $N$ and $T$ as long as $NT\rightarrow\infty
. Part (B) derives the limiting distribution of $\hat{\theta}_{M}$. It
shows, in particular, that $\hat{\rho}_{M}$ achieves the efficiency bound
\left(\mathcal{I}_{T}^{M}\left(\theta^{\ast}\right)^{-1}\right)_{11}$ for
regular invariant estimators as $N\rightarrow\infty$. Regular estimators
exclude superefficient estimators in the sense of Hodges-Le Cam and,
heuristically, a regular estimator is one whose asymptotic distribution does
not change in shrinking neighborhoods of the true parameter value (see
\citet{BickelKlaasenRitovWellner} for more details).

Part (B) of Theorem 1 finds the asymptotic distribution of $\hat{\theta}_{M}
, assuming that $\omega_{\eta}^{2}$ is fixed at $\omega_{\eta}^{\ast2}$. A
generalization of part (B) that allows for $\omega_{\eta}^{2}\rightarro
\omega_{\eta}^{\ast2}$ follows from a simple application of Le Cam's third
lemma.

\bigskip{}

{\large {}{}{}{}{}{}Lemma 2}: Assume $\omega_{\eta}^{2}=\omega_{\eta}^
\ast2}+h/\sqrt{N}$, where $\omega_{\eta}^{\ast2}>0$, $h\in\mathbb{R}$ and
\omega_{\eta}^{\ast2}$ is in the parameter space. As $N\rightarrow\infty$
with $T$ fixed,
\begin{equation*}
\sqrt{NT}\left(\hat{\theta}_{M}-\theta^{\ast}\right)\rightarrow_{d}N\left
\left
\begin{array}{c}
0 \\
0 \\

\end{array
\right),\mathcal{I}_{T}^{M}\left(\theta^{\ast}\right)^{-1}\right),
\end{equation*}
where $\mathcal{I}_{T}^{M}\left(\theta^{\ast}\right)^{-1}$ is defined in
Theorem 1, part (B).

\bigskip{}

Lemma 2 shows that the asymptotic distribution of the structural parameters
does not change, whether $\omega_{\eta}^{2}$ is fixed at
\omega_{\eta}^{\ast2}$ or $\omega_{\eta}^{2}\rightarrow\omega_{\eta}^{\ast2}
. For simplicity's sake, we assume throughout the paper that the sequence
\omega_{\eta}^{2}$ (which can depend on the sample size $N$) is fixed at
\omega_{\eta}^{\ast2}>0$.

\section{Lancaster's Estimator \label{Lancaster Estimator Sec}}

\citet{Lancaster02} proposes a Bayesian approach to estimate the structural
parameters which involves reparameterizing model \eqref{eq:Model-Matrix-Form}
so that the information matrix is block diagonal, with the common parameters
in one block and the incidental parameters in the other. Then he defines his
estimator as one of the local maxima of the integrated likelihood function,
integrating with respect to the Lebesgue measure on $\mathbb{R}^{N}$.
\citet{DhaeneJochmans16} give an alternative interpretation for Lancaster's
estimator by showing that Lancaster's posterior can be obtained by adjusting
the profile likelihood so that its score is free from asymptotic bias. In
this sense, Lancaster's estimator is a bias-corrected estimator.

Lancaster's estimator seeks to maximize the following objective function:
\begin{equation}
Q_{N}^{L}\left(\rho,\sigma^{2}\right)=-\frac{1}{2}\ln\left(\sigma^{2}\right)
\frac{1_{T}^{\prime}F_{0}1_{T}}{T\left(T-1\right)}-\frac{1}{2\sigma^{2}
\frac{tr\left(DY^{\prime}YD^{\prime}H\right)}{N\left(T-1\right)},
\label{eq:Lancaster-Object-1}
\end{equation}
where the matrix $H$ is defined as
\begin{equation*}
H=I_{T}-\frac{1}{T}1_{T}1_{T}^{\prime}.
\end{equation*}

The posterior (\ref{eq:Lancaster-Object-1}) is not a likelihood function and
$\left(\rho^{\ast},\sigma^{\ast2}\right)$ is not a global maximizer of
\underset{N\rightarrow\infty}{plim}Q_{N}^{L}\left(\rho,\sigma^{2}\right)$ as
in standard maximum likelihood theory (see \citet{DhaeneJochmans16}).
Nonetheless, Theorem 3, proved by \citet{Lancaster02}, shows that the
posterior (\ref{eq:Lancaster-Object-1}) can be used to consistently estimate
the structural parameters.

\bigskip{}

{\large {}{}{}{}{}{}Theorem 3}: Let $S_{N}^{L}\left(\rho,\sigma^{2}\right)
\frac{\partial Q_{N}^{L}\left(\rho,\sigma^{2}\right)}{\partial\left(\rho
\sigma^{2}\right)^{\prime}}$ be the score of the posterior (\re
{eq:Lancaster-Object-1}) and let $\Theta_{N}$ be the set of roots of
\begin{equation}
S_{N}^{L}\left(\rho,\sigma^{2}\right)=0  \label{eq:Lancaster-Estimation-Eq}
\end{equation}
corresponding to the local maxima. If that set is empty, set $\Theta_{N}$
equals to $\left\{ 0\right\} $. Then, there is a consistent root of equation
(\ref{eq:Lancaster-Estimation-Eq}).

\bigskip{}

The usefulness of Theorem 3 is limited by the fact that it only states that
one (of possibly many) of the local maxima of (\ref{eq:Lancaster-Object-1})
is a consistent estimator of the structural parameters. However, it does not
indicate how to find a consistent estimator. Therefore, even though the
posterior can be used to consistently estimate the structural parameters of
model \eqref{eq:Model-Matrix-Form}, Lancaster leaves unanswered the question
of how to uniquely determine the consistent root of (\re
{eq:Lancaster-Estimation-Eq}). To our knowledge, two different methodologies
to uniquely choose the consistent root of Lancaster's score have been
proposed. The first approach, by \citet{DhaeneJochmans16}, suggests as a
consistent estimator of the structural parameters, the minimizer of the norm
of Lancaster's score on an interval around the maximum likelihood estimator
obtained from the maximization of the likelihood of
\eqref{eq:Model-Matrix-Form}. The second method, by \citet{Kruiniger14},
uses as a consistent estimator, the minimizer of a quadratic form in
Lancaster's score subjected to a condition on the Hessian matrix.

Lemma 4 finds the asymptotic variance of a consistent root of (\re
{eq:Lancaster-Estimation-Eq}).

\bigskip{}

{\large {}{}{}{}{}{}Lemma 4}: Assume $\omega_{\eta}^{\ast2}>0$ and let
\left(\hat{\rho}_{L},\hat{\sigma}_{L}^{2}\right)$ be a consistent root of
S_{N}^{L}\left(\rho,\sigma^{2}\right)$. Let the Hessian matrix be

\begin{equation*}
H_{N}^{L}\left(\rho,\sigma^{2}\right)=\frac{\partial^{2}Q_{N}^{L}\left(\rho
\sigma^{2}\right)}{\partial\left(\rho,\sigma^{2}\right)^{\prime}\partia
\left(\rho,\sigma^{2}\right)},
\end{equation*}
and define the matrices
\begin{equation*}
\mathcal{I}_{T}^{L}\left(\theta^{*}\right)=\left
\begin{array}{cc}
h_{T}^{L} & -\frac{1}{\sigma^{*2}}\frac{1_{T}^{\prime}F_{1}^{*}1_{T}}
T\left(T-1\right)} \\
-\frac{1}{\sigma^{*2}}\frac{1_{T}^{\prime}F_{1}^{*}1_{T}}{T\left(T-1\right)}
& \frac{1}{2\left(\sigma^{*2}\right)^{2}
\end{array
\right]
\end{equation*}
and
\begin{equation}
\Sigma_{T}\left(\theta^{*}\right)=\frac{T}{T-1}\left
\begin{array}{cc}
a_{T}^{L} & \mathcal{I}_{T}^{L}\left(1,2\right) \\
\mathcal{I}_{T}^{L}\left(1,2\right) & \mathcal{I}_{T}^{L}\left(2,2\right
\end{array
\right],  \label{eq:Var-Score-lancaster}
\end{equation}

where
\begin{equation*}
h_{T}^{L}=-\frac{1_{T}^{\prime}F_{2}^{\ast}1_{T}}{T\left(T-1\right)}+\frac
tr\left(F_{1}^{\ast}F_{1}^{\ast^{\prime}}\right)}{T-1}+\frac
1_{T}^{\prime}F_{1}^{\ast^{\prime}}F_{1}^{\ast}1_{T}}{T}\frac
\left(\omega_{\eta}^{\ast2}T-1\right)}{T-1}-\frac{\omega_{\eta}^{\ast2}T}{T-
}\left(\frac{1_{T}^{\prime}F_{1}^{\ast^{\prime}}1_{T}}{T}\right)^{2},
\end{equation*}
\begin{equation*}
a_{T}^{L}=2h_{T}^{L}+2\frac{1_{T}^{\prime}F_{2}^{\ast}1_{T}}
T\left(T-1\right)},
\end{equation*}
and $\mathcal{I}_{T}^{L}\left(i,j\right)$ is the $\left(i,j\right)$-th entry
of $\mathcal{I}_{T}^{L}\left(\theta^{\ast}\right)$. When $\mathcal{I
_{T}^{L}\left(\theta^{\ast}\right)$ is nonsingular, as $N\rightarrow\infty$
with $T$ fixed, (i) $\sqrt{NT}S_{N}^{L}\left(\rho^{\ast},\sigma^{\ast2
\right)\rightarrow_{d}N\left(0,\Sigma_{T}\left(\theta^{\ast}\right)\right)$;
(ii) $-H_{N}^{L}\left(\rho^{\ast},\sigma^{\ast2}\right)\rightarrow_{p
\mathcal{I}_{T}^{L}\left(\theta^{\ast}\right)$; and (iii) $\sqrt{NT}\left
\begin{array}{c}
\hat{\rho}_{L}-\rho^{\ast} \\
\hat{\sigma^{2}}_{L}-\sigma^{\ast2
\end{array
\right)\rightarrow_{d}N\left(0,\mathcal{I}_{T}^{L}\left(\theta^{\ast
\right)^{-1}\Sigma_{T}\left(\theta^{\ast}\right)\mathcal{I
_{T}^{L}\left(\theta^{\ast}\right)^{-1}\right)$.

\bigskip{}

Lemma 4 follows from standard asymptotic theory because of the assumption
that the Hessian matrix $-H_{N}^{L}\left(\rho^{\ast},\sigma^{\ast2}\right)$
converges in probability to a nonsingular matrix $\mathcal{I
_{T}^{L}\left(\theta^{\ast}\right)$. \citet{DhaeneJochmans16} prove that
this assumption is violated, i.e. $\mathcal{I}_{T}^{L}\left(\theta^{\ast
\right)$ is singular, when $T=2$ or $\rho^{\ast}=1$.

\section{Lancaster's Estimator and a Decomposition of the Maximal Invariant
Statistic's Log-Likelihood \label{Decomposition Sec}}

In contrast to Lancaster's estimator, the estimator defined in (\ref{eq:MILE
), based on the maximal invariant statistic's log-likelihood, is a standard
maximum likelihood estimator in the sense that its objective function is a
likelihood function and the limit of its objective function attains a unique
global maximum at the vector of true parameters. This implies that the
estimator that maximizes the maximal invariant statistic's log-likelihood is
uniquely determined and no additional procedures are necessary to obtain a
consistent and asymptotically normal estimator of the structural parameters.
More than that, Theorem 1 shows that it attains the minimum variance bound
for invariant regular estimators of the structural parameters of model
\eqref{eq:Model-Matrix-Form}. Therefore, the estimator defined in (\re
{eq:MILE}) efficiently uses all information available in the maximal
invariant statistic $Y^{\prime}Y$.

Lancaster's estimator is also invariant to orthogonal transformations since
it depends on the data only through the maximal invariant statistic
Y^{\prime}Y$; see equation (\ref{eq:Lancaster-Object-1}). However, it is not
uniquely determined and additional steps, such as those proposed by
\citet{DhaeneJochmans16} and \citet{Kruiniger14}, are necessary to choose
the consistent root among the (possibly) many roots of Lancaster's score
function. Furthermore, the limit of probability of the Hessian of
Q_{N}^{L}\left(\rho,\sigma^{2}\right)$ is not necessarily proportional to
the asymptotic variance (\emph{AVar}) of the score. Since the information
equality does not hold, the asymptotic variance of Lancaster's consistent
estimator cannot attain the lower bound for invariant regular estimators
found in Theorem 1.

The explanation for the shortcomings of Lancaster's estimator is that, even
though it is invariant, its objective function ignores information available
in the maximal invariant by using only part of the maximal invariant
statistic's log-likelihood function. Indeed, the maximal invariant
statistic's log-likelihood function (\ref{eq:Objetive-Function-MILE1})
concentrated out of $\sigma^{2}$ and $\omega_{\eta}^{2}$ is
\begin{equation}
Q_{N}^{M}\left(\rho\right)=-\frac{1}{2}\frac{T-1}{T}\ln\left(\frac
tr\left(DY^{\prime}YD^{\prime}H\right)}{N\left(T-1\right)}\right)-\frac{1}{2
}\ln\left(\frac{1_{T}^{\prime}DY^{\prime}YD^{\prime}1_{T}}{NT}\right),
\label{eq:MILE-Concentrated1}
\end{equation}
while Lancaster's objective function concentrated out of $\sigma^{2}$ is
\begin{equation*}
Q_{N}^{L}\left(\rho\right)=\frac{1_{T}^{\prime}F_{0}1_{T}}{T\left(T-1\right)
-\frac{1}{2}\ln\left(\frac{tr\left(DY^{\prime}YD^{\prime}H\right)}
N\left(T-1\right)}\right),
\end{equation*}
which allows us to write
\begin{eqnarray}
Q_{N}^{M}\left(\rho\right) & = & \frac{T-1}{T}Q_{N}^{L}\left(\rh
\right)+Q_{N}^{M-L}\left(\rho\right),\text{ where}  \label{eq:Decomposition1}
\\
Q_{N}^{M-L}\left(\rho\right) & = & -\frac{1_{T}^{\prime}F_{0}1_{T}}{T^{2}}
\frac{1}{2T}\ln\left(\frac{1_{T}^{\prime}DY^{\prime}YD^{\prime}1_{T}}{NT
\right).  \notag
\end{eqnarray}
The respective score functions are given by
\begin{equation*}
S_{N}^{L}\left(\rho\right)=\frac{1_{T}^{\prime}F_{1}1_{T}}{T\left(T-1\right)
+\frac{tr\left(J_{T}Y^{\prime}YD^{\prime}H\right)}{tr\left(DY^{\prime}YD^
\prime}H\right)}
\end{equation*}
and
\begin{eqnarray}
S_{N}^{M}\left(\rho\right) & = & \frac{T-1}{T}S_{N}^{L}\left(\rh
\right)+S_{N}^{M-L}\left(\rho\right),\text{ where} \\
S_{N}^{M-L}\left(\rho\right) & = & -\frac{1_{T}^{\prime}F_{1}1_{T}}{T^{2}}
\frac{1}{T}\frac{1_{T}^{\prime}J_{T}Y^{\prime}YD^{\prime}1_{T}}
1_{T}^{\prime}DY^{\prime}YD^{\prime}1_{T}}.  \notag
\end{eqnarray}
The decomposition (\ref{eq:Decomposition1}) implies an orthogonal
decomposition of the concentrated score function:

\bigskip{}

{\large {}{}{}{}{}{}Theorem 5}: Let $S_{N}^{M}\left(\rho\right)$,
S_{N}^{L}\left(\rho\right)$ and $S_{N}^{M-L}\left(\rho\right)$ be the score
functions associated with $Q_{N}^{M}\left(\rho\right)$, $Q_{N}^{L}\left(\rh
\right)$ and $Q_{N}^{M-L}\left(\rho\right)$, respectively. Then,
\begin{equation}
S_{N}^{M}\left(\rho\right)=\frac{T-1}{T}S_{N}^{L}\left(\rh
\right)+S_{N}^{M-L}\left(\rho\right),  \label{eq:Decomposition2}
\end{equation}
where
\begin{equation}
S_{N}^{M-L}\left(\rho^{\ast}\right)\rightarrow_{p}0
\label{eq:Limit-Score-M-L}
\end{equation}
and
\begin{equation}
\left
\begin{array}{c}
\sqrt{NT}S_{N}^{L}\left(\rho^{\ast}\right) \\
\sqrt{NT}S_{N}^{M-L}\left(\rho^{\ast}\right
\end{array
\right)\rightarrow_{d}N\left(\left
\begin{array}{c}
0 \\

\end{array
\right),\left
\begin{array}{cc}
b_{T} & 0 \\
0 & c_{T
\end{array
\right)\right),  \label{eq:Scores-concentrated-Distribution}
\end{equation}
with
\begin{equation*}
b_{T}=a_{T}^{L}-2\frac{T}{\left(T-1\right)^{3}}\left(\frac
1_{T}^{\prime}F_{1}^{\ast}1_{T}}{T}\right)^{2}>0,
\end{equation*}
the constant $a_{T}^{L}$ as defined in Lemma 4, and
\begin{equation*}
c_{T}=\frac{2T}{\left(1+\omega_{\eta}^{\ast2}T\right)}\left(\frac
1_{T}^{\prime}F_{1}^{\ast^{\prime}}F_{1}^{\ast}1_{T}}{T}-\left(\frac
1_{T}^{\prime}F_{1}^{\ast}1_{T}}{T}\right)^{2}\right)>0.
\end{equation*}

\bigskip{}

A consistent estimator for $\rho ^{\ast }$ is expected to be more efficient,
the closer it stays to the maximum likelihood estimator $\hat{\rho}_{M}$.
This implies that the consistent estimator $\hat{\rho}_{L}$ might have low
efficiency because it only uses part of the full log-likelihood (\re
{eq:Decomposition1}). One may use $\left\vert S_{N}^{M}\left( \hat{\rho
_{L}\right) \right\vert $ as a measure of how \textquotedblleft
close\textquotedblright\ any inflection point $\hat{\rho}_{L}$ is to $\hat
\rho}_{M}$ and, because $S_{N}^{M}\left( \hat{\rho}_{L}\right)
=S_{N}^{M-L}\left( \hat{\rho}_{L}\right) $, $\hat{\rho}_{L}$ will be more
efficient, the smaller $\left\vert S_{N}^{M-L}\left( \hat{\rho}_{L}\right)
\right\vert $ is. The term $S_{N}^{M-L}\left( .\right) $ evaluated at the
true value $\rho ^{\ast }$ or at a consistent estimator, converges in
probability to zero. This suggests choosing the inflexion points based on
Lancaster's score by looking at $S_{N}^{M-L}\left( \hat{\rho}_{L}\right) $
to uniquely determine the consistent root, akin to \citet{DhaeneJochmans16}
and \citet{Kruiniger14}.

Instead, we can use the information on $S_{N}^{M-L}\left(.\right)$ to
increase the efficiency of $\hat{\rho}_{L}$. Simple algebra shows that
\begin{equation*}
\left\vert S_{N}^{M-L}\left(\hat{\rho}_{L}\right)\right\vert =\left\vert
\left(T-1\right)\frac{tr\left(J_{T}Y^{\prime}Y\hat{D}_{L}^{\prime}H\right)}
tr\left(\hat{D}_{L}Y^{\prime}Y\hat{D}_{L}^{\prime}H\right)}+\frac
1_{T}^{\prime}J_{T}Y^{\prime}Y\hat{D}_{L}^{\prime}1_{T}}{1_{T}^{\prime}\hat{
}_{L}Y^{\prime}Y\hat{D}_{L}^{\prime}1_{T}}\right\vert ,
\end{equation*}
where $\hat{D}_{L}$ is the matrix $D$ evaluated at $\hat{\rho}_{L}$. Notice
that $\hat{\rho}_{L}$ is inefficient since $\sqrt{NT}S_{N}^{M-L}\left(\hat
\rho}_{L}\right)$ does not converge in probability to zero. Instead of
choosing the consistent root following Lancaster's methodology, we could add
the information in $S_{N}^{M-L}\left(.\right)$ to $S_{N}^{L}\left(.\right)$
by using (\ref{eq:Decomposition2}). The associated estimator $\hat{\rho}_{M}$
uses the information in both $S_{N}^{M-L}\left(.\right)$ and
S_{N}^{L}\left(.\right)$ efficiently and is uniquely defined, making the
estimation problem more simple and objective.

\section{Initial Condition and Likelihood Inference\label{Initial Condition
Sec}}

Theorem 1 shows that the estimator for the structural parameters $\left(
\rho ,\sigma ^{2}\right) $ attains the efficiency bound for model (\re
{eq:Model}) when the first observation is $y_{i,1}=0$. \citet{Lancaster02}
suggests working with the differenced data $\tilde{y}_{i,t}=y_{i,t}-y_{i,1}$
so that we may continue working with an initial observation equal to zero.
Instead, we draw inference on $\left( \rho ,\sigma ^{2}\right) $ by
conditioning on the first observation $y_{i,1}$. We will show the
differencing method is indeed less efficient than the conditional method.

By using model (\ref{eq:Model}) with the variables $y_{i,t}$ in levels,
there are more observations that can be used to estimate the parameters. On
the other hand, the number of parameters to be estimated increases. In this
section we show that the estimator that maximizes the likelihood of the
maximal invariant statistic of the original model (\ref{eq:Model}) has
asymptotic variance that is no larger than the variance of the estimator
\ref{eq:MILE}) and, under very general conditions, using $y_{i,t}$ in levels
will be strictly more efficient.

When we allow for a nonzero initial condition, the model \eqref{eq:Model}
becomes
\begin{eqnarray}
Y_{T} & = & y_{1}.\rho
e_{1}^{\prime}B_{T}^{\prime}+\eta.1_{T}^{\prime}B^{\prime}+\sigma.U_{T}.B^
\prime},\text{ where}  \label{eq:Initial-Model} \\
U_{T} & \sim & N\left(0_{N\times T},I_{N}\otimes I_{T}\right),  \notag
\end{eqnarray}
$y_{1}=$ $\left(y_{1,1},\cdots,y_{N,1}\right)^{\prime}$ is the vector of
first observations and $e_{1}=\left(1,0,...,0\right)^{\prime}$ is the
canonical vector.

\citet{Lancaster02} works with a differenced version of model (\ref{eq:Model
) given by
\begin{eqnarray}
\widetilde{y}_{i,t+1} & = & \rho\widetilde{y}_{i,t}+\widetilde{\eta
_{i}+\sigma u_{i,t},\;i=1,\cdots,N;\;t=1,\cdots,T  \label{eq:Dif-Model} \\
\widetilde{y}_{i,t} & \equiv &
y_{i,t}-y_{i,1},\;i=1,\cdots,N;\;t=1,\cdots,T+1  \notag \\
\widetilde{\eta}_{i} & \equiv &
\eta_{i}-y_{i,1}\left(1-\rho\right),\;i=1,\cdots,N  \notag
\end{eqnarray}
and seeks inference based on $\widetilde{y}_{i,1}=0$ for all $i$. In matrix
form,

\begin{equation*}
Y_{T+1}-y_{1}.1_{T+1}^{\prime}=\left[0_{N\times1}:Y_{T}-y_{1}.1_{T}^{\prime
\right].
\end{equation*}
The second term equals
\begin{eqnarray*}
Y_{T}-y_{1}.1_{T}^{\prime} & = &
y_{1}.\left(\rho.e_{1}^{\prime}B^{\prime}-1_{T}^{\prime}\right)+\eta.1_{T}^
\prime}B^{\prime}+\sigma.U.B^{\prime} \\
& = & \left[\eta-\left(1-\rho\right)y_{1}\right].1_{T}^{\prime}B^{\prime}
\sigma.U.B^{\prime}.
\end{eqnarray*}

Defining differenced variables and the incidental parameters,
\begin{equation*}
\widetilde{Y}_{T}=Y_{T}-y_{1}.1_{T}^{\prime}\text{ and }\widetilde{\eta
=\eta-\left(1-\rho\right).y_{1},
\end{equation*}
we are back to the model with the first observation equal to zero:
\begin{equation}
\widetilde{Y}_{T}=\widetilde{\eta}.1_{T}^{\prime}B^{\prime}+\sigma.U_{T}.B^
\prime},\;U\sim N\left(0_{N\times T},I_{N}\otimes I_{T}\right),
\label{eq:Model-1-Matrix-Form}
\end{equation}
where the incidental parameter is $\tau=\widetilde{\eta}$.

Differencing eliminates one time period from the data. Instead, we will
condition on the initial observation $y_{1}$ itself. It is convenient to
work with the linear setup of \citet{ChamberlainMoreira09}:
\begin{equation}
Y=x.a\left( \gamma \right) +\tau .b\left( \gamma \right) +U.c\left( \gamma
\right) ,  \label{eq:General-Model}
\end{equation
where $x\in \mathbb{R}^{N\times K}$ are explanatory variables, $\tau \in
\mathbb{R}^{N\times J}$ are the incidental parameters, and $a\left( \cdot
\right) ,$ $b\left( \cdot \right) ,$ $c\left( \cdot \right) $ are given
functions of the unknown parameter of interest $\gamma $. Consider the group
of transformations $g.Y$, where $g$ are orthogonal matrices such that $g.x=x
. This group modifies the unknown incidental parameter $\tau $, but
preserves the model and the parameter $\gamma $ of interest:
\begin{eqnarray*}
g.Y &=&g.x.a\left( \gamma \right) +g.\tau .b\left( \gamma \right)
+g.U.c\left( \gamma \right) \\
&=&x.a\left( \gamma \right) +\left( g.\tau \right) .b\left( \gamma \right)
+U.c\left( \gamma \right)
\end{eqnarray*
in distribution, because the law of $g.U$ is the same as the law of $U$.
This group yields the maximal invariant statistic, which is given by the
pair
\begin{equation}
Z_{1}=\left( x^{\prime }x\right) ^{-1/2}x^{\prime }Y\text{ and
Z_{2}^{\prime }Z_{2}=Y^{\prime }M_{x}Y,  \label{eq:Maximal-Invariant-1}
\end{equation
where $Z_{2}=q_{2}^{\prime }Y$ for any matrix $q_{2}$ such that the matrix
q=\left[ x\left( x^{\prime }x\right) ^{-1/2}:q_{2}\right] $ is orthogonal.
Because $q$ is an orthogonal matrix, we have $q_{2}q_{2}^{\prime
}=I_{N}-N_{x}=M_{x}$ for $N_{x}=x\left( x^{\prime }x\right) ^{-1}x^{\prime }
. We refer the reader to \citet{ChamberlainMoreira09} for more details.

Define the parameters
\begin{equation}
\omega_{\tau,x}^{2}=\frac{\tau^{\prime}M_{x}\tau}{\sigma^{2}N}\text{ and
\delta_{\tau,x}=\left(x^{\prime}x\right)^{-1}x^{\prime}\tau.
\end{equation}
The coefficient $\delta_{\tau,x}$ is the ordinary least squares (OLS)\
estimator and $\omega_{\tau,x}^{2}$ is the standardized average of sum of
squared residuals from a fictitious regression of $\tau$ on $x$ (if there
are no covariates $x$, we define $\omega_{\tau}^{2}=\left(\sigma^{2}
\right)^{-1}\tau^{\prime}\tau$).

For the model (\ref{eq:General-Model}), the statistics $Z_{1}$ and
Z_{2}^{\prime}Z_{2}$ are independently normal and noncentral Wishart
distributed. Their distributions depend on $\left(\rho,\sigma^{2},\delta_
\tau,x}\right)$ and $\left(\rho,\sigma^{2},\omega_{\tau,x}^{2}\right)$,
respectively.

If we condition on the initial observation $y_{1}$, the model (\re
{eq:Initial-Model}) for $Y_{T}$ is a special case of (\ref{eq:General-Model
), where $x=y_{1}$, $\tau=\eta$, and the regression coefficients are
\begin{equation*}
a\left(\gamma\right)=\rho e_{1}^{\prime}B^{\prime}\text{,
b\left(\gamma\right)=1_{T}^{\prime}B^{\prime},\text{ and
c\left(\gamma\right)=\sigma.B^{\prime}.
\end{equation*}
Let $\theta_{1}=\left(\rho,\sigma^{2},\delta_{\eta,y_{1}},\omega_
\eta,y_{1}}^{2}\right)\in\mathbb{R}^{4}$ be the parameters of the joint
distribution of \eqref{eq:Maximal-Invariant-1}. The likelihood estimator
that maximizes the joint likelihood of \eqref{eq:Maximal-Invariant-1} does
not have the incidental parameter problem since it is parametrized by
\theta_{1}$, which has fixed dimension.

If we instead difference out the data, as \citet{Lancaster02} does, the
model (\ref{eq:Model-1-Matrix-Form}) for $\widetilde{Y}_{T}$ is a special
case of (\ref{eq:General-Model}), in which $x$ is absent, the incidental
parameter $\tau $ is $\widetilde{\eta }=\eta -\left( 1-\rho \right) .y_{1}$,
and the regression coefficients are
\begin{equation*}
b\left( \gamma \right) =1_{T}^{\prime }B^{\prime }\text{ and }c\left( \gamma
\right) =\sigma .B^{\prime }.
\end{equation*}

We now show that \citet{Lancaster02}'s differencing method entails
unnecessary efficiency loss. We begin by defining the estimator based on the
likelihood of $\left( Z_{1},Z_{2}^{\prime }Z_{2}\right) $. Let
\begin{equation}
\hat{\theta}_{M1}=\underset{\theta _{1}\in \varTheta}{\arg \min
Q_{N}^{M1}\left( \theta _{1}\right) ,  \label{eq:MILEG}
\end{equation
where
\begin{eqnarray}
Q_{N}^{M1}\left( \theta _{1}\right) &=&-\frac{1}{2}\ln \sigma ^{2}-\frac
\omega _{\eta ,y_{1}}^{2}}{2}-\frac{1}{2\sigma ^{2}}tr\left( \frac
DZ_{2}^{\prime }Z_{2}D^{\prime }}{NT}\right) +\frac{1}{2T}\left(
1+A_{1,N}^{2}\right) ^{1/2}  \notag \\
&&-\frac{1}{2\sigma ^{2}NT}\left( Z_{1}D^{\prime }-\left\Vert
y_{1}\right\Vert \left( \rho e_{1}^{\prime }+\delta _{\eta
,y_{1}}1_{T}^{\prime }\right) \right) \left( DZ_{1}^{\prime }-\left\Vert
y_{1}\right\Vert \left( \rho e_{1}+\delta _{\eta ,y_{1}}1_{T}\right) \right)
\notag \\
&&-\frac{1}{2T}\ln \left( 1+\left( 1+A_{1,N}^{2}\right) ^{1/2}\right) ,
\label{eq:Objective-Function-MILE-2}
\end{eqnarray
is, up to $o_{p}\left( N^{-1}\right) $ terms, proportional to the
(conditional on $y_{1}$) log-likelihood of $\left( Z_{1},Z_{2}^{\prime
}Z_{2}\right) $ and $A_{1,N}=2\sqrt{\omega _{\eta ,y_{1}}^{2}\frac
1_{T}^{\prime }DZ_{2}^{\prime }Z_{2}D^{\prime }1_{T}}{\sigma ^{2}N}}$. The
asymptotic behavior of the estimator $\hat{\theta}_{M1}$ is given next.

\bigskip{}

{\large {}{}{}{}{}{}Lemma }6: Assume that the true parameters
\delta_{\eta,y_{1}}$ and $\omega_{\eta,y_{1}}^{2}$ are fixed, respectively,
at $\delta_{\eta,y_{1}}^{\ast}$ and $\omega_{\eta,y_{1}}^{\ast2}>0$ and that
$0<\underset{N\rightarrow\infty}{\lim}\frac{\left\Vert y_{1}\right\Vert ^{2
}{N}\equiv\overline{\left\Vert y_{1}\right\Vert }^{2}<\infty$ . As
N\rightarrow\infty$ with $T$ fixed,

(A) $\hat{\theta}_{M1}\rightarrow_{p}\theta_{1}^{\ast}=\left(\rho^{\ast}
\sigma^{\ast2},\delta_{\eta,y_{1}}^{\ast},\omega_{\eta,y_{1}}^{\ast2
\right). $

(B) Let $Q_{N}^{M1}\left(\rho,\sigma^{2}\right)$ be the objective function
\ref{eq:Objective-Function-MILE-2}) concentrated out of $\delta_{\eta,y_{1}}$
and $\omega_{\eta,y_{1}}^{2}$ and denote the score statistic and the Hessian
matrix by
\begin{equation*}
S_{N}^{M1}\left(\rho,\sigma^{2}\right)=\frac{\partial
Q_{N}^{M1}\left(\rho,\sigma^{2}\right)}{\partial\left(\rho,\sigma^{2
\right)^{\prime}}\;\mathrm{and}\;H_{N}^{M1}\left(\rho,\sigma^{2}\right)
\frac{\partial^{2}Q_{N}^{M1}\left(\rho,\sigma^{2}\right)}
\partial\left(\rho,\sigma^{2}\right)^{\prime}\partial\left(\rho,\sigma^{2
\right)},
\end{equation*}
respectively. Also, define the matrix
\begin{equation*}
\mathcal{I}_{T}^{M1}\left(\theta_{1}^{\ast}\right)=\left
\begin{array}{cc}
d_{T}^{M1} & -\frac{1_{T}^{\prime}F_{1}^{\ast}1_{T}}{\sigma^{\ast2}T^{2}} \\
-\frac{1_{T}^{\prime}F_{1}^{\ast}1_{T}}{\sigma^{\ast2}T^{2}} & \frac{1}
2\left(\sigma^{\ast2}\right)^{2}}\frac{T-1}{T
\end{array
\right],
\end{equation*}
where
\begin{align*}
d_{T}^{M1} & =\frac{1}{T\left(1+\omega_{\eta,y_{1}}^{\ast2}T\right)}\left
\frac{1_{T}^{\prime}F_{1}^{\ast^{\prime}}F_{1}^{\ast}1_{T}}{T
+\omega_{\eta,y_{1}}^{\ast2}T\left(\frac{1_{T}^{\prime}F_{1}^{\ast}1_{T}}{T
\right)^{2}\right)-\frac{2}{T}\left(\frac{1_{T}^{\prime}F_{1}^{\ast}1_{T}}{T
\right)^{2} \\
& +\left(\omega_{\eta,y_{1}}^{\ast2}+\frac{\overline{\left\Vert
y_{1}\right\Vert }^{2}}{\sigma^{\ast2}}\left(\delta_{\eta,y_{1}}^{*}+\rho^
\ast}-1\right)^{2}\right)\left(\frac{1_{T}^{\prime}F_{1}^{\ast^
\prime}}F_{1}^{\ast}1_{T}}{T}-\left(\frac{1_{T}^{\prime}F_{1}^{\ast}1_{T}}{T
\right)^{2}\right) \\
& +\frac{tr\left(F_{1}^{\ast}F_{1}^{\ast^{\prime}}\right)}{T}-\frac
1_{T}^{\prime}F_{1}^{\ast^{\prime}}F_{1}^{\ast}1_{T}}{T^{2}}.
\end{align*}
Then, (i) $\sqrt{NT}S_{N}^{M1}\left(\rho^{\ast},\sigma^{\ast2}\right
\rightarrow_{d}N\left(0,\mathcal{I}_{T}^{M1}\left(\theta_{1}^{\ast}\right
\right)$; (ii) $H_{N}^{M1}\left(\rho^{\ast},\sigma^{\ast2}\right
\rightarrow_{p}-\mathcal{I}_{T}^{M1}\left(\theta_{1}^{\ast}\right)$; and
(iii) $\sqrt{NT}\left
\begin{array}{c}
\hat{\rho}_{M1}-\rho^{\ast} \\
\hat{\sigma}_{M1}^{2}-\sigma^{\ast2
\end{array
\right)\rightarrow_{d}N\left(0,\mathcal{I}_{T}^{M1}\left(\theta_{1}^{\ast
\right)^{-1}\right)$.

\bigskip{}

Lemma 7 below shows that, in general, \eqref{eq:MILEG} estimates the
structural parameters with strictly smaller variances than the original
estimator that uses differenced data, as defined in \eqref{eq:MILE}.

\bigskip{}

{\large {}{}{}{}{}{}Lemma }7: If $\delta_{\eta,y_{1}}^{\ast}+\rho^{\ast
\neq1 $, then \emph{AVar}$\left(\hat{\rho}_{M}\right)>\emph{AVar}\left(\hat
\rho}_{M1}\right)$ and \emph{AVar}$\left(\hat{\sigma}_{M}^{2}\right)>\emph
AVar}\left(\hat{\sigma}_{M1}^{2}\right)$. If $\delta_{\eta,y_{1}}^{\ast}
\rho^{\ast}=1$, then \emph{AVar}$\left(\hat{\rho}_{M}\right)=\emph{AVar
\left(\hat{\rho}_{M1}\right)$ and \emph{AVar}$\left(\hat{\sigma
_{M}^{2}\right)=\emph{AVar}\left(\hat{\sigma}_{M1}^{2}\right)$.

\section{Random Effects and Moment Conditions\label{Random Effects Sec}}

In Section \ref{Initial Condition Sec} we draw inference on the structural
parameters by conditioning or by differencing the data based on the first
\emph{observed} initial condition $y_{i,1}$. Both methods are fixed-effect
approaches, which do not rely on any further assumptions on the \emph
unobserved} data such as $y_{i,0}$. We could instead consider random-effect
approaches which rely on assumptions for the unobserved value $y_{i,0}$.
These include Chamberlain's (correlated)\ random effects and
\citet{BlundellBond98}'s stationarity assumptions for the first unobserved
value $y_{i,0}$.

In this section, we compare the conditional method with the random-effect
methods using the first moment of
\begin{equation}
\left(x^{\prime}x\right)^{-1}x^{\prime}Y\text{ and }\frac{Y^{\prime}Y}{N}.
\label{eq:Maximal-Invariant-2}
\end{equation}
Define the function
\begin{equation*}
\pi\left(\gamma,\delta_{\tau,x}\right)=a\left(\gamma\right)+\delta_
\tau,x}.b\left(\gamma\right).
\end{equation*}
The expectation of the maximal invariant is then given by
\begin{eqnarray*}
E\text{ }\left(x^{\prime}x\right)^{-1}x^{\prime}Y & = &
\pi\left(\gamma,\delta_{\tau,x}\right)\text{ and} \\
E\text{ }\frac{Y^{\prime}Y}{N} & = & \sigma^{2}\left[\pi\left(\gamma
\delta_{\tau,x}\right)^{\prime}\omega_{x}^{2}\pi\left(\gamma,\delta_{\tau,x
\right)+b\left(\gamma\right)^{\prime}\omega_{\tau,x}^{2}b\left(\gamma\right
\right]+c\left(\gamma\right)^{\prime}c\left(\gamma\right),
\end{eqnarray*}

For example, \citet{Lancaster02}'s\ differencing approach yields
\begin{equation*}
E\text{ }\frac{\widetilde{Y}_{T}^{\prime }\widetilde{Y}_{T}}{N}=\sigma
^{2}B_{T}\left[ \omega _{\widetilde{\eta }}^{2}1_{T}1_{T}^{\prime }+I_{T
\right] B_{T}^{\prime }.
\end{equation*
Hence, we have $\left( T+1\right) T/2$ moments and three unknown parameters:
$\rho $, $\sigma ^{2}$, and $\omega _{\widetilde{\eta }}^{2}$. Conditioning
on $y_{1}$ yields the following (conditional)\ moments based on the maximal
invariant:
\begin{equation}
E\left[ \left. \left( y_{1}^{\prime }y_{1}\right) ^{-1}y_{1}^{\prime
}Y_{T}\right\vert y_{1}\right] =\rho e_{1}^{\prime }B_{T}^{\prime }+\delta
_{\eta ,y_{1}}.1_{T}^{\prime }B_{T}^{\prime }  \label{eq:ConditionalMoment1}
\end{equation
and
\begin{eqnarray}
&\hspace{0.01in}&E\left[ \left. \frac{Y_{T}^{\prime }Y_{T}}{N}\right\vert
y_{1}\right] =\sigma ^{2}B_{T}\left\{ \omega _{\eta
,y_{1}}^{2}1_{T}1_{T}^{\prime }+I_{T}\right\} B_{T}^{\prime }
\label{eq:ConditionalMoment2} \\
&\hspace{0.01in}&+\text{ }\sigma ^{2}\left[ \rho B_{T}e_{1}+\delta _{\eta
,y_{1}}.B_{T}1_{T}\right] \omega _{y_{1}}^{2}\left[ \rho e_{1}^{\prime
}B_{T}^{\prime }+\delta _{\eta ,y_{1}}.1_{T}^{\prime }B_{T}^{\prime }\right]
.  \notag
\end{eqnarray
The unknown parameters are the autoregressive coefficient $\rho $, the error
variance $\sigma ^{2}$, the OLS coefficient $\delta _{\eta ,y_{1}}$, and the
standardized squared residuals $\omega _{\eta ,y_{1}}^{2}$.

Invariance reduces the information to $T+\left( T+1\right) T/2=\left(
T+1\right) \left( T+2\right) /2-1$ moment conditions which depend on only
four parameters. Conditioning on the first observation yields $T$ more
moments than \citet{Lancaster02}'s differencing method and one additional
parameter (four instead of three). This comparison helps to explain the
efficiency gains from conditioning instead of differencing.

We can then explore the conditional moments (\ref{eq:ConditionalMoment1})
and (\ref{eq:ConditionalMoment2}) to find Minimum Distance (MD) estimators.
We further note that \citet{ArellanoBond91} and \citet{AhnSchmidt95}
implicitly use linear combinations of the moments (\re
{eq:ConditionalMoment1}) and (\ref{eq:ConditionalMoment2}) to find moments
for the autoregressive parameter $\rho$.

We instead explore connections between these moments, conditional on the
first observed value $y_{i,1}$, and random effects assumptions for the
unobserved value $y_{i,0}$. To simplify comparison, we continue working with
the model without covariates and with homoskedastic errors.

\subsection{Unobserved Initial Conditions as Incidental Parameters}

As a preliminary to the correlated random effects assumption, consider the
model written recursively to the time $t=0$:
\begin{eqnarray}
Y_{T+1} & = & y_{0}.\rho
e_{1}^{\prime}B_{T+1}^{\prime}+\eta.1_{T+1}^{\prime}B_{T+1}^{\prime}
\sigma.U_{T+1}.B_{T+1}^{\prime},\text{ where}  \label{eq:Model-Levels-Matrix}
\\
U_{T+1} & \sim & N\left(0_{N\times\left(T+1\right)},I_{N}\otimes
I_{\left(T+1\right)}\right),  \notag
\end{eqnarray}
and $y_{0}$ is unobserved. This model is again a special case of the linear
setup of \citet{ChamberlainMoreira09} without covariates $x$. The incidental
parameter $\tau=\left
\begin{array}{cc}
y_{0} & \et
\end{array
\right]$ would encapsulate the individual fixed effects and the initial
conditions themselves. The regression coefficients would be given by
\begin{equation*}
b\left(\gamma\right)=\left
\begin{array}{c}
\rho e_{1}^{\prime} \\
1_{T+1}^{\prime
\end{array
\right]B^{\prime}\text{ and }c\left(\gamma\right)=\sigma.B^{\prime}
\end{equation*}
(where we again omit the subscript, here $\left(T+1\right)$, from the
matrices when there is no confusion).

The maximal invariant is $Y_{T+1}^{\prime }Y_{T+1}$, which contains $\left(
T+1\right) \left( T+2\right) /2$ nonredundant moments. Its expectation is
given by
\begin{equation*}
E\text{ }\frac{Y_{T+1}^{\prime }Y_{T+1}}{N}=\sigma ^{2}B_{T+1}\left\{ \left[
\begin{array}{cc}
\rho .e_{1} & 1_{T+1
\end{array
\right] \omega _{\tau }^{2}\left[
\begin{array}{c}
\rho .e_{1}^{\prime } \\
1_{T+1}^{\prime
\end{array
\right] +I_{T+1}\right\} B_{T+1}^{\prime }.
\end{equation*
The total number of parameters are five: the autoregressive parameter $\rho
, the error variance $\sigma ^{2}$, and the three nonredundant elements of
\omega _{\tau }^{2}$. The parameters need to be well-behaved as the sample
size $N$ grows, and inference is based on the asymptotic normality of their
respective estimators. If different series start at different points in time
(as typically happens with firms' data), then the parameter $\omega
_{y_{0}}^{2}$ may not be well-behaved or its estimator may not be
asymptotically normal. For example, moment equations depend on terms such a
\begin{equation*}
\sum_{i=1}^{N}y_{i,0}u_{i,t}.
\end{equation*
This term may not be approximately normal if the bulk of observations
y_{i,0}$ are close to zero. This happens because the variance ratio of some
terms $y_{i,0}u_{i,t}$ to their sum
\begin{equation*}
\max_{j\leq N}\frac{Var\left( y_{j,0}u_{j,t}\right) }{\sum_{i=1}^{N}Va
\left( y_{i,0}u_{i,t}\right) }
\end{equation*
may not be negligible.\footnote
The Lindeberg condition holds if and only if the series is approximately
normal and the terms are asymptotically negligible.} This problem is
mitigated by shocks over time, hence conditioning on the observations
y_{i,1}$ is more robust.

\subsection{The Correlated Random Effects Estimator}

For us to make a connection to the CRE estimator, we need to include the
vector of ones in $x=1_{N}$. The reason is that the (correlated) random
effects assumption
\begin{equation*}
\left[
\begin{array}{cc}
y_{0} & \et
\end{array
\right] |x\sim N\left( x.\iota ,I_{N}\otimes \Phi \right) \text{, where
\iota =\left( \iota _{1},\iota _{2}\right) .
\end{equation*
would need to be invariant to our group of transformations to be decomposed
into an invariant uniform prior and an additional term (and we want to allow
the random effects to have a nonzero mean even without additional
regressors).

The setup is the same as treating the initial condition as an incidental
parameter. However, we include $x=1_{N}$ and define the regression
coefficient on the vector of ones, as in Section 7 of
\citet{ChamberlainMoreira09}. The maximal invariant is
\begin{equation*}
1_{N}^{\prime}Y_{T+1}\text{ and }Y_{T+1}^{\prime}Y_{T+1}.
\end{equation*}
Their (conditional on $\tau=\left
\begin{array}{cc}
y_{0} & \et
\end{array
\right]$) expectation is given by
\begin{eqnarray*}
E\left[1_{N}^{\prime}Y_{T+1}\right] & = & \delta_{\tau,1_{N}}\left
\begin{array}{c}
\rho.e_{1}^{\prime} \\
1_{T+1}^{\prime
\end{array
\right]B_{T+1}^{\prime}\text{ and} \\
E\left[\frac{Y_{T+1}^{\prime}Y_{T+1}}{N}\right] & = & B_{T+1}\left\{ \left
\begin{array}{cc}
\rho.e_{1} & 1_{T+1
\end{array
\right]\left(\sigma^{2}\omega_{\tau,1_{N}}^{2}+\delta_{\tau,1_{N}}^{\prime
\delta_{\tau,1_{N}}\right)\left
\begin{array}{c}
\rho.e_{1}^{\prime} \\
1_{T+1}^{\prime
\end{array
\right]+\sigma^{2}.I_{T+1}\right\} B_{T+1}^{\prime}.
\end{eqnarray*}
The unknown parameters are the autoregressive coefficient $\rho$, the error
variance $\sigma^{2}$, the sample averages $\delta_{\tau,1_{N}}$, and
standardized squared deviations $\omega_{\tau,1_{N}}^{2}$. Hence, the
invariance argument reduces the data space to $T+1+\left(T+1\right)\left(T+
\right)/2$ moment conditions, which depend on seven parameters. The
parameter $\omega_{\tau,1_{N}}^{2}$ may not be well-behaved and its
estimator may not be asymptotically normal.

Under the random effects assumption, the model becomes

\begin{equation*}
Y_{T+1}=1_{N}.a\left(\gamma\right)+U_{T+1}.c\left(\gamma\right),
\end{equation*}
where the coefficients are given by
\begin{eqnarray*}
a\left(\gamma\right) & = & \iota\left
\begin{array}{c}
\rho e_{1}^{\prime} \\
1_{T+1}^{\prime
\end{array
\right]B_{T+1}^{\prime}\text{ and} \\
c\left(\gamma\right)^{\prime}c\left(\gamma\right) & = & B_{T+1}\left\{ \left
\begin{array}{cc}
\rho.e_{1} & 1_{T+1
\end{array
\right]\Phi\left
\begin{array}{c}
\rho e_{1}^{\prime} \\
1_{T+1}^{\prime
\end{array
\right]+\sigma^{2}I_{T+1}\right\} B_{T+1}^{\prime}.
\end{eqnarray*}

The (unconditional) expectation of the maximal invariant has the same
functional form as the moments conditional on $\tau=\left
\begin{array}{cc}
y_{0} & \et
\end{array
\right]$:
\begin{eqnarray*}
E\left[\frac{1_{N}^{\prime}Y_{T+1}}{N}\right] & = & \iota\left
\begin{array}{c}
\rho.e_{1}^{\prime} \\
1_{T+1}^{\prime
\end{array
\right]B_{T+1}^{\prime}\text{ and} \\
E\left[\frac{Y_{T+1}^{\prime}Y_{T+1}}{N}\right] & = & B_{T+1}\left\{ \left
\begin{array}{cc}
\rho.e_{1} & 1_{T+1
\end{array
\right]\left(\Phi+\iota^{\prime}\iota\right)\left
\begin{array}{c}
\rho.e_{1}^{\prime} \\
1_{T+1}^{\prime
\end{array
\right]+\sigma^{2}.I_{T+1}\right\} B_{T+1}^{\prime}.
\end{eqnarray*}
So the criticism on lack of robustness to the data-generating process for
y_{i,0}$ is applicable here as well.

\subsection{Blundell and Bond (1998)}

\citet{BlundellBond98} instead consider a different assumption, that $y_{0}$
does not deviate systematically from the stationary mean
\eta/\left(1-\rho\right)$. We start with the assumption
\begin{equation*}
y_{0}\sim N\left(\frac{\eta}{1-\rho},\sigma_{0}^{2}.I_{N}\right).
\end{equation*}
We assume normality for the purposed of invariance. However, only the mean
and variance are relevant for the expectation calculations derived below.

Under this assumption, the model is equivalent to
\begin{equation*}
Y_{T+1}=\eta.b\left(\gamma\right)+U_{T+1}.c\left(\gamma\right),
\end{equation*}
where the coefficients are given by
\begin{eqnarray*}
b\left(\gamma\right) & = & 1_{T+1}^{\prime}B_{T+1}^{\prime}+\frac{\rho}
1-\rho}e_{1}^{\prime}B_{T+1}^{\prime}\text{ and} \\
c\left(\gamma\right)^{\prime}c\left(\gamma\right) & = &
B_{T+1}\left(\sigma_{0}^{2}\rho^{2}e_{1}e_{1}^{\prime}+\sigma^{2}I_{T+1
\right)B_{T+1}^{\prime}.
\end{eqnarray*}
The maximal invariant is given by $Y_{T+1}^{\prime}Y_{T+1}$:
\begin{eqnarray*}
& \hspace{0.01in} & E\text{ }\frac{Y_{T+1}^{\prime}Y_{T+1}}{N
=B_{T+1}\left(\sigma_{0}^{2}\rho^{2}e_{1}e_{1}^{\prime}+\sigma^{2}I_{T+1
\right)B_{T+1}^{\prime} \\
& \hspace{0.01in} & +\sigma^{2}\omega_{\eta}^{2}B_{T+1}\left
1_{T+1}1_{T+1}^{\prime}+\frac{\rho}{1-\rho}\left(e_{1}1_{T+1}^
\prime}+1_{T+1}e_{1}^{\prime}\right)+\frac{\rho^{2}}{\left(1-\rho\right)^{2}
e_{1}e_{1}^{\prime}\right]B_{T+1}^{\prime}.
\end{eqnarray*}
This expectation depends on four parameters: $\rho$, $\sigma^{2},$
\omega_{\eta}^{2}$, and $\sigma_{0}^{2}$.

We can re-arrange some of these $\left(T+1\right)(T+2)/2$ moments so that
the expectation is zero at the true autoregressive coefficient $\rho$. For
example,
\begin{equation}
E\left[\left(y_{i,2}-y_{i,1}\right).\left(y_{i,3}-\rho y_{i,2}\right)\right
=0.  \label{eq:BB98}
\end{equation}
As \citet{BlundellBond98} also note, this expectation may fail to be zero if
the stationarity assumption breaks down. For example, this moment condition
breaks down if either $y_{i,1}=k$ or even if $y_{i,1}\overset{iid}{\sim
N\left(0,\sigma_{0}^{2}\right)$.

It is worthwhile making a connection between the stationarity assumption and
inference conditional on the first observations. Since inference is
conditional on $y_{1}$, we could stack the quantities together and look at
the (conditional) expectation of
\begin{equation*}
Y_{T+1}^{\prime}Y_{T+1}=\left
\begin{array}{cc}
y_{1}^{\prime}y_{1} & y_{1}^{\prime}Y_{T} \\
Y_{T}^{\prime}y_{1} & Y_{T}^{\prime}Y_{T
\end{array
\right].
\end{equation*}
As in the derivation based on that of \citet{BlundellBond98}, we have
exactly the same quantity $Y_{T+1}^{\prime}Y_{T+1}$. However, we have five
parameters (if we consider $y_{1}^{\prime}y_{1}$ itself to be the
parameter). Alternatively, we lose one moment (from removing
y_{1}^{\prime}y_{1}$ itself) and four parameters. If the stationarity
assumption is correct, there should be efficiency loss from making inference
conditional on the first observation. On the other hand, conditional
inference should be robust to different data-generating process for the
initial data values.

\section{Conclusion}

This paper studies the Bayesian estimator of \citet{Lancaster02}, which is
invariant to natural rotations of the data. The likelihood of the maximal
invariant can be divided into two parts, one of which is maximized to obtain
Lancaster's estimator. The second part is not asymptotically negligible and,
as such, can be used to attain a more efficient estimator. A natural
conclusion is that it is unnecessary to use the data to choose the correct
inflection point of Lancaster's score likelihood. \citet{Lancaster02}'s
theory is based on differencing the data using the first observation.
Instead, we can condition on the first observation itself to further improve
efficiency. The conditional argument is essentially a fixed-effects approach
in which we make no further assumptions on unobserved data.

Current practice in economics uses standard GMM methods based on data
differencing; e.g., \citet{AthanasoglouBrissimisDelis08} on bank
profitability, \citet{GuisoPistaferriSchivardi05} on risk allocation,
\citet{KoningsVandenbussche05} on antidumping protection effects, and
\citet{TopalovaKhandelwal11} on tariffs' change on firm productivity, among
others. Instead, we advocate using moments based on invariant statistics
from a model that allows for heteroskedasticity and factor structure. It is
relatively straightforward to extend our method to allow for covariates in
the linear model of \citet{ChamberlainMoreira09}. \citet{Bai13} allows for
general time-series heteroskedasticity as opposed to assuming specific
moving average (MA) processes and data values lagged enough as instruments.
\citet{ChamberlainMoreira09} and \citet{MoonWeidner15} suggest including
factors which generalize individual and time effects. Conditioning on the
first observation can then provide reliable inference in dynamic panel data
models using moments based on invariant statistics for more general models.

\addcontentsline{toc}{section}{\refname}
\bibliographystyle{econometrica}

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text:
Giacomo Valentini (born May 16, 1911 in Rome) was an Italian professional football player.

He played 6 games and scored 1 goal in the Serie A 1935/36 season for A.S. Roma.

1911 births
Year of death missing
Italian footballers
Serie A players
A.S. Roma players
Association football midfielders

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The typical aeroelastic analysis for a complex configuration such as a complete aircraft was done using the aerodynamic results of the wing and the structural modes of a complete aircraft; that is, the aerodynamics of a wing of a complete aircraft is assumed to be not much influenced by the body shape. Nevertheless, the body shape can cause a distortion of aerodynamic pressure on the wing surface and it is necessary to investigate the body effect in flutter analysis. In this reseasrch, MGM inverse design method is applied to include the body effect of a wing-body model which disturbs the pressure distribution on the wing surface.
Malone, J. B., Narramore, J. C., and Sanker, L. N., 1990, "Airfoil Design Method Using the Navier-Stokes Equation", Journal of Aircraft, Vol. 28, No. 3.
Soemarwoto, B. I., Labrujèere, Th.E., Laban, M., and Yansyah H., 1980, "Inverse Aerodynamic Shape Design for Improved Wing Buffet-onset Performance", NLR-TP-2000-150.
Kim, D. H., 2000, "Transonic/Supersonic Flutter Analysis of Wings with Control Surface and External Store", Ph'D Thesis, KAIST, Korea.
Lee, S. J., Lee, I., and Han, J. H., 2006, "Transonic Flutter Suppression of the 2-D Flap Wing with External Store using CFD-based Aeroservoelasticity ", KASA International Journal, Vol. 1, No. 2.
Donald, L. L. and Bruce B. E.,1951, Research Memorandum, RM151F07.
Sung, C. H., Park, S. H., and Kwon, J. H., 2001, "Multigrid Diagonalized ADI Mehod for Compressible Flows," AIAA Paper 2001-2556.


text:
Title: Kathy Baker
Subject: Primetime Emmy Award for Outstanding Lead Actress in a Drama Series, Edie Falco, Julianna Margulies, Gillian Anderson, Claire Danes
Collection: 1950 Births, 20Th-Century American Actresses, 21St-Century American Actresses, Actresses from New Mexico, Actresses from Texas, American Film Actresses, American People of French Descent, American Stage Actresses, American Television Actresses, Best Drama Actress Golden Globe (Television) Winners, Living People, Outstanding Performance by a Female Actor in a Drama Series Screen Actors Guild Award Winners, Outstanding Performance by a Lead Actress in a Drama Series Primetime Emmy Award Winners, People from Midland, Texas, University of California, Berkeley Alumni
Kathy Baker at the 45th Emmy Awards, in September 1993
Katherine Whitton Baker
(1950-06-08) June 8, 1950
Midland, Texas, U.S.
Donald Camillieri (m. 1985–99)
Steven Robman (m. 2003)
Katherine Whitton "Kathy" Baker (born June 8, 1950) is an American stage, film and television actress, known for her leading and supporting performances in film and television.
Baker began her career in theatre and made her screen debut in the 1983 drama film, The Right Stuff. She received the National Society of Film Critics Award for Best Supporting Actress and an Independent Spirit Award nomination for her performance in Street Smart (1987).[1] Baker also has appeared in over 50 films, including Edward Scissorhands (1990), The Cider House Rules (1999), Cold Mountain (2003), Nine Lives (2005), The Jane Austen Book Club (2007), Last Chance Harvey (2008), Take Shelter (2011), Saving Mr. Banks (2013) and The Age of Adaline (2015).
On television, Baker starred as Dr. Jill Brock in the CBS drama series, Picket Fences (1992-1996), for which she received three Primetime Emmy Award for Outstanding Lead Actress in a Drama Series and Golden Globe and Screen Actors Guild Awards. She later received three additional nominations and a Primetime Emmy Award for her performances in Touched by an Angel, Boston Public and Door to Door.
Personal life 3
Filmography 4
Film 4.1
Television 4.2
Awards and nominations 5
Baker was born in Midland, Texas, and raised a Quaker[2] in New Mexico, the daughter of John Seawand Baker, a geologist and educator, and his French-born wife, Helene Andree (née Whitton).[3] She studied acting at the California Institute of the Arts in the early 1970s. She later earned a B.A. degree in French in 1977 from UC Berkeley.[4]
Baker began her acting career at San Francisco's Magic Theatre, performing in several of Sam Shepard's plays before getting her break in an off-Broadway production of Fool for Love (1983) opposite Ed Harris. She won an Obie Award for this role.[4] In same year she was cast as Louise Shepard, Alan Shepard's wife, in the drama film The Right Stuff.[4] She later had dramatic performances as a prostitute in Street Smart (1987) and a recovering alcoholic and victim of domestic abuse in Clean and Sober (1988).[1] For her performance in Street Smart, Baker has won National Society of Film Critics Award for Best Supporting Actress and Boston Society of Film Critics Award for Best Supporting Actress as well as an Independent Spirit Award for Best Supporting Female nomination.[1] She later starred in films Dad, Jacknife and Edward Scissorhands.
Baker at the 2007 Toronto International Film Festival
From 1992 to 1996, Baker starred as Jill Brock, a small-town doctor, in the CBS critically acclaimed drama series, Picket Fences created by David E. Kelley. For her performance in the series she won the Primetime Emmy Award for Outstanding Lead Actress in a Drama Series three times: in 1993, 1995 and 1996, Golden Globe Award for Best Actress – Television Series Drama in 1994, and Screen Actors Guild Award for Outstanding Performance by a Female Actor in a Drama Series in 1995.[4] She later appeared in David E. Kelley's dramas Ally McBeal, The Practice and had the recurring role as Meredith Peters in Boston Public, for which she was nominated for a Primetime Emmy Award for Outstanding Guest Actress in a Drama Series in 2001. Baker also received Emmy Award nominations for her guest performance in Touched by an Angel and in the category Outstanding Supporting Actress in a Miniseries or a Movie for Door to Door.[4]
Baker has appeared in the over 50 theatrical films in supporting and leading roles. She starred in two movies directed by Rodrigo García: Things You Can Tell Just by Looking at Her (2000) and Nine Lives (2005). Her other major credits include The Cider House Rules (1999), Cold Mountain (2003), 13 Going on 30 (2004), The Jane Austen Book Club (2007), Last Chance Harvey (2008), Take Shelter (2011) and Saving Mr. Banks (2013). She starred opposite Tom Selleck in the Jesse Stone television films series in 2000s. Also on television she guest-starred on Nip/Tuck, Law & Order: Special Victims Unit, Grey's Anatomy, and Medium. She also starred in the short-lived Lifetime drama series Against the Wall in 2011.[5]
She lives in Southern California with her husband, director/producer Steven Robman. She has a son and daughter with her first husband, Donald Camillieri.[4]
1983 The Right Stuff Louise Shepard
1986 A Killing Affair Maggie Gresham
1987 Street Smart Punchy
1988 Permanent Record Martha Sinclair
1988 Clean and Sober Charlie Standers
1989 Dad Annie
1989 Jacknife Martha Flannigan
1990 The Image Marcie Guilford
1990 Edward Scissorhands Joyce
1990 Mister Frost Dr. Sarah Day
1991 One Special Victory Ellen Television film
1992 Article 99 Dr. Diana Walton
1992 Jennifer Eight Margie Ross
1993 Mad Dog and Glory Lee
1993 Lush Life Janice Oliver Television film
1996 To Gillian on Her 37th Birthday Esther Wheeler
1997 Inventing the Abbotts Helen Holt
1997 Not in This Town Tammy Schnitzer Television film
1997 Weapons of Mass Distraction Margo Powers
1998 Oklahoma City: A Survivor's Story Priscilla Salyers Television film
1999 The Cider House Rules Nurse Angela
1999 A Season for Miracles Ruth Doyle Television film
1999 A Little Inside Nancy
1999 Shake, Rattle and Roll: An American Love Story Janice Danner Television film
2000 Things You Can Tell Just by Looking at Her Rose
2000 Ratz Doris Trowbridge Television film
2001 The Glass House Nancy Ryan
2001 Ten Tiny Love Stories Ten
2002 Assassination Tango Maggie
2002 Door to Door Gladys Sullivan Television film
2003 Cold Mountain Sally Swanger
2004 13 Going on 30 Bev Rink
2005 Spring Break Shark Attack Mary Jones Television film
2005 Nine Lives Camille
2006 All the King's Men Mrs. Burden
2007 The Jane Austen Book Club Bernadette
2007 Jesse Stone: Sea Change Rose Gammon Television film
2008 Last Chance Harvey Jean
2008 Shades of Ray Janet Rehman
2009 Unstable Betty Walker Television film
2009 Jesse Stone: Thin Ice Rose Gammon Television film
2010 Miss Nobody Claire McKinney
2010 Jesse Stone: No Remorse Rose Gammon Television film
2011 Good Day for It Rose Carter
2011 Take Shelter Sarah
2011 Jesse Stone: Innocents Lost Rose Gammon Television film
2011 Too Big to Fail Wendy Paulson
2011 Seven Days in Utopia Mabel
2011 Machine Gun Preacher Daisy
2012 Big Miracle Ruth McGraw
2012 Jesse Stone: Benefit of the Doubt Rose Gammon Television film
2012 The Party Is Over Appoline
2013 Saving Mr. Banks Tommie
2013 The Trials of Cate McCall Therapist
2014 Return to Zero Kathleen Callaghan
2014 Boulevard Joy Mack
2014 Model Home Brenda
2015 The Age of Adaline Connie Jones
2015 Midnight Rider
1992–1996 Picket Fences Dr. Jill Brock 87 episodes
1997 Ally McBeal Katherine Dawson Episode: "The Affair"
1998 The Practice Evelyn Mayfield 2 episodes
1999 Hey Arnold! Dr. Bliss Episode: "Helga on the Couch"
2000 Touched by an Angel Ellen Episode: "Buy Me a Rose"
2000 Chicago Hope Mary Wyzinski Episode: "Everybody's Special at Chicago Hope"
2001–2002 Boston Public Meredith Peters 14 episodes
2004 Monk Sylvia Fairbourn Episode: "Mr. Monk Goes to Jail"
2005 Nip/Tuck Gail Pollack 2 episodes
2005–2010 Medium Marjorie Dubois 5 episodes
2007 Law & Order: Special Victims Unit Hannah Curtis Episode: "Sin"
2007 Gilmore Girls Mia Episode: "Gilmore Girls Only"
2008 Grey's Anatomy Anna Loomis 2 episodes
2009 Law & Order: Criminal Intent Camille Hayes-Fitzgerald Episode: "Playing Dead"
2010 Law & Order Camille Peterson Episode: "Crashers"
2011 Against the Wall Sheila Kowalski 12 episodes
2011 Criminal Minds Linda Collins Episode: "A Family Affair"
2012 Private Practice Dianne Episode: "I'm Fine"
2014 Those Who Kill Marie Burgess 5 episodes
2015 Big Time in Hollywood, FL Diana 10 episodes
1984 Obie Award Best Performance by an Actress Fool for Love Won
1984 Theatre World Award Theatre World Award Fool for Love Won
1988 Boston Society of Film Critics Award Best Supporting Actress Street Smart Won
1988 Independent Spirit Award Best Supporting Female Street Smart Nominated
1988 National Society of Film Critics Award Best Supporting Actress Street Smart Won
1993 Primetime Emmy Award Outstanding Lead Actress in a Drama Series Picket Fences Won
1994 Golden Globe Award Best Actress – Television Series Drama Picket Fences Won
1994 Primetime Emmy Award Outstanding Lead Actress in a Drama Series Picket Fences Nominated
1994 Viewers for Quality Television Best Actress in a Quality Drama Series Picket Fences Won
1995 Golden Globe Award Best Actress – Television Series Drama Picket Fences Nominated
1995 Screen Actors Guild Award Outstanding Performance by a Female Actor in a Drama Series Picket Fences Won
1995 Screen Actors Guild Award Outstanding Performance by an Ensemble in a Drama Series Picket Fences Nominated
2000 Primetime Emmy Award Outstanding Guest Actress in a Drama Series Touched by an Angel Nominated
2000 Screen Actors Guild Award Outstanding Performance by a Cast in a Motion Picture The Cider House Rules Nominated
2001 Primetime Emmy Award Outstanding Guest Actress in a Drama Series Boston Public Nominated
2003 Primetime Emmy Award Outstanding Supporting Actress in a Miniseries or a Movie Door to Door Nominated
2005 Gotham Awards Best Cast Nine Lives Nominated
2005 Locarno International Film Festival Best Actress Nine Lives Won
2011 Gotham Awards Best Cast Take Shelter Nominated
2013 Phoenix Film Critics Society Award Best Cast Saving Mr. Banks Nominated
^ a b c "Kathy Baker - Biography". The New York Times. Retrieved 29 July 2014.
^ Tavis Smiley interview archive site; interview date: September 25, 2007
^ "Kathy Baker profile at Film Reference.com". Filmreference.com. Retrieved 2012-03-01.
^ a b c d e f "Kathy Baker Biography". TV Guide. Retrieved 29 July 2014.
^ "Against the Wall: Cancelled; No Season Two". TVSeriesFinale.com. December 10, 2011. Retrieved 29 July 2014.
Kathy Baker at the Internet Movie Database
Kathy Baker at AllMovie
Kathy Baker 2003 Interview on Sidewalks Entertainment
Awards for Kathy Baker
Boston Society of Film Critics Award for Best Supporting Actress
Mona Washbourne (1981)
Linda Hunt (1983)
Dianne Wiest (1986)
Kathy Baker (1987)
Joan Cusack (1988)
Brenda Fricker (1989)
Jennifer Jason Leigh (1990)
Rosie Perez (1993)
Courtney Love (1996)
Sarah Polley (1997)
Chloë Sevigny (1999)
Cameron Diaz (2001)
Toni Collette (2002)
Patricia Clarkson (2003)
Laura Dern/Sharon Warren (2004)
Catherine Keener (2005)
Shareeka Epps (2006)
Amy Ryan (2007)
Mo'Nique (2009)
Juliette Lewis (2010)
June Squibb (2013)
Primetime Emmy Award for Outstanding Lead Actress in a Drama Series
Barbara Bain (1967)
Susan Hampshire (1970)
Michael Learned (1973)
Jean Marsh (1975)
Lindsay Wagner (1977)
Mariette Hartley (1979)
Barbara Bel Geddes (1980)
Barbara Babcock (1981)
Tyne Daly (1983)
Sharon Gless (1986)
Dana Delany (1989)
Patricia Wettig (1990)
Sela Ward (1994)
Gillian Anderson (1997)
Christine Lahti (1998)
Edie Falco (1999)
Allison Janney (2002)
Patricia Arquette (2005)
Mariska Hargitay (2006)
Kyra Sedgwick (2010)
Julianna Margulies (2011)
Golden Globe Award for Best Actress – Television Series Drama
Linda Cristal (1969)
Peggy Lipton (1970)
Gail Fisher (1972)
Lee Remick (1973)
Susan Blakely (1976)
Lesley Ann Warren (1977)
Natalie Wood (1979)
Yoko Shimada (1980)
Linda Evans/Barbara Bel Geddes (1981)
Joan Collins (1982)
Angela Lansbury (1984)
Susan Dey (1987)
Jill Eikenberry (1988)
Sharon Gless/Patricia Wettig (1990)
Regina Taylor (1992)
Jane Seymour (1995)
Keri Russell (1998)
Jennifer Garner (2001)
Frances Conroy (2003)
Anna Paquin (2008)
Katey Sagal (2010)
Robin Wright (2013)
Ruth Wilson (2014)
National Society of Film Critics Award for Best Supporting Actress
Marjorie Rhodes (1967)
Billie Whitelaw (1968)
Siân Phillips / Delphine Seyrig (1969)
Jeannie Berlin (1972)
Valentina Cortese (1973)
Bibi Andersson (1974)
Ann Wedgeworth (1977)
Sandra Bernhard (1983)
Jane Horrocks (1991)
Madeleine Stowe (1993)
Hanna Schygulla (2008)
Olivia Williams (2010)
Jessica Chastain (2011)
Screen Actors Guild Award for Outstanding Performance by a Female Actor in a Drama Series
Sandra Oh (2005)
Chandra Wilson (2006)
American film actresses
American stage actresses
Best Drama Actress Golden Globe (television) winners
Outstanding Performance by a Lead Actress in a Drama Series Primetime Emmy Award winners
Outstanding Performance by a Female Actor in a Drama Series Screen Actors Guild Award winners
American people of French descent
People from Midland, Texas
Actresses from Texas
Actresses from New Mexico
University of California, Berkeley alumni
Angela Lansbury, The Sopranos, Julianna Margulies, Claire Danes, 30 Rock


text:
Q: Zero downtime copy/restore of a PostgreSQL database I have a production server with static (no deletes/updates) PostgreSQL database, serving a huge amount of read queries.
Once a week or two I have to update my database with new data. Right now the process it pretty complicated, so I'm using a separate server to build new database from the scratch.
So, at some point I have live production server with outdated data and up to date database on some secondary server.
First I've tried backup/restore approach, but it resulted in 5 minutes downtime, which is not acceptable.
Then I've tried adding --single-transaction flag to pg_restore call, but it resulted in 2 minuted downtime, because all requests to PostgreSQL hanged for a couple of minutes after transaction was committed.
The best solution I came up with so far is to copy new database to production server under some new name and then switch my application to it.
But I wonder if there's any better solution, which will allow me to ship new data to production server without restarting my application.

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