I'm following a course in econometrics where we have established that for a linear regression model $$y_i=x_i'\beta+\varepsilon_i$$  If $1/N \sum_{i=1}^N x_i x_i'$ converges to a nonsingular matrix and $E[x_i \varepsilon_i]=0$ OLS will be consistent .   What I would like to know if it can happen that OLS is consistent and $E[x_i \varepsilon_i]\neq 0$ and if so this leads to some kind of degeneracy.  I'm not particular proficient in statistics so I hope I have tagged and formulated the question correctly - otherwise feel free to correct me.
Many econometrics textbooks (e.g. Wooldridge, "Econometric analysis...") simply write something similar to: "If the population model is $y = xB + u$ and (1) $\text{Cov}(X,U) = 0$; (2) $X'X$ is full rank, then OLS consistently estimate parameters $B$". I've worked out the math behind consistency but I'm a bit lost in interpreting the meaning.  We're talking about consistent estimation, but estimation of what? Please tell me if the following points are correct:   If we have random sample $X,Y$ and $X'X$ is invertible, then we can always define Best Linear Predictor of $y$ given $x$. And then OLS always consistently estimates coefficients of Best Linear Predictor (because in BLP we have $\text{Cov}(u,x)=0$ from the definition). Bottom line: we can always interpret OLS estimates as coefficients of BLP The only question is whether BLP corresponds to conditional expectation $\text{E}(y|x)$. If it does (for which we need $\text{E}(u|x) = 0$), then we can interpret OLS estimates as partial effects.   What is wrong with this or what am I missing? I don't get the point of stating the assumption of $\text{Cov}(u,x) = 0$, if this assumption is by definition fulfilled in case of Best Linear Predictor. And if we want to give structural interpretation to BLP coefficients (partial effects) we need stronger assumption of $\text{E}(u|x) = 0$ anyway. Is there any other interpretation we may have with zero covariance?
Office 2016 Advanced Setup Word for a modified version of a quote Why is this function continuous, unlike the Dirichlet function? Let $f$ be a function of bounded variation on $[0,1]$. Let $\{[a_n,b_n]\}_{n=1}^\infty$  such that $(a_n,b_n)$ are pairwise disjoint and $\cup_{n=1}^\infty [a_n,b_n]=[0,1]$. (for example, $[1/2, 1], [0,1/3], [1/3,1/3+1/3^2], \cdots$ ) Can we write  $$ \operatorname{Var}_{[0,1]} f=\sum_{k=1}^\infty \operatorname{Var}_{[a_k,b_k]} f? $$ Are there any banned subjects on Area 51? I know of the following rules for creating a proposal on Area 51:  The scope must be reasonably well-defined, not too unclear (e.g. &quot;cool stuff&quot;), too opinion-based (e.g. &quot;car reviews&quot;), too broad (e.g. &quot;life on Earth&quot;), or too narrow (e.g. &quot;The 1995 academic misconduct scandal at East Oak Mews High School&quot;). One  come to Area 51 with an already existing community rather than hope to build one on Area 51. Proposals that are subsets of existing sites are not acceptable (e.g.  — deleted). These get closed as &quot;This subject is already well-covered by a live Stack Exchange site. We generally do not split off subjects simply to give them their own space.&quot; Proposals for non-English sites .  Other than the criteria above, are there any per se restrictions on site content for an Area 51 proposal? If so, what topics are currently banned? I would expect a proposal for a &quot;Let's organize a lawsuit against Stack Exchange&quot; or &quot;[Specific SE staff member] sucks, amirite?&quot; would get closed for obvious reasons, but I'm interested if there are any non-obvious topics that are not currently allowed in proposals regardless of how well-defined, well-scoped, original, and well-supported by a community they are. Having a list of banned topics would be helpful to potential proposers as well as the communities that back them. For example, I wouldn't want to go to all the effort of organizing a community, bringing them to Area 51, etc., only to have my proposal closed with a curt &quot;Sorry, due to the political climate, we are not currently accepting proposals for sites about real-life weapons.&quot; custom closure. To be clear, I am asking solely about topic restrictions (e.g. &quot;no sites about fishing, hunting, or weapons&quot;). I'm not asking about clarity requirements, the requirement to have a pre-existing community, not duplicating an existing site, or the requirement to not use sock puppets to artificially inflate proposal stats. Those requirements are already well-defined elsewhere or else are common sense. Also to be clear, I don't have a specific proposal idea in mind right now or the community to make it happen. I'm asking in general. The question could be rephrased as &quot;What are those topics where I shouldn't even bother to try to define, gather a community, etc. because any attempt to propose a site will be summarily closed due to covering a banned topic?&quot; This is not a duplicate of  because that question is about the appropriateness of expanding a proposal to cover things that are acceptable proposal topics but arguably don't make sense in combination with the original proposal scope. My question is about topics that are completely banned from proposals regardless of whether they are included in the original proposal description or tacked on by the community later. Prove that $f(X)$ and $g(Y)$ are independent if $X$ and $Y$ are independent Let $X$ and $Y$ be independent random variables. Prove that $f(X)$ and $g(Y)$ are independent for any choice of measurable functions $f$ and $g$.   This sounds very obvious, but I have no idea how to approach it.  EDIT: Two random variables are independent if $\Pr\{X = x \text{ and } Y = y\} = \Pr\{X = x\} \cdot \Pr\{Y = y\}$ Earth moves how much under my feet when I jump? If I'm standing at the equator, jump, and land 1 second later, the  Earth does NOT move 1000mph (or .28 miles per second) relative to me,  since my velocity while jumping is also 1000mph.   However, the Earth is moving in a circle (albeit a very large one),  while I, while jumping, am moving in a straight line.   How much do I move relative to my starting point because of this? I  realize it will be a miniscule amount, and not noticeable in practise,  but I'd be interested in the theoretical answer. Export each image from a collection in Google Earth Engine Universal Chord Theorem Before Mavericks, I could use /etc/launchd.conf file to change maximum system resource consumption, for example:  limit maxfiles 16384 unlimited limit maxproc 16384 unlimited   It no longer works in Mavericks.  What is the correct way to do it in the recent version of OS X? Why is it that they decided to make String immutable in Java and .NET (and some other languages)? Why didn't they make it mutable? If $|G| = pq$,  $p,q$ primes, $p \gt q, q \nmid p-1 $, then how do I prove $G$ is cyclic without using Sylow's theorems? Probably dumb question, but I would like to ask it anyway.  I was to solve this equation:  $z^4 = -16$  At first glance, the way to solve it would be (as any other equation):  $z^2 = \sqrt{-1} * 4 \lor z^2 = -\sqrt{-1} * 4$  $z^2 = 4i \lor z^2 = -4i$  $z = 2\sqrt{i} \lor z = -2\sqrt{i} \lor z = 2i\sqrt{i} \lor z = -2i\sqrt{i}$  However, in suggested answers, there are $z_1, z_2, z_3$ and $z_4$ found using trigonometric formulas. Is there something particulary wrong in my solution and the only proper one uses trigonometric form of complex number? Cleft sentences Let's assume that John gave me a cat. I can rephrase the fact with:     What John did was to give me a cat   What John did was give me a cat   What John did was, he gave me a cat   But can I say the following phrase?  What John did was gave me a cat Points at integer distance How many points can one can place in $\mathbb{R}^n$, with the requirement that no $n+1$ points lie in the same $\mathbb{R}^{n-1}$-plane, and the euclidean distance between every two points is an integer? I use Chrome as my main browser.  I also spend a lot of time on a particular webapp that my company uses.  I'd like to keep those two things separate.  I can run the webapp in a separate window - but I'd like to go further than that.  I'd like to have it be identified by a different task in the Win 7 taskbar; ideally a different icon; and ideally a different title in Process Monitor.  How can I run a second, independent instance of Chrome, with a different title and icon, that won't merge with my main Chrome browser. Difference between $:=$ and $=$ I am sorry if this is quite elementary question. But I always think, that why we use $:=$ at some places, instead of $=$. Is there any fundamental difference between these two? Before reading Terry Tao's blog (4 months ago), I had never seen a symbol like this (:=). What caused the apocalypse in the Hunger Games? So the stories of the Hunger Games are set in , a post-apocalyptic society situated in North America. Is there anything to explain what caused the Apocalypse? Whether from the main books, any extra material or interviews? (No fanfic please.)  As well as the cause, is there any indication of when the apocalypse happened? And how long after the apocalypse are the events of the Hunger Games? Compare using Thread.Sleep and Timer for delayed execution I have a method which should be delayed running for a specified amount of time.  Should I use  Thread thread = new Thread(() =&gt; {     Thread.Sleep(millisecond);     action(); }); thread.IsBackground = true; thread.Start();   Or  Timer timer = new Timer(o =&gt; action(), null, millisecond, -1);   I had read some  about using Thread.Sleep is bad design. But I don't really understand why.  But for using Timer, Timer has dispose method. Since the execution is delayed, I don't know how to dispose Timer. Do you have any suggestions?  Or if you have alternative codes for delayed execution are also appreciate. Is password confirmation necessary on a Change Password dialog?