Why do we use a fixed design analysis of regression coefficients, even for observational data, where the design is not fixed?  For instance: $Var[\hat \beta]=(X'X)^{-1}\sigma^2$ is conditional on $X$. Since $X$ is random in observational studies, this is an under estimate of the true $Var[\hat \beta]$.   Edit: As pointed out by @christoph-hanck, $(X'X)^{-1}$ cannot be, by definition, systematically smaller than $\mathbb{E}(X'X)^{-1}$. Question remains: why do we use fixed design standard errors, instead of random design standard errors?
Sometimes we assume that regressors are fixed, i.e. they are non-stochastic. I think that means all our predictors, parameter estimates etc. are unconditional then, right? Might I even go so far that they are no longer random variables?  If on the other hand we accept that most regressors in economics say are stochastic because no outside force determined them with some experiment in mind. Econometricians then condition on these stochastic regressors.  How is this different from treating them as fixed?  I understand what conditioning is. Mathematically, it means we make all observations and inference conditional on that particular set of regressors and have no ambitions to say that inferences, parameter estimates, variance estimates etc. would have been the same had we seen a different realization of our regressors (such is the crux in time series, where each time series is only ever seen once).  However, to really grasp the difference between fixed regressors vs. conditioning on stochastic regressors, I am wondering if anyone here knows of an example of an estimation or inference procedure that is valid for say fixed regressors but breaks down when they are stochastic (and will be conditioned on).  I am looking forward to seeing those examples!
VirtualBox error kernel headers I'm with Ubuntu 15.04 and I'm trying to create an emulator (genymotion), but when run virtualbox, i got the error:     Stopping VirtualBox kernel modules ...done.   Uninstalling old VirtualBox DKMS kernel modules ...done.   Trying to register the VirtualBox kernel modules using DKMSError! Your kernel headers for kernel 3.16.0-30-generic cannot be found.   Please install the linux-headers-3.16.0-30-generic package,   or use the --kernelsourcedir option to tell DKMS where it's located    ...failed!     (Failed, trying without DKMS)   Recompiling VirtualBox kernel modules ...failed!     (Look at /var/log/vbox-install.log to find out what went wrong)   I performed   uname -a   And the result is:     Linux douglas 3.16.0-30-generic #40-Ubuntu SMP Mon Jan 12 22:06:37 UTC 2015 x86_64 x86_64 x86_64 GNU/Linux   Someone help me?Already I uninstalled and installed again. Grandi's Series; tends to $1/2$, but why is this considered a valid sum? Recording claimable expenses in GnuCash How to download Ubuntu Server backups automatically? I'm thinking of an investment in a new root server running ubuntu. Due to the fact, that the provider won't have extra backup-space, I need an other solution.  Is it possible to download the server backups automatically when my desktop machine is connected to the internet and in idle state? How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$? How to merge vertices in Blender? I am attempting to create my first 3D character just for practice. Being one of my first 3d models I going through the learning curve needless to say. In doing so I find myself creating many extra vertices. Where we will say 10 vertices on a rounded shape are spaced out on the equator and on the other north pole end of the edges they are very close. The close end could all be one vertex. Is it possible to say merge all points to one point and delete the rest? Error handling in C code Given $S \in B(Y^{*}, X^{*})$, does there exist $T\in B(X,Y)$ such that $S=T^{*}$? Prove a assertion concerning Lebesgue space Let $f_n \in L^1$ such that $f_n\rightrightarrows f$. Then I want to conclude that If $\mu(X)&lt;\infty,$ then $f\in L^1$ and $\int f_n\to\int f$, so I want to make use of the Dominated convergence theorem, the thing is that I don't know which would be my dominating function.  I was trying to figure out if the measure could play that role, but after thinking a little bit I have realize that the measure doesn't have to be bigger than the absolute value of the functions of the sequence.  Another thing I was thinking of, is to apply Fatou's lemma, but I can't ensure that my functions are positive.  So, Can someone help me to prove this theorem please ?  Thanks a lot in advance :).    NOTE: I think this theorem is taken from Folland's Real analysis, modern techniques, and their aplications 2nd Edition :), This is to have all the definitions so we can talk the same language as in the book (this is, same notation and theorems or machinery) Prove that if $V$ and $W$ are three-dimensional subspaces of $\Bbb R^{5}$, then $V$ and $W$ must have a non-zero vector in common.  So far I got $V = \{v_{1}, v_{2}, v_{3}\}$ and $W = \{ w_{1}, w_{2}, w_{3}\}$ because they are 3-dimensional subspaces. Then $V \cup W = \{v_{1}, v_{2}, v_{3}, w_{1}, w_{2}, w_{3}\}$  My prof said that now I need to prove that they are linearly independent so this is a contradiction and the statement is wrong because we are in $\Bbb R^{5}$ and we have 6 elements on it.  I don't know how to express them as a linearly independent proof, so I hope anyone can give me a good explanation on it. Is there a license / royalty free DRM available for protecting and distributing video content? Fantasy book series with non-human races and labyrinth I'm trying to identify fantasy book series I encountered some time ago. Here are things I remember:   it was a fantasy setting but with some s-f elements; with many worlds and possibility to travel between them; resembling  setting a little, the two main races were non-human and probably hostile to each other,  a labyrinth (?), which somehow played an important role in the plot, a protagonist was some kind of emissary and was sent on a mission to another worlds  time period: 80s - 90s    Does anyone know what it is the name of the series? How do you start Windows 10 in safe mode? I've been through the Windows 10 upgrade, and am experiencing a recurring "INTERNAL_POWER_ERROR", which is stopping me from logging in.  I'd like to be able to try to boot to safe mode, but it appears that pressing F8 during boot is no longer enough, so how can I boot in safe mode (or any of the other recovery options) on a Windows 10 PC that doesn't let me log on? Trigger a button click with JavaScript on the Enter key in a text box I have one text input and one button (see below). How can I use JavaScript to trigger the button's click event when the Enter key is pressed inside the text box?  There is already a different submit button on my current page, so I can't simply make the button a submit button. And, I only want the Enter key to click this specific button if it is pressed from within this one text box, nothing else.  &lt;input type="text" id="txtSearch" /&gt; &lt;input type="button" id="btnSearch" value="Search" onclick="doSomething();" /&gt; What is the use of the Dot Product of two vectors? Hair particles render not same as 3dView(Eevee) Why can't we use double pointer to represent two dimensional arrays? Test if there are files matching a pattern in order to execute a script I am trying to write an if statement to test whether there are any files matching a certain pattern. If there is a text file in a directory it should run a given script.  My code currently:  if [ -f /*.txt ]; then ./script fi   Please give some ideas; I only want to run the script if there is a .txt in the directory. Base64 Encode String in VBScript I have a web service load driver that's a Windows Script File (WSF), that includes some VBScript and JavaScript files. My web service requires that the incoming message is base64 encoded. I currently have a VBScript function that does this, but it's very inefficient (memory intensive, mostly due to VBScripts awful string concatenation)  [Aside; Yes, I've seen . The concatenation is happening in a loop across messages that are 1,000's to 10,000's bytes in size.]  I've tried using some custom string concatenation routines; one using an array and one using ADODB.Stream. These help, a little, but I think it would help more if I had some other way of encoding the message rather than via my own VBS function.  Is there some other way of encoding my message, preferebly using native Windows methods? I am trying to make a snowball that triggers a command when it hits a player in 1.15, but I don't know how to do that. When I do execute at @e[type=snowball] run gamemode spectator @a[distance=..2] it only runs the command if it goes right next to them.