$$ y_t = θy_{t−1} + u_t \\ t = 1,...,T;  $$ I need to derive a formula for $y_t$ and show that  $$ E\left[\frac{\Sigma y_{t-1}u_t}{ \Sigma(y_{t-1})^2}\right] \neq 0 $$
I am having trouble understanding the estimation of an AR process. In some textbooks, the AR(1) process is defined as follows: $y_{t}=\theta y_{t-1}+ϵ_t$ (which does not contain a constant). So the OLS estimator is biased. I am confused about the cause of the bias. It is explained that $y_{t-1}$ is dependent on $ϵ_{t-1}$ although it is independent of $ϵ_t$. However in linear regression, if the equation does not contain a constant, we cannot make sure the expectation of disturbance is zero. So the OLS estimator is bias without a constant. Does it mean that the OLS estimator is unbiased if I add a constant in AR process?
extended stars-and-bars problem(where the upper limit of the variable is bounded) I am trying to create an Apex Trigger that performs a Rollup from a custom object to Contacts.   Here is the error that I am receiving:      Didn't understand relationship 'Assignments__r' in FROM part of query   call. If you are attempting to use a custom relationship, be sure to   append the '__r' after the custom relationship name. Please reference   your WSDL or the describe call for the appropriate names.   The child relationship name is Assignments.  Below is the code:      trigger RollupActiveAssigmentLevelsToContact on pse__Assignment__c (after insert, after update, after delete) {      set&lt;Id&gt; set_id = new set&lt;Id&gt;();      List&lt;Contact&gt; con_list = new List&lt;Contact&gt;();      if(trigger.isInsert || trigger.isUpdate){         for(pse__Assignment__c myAssignment : trigger.New) {             set_id.add(myAssignment.pse__Resource__c);         }     }     else if(trigger.isDelete){         for(pse__Assignment__c myAssignment : trigger.Old){             set_id.add(myAssignment.pse__Resource__c);         }     }      if(trigger.isAfter &amp;&amp; (trigger.isInsert || trigger.isUpdate || trigger.isDelete)){         con_list = [SELECT id,Total_Levels__c, (SELECT id,name FROM Assignments__r WHERE Active_Assignment__c = TRUE) FROM Contact WHERE id IN :set_id];          for(Contact con : con_list){             if(con.Assignments__r.size()&gt;0)                 con.Total_Levels__c = con.Assignments__r.size();             else                 con.Total_Levels__c = 0;         }         if(!con_list.isEmpty())         update con_list;     } }   Below is a snap of the Child Relationship name: Moving an already-running process to Screen My office job routinely sees me connected to a Linux box via VNC.  Sometimes I start a remote job on the console, and realize later that it runs much longer than expected. (Should have started that one under Screen in the first place...)  I don't want to keep my workstation running overnight just to keep the VNC session open; I would like  to move that already-running remote job into a Screen session (on the remote box), so I can power down the workstation (and reconnect next morning).  How can this be done, if at all? How to install TP-LINK Archer T4U driver? How to force NSLocalizedString to use a specific language Blending colors in photoshop and illustrator I am working on a print for my wall and I am trying to create something like this:      I have already created the letters in Illustrator and now I am adding the colors in Illustrator and then exporting everything to Photoshop and then using the liquify tool to blend the colors to create a similar effect. I also used the healing patch tool to blend in the hard edges.  Is that the right way of doing it? Or is there another way? Unable to install "": snap "" has "install-snap" change in progress I'm travelling to country X. I want to install an application which isn't available in my country, but is available in country X; but I prefer to download and install it in advance, while the data connection is still cheap.  Unfortunately, the Play Stores does not allow me to install this app. Is there any way to circumvent this limitation? How did the ending happen from Neil's perspective? I am trying to figure out the scene near the finale of Tenet where Neil unlocks the door and gets shot in the face. I can't really see the flow where he opens and holds the door and the protagonist walks in and fights the bad guy. Can you tell the series of events as Neil sees them happen till he gets shot? UPDATE If I am not wrong, I see it like this: Neil was running down towards the door, both the protagonist and his mate are inside with the door locked fighting the bad guy, then, he unlocks the door and walks in and stands behind the door. The protagonist's mate comes out of the room, then the protagonist, then he closes the room, then he look back to get shot in the face. But this doesn't make sense, if he wanted to unlock the door when he was running down, they were already in fighting the bad guy!!!!! $L^p$ and $L^q$ space inclusion Let $(X, \mathcal B, m)$ be a measure space.  For $1 \leq p &lt; q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample in the case the condition is not satisfied? Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$. Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$.   My try :  I consider $h(z)=\frac{f(z)}{g(z)}$. If I prove that $h(z)$ is entire, using the fact that $|h(z)|\leq 1$, the result follows immediately from Liouville's theorem.  To prove that $h(z)$ is entire, I have to prove that $h(z)$ has removable singularities at possible zeros of $g(z)$.  Suppose $g(z_0)=0 \Rightarrow |g(z_0)|=0 \Rightarrow |f(z_0)|=0$, BUT what if $z_0$ be a root of of order $k$ of $g(z)$ (i.e. $f^{(n)}(z_0) = 0 $ for all $ 0 \leq n \leq k-1$) and be a zero of order $m$ of $f(z)$ and $k&gt;m$ ?  And if that's not the case, how to rigorously show that $z_0$ is a removable singularity ?  Thank you all in advance. How can I vertically align the numerators of two fractions? I am writing the following formula:  % arara: lualatex  \documentclass{article} \usepackage{mathtools} \usepackage{unicode-math}  \begin{document} \[y_\mathup{COG}=\frac{h}{3}\frac{l_\mathup{c,a}+2l_\mathup{c,i}}{l_\mathup{c,a}+l_\mathup{c,i}}\] \end{document}   which yields:    I don't like the way, the h and l_\mathup{index} is vertically aligned. Is that typical behavior? Good style? Could I avoid that? How do I make sure that values from MySQL keep their type in PHP? Cap colour in IPL If a player is the highest run scorer and at the same time is the highest wicket taker in IPL, which colour cap would he wear? terminator: permission denied for sudo Rollup Summary Trigger Help I am trying to create an Apex Trigger that performs a Rollup from a custom object to Contacts.   Here is the error that I am receiving:      Didn't understand relationship 'Assignments__r' in FROM part of query   call. If you are attempting to use a custom relationship, be sure to   append the '__r' after the custom relationship name. Please reference   your WSDL or the describe call for the appropriate names.   The child relationship name is Assignments.  Below is the code:      trigger RollupActiveAssigmentLevelsToContact on pse__Assignment__c (after insert, after update, after delete) {      set&lt;Id&gt; set_id = new set&lt;Id&gt;();      List&lt;Contact&gt; con_list = new List&lt;Contact&gt;();      if(trigger.isInsert || trigger.isUpdate){         for(pse__Assignment__c myAssignment : trigger.New) {             set_id.add(myAssignment.pse__Resource__c);         }     }     else if(trigger.isDelete){         for(pse__Assignment__c myAssignment : trigger.Old){             set_id.add(myAssignment.pse__Resource__c);         }     }      if(trigger.isAfter &amp;&amp; (trigger.isInsert || trigger.isUpdate || trigger.isDelete)){         con_list = [SELECT id,Total_Levels__c, (SELECT id,name FROM Assignments__r WHERE Active_Assignment__c = TRUE) FROM Contact WHERE id IN :set_id];          for(Contact con : con_list){             if(con.Assignments__r.size()&gt;0)                 con.Total_Levels__c = con.Assignments__r.size();             else                 con.Total_Levels__c = 0;         }         if(!con_list.isEmpty())         update con_list;     } }   Below is a snap of the Child Relationship name: How can one refute John Searle's "syntax is not semantics" argument against strong AI? How is a Kelvin Pod different than a standard Escape pod? Error when using endfloat with unicode characters I have a problem with the endfloat package and unicode characters.  When I compile the following, everything works as expected:  \documentclass{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{graphicx}  \begin{document}  \begin{figure} \caption{Ö} \end{figure}  \end{document}   However, when I also load the endfloat package (\usepackage{endfloat}), I get the following error message:  ! Package inputenc Error: Unicode char \u8:ÖÖ not set up for use with LaTeX.   This could be a bug, but my suspicion is that I'm making a mistake somewhere.  I'm hardly the first person to use special characters with the endfloat package.  However, the above example is so simple, I don't see where I could have made a mistake.  Any advice is appreciated.  Edit: The file is of course encoded using utf-8. Why are there non-visible objects in my final render?