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Take the task of fitting an a priori distribution like the ex-Gaussian to a collection of observed human response times (RT). One method is to compute the sum log likelihood of each observed RT given a set of candidate ex-Gaussian parameters, then try to find the set of parameters that maximizes this sum log likelihood...
Should the n for sample size be capitalized? Is there a difference between n and N?
I'm trying to figure out how to calculate the standard error of a mean correlation coefficient. I have 6 bilateral correlation coefficients for 4 countries. I have transformed them using the Fisher z transformation in order to calculate their mean correlation coefficient. I'm trying to figure out what the standard erro...
It is mentioned here that one of the methods to determine the optimal number of clusters in a data-set is the "elbow method". Here the percentage of variance is calculated as the ratio of the between-group variance to the total variance. I felt difficult in understanding this calculation. Can any one explain how to cal...
Heathcote, Brown & Mewhort (2002, PDF) present Quantile Maximum Probability Estimation (originally termed Quantile Maximum Likelihood Estimation but later corrected) as a method of fitting distributional data, and find that it outperforms the more traditional Continuous Maximum Likelihood Estimation approach, at least ...
Possible Duplicate: Logistic Regression in R (Odds Ratio) I need to do a logistic regression in R. My response variable is surv=0; surv=1 and I have about 18 predictor variables. After reading my model, I got the table of Coefficients below and I need to go through some steps, which I am not familiar with, until I g...
Does anybody know why offset in a Poisson regression is used? What do you achieve by this?
Suppose we have time-series $ X_t $ and it has the following decomposition $$X_t=\mu + \varepsilon_t,$$ where $\mu$ is a mean and $\varepsilon_t$ - the error term. The model complexity will increase if we divide this time-series in to some segments,say $k$, and repeat above process. As the model complexity increases ...
I received the following question by email: I was wondering should I use tick the option for pairwise exclusion of missing data when I carry out regression analyses (or any analyses for that matter) rather than using [some other missing values replacement strategy]. Julie Pallant recommends pairwise exclus...
I am working on 4 different species of tomatoes. From the data I had, I looked at the occurrence of a particular "event" in certain intervals of their genome (this interval is identical in all 4 plants) and I have a file for each of the species with their probability of occurrence. The file looks something like this: ...
This question came up in a consulting context, and I was interested in your thoughts. Context One strategy for dealing with occasional missing data when calculating scale means looks like this in the language of SPSS: COMPUTE depmean = mean.4(dep1, dep2, dep3, dep4, dep5, dep6). EXECUTE. I.e., calculate the mean of a...
I have a data frame that contains some duplicate ids. I want to remove records with duplicate ids, keeping only the row with the maximum value. So for structured like this (other variables not shown): id var_1 1 2 1 4 2 1 2 3 3 5 4 2 I want to generate this: id var_1 1 4 2 3 3 5 4 2 I know about unique() and duplica...
Today I opened two STATA windows and ran the following command in both: set obs 100 gen x = rnormal() sort x (the difference is that on the second window I generated a variable called y). Summing up: I asked STATA to give me 100 pseudo-random numbers taken from a standard normal distribution, then I sorted it. To my...
Hello I am trying to forecast using different exponential smoothing methods(Linear and Winter's). For the optimal parameters, I am getting negative values of the forecasats. I am assuming it means that the values will be zero, since it is a sales forecast. I wanted to know if negative values denote something wrong wi...
Hi I am using Linear and exponential forecasting models to do sales forecasting. In the model itself, we use the forecasts of period t to get next forecast and so on. While analyzing the accuracy of the forecast using Mean Absolute Percentage Error, I get good results. But when I compare the intermediate forecast value...
Problem is that government wants to close electronic roulette and they claim that roulette failed at statistical test. Sorry for my language but this is translated from Slovenian law as good as possible Official (by law) requirements are: frequency of each event should not differ from expected frequency by more than 3...
I have written a 3-way ANOVA in C++. I have 3 factors, lets say A, B and C and my aim is to check the strength of all possible interactions and main effects. The result of my code is the same as in MATLAB when I use type-I sum of squares. But when I change the data so that the number of replicates/samples is high in ...
I appreciate the usefulness of the bootstrap in obtaining uncertainty estimates, but one thing that's always bothered me about it is that the distribution corresponding to those estimates is the distribution defined by the sample. In general, it seems like a bad idea to believe that our sample frequencies look exactly ...
I have been reading about appropriate measures of central tendency for ordinal level data. So far I have learned that the median and mode can be used but that the latter can only be used in some cases. Some sources state that the median can only be used with Likert questions when there is an odd number of scores. It i...
I have data from a load test of a web site with several thousand data points spread out over roughly 30 minutes (the values are the response time of the site in milliseconds). The values are spread out among the 30 minute range, but not at a constant rate (i.e. there may be a few milliseconds between some points, other...
I need to calculate an exponential moving average for a series of data. The intended sampling interval is fixed (say 1s) but the data stream has varying intervals (data intervals vary from 0.01s to 10s or so). The data is somewhat noisy (a random data sample would virtually never be on the average). My impression is th...
Say I have 2 sets, $A$ and $B$ with $n_{A}$ and $n_{B}$ elements respectively, which I assume is known. I would like to estimate $| A \bigcup B |$ using samples of $\tilde{A} \subset A$ and $ \tilde{B} \subset B$. That is if $\tilde{A}$'s elements are uniformly sampled from $A$, and likewise for $\tilde{B}$, will $ ...
I am investigating many different kinds of PCA versions, I am trying to find out whether PCR will apply to my analysis thus the question on use of PCR.
I am trying to compare the difference between two means with two pairwise samples. Unfortunately, my data are very far of being normal. What test would you recommend to use in this situation? Should I revert to a nonparametric test?
Assume, I have a data set, which is similar to require(nlme) ?Orthodont and my model is fm2 <- lme(distance ~ age + Sex, data = Orthodont, random = ~ 1) How can I use the model fit object fm2 to generate several datasets, which have sample sizes 300, 400, 500, ... ? I read this great answer on r-sig-mixed-models...
I've been using R's lm to do some linear regression, but decided to give MCMCregress a try to get a feel for how it works. As expected, I got basically the same coefficients, but the extra sigma2 value puzzles me. When I do a qqmath plot of the coefficients, I get the following graph, and I'm puzzled by the sigma2 plot...
This is a question strongly related to Cauchy "characters". I'm constructing a 4 question canvassing questionnaire that will tell the likely voter being contacted which of the presidential candidates most closely matches them. The advantage of this approach for a dark horse presidential candidate is obvious, presuming...
Context I have a regression framework and two sets of data. Using leave-one-out cross-validation, the first set gives very good performance and the second set gives rather poor performance. I need to explain the reason for this difference in performance. Having looked at the data, it is clear that the first set is a mu...
The last line is an example of what I'm looking for: data(airquality) attach(airquality) lm1 <- lm(Ozone ~ Solar.R+Wind) lm2 <- lm(Ozone ~ Solar.R+Wind+Temp) anova(lm1 , lm2) require(rpart) rp1 <- rpart(Ozone ~ Solar.R+Wind) rp2 <- rpart(Ozone ~ Solar.R+Wind+Temp) anova(rp1 , rp2) # this doesn't exist - is there some...
I'm using Gibbs sampling to learn the distributions of coefficients for a multinomial logistic regression model. At the end, I end up using the mean values of distributions of coefficients, and the resulting logistic regression is used as a classifier. I'm trying to find out advantages of having probability distributi...
I have a large data set which is in .dbf format right now and what I would like to do is be able to manipulate it easily in Excel and do something like subtotal and calculate stdev and ratios. Details of the data set; This data set contains shopper information. It has 1.2 million rows and 20 columns where the rows are ...
I received a question today that I wasn't exactly sure how to answer. I have built a predictive model using a fairly basic logistic regression that works pretty well and fits our business needs. Recently, we purchased a CRM tool that allows us to build "probability" scores, but only allows the end users to give inte...
If I take a set of measurements and test correlation of variable $A$ vs variable $B$ and get a significant correlation, that makes sense to me. But what if further analysis reveals that of those factors, there is only a significant positive correlation within one group, and that group is over-represented. Is the glob...
I've noticed this issue coming up a lot in statistical consulting settings and i was keen to get your thoughts. Context I often speak to research students that have conducted a study approximately as follows: Observational study Sample size might be 100, 200, 300, etc. Multiple psychological scales have been measured...
I need to do a regression with a non-normal DV for which no proper non-linear transformation (that I know of) exists: It is a score ranging from 10 to 50, with a high peak at 10, a drop at 11 and a regular decline from 11 to 50. The distribution of residuals is not normal. There are more or less 500 subjects in the s...
Context: I am analysing some impact assessment data (measuring invertebrate richness in response to pollution), but they are unbalanced - there are not data for every site at sampling occasion, and there were more datapoints recorded after the impact than before the impact. I am a new user to R, and have gathered thro...
Related to my earlier question, I need to perform regression on a skewed dependent variable (n = 500). Since the residuals weren't normally distributed, I was able to transform the DV non-linearly in a way that it now approaches normality. Residuals are normal when using this transformed variable as a dependent variabl...
This question is motivated by a issue regarding network motifs. To determine if a (connected, induced) subgraph $H$ occurs with significantly high frequency in an input network $G$, we generate an ensemble of comparison networks similar to $G$ and count the number of occurences of $H$ in them. Thus we obtain the numb...
I'm sure this is a pretty standard statistics question, but I'm no expert... I'm running an A/B test on my website to see if a change results in users adding more content. So there are 2 basic things I'm looking at; the # of users adding at least 1 piece of content and the total # of pieces of content added by all us...
I would like (in the distant future) to teach statistics to kids. For that matter, I'd be happy to know of software (obviously I am tending towards FOSS), or webapps, that are helpful in explaining statistical/probabilistic ideas to kids (or adults for that matter). This can be used either by the instructor, the kids,...
My study is about customers’ perception of the effectiveness for a Malaysia corporate Weblog. In my study, customers’ perception of the effectiveness defined as perceived ease of use, perceived interactivity, and perceived trustworthiness; whereas the corporate Weblog defined as Weblog publishing software, Weblog comme...
In principle a simple question: What is the pull distribution? (All I could find out is that it is the error-weighted distribution of estimators around the true value.) I'd be interested in the precise mathematical definition, how, why, and when to use it, what is it expected to look like, and if both estimator values ...
In a nutshell, here's what I have: Annual population estimates for the State Periodical (5 years) age, population, and basic census data per zones Here's what I want to do: Create a simplistic model to generate the data for the missing years between the period for each zone, and have the total sums add up to where t...
I am new to R and some help would be of great use to me. Basically, I need to perform a GLM analysis with negative binomial errors and with fixed factors, no covariates and no random effects. My factors are of type: year (1-4), site (1-3), sex (1-2), age (1-3), with a sample size of around 5000. Currently I am fitting ...
I am running a logistic model on insurance data. I have a field agent gender which matters for a channel A (say) and doesn't matter for B. I want to put null values in case of B. The only thing I risk is exclusion by SAS (as SAS excludes every missing case by default). I heard that pairwise exclusion can solve my probl...
I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts. My problem is to find an exponential upper bound over the probability that the sum of unbounded i.i.d. random variables, which are in fact the multiplication of two...
I have come across the sampling method called "Propensity Weighting Sampling/RIM", but I do not have a good idea of what these survey methods are all about. What references in the literature cover this topic?
I am performing a retrospective study on patients looking at the size of their nostril (continuous variable measured in millimetres) and the need for treatment which is either conservative or surgical (this is a categorical variable). Sample size is only 15. What would be the right test to compare groups to determine i...
I have a table with four groups (4 BMI groups) as the independent variable (factor). I have a dependent variable that is "percent mother smoking in pregnancy". Is it permissible to use ANOVA for this or do I have to use chi-square or some other test?
Context: I'm investigating behaviour in a clinical study involving children. I had both parents and teachers completing questionnaires to inform an understanding of the same underlying constructs, for example reactive aggression. At the conclusion of data collection I have parent data in all cases, n=55, and teacher ...
I have a particular semiparametric model which I'm fitting via MCMC. One of the model parameters I have "semiparametric'ed" away (say $\alpha$) is known to lie between two other parameters, $\theta_1$ and $\theta_2$. Since I have a series of samples $(\theta_1^t, \theta_2^t)$ I also have a series of interval estimates ...
I am wondering what the proper term is, for when a table like this (where values that did not occur are omitted entirely): ________ _______ | Length | Count | |--------|-------| | 1 | 5 | | 3 | 2 | | 6 | 12 | |________|_______| Is rewritten like this (where values that did not occur are note...
I've found critical values for the Anderson Darling test for a Normal Distribution at 1%, 2.5%, 5%, 10% and 15% significance levels from various sources, including wikipedia: http://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test I'd really like a critical value for a 0.1% significance level (in "case 4" - neither ...
So when I assume that the error terms are normally distributed in a linear regression, what does it mean for the response variable, $y$?
I have an experiment where people click on different ads online. My measure is click counts. I end up finding that I should use models for count data such as Poisson, Quasi-Poisson, or Negative Binomial regression. Is there a standard in marketing regarding what model should be used for click counts? Thanks
Here is a recent Google correlate query: http://www.google.com/trends/correlate/search?e=internet+usage&t=weekly# As you can see in the search box at that link, I entered "internet usage" and Google did the rest. It shows a value of 0.9298 as the "correlation" with the query "data mining". However, when I read page 2...
We've created a survey asking students, among other things, their GPA (=weighted average of grades) and their marks in some specific courses (which count towards GPA). We wanted to see which regressors influence the GPA using a simple OLS model. Is it sensible to use a formula like this? GPA ~ grade_maths + grade_stati...
Possible Duplicate: Wrong results using ANOVA with repeated measures Hello everybody, I did an experiment and I need to understand how to detect, by means of an ANOVA (repeated measures), the differences between males and females evaluations at stimulus level. In the experiment, participants had to evaluate 7 stimul...
Suppose we have a set $S$ consisting of $p$ features, and a subset $S_+$ of the features are positive. If $Q$ is any subset of $S$, define the false positive rate as the proportion of features in $Q$ which are not positive: $$FPR[Q] = 1 - \frac{|Q \cap S_+|}{|Q|}$$ where $|\cdot|$ denotes cardinality. If $Q$ is a fun...
I asked this on the mathematics site, but now I think this is a better place. Sorry for the cross-post. Given any line graph, is there a reliable way to identify any sort of regular oscillation? Let's assume I'm charting the prevalence of different species of animals in a single location, over the span of several year...
I have a multinomial model estimated with the zelig package in R. Whenever I try to use the setx() command, I get an error message saying there is more than one mode. So in stead of using Zelig, I thought I would do it the hard way. I used the instructions here, but I am not sure if I trust these instructions as they g...
A six-sided die is rolled 100 times. Using the normal approximation, find the probability that the face showing six turns up between 15 and 20 times. Find the probability that the sum of the face values of the 100 trials is less than 300. For the first part of the question, I did the following: $P(15 \le X \le 20) = \s...
Is there a way to give (in R or Minitab or Statgraphics) a fractional factorial design like that and inspect the generators and the complete defining relation ($2^4 - 1$ relations)? A B C D E F G H -1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 1 -1 1 1 -1 -1 -1 -1 -1 ...
This is probably a pretty simple question, but I have been having some trouble interpreting the documentation for the predict function. I am generating a simple linear model from a data frame containing (X, Y) pairs, which I would then like to use to predict Y given new X. My code looks something like this: my_lm = lm(...
I want to jointly estimate a very simple MV-Normal two-dimensional AR[1] process, $[x_t,y_t]=[x_{t-1},y_{t-1}]+\text{[Bivariate Gaussian error]}$, in BUGS. But the syntax has been impossible to figure out. Here's the problem part of the code: ## transition model (aka random walk prior) for(i in 2:NPERIODS1){ ...
This is pretty hard for me to describe, but I'll try to make my problem understandable. So first you have to know that I've done a very simple linear regression so far. Before I estimated the coefficient, I watched the distribution of my $y$. It is heavy left skewed. After I estimated the model, I was quite sure to obs...
I have hundreds of explanatory variables and under 100 observations (saturated data set). I'd like to create a linear model in which I have two or so composite variables made up of a dozen of the explanatory variables each. How do I find the best variables to use for the composites without going through every combin...
What is the primary reason that someone would apply the square root transformation to their data? I always observe that doing this always increases the $R^2$. However, this is probably just due to centering the data. Any thoughts are appreciated!
Suppose I have the following model $$y_i=f(x_i,\theta)+\varepsilon_i$$ where $y_i\in \mathbb{R}^K$ , $x_i$ is a vector of explanatory variables, $\theta$ is the parameters of non-linear function $f$ and $\varepsilon_i\sim N(0,\Sigma)$, where $\Sigma$ naturally is $K\times K$ matrix. The goal is the usual to estimate...
I have data from an experiment that is testing how the order of two studying methods (visual or auditory) affects word recall. For analysis a multi-factor anova with a repeated measure is appropriate, but I am not sure if I am structuring my data correctly. This is the command I'm using: aov(recalled_items~task*order)...
I have to run a factor analysis on a dataset made up of dichotomous variables (0=yes, 1= no) and I don´t know if I'm on the right track. Using tetrachoric() I create a correlation matrix, on which I run fa(data,factors=1). The result is quite near to the results i receive when using MixFactor, but it´s not the same. ...
I'm a statistics newbie and I would like to make sense of the p-value with ANOVA, hopefully with a visual presentation. All online visual tools I've found so far only have MSB, MSE and F values visually presented (such as this here. I'm a very visual person and I would like to see how it works before I learn more about...
I have a question for adjusted $R^2$ given a specific regression model. I am doing a project on January effect and I have a model from some journal using $$R_i = a_0 + a_1D_{\mathrm{Jan}} + \varepsilon_i$$ where $R_i$ is daily return of portfolio/index, $a_0$ is non-January daily returns, $a_1$ is January returns...
I am attempting to perform a two-group confirmatory factor analysis (CFA) of one continuous factor on six ordinal predictors in OpenMx (R) using (robust) Weighted Least Squares (WLS) estimation. While I am completely new to OpenMx (I only used sem and lavaan ) I think I have most things now: I have a way of estimating ...
I've calculated the proportion of chicks fledged out of the number of eggs hatched in each year using prop.test() in R. I see that it gives me the proportion fledged, but also the 95% confidence interval, which is what I'm after. Having read the excellent information from another question on this site here, I understa...
I want to apply a PCA on a dataset, which consists of mixed type variables (continuous and binary). To illustrate the procedure, I paste a minimal reproducible example in R below. # Generate synthetic dataset set.seed(12345) n <- 100 x1 <- rnorm(n) x2 <- runif(n, -2, 2) x3 <- x1 + x2 + rnorm(n) x4 <- rbinom(n, 1, 0.5) ...
I am trying to test whether my regression has an issue of heteroscedasticity. After running a regression, I can clearly see that the residual plot has a pattern. After taking a log of the dependent variable the pattern is much, much reduced. The White's test on the original formula returns a p-value of 0.0004 before th...
Let's say i have sequences of symbols which can have five values : A, B, C, X, Y. The average length of sequences is around 7. It is important that the symbols A, B, C have a bigger importance than X and Y which may be consider as 'whatever different from A, B or C' I need to classify those data among two classes : pos...
I am working on a new webpage for my part-time job as a methodological/statistical consultant for (psychology) students at my university. On this website I would like to place several links to online recourses for clients to consult themselves. So I am looking for links to websites that offer a lot of statistical infor...
I'm trying to implement logNormal distribution into my java program because lognormal dist doesn't exist into apache commons math library. I have no problem to re-write density and cumulative probability function, extending abstract classes of apache commons math library, like this : public double cumulativeProbabili...
In case you want to compare the average income of a group of male employees against the average income of a group of female employees, the observations are clearly independent. Now, I have a network of a certain number of nodes. These nodes are linked by edges and I can characterize each node by the number of links it ...
My nonparametric text, Practical Nonparametric Statistics, often gives clean formulas for expectations, variances, test statistics, and the like, but includes the caveat that this only works if we ignore ties. When calculating the Mann-Whitney U Statistic, it is encouraged that you throw out tied pairs when comparing w...
Does anybody know where to find good application and examples (besides the manual and the book applied econometrics with R) using the tobit model with the packages AER? Edit I'm searching for a command to compute the marginal effects for y (not for the latent variable y*). It seems to be $\phi(x\beta/\sigma)\beta$, whe...
I am very new to R and to any packages in R. I looked at the ggplot2 documentation but could not find this. I want a box plot of variable boxthis with respect to two factors f1 and f2. That is suppose both f1 and f2 are factor variables and each of them takes two values and boxthis is a continuous variable. I want to g...
Possible Duplicate: Cross-correlation significance in R How can I test the significance of the correlation coefficient? I have two time series and I want to test if they are cross correlation. Should I do prewhitening the two series before comuputing the ccf or there are an easy way?
I've used a wide array of tests for my thesis data, from parametric ANOVAs and t-tests to non-parametric Kruskal-Wallis tests and Mann-Whitneys, as well as rank-transformed 2-way ANOVAs, and GzLMs with binary, poisson and proportional data. Now I need to report everything as I write all of this up in my results. I've a...
In plain English: I have a multiple regression or ANOVA model but the response variable for each individual is a curvilinear function of time. How can I tell which of the right-hand-side variables are responsible for significant differences in the shapes or vertical offsets of the curves? Is this a time-series pr...
I know, this may sound like it is off-topic, but hear me out. At Stack Overflow and here we get votes on posts, this is all stored in a tabular form. E.g.: post id voter id vote type datetime ------- -------- --------- -------- 10 1 2 2000-1-1 10:00:01 11 ...
I am working through a problem on CI for a sample mean and I cannot get the answer listed in the text I have. I am wondering if I am missing something or whether the text may be incorrect. We are given a sample of 10 scores: 45,38,52,48,25,39,51,46,55,46 I get a mean of 44.5 and SD of 8.68 which is the same as the s...
this question started as "Clustering spatial data in R" and now has moved to DBSCAN question. As the responses to the first question suggested I searched information about DBSCAN and read some docs about. New questions have arisen. DBSCAN requires some parameters, one of them is "distance". As my data are three dimensi...
is anybody familiar with the tobit() command using the package AER? I'm searching for a command to compute the marginal effects for y (not for the latent variable y*). It seems to be $\phi(x\beta/\sigma)\beta$, where $\phi$ is the std.normal cumulative distribution function. But how can I compute those effects with R?
I prepared a model which had very good accuracy (80.5%) on my out of sample data. However when I ran that model on a population which is some 6mths old the accuracy went down to abysmal 33%. I am talking here about percentage detection of event (say defaulters). So right now my model is only detecting 33 out of 100 def...
As far as I know variance is calculated as $$\text{variance} = \frac{(x-\text{mean})^2}{n}$$ while $$\text{Empirical Variance} = \frac{(x-\text{mean})^2}{n(n-1)} $$ Is it correct? Or is there some other definition? Kindly explain with example or any refence for reading on this topic
I want to evaluate the implications of increasing fine prices. I will have a few different scenarios ranging from business as usual, minor increase, proportional increase, categorical increase, to extreme-increases. Each scenario will have different levels of monetary increase depending on the fine details and subprogr...
Does anyone know such? I tried to find such procedures in Gretl, but there you can use either hsk procedure for heteroskedasticity correction or ar1 procedure for serial correlation correction. I need the GLS procedure that will deal with both. Thanks
I know that there are various posts regarding variable selection but I am asking something particular. With respect to the question that I posted today in the following link: Low accuracy in out of time validation If you had a look at the above link, you have seen that my problem is low detection rate in out of time da...
Suppose $x_{1}, x_{2} \dots x_{N}$ are gaussian RVs with variance $S$ and mean $1$. What is the density function of $$\frac{ |\sum_{n=1}^{N}x_{n}|^{2}}{\sum_{n=1}^{N}|x_{n}|^{2}}\text{?}$$
Does anyone use the $L_1$ or $L_.5$ metrics for clustering, rather than $L_2$ ? Aggarwal et al., On the surprising behavior of distance metrics in high dimensional space said (in 2001) that $L_1$ is consistently more preferable then the Euclidean distance metric $L_2$ for high dimensional data mining application...
I am interested in fitting a Bayesian Two Factor ANOVA in BUGS or by utilizing some R package. Unfortunately I am having a hard time finding resources on this topic. Any suggestions? Even an article describing the approach would be helpful.
I have a model $$Y=\beta_0 + \beta_1 x_1 + \beta_2x_2 +\epsilon$$ I would like the minimum variance unbiased estimate of $\gamma=\beta_1 + \beta_2$. Assuming the Gauss Markov conditions hold, but $x_1$ and $x_2$ are correlated, is there a more efficient way to estimate $\gamma$ than running OLS and adding the estimat...