Questions tagged [regression]

Techniques for analyzing the relationship between one (or more) "dependent" variables and "independent" variables.

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How does a Relevance Vector Machine (RVM) work?

Relevance Vector Machines (RVMs) are really interesting models when contrasted with the highly geometrical (and popular) SVMs. In the light of a question like How does a Support Vector Machine (SVM) ...
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Gamma hurdle model for continuous response?

I am modelling invertebrate.biomass ~ habitat.type * calendar.day + habitat.type * calendar.day ^ 2, with a random intercept of ...
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Asymptotic property of tuning parameter in penalized regression

I'm currently working on asymptotic properties of penalized regression. I've read a myriad of papers by now, but there is an essential issue that I cannot get my head around. To keep things simple, I'...
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Adjustments to (Linear Regression) Forecast

Full disclosure: I am not a statistician, nor do I claim to be one. I am a lowly IT administrator. Please play gentle with me. :) I am responsible for collecting and forecasting disk storage use ...
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Time series regression with overlapping data

I am seeing a regression model which is regressing Year-on-Year stock index returns on lagged (12 months) Year-on-Year returns of the same stock index, credit spread (difference between monthly mean ...
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Is autocorrelation not worth addressing with small N?

Consider a simple regression context in which there is a small set of response values, $Y$, and corresponding dates, $X$. (For simplicity, we can assume the dates are equally spaced.) We would like ...
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11 votes
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Interpreting regression coefficients based on Andrew Gelman's re-scaling method

I have two predictors in a binary logistic regression model: One binary and one continuous. My primary goal is to compare the coefficients of the two predictors within the same model. I have come ...
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Logistic regression for classification: are there any analytical solutions for the out-of-sample accuracy?

I run a binary logistic regression, with a binary dependent variable and a continuous independent one. Now I want to evaluate the out-of-sample performance of the classification algorithm so obtained. ...
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Pope effect on pizza - Regression with presence absence and similarity data as dependent variables

I'm trying to figure out the right way to set up a regression when the dependent variables are presence absence data (of pizzas), and the similarity between the present pizzas. Bear with the story: ...
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10 votes
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Bootstrap Prediction Interval: which residuals to use and which method?

I ask this question referring to the post: Bootstrap prediction interval, where a step by step method for calculating the prediction interval for linear regression models is explained. In the ...
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571 views

Does there exist an analogous statement to BLUE (Gauss-Markov) for GLMs?

I recall from my graduate school days that the Gauss-Markov (GM) theorem states that the Best Linear Unbiased Estimator (BLUE) in a linear regression is $\vec{\beta}=(X^TX)^{-1}X^T\vec{y}$. An amazing ...
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does serial correlation have something to do with endogeneity?

I'm a beginner of econometrics, and I've construed that endogeneity is caused by omitted variable bias, measurement error, and reverse causality, and it makes OLS estimator be biased. And also I've ...
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Regress residuals in second regression

I am wondering if anyone can point me to a paper/lecture notes on the rationale behind first running an OLS on a set of variables, and then in a second regression using the residuals of that ...
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Is the OLS estimator the UMVUE (assuming Normality)?

Suppose $$ \mathbf{y} = \mathbf{X} \mathbf{b} + \mathbf{e} \, , \\ \mathbf{e} \sim \mathcal{N}(0,\mathbf{I}_P) \, . $$ We know that $\mathbf{\hat{b}} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \...
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Efficient nonparametric estimation of confidence intervals and p-values for nonlinear regression

I'm estimating parameters for a complex, "implicit" nonlinear model $f(\mathbf{x}, \boldsymbol{\theta})$. It's "implicit" in the sense that I don't have an explicit formula for $f$: its value is the ...
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How do sufficiency statistics help in the interpretation of regression results?

One of the results why canonical link functions are widely used in GLMs is the existence of sufficiency statistics for the regression parameters, which in turn allow for: ... minimal sufficient ...
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Using Fieller's theorem to calculate the confidence interval of a ratio (paired measurements)

If you have two means (with their own confidence intervals) and want to represent them as a ratio, how do calculate the confidence interval for the ratio? An answer that was given to me, mentions ...
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When using L2 regularization outside of linear regression, do the same MAP estimation assumptions hold?

Some context is shared below, and my question is bolded at the end. In the linear regression setting, we learn model weights $\hat{\mathbf{w}}$ to make predictions $\mathbf{\hat{y}}$ from new samples ...
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Calculating R-squared using standard errors

I have the following estimated model: $\hat{y} = 0.2857 + 0.8019x_1 - 0.0741x_2$ (the $t$-statistics are $1.8959$, $8.4198$, and $-3.7017$, respectively). Furthermore, I know the sample size $N = 92$,...
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When is oversampling poor practice?

For my particular domain and problem, I have data on the entire population. However, my "event" only occurs in 0.5% of the cases. I want my model to be able to pick up on significant characteristics ...
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Assumptions for PCR and PLS

I am writing up a report on fitting Principal Component Regression (PCR) and Partial Least Squares (PLS) to my data-set. A similar question: Model assumptions of partial least squares (PLS) ...
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Logistic Regression with (Normal) Distributions for Independent Variables

Consider the logistic regression where $Y_i \in {0,1}$ are dependent variable observations and $X_i \in \mathbb{R}$ are the independent variables. However we do not observe the $X_i$ themselves. ...
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Why would I use ratio estimation instead of regression estimation to estimate means?

I am taking a graduate course on survey data analysis. I was recently introduced to ratio estimation and regression estimation. I understand that using ratio estimator may be easier if we are ...
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8 votes
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Assumptions of correlation test vs regression slope test (significance testing)

If my understanding is correct, then the test on a regression slope in a simple bivariate regression - i.e. the test of $\mathcal{H}_0$: $b = 0$ in $Y' = a + bX$ and the test of a correlation, i.e. $...
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Fixed Effects vs Lagged DV vs. First Differences Regression

What are the differences between using unit fixed effects, unit fixed effects and time fixed effects, lagged DV, or first differences to analyze a time series with 4-5 time periods and 35-50 units per ...
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1 answer
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Calculating and plotting confidence interval for Theil-Sen estimator

I'm using Wilcox's R functions (specifically, regplot) to plot a Theil-Sen estimator with a single predictor. However, regplot ...
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8 votes
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908 views

Dantzig Selector, LASSO, LAD LASSO

I am wondering about this. When is it best to use Dantzig Selector (the infinity normed error measure plus the L1 regularizer) , the LASSO (the mean square error measure plus the L1 regularizer), and ...
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Poisson regression for binary data

I've been trying to read up on Poisson regression models, and it looks like it is possible to estimate such a model with a binary outcome. This has come up before on this site here (and somewhat here ...
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7 votes
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Identifying non-linearities in relationship between variables

Logistic regression is often used to identify the effect of $x$ on a binary variable $y$ after adjusting for potential confounders $x_1,...,x_n$. In the medical literature, I will sometimes encounter ...
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7 votes
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309 views

Robust Gamma Regression

I am modeling some spectroscopic data where the response of the instrument to the size of the input is strictly positive and non-linear. Gamma regression seems like a good choice to explain the data, ...
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Regression with dependent data with low dependence

Suppose you have data that is grouped in one way or another and therefore the assumption of independence is suspect. But you look at the intraclass correlation (or autocorrelation) and it is very ...
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7 votes
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Intuition: What is the difference between linear factor models and regular linear regression?

So, I have a very vexing theoretical question that I hope some experienced econometrician can help me with. Being in finance, I have recently been exposed to linear factor models, which are models ...
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7 votes
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916 views

Geometrical interpretation of L1 regression

I have found the following image (or a similar version) in a lot of books related to penalized linear models. I get the insight of this image. The ellipsoids are the solution of the linear regression ...
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7 votes
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235 views

Is there any geometric intuition on least absolute deviation regression?

There are a lot of geometric intuitions for regression with least square, e.g., projection, orthogonal, etc. (This and this answers are good examples.) Is there similar geometric intuition for least ...
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691 views

Is lasso always outperformed by adaptive lasso?

I have been reading some papers and I understood that adaptive lasso has the Oracle properties which lasso lacks. Does that mean adaptive lasso always better than lasso (let's focus on the simple ...
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Can I use bootstrap results at the observation level?

I have read quite a bit of bootstrapping, but the issue I want to address seem not to appear. Consider a simple regression model: $$ y_{i} = \beta_{0} + \beta_{1}x_{i} + e_{i}$$ I am aware that ...
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7 votes
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Interpretation of smoothing spline

This question is about interpreting the results from non-linear regression models, especially when using regression splines. The numerical output is not very informative when interpreting the effects, ...
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Can I calculate Cohen's $d$ from multiple regression coefficient?

Question: Is it appropriate to calculate Cohen's $d$ (effect size) from the regression coefficient of an independent categorical variable? Background: My regression coefficient represents ...
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7 votes
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What techniques are there to measure goodness of fit of Deming (orthogonal) regression?

Questions: Even if there is no "widely accepted" technique, is there a useful-and-above-average technique for estimating goodness of fit in orthogonal regressions? What are the pros/cons of this ...
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Rule of thumb for excluded variable in Heckman selection model?

I'm working on a project that involves the use of a Heckman selection model (more specifically a Roy or move-stay model, which is essentially a two-sided Heckman) of the following form: $$ Y_{i1} = ...
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390 views

minimizer weighted linear regression

In a regression problem, with $y=X\theta+\epsilon$ and $X$ is an $n$ by $p$ matrix the ‘weighted least squares estimate is the minimizer $\theta^{*}$ of $f(\theta)=\sum_{i=1}^{n}\omega_{i}(y_i-x_i^{'}\...
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Bias Variance tradeoff from a Bayesian perspective

I know the general question about bias variance has been asked before. I understand the frequentist approach and the concept of model selection and the impact of bias and variance on "accuracy" of a ...
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7 votes
0 answers
141 views

R packages that work with biased samples

I'm working with a biased sample of web users. I'm only able to track responses of users who have navigated my site in a certain way, and I'd like to run an analysis to determine how certain factors (...
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7 votes
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Logistic regression and maximum entropy

I have read (e.g. here) that a (multinomial) logistic regressor corresponds to a maximum entropy classifier. My question is, how does one end up with the formula for logistic regression starting with ...
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Geometric Interpretation of Softmax Regression

I'm writing a series of blog posts on the basics of machine learning, just for fun, mostly to validate my understanding of Andrew Ng's class. As I'm currently studying generalized linear models (GLMs),...
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Generalization of cumulative probability models for ordinal Y

There are many models in existence for ordinal $Y$, for example the proportional odds ordinal logistic model, the continuation ratio model, and the cumulative probit model. The first and third of ...
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7 votes
0 answers
110 views

Zero values and discontinuity in explanatory variable

One of my independent variables measures worker productivity through the variable $\frac{\log{sales}}{\text{# of workers}}$, and I'm creating one variable for skilled and another for unskilled workers....
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Studentized residuals and goodness-of-fit with robust linear regression

Could you please advise whether studentized residuals are meaningful when computed on a robust linear regression model using an M-estimator? I'd like to use it to detect outliers by doing something ...
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7 votes
1 answer
779 views

Average Structural Function Calculation

EDIT: I have solved this problem myself. The problem with the simulation below is that the omitted variable should not be included in the 'true model'. I have written a blog post with a more detailed ...
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7 votes
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295 views

Making new variable instead of correcting for temporal autocorrelation in a GLMM. Is it a valid alternative?

I am doing some forest disturbance research, in which the aim is to predict the probabilities of wind damage occurrence in forest stands of different site (altitude, slope steepness) and stand ...
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