![]() A Real-World Example for Heteroskedasticity.5.4 Heteroskedasticity and Homoskedasticity.5.3 Regression when X is a Binary Variable.5.2 Confidence Intervals for Regression Coefficients.5.1 Testing Two-Sided Hypotheses concerning the Slope Coefficient.5 Hypothesis Tests and Confidence Intervals in SLR Model.4.5 The Sampling Distribution of the OLS Estimator.Assumption 3: Large Outliers are Unlikely.Assumption 2: Independently and Identically Distributed Data.Assumption 1: The Error Term has Conditional Mean of Zero.4.2 Estimating the Coefficients of the Linear Regression Model.3.7 Scatterplots, Sample Covariance and Sample Correlation.3.6 An Application to the Gender Gap of Earnings.3.5 Comparing Means from Different Populations.3.4 Confidence Intervals for the Population Mean.Hypothesis Testing with a Prespecified Significance Level.Calculating the p-value When the Standard Deviation is Unknown.Sample Variance, Sample Standard Deviation and Standard Error.Calculating the p-Value when the Standard Deviation is Known.3.3 Hypothesis Tests concerning the Population Mean.Large Sample Approximations to Sampling Distributions.2.2 Random Sampling and the Distribution of Sample Averages.Probability Distributions of Continuous Random Variables.Probability Distributions of Discrete Random Variables.2.1 Random Variables and Probability Distributions.1.2 A Very Short Introduction to R and RStudio.Which I can also control in the population. We just need to remember that the SEs come from the diagonal of:Īgain, I have control over and all I need to remember is that for (centered) predictors, is really just the inverse of the variance-covariance matrix of the predictors. But what about multiple OLS regression? Well, I think the multiple regression case is even simpler. The code line e sqrt(var.e/ssq) #population SE There are two elements here that we need to have control over: and. Of the various ways to calculate the SE for the regression coefficient in simple linear regression, I prefer this one for our purposes: Let’s start from the beginning, simple linear regression. Therefore, I reasoned, it makes sense to set these in simulation studies whenever possible so we have a version of what the ‘true’ value would be. Well, the fact of the matter is that the way standard errors are defined in OLS multiple linear regression do involve population-level quantities. So the question comes to mind: We are well-acquainted with how to simulate data where the parameters are known in the population…but what about their variabilities? Does it even make sense to talk about “population standard errors”? Like… you need to simulate what the population may look like and then you need to simulate what the sampling behaviour may be… too much simulating, LoL. But this approach has always made me feel kinda awkward because it’s like you’re “double-dipping” in your simulation. So it stands to reason that the average SE should be close to this quantity, even if it was derived empirically through simulation and not analytically. In layperson’s terms, the standard deviation of the sampling distribution of the coefficients is the standard error. (3) Compare the mean SE to the SD of the coefficients. (2) Calculate the mean of the SEs and the standard deviation of the coefficients. (1) Run your simulation and save both regression coefficients and their standard errors. So…the only way I knew at first about how to assess SEs was in three steps: For the remaining of this article every time I talk about “standard errors” I’m most likely referring to the standard errors of regression coefficients within OLS multiple regression. If they shrink or increase under violations of assumptions then we know something is going to be wrong with our p-values and Type I error rates and yadda-yadda. And a simple way to assess this efficiency is by looking at the estimated standard errors (SEs) of our parameter estimates. When working on ‘robustness-type’ simulations, we are usually not only interested in the unbiasedness and consistency but also the efficiency of our estimators. This is an interesting insight that I had a couple of weeks ago when I was preparing to teach my class in Monte Carlo simulations.
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