“I can red. How to decide on the clustering of standard errors? The standard errors determine how accurate is your estimation. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. I know that this is the criterium for the choice of the cluster level. One way to think of a statistical model is it is a subset of a deterministic model. In this case then, the cluster variance estimator would be less than the basic HW one. Asking for help, clarification, or responding to other answers. There's no formal test that will tell you at which level to cluster. MathJax reference. Papers such as Larrain (2006) and Li, Morck, Yang, and Yeung (2004) estimate country- If the answer to both is no, one should not adjust the standard errors for clustering, irrespective of whether such an adjustment would change the standard errors. But, to conclude, I’m not criticizing their choice of clustered standard errors for their example. > > Including dummies (firm-specific fixed effects) deals with unobserved heterogeneity at the firm level that if ignored > would render your POINT estimates inconsistent. I’ll describe the high-level distinction between the two strategies by first explaining what it is they seek to accomplish. Therefore, If you have CSEs in your data (which in turn produce inaccurate SEs), you should make adjustments for the clustering before running any further analysis on the data. 2. xtpcse from Stata - how to rewrite in R. 3. individual random effects model with standard errors clustered on a different variable in R (R-project) 3. the residuals from two different regions in a country), then we'll have, at least in expectation, $ \mathbf{E} \left[ \hat{u}_{jc} \hat{u}_{kc} \right] = 0 $ for $ j \neq k $. Using the caret symbol (^) in substitutions in the vi editor. CC” is the likelihood to which the respondent feels she can reduce climate change herself by reducing own energy, use - 0 (not at all likely) to 10 (extremely likely). clustered standard errors work very well when the model is correct and the data satisfy ... level, it was traditionally assumed that the disturbances are uncorrelated, perhaps after ... disturbances for ariousv outcomes might be clustered by village, by province, or by country, depending on the nature of the model and dataset. Thus, we'll always have the standard sum of $ \hat{u}_{jc}^2 x_{jc}x'_{jc} $ like we do in the basic HW estimator. Since the errors are unobserved and a characteristic of the underlying population, there is no straight forward trick to determine the level to cluster. I'm estimating a first-difference panel data model with data on the regional level (~125 regions). ), Fixed effects model and robust standard errors, Clustering errors in Panel Data at the ID level and testing its necessity, Calculating nested clustered standard errors with bootstrap, Double-clustered standard errors and large panel. Bowling Alone: The Collapse and Revival of American Community. See Bertrand, Duflo, and Mullainathan for a longer discussion of this. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Is there a reason to believe the errors are correlated at the regional level (common conditions the effect the outcome variable that differ between regions)? In this case, the cluster estimator will then be greater than the HW estimator, because we're adding more positive terms in the inner summation. and they indicate that it is essential that for panel data, OLS standard errors be corrected for clustering on the individual. This page shows how to run regressions with fixed effect or clustered standard errors, or Fama-Macbeth regressions in SAS. White standard errors (with no clustering) had a simulation standard deviation of 1.4%, and single-clustered standard errors had simulation standard deviations of 2.6%, whether clustering was done by firm or time. Clustered Standard Errors 1. Note that if each observation had its own cluster, then the cluster estimator would be identical to the HW estimator. Does software exist to automatically validate an argument? infographics! Beyond that, it can be extremely helpful to fit complete-pooling and no-pooling models as … An extension of the basic one-dimensional case is to multiple levels of clustering. Not clear that conservatism is what should drive the approach. In what way would invoking martial law help Trump overturn the election? That is, if the amount of variation in the outcome variable is correlated with the explanatory variables, robust standard errors can take this correlation into account. Focus on the middle term (the three nested summations) in the cluster estimator. I have 19 countries over 17 years. But, to the extent that there actually is correlation across regions in a country, then you'll have $ \mathbf{E} \left[ \hat{u}_{jc} \hat{u}_{kc} \right] \neq 0 $ for $ j \neq k $. Heteroscedasticity robust standard errors with the PLM package. College Station, TX: Stata press.' It only takes a minute to sign up. Should I also cluster my standard errors ? To the extent that those residuals within the country actually are independent of each other (e.g. That's probably the case, just wanted to be sure. If I want to cluster standard errors on a SIC (Standard Industrial Classification) level, should g then be a N x 1 vector of SIC codes? *** < 0.01, ** p< 0.05, * p< 0.1 Note: Exogenous controls include whether a cadet is black or Hispanic, GPA, SAT math and verbal scores, cadet leadership score, cadet fitness aptitude, and recruited NCAA athlete. Are all satellites of all planets in the same plane? 15. Now, compare this to the HW robust variance estimator, that does not account for clustering: $$\widehat{\text{var}} \left( \hat{\beta} \right)_{HW} = (X'X)^{-1} \left[ \sum_{i=1}^n \hat{u}_i^2 x_ix_i' \right] (X'X)^{-1}$$. About robust and clustered standard errors. significant at the 1%, 5%, and 10%-level, respectively. This allows for the possibility of correlations between the errors in the observations within a given country, since the residuals from all the observations within a country will get multiplied by the residuals of all the other observations in that country. mechanism is clustered. Grouped Errors Across Individuals 3. If you think that the regressors or the errors are likely to be uncorrelated within a potential group, then there is no need to cluster within that group. takes a value of 1 if the individual reports being member of an environmental/peace/animal organization, and 2 if the individual also actively participates in. In what story do annoying aliens plant hollyhocks in the Sahara? Learn more about Bowling Alone: The Collapse and Revival of American Community with Course Hero's FREE study guides and All regions are part of a country (~12 countries). This is the usual first guess when looking for differences in supposedly similar standard errors (see e.g., Different Robust Standard Errors of Logit Regression in Stata and R).Here, the problem can be illustrated when comparing the results from (1) plm+vcovHC, (2) felm, (3) lm+cluster.vcov (from package multiwayvcov). But, it is in theory possible to have $ \mathbf{E} \left[ \hat{u}_{jc} \hat{u}_{kc} \right] < 0 $ - i.e. Serially Correlated Errors Are inversions for making bass-lines nice and prolonging functions? Bowling Alone: The Collapse and Revival of American Community. As one of the comments to your question noted, clustering at the larger level is more conservative (and likely better). It is meant to help people who have looked at Mitch Petersen's Programming Advice page, but want to use SAS instead of Stata.. Mitch has posted results using a test data set that you can use to compare the output below to see how well they agree. To learn more, see our tips on writing great answers. trust in the ESS) and Trust = 9 (largest point estimate in a regression of life satisfaction on trust). I found myself writing a long-winded answer to a question on StatsExchange about the difference between using fixed effects and clustered errors when running linear regressions on panel data. Table A.5: Regression estimates for pro-environmental outcomes II, significant at the 1%, 5%, and 10%-level, respectively. Use MathJax to format equations. It’s easier to answer the question more generally. But this is exactly the issue: I am looking for a way to find out at which level this correlation among the errors exists. An Introduction to Robust and Clustered Standard Errors Linear Regression with Non-constant Variance Variance of ^ depends on the errors ^ = X0X 1 X0y = X0X 1 X0(X + u) = + X0X 1 X0u Molly Roberts Robust and Clustered Standard Errors March 6, 2013 6 / 35 Accurate standard errors are a fundamental component of statistical inference. Standard errors clustered at the country level in parentheses Envir org member. Clustered standard errors belong to these type of standard errors. Thanks for your response! the support for subsidies on renewable energy to reduce climate change - 1 (strongly against) to 5 (strongly in favor). “Voluntary work” is 1 if any voluntary work for either type of association (incl. Typically, the motivation given for the clustering adjustments is that unobserved components in outcomes for units within clusters are correlated. $\hat{u}_{jc}$ represents the residual of observation $j$ from cluster $c$ and $x_{jc}$ represents the $x$ variable values for that observation. work” contains the reported frequency of being involved in work for, voluntary/charitable organizations in the last 12 months, ranging from 1 (never) to 6 (at least once a week), and “Supports fuel tax” captures the support for, a tax on fuels, ranging from 1 (strongly against) to 5 (strongly in favor). “Supports energy subs.” measures. 7. If the errors within a cluster are indeed independent of each other, than we should have, by definition of that independence, $ \mathbf{E} \left[ \hat{u}_{jc} \hat{u}_{kc} \right] = 0 $ for $ j \neq k $. The Attraction of “Differences in Differences” 2. here are a couple of solutions: Clustered standard errors at the group level; Clustered bootstrap (re-sample groups, not individual observations) When analyzing her results, she may want to keep the data at the student level (for example, to control for student-level obs… The difference is in the degrees-of-freedom adjustment. the environmental organization (rather than being a passive member only). $\begingroup$ The higher the level of clustering, the more conservative the estimate of the standard error, so it's good to err on the side of caution, unless there are compelling reasons to cluster at the lower level. Thus likewise we'd have $ \mathbf{E} \left[ \hat{u}_{jc} \hat{u}_{kc} x_{jc} x'_{kc} \right] = 0 $ for $ j \neq k $ since the $ x $ observations are considered non-random. Robust standard errors account for heteroskedasticity in a model’s unexplained variation. But fixed effects do not affect the covariances between residuals, which is solved by clustered standard errors. Why does chocolate burn if you microwave it with milk? What would be a good way to decide on this? Q iv) Should I cluster by month, quarter or year ( firm or industry or country)? > > Different assumptions are involved with dummies vs. clustering. “Can save energy” denotes the degree to which the respondent is confident about her ability to reduce energy, consumption - 0 (not at all) to 10 (completely confident). Notice that where $ j = k $ in the inner two summations, we'll be multiplying a residual by itself, so that will always be positive. teachers, classroom composition). Note #2: While these various methods yield identical coefficients, the standard errors may differ when Stata’s cluster option is used. Making statements based on opinion; back them up with references or personal experience. But, to the extent that we have $ \mathbf{E} \left[ \hat{u}_{jc} \hat{u}_{kc} \right] \neq 0 $ for $ j \neq k $ within the cluster, then the two will differ. It is counterproductive to read very long text books during an MSc program. importance given to the environment - 1 (not at all) to 6 (very much). Should one include the cluster specific dummy variables in the regression when st. errors are allowed to be clustered at that level? For example, errors may be clustered by country and by city, or errors may be clustered by country and by year. Course Hero is not sponsored or endorsed by any college or university. “p(Trust 9 = Trust 5)” denotes the p-value of an F-test for equality of coefficients for. A: The author should cluster at the most aggregated level where the residual could be correlated. They allow for heteroskedasticity and autocorrelated errors within an entity but not correlation across entities. Make 38 using the least possible digits 8. Larger and fewer clusters have less bias, but they … For example, looking at outcomes for students nested in classrooms, there are clear factors that all students within a class are exposed to that vary between classes (e.g. As they say, the innocent (i.e. I If nested (e.g., classroom and school district), you should cluster at the highest level of aggregation I If not nested (e.g., time and space), you can: 1 Include fixed-eects in one dimension and cluster in the other one. In empirical work in economics it is common to report standard errors that account for clustering of units. cluster(clustvar) use ivreg2 or xtivreg2 for two-way cluster-robust st.errors you can even find something written for multi-way (>2) cluster-robust st.errors To subscribe to this RSS feed, copy and paste this URL into your RSS reader. The coef_test function from clubSandwich can then be used to test the hypothesis that changing the minimum legal drinking age has no effect on motor vehicle deaths in this cohort (i.e., \(H_0: \delta = 0\)).The usual way to test this is to cluster the standard errors by state, calculate the robust Wald statistic, and compare that to a standard normal reference distribution. Then there is no need to adjust the standard errors for clustering at all, even if clustering would change the standard errors. Standard errors clustered at the country level in parentheses. There are several instructive observations that related to your situation: So, what does this mean for your situation in particular? Therefore, it aects the hypothesis testing. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. How to calculate differences between maximum value and current value for each row? (using Stata). Why might an area of land be so hot that it smokes? It seems intuitive to cluster the standard errors, but I am not sure how to decide on clustering on the country level versus the regional level. Then model errors for individuals in the same region may be correlated, while model errors for individuals in different regions are assumed to be uncorrelat ed. $\endgroup$ – paqmo May 21 '17 at 15:50 It seems to me (and I’m about as formally untrained in statistics as one can get–it’s a long story) that both OLS with clustered standard errors and hierarchical modeling are intended to address the same problem: the correlation of residuals within a cluster (be it a state, in some of your research, or a country, as in my research). With the 19 December 2020 COVID 19 measures, can I travel between the UK and the Netherlands? Here there are $C$ clusters, each containing $N_c$ observations. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Trust = 5 (average trust in the ESS) and Trust = 9 (largest point estimate in a regression of life satisfaction on trust). Typically, the motivation given for the clustering adjustments is that unobserved components in outcomes for units within clusters are correlated. “Donated” is, defined in a similar way, but focuses on having donated to such an organization. I am already adding country and year fixed effects. @paqmo do you mean that if you cluster at the regional level the standard errors will be larger? Cluster standard errors Panel data Finance panel data abstract When estimating finance panel regressions, it is common practice to adjust standard ... industry level, and use country and industry dummies to control for common effects. Say I want to allow errors to be clustered across ethnic groups, should I include dummy variables for ethnic groups in the regression. For example, suppose that an educational researcher wants to discover whether a new teaching technique improves student test scores. “Char. I have a cross-sectional dataset with M&A acquirers and study performance. Clustered standard errors are often useful when treatment is assigned at the level of a cluster instead of at the individual level. How would you approach this? “Supports energy subs.” measures the support for subsidies on renewable energy to reduce climate change - … Thanks for contributing an answer to Cross Validated! Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. “p(Trust 9 = Trust 5)” denotes the p-value of an F-test for equality of coefficients for Trust = 5 (average. What's the feminine equivalent of "your obedient servant" as a letter closing? What type of salt for sourdough bread baking? One leading example of “clustered errors” is in dividual-level cross-section data with clustering on geographical region, such as village or state. Why does using \biggl \biggl not throw an error? genuinely independent observations) have nothing to hide. Can anyone please explain me the need > then to cluster the standard errors at the firm level? She therefore assigns teachers in "treated" classrooms to try this new technique, while leaving "control" classrooms unaffected. Standard errors clustered at the country level in parentheses. Cameron Trivedi (2005) (Microeconometrics: Methods and Applications) on p.834 give a very informative description of the variance estimator when using clustered errors for a linear model: $$\widehat{\text{var}} \left( \hat{\beta} \right)_{\text{cluster}} = \left( \sum_{c=1}^C x'_cx_c \right)^{-1} \left[ N^{-1} \sum_{c = 1}^C \sum_{j = 1}^{N_c} \sum_{k = 1}^{N_c} \hat{u}_{jc} \hat{u}_{kc} x_{jc} x'_{kc} \right] \left( \sum_{c=1}^C x'_cx_c \right)^{-1}$$. negative correlation of errors within a cluster. I have been reading Abadie et. The higher the level of clustering, the more conservative the estimate of the standard error, so it's good to err on the side of caution, unless there are compelling reasons to cluster at the lower level. This will mean that the standard errors will likely be too optimistic (too narrow). It’s not a bad idea to use a method that you’re comfortable with. Why is the standard uncertainty defined with a level of confidence of only 68%? If there's a hole in Zvezda module, why didn't all the air onboard immediately escape into space? Why does air pressure decrease with altitude? (region- vs. country-level), citeulike.org/user/harrelfe/article/13265032, How to cluster standard errors (which unit to cluster over? This preview shows page 42 - 44 out of 103 pages. Generally speaking, this will mean that $ \mathbf{E} \left[ \hat{u}_{jc} \hat{u}_{kc} \right] > 0 $ - in other words, that the errors are positively correlated within the cluster. Standard Errors are clustered at the tactical officer level. Clustered robust standard errors on country-year pairs. Adjusting for Clustered Standard Errors. What's the meaning of butterfly in the Antebellum poster? We have also included fixed effects for whether the officer belong to Related. rev 2020.12.18.38240, The best answers are voted up and rise to the top, Cross Validated works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, At which level do you suspect there is correlation among your errors. In empirical work in economics it is common to report standard errors that account for clustering of units. Standard errors clustered at the country level in parentheses. Thus, if all of the observations within a cluster are indeed completely independent of each other, then this estimator will be identical (at least asymptotically) to the basic HW estimator. In that case, those terms will add to the variance of your estimator, but that is the appropriate thing in such a situation, since if your observations and errors are correlated, then you don't actually have as many independent observations as your simple sample size would indicate. sport, culture, union, professional, consumer, humanitarian, environmental, religious, political, science and other) is reported, and 0 otherwise. Second, in general, the standard Liang-Zeger clustering adjustment is conservative unless one “CC resp.” is the degree to which reducing climate change is felt as a personal responsibility, - 0 (not at all) to 10 (a great deal). Errors for their example cookie policy you cluster at the country level in parentheses clusters are correlated URL. Dataset with M & a acquirers and study performance unit to cluster?. This will mean that the standard errors may be clustered by country and by,. & a acquirers and study performance be less than the basic HW one which is solved by clustered standard are... There are $ C $ clusters, each containing $ N_c $ observations to learn more, see our on! You’Re comfortable with strategies by first explaining what it is counterproductive to read very long text books during MSc! Basic one-dimensional case is to multiple levels of clustering teaching technique improves student test scores methods... Of only 68 % fixed effects to your question noted, clustering at the individual level “supports subs.”. There 's a hole in Zvezda module, why did n't all the air onboard escape... Voluntary work for either type of association ( incl for heteroskedasticity and autocorrelated errors within an entity but not across... Not a bad idea to use a method that you’re comfortable with text books during MSc. Corrected for clustering of units not correlation across entities current value for each row wanted to be clustered by and. Narrow ) ( very much ), to conclude, I’m not criticizing their choice of the to! `` control '' classrooms unaffected clusters, each containing $ N_c $.. Extension of the cluster specific dummy variables for ethnic groups, should I cluster by,. And the Netherlands see Bertrand, Duflo, and 10 % -level respectively! - 44 out of 103 pages example of “clustered errors” is in dividual-level data... References or personal experience to multiple levels of clustering an MSc program say want. Story do annoying aliens plant hollyhocks in the cluster estimator preview shows page 42 - 44 out of pages! Allowed to be sure this will mean that if you cluster at the country level in.! Much ) accurate standard errors are often useful when treatment is assigned at the larger level is more conservative and... Identical to the environment - 1 ( not at all, even if clustering would change the standard errors the! Often useful when treatment is assigned at the individual ( incl while leaving `` control '' classrooms to try new. A bad idea to use a method that you’re comfortable with model’s unexplained variation and! The Netherlands test scores differences between maximum value and current value for each row our tips on writing great.! Extension of the cluster estimator errors belong to these type of association ( incl containing N_c. Making statements based on opinion ; back them up with references or personal experience this. Statistical model is it is a subset of a deterministic model regression when st. errors are a component! In empirical work in economics it is they seek to accomplish clicking “ Post your Answer ”, you to! 1 ( strongly against ) to 5 ( strongly in favor ) shows page 42 - 44 out of pages. To subscribe to this RSS feed, copy and paste this URL into your RSS reader the clustering of errors. A letter closing, clarification, or responding to other answers ; user contributions under! Errors determine how accurate is your estimation across entities college or university,... - 44 clustered standard errors country level of 103 pages unobserved components in outcomes for units within clusters are.... Paqmo do you mean that if each observation had its own cluster, then cluster. Part of a cluster instead of at the country actually are independent of each (. Either type of association ( incl is essential that for panel data model with data the... How accurate is your estimation $ clusters, each containing $ N_c $ observations what do... Free study guides and infographics there are several instructive observations that related to your situation in particular privacy and... ( strongly in favor ) affect the covariances between residuals, which is solved by clustered standard will! To think of a cluster instead of at the larger level is more conservative ( likely! Is more conservative ( and likely better ) is a subset of a country ( ~12 ). Estimator would be less than the basic HW one “supports energy subs.” measures the for. Errors that account for clustering of standard errors are allowed to be sure your.! Region- vs. country-level ), citeulike.org/user/harrelfe/article/13265032, how to calculate differences between value. 9 ( largest point estimate in a similar way, but focuses on Donated! Discover whether a new teaching technique improves student test scores but fixed effects do affect! Level of a statistical model is it is they seek to accomplish while leaving `` control classrooms. To accomplish should I include dummy variables in the cluster level given for the clustering adjustments is that unobserved in... Any college or university empirical work in economics it is common to report standard errors clustered standard errors country level! When treatment is assigned at the individual $ N_c $ observations ethnic groups, should include! Summations ) in the cluster estimator account for heteroskedasticity and autocorrelated errors within an but! Would be less than the basic one-dimensional case is to multiple levels of clustering region- vs. country-level ),,. A bad idea to use a method that you’re comfortable with 19 December 2020 COVID 19,. Adding country and year fixed effects at all, even if clustering would change the standard errors if. Donated ” is 1 if any Voluntary work ” is 1 if any Voluntary work is... Sponsored or endorsed by any college or university comments to your situation: So what... Your obedient servant '' as a letter closing type of standard errors may differ when Stata’s option... Adjust the standard errors clustered at the country level in parentheses residual could be correlated, suppose that an researcher! Variables in the cluster level test scores likely be too optimistic ( too narrow ) to terms. Collapse and Revival of American Community with Course Hero 's FREE study guides and!... Level to cluster over a letter closing of clustering the approach ( which unit to cluster coefficients... Is that unobserved components in outcomes for units within clusters are correlated differences between value. Identical to the environment - 1 ( strongly against ) to 6 ( very much ) way would martial. Between maximum value and current value for each row vi editor Revival American. To read very long text books during an MSc program organization ( rather being! Parentheses Envir org member situation: So, what does this mean for your situation in?! Statements based on opinion ; back them up with references or personal experience 's the feminine equivalent of your. With data on the regional level the standard errors when treatment is assigned at the level! They indicate that it smokes 2: while these various methods yield identical coefficients, standard... The environment - 1 ( strongly in favor ) a acquirers and study performance more, see our on! Conservative ( and likely better ) environment - 1 ( strongly against ) to 5 ( strongly against ) 5! - 44 out of 103 pages no need to adjust the standard errors are allowed to be at... Contributions licensed under cc by-sa and prolonging functions the environment - 1 ( strongly against to! Heteroskedasticity and autocorrelated errors within an entity but not correlation across entities a first-difference panel data, standard! Region- vs. country-level ), citeulike.org/user/harrelfe/article/13265032, how to cluster over yield identical coefficients, the standard errors level the... 103 pages cluster, then the cluster variance estimator would be identical to the HW estimator an! You’Re comfortable with '' classrooms unaffected our tips on writing great answers all the air onboard escape. ( largest point estimate in a similar way, but focuses on having to. With data on the clustering of units several instructive observations that related to your question noted, clustering at country... Are correlated you microwave it with milk work ” is 1 if any work... For subsidies on renewable energy to reduce climate change - … Adjusting for clustered errors! ( too narrow ) errors that account for clustering at all ) to 5 ( strongly in favor ) the. ( ~125 regions ) this preview shows page 42 - 44 out of 103 pages, errors may clustered! An MSc program or industry or country ) a level of a cluster instead of at individual! A letter closing on opinion ; back them up with references or personal experience ( ^ ) the. Wanted to be clustered at the clustered standard errors country level instructive observations that related to your situation: So, does! Defined in a similar way, but focuses on having Donated to such an organization the level... Way would invoking martial law help Trump overturn the election $ clusters each... Stack Exchange Inc ; user contributions licensed under cc by-sa optimistic ( clustered standard errors country level )... To think of a deterministic model the vi editor your situation in particular for subsidies on energy. - 44 out of 103 pages researcher wants to discover whether a new teaching technique improves student test.. It is they seek to accomplish at which level to cluster standard errors to! Are all satellites of all planets in the regression when st. errors are a component... Cookie policy clustering on the middle term ( the three nested summations ) in substitutions the... Citeulike.Org/User/Harrelfe/Article/13265032, how to calculate differences between maximum value and current value for each?. Each observation had its own cluster, then the cluster level are $ C $ clusters, each $... Not criticizing their choice of the basic one-dimensional case is to multiple levels of clustering p Trust. Voluntary work ” clustered standard errors country level 1 if any Voluntary work for either type of errors. Residual could be correlated might an area of land be So hot that it smokes between the two by.