[R-sig-Geo] Spatial Autocorrelation Estimation Method

Tue Nov 26 20:48:14 CET 2019

Sorry for late reply, am indisposed and unable to help further. I feel 
that there is so much noise in your data (differences in offers, rental 
lengths, repeats or not, etc.), that you will certainly have to subset 
vigorously first to isolate response cases that are comparable. What you 
are trying to disentangle are the hedonic components in the bundle where 
you just have price as response, but lots of other bundle characteristics 
on the right hand side (days, etc.). I feel you'd need to try to get to a 
response index of price per day per rental area or some such. I'd 
certainly advise examining responses to a specific driver (major concert 
or sports event) to get a feel for how the market responds, and return to 
spatial hedonic after finding an approach that gives reasonable aspatial 
outcomes.

Roger

On Sun, 17 Nov 2019, Robert R wrote:

> Dear Roger,
>
> Thank you for your message and sorry for my late answer.
>
> Regarding the number of listings (lettings) for my data set (2.216.642 observations), each listing contains an individual id:
>
> unique ids: 180.004
> time periods: 54 (2015-01 to 2019-09)
> number of ids that appear only once: 28.486 (of 180.004 ids) (15,8%)
> number of ids that appear/repeat 2-10 times: 82.641 (of 180.004 ids) (45,9%)
> number of ids that appear/repeat 11-30 times: 46.465 (of 180.004 ids) (25,8%)
> number of ids that appear/repeat 31-54 times: 22.412 (of 180.004 ids) (12,5%)
>
> Important to notice is that hosts can change the room_category (between entire/home apt, private room and shared room) keeping the same listing id number. In my data, the number of unique ids that in some point changed the room_type is of 7.204 ids.
>
> --
>
> For the OLS model, I was using only a fixed effect model, where each time period (date_compiled) (54 in total) is a time dummy.
>
> plm::plm(formula = model, data = listings, model = "pooling", index = c("id", "date_compiled"))
>
>
> --
> Osland et al. (2016) (https://doi.org/10.1111/jors.12281) use a spatial fixed effects (SFE) hedonic model, where each defined neighborhood zone in the study area is represented by dummy variables.
>
> Dong et al. (2015) (https://doi.org/10.1111/gean.12049) outline four model specifications to accommodate geographically hierarchical data structures: (1) groupwise W and fixed regional effects; (2) groupwise W and random regional effects; (3) proximity-based W and fixed regional effects; and (4) proximity-based W and random regional effects.
> --
>
> I created a new column/variable containing the borough where the zipcode is found (Manhattan, Brooklyn, Queens, Bronx, Staten Island).
>
> If I understood it right, the (two-level) Hierarchical Spatial Simultaneous Autoregressive Model (HSAR) considers the occurrence of spatial relations at the (lower) individual (geographical coordinates - in my case, the listing location) and (higher) group level (territorial units - in my case, zipcodes).
>
> According to Bivand et al. (2017): "(...) W is a spatial weights matrix. The HSAR model may also be estimated without this component.". So, in this case I only estimate the Hierarchical Spatial Simultaneous Autoregressive Model (HSAR) in a "one-level" basis, i.e., at the higher-level.
>
> HSAR::hsar(model, data = listings, W = NULL, M = M, Delta = Delta, burnin = 5000, Nsim = 10000, thinning = 1, parameters.start = pars)
>
> (Where the "model" formula contains the 54 time dummy variables)
>
> Do you think I can proceed with this model? I was able to calculate it.
>
> If I remove all observations/rows with NAs in one of the chosen variables/observations, 884.183 observations remain. If I would create a W matrix for HSAR::hsar, I would have a gigantic 884.183 by 884.183 matrix. This is the reason why I put W = NULL.
>
>
> Thank you and best regards
>
> ________________________________________
> From: Roger Bivand <Roger.Bivand using nhh.no>
> Sent: Monday, November 11, 2019 11:31
> To: Robert R
> Cc: r-sig-geo using r-project.org
> Subject: Re: [R-sig-Geo] Spatial Autocorrelation Estimation Method
>
> On Sun, 10 Nov 2019, Robert R wrote:
>
>> Dear Roger,
>>
>> Again, thank you for your answer. I read the material provided and
>> decided that Hierarchical Spatial Autoregressive (HSAR) could be the
>> right model for me.
>>
>> I indeed have the precise latitude and longitude information for all my
>> listings for NYC.
>>
>> I created a stratified sample (group = zipcode) with 22172 (1%) of my
>> observations called listings_sample and tried to replicate the hsar
>> model, please see below.
>>
>> For now W = NULL, because otherwise I would have a 22172 x 22172 matrix.
>
> Unless you know definitely that you want to relate the response to its
> lagged value, you do not need this. Do note that the matrix is very
> sparse, so could be fitted without difficulty with ML in a cross-sectional
> model.
>
>>
>> You recommended then to introduce a Markov random field (MRF) random
>> effect (RE) at the zipcode level, but I did not understand it so well.
>> Could you develop a litte more?
>>
>
> Did you read the development in
> https://doi.org/10.1016/j.spasta.2017.01.002? It is explained there, and
> includes code for fitting the Beijing housing parcels data se from HSAR
> with many other packages (MCMC, INLA, hglm, etc.). I guess that you should
> try to create a model that works on a single borough, sing the zipcodes
> in that borough as a proxy for unobserved neighbourhood effects. Try for
> example using lme4::lmer() with only a zipcode IID random effect, see if
> the hedonic estimates are similar to lm(), and leave adding an MRF RE
> (with for example mgcv::gam() or hglm::hglm()) until you have a working
> testbed. Then advance step-by-step from there.
>
> You still have not said how many repeat lettings you see - it will affect
> the way you specify your model.
>
> Roger
>
>> ##############
>> library(spdep)
>> library(HSAR)
>> library(dplyr)
>> library(splitstackshape)
>>
>>
>> # Stratified sample per zipcode (size = 1%) listings_sample <-
>> splitstackshape::stratified(indt = listings, group = "zipcode", size =
>> 0.01)
>>
>> # Removing zipcodes from polygon_nyc which are not observable in
>> listings_sample polygon_nyc_listings <- polygon_nyc %>% filter(zipcode
>> %in% c(unique(as.character(listings_sample$zipcode))))
>>
>>
>> ## Random effect matrix (N by J)
>>
>> # N: 22172
>> # J: 154
>>
>> # Arrange listings_sample by zipcode (ascending)
>> listings_sample <- listings_sample %>% arrange(zipcode)
>>
>> # Count number of listings per zipcode
>> MM <- listings_sample %>% st_drop_geometry() %>% group_by(zipcode) %>% summarise(count = n()) %>% as.data.frame()
>> # sum(MM$count)
>>
>> # N by J nulled matrix creation
>> Delta <- matrix(data = 0, nrow = nrow(listings_sample), ncol = dim(MM)[1])
>>
>> # The total number of neighbourhood
>> Uid <- rep(c(1:dim(MM)[1]), MM[,2])
>>
>> for(i in 1:dim(MM)[1]) {
>>  Delta[Uid==i,i] <- 1
>> }
>> rm(i)
>>
>> Delta <- as(Delta,"dgCMatrix")
>>
>>
>> ## Higher-level spatial weights matrix or neighbourhood matrix (J by J)
>>
>> # Neighboring polygons: list of neighbors for each polygon (queen contiguity neighbors)
>> polygon_nyc_nb <- poly2nb(polygon_nyc_listings, row.names = polygon_nyc$zipcode, queen = TRUE)
>>
>> # Include neighbour itself as a neighbour
>> polygon_nyc_nb <- include.self(polygon_nyc_nb)
>>
>> # Spatial weights matrix for nb
>> polygon_nyc_nb_matrix <- nb2mat(neighbours = polygon_nyc_nb, style = "W", zero.policy = NULL)
>> M <- as(polygon_nyc_nb_matrix,"dgCMatrix")
>>
>>
>> ## Fit HSAR SAR upper level random effect
>> model <- as.formula(log_price ~ guests_included + minimum_nights)
>>
>> betas = coef(lm(formula = model, data = listings_sample))
>> pars = list(rho = 0.5, lambda = 0.5, sigma2e = 2.0, sigma2u = 2.0, betas = betas)
>>
>> m_hsar <- hsar(model, data = listings_sample, W = NULL, M = M, Delta = Delta, burnin = 5000, Nsim = 10000, thinning = 1, parameters.start = pars)
>>
>> ##############
>>
>> Thank you and best regards
>> Robert
>>
>> ________________________________________
>> From: Roger Bivand <Roger.Bivand using nhh.no>
>> Sent: Friday, November 8, 2019 13:29
>> To: Robert R
>> Cc: r-sig-geo using r-project.org
>> Subject: Re: [R-sig-Geo] Spatial Autocorrelation Estimation Method
>>
>> On Fri, 8 Nov 2019, Robert R wrote:
>>
>>> Dear Roger,
>>>
>>> Thank you for your answer.
>>>
>>> I successfully used the function nb2blocknb() for a smaller dataset.
>>>
>>> But for a dataset of over 2 million observations, I get the following
>>> error: "Error: cannot allocate vector of size 840 Kb".
>>
>> I don't think the observations are helpful. If you have repeat lets in the
>> same property in a given month, you need to handle that anyway. I'd go for
>> making the modelling exercise work (we agree that this is not panel data,
>> right?) on a small subset first. I would further argue that you need a
>> multi-level approach rather than spdep::nb2blocknb(), with a zipcode IID
>> RE. You could very well take (stratified) samples per zipcode to represent
>> your data. Once that works, introduce an MRF RE at the zipcode level,
>> where you do know relative position. Using SARAR is going to be a waste of
>> time unless you can geocode the letting addresses. A multi-level approach
>> will work. Having big data in your case with no useful location
>> information per observation is just adding noise and over-smoothing, I'm
>> afraid. The approach used in https://doi.org/10.1016/j.spasta.2017.01.002
>> will work, also when you sample the within zipcode lets, given a split
>> into training and test sets, and making CV possible.
>>
>> Roger
>>
>>>
>>> I am expecting that at least 500.000 observations will be dropped due
>>> the lack of values for the chosen variables for the regression model, so
>>> probably I will filter and remove the observations/rows that will not be
>>> used anyway - do you know if there is any package that does this
>>> automatically, given the variables/columns chosed by me?
>>>
>>> Or would you recommend me another approach to avoid the above mentioned
>>> error?
>>>
>>> Thank you and best regards,
>>> Robert
>>>
>>> ________________________________________
>>> From: Roger Bivand <Roger.Bivand using nhh.no>
>>> Sent: Thursday, November 7, 2019 10:13
>>> To: Robert R
>>> Cc: r-sig-geo using r-project.org
>>> Subject: Re: [R-sig-Geo] Spatial Autocorrelation Estimation Method
>>>
>>> On Thu, 7 Nov 2019, Robert R wrote:
>>>
>>>> Dear Roger,
>>>>
>>>> Many thanks for your help.
>>>>
>>>> I have an additional question:
>>>>
>>>> Is it possible to create a "separate" lw (nb2listw) (with different
>>>> rownumbers) from my data set? For now, I am taking my data set and
>>>> merging with the sf object polygon_nyc with the function
>>>> "merge(polygon_nyc, listings, by=c("zipcode" = "zipcode"))", so I create
>>>> a huge n x n matrix (depending of the size of my data set).
>>>>
>>>> Taking the polygon_nyc alone and turning it to a lw (weights list)
>>>> object has only n = 177.
>>>>
>>>> Of course running
>>>>
>>>> spatialreg::lagsarlm(formula=model, data = listings_sample,
>>>> spatialreg::polygon_nyc_lw, tol.solve=1.0e-10)
>>>>
>>>> does not work ("Input data and weights have different dimensions").
>>>>
>>>> The only option is to take my data set, merge it to my polygon_nyc (by
>>>> zipcode) and then create the weights list lw? Or there another option?
>>>
>>> I think we are getting more clarity. You do not know the location of the
>>> lettings beyond their zipcode. You do know the boundaries of the zipcode
>>> areas, and can create a neighbour object from these boundaries. You then
>>> want to treat all the lettings in a zipcode area i as neighbours, and
>>> additionally lettings in zipcode areas neighbouring i as neighbours of
>>> lettings in i. This is the data structure that motivated the
>>> spdep::nb2blocknb() function:
>>>
>>> https://r-spatial.github.io/spdep/reference/nb2blocknb.html
>>>
>>> Try running the examples to get a feel for what is going on.
>>>
>>> I feel that most of the variability will vanish in the very large numbers
>>> of neighbours, over-smoothing the outcomes. If you do not have locations
>>> for the lettings themselves, I don't think you can make much progress.
>>>
>>> You could try a linear mixed model (or gam with a spatially structured
>>> random effect) with a temporal and a spatial random effect. See the HSAR
>>> package, articles by Dong et al., and maybe
>>> https://doi.org/10.1016/j.spasta.2017.01.002 for another survey. Neither
>>> this nor Dong et al. handle spatio-temporal settings. MRF spatial random
>>> effects at the zipcode level might be a way forward, together with an IID
>>> random effect at the same level (equivalent to sef-neighbours).
>>>
>>> Hope this helps,
>>>
>>> Roger
>>>
>>>>
>>>> Best regards,
>>>> Robert
>>>>
>>>> ________________________________________
>>>> From: Roger Bivand <Roger.Bivand using nhh.no>
>>>> Sent: Wednesday, November 6, 2019 15:07
>>>> To: Robert R
>>>> Cc: r-sig-geo using r-project.org
>>>> Subject: Re: [R-sig-Geo] Spatial Autocorrelation Estimation Method
>>>>
>>>> On Tue, 5 Nov 2019, Robert R wrote:
>>>>
>>>>> Dear Roger,
>>>>>
>>>>> Thank you for your reply. I disabled HTML; my e-mails should be now in
>>>>> plain text.
>>>>>
>>>>> I will give a better context for my desired outcome.
>>>>>
>>>>> I am taking Airbnb's listings information for New York City available
>>>>> on: http://insideairbnb.com/get-the-data.html
>>>>>
>>>>> I save every listings.csv.gz file available for NYC (2015-01 to 2019-09)
>>>>> - in total, 54 files/time periods - as a YYYY-MM-DD.csv file into a
>>>>> Listings/ folder. When importing all these 54 files into one single data
>>>>> set, I create a new "date_compiled" variable/column.
>>>>>
>>>>> In total, after the data cleansing process, I have a little more 2
>>>>> million observations.
>>>>
>>>> You have repeat lettings for some, but not all properties. So this is at
>>>> best a very unbalanced panel. For those properties with repeats, you may
>>>> see temporal movement (trend/seasonal).
>>>>
>>>> I suggest (strongly) taking a single borough or even zipcode with some
>>>> hindreds of properties, and working from there. Do not include the
>>>> observation as its own neighbour, perhaps identify repeats and handle them
>>>> specially (create or use a property ID). Unbalanced panels may also create
>>>> a selection bias issue (why are some properties only listed sometimes?).
>>>>
>>>> So this although promising isn't simple, and getting to a hedonic model
>>>> may be hard, but not (just) because of spatial autocorrelation. I wouldn't
>>>> necessarily trust OLS output either, partly because of the repeat property
>>>> issue.
>>>>
>>>> Roger
>>>>
>>>>>
>>>>> I created 54 timedummy variables for each time period available.
>>>>>
>>>>> I want to estimate using a hedonic spatial timedummy model the impact of
>>>>> a variety of characteristics which potentially determine the daily rate
>>>>> on Airbnb listings through time in New York City (e.g. characteristics
>>>>> of the listing as number of bedrooms, if the host if professional,
>>>>> proximity to downtown (New York City Hall) and nearest subway station
>>>>> from the listing, income per capita, etc.).
>>>>>
>>>>> My dependent variable is price (log price, common in the related
>>>>> literature for hedonic prices).
>>>>>
>>>>> The OLS model is done.
>>>>>
>>>>> For the spatial model, I am assuming that hosts, when deciding the
>>>>> pricing of their listings, take not only into account its structural and
>>>>> location characteristics, but also the prices charged by near listings
>>>>> with similar characteristics - spatial autocorrelation is then present,
>>>>> at least spatial dependence is present in the dependent variable.
>>>>>
>>>>> As I wrote in my previous post, I was willing to consider the neighbor
>>>>> itself as a neighbor.
>>>>>
>>>>> Parts of my code can be found below:
>>>>>
>>>>> ########
>>>>>
>>>>> ## packages
>>>>>
>>>>> packages_install <- function(packages){
>>>>> new.packages <- packages[!(packages %in% installed.packages()[, "Package"])]
>>>>> if (length(new.packages))
>>>>> install.packages(new.packages, dependencies = TRUE)
>>>>> sapply(packages, require, character.only = TRUE)
>>>>> }
>>>>>
>>>>> packages_required <- c("bookdown", "cowplot", "data.table", "dplyr", "e1071", "fastDummies", "ggplot2", "ggrepel", "janitor", "kableExtra", "knitr", "lubridate", "nngeo", "plm", "RColorBrewer", "readxl", "scales", "sf", "spdep", "stargazer", "tidyverse")
>>>>> packages_install(packages_required)
>>>>>
>>>>> # Working directory
>>>>> setwd("C:/Users/User/R")
>>>>>
>>>>>
>>>>>
>>>>> ## shapefile_us
>>>>>
>>>>> # Shapefile zips import and Coordinate Reference System (CRS) transformation
>>>>> # Shapefile download: https://www2.census.gov/geo/tiger/GENZ2018/shp/cb_2018_us_zcta510_500k.zip
>>>>> shapefile_us <- sf::st_read(dsn = "Shapefile", layer = "cb_2018_us_zcta510_500k")
>>>>>
>>>>> # Columns removal
>>>>> shapefile_us <- shapefile_us %>% select(-c(AFFGEOID10, GEOID10, ALAND10, AWATER10))
>>>>>
>>>>> # Column rename: ZCTA5CE10
>>>>> setnames(shapefile_us, old=c("ZCTA5CE10"), new=c("zipcode"))
>>>>>
>>>>> # Column class change: zipcode
>>>>> shapefile_us$zipcode <- as.character(shapefile_us$zipcode)
>>>>>
>>>>>
>>>>>
>>>>> ## polygon_nyc
>>>>>
>>>>> # Zip code not available in shapefile: 11695
>>>>> polygon_nyc <- shapefile_us %>% filter(zipcode %in% zips_nyc)
>>>>>
>>>>>
>>>>>
>>>>> ## weight_matrix
>>>>>
>>>>> # Neighboring polygons: list of neighbors for each polygon (queen contiguity neighbors)
>>>>> polygon_nyc_nb <- poly2nb((polygon_nyc %>% select(-borough)), queen=TRUE)
>>>>>
>>>>> # Include neighbour itself as a neighbour
>>>>> # for(i in 1:length(polygon_nyc_nb)){polygon_nyc_nb[[i]]=as.integer(c(i,polygon_nyc_nb[[i]]))}
>>>>> polygon_nyc_nb <- include.self(polygon_nyc_nb)
>>>>>
>>>>> # Weights to each neighboring polygon
>>>>> lw <- nb2listw(neighbours = polygon_nyc_nb, style="W", zero.policy=TRUE)
>>>>>
>>>>>
>>>>>
>>>>> ## listings
>>>>>
>>>>> # Data import
>>>>> files <- list.files(path="Listings/", pattern=".csv", full.names=TRUE)
>>>>> listings <- setNames(lapply(files, function(x) read.csv(x, stringsAsFactors = FALSE, encoding="UTF-8")), files)
>>>>> listings <- mapply(cbind, listings, date_compiled = names(listings))
>>>>> listings <- listings %>% bind_rows
>>>>>
>>>>> # Characters removal
>>>>> listings$date_compiled <- gsub("Listings/", "", listings$date_compiled)
>>>>> listings$date_compiled <- gsub(".csv", "", listings$date_compiled)
>>>>> listings$price <- gsub("\\$", "", listings$price)
>>>>> listings$price <- gsub(",", "", listings$price)
>>>>>
>>>>>
>>>>>
>>>>> ## timedummy
>>>>>
>>>>> timedummy <- sapply("date_compiled_", paste, unique(listings$date_compiled), sep="")
>>>>> timedummy <- paste(timedummy, sep = "", collapse = " + ")
>>>>> timedummy <- gsub("-", "_", timedummy)
>>>>>
>>>>>
>>>>>
>>>>> ## OLS regression
>>>>>
>>>>> # Pooled cross-section data - Randomly sampled cross sections of Airbnb listings price at different points in time
>>>>> regression <- plm(formula=as.formula(paste("log_price ~ #some variables", timedummy, sep = "", collapse = " + ")), data=listings, model="pooling", index="id")
>>>>>
>>>>> ########
>>>>>
>>>>> Some of my id's repeat in multiple time periods.
>>>>>
>>>>> I use NYC's zip codes to left join my data with the neighborhood zip code specific characteristics, such as income per capita to that specific zip code, etc.
>>>>>
>>>>> Now I want to apply the hedonic model with the timedummy variables.
>>>>>
>>>>> Do you know how to proceed? 1) Which package to use (spdep/splm)?; 2) Do I have to join the polygon_nyc (by zip code) to my listings data set, and then calculate the weight matrix "lw"?
>>>>>
>>>>> Again, thank you very much for the help provided until now.
>>>>>
>>>>> Best regards,
>>>>> Robert
>>>>>
>>>>> ________________________________________
>>>>> From: Roger Bivand <Roger.Bivand using nhh.no>
>>>>> Sent: Tuesday, November 5, 2019 15:30
>>>>> To: Robert R
>>>>> Cc: r-sig-geo using r-project.org
>>>>> Subject: Re: [R-sig-Geo] Spatial Autocorrelation Estimation Method
>>>>>
>>>>> On Tue, 5 Nov 2019, Robert R wrote:
>>>>>
>>>>>> I have a large pooled cross-section data set. ?I would like to
>>>>>> estimate/regress using spatial autocorrelation methods. I am assuming
>>>>>> for now that spatial dependence is present in both the dependent
>>>>>> variable and the error term.? ?My data set is over a period of 4 years,
>>>>>> monthly data (54 periods). For this means, I've created a time dummy
>>>>>> variable for each time period.? ?I also created a weight matrix using the
>>>>>> functions "poly2nb" and "nb2listw".? ?Now I am trying to figure out a way
>>>>>> to estimate my model which contains a really big data set.? ?Basically, my
>>>>>> model is as follows: y = ?D + ?W1y + X? + ?W2u + ?? ?My questions are:? ?1)
>>>>>> My spatial weight matrix for the whole data set will be probably a
>>>>>> enormous matrix with submatrices for each time period itself. I don't
>>>>>> think it would be possible to calculate this.? What I would like to know
>>>>>> is a way to estimate each time dummy/period separately (to compare
>>>>>> different periods alone). How to do it?? ?2) Which package to use: spdep
>>>>>> or splm?? ?Thank you and best regards,? Robert?
>>>>>
>>>>> Please do not post HTML, only plain text. Almost certainly your model
>>>>> specification is wrong (SARAR/SAC is always a bad idea if alternatives are
>>>>> untried). What is your cross-sectional size? Using sparse kronecker
>>>>> products, the "enormous" matrix may not be very big. Does it make any
>>>>> sense using time dummies (54 x N x T will be mostly zero anyway)? Are most
>>>>> of the covariates time-varying? Please provide motivation and use area
>>>>> (preferably with affiliation (your email and user name are not
>>>>> informative) - this feels like a real estate problem, probably wrongly
>>>>> specified. You should use splm if time make sense in your case, but if it
>>>>> really doesn't, simplify your approach, as much of the data will be
>>>>> subject to very large temporal autocorrelation.
>>>>>
>>>>> If this is a continuation of your previous question about using
>>>>> self-neighbours, be aware that you should not use self-neighbours in
>>>>> modelling, they are only useful for the Getis-Ord local G_i^* measure.
>>>>>
>>>>> Roger
>>>>>
>>>>>>
>>>>>>       [[alternative HTML version deleted]]
>>>>>>
>>>>>> _______________________________________________
>>>>>> R-sig-Geo mailing list
>>>>>> R-sig-Geo using r-project.org
>>>>>> https://stat.ethz.ch/mailman/listinfo/r-sig-geo
>>>>>
>>>>> --
>>>>> Roger Bivand
>>>>> Department of Economics, Norwegian School of Economics,
>>>>> Helleveien 30, N-5045 Bergen, Norway.
>>>>> voice: +47 55 95 93 55; e-mail: Roger.Bivand using nhh.no
>>>>> https://orcid.org/0000-0003-2392-6140
>>>>> https://scholar.google.no/citations?user=AWeghB0AAAAJ&hl=en
>>>>>
>>>>
>>>> --
>>>> Roger Bivand
>>>> Department of Economics, Norwegian School of Economics,
>>>> Helleveien 30, N-5045 Bergen, Norway.
>>>> voice: +47 55 95 93 55; e-mail: Roger.Bivand using nhh.no
>>>> https://orcid.org/0000-0003-2392-6140
>>>> https://scholar.google.no/citations?user=AWeghB0AAAAJ&hl=en
>>>>
>>>
>>> --
>>> Roger Bivand
>>> Department of Economics, Norwegian School of Economics,
>>> Helleveien 30, N-5045 Bergen, Norway.
>>> voice: +47 55 95 93 55; e-mail: Roger.Bivand using nhh.no
>>> https://orcid.org/0000-0003-2392-6140
>>> https://scholar.google.no/citations?user=AWeghB0AAAAJ&hl=en
>>>
>>
>> --
>> Roger Bivand
>> Department of Economics, Norwegian School of Economics,
>> Helleveien 30, N-5045 Bergen, Norway.
>> voice: +47 55 95 93 55; e-mail: Roger.Bivand using nhh.no
>> https://orcid.org/0000-0003-2392-6140
>> https://scholar.google.no/citations?user=AWeghB0AAAAJ&hl=en
>>
>
> --
> Roger Bivand
> Department of Economics, Norwegian School of Economics,
> Helleveien 30, N-5045 Bergen, Norway.
> voice: +47 55 95 93 55; e-mail: Roger.Bivand using nhh.no
> https://orcid.org/0000-0003-2392-6140
> https://scholar.google.no/citations?user=AWeghB0AAAAJ&hl=en
>

-- 
Roger Bivand
Department of Economics, Norwegian School of Economics,
Helleveien 30, N-5045 Bergen, Norway.
voice: +47 55 95 93 55; e-mail: Roger.Bivand using nhh.no
https://orcid.org/0000-0003-2392-6140
https://scholar.google.no/citations?user=AWeghB0AAAAJ&hl=en