Hedonic Home Price Prediction for Boulder, CO
Anran Zheng, Ran Wang
2021/10/18
knitr::opts_chunk$set(warning = FALSE, message = FALSE)
##Set Up
rm(list=ls())
library(tidyverse)
library(sf)
library(spdep)
library(caret)
library(ckanr)
library(FNN)
library(grid)
library(gridExtra)
library(ggcorrplot)
library(kableExtra)
library(jtools)
library(ggstance)
library(mapview)
library(tigris)
library(leaflet)
library(osmdata)
library(tidycensus)
library(stargazer)
root.dir = "https://raw.githubusercontent.com/urbanSpatial/Public-Policy-Analytics-Landing/master/DATA/"
source("https://raw.githubusercontent.com/urbanSpatial/Public-Policy-Analytics-Landing/master/functions.r")
palette5 <- c("#25CB10", "#5AB60C", "#8FA108", "#C48C04", "#FA7800")
qBr <- function(df, variable, rnd) {
if (missing(rnd)) {
as.character(quantile(round(df[[variable]],0),
c(.01,.2,.4,.6,.8), na.rm=T))
} else if (rnd == FALSE | rnd == F) {
as.character(formatC(quantile(df[[variable]]), digits = 3),
c(.01,.2,.4,.6,.8), na.rm=T)
}
}
st_c <- st_coordinates1 Introduction
This project develops model to predict the home price for single-family housing in Boulder county, Colorado. Despite powerful models that Zillow possesses for home price prediction, the always unique context of places requires more refined and site-specific models that take local characteristics into consideration for price prediction. However, such a site-specific model is difficult to build because factors that influence home price tend to be correlate with each other within the system of real estate. Moreover, a site-specific model to Boulder has limited data for model training for Boulder is not a large county, a limitation that will harm the accuracy of the model. Responding to the request for a site-specific model, we develops a Hedonic model that deconstructs the home price into the values of constituent parts including internal characteristics such as housing quality, public services and amenities such as public transit, and spatial process such as the clustering of home price. We then use the model to predict the home price in relation to changes in aforementioned features.
The final model that we develop to predict home price in Boulder explains around 72% of the fluctuation of price with a 19% range of accuracy for each price that the model predicts. Our model has o average an absolute error of 130,000 for each price prediction; comparing to the mean of housing price ($739,555), our model has a decent accuracy for prediction. The mean absolute percentage error for all prediction is nearly randomly distributed in spatial term. While our model has a slightly higher mean absolute percentage error for neighborhoods with high income, the relatively small gap (3%) between the MAPE for high-income and low-income neighborhoods suggest that the model is generalizable. Thus, we believe that our model with good accuracy and generalizability will be useful for Zillow to predict home price in Boulder county, CO.
2 Data
2.1 Data Wrangling and Feature Engineering
We first download the county boundary data from the open data portal of Boulder county. We then use the county boundary to download key amenities–including parks, playgrounds, waters, restaurants, fast food, companies, and bus stations–spatial information from OpenStreetMap. We download spatial information of census tracts within Boulder and urban areas that the Census defines as densely developed areas with a population at least greater than 2,500 from the Census Bureau. We also manually collect k-12 school information from the website of Boulder Valley School District and clean it into a csv file.
We then convert raw variables into useful predictive features. We first eliminate outlinears that are significantly more expensive than the general housing stock in Boulder. To develop indicators, we calculate the age of homes and the distance to the nearest 1 ~ 3 amenities. We also create buffers to calculate how many amenities exist within certain miles of a house. We categorize numeric variables such as the number of bedrooms into categorical variables that divide the data into several categories such as house with more than 3 bedrooms. We spatial join the census tracts and urban areas to the housing characteristics so that each house is assigned to its census tract and urban status as either urban or non-urban. At last, we create lag price variable that represents the mean of the nearest 5 houses to account for the clustering effects of houses.
Boulder_city <- st_read("C:/Users/zheng/Desktop/MUSA 508/508 midterm/City_of_Boulder_City_Limits.kml") %>%
st_transform('EPSG:26913')%>%
select(-Name, -Description)
Boulder_city <- st_union(Boulder_city)
Boulder.county=st_read("https://opendata.arcgis.com/api/v3/datasets/964b8f3b3dbe401bb28d49ac93d29dc4_0/downloads/data?format=kml&spatialRefId=4326")%>%
st_as_sf()
Boulder.county.reproject<-Boulder.county %>%
st_transform('EPSG:26913')%>%
dplyr::select(-Name,-Description)
q0 <- opq(bbox = c(-105.6945,39.91297,-105.0528,40.26396))
st_bbox(Boulder.county.reproject)
housing <- st_read("C:/Users/zheng/Desktop/MUSA 508/508 midterm/studentData.geojson", crs = 'ESRI:102254')%>%
st_transform('EPSG:26913')
housing <- housing %>%
select(-Stories, -UnitCount)%>%
mutate(age = 2021- EffectiveYear)%>%
filter(nbrBedRoom < 10, carStorageSF < 3000)
housing <- housing[-2637,]
##Public Facilities
#parks
park <- add_osm_feature(opq = q0, key = 'leisure', value = "park") %>%
osmdata_sf(.)
park.sf <- st_geometry(park$osm_points) %>%
st_transform(4326) %>%
st_sf() %>%
cbind(., park$osm_points$name) %>%
rename(NAME = park.osm_points.name)%>%
st_transform('EPSG:26913')%>%
st_intersection(Boulder.county.reproject,park.sf)%>%
dplyr::select(geometry)
housing <- housing%>%
mutate(park=nn_function(st_c(housing),st_c(park.sf),1))
#fast food
fast_food <- add_osm_feature(opq = q0, key = 'amenity', value = "fast_food") %>%
osmdata_sf(.)
fast_food.sf <- st_geometry(fast_food$osm_points) %>%
st_transform(4326) %>%
st_sf() %>%
cbind(., fast_food$osm_points$amenity) %>%
rename(NAME = fast_food.osm_points.amenity)%>%
st_transform('EPSG:26913')%>%
st_intersection(Boulder.county.reproject,park.sf)
fast_food.sf<-
fast_food.sf%>%
dplyr::select(geometry)
housing$fastfood_buffer =
st_buffer(housing, 800) %>% ##WHAT IS THE RIGHT PARAMETER
aggregate(mutate(fast_food.sf, counter = 1),., sum) %>%
pull(counter)
#water
water <- add_osm_feature(opq = q0, key = 'natural', value = "water") %>%
osmdata_sf(.)
water.sf <- water$osm_points %>%
dplyr::select(geometry) %>%
st_as_sf(crs = 4326, agr = "constant") %>%
distinct() %>%
st_transform('EPSG:26913')%>%
st_intersection(Boulder.county.reproject,park.sf)
housing <- housing%>%
mutate(water_nn1=nn_function(st_c(housing),st_c(water.sf),1),
water_nn2=nn_function(st_c(housing),st_c(water.sf),2),
water_nn3=nn_function(st_c(housing),st_c(water.sf),3))
#playground
playground <- add_osm_feature(opq = q0, key = 'leisure', value = "playground") %>%
osmdata_sf(.)
playground.sf <- st_geometry(playground$osm_points) %>%
st_transform(4326) %>%
st_sf() %>%
cbind(., playground$osm_points$name) %>%
rename(NAME = playground.osm_points.name)%>%
st_transform('EPSG:26913')%>%
st_intersection(Boulder.county.reproject,park.sf)%>%
dplyr::select(geometry)
housing <- housing%>%
mutate(playground=nn_function(st_c(housing),st_c(playground.sf),1))
#restaurant
restaurant <- add_osm_feature(opq = q0, key = 'amenity', value = "restaurant") %>%
osmdata_sf(.)
restaurant.sf <- st_geometry(restaurant$osm_points) %>%
st_transform(4326) %>%
st_sf() %>%
cbind(., restaurant$osm_points$amenity) %>%
rename(NAME = restaurant.osm_points.amenity)%>%
st_transform('EPSG:26913')%>%
st_intersection(Boulder.county.reproject,park.sf)%>%
dplyr::select(geometry)%>%
distinct()
housing<-housing%>%
mutate(
restaurant_nn1 = nn_function(st_c(housing), st_c(restaurant.sf), 1),
restaurant_nn2 = nn_function(st_c(housing), st_c(restaurant.sf), 2),
restaurant_nn3 = nn_function(st_c(housing), st_c(restaurant.sf), 3))
##School District
school <- read.csv("C:/Users/zheng/Desktop/MUSA 508/508 midterm/school.csv")%>%
st_as_sf(coords = c("LONGITUDE", "LATITUDE"), crs = 4326, agr = "constant")%>%
st_transform('EPSG:26913')
school.sf <- school%>%
dplyr::select(geometry) %>%
st_as_sf(crs = 4326, agr = "constant")%>%
st_transform('EPSG:26913')
housing$school_buffer =
st_buffer(housing, 8000) %>% ##WHAT IS THE RIGHT PARAMETER
aggregate(mutate(school.sf, counter = 1),., sum) %>%
pull(counter)
housing <- housing%>%
mutate(school=nn_function(st_c(housing),st_c(school.sf),1))
#Company
company <- add_osm_feature(opq = q0, key = 'office', value = "company") %>%
osmdata_sf(.)
company.sf <- st_geometry(company$osm_points) %>%
st_transform(4326) %>%
st_sf() %>%
cbind(., company$osm_points$office) %>%
rename(NAME = company.osm_points.office)%>%
st_transform('EPSG:26913')%>%
st_intersection(Boulder.county.reproject,park.sf)%>%
na.omit()
housing <- housing %>%
mutate(company=nn_function(st_c(housing),st_c(company.sf),1))
#Bus station
bus_station <- add_osm_feature(opq = q0, key = 'amenity', value = "bus_station") %>%
osmdata_sf(.)
bus_station.sf <- st_geometry(bus_station$osm_points) %>%
st_transform(4326) %>%
st_sf() %>%
cbind(., bus_station$osm_points$amenity) %>%
rename(NAME = bus_station.osm_points.amenity)%>%
st_transform('EPSG:26913')%>%
dplyr::select(geometry)%>%
st_intersection(Boulder.county.reproject,park.sf)
housing <- housing%>%
mutate(bus_stop_nn1=nn_function(st_c(housing),st_c(bus_station.sf),1),
bus_stop_nn2=nn_function(st_c(housing),st_c(bus_station.sf),2),
bus_stop_nn3=nn_function(st_c(housing),st_c(bus_station.sf),3))
##Internal Characteristics
housing <-
housing %>%
mutate(nbrBedRoom.cat = case_when(
nbrBedRoom >= 0 & nbrBedRoom < 4 ~ "Up to 3 Bedrooms",
nbrBedRoom >= 4 & nbrBedRoom < 5 ~ "4 Bedrooms",
nbrBedRoom > 4 ~ "5+ Bedrooms"))
##Spatial Process
#neighborhood
neighborhood <- get_acs(geography = "tract",
year = 2019,
variables = c("B06011_001E", #Median income in the past 12 months
"B19013_001E"), #Median household income in the past 12 months
geometry = T,
state = "CO",
county = "Boulder",
output = "wide") %>%
st_transform('EPSG:26913')%>%
dplyr::select(-NAME,-B06011_001M,-B19013_001M,-B06011_001E,-B19013_001E)
#urban status
urban_area <- urban_areas(cb = FALSE, year = NULL)%>%
st_transform('EPSG:26913')%>%
select(NAME10, geometry)
urban_area <- st_intersection(Boulder.county.reproject, urban_area)%>%
filter(NAME10 != "Denver--Aurora, CO")
housing <- st_join(housing, neighborhood, join = st_intersects)
housing <- st_join(housing, urban_area, join = st_intersects)
housing$NAME10[is.na(housing$NAME10)] <- 0
housing <- housing %>%
rename(urban_status = NAME10)
housing$urban_status <- ifelse(housing$urban_status == "0", "non-urban", "urban")#the spatial lag of housing
Boulder <- housing %>%
filter(toPredict == 0) %>%
dplyr::select(-toPredict)
coords <- st_coordinates(Boulder)
neighborList <- knn2nb(knearneigh(coords, 5))
spatialWeights <- nb2listw(neighborList, style="W")
Boulder$lagPrice <- lag.listw(spatialWeights,
Boulder$price)
housing$price[is.na(housing$price)] <- 0
coords.all <- st_coordinates(housing)
neighborList.all <- knn2nb(knearneigh(coords.all, 5))
spatialWeights.all <- nb2listw(neighborList.all, style="W")
housing$lagPrice <- lag.listw(spatialWeights.all,
housing$price)
housing$price[is.na(housing$price)] <- 0
coords.all <- st_coordinates(housing)
neighborList.all <- knn2nb(knearneigh(coords.all, 5))
spatialWeights.all <- nb2listw(neighborList.all, style="W")
housing$lagPrice <- lag.listw(spatialWeights.all,
housing$price)2.2 Variable Descriptions
For variables in our final model, we select the following variables:
Internal Characteristics
- age: age of the house calculated by 2021 minus the effective year of the house.
- designCodeDscr: description of building’s design type
- qualityCodeDscr: description of the quality of the house as determined by the government staff
- TotalFinishedSF: total number of finished square feet without the basement squre feed
- nbrBedRoom.cat : whether the house has more than 3 bedrooms
- HeatingDscr: description of the type of heating system in the house
- Roof_CoverDscr: the material used on the roof
Public Service/Amenities
- park: distance from the house to the nearest park
- school: distance from the house to the nearest school
- restaurant_nn1: distance from the house to the nearest restaurant
- bus_stop_nn1: distance from the house to the nearest bus station
- company: distance from the house to the nearest company
- water_nn1: distance from the house to the nearest water
Spatial Process
- GEOID: census tract that the house belongs to
- urban_status: whether the house is located in an urban area defined by the Census
- lagPrice: the average sale price of the house’s 5 nearest neighbors
The summary statistics for all numerical varibles are presented below.
final_vars<-st_drop_geometry(Boulder)%>%
select(
price,
age,
GEOID,
urban_status,
designCodeDscr,
qualityCodeDscr,
TotalFinishedSF,
nbrBedRoom.cat,
HeatingDscr,
Roof_CoverDscr,
park,
school,
restaurant_nn1,
bus_stop_nn1,
company,
water_nn1,
lagPrice)%>% na.omit()
stargazer(final_vars,type = "text",title = "Table 1. summary statistics with chosen variable descriptions")##
## Table 1. summary statistics with chosen variable descriptions
## ======================================================================================
## Statistic N Mean St. Dev. Min Pctl(25) Pctl(75) Max
## --------------------------------------------------------------------------------------
## price 11,259 746,123.700 543,627.200 10,000 452,600 831,100 7,350,000
## age 11,259 23.300 17.476 0 11 32 142
## TotalFinishedSF 11,259 1,953.673 889.723 0 1,276 2,442.5 10,188
## park 11,259 669.182 1,375.010 7.590 154.185 501.759 10,712.720
## school 11,259 6,110.231 6,185.961 12.666 849.973 12,282.800 22,328.780
## restaurant_nn1 11,259 1,444.577 1,473.535 10.489 581.046 1,723.857 12,111.570
## bus_stop_nn1 11,259 11,891.590 7,875.543 105.256 4,588.042 19,497.640 35,165.250
## company 11,259 6,830.712 6,154.506 50.168 1,364.029 12,533.640 31,577.050
## water_nn1 11,259 675.573 520.170 11.318 340.253 869.275 5,028.900
## lagPrice 11,259 733,469.600 427,418.800 137,900 461,870 839,880 3,849,000
## --------------------------------------------------------------------------------------2.3 Data Selection: Correlation Matrix and Correlation Plots
Below if the correlation matrix of all variables that we develop. We select and delete variables based on the correlation matrix. If two variables have a high correlation, we only choose one to include in our model.
numericVars <-
select_if(st_drop_geometry(housing), is.numeric) %>% na.omit()
ggcorrplot(
round(cor(numericVars), 1),
p.mat = cor_pmat(numericVars),
colors = c("#25CB10", "white", "#FA7800"),
type="lower",
insig = "blank") +
labs(title = "Figure 1. Correlation across numeric variables")
For categorical variables that cannot be incorporated into the correlation matrix, we plot price as a function of categorical variables to see if there is a correlation. Below are the plots for variables of interest that show a correlation to the home price.
st_drop_geometry(housing) %>%
dplyr::select(price,
designCodeDscr,
qualityCodeDscr,
HeatingDscr,
Roof_CoverDscr) %>%
gather(Variable, Value, -price) %>%
ggplot(aes(Value, price)) +
geom_bar(position = "dodge", stat = "summary", fun.y = "mean") +
facet_wrap(~Variable, ncol = 2, scales = "free") +
labs(title = "Price as a function of\ncategorical variables", y = "Mean_Price") +
plotTheme() + theme(axis.text.x = element_text(angle = 45, hjust = 1))
2.4 Price Correlation Scatter Plots
The scatter plots below show the home price correlated with 4 different variables of interest. The age of house is negatively correlated with the home price, meaning that the older a house is, the lower the sale price will be; yet the correlation line is rather gradual, potentially skewed by a few old houses with high value. The distance to the nearest bus stop is negatively correlate with home price, suggesting that the home price is likely to decrease with the decrease in accessibility to public transit. The distance to the nearest company site is negatively correlate with home price, suggesting that more far away a house is from company sites, the lower the sale price will be; this correlation indicates the relationship between home price and potential job opportunity. The distance to the nearest school site is also negatively correlate with home price, meaning that more distant a house is to school, the lower the sale price will be.
st_drop_geometry(housing) %>%
dplyr::select(price, company, school, age, bus_stop_nn1) %>%
gather(Variable, Value, -price) %>%
ggplot(aes(Value, price)) +
geom_point(size = .5) +
geom_smooth(method = "lm", se=T, colour = "#FA7800") +
facet_wrap(~Variable, ncol = 2, scales = "free") +
labs(title = "Figure 2. home price correlation scatter plots") +
plotTheme()
2.5 Map Home Sale Price
The map below visualizes the distribution of sale prices in Boulder county. The higher price tends to concentrate in the Boulder city, and the lower price tends to concentrate in Northeast Boulder and to scatter over the less populated parts in the county.
ggplot() +
geom_sf(data = Boulder.county.reproject, fill = "grey40") +
geom_sf(data = Boulder_city)+
geom_sf(data = housing, aes(colour = q5(price)),
show.legend = "point", size = .75) +
scale_colour_manual(values = palette5,
labels=qBr(housing,"price"),
name="Quintile\nBreaks") +
labs(title="Figure 3. Sales price distribution in Boulder") +
mapTheme()
2.6 Maps of Interesting Independent Variables
The following maps show the spatial distribution of three independent variables: urban status, distance to nearest bus station, and the design style of house. Each variable has its own distribution pattern, all with certain degree of clustering that does not resonate with a single spatial process, say, urban status. The different patterns of clustering for independent variables suggest the importance to include all of them into the model.
#urban status
ggplot() +
geom_sf(data = Boulder.county.reproject, fill = "grey40") +
geom_sf(data = housing, aes(colour = urban_status),
show.legend = "point", size = .75) +
labs(title="Figure 4.1. Urban status in Boulder") +
mapTheme()
#distance to bus station
ggplot() +
geom_sf(data = Boulder.county.reproject, fill = "grey40") +
geom_sf(data = housing, aes(colour = q5(bus_stop_nn1)),
show.legend = "point", size = .75) +
scale_colour_manual(values = palette5,
labels=qBr(housing,"bus_stop_nn1"),
name="Quintile\nBreaks") +
labs(title="Figure 4.2. Distance to the Nearest Bus Station, Boulder") +
mapTheme()
#housing design
ggplot() +
geom_sf(data = Boulder.county.reproject, fill = "grey40") +
geom_sf(data = housing, aes(colour = designCodeDscr),
show.legend = "point", size = .75) +
labs(title="Figure 5.2 Style of Design for Houses in Boulder") +
mapTheme()
3 Methods
3.1 Split the training and testing datasets
We first split our data into training for 75% and testing sets for 25%. According to the correlation matrix above (Figure 1), we should eliminate the multicollinearity as possible to ensure that all these variables should be independent.
set.seed(645)
inTrain <- createDataPartition(
y = paste(Boulder$designCodeDscr,
Boulder$nbrBedRoom.cat,
Boulder$HeatingDscr,
Boulder$Roof_CoverDscr,
Boulder$GEOID),
p = .75, list = FALSE)
Boulder.training <- Boulder[inTrain,]
Boulder.testing <- Boulder[-inTrain,]
housing.test.nhood <- Boulder.testing3.2 OLS regression model
After selecting the variables from correlation matrix, we conducted OLS regression analysis on these variables with different combinations. We compare the results of them by looking at the following indexes: 1. P-value and estimated coefficients of each variables. P-values should be significant as possible.
2. The final F-statistics
3. The adjusted R square, which should be greater as possible.
After several trials, we finally selected the following variables for our training datasets.
Internal characteristics
- Age
- designCodeDscr
- qualityCodeDscr
- TotalFinishedSF
- nbrBedRoom.cat
- HeatingDscr
- Roof_CoverDscr
Public services
-park
-school
-restaurant_nn1
-bus_stop_nn1
-company
-water_nn1
Spatial structure
-neighbourhoods
-urban status
-price lag
##Regression 1
reg1 <- lm(price ~ ., data = st_drop_geometry(housing) %>%
dplyr::select(price,
age,
urban_status,
GEOID,
designCodeDscr,
qualityCodeDscr,
TotalFinishedSF,
nbrBedRoom.cat,
HeatingDscr,
Roof_CoverDscr,
park,
school,
restaurant_nn1,
bus_stop_nn1,
company,
water_nn1,
lagPrice))
summary(reg1)##
## Call:
## lm(formula = price ~ ., data = st_drop_geometry(housing) %>%
## dplyr::select(price, age, urban_status, GEOID, designCodeDscr,
## qualityCodeDscr, TotalFinishedSF, nbrBedRoom.cat, HeatingDscr,
## Roof_CoverDscr, park, school, restaurant_nn1, bus_stop_nn1,
## company, water_nn1, lagPrice))
##
## Residuals:
## Min 1Q Median 3Q Max
## -3176880 -78418 -1262 76986 6300544
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.890e+05 5.113e+04 19.344 < 2e-16
## age -1.742e+03 2.428e+02 -7.175 7.70e-13
## urban_statusurban -3.270e+04 1.490e+04 -2.195 0.028213
## GEOID08013012102 -3.028e+05 3.100e+04 -9.766 < 2e-16
## GEOID08013012103 -2.478e+05 3.792e+04 -6.533 6.72e-11
## GEOID08013012104 -2.119e+05 4.005e+04 -5.291 1.24e-07
## GEOID08013012105 -3.466e+05 3.800e+04 -9.120 < 2e-16
## GEOID08013012201 -2.066e+04 4.086e+04 -0.506 0.613155
## GEOID08013012202 -3.244e+05 5.293e+04 -6.129 9.12e-10
## GEOID08013012203 -5.696e+05 4.176e+04 -13.638 < 2e-16
## GEOID08013012204 2.745e+05 7.596e+04 3.613 0.000304
## GEOID08013012300 -1.079e+05 2.125e+05 -0.508 0.611730
## GEOID08013012401 -2.236e+05 4.252e+04 -5.258 1.48e-07
## GEOID08013012501 -4.875e+05 5.215e+04 -9.349 < 2e-16
## GEOID08013012505 -2.382e+05 3.573e+04 -6.666 2.75e-11
## GEOID08013012507 -4.986e+05 4.128e+04 -12.077 < 2e-16
## GEOID08013012508 -4.697e+05 5.064e+04 -9.276 < 2e-16
## GEOID08013012509 -3.712e+05 4.137e+04 -8.974 < 2e-16
## GEOID08013012510 -1.952e+05 4.012e+04 -4.866 1.16e-06
## GEOID08013012511 -5.052e+05 5.134e+04 -9.839 < 2e-16
## GEOID08013012603 -4.889e+05 3.979e+04 -12.287 < 2e-16
## GEOID08013012605 -5.335e+05 1.251e+05 -4.265 2.01e-05
## GEOID08013012607 -5.077e+05 7.448e+04 -6.817 9.80e-12
## GEOID08013012608 -5.209e+05 4.435e+04 -11.746 < 2e-16
## GEOID08013012701 -5.422e+05 3.605e+04 -15.043 < 2e-16
## GEOID08013012705 -5.717e+05 5.437e+04 -10.514 < 2e-16
## GEOID08013012707 -3.214e+05 6.092e+04 -5.275 1.35e-07
## GEOID08013012708 -6.100e+05 4.344e+04 -14.042 < 2e-16
## GEOID08013012709 -5.885e+05 5.423e+04 -10.852 < 2e-16
## GEOID08013012710 -4.302e+05 4.022e+04 -10.698 < 2e-16
## GEOID08013012800 -7.618e+05 5.180e+04 -14.704 < 2e-16
## GEOID08013012903 -6.896e+05 5.202e+04 -13.255 < 2e-16
## GEOID08013012904 -6.235e+05 4.272e+04 -14.596 < 2e-16
## GEOID08013012905 -5.661e+05 5.744e+04 -9.857 < 2e-16
## GEOID08013012907 -5.845e+05 5.260e+04 -11.113 < 2e-16
## GEOID08013013003 -5.708e+05 3.958e+04 -14.423 < 2e-16
## GEOID08013013004 -5.575e+05 4.280e+04 -13.025 < 2e-16
## GEOID08013013005 -5.497e+05 4.279e+04 -12.845 < 2e-16
## GEOID08013013006 -5.322e+05 3.702e+04 -14.377 < 2e-16
## GEOID08013013201 -5.732e+05 7.412e+04 -7.733 1.14e-14
## GEOID08013013202 -2.430e+05 6.970e+04 -3.486 0.000492
## GEOID08013013205 -6.322e+05 4.950e+04 -12.770 < 2e-16
## GEOID08013013207 -6.597e+05 6.345e+04 -10.397 < 2e-16
## GEOID08013013208 -6.547e+05 5.951e+04 -11.002 < 2e-16
## GEOID08013013210 -6.200e+05 5.629e+04 -11.013 < 2e-16
## GEOID08013013211 -7.046e+05 5.303e+04 -13.286 < 2e-16
## GEOID08013013212 -6.305e+05 5.396e+04 -11.685 < 2e-16
## GEOID08013013213 -7.484e+05 5.041e+04 -14.844 < 2e-16
## GEOID08013013302 -5.394e+05 5.822e+04 -9.266 < 2e-16
## GEOID08013013305 -6.223e+05 6.288e+04 -9.897 < 2e-16
## GEOID08013013306 -6.022e+05 6.614e+04 -9.104 < 2e-16
## GEOID08013013307 -6.195e+05 6.285e+04 -9.857 < 2e-16
## GEOID08013013308 -5.745e+05 6.277e+04 -9.153 < 2e-16
## GEOID08013013401 -5.668e+05 6.307e+04 -8.986 < 2e-16
## GEOID08013013402 -6.489e+05 6.054e+04 -10.720 < 2e-16
## GEOID08013013503 -5.813e+05 6.330e+04 -9.183 < 2e-16
## GEOID08013013505 -5.982e+05 7.173e+04 -8.339 < 2e-16
## GEOID08013013506 -6.755e+05 6.413e+04 -10.533 < 2e-16
## GEOID08013013507 -6.494e+05 6.469e+04 -10.038 < 2e-16
## GEOID08013013508 -6.858e+05 6.292e+04 -10.899 < 2e-16
## GEOID08013013601 -5.919e+05 5.749e+04 -10.296 < 2e-16
## GEOID08013013602 -5.800e+05 6.065e+04 -9.563 < 2e-16
## GEOID08013013701 -5.623e+05 3.798e+04 -14.806 < 2e-16
## GEOID08013013702 -6.287e+05 4.438e+04 -14.165 < 2e-16
## GEOID08013060600 -7.354e+05 3.486e+04 -21.097 < 2e-16
## GEOID08013060700 -6.188e+05 4.172e+04 -14.834 < 2e-16
## GEOID08013060800 -5.912e+05 5.033e+04 -11.747 < 2e-16
## GEOID08013060900 -6.908e+05 4.312e+04 -16.021 < 2e-16
## GEOID08013061300 -7.412e+05 4.218e+04 -17.570 < 2e-16
## GEOID08013061400 -7.252e+05 4.066e+04 -17.837 < 2e-16
## designCodeDscr2-3 Story -2.440e+04 7.983e+03 -3.057 0.002240
## designCodeDscrBi-level 4.288e+04 1.791e+04 2.394 0.016676
## designCodeDscrMULTI STORY- TOWNHOUSE -1.237e+05 1.174e+04 -10.538 < 2e-16
## designCodeDscrSplit-level 1.142e+04 1.195e+04 0.956 0.339211
## qualityCodeDscrAVERAGE + -4.153e+04 1.299e+04 -3.196 0.001397
## qualityCodeDscrAVERAGE ++ -4.504e+03 1.341e+04 -0.336 0.737060
## qualityCodeDscrEXCELLENT 9.101e+05 3.067e+04 29.676 < 2e-16
## qualityCodeDscrEXCELLENT + 1.168e+06 6.528e+04 17.888 < 2e-16
## qualityCodeDscrEXCELLENT++ 1.687e+06 5.321e+04 31.703 < 2e-16
## qualityCodeDscrEXCEPTIONAL 1 8.688e+05 6.652e+04 13.061 < 2e-16
## qualityCodeDscrEXCEPTIONAL 2 1.519e+06 1.799e+05 8.443 < 2e-16
## qualityCodeDscrFAIR -4.417e+04 3.470e+04 -1.273 0.203083
## qualityCodeDscrGOOD 2.901e+04 9.501e+03 3.053 0.002271
## qualityCodeDscrGOOD + 4.235e+04 1.566e+04 2.705 0.006848
## qualityCodeDscrGOOD ++ 1.266e+05 1.467e+04 8.632 < 2e-16
## qualityCodeDscrLOW -1.273e+05 7.070e+04 -1.800 0.071860
## qualityCodeDscrVERY GOOD 1.764e+05 1.565e+04 11.273 < 2e-16
## qualityCodeDscrVERY GOOD + 3.542e+05 2.670e+04 13.268 < 2e-16
## qualityCodeDscrVERY GOOD ++ 4.734e+05 2.255e+04 20.991 < 2e-16
## TotalFinishedSF 1.261e+02 6.115e+00 20.630 < 2e-16
## nbrBedRoom.cat5+ Bedrooms 4.466e+04 9.598e+03 4.654 3.30e-06
## nbrBedRoom.catUp to 3 Bedrooms -9.917e+03 7.193e+03 -1.379 0.168004
## HeatingDscrElectric -4.767e+04 3.523e+04 -1.353 0.176066
## HeatingDscrElectric Wall Heat (1500W) 8.863e+04 2.138e+05 0.414 0.678531
## HeatingDscrForced Air -2.240e+04 3.111e+04 -0.720 0.471528
## HeatingDscrGravity -3.479e+04 6.768e+04 -0.514 0.607228
## HeatingDscrHeat Pump -2.241e+05 1.110e+05 -2.019 0.043510
## HeatingDscrHot Water 2.944e+04 3.268e+04 0.901 0.367586
## HeatingDscrNo HVAC 2.040e+05 2.142e+05 0.952 0.341082
## HeatingDscrPackage Unit 5.780e+03 3.012e+05 0.019 0.984690
## HeatingDscrRadiant Floor 2.568e+05 4.453e+04 5.768 8.25e-09
## HeatingDscrVentilation Only -4.648e+05 3.024e+05 -1.537 0.124299
## HeatingDscrWall Furnace 7.982e+04 4.563e+04 1.749 0.080303
## Roof_CoverDscrAsphalt -1.612e+04 7.115e+03 -2.265 0.023523
## Roof_CoverDscrBuilt-Up 2.699e+05 1.163e+05 2.321 0.020299
## Roof_CoverDscrClay Tile -3.931e+04 4.716e+04 -0.834 0.404547
## Roof_CoverDscrConcrete Tile -1.098e+05 2.384e+04 -4.606 4.16e-06
## Roof_CoverDscrMetal 1.697e+05 2.562e+04 6.623 3.68e-11
## Roof_CoverDscrRoll -1.451e+05 2.995e+05 -0.485 0.627925
## Roof_CoverDscrRubber Membrane 8.147e+04 2.670e+04 3.051 0.002288
## Roof_CoverDscrShake -7.813e+04 4.622e+04 -1.691 0.090935
## Roof_CoverDscrTar and Gravel 8.104e+04 1.512e+05 0.536 0.591916
## park -1.127e+01 4.106e+00 -2.745 0.006053
## school 2.396e+00 3.081e+00 0.778 0.436822
## restaurant_nn1 8.145e+00 3.653e+00 2.229 0.025808
## bus_stop_nn1 -1.094e+01 5.324e+00 -2.054 0.039992
## company 3.826e+00 5.299e+00 0.722 0.470281
## water_nn1 -3.679e+00 7.696e+00 -0.478 0.632628
## lagPrice 3.147e-01 1.282e-02 24.552 < 2e-16
##
## (Intercept) ***
## age ***
## urban_statusurban *
## GEOID08013012102 ***
## GEOID08013012103 ***
## GEOID08013012104 ***
## GEOID08013012105 ***
## GEOID08013012201
## GEOID08013012202 ***
## GEOID08013012203 ***
## GEOID08013012204 ***
## GEOID08013012300
## GEOID08013012401 ***
## GEOID08013012501 ***
## GEOID08013012505 ***
## GEOID08013012507 ***
## GEOID08013012508 ***
## GEOID08013012509 ***
## GEOID08013012510 ***
## GEOID08013012511 ***
## GEOID08013012603 ***
## GEOID08013012605 ***
## GEOID08013012607 ***
## GEOID08013012608 ***
## GEOID08013012701 ***
## GEOID08013012705 ***
## GEOID08013012707 ***
## GEOID08013012708 ***
## GEOID08013012709 ***
## GEOID08013012710 ***
## GEOID08013012800 ***
## GEOID08013012903 ***
## GEOID08013012904 ***
## GEOID08013012905 ***
## GEOID08013012907 ***
## GEOID08013013003 ***
## GEOID08013013004 ***
## GEOID08013013005 ***
## GEOID08013013006 ***
## GEOID08013013201 ***
## GEOID08013013202 ***
## GEOID08013013205 ***
## GEOID08013013207 ***
## GEOID08013013208 ***
## GEOID08013013210 ***
## GEOID08013013211 ***
## GEOID08013013212 ***
## GEOID08013013213 ***
## GEOID08013013302 ***
## GEOID08013013305 ***
## GEOID08013013306 ***
## GEOID08013013307 ***
## GEOID08013013308 ***
## GEOID08013013401 ***
## GEOID08013013402 ***
## GEOID08013013503 ***
## GEOID08013013505 ***
## GEOID08013013506 ***
## GEOID08013013507 ***
## GEOID08013013508 ***
## GEOID08013013601 ***
## GEOID08013013602 ***
## GEOID08013013701 ***
## GEOID08013013702 ***
## GEOID08013060600 ***
## GEOID08013060700 ***
## GEOID08013060800 ***
## GEOID08013060900 ***
## GEOID08013061300 ***
## GEOID08013061400 ***
## designCodeDscr2-3 Story **
## designCodeDscrBi-level *
## designCodeDscrMULTI STORY- TOWNHOUSE ***
## designCodeDscrSplit-level
## qualityCodeDscrAVERAGE + **
## qualityCodeDscrAVERAGE ++
## qualityCodeDscrEXCELLENT ***
## qualityCodeDscrEXCELLENT + ***
## qualityCodeDscrEXCELLENT++ ***
## qualityCodeDscrEXCEPTIONAL 1 ***
## qualityCodeDscrEXCEPTIONAL 2 ***
## qualityCodeDscrFAIR
## qualityCodeDscrGOOD **
## qualityCodeDscrGOOD + **
## qualityCodeDscrGOOD ++ ***
## qualityCodeDscrLOW .
## qualityCodeDscrVERY GOOD ***
## qualityCodeDscrVERY GOOD + ***
## qualityCodeDscrVERY GOOD ++ ***
## TotalFinishedSF ***
## nbrBedRoom.cat5+ Bedrooms ***
## nbrBedRoom.catUp to 3 Bedrooms
## HeatingDscrElectric
## HeatingDscrElectric Wall Heat (1500W)
## HeatingDscrForced Air
## HeatingDscrGravity
## HeatingDscrHeat Pump *
## HeatingDscrHot Water
## HeatingDscrNo HVAC
## HeatingDscrPackage Unit
## HeatingDscrRadiant Floor ***
## HeatingDscrVentilation Only
## HeatingDscrWall Furnace .
## Roof_CoverDscrAsphalt *
## Roof_CoverDscrBuilt-Up *
## Roof_CoverDscrClay Tile
## Roof_CoverDscrConcrete Tile ***
## Roof_CoverDscrMetal ***
## Roof_CoverDscrRoll
## Roof_CoverDscrRubber Membrane **
## Roof_CoverDscrShake .
## Roof_CoverDscrTar and Gravel
## park **
## school
## restaurant_nn1 *
## bus_stop_nn1 *
## company
## water_nn1
## lagPrice ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 298400 on 11240 degrees of freedom
## Multiple R-squared: 0.7042, Adjusted R-squared: 0.7011
## F-statistic: 226.8 on 118 and 11240 DF, p-value: < 2.2e-163.3 Accounting for neighbourhoods effects
In this regression model, we include all the variables above except the spatial structure, since we want to compare the neighbouhood effects in the later analysis. The final adjusted R square is 0.7232, which means that 72.32% of sales price can be explained by these variables we picked. We also conducted the cross validation and saw that the mean absolute error (MAE) is 155485.5. We hypothesized that introducing neighbourhoods to this model will enhance the accuracy, then we tested our hypothesis in the following results parts.
reg.training <-
lm(price ~ ., data = as.data.frame(Boulder.training) %>%
dplyr::select(price,
age,
designCodeDscr,
qualityCodeDscr,
TotalFinishedSF,
nbrBedRoom.cat,
HeatingDscr,
Roof_CoverDscr,
park,
school,
restaurant_nn1,
bus_stop_nn1,
company,
water_nn1))
summary(reg.training)##
## Call:
## lm(formula = price ~ ., data = as.data.frame(Boulder.training) %>%
## dplyr::select(price, age, designCodeDscr, qualityCodeDscr,
## TotalFinishedSF, nbrBedRoom.cat, HeatingDscr, Roof_CoverDscr,
## park, school, restaurant_nn1, bus_stop_nn1, company,
## water_nn1))
##
## Residuals:
## Min 1Q Median 3Q Max
## -2735250 -126430 -21526 85208 6692192
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.858e+05 4.396e+04 17.876 < 2e-16
## age 1.215e+03 2.876e+02 4.225 2.41e-05
## designCodeDscr2-3 Story -6.035e+04 9.885e+03 -6.105 1.07e-09
## designCodeDscrBi-level 4.440e+04 2.162e+04 2.054 0.040045
## designCodeDscrMULTI STORY- TOWNHOUSE -2.425e+05 1.414e+04 -17.156 < 2e-16
## designCodeDscrSplit-level 8.595e+03 1.482e+04 0.580 0.561841
## qualityCodeDscrAVERAGE + -7.745e+03 1.633e+04 -0.474 0.635247
## qualityCodeDscrAVERAGE ++ 3.312e+04 1.671e+04 1.982 0.047551
## qualityCodeDscrEXCELLENT 1.306e+06 3.736e+04 34.944 < 2e-16
## qualityCodeDscrEXCELLENT + 1.545e+06 8.511e+04 18.151 < 2e-16
## qualityCodeDscrEXCELLENT++ 2.214e+06 6.711e+04 32.996 < 2e-16
## qualityCodeDscrEXCEPTIONAL 1 1.328e+06 8.559e+04 15.516 < 2e-16
## qualityCodeDscrEXCEPTIONAL 2 1.719e+06 2.102e+05 8.177 3.30e-16
## qualityCodeDscrFAIR -1.226e+05 4.256e+04 -2.880 0.003980
## qualityCodeDscrGOOD 7.516e+04 1.148e+04 6.549 6.10e-11
## qualityCodeDscrGOOD + 1.040e+05 1.964e+04 5.293 1.23e-07
## qualityCodeDscrGOOD ++ 2.094e+05 1.788e+04 11.711 < 2e-16
## qualityCodeDscrLOW -2.305e+05 9.230e+04 -2.498 0.012515
## qualityCodeDscrVERY GOOD 3.090e+05 1.913e+04 16.159 < 2e-16
## qualityCodeDscrVERY GOOD + 7.182e+05 3.387e+04 21.208 < 2e-16
## qualityCodeDscrVERY GOOD ++ 7.360e+05 2.784e+04 26.435 < 2e-16
## TotalFinishedSF 1.489e+02 7.416e+00 20.080 < 2e-16
## nbrBedRoom.cat5+ Bedrooms 4.044e+04 1.225e+04 3.301 0.000968
## nbrBedRoom.catUp to 3 Bedrooms -9.786e+03 9.203e+03 -1.063 0.287619
## HeatingDscrElectric -7.088e+04 4.325e+04 -1.639 0.101257
## HeatingDscrElectric Wall Heat (1500W) 2.158e+03 2.501e+05 0.009 0.993113
## HeatingDscrForced Air -1.082e+05 3.835e+04 -2.820 0.004810
## HeatingDscrGravity -8.029e+04 7.984e+04 -1.006 0.314570
## HeatingDscrHeat Pump -1.105e+05 1.300e+05 -0.850 0.395177
## HeatingDscrHot Water 5.435e+04 4.026e+04 1.350 0.177021
## HeatingDscrNo HVAC 9.983e+03 2.507e+05 0.040 0.968234
## HeatingDscrPackage Unit -7.413e+04 3.513e+05 -0.211 0.832888
## HeatingDscrRadiant Floor 3.198e+05 5.343e+04 5.984 2.25e-09
## HeatingDscrVentilation Only -5.377e+05 3.530e+05 -1.523 0.127785
## HeatingDscrWall Furnace 4.917e+04 5.616e+04 0.876 0.381277
## Roof_CoverDscrAsphalt 2.786e+04 8.917e+03 3.124 0.001790
## Roof_CoverDscrBuilt-Up 4.322e+05 1.472e+05 2.935 0.003341
## Roof_CoverDscrClay Tile 2.492e+04 5.638e+04 0.442 0.658513
## Roof_CoverDscrConcrete Tile -2.529e+05 2.733e+04 -9.254 < 2e-16
## Roof_CoverDscrMetal 2.056e+05 3.077e+04 6.681 2.51e-11
## Roof_CoverDscrRoll -2.553e+04 3.498e+05 -0.073 0.941835
## Roof_CoverDscrRubber Membrane 8.521e+04 3.239e+04 2.631 0.008540
## Roof_CoverDscrShake -2.329e+04 5.456e+04 -0.427 0.669467
## Roof_CoverDscrTar and Gravel 2.429e+05 1.761e+05 1.379 0.167798
## park 1.499e-01 4.242e+00 0.035 0.971812
## school 1.769e+01 1.600e+00 11.055 < 2e-16
## restaurant_nn1 -1.574e+01 3.598e+00 -4.375 1.23e-05
## bus_stop_nn1 -5.098e+01 1.684e+00 -30.268 < 2e-16
## company 2.203e+01 2.067e+00 10.659 < 2e-16
## water_nn1 2.451e+01 7.898e+00 3.103 0.001921
##
## (Intercept) ***
## age ***
## designCodeDscr2-3 Story ***
## designCodeDscrBi-level *
## designCodeDscrMULTI STORY- TOWNHOUSE ***
## designCodeDscrSplit-level
## qualityCodeDscrAVERAGE +
## qualityCodeDscrAVERAGE ++ *
## qualityCodeDscrEXCELLENT ***
## qualityCodeDscrEXCELLENT + ***
## qualityCodeDscrEXCELLENT++ ***
## qualityCodeDscrEXCEPTIONAL 1 ***
## qualityCodeDscrEXCEPTIONAL 2 ***
## qualityCodeDscrFAIR **
## qualityCodeDscrGOOD ***
## qualityCodeDscrGOOD + ***
## qualityCodeDscrGOOD ++ ***
## qualityCodeDscrLOW *
## qualityCodeDscrVERY GOOD ***
## qualityCodeDscrVERY GOOD + ***
## qualityCodeDscrVERY GOOD ++ ***
## TotalFinishedSF ***
## nbrBedRoom.cat5+ Bedrooms ***
## nbrBedRoom.catUp to 3 Bedrooms
## HeatingDscrElectric
## HeatingDscrElectric Wall Heat (1500W)
## HeatingDscrForced Air **
## HeatingDscrGravity
## HeatingDscrHeat Pump
## HeatingDscrHot Water
## HeatingDscrNo HVAC
## HeatingDscrPackage Unit
## HeatingDscrRadiant Floor ***
## HeatingDscrVentilation Only
## HeatingDscrWall Furnace
## Roof_CoverDscrAsphalt **
## Roof_CoverDscrBuilt-Up **
## Roof_CoverDscrClay Tile
## Roof_CoverDscrConcrete Tile ***
## Roof_CoverDscrMetal ***
## Roof_CoverDscrRoll
## Roof_CoverDscrRubber Membrane **
## Roof_CoverDscrShake
## Roof_CoverDscrTar and Gravel
## park
## school ***
## restaurant_nn1 ***
## bus_stop_nn1 ***
## company ***
## water_nn1 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 349100 on 9069 degrees of freedom
## Multiple R-squared: 0.6059, Adjusted R-squared: 0.6038
## F-statistic: 284.6 on 49 and 9069 DF, p-value: < 2.2e-16Boulder.testing <-
Boulder.testing %>%
mutate(Regression = "Baseline Regression",
price.Predict = predict(reg.training, Boulder.testing),
price.Error = price.Predict - price,
price.AbsError = abs(price.Predict - price),
price.APE = (abs(price.Predict - price)) /
price.Predict)%>%
filter(price < 8000000)
##Generalizibility
fitControl <- trainControl(method = "cv", number = 100)
set.seed(825)
reg.cv <-
train(price ~ ., data = st_drop_geometry(Boulder.training) %>%
dplyr::select(price,
age,
urban_status,
designCodeDscr,
qualityCodeDscr,
TotalFinishedSF,
nbrBedRoom.cat,
HeatingDscr,
Roof_CoverDscr,
park,
school,
restaurant_nn1,
bus_stop_nn1,
company,
water_nn1,
lagPrice),
method = "lm", trControl = fitControl, na.action = na.pass)
reg.cv## Linear Regression
##
## 9119 samples
## 15 predictor
##
## No pre-processing
## Resampling: Cross-Validated (100 fold)
## Summary of sample sizes: 9029, 9028, 9027, 9028, 9028, 9027, ...
## Resampling results:
##
## RMSE Rsquared MAE
## 293539.7 0.723245 155487.2
##
## Tuning parameter 'intercept' was held constant at a value of TRUEmean(reg.cv$resample[,3])## [1] 155487.24 Results
4.1 Results of training dataset
Table 2 shows the summary results of our training dataset. According to the P-value and t-value, the majority of factors are significant. Variables categorized as internal characteristics (total finished size, quality code, age, etc.) and spatial structure (most neighbourhoods and price lag) play most significant roles in the OLS regression model of our training dataset. According to the coefficients, houses with younger built ages, high quality, larger size, more bedrooms are associated with higher salesprice, which accords with our expectations. Except for the distance to the restaurant, public services are not statistically significant in our designed model. The is adjusted R square is 0.7267 , which means that 72.67% of variance in this model can be explained by all these predictors we chose.
reg.nhood <- lm(price ~ ., data = st_drop_geometry(Boulder.training) %>%
dplyr::select(price,
age,
GEOID,
urban_status,
designCodeDscr,
qualityCodeDscr,
TotalFinishedSF,
nbrBedRoom.cat,
HeatingDscr,
Roof_CoverDscr,
park,
school,
restaurant_nn1,
bus_stop_nn1,
company,
water_nn1,
lagPrice))
kable(tidy(reg.nhood),caption = "Table 2. summary results of training dataset") %>% kable_styling()| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| (Intercept) | 8.893211e+05 | 5.492575e+04 | 16.1913341 | 0.0000000 |
| age | -1.694320e+03 | 2.611500e+02 | -6.4879161 | 0.0000000 |
| GEOID08013012102 | -2.577510e+05 | 3.369440e+04 | -7.6496687 | 0.0000000 |
| GEOID08013012103 | -2.123701e+05 | 4.054170e+04 | -5.2383127 | 0.0000002 |
| GEOID08013012104 | -1.174276e+05 | 4.256825e+04 | -2.7585731 | 0.0058172 |
| GEOID08013012105 | -2.932291e+05 | 4.040904e+04 | -7.2565233 | 0.0000000 |
| GEOID08013012201 | 3.267974e+04 | 4.249490e+04 | 0.7690275 | 0.4418972 |
| GEOID08013012202 | -2.710871e+05 | 5.489525e+04 | -4.9382611 | 0.0000008 |
| GEOID08013012203 | -4.985736e+05 | 4.551599e+04 | -10.9538128 | 0.0000000 |
| GEOID08013012204 | 4.151615e+05 | 7.882404e+04 | 5.2669402 | 0.0000001 |
| GEOID08013012300 | -4.833522e+04 | 2.068784e+05 | -0.2336407 | 0.8152693 |
| GEOID08013012401 | -1.453286e+05 | 4.542295e+04 | -3.1994531 | 0.0013817 |
| GEOID08013012501 | -4.249189e+05 | 5.496778e+04 | -7.7303265 | 0.0000000 |
| GEOID08013012505 | -1.728226e+05 | 3.823529e+04 | -4.5199752 | 0.0000063 |
| GEOID08013012507 | -4.310379e+05 | 4.498063e+04 | -9.5827461 | 0.0000000 |
| GEOID08013012508 | -4.103342e+05 | 5.427016e+04 | -7.5609540 | 0.0000000 |
| GEOID08013012509 | -3.000941e+05 | 4.405788e+04 | -6.8113613 | 0.0000000 |
| GEOID08013012510 | -1.502248e+05 | 4.255698e+04 | -3.5299688 | 0.0004177 |
| GEOID08013012511 | -4.559635e+05 | 5.332887e+04 | -8.5500310 | 0.0000000 |
| GEOID08013012603 | -4.269271e+05 | 4.257071e+04 | -10.0286588 | 0.0000000 |
| GEOID08013012605 | -4.726432e+05 | 1.223867e+05 | -3.8618850 | 0.0001133 |
| GEOID08013012607 | -4.386106e+05 | 7.926728e+04 | -5.5333128 | 0.0000000 |
| GEOID08013012608 | -4.541142e+05 | 4.748705e+04 | -9.5629048 | 0.0000000 |
| GEOID08013012701 | -5.160196e+05 | 3.922808e+04 | -13.1543424 | 0.0000000 |
| GEOID08013012705 | -5.125036e+05 | 5.884075e+04 | -8.7100105 | 0.0000000 |
| GEOID08013012707 | -2.841530e+05 | 6.286795e+04 | -4.5198394 | 0.0000063 |
| GEOID08013012708 | -5.351398e+05 | 4.693536e+04 | -11.4016342 | 0.0000000 |
| GEOID08013012709 | -5.198687e+05 | 5.813934e+04 | -8.9417715 | 0.0000000 |
| GEOID08013012710 | -3.547513e+05 | 4.282652e+04 | -8.2834511 | 0.0000000 |
| GEOID08013012800 | -6.966864e+05 | 5.593500e+04 | -12.4552863 | 0.0000000 |
| GEOID08013012903 | -6.069328e+05 | 5.651994e+04 | -10.7383844 | 0.0000000 |
| GEOID08013012904 | -5.492873e+05 | 4.630304e+04 | -11.8628780 | 0.0000000 |
| GEOID08013012905 | -4.903037e+05 | 6.059436e+04 | -8.0915721 | 0.0000000 |
| GEOID08013012907 | -4.970911e+05 | 5.681057e+04 | -8.7499759 | 0.0000000 |
| GEOID08013013003 | -4.931513e+05 | 4.284049e+04 | -11.5113353 | 0.0000000 |
| GEOID08013013004 | -4.845179e+05 | 4.603950e+04 | -10.5239606 | 0.0000000 |
| GEOID08013013005 | -4.769830e+05 | 4.611979e+04 | -10.3422635 | 0.0000000 |
| GEOID08013013006 | -4.596758e+05 | 4.028766e+04 | -11.4098403 | 0.0000000 |
| GEOID08013013201 | -5.352633e+05 | 7.845758e+04 | -6.8223280 | 0.0000000 |
| GEOID08013013202 | -2.452556e+05 | 7.215767e+04 | -3.3988843 | 0.0006795 |
| GEOID08013013205 | -5.517420e+05 | 5.326846e+04 | -10.3577618 | 0.0000000 |
| GEOID08013013207 | -6.147870e+05 | 6.809967e+04 | -9.0277525 | 0.0000000 |
| GEOID08013013208 | -6.115623e+05 | 6.386656e+04 | -9.5756274 | 0.0000000 |
| GEOID08013013210 | -5.655319e+05 | 6.056723e+04 | -9.3372583 | 0.0000000 |
| GEOID08013013211 | -6.565803e+05 | 5.694002e+04 | -11.5310863 | 0.0000000 |
| GEOID08013013212 | -5.681172e+05 | 5.797262e+04 | -9.7997508 | 0.0000000 |
| GEOID08013013213 | -6.935187e+05 | 5.432134e+04 | -12.7669677 | 0.0000000 |
| GEOID08013013302 | -4.928778e+05 | 6.236340e+04 | -7.9033181 | 0.0000000 |
| GEOID08013013305 | -5.727005e+05 | 6.737697e+04 | -8.4999444 | 0.0000000 |
| GEOID08013013306 | -5.587540e+05 | 7.060516e+04 | -7.9137841 | 0.0000000 |
| GEOID08013013307 | -5.736340e+05 | 6.715546e+04 | -8.5418812 | 0.0000000 |
| GEOID08013013308 | -5.143094e+05 | 6.744503e+04 | -7.6256088 | 0.0000000 |
| GEOID08013013401 | -5.023131e+05 | 6.771046e+04 | -7.4185449 | 0.0000000 |
| GEOID08013013402 | -6.149091e+05 | 6.500894e+04 | -9.4588388 | 0.0000000 |
| GEOID08013013503 | -5.542901e+05 | 6.776112e+04 | -8.1800614 | 0.0000000 |
| GEOID08013013505 | -5.564080e+05 | 7.658938e+04 | -7.2648195 | 0.0000000 |
| GEOID08013013506 | -6.398578e+05 | 6.880859e+04 | -9.2990976 | 0.0000000 |
| GEOID08013013507 | -5.954870e+05 | 6.947707e+04 | -8.5709858 | 0.0000000 |
| GEOID08013013508 | -6.420151e+05 | 6.739493e+04 | -9.5261635 | 0.0000000 |
| GEOID08013013601 | -5.484979e+05 | 6.135548e+04 | -8.9396731 | 0.0000000 |
| GEOID08013013602 | -5.254734e+05 | 6.502775e+04 | -8.0807559 | 0.0000000 |
| GEOID08013013701 | -5.023326e+05 | 4.138079e+04 | -12.1392707 | 0.0000000 |
| GEOID08013013702 | -5.714023e+05 | 4.805335e+04 | -11.8909983 | 0.0000000 |
| GEOID08013060600 | -6.629024e+05 | 3.793964e+04 | -17.4725542 | 0.0000000 |
| GEOID08013060700 | -5.477331e+05 | 4.552775e+04 | -12.0307538 | 0.0000000 |
| GEOID08013060800 | -5.110172e+05 | 5.425024e+04 | -9.4196300 | 0.0000000 |
| GEOID08013060900 | -6.146713e+05 | 4.689233e+04 | -13.1081420 | 0.0000000 |
| GEOID08013061300 | -6.882637e+05 | 4.609559e+04 | -14.9312258 | 0.0000000 |
| GEOID08013061400 | -6.592551e+05 | 4.410467e+04 | -14.9475130 | 0.0000000 |
| urban_statusurban | -2.784517e+04 | 1.618221e+04 | -1.7207272 | 0.0853347 |
| designCodeDscr2-3 Story | -2.898402e+04 | 8.593303e+03 | -3.3728617 | 0.0007470 |
| designCodeDscrBi-level | 4.715352e+04 | 1.835847e+04 | 2.5684893 | 0.0102303 |
| designCodeDscrMULTI STORY- TOWNHOUSE | -1.245953e+05 | 1.271913e+04 | -9.7958998 | 0.0000000 |
| designCodeDscrSplit-level | 2.123469e+04 | 1.268721e+04 | 1.6737075 | 0.0942228 |
| qualityCodeDscrAVERAGE + | -4.314465e+04 | 1.402002e+04 | -3.0773592 | 0.0020947 |
| qualityCodeDscrAVERAGE ++ | -6.814125e+03 | 1.431565e+04 | -0.4759912 | 0.6340922 |
| qualityCodeDscrEXCELLENT | 9.129136e+05 | 3.247932e+04 | 28.1075357 | 0.0000000 |
| qualityCodeDscrEXCELLENT + | 1.107320e+06 | 7.173182e+04 | 15.4369400 | 0.0000000 |
| qualityCodeDscrEXCELLENT++ | 1.745379e+06 | 5.708364e+04 | 30.5758123 | 0.0000000 |
| qualityCodeDscrEXCEPTIONAL 1 | 9.275892e+05 | 7.246713e+04 | 12.8001376 | 0.0000000 |
| qualityCodeDscrEXCEPTIONAL 2 | 1.386462e+06 | 1.761342e+05 | 7.8716240 | 0.0000000 |
| qualityCodeDscrFAIR | -4.974818e+04 | 3.615091e+04 | -1.3761253 | 0.1688171 |
| qualityCodeDscrGOOD | 2.265787e+04 | 1.033216e+04 | 2.1929463 | 0.0283367 |
| qualityCodeDscrGOOD + | 3.438810e+04 | 1.704543e+04 | 2.0174386 | 0.0436794 |
| qualityCodeDscrGOOD ++ | 1.184495e+05 | 1.592375e+04 | 7.4385438 | 0.0000000 |
| qualityCodeDscrLOW | -1.042651e+05 | 7.765507e+04 | -1.3426700 | 0.1794127 |
| qualityCodeDscrVERY GOOD | 1.738849e+05 | 1.687169e+04 | 10.3063130 | 0.0000000 |
| qualityCodeDscrVERY GOOD + | 3.801528e+05 | 2.930885e+04 | 12.9705837 | 0.0000000 |
| qualityCodeDscrVERY GOOD ++ | 4.978339e+05 | 2.446135e+04 | 20.3518514 | 0.0000000 |
| TotalFinishedSF | 1.367235e+02 | 6.530435e+00 | 20.9363478 | 0.0000000 |
| nbrBedRoom.cat5+ Bedrooms | 4.239169e+04 | 1.029384e+04 | 4.1181612 | 0.0000385 |
| nbrBedRoom.catUp to 3 Bedrooms | -1.090800e+04 | 7.728250e+03 | -1.4114448 | 0.1581481 |
| HeatingDscrElectric | -4.604484e+04 | 3.648099e+04 | -1.2621598 | 0.2069240 |
| HeatingDscrElectric Wall Heat (1500W) | 8.146415e+04 | 2.082973e+05 | 0.3910956 | 0.6957358 |
| HeatingDscrForced Air | -2.047118e+04 | 3.220163e+04 | -0.6357187 | 0.5249760 |
| HeatingDscrGravity | -4.472803e+04 | 6.669198e+04 | -0.6706658 | 0.5024506 |
| HeatingDscrHeat Pump | -2.556164e+05 | 1.086514e+05 | -2.3526281 | 0.0186627 |
| HeatingDscrHot Water | 2.679087e+04 | 3.378022e+04 | 0.7930936 | 0.4277442 |
| HeatingDscrNo HVAC | 2.036438e+05 | 2.087373e+05 | 0.9755985 | 0.3292897 |
| HeatingDscrPackage Unit | 1.350673e+04 | 2.931802e+05 | 0.0460697 | 0.9632557 |
| HeatingDscrRadiant Floor | 2.357667e+05 | 4.485083e+04 | 5.2566857 | 0.0000002 |
| HeatingDscrVentilation Only | -4.257058e+05 | 2.945207e+05 | -1.4454191 | 0.1483750 |
| HeatingDscrWall Furnace | 8.825565e+04 | 4.702262e+04 | 1.8768766 | 0.0605673 |
| Roof_CoverDscrAsphalt | -1.344134e+04 | 7.642054e+03 | -1.7588647 | 0.0786345 |
| Roof_CoverDscrBuilt-Up | 4.246797e+05 | 1.229092e+05 | 3.4552295 | 0.0005524 |
| Roof_CoverDscrClay Tile | -5.990298e+04 | 4.720520e+04 | -1.2689911 | 0.2044771 |
| Roof_CoverDscrConcrete Tile | -1.299805e+05 | 2.495116e+04 | -5.2093975 | 0.0000002 |
| Roof_CoverDscrMetal | 1.261479e+05 | 2.583480e+04 | 4.8828685 | 0.0000011 |
| Roof_CoverDscrRoll | -1.700284e+05 | 2.913044e+05 | -0.5836794 | 0.5594506 |
| Roof_CoverDscrRubber Membrane | 9.947722e+04 | 2.776302e+04 | 3.5830833 | 0.0003414 |
| Roof_CoverDscrShake | -8.088747e+04 | 4.558041e+04 | -1.7746105 | 0.0759960 |
| Roof_CoverDscrTar and Gravel | 3.378294e+04 | 1.471402e+05 | 0.2295970 | 0.8184101 |
| park | -1.087381e+01 | 4.340866e+00 | -2.5049866 | 0.0122631 |
| school | 3.584541e+00 | 3.286894e+00 | 1.0905557 | 0.2754976 |
| restaurant_nn1 | 9.420545e+00 | 3.952131e+00 | 2.3836620 | 0.0171619 |
| bus_stop_nn1 | -1.233675e+01 | 5.685006e+00 | -2.1700500 | 0.0300291 |
| company | 6.237236e+00 | 5.658850e+00 | 1.1022091 | 0.2704003 |
| water_nn1 | -9.872340e-01 | 8.266886e+00 | -0.1194203 | 0.9049450 |
| lagPrice | 3.309460e-01 | 1.391190e-02 | 23.7887729 | 0.0000000 |
summary(reg.nhood)##
## Call:
## lm(formula = price ~ ., data = st_drop_geometry(Boulder.training) %>%
## dplyr::select(price, age, GEOID, urban_status, designCodeDscr,
## qualityCodeDscr, TotalFinishedSF, nbrBedRoom.cat, HeatingDscr,
## Roof_CoverDscr, park, school, restaurant_nn1, bus_stop_nn1,
## company, water_nn1, lagPrice))
##
## Residuals:
## Min 1Q Median 3Q Max
## -2230996 -84580 -5029 75868 6271992
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.893e+05 5.493e+04 16.191 < 2e-16
## age -1.694e+03 2.612e+02 -6.488 9.16e-11
## GEOID08013012102 -2.578e+05 3.369e+04 -7.650 2.22e-14
## GEOID08013012103 -2.124e+05 4.054e+04 -5.238 1.66e-07
## GEOID08013012104 -1.174e+05 4.257e+04 -2.759 0.005817
## GEOID08013012105 -2.932e+05 4.041e+04 -7.257 4.30e-13
## GEOID08013012201 3.268e+04 4.249e+04 0.769 0.441897
## GEOID08013012202 -2.711e+05 5.490e+04 -4.938 8.02e-07
## GEOID08013012203 -4.986e+05 4.552e+04 -10.954 < 2e-16
## GEOID08013012204 4.152e+05 7.882e+04 5.267 1.42e-07
## GEOID08013012300 -4.834e+04 2.069e+05 -0.234 0.815269
## GEOID08013012401 -1.453e+05 4.542e+04 -3.199 0.001382
## GEOID08013012501 -4.249e+05 5.497e+04 -7.730 1.19e-14
## GEOID08013012505 -1.728e+05 3.824e+04 -4.520 6.26e-06
## GEOID08013012507 -4.310e+05 4.498e+04 -9.583 < 2e-16
## GEOID08013012508 -4.103e+05 5.427e+04 -7.561 4.39e-14
## GEOID08013012509 -3.001e+05 4.406e+04 -6.811 1.03e-11
## GEOID08013012510 -1.502e+05 4.256e+04 -3.530 0.000418
## GEOID08013012511 -4.560e+05 5.333e+04 -8.550 < 2e-16
## GEOID08013012603 -4.269e+05 4.257e+04 -10.029 < 2e-16
## GEOID08013012605 -4.726e+05 1.224e+05 -3.862 0.000113
## GEOID08013012607 -4.386e+05 7.927e+04 -5.533 3.23e-08
## GEOID08013012608 -4.541e+05 4.749e+04 -9.563 < 2e-16
## GEOID08013012701 -5.160e+05 3.923e+04 -13.154 < 2e-16
## GEOID08013012705 -5.125e+05 5.884e+04 -8.710 < 2e-16
## GEOID08013012707 -2.842e+05 6.287e+04 -4.520 6.27e-06
## GEOID08013012708 -5.351e+05 4.694e+04 -11.402 < 2e-16
## GEOID08013012709 -5.199e+05 5.814e+04 -8.942 < 2e-16
## GEOID08013012710 -3.548e+05 4.283e+04 -8.283 < 2e-16
## GEOID08013012800 -6.967e+05 5.593e+04 -12.455 < 2e-16
## GEOID08013012903 -6.069e+05 5.652e+04 -10.738 < 2e-16
## GEOID08013012904 -5.493e+05 4.630e+04 -11.863 < 2e-16
## GEOID08013012905 -4.903e+05 6.059e+04 -8.092 6.65e-16
## GEOID08013012907 -4.971e+05 5.681e+04 -8.750 < 2e-16
## GEOID08013013003 -4.932e+05 4.284e+04 -11.511 < 2e-16
## GEOID08013013004 -4.845e+05 4.604e+04 -10.524 < 2e-16
## GEOID08013013005 -4.770e+05 4.612e+04 -10.342 < 2e-16
## GEOID08013013006 -4.597e+05 4.029e+04 -11.410 < 2e-16
## GEOID08013013201 -5.353e+05 7.846e+04 -6.822 9.54e-12
## GEOID08013013202 -2.453e+05 7.216e+04 -3.399 0.000680
## GEOID08013013205 -5.517e+05 5.327e+04 -10.358 < 2e-16
## GEOID08013013207 -6.148e+05 6.810e+04 -9.028 < 2e-16
## GEOID08013013208 -6.116e+05 6.387e+04 -9.576 < 2e-16
## GEOID08013013210 -5.655e+05 6.057e+04 -9.337 < 2e-16
## GEOID08013013211 -6.566e+05 5.694e+04 -11.531 < 2e-16
## GEOID08013013212 -5.681e+05 5.797e+04 -9.800 < 2e-16
## GEOID08013013213 -6.935e+05 5.432e+04 -12.767 < 2e-16
## GEOID08013013302 -4.929e+05 6.236e+04 -7.903 3.04e-15
## GEOID08013013305 -5.727e+05 6.738e+04 -8.500 < 2e-16
## GEOID08013013306 -5.588e+05 7.061e+04 -7.914 2.79e-15
## GEOID08013013307 -5.736e+05 6.716e+04 -8.542 < 2e-16
## GEOID08013013308 -5.143e+05 6.745e+04 -7.626 2.68e-14
## GEOID08013013401 -5.023e+05 6.771e+04 -7.419 1.29e-13
## GEOID08013013402 -6.149e+05 6.501e+04 -9.459 < 2e-16
## GEOID08013013503 -5.543e+05 6.776e+04 -8.180 3.22e-16
## GEOID08013013505 -5.564e+05 7.659e+04 -7.265 4.05e-13
## GEOID08013013506 -6.399e+05 6.881e+04 -9.299 < 2e-16
## GEOID08013013507 -5.955e+05 6.948e+04 -8.571 < 2e-16
## GEOID08013013508 -6.420e+05 6.739e+04 -9.526 < 2e-16
## GEOID08013013601 -5.485e+05 6.136e+04 -8.940 < 2e-16
## GEOID08013013602 -5.255e+05 6.503e+04 -8.081 7.27e-16
## GEOID08013013701 -5.023e+05 4.138e+04 -12.139 < 2e-16
## GEOID08013013702 -5.714e+05 4.805e+04 -11.891 < 2e-16
## GEOID08013060600 -6.629e+05 3.794e+04 -17.473 < 2e-16
## GEOID08013060700 -5.477e+05 4.553e+04 -12.031 < 2e-16
## GEOID08013060800 -5.110e+05 5.425e+04 -9.420 < 2e-16
## GEOID08013060900 -6.147e+05 4.689e+04 -13.108 < 2e-16
## GEOID08013061300 -6.883e+05 4.610e+04 -14.931 < 2e-16
## GEOID08013061400 -6.593e+05 4.410e+04 -14.948 < 2e-16
## urban_statusurban -2.785e+04 1.618e+04 -1.721 0.085335
## designCodeDscr2-3 Story -2.898e+04 8.593e+03 -3.373 0.000747
## designCodeDscrBi-level 4.715e+04 1.836e+04 2.568 0.010230
## designCodeDscrMULTI STORY- TOWNHOUSE -1.246e+05 1.272e+04 -9.796 < 2e-16
## designCodeDscrSplit-level 2.123e+04 1.269e+04 1.674 0.094223
## qualityCodeDscrAVERAGE + -4.314e+04 1.402e+04 -3.077 0.002095
## qualityCodeDscrAVERAGE ++ -6.814e+03 1.432e+04 -0.476 0.634092
## qualityCodeDscrEXCELLENT 9.129e+05 3.248e+04 28.108 < 2e-16
## qualityCodeDscrEXCELLENT + 1.107e+06 7.173e+04 15.437 < 2e-16
## qualityCodeDscrEXCELLENT++ 1.745e+06 5.708e+04 30.576 < 2e-16
## qualityCodeDscrEXCEPTIONAL 1 9.276e+05 7.247e+04 12.800 < 2e-16
## qualityCodeDscrEXCEPTIONAL 2 1.386e+06 1.761e+05 7.872 3.91e-15
## qualityCodeDscrFAIR -4.975e+04 3.615e+04 -1.376 0.168817
## qualityCodeDscrGOOD 2.266e+04 1.033e+04 2.193 0.028337
## qualityCodeDscrGOOD + 3.439e+04 1.705e+04 2.017 0.043679
## qualityCodeDscrGOOD ++ 1.184e+05 1.592e+04 7.439 1.11e-13
## qualityCodeDscrLOW -1.043e+05 7.766e+04 -1.343 0.179413
## qualityCodeDscrVERY GOOD 1.739e+05 1.687e+04 10.306 < 2e-16
## qualityCodeDscrVERY GOOD + 3.802e+05 2.931e+04 12.971 < 2e-16
## qualityCodeDscrVERY GOOD ++ 4.978e+05 2.446e+04 20.352 < 2e-16
## TotalFinishedSF 1.367e+02 6.530e+00 20.936 < 2e-16
## nbrBedRoom.cat5+ Bedrooms 4.239e+04 1.029e+04 4.118 3.85e-05
## nbrBedRoom.catUp to 3 Bedrooms -1.091e+04 7.728e+03 -1.411 0.158148
## HeatingDscrElectric -4.604e+04 3.648e+04 -1.262 0.206924
## HeatingDscrElectric Wall Heat (1500W) 8.146e+04 2.083e+05 0.391 0.695736
## HeatingDscrForced Air -2.047e+04 3.220e+04 -0.636 0.524976
## HeatingDscrGravity -4.473e+04 6.669e+04 -0.671 0.502451
## HeatingDscrHeat Pump -2.556e+05 1.087e+05 -2.353 0.018663
## HeatingDscrHot Water 2.679e+04 3.378e+04 0.793 0.427744
## HeatingDscrNo HVAC 2.036e+05 2.087e+05 0.976 0.329290
## HeatingDscrPackage Unit 1.351e+04 2.932e+05 0.046 0.963256
## HeatingDscrRadiant Floor 2.358e+05 4.485e+04 5.257 1.50e-07
## HeatingDscrVentilation Only -4.257e+05 2.945e+05 -1.445 0.148375
## HeatingDscrWall Furnace 8.826e+04 4.702e+04 1.877 0.060567
## Roof_CoverDscrAsphalt -1.344e+04 7.642e+03 -1.759 0.078634
## Roof_CoverDscrBuilt-Up 4.247e+05 1.229e+05 3.455 0.000552
## Roof_CoverDscrClay Tile -5.990e+04 4.721e+04 -1.269 0.204477
## Roof_CoverDscrConcrete Tile -1.300e+05 2.495e+04 -5.209 1.94e-07
## Roof_CoverDscrMetal 1.261e+05 2.583e+04 4.883 1.06e-06
## Roof_CoverDscrRoll -1.700e+05 2.913e+05 -0.584 0.559451
## Roof_CoverDscrRubber Membrane 9.948e+04 2.776e+04 3.583 0.000341
## Roof_CoverDscrShake -8.089e+04 4.558e+04 -1.775 0.075996
## Roof_CoverDscrTar and Gravel 3.378e+04 1.471e+05 0.230 0.818410
## park -1.087e+01 4.341e+00 -2.505 0.012263
## school 3.585e+00 3.287e+00 1.091 0.275498
## restaurant_nn1 9.421e+00 3.952e+00 2.384 0.017162
## bus_stop_nn1 -1.234e+01 5.685e+00 -2.170 0.030029
## company 6.237e+00 5.659e+00 1.102 0.270400
## water_nn1 -9.872e-01 8.267e+00 -0.119 0.904945
## lagPrice 3.310e-01 1.391e-02 23.789 < 2e-16
##
## (Intercept) ***
## age ***
## GEOID08013012102 ***
## GEOID08013012103 ***
## GEOID08013012104 **
## GEOID08013012105 ***
## GEOID08013012201
## GEOID08013012202 ***
## GEOID08013012203 ***
## GEOID08013012204 ***
## GEOID08013012300
## GEOID08013012401 **
## GEOID08013012501 ***
## GEOID08013012505 ***
## GEOID08013012507 ***
## GEOID08013012508 ***
## GEOID08013012509 ***
## GEOID08013012510 ***
## GEOID08013012511 ***
## GEOID08013012603 ***
## GEOID08013012605 ***
## GEOID08013012607 ***
## GEOID08013012608 ***
## GEOID08013012701 ***
## GEOID08013012705 ***
## GEOID08013012707 ***
## GEOID08013012708 ***
## GEOID08013012709 ***
## GEOID08013012710 ***
## GEOID08013012800 ***
## GEOID08013012903 ***
## GEOID08013012904 ***
## GEOID08013012905 ***
## GEOID08013012907 ***
## GEOID08013013003 ***
## GEOID08013013004 ***
## GEOID08013013005 ***
## GEOID08013013006 ***
## GEOID08013013201 ***
## GEOID08013013202 ***
## GEOID08013013205 ***
## GEOID08013013207 ***
## GEOID08013013208 ***
## GEOID08013013210 ***
## GEOID08013013211 ***
## GEOID08013013212 ***
## GEOID08013013213 ***
## GEOID08013013302 ***
## GEOID08013013305 ***
## GEOID08013013306 ***
## GEOID08013013307 ***
## GEOID08013013308 ***
## GEOID08013013401 ***
## GEOID08013013402 ***
## GEOID08013013503 ***
## GEOID08013013505 ***
## GEOID08013013506 ***
## GEOID08013013507 ***
## GEOID08013013508 ***
## GEOID08013013601 ***
## GEOID08013013602 ***
## GEOID08013013701 ***
## GEOID08013013702 ***
## GEOID08013060600 ***
## GEOID08013060700 ***
## GEOID08013060800 ***
## GEOID08013060900 ***
## GEOID08013061300 ***
## GEOID08013061400 ***
## urban_statusurban .
## designCodeDscr2-3 Story ***
## designCodeDscrBi-level *
## designCodeDscrMULTI STORY- TOWNHOUSE ***
## designCodeDscrSplit-level .
## qualityCodeDscrAVERAGE + **
## qualityCodeDscrAVERAGE ++
## qualityCodeDscrEXCELLENT ***
## qualityCodeDscrEXCELLENT + ***
## qualityCodeDscrEXCELLENT++ ***
## qualityCodeDscrEXCEPTIONAL 1 ***
## qualityCodeDscrEXCEPTIONAL 2 ***
## qualityCodeDscrFAIR
## qualityCodeDscrGOOD *
## qualityCodeDscrGOOD + *
## qualityCodeDscrGOOD ++ ***
## qualityCodeDscrLOW
## qualityCodeDscrVERY GOOD ***
## qualityCodeDscrVERY GOOD + ***
## qualityCodeDscrVERY GOOD ++ ***
## TotalFinishedSF ***
## nbrBedRoom.cat5+ Bedrooms ***
## nbrBedRoom.catUp to 3 Bedrooms
## HeatingDscrElectric
## HeatingDscrElectric Wall Heat (1500W)
## HeatingDscrForced Air
## HeatingDscrGravity
## HeatingDscrHeat Pump *
## HeatingDscrHot Water
## HeatingDscrNo HVAC
## HeatingDscrPackage Unit
## HeatingDscrRadiant Floor ***
## HeatingDscrVentilation Only
## HeatingDscrWall Furnace .
## Roof_CoverDscrAsphalt .
## Roof_CoverDscrBuilt-Up ***
## Roof_CoverDscrClay Tile
## Roof_CoverDscrConcrete Tile ***
## Roof_CoverDscrMetal ***
## Roof_CoverDscrRoll
## Roof_CoverDscrRubber Membrane ***
## Roof_CoverDscrShake .
## Roof_CoverDscrTar and Gravel
## park *
## school
## restaurant_nn1 *
## bus_stop_nn1 *
## company
## water_nn1
## lagPrice ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 290000 on 9000 degrees of freedom
## Multiple R-squared: 0.7302, Adjusted R-squared: 0.7266
## F-statistic: 206.4 on 118 and 9000 DF, p-value: < 2.2e-164.2 Results of test dataset
Table 3 shows the mean absolute error (AbsError) and the mean absolute percentage error (MAPE) for a single test set. Here we compare the test set with and without neighbourhood effects. Notably model with neighbourhood effects becomes more accurate, since both MAPE and AbsError decreased as we introduced neighbourhoods to our model. However, a model with a MAPE of 18.9% still needs future improvement.
reg.nhood.training <- lm(price ~ ., data = st_drop_geometry(Boulder.training) %>%
dplyr::select(price,
age,
GEOID,
urban_status,
designCodeDscr,
qualityCodeDscr,
TotalFinishedSF,
nbrBedRoom.cat,
HeatingDscr,
Roof_CoverDscr,
park,
school,
restaurant_nn1,
bus_stop_nn1,
company,
water_nn1,
lagPrice))
housing.test.nhood <-
housing.test.nhood %>%
mutate(Regression = "Neighbourhood effects",
price.Predict = predict(reg.nhood.training, housing.test.nhood),
price.Error = price.Predict - price,
price.AbsError = abs(price.Predict - price),
price.APE = (abs(price.Predict - price)) /
price.Predict)%>%
filter(price < 8000000)
coords_1 <- st_coordinates(housing.test.nhood)
neighborList_1 <- knn2nb(knearneigh(coords_1, 5))
spatialWeights_1 <- nb2listw(neighborList_1, style="W")
housing.test.nhood$lagPriceError <- lag.listw(spatialWeights_1, housing.test.nhood$price.AbsError)
coords_2 <- st_coordinates(Boulder.testing)
neighborList_2 <- knn2nb(knearneigh(coords_2, 5))
spatialWeights_2 <- nb2listw(neighborList_2, style="W")
Boulder.testing$lagPriceError <- lag.listw(spatialWeights_2,
Boulder.testing$price.AbsError)
comparison <-
rbind(
dplyr::select(Boulder.testing, starts_with("price"), Regression) %>%
mutate(lagPriceError = lag.listw(spatialWeights_2, price.Error)),
dplyr::select(housing.test.nhood, starts_with("price"), Regression) %>%
mutate(lagPriceError = lag.listw(spatialWeights_1, price.Error)))
st_drop_geometry(comparison) %>%
gather(Variable, Value, -Regression) %>%
filter(Variable == "price.AbsError" | Variable == "price.APE") %>%
group_by(Regression, Variable) %>%
summarize(meanValue = mean(Value, na.rm = T)) %>%
spread(Variable, meanValue) %>%
kable() %>%
kable_styling("striped", full_width = F) %>%
row_spec(1, color = "black", background = "#25CB10") %>%
row_spec(2, color = "black", background = "#FA7800") %>%
footnote(general_title = "\n",
general = "Table 3. Table of mean absolute error and MAPE for a single test set")| Regression | price.AbsError | price.APE |
|---|---|---|
| Baseline Regression | 175798.4 | 0.2662743 |
| Neighbourhood effects | 130056.4 | 0.1894812 |
|
|
||
| Table 3. Table of mean absolute error and MAPE for a single test set |
4.3 Cross validation
100-folds cross validation is then conducted to identify the optimal parameters in our training dataset. The MAE is 149189.9 and its standard deviation is 23366.62. The root mean square (RMSE) is 283864.8 and the R square is 0.7434937, which suggests that about 74.75% of variance can be explained in validation dataset. According to this histogram, MAE, RMSE and R square are not distributed normally. Further analysis should be conducted to test the generalizability of the training dataset.
fitControl <- trainControl(method = "cv", number = 100)
set.seed(825)
reg_neighbourhood.cv <-
train(price ~ ., data = st_drop_geometry(Boulder.training) %>%
dplyr::select(price,
age,
GEOID,
designCodeDscr,
qualityCodeDscr,
TotalFinishedSF,
nbrBedRoom.cat,
HeatingDscr,
Roof_CoverDscr,
urban_status,
park,
school,
restaurant_nn1,
bus_stop_nn1,
company,
water_nn1,
lagPrice),
method = "lm", trControl = fitControl, na.action = na.pass)
reg_neighbourhood.cv## Linear Regression
##
## 9119 samples
## 16 predictor
##
## No pre-processing
## Resampling: Cross-Validated (100 fold)
## Summary of sample sizes: 9029, 9028, 9027, 9028, 9028, 9027, ...
## Resampling results:
##
## RMSE Rsquared MAE
## 280149.2 0.747521 149194.7
##
## Tuning parameter 'intercept' was held constant at a value of TRUEreg_neighbourhood.cv$resample %>%
pivot_longer(-Resample) %>%
mutate(name = as.factor(name)) %>%
ggplot(., aes(x = value, color = name)) +
geom_histogram(bins = 30, colour="black", fill = "#FDE725FF") +
facet_wrap(~name, ncol = 3, scales = "free") +
theme_bw() +
theme(
legend.position = "none"
)
MAE_mean<- mean(reg_neighbourhood.cv$resample[,3])
MAE_SD<- sd(reg_neighbourhood.cv$resample[,3])
compare<- data.frame(MAE_mean=MAE_mean,
MAE_SD=MAE_SD)
compare## MAE_mean MAE_SD
## 1 149194.7 23373.694.4 Predicted sale price as function of observed price
Figure 6.1 plotted the predicted sale price as function of observed price. By comparing the gaps between the prediction (green line) and the perfect prediction (orange line), our predicted prices were close for majority. However, as prices increased, more errors spread out. By comparing the baseline regression and the neighbourhoods effects model in Figure 6.2, we noticed that the previous baseline regression was already close to the prefect prediction, but neighbourhoods effect even fits better slightly.
housing.test.nhood1<-housing.test.nhood%>%
filter(price<5500000, price.Predict<5500000&price.Predict>0)
ggplot(housing.test.nhood1,aes(price, price.Predict)) +
geom_point() +
stat_smooth(data=housing.test.nhood1,aes(price, price),
method = "lm", se = FALSE, size = 1, colour="#FA7800") +
stat_smooth(data=housing.test.nhood1,aes(price,price.Predict),
method = "lm", se = FALSE, size = 1, colour="#25CB10") +
facet_wrap(~Regression) +
labs(title="Figure 6.1 Predicted sale price as a function of observed price",
subtitle="Orange line represents a perfect prediction;
Green line represents prediction") +
plotTheme()
comparison %>%
dplyr::select(price.Predict, price, Regression) %>%
ggplot(aes(price, price.Predict)) +
geom_point() +
stat_smooth(aes(price, price),
method = "lm", se = FALSE, size = 1, colour="#FA7800") +
stat_smooth(aes(price.Predict, price),
method = "lm", se = FALSE, size = 1, colour="#25CB10") +
facet_wrap(~Regression) +
labs(title="Figure 6.2 Predicted sale price as a function of observed price in comparison",
subtitle="Orange line represents a perfect prediction;
Green line represents prediction") +
plotTheme()
4.5 Residuals of test sets
Figure 7.1 shows the distribution of residuals for the test set. We observed some clusters in the east, southeastern area and city of Boulder, suggesting the positive spatial autocorrelation exists. Figure 7.2 indicates that the observed Moran’s I (represented by the orange line) is significantly higher than all 999 randomly permuted I, with a positive value of about 0.25, which again confirmed that positive spatial autocorrelation exists in these residuals. Then according to Figure 7.3, generally the prices increase as the spatial lag of errors increase. Seemingly more house prices, as well as outliers, are above the best-fit line, which suggest a clustering pattern of house prices.
Map of residuals
ggplot() +
geom_sf(data = Boulder.county.reproject, fill = "grey40") +
geom_sf(data = housing.test.nhood, aes(colour = q5(price.AbsError)),
show.legend = "point", size = 1) +
scale_colour_manual(values = palette5,
labels=qBr(housing.test.nhood,"price.AbsError"),
name="Quintile\nBreaks") +
labs(title="Figure 7.1. Map of Residuals for the test set") +
mapTheme()
Moran’s I
moranTest <- moran.mc(housing.test.nhood$price.AbsError,
spatialWeights_1, nsim = 999)
ggplot(as.data.frame(moranTest$res[c(1:999)]),
aes(moranTest$res[c(1:999)])) +
geom_histogram(binwidth = 0.01) +
geom_vline(aes(xintercept = moranTest$statistic),
colour = "#FA7800",size=1) +
scale_x_continuous(limits = c(-1, 1)) +
labs(title="Figure 7.2. Observed and permuted Moran's I",
subtitle= "Observed Moran's I in orange",
x="Moran's I",
y="Count") +
plotTheme()
plot of the spatial lag in errors
ggplot(housing.test.nhood, aes(x=lagPriceError, y=price)) +
geom_point(colour = "#FA7800") +
geom_smooth(method = "lm", se = FALSE, colour = "#25CB10") +
labs(title = "Figure 7.3. plot of the spatial lag in errors",
x = "Spatial lag of errors",
y = "price") +
plotTheme()
4.6 Mapping predicted values
With the regression model we predicted sales prices where to predict equals 0 in the original housing dataset. Then we plotted our results for all sales price where toPredict is both 0 and 1 as Figure 8 shows. To our prediction, the most expensive houses are concentrated in Boulder city and southeast corner of Boulder county, while the cheapest prices are clustered close to the northeast corner of Boulder city.
housing.test.all <- housing
housing.test.all <- housing.test.all%>%
mutate(Regression = "Neighbourhood effects",
price.Predict = predict(reg.nhood.training, housing.test.all),
price.Error = price.Predict - price,
price.AbsError = abs(price.Predict - price),
price.APE = (abs(price.Predict - price)) /
price.Predict)%>%
filter(price < 8000000)
ggplot() +
geom_sf(data = Boulder.county.reproject, fill = "grey40") +
geom_sf(data = housing.test.all, aes(colour = q5(price.Predict)),
show.legend = "point", size = 1) +
scale_colour_manual(values = palette5,
labels=qBr(housing.test.all,"price.Predict"),
name="Quintile\nBreaks") +
labs(title="Figure 8. map of predicted values for where toPredict is both 0 and 1") +
mapTheme()
4.7 MAPE by neighborhood
Using the test set prediction, mean absolute percentage error (MAPE) by neighborhood is mapped as Figure 9. Most neighhourhoods have low value of MAPE, which is below 0.4. Notably, in the central Boulder county, one neighbourhood was significantly high, which exceeds 100%. Probably because this area has limited houses and more easily caused distortion.
st_drop_geometry(housing.test.nhood)%>%
group_by(GEOID) %>%
summarise(MAPE = mean(price.APE, na.rm = T))%>%
ungroup()%>%
left_join(neighborhood)%>%
st_sf()%>%
ggplot() +
geom_sf(aes(fill = MAPE)) +
geom_sf(data = housing.test.nhood, colour ="black", size =.5) +
scale_fill_gradient(low = palette5[1], high = palette5[5],
name = "MAPE") +
labs(title = "Figure 9. map of mean absolute percentage error(MAPE) by neighborhood") +
mapTheme()
Figure 10 plots the mean MAPE by neighborhood as a function of mean price by neighborhood. Except for few outliers, the MAPE is relatively stable and is below 0.4, which suggests good generalizability of our model.
nhood.summary <- housing.test.nhood %>%
group_by(GEOID) %>%
summarize(meanPrice = mean(price, na.rm = T),
meanPrediction = mean(price.Predict, na.rm = T),
meanMAE = mean(price.AbsError, na.rm = T),
MAPE=mean(price.APE, na.rm = T))
nhood.summary %>%
st_drop_geometry %>%
arrange(desc(meanMAE)) %>%
knitr::kable() %>% kable_styling()| GEOID | meanPrice | meanPrediction | meanMAE | MAPE |
|---|---|---|---|---|
| 08013012707 | 2120500.0 | 1166850.7 | 1273518.75 | 1.1238463 |
| 08013013202 | 2226250.0 | 1776044.6 | 918029.34 | 0.4859673 |
| 08013012204 | 798350.0 | 1621731.4 | 823381.40 | 0.5106646 |
| 08013012101 | 2290516.7 | 2067379.8 | 569175.45 | 0.2693979 |
| 08013012103 | 1404588.2 | 1337481.9 | 440614.70 | 0.2642201 |
| 08013012102 | 1406130.0 | 1278106.9 | 398836.43 | 0.2870487 |
| 08013012104 | 1014168.7 | 1301185.4 | 394024.46 | 0.3140760 |
| 08013012505 | 1398710.5 | 1359648.7 | 304612.49 | 0.2173348 |
| 08013012701 | 1224512.9 | 1116642.0 | 295421.07 | 0.2183454 |
| 08013012201 | 2014471.4 | 1876777.7 | 283350.50 | 0.1423570 |
| 08013012510 | 1260266.7 | 1180778.0 | 276279.93 | 0.2093141 |
| 08013012401 | 1132791.7 | 1280710.3 | 274719.39 | 0.2138087 |
| 08013012710 | 1079828.6 | 1228024.5 | 251959.46 | 0.1995180 |
| 08013013205 | 1162228.5 | 1255044.4 | 235572.86 | 0.1791924 |
| 08013012105 | 1382153.8 | 1330401.1 | 222853.53 | 0.1573124 |
| 08013012903 | 770633.3 | 853343.6 | 201922.28 | 0.1961022 |
| 08013013701 | 911678.8 | 873731.8 | 201628.80 | 0.2328420 |
| 08013013201 | 739500.0 | 872439.4 | 175776.96 | 0.2174667 |
| 08013013503 | 467741.7 | 338356.2 | 173040.03 | 1.0628816 |
| 08013012511 | 1139250.0 | 970878.5 | 168994.98 | 0.1695412 |
| 08013013602 | 350129.4 | 373474.0 | 151727.96 | 0.4345501 |
| 08013012202 | 1070925.0 | 1003688.6 | 134416.38 | 0.1345300 |
| 08013013402 | 506371.4 | 467595.4 | 130221.07 | 0.3728257 |
| 08013061400 | 721897.9 | 741355.8 | 129741.97 | 0.1825475 |
| 08013012800 | 663073.8 | 683950.5 | 127966.92 | 0.1950462 |
| 08013013213 | 704024.5 | 695430.2 | 121345.35 | 0.1947325 |
| 08013013003 | 748582.1 | 820966.9 | 121063.08 | 0.1287529 |
| 08013012708 | 728356.9 | 784143.6 | 119787.89 | 0.1424354 |
| 08013060700 | 805805.6 | 826428.3 | 116788.44 | 0.1280086 |
| 08013012603 | 996720.0 | 963564.5 | 115047.43 | 0.1122261 |
| 08013060600 | 623158.3 | 685646.4 | 110131.48 | 0.1553547 |
| 08013012509 | 884500.0 | 914385.7 | 110110.16 | 0.1193184 |
| 08013013702 | 578085.9 | 588847.6 | 106741.35 | 0.2055834 |
| 08013013005 | 862844.4 | 830690.1 | 104235.80 | 0.1480960 |
| 08013060900 | 665766.7 | 659528.2 | 103482.74 | 0.1833571 |
| 08013060800 | 533064.3 | 519844.9 | 93364.76 | 0.1825616 |
| 08013013601 | 665925.0 | 696686.9 | 88760.92 | 0.1414480 |
| 08013013004 | 752631.2 | 756682.4 | 86471.70 | 0.1116169 |
| 08013013006 | 723474.4 | 742955.9 | 84629.70 | 0.1091599 |
| 08013013302 | 487274.4 | 464067.2 | 84497.17 | 0.1930581 |
| 08013013211 | 560723.9 | 532946.5 | 83426.29 | 0.1933349 |
| 08013012608 | 651090.9 | 619179.0 | 82783.69 | 0.1293464 |
| 08013013508 | 492277.0 | 508276.7 | 77933.09 | 0.2010885 |
| 08013012203 | 552178.9 | 552744.1 | 77187.08 | 0.1454108 |
| 08013061300 | 760837.0 | 716928.6 | 77107.80 | 0.1214151 |
| 08013013208 | 479793.5 | 471337.1 | 76043.49 | 0.1742847 |
| 08013013401 | 361942.9 | 425785.0 | 74706.87 | 0.1682240 |
| 08013012905 | 483333.3 | 484521.0 | 73558.65 | 0.1673199 |
| 08013012907 | 483635.6 | 530540.3 | 73424.42 | 0.1354508 |
| 08013012705 | 757411.1 | 745427.4 | 73238.90 | 0.1011871 |
| 08013012507 | 671218.2 | 685611.4 | 72625.75 | 0.1048037 |
| 08013012501 | 804266.7 | 794134.6 | 72184.49 | 0.0923170 |
| 08013012508 | 735550.0 | 694082.5 | 72159.15 | 0.1068274 |
| 08013013308 | 389150.0 | 425392.5 | 72043.64 | 0.1527908 |
| 08013012904 | 596509.5 | 585442.7 | 71110.95 | 0.1317329 |
| 08013013207 | 534327.3 | 539925.1 | 70813.58 | 0.1479728 |
| 08013012607 | 617433.3 | 640693.1 | 70589.73 | 0.1103053 |
| 08013013506 | 476996.4 | 454730.9 | 61513.88 | 0.1681004 |
| 08013012709 | 663875.0 | 640852.1 | 59984.50 | 0.0863425 |
| 08013013305 | 368845.6 | 380597.2 | 44313.41 | 0.1177579 |
| 08013013307 | 389977.8 | 384569.1 | 42429.32 | 0.1101306 |
| 08013013210 | 385119.0 | 389124.1 | 41046.18 | 0.1020875 |
| 08013013212 | 460527.8 | 474533.4 | 39894.48 | 0.0844196 |
| 08013013505 | 354450.0 | 345362.8 | 34131.12 | 0.1010426 |
| 08013013507 | 390552.0 | 394478.7 | 32998.27 | 0.0790556 |
| 08013013306 | 375133.3 | 367951.7 | 32391.78 | 0.0904617 |
map_MAPE <- housing.test.nhood %>%
group_by(GEOID) %>%
summarise(MAPE = mean(price.APE, na.rm = T))
plot(nhood.summary$meanPrice,
nhood.summary$MAPE,
main="Figure 10. MAPE by neighborhood as a function of mean price by neighbourhood",
cex.main=0.75, ylim=range(0:1))
4.8 Test generalizability under income context
To further test our model’s generalizability, we split our cities into high and low income groups. We collected income data from 15-19 ACS using tidycensus and calculated the median income of Boulder county (40453). Figure 11 shows the test of generalizability under income context, where most neighbourhoods are wealthier. MAPE is calculated across the Baseline regression and neighbourhoods effects under income contexts in Table 3, where we observed lower MAPE of models with neighbourhoods effects. Therefore, neighourhood effects make a more generalizable prediction model.
tracts19<- get_acs(geography = "tract", variables = c("B01001_001E","B01001A_001E","B06011_001"),
year = 2019,
geometry = T,
state = "CO",
county = "Boulder",
output = "wide") %>%
st_transform('EPSG:26913') %>%
rename(Median_Income = B06011_001E) %>%
mutate(incomeContext = ifelse(Median_Income > 40453, "High Income", "Low income"))
ggplot() +
geom_sf(data = na.omit(tracts19),
aes(fill = incomeContext)) +
scale_fill_manual(values = c("#25CB10", "#FA7800"),
name="Income Context") +
labs(title = "Figure 11. Test of generalizability under income context") +
mapTheme() + theme(legend.position="bottom")
st_join(comparison, tracts19) %>%
group_by(Regression, incomeContext) %>%
summarize(mean.MAPE = scales::percent(mean(price.APE, na.rm = T))) %>%
st_drop_geometry() %>%
spread(incomeContext, mean.MAPE) %>%
kable(caption = "Table 4. Table set MAPE by neighbourhood income context") %>%
kable_styling("striped", full_width = F) %>%
row_spec(1, color = "black", background = "#25CB10") %>%
row_spec(2, color = "black", background = "#FA7800")| Regression | High Income | Low income |
|---|---|---|
| Baseline Regression | 26% | 27% |
| Neighbourhood effects | 18% | 21% |
5 Discussion
Our model is an effective model with the p-value of the F-statistic small smaller than 0.001 (p-value < 2.2e-16). Our model predicts around 72% of the variation in prices.
Some of the more interesting variables include distance to nearest school, distance to nearest bus station, and distance to nearest company that are statistically significant in the baseline model but lose significance once we introduce the spatial process variables including the neighborhood variable and the urban status.
Important features for home price prediction include age, neighborhood, urban status, design style, quality, built-up area, number of bedrooms, heating system, roof material, distance to park, distance to restaurant, and the mean price of the nearest 5 houses. Specifically, the neighborhood, urban status, and lag price variables represent the spatial process of price clustering in which houses in neighborhoods with higher home price tend to also have higher price. The design style and roof material variables are distributed spatially correlated to the clustering process of homes and are corresponding with the urban/non-urban division of spatial context; specific design styles are concentrated in urban neighborhoods that tend to have higher home price. While the distance to amenities such as parks and restaurants stay significant after we introduce spatial process variables, the significance of them decrease because they too are clustered in relation to the neighborhood effects. Thus, the correlation of variables are complicated in spatial sense, reflecting a clustering distribution that is crucial for home price. The model has a mean of absolute error of 149189.9 and a mean absolute percentage error of 19%. While our model has a slightly higher mean absolute percentage error for neighborhoods with high income, the relatively small gap between the MAPE for high-income and low-income neighborhoods suggest that the model can account the spatial variation in prices. The residuals, although largely randomly distributed, tend to be higher in Boulder city; however, the residuals tend to be lower in Longmont, also an urban region. Thus, what causes the error can be spatial characteristics specific to Boulder city that our model fails to capture. For instance, the large young population of students of University of Colorado in Boulder city can skew the home price. In more remote regions where few observations exist, the results of residuals are mixed, indicating that the model might require more observation from the rural context to reduce the residual.
6 Conclusion
We would recommend our model to Zillow to include more local intelligence into their home price prediction model. Our model has a decent degree of accuracy and generalizability to predict home price across neighborhoods and to account for the variation of prices in a spatial sense. We can improve the model more by using more training data as the training data is rather limited fro Boulder county and we lack some key data such as crime data to train our model better. Also, the socioeconomic data including income, racial composition, and educational attainment that are accessible to us is census data that has a high collinearity with the neighborhood data that we use census tracts to approximate for; thus, we are unable to include socioeconomic data that is likely to be influential into our model. More feature engineering with using log rather than the raw data might also reshapes the distribution of independent variables and builds a fitter model. We believe that Zillow can use the rich resources and data that they already possess to train and improve our model with more available variables to get even better prediction results.