repsale: Repeat Sales Estimation
In McSpatial: Nonparametric spatial data analysis
Description
Usage
Arguments
Details
Value
References
See Also
Examples
Description
Standard and Weighted Least Squares Repeat Sales Estimation
Usage
1
2
3
Arguments
price0
Earlier price in repeat sales pair
time0
Earlier time in repeat sales pair
price1
Later price in repeat sales pair
time1
Later time in repeat sales pair
mergefirst
Number of initial periods with coefficients constrained to zero. Default: mergefirst=1
graph
If TRUE, graph results. Default: graph=T
graph.conf
If TRUE, add confidence intervals to graph. Default: graph.conf=T
conf
Confidence level for intervals. Default: .95
stage3
If stage3 = NULL, no corrections for heteroskedasticty. If stage3="abs", uses the absolute value of the
first-stage residuals as the dependent variable in the second-stage regression. If stage3="square", uses the square of the
first-stage residuals as the dependent variable. Default: stage3=NULL.
stage3_xlist
List of explanatory variables for heteroskedasticity. By default, the single variable timesale = time1-time0 is
constructed and used as the explanatory variable when stage3="abs" or stage3="square". Alternatively, a formula can be provided for
a user-specified list of explanatory variables, e.g., stage3_xlist=~x1+x2. Important: note the "~" before the variable list.
print
If print=T, prints the regression results. Prints one stage only – the first stage when stage=NULL and
the final stage when stage3="square" or stage3="abs". Default: print=T.
Details
The repeat sales model is
y(t) - y(s) = δ(t) - δ(s) + u(t) - u(s)
where y is the log of sales price, s denotes the earlier sale in a repeat sales pair, and t denotes the later sale.
Each entry of the data set should represent a repeat sales pair, with price0 = y(s), price1 = y(t),
time0 = s, and time1 = t. The function repsaledata can help transfer a standard hedonic data set to a set of
repeat sales pairs.
Repeat sales estimates are sometimes very sensitive to sales from the first few time periods, particularly when the sample size is small.
The option mergefirst indicates the number of time periods for which the price index is constrained to equal zero. The default is
mergefirst = 1, meaning that the price index equals zero for just the first time period. The repsale command does not have an
option for including an intercept in the model.
Following Case and Shiller (1987), many authors use a three-stage procedure to construct repeat sales price indexes that are adjusted for
heteroskedasticity related to the length of time between sales. Common specifications for the second-stage function are
e^2 = α0 + α1 (t-s) or |e| = α0 + α1 (t-s), where e represents the first-stage residuals.
The first equation implies an error variance of σ^2 = e^2 and the second equation leads to
σ^2 = |e|^2. The repsale function uses a standard F test to
determine whether the slope cofficients are significant in the second-stage regression. The results are reported if print=T.
The third-stage equation is
(y(t) - y(s))/σ = (δ(t) - δ(s))/σ + (u(t) - u(s))/σ
This equation is estimated by regressing y(t) - y(s) on the series of indicator variables implied by δ(t) - δ(s)
using the weights option in lm with weights = 1/sigma^2
Value
fit
Full regression model.
pindex
The estimated price index.
lo
The lower bounds for the price index confidence intervals.
hi
The upper bounds for the price index confidence intervals.
dy
The dependent variable for the repeat sales regression, dy = price1-price0.
xmat
The matrix of explanatory variables for the repeat sales regressions. dim(xmat) = nt - mergefirst, where
nt = the number of time periods and mergefirst is specified in the call to repsale.
References
Case, Karl and Robert Shiller, "Prices of Single-Family Homes since 1970: New Indexes for Four Cities," New England Economic Review (1987), 45-56.
See Also
repsaledata
repsalefourier
repsaleqreg
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
set.seed(189)
n = 2000
# sale dates range from 0-10
# drawn uniformly from all possible time0, time1 combinations with time0<time1
tmat <- expand.grid(seq(0,10), seq(0,10))
tmat <- tmat[tmat[,1]<tmat[,2], ]
tobs <- sample(seq(1:nrow(tmat)),n,replace=TRUE)
time0 <- tmat[tobs,1]
time1 <- tmat[tobs,2]
timesale <- time1-time0
table(timesale)
# constant variance; index ranges from 0 at time 0 to 1 at time 10
y0 <- time0/10 + rnorm(n,0,.2)
y1 <- time1/10 + rnorm(n,0,.2)
fit <- repsale(price0=y0, price1=y1, time0=time0, time1=time1)
# variance rises with timesale
# var(u0) = .2^2; var(u1) = (.2 + timesale/10)^2
# var(u1-u0) = var(u0) + var(u1) = 2*(.2^2) + .4*timesale/10 + (timesale^2)/100
y0 <- time0/10 + rnorm(n,0,.2)
y1 <- time1/10 + rnorm(n,0,.2+timesale/10)
par(ask=TRUE)
fit <- repsale(price0=y0, price1=y1, time0=time0, time1=time1)
summary(fit$pindex)
fit <- repsale(price0=y0, price1=y1, time0=time0, time1=time1, stage3="abs")
summary(fit$pindex)
timesale2 <- timesale^2
fit <- repsale(price0=y0, price1=y1, time0=time0, time1=time1, stage3="square",
stage3_xlist=~timesale+timesale2)
Example output
Loading required package: lattice
Loading required package: locfit
locfit 1.5-9.1 2013-03-22
Loading required package: maptools
Loading required package: sp
Checking rgeos availability: TRUE
Loading required package: quantreg
Loading required package: SparseM
Attaching package: 'SparseM'
The following object is masked from 'package:base':
backsolve
Loading required package: RANN
timesale
1 2 3 4 5 6 7 8 9 10
368 349 264 253 223 178 167 104 56 38
Call:
lm(formula = dy ~ xmat + 0)
Residuals:
Min 1Q Median 3Q Max
-0.98034 -0.18923 -0.00071 0.19144 0.90325
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Time 2 0.07128 0.01999 3.567 0.00037 ***
Time 3 0.18775 0.01969 9.533 < 2e-16 ***
Time 4 0.29225 0.01981 14.753 < 2e-16 ***
Time 5 0.39466 0.02012 19.613 < 2e-16 ***
Time 6 0.48243 0.01961 24.600 < 2e-16 ***
Time 7 0.60606 0.01990 30.462 < 2e-16 ***
Time 8 0.66532 0.02010 33.104 < 2e-16 ***
Time 9 0.79412 0.01934 41.053 < 2e-16 ***
Time 10 0.89219 0.01985 44.943 < 2e-16 ***
Time 11 1.00907 0.01962 51.422 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2781 on 1990 degrees of freedom
Multiple R-squared: 0.742, Adjusted R-squared: 0.7407
F-statistic: 572.4 on 10 and 1990 DF, p-value: < 2.2e-16
Call:
lm(formula = dy ~ xmat + 0)
Residuals:
Min 1Q Median 3Q Max
-3.3314 -0.4337 -0.0131 0.3928 3.0717
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Time 2 0.20984 0.04976 4.217 2.59e-05 ***
Time 3 0.26246 0.04904 5.352 9.69e-08 ***
Time 4 0.36714 0.04932 7.443 1.45e-13 ***
Time 5 0.49125 0.05010 9.805 < 2e-16 ***
Time 6 0.60275 0.04883 12.344 < 2e-16 ***
Time 7 0.70359 0.04954 14.203 < 2e-16 ***
Time 8 0.85174 0.05004 17.021 < 2e-16 ***
Time 9 0.89148 0.04816 18.509 < 2e-16 ***
Time 10 0.99968 0.04943 20.224 < 2e-16 ***
Time 11 1.09254 0.04886 22.360 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.6925 on 1990 degrees of freedom
Multiple R-squared: 0.3403, Adjusted R-squared: 0.337
F-statistic: 102.6 on 10 and 1990 DF, p-value: < 2.2e-16
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0000 0.3148 0.6027 0.5884 0.8716 1.0925
F-value for heteroskedasticity test = 415.8353
p-value = 1
Call:
lm(formula = dy ~ xmat + 0, weights = wgt)
Weighted Residuals:
Min 1Q Median 3Q Max
-4.8921 -0.8596 -0.0084 0.8311 3.8085
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Time 2 0.19027 0.03970 4.792 1.77e-06 ***
Time 3 0.24730 0.04043 6.117 1.14e-09 ***
Time 4 0.33152 0.04206 7.882 5.27e-15 ***
Time 5 0.45119 0.04308 10.472 < 2e-16 ***
Time 6 0.55364 0.04291 12.902 < 2e-16 ***
Time 7 0.68489 0.04405 15.547 < 2e-16 ***
Time 8 0.79288 0.04514 17.563 < 2e-16 ***
Time 9 0.84239 0.04455 18.908 < 2e-16 ***
Time 10 0.95772 0.04661 20.550 < 2e-16 ***
Time 11 1.04569 0.04934 21.195 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.247 on 1990 degrees of freedom
Multiple R-squared: 0.2605, Adjusted R-squared: 0.2568
F-statistic: 70.11 on 10 and 1990 DF, p-value: < 2.2e-16
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0000 0.2894 0.5536 0.5543 0.8176 1.0457
F-value for heteroskedasticity test = 185.2463
p-value = 1
Call:
lm(formula = dy ~ xmat + 0, weights = wgt)
Weighted Residuals:
Min 1Q Median 3Q Max
-3.9776 -0.6828 -0.0080 0.6746 2.8256
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Time 2 0.18887 0.04096 4.611 4.26e-06 ***
Time 3 0.24581 0.04088 6.014 2.15e-09 ***
Time 4 0.33034 0.04228 7.813 8.98e-15 ***
Time 5 0.45061 0.04316 10.441 < 2e-16 ***
Time 6 0.55518 0.04297 12.921 < 2e-16 ***
Time 7 0.67922 0.04413 15.391 < 2e-16 ***
Time 8 0.79412 0.04532 17.524 < 2e-16 ***
Time 9 0.83978 0.04472 18.778 < 2e-16 ***
Time 10 0.95210 0.04692 20.293 < 2e-16 ***
Time 11 1.04344 0.04950 21.078 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9989 on 1990 degrees of freedom
Multiple R-squared: 0.2633, Adjusted R-squared: 0.2596
F-statistic: 71.14 on 10 and 1990 DF, p-value: < 2.2e-16
McSpatial documentation built on May 2, 2019, 9:32 a.m.
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Description Usage Arguments Details Value References See Also Examples
Description
Standard and Weighted Least Squares Repeat Sales Estimation
Usage
1 2 3 |
Arguments
price0 |
Earlier price in repeat sales pair |
time0 |
Earlier time in repeat sales pair |
price1 |
Later price in repeat sales pair |
time1 |
Later time in repeat sales pair |
mergefirst |
Number of initial periods with coefficients constrained to zero. Default: mergefirst=1 |
graph |
If TRUE, graph results. Default: graph=T |
graph.conf |
If TRUE, add confidence intervals to graph. Default: graph.conf=T |
conf |
Confidence level for intervals. Default: .95 |
stage3 |
If stage3 = NULL, no corrections for heteroskedasticty. If stage3="abs", uses the absolute value of the first-stage residuals as the dependent variable in the second-stage regression. If stage3="square", uses the square of the first-stage residuals as the dependent variable. Default: stage3=NULL. |
stage3_xlist |
List of explanatory variables for heteroskedasticity. By default, the single variable timesale = time1-time0 is constructed and used as the explanatory variable when stage3="abs" or stage3="square". Alternatively, a formula can be provided for a user-specified list of explanatory variables, e.g., stage3_xlist=~x1+x2. Important: note the "~" before the variable list. |
print |
If print=T, prints the regression results. Prints one stage only – the first stage when stage=NULL and the final stage when stage3="square" or stage3="abs". Default: print=T. |
Details
The repeat sales model is
y(t) - y(s) = δ(t) - δ(s) + u(t) - u(s)
where y is the log of sales price, s denotes the earlier sale in a repeat sales pair, and t denotes the later sale. Each entry of the data set should represent a repeat sales pair, with price0 = y(s), price1 = y(t), time0 = s, and time1 = t. The function repsaledata can help transfer a standard hedonic data set to a set of repeat sales pairs.
Repeat sales estimates are sometimes very sensitive to sales from the first few time periods, particularly when the sample size is small. The option mergefirst indicates the number of time periods for which the price index is constrained to equal zero. The default is mergefirst = 1, meaning that the price index equals zero for just the first time period. The repsale command does not have an option for including an intercept in the model.
Following Case and Shiller (1987), many authors use a three-stage procedure to construct repeat sales price indexes that are adjusted for heteroskedasticity related to the length of time between sales. Common specifications for the second-stage function are e^2 = α0 + α1 (t-s) or |e| = α0 + α1 (t-s), where e represents the first-stage residuals. The first equation implies an error variance of σ^2 = e^2 and the second equation leads to σ^2 = |e|^2. The repsale function uses a standard F test to determine whether the slope cofficients are significant in the second-stage regression. The results are reported if print=T.
The third-stage equation is
(y(t) - y(s))/σ = (δ(t) - δ(s))/σ + (u(t) - u(s))/σ
This equation is estimated by regressing y(t) - y(s) on the series of indicator variables implied by δ(t) - δ(s) using the weights option in lm with weights = 1/sigma^2
Value
fit |
Full regression model. |
pindex |
The estimated price index. |
lo |
The lower bounds for the price index confidence intervals. |
hi |
The upper bounds for the price index confidence intervals. |
dy |
The dependent variable for the repeat sales regression, dy = price1-price0. |
xmat |
The matrix of explanatory variables for the repeat sales regressions. dim(xmat) = nt - mergefirst, where nt = the number of time periods and mergefirst is specified in the call to repsale. |
References
Case, Karl and Robert Shiller, "Prices of Single-Family Homes since 1970: New Indexes for Four Cities," New England Economic Review (1987), 45-56.
See Also
repsaledata
repsalefourier
repsaleqreg
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | set.seed(189)
n = 2000
# sale dates range from 0-10
# drawn uniformly from all possible time0, time1 combinations with time0<time1
tmat <- expand.grid(seq(0,10), seq(0,10))
tmat <- tmat[tmat[,1]<tmat[,2], ]
tobs <- sample(seq(1:nrow(tmat)),n,replace=TRUE)
time0 <- tmat[tobs,1]
time1 <- tmat[tobs,2]
timesale <- time1-time0
table(timesale)
# constant variance; index ranges from 0 at time 0 to 1 at time 10
y0 <- time0/10 + rnorm(n,0,.2)
y1 <- time1/10 + rnorm(n,0,.2)
fit <- repsale(price0=y0, price1=y1, time0=time0, time1=time1)
# variance rises with timesale
# var(u0) = .2^2; var(u1) = (.2 + timesale/10)^2
# var(u1-u0) = var(u0) + var(u1) = 2*(.2^2) + .4*timesale/10 + (timesale^2)/100
y0 <- time0/10 + rnorm(n,0,.2)
y1 <- time1/10 + rnorm(n,0,.2+timesale/10)
par(ask=TRUE)
fit <- repsale(price0=y0, price1=y1, time0=time0, time1=time1)
summary(fit$pindex)
fit <- repsale(price0=y0, price1=y1, time0=time0, time1=time1, stage3="abs")
summary(fit$pindex)
timesale2 <- timesale^2
fit <- repsale(price0=y0, price1=y1, time0=time0, time1=time1, stage3="square",
stage3_xlist=~timesale+timesale2)
|
Example output
Loading required package: lattice
Loading required package: locfit
locfit 1.5-9.1 2013-03-22
Loading required package: maptools
Loading required package: sp
Checking rgeos availability: TRUE
Loading required package: quantreg
Loading required package: SparseM
Attaching package: 'SparseM'
The following object is masked from 'package:base':
backsolve
Loading required package: RANN
timesale
1 2 3 4 5 6 7 8 9 10
368 349 264 253 223 178 167 104 56 38
Call:
lm(formula = dy ~ xmat + 0)
Residuals:
Min 1Q Median 3Q Max
-0.98034 -0.18923 -0.00071 0.19144 0.90325
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Time 2 0.07128 0.01999 3.567 0.00037 ***
Time 3 0.18775 0.01969 9.533 < 2e-16 ***
Time 4 0.29225 0.01981 14.753 < 2e-16 ***
Time 5 0.39466 0.02012 19.613 < 2e-16 ***
Time 6 0.48243 0.01961 24.600 < 2e-16 ***
Time 7 0.60606 0.01990 30.462 < 2e-16 ***
Time 8 0.66532 0.02010 33.104 < 2e-16 ***
Time 9 0.79412 0.01934 41.053 < 2e-16 ***
Time 10 0.89219 0.01985 44.943 < 2e-16 ***
Time 11 1.00907 0.01962 51.422 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.2781 on 1990 degrees of freedom
Multiple R-squared: 0.742, Adjusted R-squared: 0.7407
F-statistic: 572.4 on 10 and 1990 DF, p-value: < 2.2e-16
Call:
lm(formula = dy ~ xmat + 0)
Residuals:
Min 1Q Median 3Q Max
-3.3314 -0.4337 -0.0131 0.3928 3.0717
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Time 2 0.20984 0.04976 4.217 2.59e-05 ***
Time 3 0.26246 0.04904 5.352 9.69e-08 ***
Time 4 0.36714 0.04932 7.443 1.45e-13 ***
Time 5 0.49125 0.05010 9.805 < 2e-16 ***
Time 6 0.60275 0.04883 12.344 < 2e-16 ***
Time 7 0.70359 0.04954 14.203 < 2e-16 ***
Time 8 0.85174 0.05004 17.021 < 2e-16 ***
Time 9 0.89148 0.04816 18.509 < 2e-16 ***
Time 10 0.99968 0.04943 20.224 < 2e-16 ***
Time 11 1.09254 0.04886 22.360 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.6925 on 1990 degrees of freedom
Multiple R-squared: 0.3403, Adjusted R-squared: 0.337
F-statistic: 102.6 on 10 and 1990 DF, p-value: < 2.2e-16
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0000 0.3148 0.6027 0.5884 0.8716 1.0925
F-value for heteroskedasticity test = 415.8353
p-value = 1
Call:
lm(formula = dy ~ xmat + 0, weights = wgt)
Weighted Residuals:
Min 1Q Median 3Q Max
-4.8921 -0.8596 -0.0084 0.8311 3.8085
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Time 2 0.19027 0.03970 4.792 1.77e-06 ***
Time 3 0.24730 0.04043 6.117 1.14e-09 ***
Time 4 0.33152 0.04206 7.882 5.27e-15 ***
Time 5 0.45119 0.04308 10.472 < 2e-16 ***
Time 6 0.55364 0.04291 12.902 < 2e-16 ***
Time 7 0.68489 0.04405 15.547 < 2e-16 ***
Time 8 0.79288 0.04514 17.563 < 2e-16 ***
Time 9 0.84239 0.04455 18.908 < 2e-16 ***
Time 10 0.95772 0.04661 20.550 < 2e-16 ***
Time 11 1.04569 0.04934 21.195 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.247 on 1990 degrees of freedom
Multiple R-squared: 0.2605, Adjusted R-squared: 0.2568
F-statistic: 70.11 on 10 and 1990 DF, p-value: < 2.2e-16
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0000 0.2894 0.5536 0.5543 0.8176 1.0457
F-value for heteroskedasticity test = 185.2463
p-value = 1
Call:
lm(formula = dy ~ xmat + 0, weights = wgt)
Weighted Residuals:
Min 1Q Median 3Q Max
-3.9776 -0.6828 -0.0080 0.6746 2.8256
Coefficients:
Estimate Std. Error t value Pr(>|t|)
Time 2 0.18887 0.04096 4.611 4.26e-06 ***
Time 3 0.24581 0.04088 6.014 2.15e-09 ***
Time 4 0.33034 0.04228 7.813 8.98e-15 ***
Time 5 0.45061 0.04316 10.441 < 2e-16 ***
Time 6 0.55518 0.04297 12.921 < 2e-16 ***
Time 7 0.67922 0.04413 15.391 < 2e-16 ***
Time 8 0.79412 0.04532 17.524 < 2e-16 ***
Time 9 0.83978 0.04472 18.778 < 2e-16 ***
Time 10 0.95210 0.04692 20.293 < 2e-16 ***
Time 11 1.04344 0.04950 21.078 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9989 on 1990 degrees of freedom
Multiple R-squared: 0.2633, Adjusted R-squared: 0.2596
F-statistic: 71.14 on 10 and 1990 DF, p-value: < 2.2e-16
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