Create a function to compute the \(Z\), \(X\), \(y\), and \(Y\) matrices in Shiller (1991, sections I-II) from sales-pair data in order to calculate a repeat-sales price index.
Arguments
- t2, t1
A pair of vectors giving the time period of the second and first sale, respectively. Usually a vector of dates, but other values are possible if they can be coerced to character vectors and sorted in chronological order (i.e., with
order()).- p2, p1
A pair of numeric vectors giving the price of the second and first sale, respectively.
- f
An optional factor the same length as
t1andt2, or a vector to be turned into a factor, that is used to group sales.- sparse
Should sparse matrices from the Matrix package be used (faster for large datasets), or regular dense matrices (the default)?
Value
A function that takes a single argument naming the desired matrix.
It returns one of two matrices (\(Z\) and \(X\)) or two vectors
(\(y\) and \(Y\)), either regular matrices if sparse = FALSE, or sparse
matrices of class dgCMatrix if sparse = TRUE.
Details
The function returned by rs_matrix() computes a generalization of the
matrices in Shiller (1991, sections I-II) that are applicable to grouped
data. These are useful for calculating separate indexes for many, say,
cities without needing an explicit loop.
The \(Z\), \(X\), and \(Y\) matrices are not well defined if either
t1 or t2 have missing values, and an error is thrown in this
case. Similarly, it should always be the case that t2 > t1, otherwise
a warning is given.
References
Bailey, M. J., Muth, R. F., and Nourse, H. O. (1963). A regression method for real estate price index construction. Journal of the American Statistical Association, 53(304):933-942.
Shiller, R. J. (1991). Arithmetic repeat sales price estimators. Journal of Housing Economics, 1(1):110-126.
See also
rs_pairs() for turning sales data into sales pairs.
Examples
# Make some data
x <- data.frame(
date = c(3, 2, 3, 2, 3, 3),
date_prev = c(1, 1, 2, 1, 2, 1),
price = 6:1,
price_prev = 1
)
# Calculate matrices
mat <- with(x, rs_matrix(date, date_prev, price, price_prev))
Z <- mat("Z") # Z matrix
X <- mat("X") # X matrix
y <- mat("y") # y vector
Y <- mat("Y") # Y vector
# Calculate the GRS index in Bailey, Muth, and Nourse (1963)
b <- solve(crossprod(Z), crossprod(Z, y))[, 1]
# or b <- qr.coef(qr(Z), y)
(grs <- exp(b) * 100)
#> 2 3
#> 235.0755 403.5654
# Standard errors
vcov <- rs_var(y - Z %*% b, Z)
sqrt(diag(vcov)) * grs # delta method
#> 2 3
#> 111.0797 257.6581
# Calculate the ARS index in Shiller (1991)
b <- solve(crossprod(Z, X), crossprod(Z, Y))[, 1]
# or b <- qr.coef(qr(crossprod(Z, X)), crossprod(Z, Y))
(ars <- 100 / b)
#> 2 3
#> 310.5263 491.6667
# Standard errors
vcov <- rs_var(Y - X %*% b, Z, X)
sqrt(diag(vcov)) * ars^2 / 100 # delta method
#> 2 3
#> 100.0316 232.3111
# Works with grouped data
x <- data.frame(
date = c(3, 2, 3, 2),
date_prev = c(2, 1, 2, 1),
price = 4:1,
price_prev = 1,
group = c("a", "a", "b", "b")
)
mat <- with(x, rs_matrix(date, date_prev, price, price_prev, group))
b <- solve(crossprod(mat("Z"), mat("X")), crossprod(mat("Z"), mat("Y")))[, 1]
100 / b
#> a.2 b.2 a.3 b.3
#> 300 100 1200 200