rs_matrix {rsmatrix}

Shiller's repeat-sales matrices

Description

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.

Usage

rs_matrix(t2, t1, p2, p1, f = NULL, sparse = FALSE)

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 t1 and t2, 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)?

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.

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.

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)

# Standard errors
vcov <- rs_var(y - Z %*% b, Z)
sqrt(diag(vcov)) * grs # delta method

# 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)

# Standard errors
vcov <- rs_var(Y - X %*% b, Z, X)
sqrt(diag(vcov)) * ars^2 / 100 # delta method

# 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