Package '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

---

Sales pairs

Description

Turn repeat-sales data into sales pairs that are suitable for making repeat-sales matrices.

Usage

rs_pairs(period, product)

Arguments

period	A vector that gives the time period for each sale. Usually a date vector, or a factor with the levels in chronological order, but other values are possible if they can be sorted in chronological order (i.e., with order()).
product	A vector that gives the product identifier for each sale. Usually a factor or vector of integer codes for each product.

Value

A numeric vector of indices giving the position of the previous sale for each product, with the convention that the previous sale for the first sale is itself. Ties are resolved according to the order they appear in period.

Note

order() is the workhorse of rs_pairs(), so performance can be sensitive to the types of period and product, and can be slow for large character vectors.

See Also

rs_matrix() for using sales pairs to make a repeat-sales index.

rtCreateTrans() in the hpiR package for a feature-rich but slower and less flexible function to make sales pairs.

Examples

# Make sales pairs
x <- data.frame(
  id = c(1, 1, 1, 3, 2, 2, 3, 3),
  date = c(1, 2, 3, 2, 1, 3, 4, 1),
  price = c(1, 3, 2, 3, 1, 1, 1, 2)
)

pairs <- rs_pairs(x$date, x$id)

x[c("date_prev", "price_prev")] <- x[c("date", "price")][pairs, ]

x

---

Robust variance matrix for repeat-sales indexes

Description

Convenience function to compute a cluster-robust variance matrix for a linear regression, with or without instruments, where clustering occurs along one dimension. Useful for calculating a variance matrix when a regression is calculated manually.

Usage

rs_var(u, Z, X = Z, ids = seq_len(nrow(X)), df = NULL)

Arguments

u	An $n \times 1$ vector of residuals from a linear regression.
Z	An $n \times k$ matrix of instruments.
X	An $n \times k$ matrix of covariates.
ids	A factor of length $n$, or something that can be coerced into one, that groups observations in u. By default each observation belongs to its own group.
df	An optional degrees of freedom correction. Default is Stata's small sample degrees of freedom correction.

Details

This function calculates the standard robust variance matrix for a linear regression, as in Manski (1988, section 8.1.2) or White (2001, Theorem 6.3); that is, $(Z'X)^{-1} V (X'Z)^{-1}$. It is useful when a regression is calculated by hand. This generalizes the variance matrix proposed by Shiller (1991, section II) when a property sells more than twice.

This function gives the same result as vcovHC(x, type = 'sss', cluster = 'group') from the plm package.

Value

A $k \times k$ covariance matrix.

References

Manski, C. (1988). Analog Estimation Methods in Econometrics. Chapman and Hall.

Shiller, R. J. (1991). Arithmetic repeat sales price estimators. Journal of Housing Economics, 1(1):110-126.

White, H. (2001). Asymptotic Theory for Econometricians (revised edition). Emerald Publishing.

Examples

# Makes some groups in mtcars
mtcars$clust <- letters[1:4]

# Matrices for regression
x <- model.matrix(~ cyl + disp, mtcars)
y <- matrix(mtcars$mpg)

# Regression coefficients
b <- solve(crossprod(x), crossprod(x, y))

# Residuals
r <- y - x %*% b

# Robust variance matrix
vcov <- rs_var(r, x, ids = mtcars$clust)

## Not run: 
# Same as plm
library(plm)
mdl <- plm(mpg ~ cyl + disp, mtcars, model = "pooling", index = "clust")
vcov2 <- vcovHC(mdl, type = "sss", cluster = "group")
vcov - vcov2

## End(Not run)

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