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sketching

The package “sketching” is an R package that provides a variety of random sketching methods via random subspace embeddings Researchers may perform regressions using a sketch of data of size m instead of the full sample of size n for a variety of reasons. Lee and Ng (2022) considers the case when the regression errors do not have constant variance and heteroskedasticity robust standard errors would normally be needed for test statistics to provide accurate inference. It is shown in Lee and Ng (2022) that estimates using data sketched by random projections will behave as if the errors were homoskedastic. Estimation by random sampling would not have this property.

For more details, see the following papers.

• Lee, S. and Ng, S. (2022). “Least Squares Estimation Using Sketched Data with Heteroskedastic Errors,” arXiv:2007.07781, accepted for presentation at the Thirty-ninth International Conference on Machine Learning (ICML 2022).
• Lee, S. and Ng, S. (2020). “An Econometric Perspective on Algorithmic Subsampling,” Annual Review of Economics, 12(1): 45–80.

Installation

You can install the released version of sketching from CRAN with:

install.packages("sketching")

Alternatively, you can install the development version from GitHub with:

# install.packages("devtools") if devtools is not installed
devtools::install_github("https://github.com/sokbae/sketching")

Numerical Illustration

We begin by calling the sketching package and fix the seed for reproducibility.

library(sketching)
seed <- 220526  
set.seed(seed)  

Estimating the Return to Education

To illustrate the usefulness of the package, we use the well-known Angrist and Krueger (1991) dataset. A particular extract of their dataset is included in the package. Specifically, we look at the ordinary least squares (OLS) and two stage least squares (2SLS) estimates of the return to education in columns (1) and (2) of Table IV in their paper. The dependent variable y is the log weekly wages, the covariates X include years of education, the intercept term and 9 year-of-birth dummies (p=11). Following Angrist and Krueger (1991), the instruments Z are a full set of quarter-of-birth times year-of-birth interactions (q=30). Their idea was that season of birth is unlikely to be correlated with workers’ ability but can affect educational attainment because of compulsory schooling laws. The full sample size is n = 247, 199.

We now define the variables accordingly.

Y <- AK$LWKLYWGE
intercept <- AK$CNST
X_end <- AK$EDUC
X_exg <- AK[,3:11]
X <- cbind(X_exg, X_end)
Z_inst <- AK[,12:(ncol(AK)-1)]
Z <- cbind(X_exg, Z_inst)
fullsample <- cbind(Y,intercept,X)
n <- nrow(fullsample)
d <- ncol(X)

How to Choose m

We start with how to choose m in this application. Lee and Ng (2020) highlights the tension between a large m required for accurate inference, and a small m for computation efficiency. From the algorithmic perspective, m needs to be chosen as small as possible to achieve computational efficiency. However, statistical analysis often cares about the variability of the estimates in repeated sampling and a larger m may be desirable from the perspective of statistical efficiency. An inference-conscious guide m₂ can be obtained as in Lee and Ng (2020) by targeting the power at γ̄ of a one-sided t-test for given nominal size. Alternatively, a data-oblivious sketch size m₃ for a pre-specified τ₂(∞) can be used.

We focus on the data-oblivious sketch size m₃, as it is simpler to use. We set the target size α = 0.05 and the target power γ = 0.8. Then, S²(ᾱ,γ̄) = 6.18. It remains to specify τ₂(∞), which can be interpreted as the value of t-statistic when the sample size is really large.

OLS Estimation Results

For OLS, we take τ₂(∞) = 10, resulting in m = 15, 283 (about 6% of n).

# choice of m (data-oblivious sketch size)
target_size <- 0.05
target_power <- 0.8
S_constant <- (stats::qnorm(1-target_size) + stats::qnorm(target_power))^2
tau_limit <- 10
m_ols <- floor(n*S_constant/tau_limit^2) 
print(m_ols)
#> [1] 15283

As a benchmark, we first obtain the OLS estimate using the full sample.

ys <- fullsample[,1]
reg <- as.matrix(fullsample[,-1])
fullmodel <- lm(ys ~ reg - 1)
# use homoskedasticity-only asymptotic variance
ztest <- lmtest::coeftest(fullmodel, df = Inf)
est <- ztest[(d+1),1] 
se <- ztest[(d+1),2]
print(c(est,se))
#> [1] 0.0801594610 0.0003552066
# use heteroskedasticity-robust asymptotic variance
ztest_hc <- lmtest::coeftest(fullmodel, df = Inf, 
            vcov = sandwich::vcovHC, type = "HC0")
est_hc <- ztest_hc[(d+1),1] 
se_hc <- ztest_hc[(d+1),2]
print(c(est_hc,se_hc))
#> [1] 0.0801594610 0.0003946747

We now obtain the OLS estimates using a Bernoulli subsampling.

subsample <- sketch(fullsample, m_ols, method = "bernoulli")
ys <- subsample[,1]
reg <- subsample[,-1]
submodel <- lm(ys ~ reg - 1) 
# use homoskedasticity-only asymptotic variance
ztest <- lmtest::coeftest(submodel, df = Inf)
est <- ztest[(d+1),1] 
se <- ztest[(d+1),2]
print(c(est,se))
#> [1] 0.080631991 0.001443075
# use heteroskedasticity-robust asymptotic variance
ztest_hc <- lmtest::coeftest(submodel, df = Inf, 
            vcov = sandwich::vcovHC, type = "HC0")
est_hc <- ztest_hc[(d+1),1] 
se_hc <- ztest_hc[(d+1),2]
print(c(est_hc,se_hc))
#> [1] 0.080631991 0.001603487

As another example of random sampling, we now consider uniform sampling.

subsample <- sketch(fullsample, m_ols, method = "unif")
ys <- subsample[,1]
reg <- subsample[,-1]
submodel <- lm(ys ~ reg - 1) 
# use homoskedasticity-only asymptotic variance
ztest <- lmtest::coeftest(submodel, df = Inf)
est <- ztest[(d+1),1] 
se <- ztest[(d+1),2]
print(c(est,se))
#> [1] 0.078158594 0.001467223
# use heteroskedasticity-robust asymptotic variance
ztest_hc <- lmtest::coeftest(submodel, df = Inf, 
            vcov = sandwich::vcovHC, type = "HC0")
est_hc <- ztest_hc[(d+1),1] 
se_hc <- ztest_hc[(d+1),2]
print(c(est_hc,se_hc))
#> [1] 0.078158594 0.001630221

We now move to random projection schemes. First, we consider countsketch.

subsample <- sketch(fullsample, m_ols, method = "countsketch")
ys <- subsample[,1]
reg <- subsample[,-1]
submodel <- lm(ys ~ reg - 1) 
# use homoskedasticity-only asymptotic variance
ztest <- lmtest::coeftest(submodel, df = Inf)
est <- ztest[(d+1),1] 
se <- ztest[(d+1),2]
print(c(est,se))
#> [1] 0.07818373 0.00142449
# use heteroskedasticity-robust asymptotic variance
ztest_hc <- lmtest::coeftest(submodel, df = Inf, 
            vcov = sandwich::vcovHC, type = "HC0")
est_hc <- ztest_hc[(d+1),1] 
se_hc <- ztest_hc[(d+1),2]
print(c(est_hc,se_hc))
#> [1] 0.07818373 0.00146307

Next, we consider Subsampled Randomized Hadamard Transform (SRHT).

subsample <- sketch(fullsample, m_ols, method = "srht")
ys <- subsample[,1]
reg <- subsample[,-1]
submodel <- lm(ys ~ reg - 1) 
# use homoskedasticity-only asymptotic variance
ztest <- lmtest::coeftest(submodel, df = Inf)
est <- ztest[(d+1),1] 
se <- ztest[(d+1),2]
print(c(est,se))
#> [1] 0.077694934 0.001417144
# use heteroskedasticity-robust asymptotic variance
ztest_hc <- lmtest::coeftest(submodel, df = Inf, 
            vcov = sandwich::vcovHC, type = "HC0")
est_hc <- ztest_hc[(d+1),1] 
se_hc <- ztest_hc[(d+1),2]
print(c(est_hc,se_hc))
#> [1] 0.077694934 0.001420585

For each sketching scheme, only one random sketch is drawn; hence, the results can change if we redraw sketches. Note that the intercept term is included in the full sample data before applying sketching methods. This is important for random sketching schemes as the observations across different rows are randomly combined.

Remarkably, all sketched estimates are 0.08, reproducing the full sample estimate up to the second digit. The sketched homoskedasticity-only standard errors are also very much the same across different methods. The Eicker-Huber-White standard error (i.e., heteroskedasticity-robust standard error) is a bit larger than the homoskedastic standard error with the full sample. As expected, the same pattern is observed for Bernoulli and uniform sampling, as these sampling schemes preserve conditional heteroskedasticity.

2SLS Estimation Results

We now move to 2SLS estimation. For 2SLS, as it is more demanding to achieve good precision, we take τ₂(∞) = 5, resulting in m = 61, 132 (about 25% of n).

fullsample <- cbind(Y,intercept,X,intercept,Z)
n <- nrow(fullsample)
p <- ncol(X)
q <- ncol(Z)
# choice of m (data-oblivious sketch size)
target_size <- 0.05
target_power <- 0.8
S_constant <- (qnorm(1-target_size) + qnorm(target_power))^2
tau_limit <- 5
m_2sls <- floor(n*S_constant/tau_limit^2) 
print(m_2sls)
#> [1] 61132

As before, we first obtain the 2SLS estimate using the full sample.

ys <- fullsample[,1]
reg <- as.matrix(fullsample[,2:(p+2)])
inst <- as.matrix(fullsample[,(p+3):ncol(fullsample)]) 
fullmodel <- ivreg::ivreg(ys ~ reg - 1 | inst - 1) 
# use homoskedasticity-only asymptotic variance
ztest <- lmtest::coeftest(fullmodel, df = Inf)
est <- ztest[(d+1),1] 
se <- ztest[(d+1),2]
print(c(est,se))
#> [1] 0.07685568 0.01504165
# use heteroskedasticity-robust asymptotic variance
ztest_hc <- lmtest::coeftest(fullmodel, df = Inf, 
            vcov = sandwich::vcovHC, type = "HC0")
est_hc <- ztest_hc[(d+1),1] 
se_hc <- ztest_hc[(d+1),2]
print(c(est_hc,se_hc))
#> [1] 0.07685568 0.01512252

The 2SLS estimate of the return to education is 0.769 and both types of standard errors are almost the same.

We now consider a variety of sketching schemes.

# sketching methods for 2SLS
methods <- c("bernoulli","unif","countsketch","srht")
results_2sls <- array(NA, dim = c(length(methods),3))
for (met in 1:length(methods)){
  method <- methods[met]
    # generate a sketch
    subsample <- sketch(fullsample, m_2sls, method = method)
    ys <- subsample[,1]
    reg <- as.matrix(subsample[,2:(p+2)])
    inst <- as.matrix(subsample[,(p+3):ncol(subsample)]) 
    submodel <- ivreg::ivreg(ys ~ reg - 1 | inst - 1) 
    # use homoskedasticity-only asymptotic variance
    ztest <- lmtest::coeftest(submodel, df = Inf)
    est <- ztest[(d+1),1] 
    se <- ztest[(d+1),2]
    # use heteroskedasticity-robust asymptotic variance
    ztest_hc <- lmtest::coeftest(submodel, df = Inf, 
            vcov = sandwich::vcovHC, type = "HC0")
    est_hc <- ztest_hc[(d+1),1] 
    se_hc <- ztest_hc[(d+1),2]
  results_2sls[met,] <- c(est, se, se_hc)
}
rownames(results_2sls) <- methods
colnames(results_2sls) <- c("est", "non-robust se","robust se")
print(results_2sls)
#>                    est non-robust se  robust se
#> bernoulli   0.08090466    0.02358229 0.02362023
#> unif        0.05960437    0.02412623 0.02467241
#> countsketch 0.10289271    0.02297033 0.02615567
#> srht        0.08491405    0.02403900 0.02412490

The sketched 2SLS estimates vary more than the sketched OLS estimates, reflecting that the 2SLS estimates are less precisely estimated than the OLS estimates. As in the full sample case, both types of standard errors are similar across all sketches for 2SLS.

References

• Angrist, J. D., and A. B. Krueger (1991). “Does Compulsory School Attendance Affect Schooling and Earnings?” Quarterly Journal of Economics 106, no. 4 (1991): 979–1014. https://doi.org/10.2307/2937954.

• Lee, S. and Ng, S. (2022). “Least Squares Estimation Using Sketched Data with Heteroskedastic Errors,” arXiv:2007.07781, accepted for presentation at the Thirty-ninth International Conference on Machine Learning (ICML 2022).

• Lee, S. and Ng, S. (2020). “An Econometric Perspective on Algorithmic Subsampling,” Annual Review of Economics, 12(1): 45–80. https://doi.org/10.1146/annurev-economics-022720-114138

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