# Inference for Rank-Rank Regressions

The following example illustrates how the csranks package can be used for estimation and inference in rank-rank regressions. These are commonly used for studying intergenerational mobility.

## Example: Intergenerational Mobility

In this example, we want to intergenerational income mobility by estimating and performing inference on the rank correlation between parents and their children’s incomes. The csranks package contains an artificial dataset with data on children’s and parents’ household incomes, the child’s gender and race (black, hisp or neither).

First, load the package csranks. Second, load the data and take a quick look at it:

library(csranks)
data(parent_child_income)
#>    c_faminc  p_faminc gender    race
#> 1  78771.72 77127.484 female neither
#> 2  79268.33 62723.303 female neither
#> 3  45405.98 65751.340   male neither
#> 4  81951.64 58723.050   male neither
#> 5  88350.33  6381.047   male neither
#> 6 161331.33 40325.466 female neither

### Rank-rank regression

In economics, it is common to estimate measures of mobility by running rank-rank regressions. For instance, the rank correlation between parents’ and children’s incomes can be estimated by running a regression of a child’s income rank on the parent’s income rank:

lmr_model <- lmranks(r(c_faminc) ~ r(p_faminc), data=parent_child_income)
summary(lmr_model)
#> The number of residual degrees of freedom is not correct.
#> Also, z-value, not t-value, since the distribution used for p-value calculation
#> is standard normal.
#>
#> Call:
#> lmranks(formula = r(c_faminc) ~ r(p_faminc), data = parent_child_income)
#>
#> Residuals:
#>      Min       1Q   Median       3Q      Max
#> -0.65601 -0.21986 -0.00376  0.22088  0.66495
#>
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.312311   0.007161   43.61   <2e-16 ***
#> r(p_faminc) 0.375538   0.014319   26.23   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: NA on 3892 degrees of freedom

This regression specification takes each child’s income (c_faminc), computes its rank among all children’s incomes, then takes each parent’s income (p_faminc) and computes its rank among all parents’ incomes. Then the child’s rank is regressed on the parent’s rank using OLS. The lmranks function computes standard errors, t-values and p-values according to the asymptotic theory developed in Chetverikov and Wilhelm (2023).

A naive approach, which does not lead to valid inference, would compute the children’s and parents’ ranks first and the run a standard OLS regression afterwards:

c_faminc_rank <- frank(parent_child_income$c_faminc, omega=1, increasing=TRUE) p_faminc_rank <- frank(parent_child_income$p_faminc, omega=1, increasing=TRUE)
lm_model <- lm(c_faminc_rank ~ p_faminc_rank)
summary(lm_model)
#>
#> Call:
#> lm(formula = c_faminc_rank ~ p_faminc_rank)
#>
#> Residuals:
#>      Min       1Q   Median       3Q      Max
#> -0.65601 -0.21986 -0.00376  0.22088  0.66495
#>
#> Coefficients:
#>               Estimate Std. Error t value Pr(>|t|)
#> (Intercept)   0.312311   0.008579   36.41   <2e-16 ***
#> p_faminc_rank 0.375538   0.014856   25.28   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.2676 on 3892 degrees of freedom
#> Multiple R-squared:  0.141,  Adjusted R-squared:  0.1408
#> F-statistic:   639 on 1 and 3892 DF,  p-value: < 2.2e-16

Notice that the point estimates of the intercept and slope are the same as those of the lmranks function. However, the standard errors, t-values and p-values differ. This is because the usual OLS formulas for standard errors do not take into account the estimation uncertainty in the ranks.

One can also run the rank-rank regression with additional covariates, e.g.:

lmr_model_cov <- lmranks(r(c_faminc) ~ r(p_faminc) + gender + race, data=parent_child_income)
summary(lmr_model_cov)
#> The number of residual degrees of freedom is not correct.
#> Also, z-value, not t-value, since the distribution used for p-value calculation
#> is standard normal.
#>
#> Call:
#> lmranks(formula = r(c_faminc) ~ r(p_faminc) + gender + race,
#>     data = parent_child_income)
#>
#> Residuals:
#>      Min       1Q   Median       3Q      Max
#> -0.66140 -0.20654 -0.00343  0.21421  0.72917
#>
#> Coefficients:
#>              Estimate Std. Error t value Pr(>|t|)
#> (Intercept)  0.299823   0.018155  16.514  < 2e-16 ***
#> r(p_faminc)  0.323785   0.015123  21.411  < 2e-16 ***
#> gendermale   0.010862   0.008484   1.280  0.20042
#> raceblack   -0.088215   0.020910  -4.219 2.46e-05 ***
#> raceneither  0.055726   0.018781   2.967  0.00301 **
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: NA on 3889 degrees of freedom

### Grouped rank-rank regression

In some economic applications, it is desired to run rank-rank regressions separately in subgroups of the population, but compute the ranks in the whole population. For instance, we might want to estimate rank-rank regression slopes as measures of intergenerational mobility separately for males and females, but the ranking of children’s incomes is formed among all children (rather than form separate rankings for males and females).

Such regressions can easily be run using the lmranks function and interaction notation:

grouped_lmr_model_simple <- lmranks(r(c_faminc) ~ r(p_faminc_rank):gender,
data=parent_child_income)
summary(grouped_lmr_model_simple)
#> The number of residual degrees of freedom is not correct.
#> Also, z-value, not t-value, since the distribution used for p-value calculation
#> is standard normal.
#>
#> Call:
#> lmranks(formula = r(c_faminc) ~ r(p_faminc_rank):gender, data = parent_child_income)
#>
#> Residuals:
#>      Min       1Q   Median       3Q      Max
#> -0.65016 -0.21977 -0.00308  0.21750  0.68833
#>
#> Coefficients:
#>                               Estimate Std. Error t value Pr(>|t|)
#> genderfemale                   0.28796    0.01119   25.74   <2e-16 ***
#> gendermale                     0.33506    0.01107   30.27   <2e-16 ***
#> r(p_faminc_rank):genderfemale  0.40800    0.02046   19.94   <2e-16 ***
#> r(p_faminc_rank):gendermale    0.34516    0.02044   16.89   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: NA on 3890 degrees of freedom

In this example, we have run a separate OLS regression of children’s ranks on parents’ ranks among the female and male children. However, incomes of children are ranked among all children and incomes of parents are ranked among all parents. The standard errors, t-values and p-values are implemented according to the asymptotic theory developed in Chetverikov and Wilhelm (2023), where it is shown that the asymptotic distribution of the estimators now need to not only account for the fact that ranks are estimated, but also for the fact that estimators are correlated across gender subgroups because they use the same estimated ranking.

A naive application of the lm function would produce the same point estimates, but not the correct standard errors:

grouped_lm_model_simple <- lm(c_faminc_rank ~ p_faminc_rank:gender + gender - 1, #group-wise intercept
data=parent_child_income)
summary(grouped_lm_model_simple)
#>
#> Call:
#> lm(formula = c_faminc_rank ~ p_faminc_rank:gender + gender -
#>     1, data = parent_child_income)
#>
#> Residuals:
#>      Min       1Q   Median       3Q      Max
#> -0.65016 -0.21977 -0.00308  0.21750  0.68833
#>
#> Coefficients:
#>                            Estimate Std. Error t value Pr(>|t|)
#> genderfemale                0.28796    0.01233   23.35   <2e-16 ***
#> gendermale                  0.33506    0.01192   28.10   <2e-16 ***
#> p_faminc_rank:genderfemale  0.40800    0.02135   19.11   <2e-16 ***
#> p_faminc_rank:gendermale    0.34516    0.02066   16.71   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.2674 on 3890 degrees of freedom
#> Multiple R-squared:  0.7858, Adjusted R-squared:  0.7855
#> F-statistic:  3567 on 4 and 3890 DF,  p-value: < 2.2e-16

One can also create more granular subgroups by interacting several characteristics such as gender and race:

parent_child_income$subgroup <- interaction(parent_child_income$gender, parent_child_income\$race)
grouped_lmr_model <- lmranks(r(c_faminc) ~ r(p_faminc_rank):subgroup,
data=parent_child_income)
summary(grouped_lmr_model)
#> The number of residual degrees of freedom is not correct.
#> Also, z-value, not t-value, since the distribution used for p-value calculation
#> is standard normal.
#>
#> Call:
#> lmranks(formula = r(c_faminc) ~ r(p_faminc_rank):subgroup, data = parent_child_income)
#>
#> Residuals:
#>      Min       1Q   Median       3Q      Max
#> -0.65574 -0.20769 -0.00387  0.21297  0.73248
#>
#> Coefficients:
#>                                         Estimate Std. Error t value Pr(>|t|)
#> subgroupfemale.hisp                      0.26280    0.03268   8.041 8.94e-16
#> subgroupmale.hisp                        0.35313    0.04516   7.820 5.26e-15
#> subgroupfemale.black                     0.19741    0.01968  10.031  < 2e-16
#> subgroupmale.black                       0.26496    0.02702   9.805  < 2e-16
#> subgroupfemale.neither                   0.35096    0.01525  23.006  < 2e-16
#> subgroupmale.neither                     0.36603    0.01322  27.694  < 2e-16
#> r(p_faminc_rank):subgroupfemale.hisp     0.41733    0.07602   5.490 4.02e-08
#> r(p_faminc_rank):subgroupmale.hisp       0.21780    0.09697   2.246 0.024694
#> r(p_faminc_rank):subgroupfemale.black    0.30798    0.05154   5.976 2.29e-09
#> r(p_faminc_rank):subgroupmale.black      0.25528    0.07234   3.529 0.000417
#> r(p_faminc_rank):subgroupfemale.neither  0.33908    0.02532  13.390  < 2e-16
#> r(p_faminc_rank):subgroupmale.neither    0.31818    0.02270  14.018  < 2e-16
#>
#> subgroupfemale.hisp                     ***
#> subgroupmale.hisp                       ***
#> subgroupfemale.black                    ***
#> subgroupmale.black                      ***
#> subgroupfemale.neither                  ***
#> subgroupmale.neither                    ***
#> r(p_faminc_rank):subgroupfemale.hisp    ***
#> r(p_faminc_rank):subgroupmale.hisp      *
#> r(p_faminc_rank):subgroupfemale.black   ***
#> r(p_faminc_rank):subgroupmale.black     ***
#> r(p_faminc_rank):subgroupfemale.neither ***
#> r(p_faminc_rank):subgroupmale.neither   ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: NA on 3882 degrees of freedom

Let’s compare the confidence intervals for regression coefficients produced by lmranks and naive approaches.

grouped_lm_model <- lm(c_faminc_rank ~ p_faminc_rank:subgroup + subgroup - 1, #group-wise intercept
data=parent_child_income)
summary(grouped_lm_model)
#>
#> Call:
#> lm(formula = c_faminc_rank ~ p_faminc_rank:subgroup + subgroup -
#>     1, data = parent_child_income)
#>
#> Residuals:
#>      Min       1Q   Median       3Q      Max
#> -0.65574 -0.20769 -0.00387  0.21297  0.73248
#>
#> Coefficients:
#>                                      Estimate Std. Error t value Pr(>|t|)
#> subgroupfemale.hisp                   0.26280    0.03531   7.443 1.20e-13 ***
#> subgroupmale.hisp                     0.35313    0.04083   8.649  < 2e-16 ***
#> subgroupfemale.black                  0.19741    0.02397   8.236 2.41e-16 ***
#> subgroupmale.black                    0.26496    0.02731   9.701  < 2e-16 ***
#> subgroupfemale.neither                0.35096    0.01561  22.481  < 2e-16 ***
#> subgroupmale.neither                  0.36603    0.01394  26.265  < 2e-16 ***
#> p_faminc_rank:subgroupfemale.hisp     0.41733    0.07490   5.572 2.69e-08 ***
#> p_faminc_rank:subgroupmale.hisp       0.21780    0.08722   2.497 0.012563 *
#> p_faminc_rank:subgroupfemale.black    0.30798    0.06121   5.032 5.08e-07 ***
#> p_faminc_rank:subgroupmale.black      0.25528    0.06926   3.686 0.000231 ***
#> p_faminc_rank:subgroupfemale.neither  0.33908    0.02536  13.370  < 2e-16 ***
#> p_faminc_rank:subgroupmale.neither    0.31818    0.02300  13.831  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.2633 on 3882 degrees of freedom
#> Multiple R-squared:  0.7927, Adjusted R-squared:  0.792
#> F-statistic:  1237 on 12 and 3882 DF,  p-value: < 2.2e-16
library(ggplot2)
theme_set(theme_minimal())
ci_data <- data.frame(estimate=coef(lmr_model),
parameter=c("Intercept", "slope"),
group="Whole sample",
method="csranks",
lower=confint(lmr_model)[,1],
upper=confint(lmr_model)[,2])

ci_data <- rbind(ci_data, data.frame(
estimate = coef(grouped_lmr_model),
parameter = rep(c("Intercept", "slope"), each=6),
group = rep(c("Hispanic female", "Hispanic male", "Black female", "Black male",
"Other female", "Other male"), times=2),
method="csranks",
lower=confint(grouped_lmr_model)[,1],
upper=confint(grouped_lmr_model)[,2]
))

ci_data <- rbind(ci_data, data.frame(
estimate = coef(lm_model),
parameter = c("Intercept", "slope"),
group = "Whole sample",
method="naive",
lower=confint(lm_model)[,1],
upper=confint(lm_model)[,2]
))

ci_data <- rbind(ci_data, data.frame(
estimate = coef(grouped_lm_model),
parameter = rep(c("Intercept", "slope"), each=6),
group = rep(c("Hispanic female", "Hispanic male", "Black female", "Black male",
"Other female", "Other male"), times=2),
method="naive",
lower=confint(grouped_lm_model)[,1],
upper=confint(grouped_lm_model)[,2]
))

ggplot(ci_data, aes(y=estimate, x=group, ymin=lower, ymax=upper,col=method, fill=method)) +
geom_point(position=position_dodge2(width = 0.9)) +
geom_errorbar(position=position_dodge2(width = 0.9)) +
geom_hline(aes(yintercept=estimate), data=subset(ci_data, group=="Whole sample"),
linetype="dashed",
col="gray") +
coord_flip() +
labs(title="95% confidence intervals of intercept and slope\nin rank-rank regression")+
facet_wrap(~parameter)

The coefficient calculated for the whole sample has a narrow confidence interval, which is expected. In this example, there are some differences in the correct (csranks) confidence intervals and the incorrect (naive) confidence intervals, but they are rather small. The paper by Chetverikov and Wilhelm (2023), however, provides empirical examples in which the differences can be quite large.

## Reference & further reading

Chetverikov and Wilhelm (2023), “Inference for Rank-Rank Regressions”. arXiv preprint arXiv:2310.15512

Check out the documentation of individual functions at the package’s website and further examples in the package’s Github repository.