From 86703cfdf3b51ffb5757a899835e01d62b88acc9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Michal=20Koles=C3=A1r?= Date: Sat, 13 Apr 2019 11:07:10 -0400 Subject: [PATCH] Add covr --- .Rbuildignore | 1 + .travis.yml | 4 + codecov.yml | 12 + doc/RDHonest.R | 87 +++++ vignettes/RDhonest.Rmd => doc/RDHonest.Rmd | 0 doc/RDHonest.pdf | Bin 0 -> 356744 bytes doc/lpkernels.pdf | Bin 238853 -> 239151 bytes readme.md | 6 +- vignettes/RDHonest.Rmd | 392 +++++++++++++++++++++ 9 files changed, 499 insertions(+), 3 deletions(-) create mode 100644 codecov.yml create mode 100644 doc/RDHonest.R rename vignettes/RDhonest.Rmd => doc/RDHonest.Rmd (100%) create mode 100644 doc/RDHonest.pdf create mode 100644 vignettes/RDHonest.Rmd diff --git a/.Rbuildignore b/.Rbuildignore index bfee83b..977b90b 100644 --- a/.Rbuildignore +++ b/.Rbuildignore @@ -7,3 +7,4 @@ readme.md notes.org .lintr +^codecov\.yml$ diff --git a/.travis.yml b/.travis.yml index a5c9cc0..5c08990 100644 --- a/.travis.yml +++ b/.travis.yml @@ -7,3 +7,7 @@ addons: apt: packages: - ghostscript +r_packages: + - covr +after_success: + - Rscript -e 'covr::codecov()' diff --git a/codecov.yml b/codecov.yml new file mode 100644 index 0000000..8f36b6c --- /dev/null +++ b/codecov.yml @@ -0,0 +1,12 @@ +comment: false + +coverage: + status: + project: + default: + target: auto + threshold: 1% + patch: + default: + target: auto + threshold: 1% diff --git a/doc/RDHonest.R b/doc/RDHonest.R new file mode 100644 index 0000000..31cf335 --- /dev/null +++ b/doc/RDHonest.R @@ -0,0 +1,87 @@ +## ---- include=FALSE, cache=FALSE----------------------------------------- +library("knitr") +knitr::opts_knit$set(self.contained = FALSE) +knitr::opts_chunk$set(tidy = TRUE, collapse=TRUE, comment = "#>", + tidy.opts=list(blank=FALSE, width.cutoff=55)) + +## ------------------------------------------------------------------------ +library("RDHonest") +## Assumes first column in the data frame corresponds to outcome, +## and second to running variable +dl <- RDData(lee08, cutoff = 0) + +## Transform earnings to log earnings +do <- RDData(data.frame(logearn=log(cghs$earnings), + year14=cghs$yearat14), cutoff = 1947) + +## ---- fig.width=4.5, fig.height=3.5, fig.cap="Lee (2008) data"----------- +## plot 25-bin averages in for observations 50 at most points away from the cutoff. +## See Figure 1 +plot_RDscatter(dl, avg=25, window = 50, xlab="Margin of victory", + ylab="Vote share in next election") + +## ---- fig.width=4.5, fig.height=3.5, fig.cap="Oreopoulos (2006) data"---- +## see Figure 2 +f2 <- plot_RDscatter(do, avg=Inf, xlab="Year aged 14", ylab="Log earnings", + propdotsize=TRUE) +## Adjust size of dots if they are too big +f2 + ggplot2::scale_size_area(max_size = 4) + +## ------------------------------------------------------------------------ +## Usual critical value +CVb(0, alpha=0.05) # returns a list +CVb(1/2, alpha=0.05)$cv # extract critical value + +## Tabulate critical values for different significance levels +## when bias-sd ratio equals 1/4 +knitr::kable(CVb(1/4, alpha=c(0.01, 0.05, 0.1)), caption="Critical values") + +## ------------------------------------------------------------------------ +RDHonest(voteshare ~ margin, data=lee08, kern="uniform", M=0.1, hp=10, sclass="T") +RDHonest(voteshare ~ margin, data=lee08, kern="uniform", M=0.1, hp=10, sclass="H") + +## ------------------------------------------------------------------------ +RDHonest(voteshare ~ margin, data=lee08, kern="triangular", + M=0.1, opt.criterion="MSE", sclass="H") +## Choose bws optimal for length of CI, allowing for different bws +## on either side of cutoff +RDHonest(voteshare ~ margin, data=lee08, kern="triangular", M=0.1, + opt.criterion="FLCI", sclass="H", bw.equal=FALSE) + +## ------------------------------------------------------------------------ +RDOptBW(voteshare ~ margin, data=lee08, kern="triangular", + M=0.1, opt.criterion="MSE", sclass="H") + +## ------------------------------------------------------------------------ +## Replicate Table 2, column (10) +RDHonest(log(earnings) ~ yearat14, cutoff=1947, + data=cghs, kern="uniform", M=0.04, opt.criterion="FLCI", sclass="H") +## Triangular kernel generally gives tigher CIs +RDHonest(log(earnings) ~ yearat14, cutoff=1947, + data=cghs, kern="triangular", M=0.04, opt.criterion="FLCI", sclass="H") + +## ------------------------------------------------------------------------ +## Replicate Table 2, column (6), run local linear regression (order=1) +## with a uniform kernel (other kernels are not yet implemented) +RDHonestBME(log(earnings) ~ yearat14, cutoff=1947, + data=cghs, hp=3, order=1) + +## ------------------------------------------------------------------------ +2*RDHonest(voteshare ~ margin, data=lee08, kern="optimal", M=0.1, opt.criterion="FLCI", se.initial="Silverman", se.method="nn")$hl + +2*RDHonest(voteshare ~ margin, data=lee08, kern="triangular", M=0.1, opt.criterion="FLCI", se.initial="Silverman", se.method="nn", sclass="T")$hl + + +## ------------------------------------------------------------------------ +## Add variance estimate to the lee data so that the RDSmoothnessBound +## function doesn't have to compute them each time +dl <- RDHonest::RDPrelimVar(dl, se.initial="NN") + +### Only use three point-average for averages of a 100 points closest to cutoff, +### and report results separately for points above and below cutoff +RDSmoothnessBound(dl, s=100, separate=TRUE, multiple=FALSE, sclass="T") + +### Pool estimates based on observations below and above cutoff, and +### use three-point averages over the entire support of the running variable +RDSmoothnessBound(dl, s=100, separate=FALSE, multiple=TRUE, sclass="H") + diff --git a/vignettes/RDhonest.Rmd b/doc/RDHonest.Rmd similarity index 100% rename from vignettes/RDhonest.Rmd rename to doc/RDHonest.Rmd diff --git a/doc/RDHonest.pdf b/doc/RDHonest.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0803305d5ac3fc608a58bb979c44697bf7604043 GIT 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zyF36(s#p#HE4oEt4)AUJV!$5Yd-uhFJHR*Xm-t%jw;)=ulocVi0Px7PBOLm8;lG4DKto+;49!+!8g_E=(^%>rd4BXXX9%4m;lB9z#wn- zZ+IXk+m{QZ19DL|HaJExF+oE!F+oB&Fhw^qG%+wnI5sjeG(}uF)&6rHZn3aL^DP;Gd^7kFHB`_XLM*FGB-0IlTi;SeMuntMshCRr_8tg*`(y$AwPzBX+ z0F6+i;wS8kTG)gQ*a9Ewf1v?#&;m^=LBYl%&f@O%p5_Cfk z^g;lA&<_JJ2r(E^iC@|c;wm?#*Bw^5`>+~Oc?zsXRbC=iV=8YUt8vxjtJQ>Rv1m1^ z`d->Mr7rbpHLdPCXEmel_Qz^g-9y@HPTkkD)x3HaQL6>W!!ew|5h + %\VignetteIndexEntry{Honest inference in Sharp Regression Discontinuity} + %\VignetteEngine{knitr::rmarkdown} + %\VignetteEncoding{UTF-8} +--- + +```{r, include=FALSE, cache=FALSE} +library("knitr") +knitr::opts_knit$set(self.contained = FALSE) +knitr::opts_chunk$set(tidy = TRUE, collapse=TRUE, comment = "#>", + tidy.opts=list(blank=FALSE, width.cutoff=55)) +``` + +The package `RDHonest` implements confidence intervals for the regression +discontinuity parameter considered in @ArKo16optimal, @ArKo16honest, and +@KoRo16. In this vignette, we demonstrate the implementation of these confidence +intervals using datasets from @lee08 and @oreopoulos06, which are included in +the package as a data frame `lee08` and `cghs`. The datasets from @lalive08 and +@LuMi07 that are used in @ArKo16honest, and @KoRo16 are also included in the +package as data frames `rebp` and `headst`. + +# Sharp RD model + +In the sharp regression discontinuity model, we observe units $i=1,\dotsc,n$, +with the outcome $y_i$ for the $i$th unit given by $$ y_i = f(x_i) + u_i, $$ +where $f(x_i)$ is the expectation of $y_i$ conditional on the running variable +$x_i$ and $u_i$ is the regression error. A unit is treated if and only if the +running variable $x_{i}$ lies above a known cutoff $c_{0}$. The parameter of +interest is given by the jump of $f$ at the cutoff, $$ \beta=\lim_{x\downarrow +c_{0}}f(x)-\lim_{x\uparrow c_{0}}f(x).$$ Let $\sigma^2(x_i)$ denote the +conditional variance of $u_i$. + +In the Lee dataset, the running variable corresponds to the margin of victory of +a Democratic candidate in a US House election, and the treatment corresponds to +winning the election. Therefore, the cutoff is zero. The outcome of interest is +the Democratic vote share in the following election. + +The Oreopoulos dataset consists of a subsample of British workers, and it +exploits a change in minimum school leaving age in the UK from 14 to 15, which +occurred in 1947. The running variable is the year in which the individual turned +14, with the cutoff equal to 1947 so that the "treatment" is being subject to a +higher minimum school-leaving age. The outcome is log earnings in 1998. + +Some of the functions in the package require the data to be transformed into a custom `RDData` format. This can be accomplished with the `RDData` function: + +```{r} +library("RDHonest") +## Assumes first column in the data frame corresponds to outcome, +## and second to running variable +dl <- RDData(lee08, cutoff = 0) + +## Transform earnings to log earnings +do <- RDData(data.frame(logearn=log(cghs$earnings), + year14=cghs$yearat14), cutoff = 1947) +``` + +# Plots + +The package provides a function `plot_RDscatter` to plot the raw data. To remove +some noise, the function plots averages over `avg` number of observations. The +function takes an `RDData` object as an argument + +```{r, fig.width=4.5, fig.height=3.5, fig.cap="Lee (2008) data"} +## plot 25-bin averages in for observations 50 at most points away from the cutoff. +## See Figure 1 +plot_RDscatter(dl, avg=25, window = 50, xlab="Margin of victory", + ylab="Vote share in next election") +``` + +The running variable in the Oreopoulos dataset is discrete. It is therefore +natural to plot the average outcome by each value of the running variable, which +is achieved using by setting `avg=Inf`. The option `dotsize="count"` makes the +size of the points proportional to the number of observations that the point +averages over. + +```{r, fig.width=4.5, fig.height=3.5, fig.cap="Oreopoulos (2006) data"} +## see Figure 2 +f2 <- plot_RDscatter(do, avg=Inf, xlab="Year aged 14", ylab="Log earnings", + propdotsize=TRUE) +## Adjust size of dots if they are too big +f2 + ggplot2::scale_size_area(max_size = 4) +``` + + +# Inference based on local polynomial estimates + +The function `RDHonest` constructs one- and two-sided confidence intervals (CIs) +around local linear and local quadratic estimators using either a user-supplied +bandwidth (which is allowed to differ on either side of the cutoff), or +bandwidth that is optimized for a given performance criterion. The sense of +honesty is that, if the regression errors are normally distributed with known +variance, the CIs are guaranteed to achieve correct coverage _in finite +samples_, and achieve correct coverage asymptotically uniformly over the +parameter space otherwise. Furthermore, because the CIs explicitly take into +account the possible bias of the estimators, the asymptotic approximation +doesn't rely on the bandwidth to shrink to zero at a particular rate. + +To describe the form of the CIs, let $\hat{\beta}_{h_{+},h_{-}}$ denote a a +local polynomial estimator with bandwidth equal to $h_{+}$ above the cutoff and +equal to $h_{-}$ below the cutoff. Let $\beta_{h_{+},h_{-}}(f)$ denote its +expectation conditional on the covariates when the regression function equals +$f$. Then the bias of the estimator is given by $\beta_{h_{+},h_{-}}(f)-\beta$. +Let $$ +B(\hat{\beta}_{h_{+},h_{-}})=\sup_{f\in\mathcal{F}}|\beta_{h_{+},h_{-}}(f)-\beta| +$$ denote the worst-case bias over the parameter space $\mathcal{F}$. Then the +lower limit of a one-sided CI is given by $$\hat{\beta}_{h_{+},h_{-}}- +B(\hat{\beta}_{h_{+},h_{-}})-z_{1-\alpha}\widehat{se}(\hat{\beta}_{h_{+},h_{-}}), +$$ where $z_{1-\alpha}$ is the $1-\alpha$ quantile of a standard normal +distribution, and $\widehat{se}(\hat{\beta}_{h_{+},h_{-}})$ is the standard +error (an estimate of the standard deviation of the estimator). Subtracting the +worst-case bias in addition to the usual critical value times standard error +ensures correct coverage at all points in the parameter space. + +A two-sided CI is given by $$ \hat{\beta}_{h_{+},h_{-}} \pm +cv_{1-\alpha}(B(\hat{\beta}_{h_{+},h_{-}})/\widehat{se}(\hat{\beta}_{h_{+},h_{-}}))\times +\widehat{se}(\hat{\beta}_{h_{+},h_{-}}),$$ where the critical value function +$cv_{1-\alpha}(b)$ corresponds to the $1-\alpha$ quantile of the $|N(b,1)|$ +distribution. To see why using this critical value ensures honesty, decompose +the $t$-statistic as $$ +\frac{\hat{\beta}_{h_{+},h_{-}}-\beta}{\widehat{se}(\hat{\beta}_{h_{+},h_{-}})} += +\frac{\hat{\beta}_{h_{+},h_{-}}-\beta_{h_{+},h_{-}}(f)}{\widehat{se}(\hat{\beta}_{h_{+},h_{-}})} ++\frac{\beta_{h_{+},h_{-}}(f)-\beta}{\widehat{se}(\hat{\beta}_{h_{+},h_{-}})} $$ +By a central limit theorem, the first term on the right-hand side will by +distributed standard normal, irrespective of the bias. The second term is +bounded in absolute value by +$B(\hat{\beta}_{h_{+},h_{-}})/\widehat{se}(\hat{\beta}_{h_{+},h_{-}})$, so that, +in large samples, the $1-\alpha$ quantile of the absolute value of the +$t$-statistic will be bounded by +$cv_{1-\alpha}(B(\hat{\beta}_{h_{+},h_{-}})/\widehat{se}(\hat{\beta}_{h_{+},h_{-}}))$. +This approach gives tighter CIs than simply adding and subtracting +$B(\hat{\beta}_{h_+,h_-})$ from the point estimate, in addition to adding and subtracting $z_{1-\alpha}\widehat{se}(\hat{\beta}_{h_+,h_-})$ + +The function `CVb` gives these critical values: + +```{r, } +## Usual critical value +CVb(0, alpha=0.05) # returns a list +CVb(1/2, alpha=0.05)$cv # extract critical value + +## Tabulate critical values for different significance levels +## when bias-sd ratio equals 1/4 +knitr::kable(CVb(1/4, alpha=c(0.01, 0.05, 0.1)), caption="Critical values") +``` + +The field `TeXDescription` is useful for plotting, or for exporting to \LaTeX, +as in the table above. + +## Parameter space + +To implement the honest CIs, one needs to specify the parameter space +$\mathcal{F}$. The function `RDHonest` computes honest CIs when the parameter +space $\mathcal{F}$ corresponds to a second-order Taylor or second-order Hölder +smoothness class, which capture two different types of smoothness restrictions. +The second-order Taylor class assumes that $f$ lies in the the class of +functions $$\mathcal{F}_{\text{Taylor}}(M)= \left \{f_{+}-f_{-}\colon +f_{+}\in\mathcal{F}_{T}(M; [c_{0} +%] %( +,\infty)),\; +f_{-}\in\mathcal{F}_{T}(M;(-\infty, c_{0})) \right\},$$ + +where $\mathcal{F}_{T}(M;\mathcal{X})$ consists of functions $f$ such that the +approximation error from second-order Taylor expansion of $f(x)$ about $c_{0}$ +is bounded by $M|x|^{2}/2$, uniformly over $\mathcal{X}$: \begin{align*} +\mathcal{F}_{T}(M;\mathcal{X}) =\left\{f\colon \left| +f(x)-f(c_0)-f'(c_0)x\right| \leq M|x|^2/2\text{ all }x\in\mathcal{X}\right\}. +\end{align*} The class $\mathcal{F}_{T}(M;\mathcal{X})$ formalizes the idea that +the second derivative of $f$ at zero should be bounded by $M$. See Section 2 in +@ArKo16optimal (note the constant $C$ in that paper equals $C=M/2$ here). This +class is doesn't impose smoothness away from boundary, which may be undesirable +in some empirical applications. The Hölder class addresses this problem by +bounding the second derivative globally. In particular, it assumes that $f$ lies +in the class of functions $$\mathcal{F}_{\text{Hölder}}(M)= \left +\{f_{+}-f_{-}\colon f_{+}\in\mathcal{F}_{H}(M;[c_{0}%] %( ,\infty)),\; +f_{-}\in\mathcal{F}_{H}(M;(-\infty, c_{0})) \right\},$$ + +where $$ \mathcal{F}_{H}(M;\mathcal{X})=\{f\colon |f'(x)-f'(y)|\leq M|x-y| +\;\;x,y\in\mathcal{X}\}.$$ + +The smoothness class is specified using the option `sclass`. CIs around a local +linear estimator with bandwidth that equals to 10 on either side of the cutoff +when the parameter space is given by a Taylor and Hölder smoothness class, +respectively, with $M=0.1$: + +```{r} +RDHonest(voteshare ~ margin, data=lee08, kern="uniform", M=0.1, hp=10, sclass="T") +RDHonest(voteshare ~ margin, data=lee08, kern="uniform", M=0.1, hp=10, sclass="H") +``` + +The confidence intervals use the nearest-neighbor method to estimate the +standard error by default (this can be changed using the option `se.method`, see +help file). The package reports two-sided as well one-sided CIs (with lower as +well as upper limit) by default. + +Instead of specifying a bandwidth, one can just specify the smoothness class and +smoothness constant $M$, and the bandwidth will be chosen optimally for a given +optimality criterion: + +```{r} +RDHonest(voteshare ~ margin, data=lee08, kern="triangular", + M=0.1, opt.criterion="MSE", sclass="H") +## Choose bws optimal for length of CI, allowing for different bws +## on either side of cutoff +RDHonest(voteshare ~ margin, data=lee08, kern="triangular", M=0.1, + opt.criterion="FLCI", sclass="H", bw.equal=FALSE) +``` + +It is also possible to compute the optimal bandwidths directly using the function +`RDOptBW` + +```{r} +RDOptBW(voteshare ~ margin, data=lee08, kern="triangular", + M=0.1, opt.criterion="MSE", sclass="H") +``` + + + + +## Inference when running variable is discrete + +The confidence intervals described above can also be used when the running +variable is discrete, with $G$ support points: their construction makes no +assumptions on the nature of the running variable (see Section 5.1 in @KoRo16 +for more detailed discussion). + +As an example, consider the @oreopoulos06 data, in which the running variable is age in years: +```{r} +## Replicate Table 2, column (10) +RDHonest(log(earnings) ~ yearat14, cutoff=1947, + data=cghs, kern="uniform", M=0.04, opt.criterion="FLCI", sclass="H") +## Triangular kernel generally gives tigher CIs +RDHonest(log(earnings) ~ yearat14, cutoff=1947, + data=cghs, kern="triangular", M=0.04, opt.criterion="FLCI", sclass="H") +``` + +In addition, the package provides function `RDHonestBME` that calculates honest +confidence intervals under the assumption that the specification bias at zero is +no worse at the cutoff than away from the cutoff as in Section 5.2 in @KoRo16. + +```{r} +## Replicate Table 2, column (6), run local linear regression (order=1) +## with a uniform kernel (other kernels are not yet implemented) +RDHonestBME(log(earnings) ~ yearat14, cutoff=1947, + data=cghs, hp=3, order=1) +``` + + +Let us describe the implementation of the variance estimator $\hat{V}(W)$ used to construct the CI as described in in Section 5.2 in @KoRo16. Suppose the point +estimate is given by the first element of the regression of the outcome $y_i$ on +$m(x_i)$. For instance, local linear regression with uniform kernel and +bandwidth $h$ corresponds to $m(x)=I(|x|\leq h)\cdot(I(x>c_0),1,x, x\cdot +I(x>c_0))'$. Let $\theta=Q^{-1}E[m(x_i)y_i]$, where $Q=E[m(x_i)m(x_i)']$, denote +the estimand for this regression (treating the bandwidth as fixed), and let +$\delta(x)=f(x)-m(x)'\theta$ denote the specification error at $x$. The RD +estimate is given by first element of the least squares estimator +$\hat{\theta}=\hat{Q}^{-1}\sum_i m(x_i)y_i$, where $\hat{Q}=\sum_i +m(x_i)m(x_i)'$. + +Let $w(x_i)$ denote a vector of indicator (dummy) variables for all support +points of $x_i$ within distance $h$ of the cutoff, so that $\mu(x_g)$, where +$x_g$ is the $g$th support point of $x_i$, is given by the $g$th element of the +regression estimand $S^{-1}E[w(x_i)y_i]$, where $S=E[w(x_i)w(x_i)']$. Let +$\hat{\mu}=\hat{S}^{-1}\sum_i w(x_i)y_i$, where $\hat{S}=\sum_i w(x_i)w(x_i)'$ +denote the least squares estimator. Then an estimate of +$(\delta(x_1),\dotsc,\delta(x_G))'$ is given by $\hat{\delta}$, the vector with +elements $\hat{\mu}_g-x_g\hat{\theta}$. + +By standard regression results, the asymptotic distribution of $\hat{\theta}$ +and $\hat{\mu}$ is given by +\begin{equation*} +\sqrt{n} +\begin{pmatrix} +\hat{\theta}-\theta\\ +\hat{\mu}-\mu +\end{pmatrix}\overset{d}{\to} +\mathcal{N}\left( +0, +\Omega +\right), +\end{equation*} +where +\begin{equation*} +\Omega=\begin{pmatrix} +Q^{-1}E[(\epsilon_i^2+\delta(x_i)^2)m(x_i)m(x_i)']Q^{-1}& +Q^{-1}E[\epsilon_i^2 m(x_i)w(x_i)']S^{-1}\\ +S^{-1}E[\epsilon_i^2 w(x_i)m(x_i)']Q^{-1}& +S^{-1}E[\epsilon_i^2 w(x_i)w(x_i)']S^{-1}\\ +\end{pmatrix}. +\end{equation*} + +Let $\hat{u}_i$ denote the regression residual from the regression of $y_i$ on +$m(x_i)$, and let $\hat{\epsilon}_i$ denote the regression residuals from the +regression of $y_i$ on $w(x_i)$. Then a consistent estimator of the asymptotic +variance $\Omega$ is given by +\begin{equation*} +\hat{\Omega}=n\sum_i T_i T_i', +\qquad +T_i'=\begin{pmatrix} +\hat{u}_i m(x_i)'\hat{Q}^{-1}& +\hat{\epsilon}_i w(x_i)'\hat{S}^{-1} +\end{pmatrix}. +\end{equation*} + +Note that the upper left block and lower right block correspond simply to the +Eicker-Huber-White estimators of the asymptotic variance of $\hat{\theta}$ and +$\hat{\mu}$. By the delta method, a consistent estimator of the asymptotic +variance of $(\hat\delta,\hat{\theta}_1)$ is given by +\begin{equation*} + \hat{\Sigma}= +\begin{pmatrix} +-X & I\\ +e_1'& 0\\ +\end{pmatrix}\hat{\Omega}\begin{pmatrix} +-X & I\\ +e_1'& 0\\ +\end{pmatrix}', +\end{equation*} +where $X$ is a matrix with $g$th row equal to $x_g'$, and $e_1$ is the first +unit vector. + +Recall that in the notation of @KoRo16, $W=(g^-,g^+,s^-,s^+)$, and $g^{+}$ and +$g^{-}$ are such that $x_{g^{-}}< c_0\leq x_{g^{+}}$, and +$s^{+},s^{-}\in\{-1,1\}$. An upper limit for a right-sided CI for +$\theta_1+b(W)$ is then given by + +\begin{equation*} +\hat{\theta}_{1}+s^{+}\hat\delta(x_{g^+})+ +s^{-}\hat\delta(x_{g^-})+z_{1-\alpha}\hat{V}(W), +\end{equation*} + +where $\hat{V}(W)=a(W)'\hat{\Sigma}a(W)$, and $a(W)\in\mathbb{R}^{G_{h}+1}$ +denotes a vector with the $g_{-}$th element equal to $s^{-}$, +$(G_{h}^{-}+g_{+})$th element equal to $s^{+}$, the last element equal to one, +and the remaining elements equal to zero. The rest of the construction then +follows the description in Section 5.2 in @KoRo16. + +# Optimal inference + +For the second-order Taylor smoothness class, the function `RDHonest`, with +`kernel="optimal"`, computes finite-sample optimal estimators and confidence +intervals, as described in Section 2.2 in @ArKo16optimal. This typically yields +tighter CIs. Comparing the lengths of two-sided CIs with optimally chosen +bandwidths, using Silverman's rule of thumb to estimate the preliminary variance +estimate used to compute optimal bandwidths: + +```{r} +2*RDHonest(voteshare ~ margin, data=lee08, kern="optimal", M=0.1, opt.criterion="FLCI", se.initial="Silverman", se.method="nn")$hl + +2*RDHonest(voteshare ~ margin, data=lee08, kern="triangular", M=0.1, opt.criterion="FLCI", se.initial="Silverman", se.method="nn", sclass="T")$hl + +``` + + +# Specification testing + +The package also implements lower-bound estimates for the smoothness constant +$M$ for the Taylor and Hölder smoothness class, as described in the supplements to @KoRo16 and @ArKo16optimal + +```{r} +## Add variance estimate to the lee data so that the RDSmoothnessBound +## function doesn't have to compute them each time +dl <- RDHonest::RDPrelimVar(dl, se.initial="NN") + +### Only use three point-average for averages of a 100 points closest to cutoff, +### and report results separately for points above and below cutoff +RDSmoothnessBound(dl, s=100, separate=TRUE, multiple=FALSE, sclass="T") + +### Pool estimates based on observations below and above cutoff, and +### use three-point averages over the entire support of the running variable +RDSmoothnessBound(dl, s=100, separate=FALSE, multiple=TRUE, sclass="H") +``` + + + + + + + + + + +# References