# Two closed-form evaluations for the generalized hypergeometric function
$\bm{{}_{4}F_{3}(\frac{1}{16})}$

Arjun K. Rathiea and Mykola A. Shpotb

${}^{a}\\!$Department of Mathematics,
Vedant College of Engineering and Technology (Rajasthan Technical University),
Tulsi, Bundi District, India
E-Mail<EMAIL_ADDRESS>
${}^{b}\\!$Institute for Condensed Matter Physics, 79011 Lviv, Ukraine
E-Mail<EMAIL_ADDRESS>
###### Abstract

The objective of this short note is to provide two closed-form evaluations for
the generalized hypergeometric function ${}_{4}F_{3}$ of the argument
$\frac{1}{16}$ . This is achieved by means of separating a generalized
hypergeometric function ${}_{3}F_{2}$ into even and odd components, together
with the use of two known results for ${}_{3}F_{2}(\pm\frac{1}{4})$ available
in the literature. As an application, we obtain an interesting infinite-sum
representation for the number $\pi^{2}$. Certain connections with the work of
Ramanujan and other authors are discussed, involving other special functions
and binomial sums of different kinds.

Keywords. Generalized hypergeometric functions, summation formulas

2020 Mathematics Subject Classification. Primary 33C20; Secondary 33C05,
33C15.

##  1 Introduction

The generalized hypergeometric function ${}_{p}F_{q}(z)$ with $p$ numerator
and $q$ denominator parameters is defined by the series expansion

${}_{p}F_{q}\Big{(}\begin{array}[]{c}a_{1},a_{2},...,a_{p}\\\
b_{1},b_{2},...,b_{q}\end{array}\Big{|}\,z\Big{)}=\sum_{n\geq
0}\dfrac{(a_{1})_{n}(a_{2})_{n}...(a_{p})_{n}}{(b_{1})_{n}(b_{2})_{n}...(b_{q})_{n}}\;\dfrac{z^{n}}{n!}\,,$
(1)

where $(a)_{n}$ is the Pochhammer symbol [1] (shifted factorial, with
$(1)_{n}=n!$),

$(a)_{n}=\dfrac{\Gamma(a+n)}{\Gamma(a)}=\begin{cases}a(a+1)...(a+n-1)&n\in\mathbb{N};\;a\in\mathbb{C}\setminus\\{0\\}\\\
1&n=0;\,a\in\mathbb{C}\end{cases}\,.$

As usual, $p$ and $q$ are non-negative integers, and the parameters
$a_{j}\in\mathbb{C},j=1,...,p$ and
$b_{k}\in\mathbb{C}\setminus\mathbb{Z}_{0}^{-},k=1,...,q$, where $\mathbb{C}$
denotes the set of complex numbers, $\mathbb{Z}_{0}^{-}$ is the set of
negative integers along with zero.

When $p\leq q$, the power series (1) converges for all finite values of $z$
with $|z|<\infty$. In the case $p=q+1$, it converges for $|z|<1$ and diverges
for $|z|>1$. On the unit circle $|z|=1$, the sufficient condition for
convergence of (1) with $p=q+1$ is $\Re(s)>0$ where $s$ is the parametric
excess

$s:=\sum_{k=1}^{q}b_{k}-\sum_{j=1}^{p}a_{j}\,.$

Also, when $p=q+1$ and $\\{|z|=1,\;z\neq 1\\}$, the series (1) converges with
$-1<\Re(s)\leq 0$. Analytic continuation can be employed to define the
function ${}_{p}F_{q}(z)$ for $z$ values outside the convergence region of the
series (1).111Several interesting instances of the analytic continuation of
the ${}_{3}F_{2}$ function are provided in [2].

For more details on generalized hypergeometric function ${}_{p}F_{q}(z)$ we
refer to standard textbooks [3, 4, 5, 6, 7].

##  2 Even and odd components of ${}_{p}F_{q}(z)$

By decomposing the generalized hypergeometric function ${}_{p}F_{q}(z)$ into
even and odd components and making use of standard identities for the
Pochhammer symbol [1],

$(a)_{2n}=2^{2n}\Big{(}\frac{a}{2}\Big{)}_{n}\Big{(}\frac{a}{2}+\frac{1}{2}\Big{)}_{n}\qquad\mbox{and}\qquad(a)_{2n+1}=a2^{2n}\Big{(}\frac{a}{2}+\frac{1}{2}\Big{)}_{n}\Big{(}\frac{a}{2}+1\Big{)}_{n},$

one can easily obtain the following two general results, recorded, for
example, in the reference book by Prudnikov et. al. [8, 7.2.3.42], viz.

$\displaystyle{}_{q+1}F_{q}\Big{(}\begin{array}[]{c}a_{1},a_{2},...,a_{q+1}\\\
b_{1},b_{2},...,b_{q}\end{array}\Big{|}\,z\Big{)}+{}_{q+1}F_{q}\Big{(}\begin{array}[]{c}a_{1},a_{2},...,a_{q+1}\\\
b_{1},b_{2},...,b_{q}\end{array}\Big{|}-z\Big{)}$ (6)
$\displaystyle=2\,{}_{2q+2}F_{2q+1}\Big{(}\begin{array}[]{c}\frac{a_{1}}{2}\,,\frac{a_{1}}{2}+\frac{1}{2}\,,...,\frac{a_{q+1}}{2},\frac{a_{q+1}}{2}+\frac{1}{2}\\\
\frac{1}{2}\,,\frac{b_{1}}{2}\,,\frac{b_{1}}{2}+\frac{1}{2}\,,...,\frac{b_{q}}{2}\,,\frac{b_{q}}{2}+\frac{1}{2}\end{array}\Big{|}\,z^{2}\Big{)}$
(9)

and

$\displaystyle{}_{q+1}F_{q}\Big{(}\begin{array}[]{c}a_{1},a_{2},...,a_{q+1}\\\
b_{1},b_{2},...,b_{q}\end{array}\Big{|}\,z\Big{)}-{}_{q+1}F_{q}\Big{(}\begin{array}[]{c}a_{1},a_{2},...,a_{q+1}\\\
b_{1},b_{2},...,b_{q}\end{array}\Big{|}-z\Big{)}$ (14)
$\displaystyle=2z\,\frac{a_{1}a_{2}...a_{q+1}}{b_{1}b_{2}...b_{q}}\,{}_{2q+2}F_{2q+1}\Big{(}\begin{array}[]{c}\frac{a_{1}}{2}+\frac{1}{2}\,,\frac{a_{1}}{2}+1,...,\frac{a_{q+1}}{2}+\frac{1}{2}\,,\frac{a_{q+1}}{2}+1\\\
\frac{3}{2}\,,\frac{b_{1}}{2}+\frac{1}{2}\,,\frac{b_{1}}{2}+1,...,\frac{b_{q+1}}{2}+\frac{1}{2}\,,\frac{b_{q+1}}{2}+1\end{array}\Big{|}\,z^{2}\Big{)}.$
(17)

##  3 Two closed-form evaluations for ${}_{4}F_{3}(\frac{1}{16})$

In this section, we employ the above procedure to the two following results:

${}_{3}F_{2}\\!\left(\\!\\!\begin{array}[]{c}\frac{1}{2},\;\frac{1}{2},\;\frac{1}{2}\vspace{1mm}\\\
\frac{3}{2},\;\frac{3}{2}\end{array}\Big{|}\,\\!\\!-\frac{1}{4}\right)=\frac{\pi^{2}}{10}\,,$
(18)

well known from [8, 7.4.6.1], and

${}_{3}F_{2}\\!\left(\\!\\!\begin{array}[]{c}\frac{1}{2},\;\frac{1}{2},\;\frac{1}{2}\vspace{1mm}\\\
\frac{3}{2},\;\frac{3}{2}\end{array}\Big{|}\,\frac{1}{4}\right)=\frac{1}{2\sqrt{3}}\;\psi^{\prime}\Big{(}\frac{1}{3}\Big{)}-\frac{\pi^{2}}{3\sqrt{3}}\,,$
(19)

obtained in [9, (26), (41)] in the course of Feynman-graph calculations. This
leads us to two new closed-form summation formulas for the ${}_{4}F_{3}$
series with argument $\frac{1}{16}$, asserted in the following theorem.

######  Theorem 1.

The following two closed-form evaluations for the function
${}_{4}F_{3}(\frac{1}{16})$ hold true:

${}_{4}F_{3}\\!\left(\\!\\!\begin{array}[]{c}\frac{1}{4},\;\frac{1}{4},\;\frac{1}{4},\;\frac{3}{4}\vspace{1mm}\\\
\frac{1}{2},\;\frac{5}{4},\;\frac{5}{4}\end{array}\Big{|}\,\frac{1}{16}\right)=\frac{1}{2}\left[\pi^{2}\Big{(}\frac{1}{10}-\frac{1}{3\sqrt{3}}\Big{)}+\frac{1}{2\sqrt{3}}\,\psi^{\prime}\Big{(}\frac{1}{3}\Big{)}\right]$
(20)

and

${}_{4}F_{3}\\!\left(\\!\\!\begin{array}[]{c}\frac{3}{4},\;\frac{3}{4},\;\frac{3}{4},\;\frac{5}{4}\vspace{1mm}\\\
\frac{3}{2},\;\frac{7}{4},\;\frac{7}{4}\end{array}\Big{|}\,\frac{1}{16}\right)=36\left[\frac{1}{2\sqrt{3}}\,\psi^{\prime}\Big{(}\frac{1}{3}\Big{)}-\pi^{2}\Big{(}\frac{1}{10}+\frac{1}{3\sqrt{3}}\Big{)}\right],$
(21)

where $\psi(z)$ is the psi-function, the logarithmic derivative of the Euler
gamma function $\Gamma(z)$.

###### Proof.

The derivation of the results (20) and (21) asserted by the Theorem 1 are
straightforward and based upon identities (6) and (14). We set therein $q=2$
and substitute $a_{1}=a_{2}=a_{3}=\frac{1}{2}$ and $b_{1}=b_{2}=\frac{3}{2}$,
along with $z=\frac{1}{4}$. After some simplifications in resulting
hypergeometric ${}_{6}F_{5}$ functions, we make use of explicit expressions
from (18) and (19) and arrive at closed-form evaluations (20) and (21). This
completes the proof of the Theorem 1. ∎

Application. The following new hypergeometric representation of $\pi^{2}$ can
be easily obtained from (20) and (21) by subtracting the appropriately
weighted equations, so that the terms containing the derivative of the $\psi$
function disappear:

$\frac{\pi^{2}}{10}={}_{4}F_{3}\\!\left(\\!\\!\begin{array}[]{c}\frac{1}{4},\;\frac{1}{4},\;\frac{1}{4},\;\frac{3}{4}\vspace{1mm}\\\
\frac{1}{2},\;\frac{5}{4},\;\frac{5}{4}\end{array}\Big{|}\,\frac{1}{16}\right)-\frac{1}{72}\,{}_{4}F_{3}\\!\left(\\!\\!\begin{array}[]{c}\frac{3}{4},\;\frac{3}{4},\;\frac{3}{4},\;\frac{5}{4}\vspace{1mm}\\\
\frac{3}{2},\;\frac{7}{4},\;\frac{7}{4}\end{array}\Big{|}\,\frac{1}{16}\right).$
(22)

Actually, the last formula directly reproduces the original result of (18) if
we recognize here the difference of even and odd parts of
${}_{3}F_{2}(\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{1}{4})$.
However, we can alternatively consider the coefficients of ${}_{4}F_{3}$
functions in (22) as

$c_{1}(n)=\frac{\left(\frac{1}{4}\right)_{n}^{3}\left(\frac{3}{4}\right)_{n}}{\left(\frac{5}{4}\right)_{n}^{2}\left(\frac{1}{2}\right)_{n}n!}=\frac{2^{-2n}\Gamma\left(2n+\frac{1}{2}\right)}{(4n+1)^{2}\Gamma\left(n+\frac{1}{2}\right)n!}$
(23)

and

$c_{2}(n)=\frac{\left(\frac{3}{4}\right)_{n}^{3}\left(\frac{5}{4}\right)_{n}}{\left(\frac{7}{4}\right)_{n}^{2}\left(\frac{3}{2}\right)_{n}n!}=9\,\frac{2^{-2n}\Gamma\left(2n+\frac{3}{2}\right)}{(4n+3)^{2}\Gamma\left(n+\frac{3}{2}\right)n!},$
(24)

and produce thereof the coefficients of the linear combination in (22) via

$c(n)=c_{1}(n)-\frac{1}{72}\,c_{2}(n)=\frac{2^{-2n}(4n(4n(12n+29)+81)+71)\Gamma\left(2n+\frac{1}{2}\right)}{16n!(4n+1)^{2}(4n+3)^{2}\Gamma\left(n+\frac{3}{2}\right)}\,.$
(25)

Now, summing up the whole combination from the right-hand side of (22) with
coefficients $c(n)$, we obtain the following representation of this identity
in the form of the infinite single sum:

$\frac{\pi^{2}}{10}=\frac{1}{16}\sum_{n=0}^{\infty}\frac{4n(4n(12n+29)+81)+71}{(4n+1)^{2}(4n+3)^{2}n!}\,\frac{\Gamma\left(2n+\frac{1}{2}\right)}{\Gamma\left(n+\frac{3}{2}\right)}\,\Big{(}\frac{1}{64}\Big{)}^{n}.$
(26)

We note that the convergence of the sum in (26) is essentially enhanced
compared to that in (18): As $n\to\infty$, the terms in the last sum decay
$\propto 4^{-n}$ times faster as in the original ${}_{3}F_{2}$ function in
(18).

The results (20) through (26) have been verified using the MATHEMATICA
software [10].

Remark 1. The method of splitting the generalized hypergeometric series (1)
into its even and odd parts to generate new evaluations of higher-order
hypergeometric functions is not new. It goes back at least to Krupnikov and
Kölbig [11, p. 84] and has been used recently in a number of publications,
among them [12, 13, 14].

##  4 Relation to Ramanujan’s notebooks, other functions, binomial and
harmonic sums

Remark 2. Due to the relation [15, (15)], [16, p. 329], [17, (1.10)]

$\mbox{Cl}_{2}\Big{(}\frac{\pi}{3}\Big{)}=\frac{1}{2\sqrt{3}}\left[\psi^{\prime}\Big{(}\frac{1}{3}\Big{)}-\frac{2}{3}\,\pi^{2}\right],$
(27)

the results (19), (20) and (21) can be alternatively expressed in terms of the
maximum value $\mbox{Cl}_{2}(\frac{\pi}{3})$ of the Clausen function
$\mbox{Cl}_{2}(\theta)$ [18, Ch. 4], [17]

$\mbox{Cl}_{2}(\theta)=-\int_{0}^{\theta}d\theta\ln\Big{|}2\sin\frac{\theta}{2}\Big{|}=\sum_{n\geq
1}\frac{\sin n\theta}{n^{2}}=\mbox{Im}\mbox{Li}_{2}(\mbox{e}^{i\theta})\,.$
(28)

This function, along with its natural extensions, is very important in the
theory of dilogarithm, $\mbox{Li}_{2}(z)$, and polylogarithm functions [18],
and in computations of the Feynman integrals [19, 20].

Remark 3. A few results, equivalent to the one in (19), have appeared in the
literature before [9, (26), (41)], though in quite different forms.

The first one goes back to Ramanujan, and appears in Example $(iii)$ of Entry
16 in [21, p. 40] and [22, p. 264], as an evaluation of the binomial sum

$\sum_{k\geq 0}\Big{(}{2k\atop
k}\Big{)}\frac{1}{2^{4k+1}(2k+1)^{2}}=\frac{3\sqrt{3}}{4}\sum_{k\geq
0}\frac{1}{(3k+1)^{2}}-\frac{\pi^{2}}{6\sqrt{3}}$ (29)

where

$2^{-2k}\Big{(}{2k\atop
k}\Big{)}=\frac{1}{k!}\Big{(}\frac{1}{2}\Big{)}_{k}=2^{-2k}\,\frac{(2k)!}{(k!)^{2}}=\frac{(2k-1)!!}{(2k)!!}:=\mu_{k}$
(30)

is the normalized binomial mid-coefficient (see e. g. [23, (2.2)] and related
references). Indeed, using for the function
${}_{3}F_{2}(\frac{1}{2},\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{3}{2};\frac{1}{4})$
from (19) its series definition (1) along with the first equality from (30)
and the simple relation $(\frac{1}{2})_{k}/(\frac{3}{2})_{k}=(2k+1)^{-1}$
between Pochhammer symbols, we see that it matches the binomial sum on the
left-hand side of (29) up to an extra factor of $\frac{1}{2}$ in (29). To
match the right-hand sides of (19) and (29), we have to take into account that
[7, 5.15.1], [24, 5.1.7.16]

$\sum_{k\geq
0}\frac{1}{(3k+1)^{2}}=\frac{1}{9}\,\psi^{\prime}\Big{(}\frac{1}{3}\Big{)}.$

The same result has been reproduced by Zucker [25, (2.13)] in the form

$\sum_{k\geq 0}\Big{(}{2k\atop
k}\Big{)}\frac{1}{4^{2k}(2k+1)^{2}}=\frac{3\sqrt{3}}{4}\,L_{-3}(2).$ (31)

In notations of Zucker, $\frac{3\sqrt{3}}{4}\,L_{-3}(2)$ means the imaginary
part of dilogarithm $\mbox{Li}_{2}(\mbox{e}^{i\pi/3})$ via [25, (1.11)] and
[25, (1.8)]. Hence we see, by noticing (28) and (27), that his equation (31)
is tantamount just to (19) expressed as

${}_{3}F_{2}\\!\left(\\!\\!\begin{array}[]{c}\frac{1}{2},\;\frac{1}{2},\;\frac{1}{2}\vspace{1mm}\\\
\frac{3}{2},\;\frac{3}{2}\end{array}\Big{|}\,\frac{1}{4}\right)=\mbox{Cl}_{2}\Big{(}\frac{\pi}{3}\Big{)}.$
($5^{*}$)

As noticed after (19), this relation appeared in [9] as a result of a
calculation of a Feynman integral. A slightly different treatment of the same
integral in [26] yielded an alternative relation,

$\frac{3\sqrt{\pi}}{8}\sum_{n=0}^{\infty}\frac{\Gamma(n+1)\psi(n+1)}{\Gamma(n+\frac{3}{2})}\,4^{-n}+\frac{\pi\sqrt{3}}{6}\left(\gamma_{E}+\ln
3\right)=\frac{2}{\sqrt{3}}\,\mbox{Cl}_{2}\Big{(}\frac{\pi}{3}\Big{)}=\frac{1}{3}\,\psi^{\prime}\Big{(}\frac{1}{3}\Big{)}-\frac{2\pi^{2}}{9}\,,$
(32)

which is obtained by combining (A13) and (A.15) of [26]222The reference to
[26] given in [9] is incorrect: Ref. [13] in [9] has to be replaced by [26]..
In (32), $\gamma_{E}$ is the Euler constant, given by the limiting behavior of
harmonic numbers333For more related information, see e. g. [23, p. 147].

$H_{n}:=\sum_{k=1}^{n}\frac{1}{k}=\gamma_{E}+\psi(n+1)\qquad\mbox{via}\qquad\gamma_{E}=\lim_{n\to\infty}\left(H_{n}-\ln
n\right).$ (33)

Using the first equation in (33) and [24, Annex 1.6]

$\Gamma\Big{(}n+\frac{3}{2}\Big{)}=\frac{\sqrt{\pi}}{2}\,2^{-n}\,\frac{(2n+1)!}{n!},$

we can rewrite (32) in a more instructive form:

$\sum_{n=0}^{\infty}\frac{H_{n}}{\left({2n\atop
n}\right)(2n+1)}=\frac{8}{3\sqrt{3}}\,\mbox{Cl}_{2}\Big{(}\frac{\pi}{3}\Big{)}-\frac{2\pi}{3\sqrt{3}}\,\ln
3.$ (34)

Now, combining (29), (31), ($5^{*}$), and (34), we obtain a nice chain of
equalities,

${}_{3}F_{2}\\!\left(\\!\\!\begin{array}[]{c}\frac{1}{2},\;\frac{1}{2},\;\frac{1}{2}\vspace{1mm}\\\
\frac{3}{2},\;\frac{3}{2}\end{array}\Big{|}\,\frac{1}{4}\right)=\sum_{k=0}^{\infty}\Big{(}{2k\atop
k}\Big{)}\frac{4^{-2k}}{(2k+1)^{2}}=\frac{3\sqrt{3}}{8}\,\sum_{k=0}^{\infty}\frac{H_{k}}{\left({2k\atop
k}\right)(2k+1)}+\frac{\pi}{4}\,\ln
3=\mbox{Cl}_{2}\Big{(}\frac{\pi}{3}\Big{)},$ (35)

relating the generalized hypergeometric function from (19) and ($5^{*}$) with
corresponding binomial and inverse binomial harmonic sums.

It seems that identifications of results (19) and ($5^{*}$) in terms of a
generalized hypergeometric function ${}_{3}F_{2}$ of argument $\frac{1}{4}$
appear only in equations (26) and (41) of reference [9].

Remark 4. Finally, we mention a similar connection of the ${}_{3}F_{2}$
function of argument $-\frac{1}{4}$ from (18) with the sum $f(x)$ considered
by Ramanujan in Example $(ii)$ of Entry 8 (see [21, p. 24], [22, p. 250]), and
that from [25, (3.9)].

Similarly as before, we easily establish the equivalence of (18) with [25,
(3.9)], namely

$\sum_{k\geq 0}\Big{(}{2k\atop
k}\Big{)}\frac{(-1)^{k}}{2^{4k}(2k+1)^{2}}=\frac{\pi^{2}}{10}.$ (36)

Moreover, Zucker [25, p. 101] notices the equivalence of his equation (3.9)
with the special result $f(1/\sqrt{5})=\pi^{2}/20$ for the Ramanujan’s sum

$f(x)=\sum_{k=1}^{\infty}\frac{h_{k}\,x^{2k-1}}{2k-1},\qquad\mbox{where}\qquad
h_{k}=\sum_{j=1}^{k}\frac{1}{2j-1}.$ (37)

Hence, combining (18), (36), and (37), we may write

${}_{3}F_{2}\\!\left(\\!\\!\begin{array}[]{c}\frac{1}{2},\;\frac{1}{2},\;\frac{1}{2}\vspace{1mm}\\\
\frac{3}{2},\;\frac{3}{2}\end{array}\Big{|}\,-\frac{1}{4}\right)=\sum_{k=0}^{\infty}\Big{(}{2k\atop
k}\Big{)}\frac{(-16)^{-k}}{(2k+1)^{2}}=2\sum_{k=1}^{\infty}\frac{h_{k}\,5^{-k+\frac{1}{2}}}{2k-1}=\frac{\pi^{2}}{10}.$
(38)

The relations of the odd harmonic number $h_{k}$ to harmonic numbers $H$ and
digamma function $\psi$ are given by [23, (2.3)], viz.

$h_{k}=H_{2k}-\frac{1}{2}\,H_{k}=\frac{1}{2}\left[\psi\Big{(}k+\frac{1}{2}\Big{)}-\psi\Big{(}\frac{1}{2}\Big{)}\right].$

We are quite sure that the relations (35) and (38) deserve further analytical
investigation, in particular towards generalizations to other arguments of
${}_{3}F_{2}$ functions, apart from $\pm\frac{1}{4}$.

##  5 Concluding remarks

In this note, we have provided two closed-form evaluations for the generalized
hypergeometric function ${}_{4}F_{3}$ of the argument $\frac{1}{16}$ . A
combination of these results yields an interesting representation of the
transcendental number $\pi^{2}$ in terms of an infinite sum with an enhanced
convergence rate.

There are many theoretical and practical applications of ${}_{p}F_{q}$
functions in different fields of mathematics [27, 28, 29, 30, 31], statistics
and combinatorics [29, 32, 13], theoretical physics [33, 2, 34] and
engineering [35, 36].444It is quite impossible to give an exhaustive list of
relevant references here; our choice is motivated by scientific interests of
the authors. Hopefully, our results could be potentially useful in some of
these areas. To the best of our knowledge, the established summation formulas
(20) and (21) did not appear in the literature before; we believe, they
represent a definite contribution to the theory of generalized hypergeometric
functions.

Moreover, we have established certain connections of functions under
consideration with the work of Ramanujan and other authors, with other special
functions, and with binomial sums of different kinds, some of them including
harmonic numbers $H_{k}$ and $h_{k}$.

A large number of results closely related to that of (20), (21) and (26) are
under investigation and will be subject of a subsequent publication.

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# A Constructive GAN-based Approach to Exact Estimate Treatment Effect without
Matching

Boyang You, Kerry Papps Ph.D. Student, Department of Economics, University of
Bath. Email<EMAIL_ADDRESS>Lecturer, Department of Economics,
University of Bath. Centre for Analysis of Social Policy (CASP) Labour,
Education and Health Economics. Email<EMAIL_ADDRESS>
###### Abstract

Matching has become the mainstream in counterfactual inference, with which
selection bias between sample groups can be significantly eliminated. However
in practice, when estimating average treatment effect on the treated (ATT) via
matching, no matter which method, the trade-off between estimation accuracy
and information loss constantly exist. Attempting to completely replace the
matching process, this paper proposes the GAN-ATT estimator that integrates
generative adversarial network (GAN) into counterfactual inference framework.
Through GAN machine learning, the probability density functions (PDFs) of
samples in both treatment group and control group can be approximated. By
differentiating conditional PDFs of the two groups with identical input
condition, the conditional average treatment effect (CATE) can be estimated,
and the ensemble average of corresponding CATEs over all treatment group
samples is the estimate of ATT. Utilizing GAN-based infinite sample
augmentations, problems in the case of insufficient samples or lack of common
support domains can be easily solved. Theoretically, when GAN could perfectly
learn the PDFs, our estimators can provide exact estimate of ATT.

To check the performance of the GAN-ATT estimator, three sets of data are used
for ATT estimations: 1) A linear toy data set with 1-dimensional input and
constant treatment effect is tested. The GAN-ATT estimate is 0.51% away from
the preset ground truth. 2) A non-linear toy data set with 2-dimensional input
and covariate-dependent treatment effect is tested. The GAN-ATT estimate is
1.70% away from the preset ground truth, which is better than traditional
matching approaches including propensity score matching (PSM) and coarsened
exact matching (CEM). 3) A real firm-level data set with high-dimensional
input is tested and the applicability towards real data sets is evaluated by
comparing the other two matching methods. Through the evidences obtained from
the three tests, we believe that the GAN-ATT estimator has significant
advantages over traditional matching methods in estimating ATT.

##### Key words:

Counterfactual inference, Generative adversarial networks, Average treatment
effect on the treated, Machine learning.

##  1 Introduction

###  1.1 Aims and Motivations

Counterfactual inference is a causal inference research method that constructs
hypotheses of no treatment happened in the past, making the effect that
attribute to the treatment available to estimate. This method is extensively
used in the evaluation of medical treatment, policy performance, and corporate
decisions etc. Since counterfactuals did not really happen, one of the classic
approaches for counterfactual inference is to search for non-treated samples
that are as close as possible to the real samples that are treated, which can
be considered as a matching process. In early studies, matching are usually
proceed based on human experiences, in which selection bias may exist
inevitably. In order to better eliminate selection bias, some advanced
matching methods are further proposed, such as propensity score matching
(PSM), coarsened exact matching (CEM).

In our previous study (You, Papps, 2022), both the two advanced matching
methods are used to estimate the average treatment effect on the treated (ATT)
of a firm-level data set. Evidence shows that the selection bias of covariates
are significantly reduced. However, we also found that in high-dimensional
sample matching processes, due to the existence of dimensionality reduction,
sample dropping, or the lack of common support domains, the estimation of
average treatment effect on the treated (ATT) may be inaccurate.

To find a solution, this paper proposes a constructive ATT estimator that
integrates classic causal inference framework with a machine learning
approach: generative adversarial networks (GAN), which can approximate the
probability density function (PDF) of sample sets. By giving identical input
condition into the PDFs of both treatment group and control group, the target
conditional average treatment effect (CATE) can be estimated with no need of
matching, and the ensemble average of the CATEs corresponding to all treated
samples is the estimate of ATT. Using GAN-based infinite sample augmentations,
problems caused by traditional matching can be easily solved. Theoretically,
when GAN could perfectly learn the PDFs, our estimators can provide exact
estimate of ATT.

Furthermore, in order to ensure the effectiveness of this approach, we
constructed two sets of toy data with ATT ground truth to check the
performance of the estimator. Finally, we apply the estimator to re-estimate
the ATT of the previously-used firm-level data set.

###  1.2 Research Background and Literature Review

Generative Adversarial Networks (GAN) is systematically proposed by Goodfellow
(2014), which is a latest deep learning approach involving dual neural network
games. In early stages, GAN is mostly used for image processing. Through
trained GAN models, images with certain given features can be generated
arbitrarily. For example, with a few strokes on the screen, images of trees,
rivers, or beaches can be randomly generated. With the popularity of GAN, this
deep learning approach has gradually extended to other fields.

One of the innovative uses of GAN is for privacy protection. Huang et al.
(2018) systematically proposed a generative network that can be used for
privacy protection. Liu et al. (2018), Bae et al. (2019), Ponte (2020) use GAN
for privacy control in the data of geolocation, medical information and
marketing. Cai et al. (2021) systematically summarize all types of generative
network used for privacy control in different perspectives.

While using GAN for privacy protection, scholars find that the function of GAN
that generate identically distributed data sets can be further used for
optimizing casual inference issues. Joint with counterfactual framework, Yoon
& Van (2018) proposed GANITE to infer individualized treatment effect (ITE).
With this method, Chu et al. (2019), Ge et al. (2020) estimate ITE based on
medical data sets. Ghosh et al. (2021) use GAN to optimize the performance of
propensity score matching.

However, current machine learning based counterfactual studies still focus on
the idea of matching. The popular idea of Yoon & Van (2018) can be extended to
the estimation of ATT, yet in their work GAN learning is used for inferring
the counterfactual outcome of individual factual sample, which did not in fact
jump out of the matching idea. In order to supplement the literature in this
area, this paper proposes a constructive ATT estimator based on GAN deep
learning approach: GAN-ATT, which is able to theoretically exact estimate ATT
without matching. In details, GAN-ATT estimator includes four steps in
estimation:

1) The two joint PDFs of samples in treatment group and control group are both
learned from two GAN training;

2) Two synthesized data sets with ideally the same PDFs of real sample sets
are generated based on the two trained models;

3) The conditional average treatment effect (CATE) is estimated by
differentiating the two conditional PDFs with identical input condition;

4) ATT is estimated by the ensemble average of CATE over all treatment group
samples.

There are several advantages of this estimator:

First, problems caused by insufficient samples or lack of common support
domains can be completely solved by performing GAN-based sample augmentations,
which can generate infinite samples of the two groups for PDF approximating.
Second, the estimation of ATT can be theoretically proved 100% accurate in the
ideal case that GAN generates samples with exact the same PDFs as the real
data sets. Third, no matching performed in the estimator for ATT estimation.
So the problem caused by matching, such as the information loss caused by
dimensionality reduction in PSM (Marco & Sabine, 2008), or the sample loss
caused by high-dimensional common support domain shortage in CEM (Iacus et
al., 2012) can be bypassed. Fourth, incorporating machine learning approaches,
the estimation results are more objective to the fact, the influence of human
factors are reduced. Last, the privacy of sample providers can be protected if
required.

In addition to proposing the estimator, we further construct two sets of toy
data sets referring to the idea of Yoon & Van (2018), together with a real
firm-level data set we previously used (You, Papps, 2022) to check the
effectiveness of the GAN-ATT estimator. Evidence proves that the estimator is
feasible and accurate.

##  2 Classic Counterfactual Inference Framework

###  2.1 Assumptions to Meet

To fit our estimator to the casual inference framework, according to
(Rosenbaum and Rubin, 1983; Imbens and Rubin, 2015), the data set used for
estimation should meet two assumptions: unconfoundedness (or conditional
independence assumption) and overlap (or common support conditions).

Assumption

1\. Unconfoundedness: $Y(d=0,1)\amalg d\mid X$

In the assumption, $d$ is the treatment dummy, $Y(d=0)$ is the outcome
variable of the control group, $Y(d=1)$ is the outcome variable of the
treatment group, $X$ is covariate and $\amalg$ denotes independence. This
assumption can be understand as the difference between $Y(d=1)$ and $Y(d=0)$
under same $X$ should be attribute to the treatment (Caliendo & Kopeinig,
2008).

Assumption

2\. Overlap: $0<P(d=1\mid X)<1$

This assumption indicates that for samples under any $X$, the probability of
being treated or not being treated should not be equal to zero (Heckman et
al., 1999). Otherwise, there will be a lack of common support domains for CATE
calculations.

Note that the unconfoundedness assumption is relatively strong in practice.
When focusing on estimating ATT only, the two assumptions can be weaken to
’unconfoundedness for controls’ and ’weak overlap’. See Caliendo & Kopeinig
(2008) for details.

###  2.2 Estimation of Average Treatment Effect on the Treated

Let $\left\\{y_{d},x_{d}\right\\}$ be the two sets of sample data sets that
have been observed, where $y_{d}$ are outcome variables and $x_{d}$ are
covariates; $d$ is the treatment dummy, $d=1$ represents treatment group and
$d=0$ represents control group. Let the PDFs of the covariates $x_{0}$ and
$x_{1}$ be $f_{d}\left(x\right)$, that is:

$x_{d}\sim f_{d}\left(x\right),\ d=0,1$ (1)

Assumes that with given covariate $x_{d}$, the potential outcome variable
$y_{d}$ conforms to the conditional probability density functions as follows:

$y_{d}\mid x_{d}\sim g_{d}\left(y\mid x\right),\>d=0,1,$ (2)

where $y\mid x$ means the value of the potential outcome variable $y$ with
given covariate $x$.

Note that when $f_{0}\left(x\right)$ and $f_{1}\left(x\right)$ share the same
probability distribution, or share a common trend, the classic difference-in-
difference (DID) method can be used for ATT estimation. However in practice,
sample data sets are usually collected from observational experiments
directly. The selection bias exists in covariates may lead to significant
distribution differences between $f_{0}\left(x\right)$ and
$f_{1}\left(x\right)$. In order to eliminate such selection bias, matching
methods are usually required before ATT estimations.

Next, according to Heckman et al. (1997), ATT can be estimated as follows:

$ATT=E_{x_{1}\sim f_{1}(x)}\left\\{E_{y_{1}\sim g_{1}\left(y\mid
x\right)}\left\\{y_{1}^{f}\mid x_{1}\right\\}-E_{y_{1}\sim g_{1}\left(y\mid
x\right)}\left\\{y_{1}^{cf}\mid x_{1}\right\\}\right\\}\text{,}$ (3)

where $y_{1}^{f}$ is the outcome variable of samples in the treatment group
that are in real treated based on the fact, while $y_{1}^{cf}$ is the
counterfactual outcome variable of samples in the treatment group that is
assumed not being treated.

Since the $y_{1}^{cf}$ cannot be observed, to estimate the treatment effect,
it is necessary to find samples with ideally the same features in the control
group. That is to say, assuming no unobserved confounding covariates exist,
$y_{0}^{f}\mid x$ and $y_{1}^{cf}\mid x$ should have very similar statistical
characteristics under same covairate $x$ (Heckman et al., 1997). This
indicates that in the control group, $y_{0}^{f}\mid x_{1}$ with given
covariate $x_{1}$ can be used as an estimate of $y_{1}^{cf}\mid x_{1}$. In
this case Equation (3) can be rewritten as:

$ATT\cong E_{x_{1}\sim f_{1}(x)}\left\\{E_{y_{1}\sim g_{1}\left(y\mid
x\right)}\left\\{y_{1}^{f}\mid x_{1}\right\\}-E_{y_{0}\sim g_{0}\left(y\mid
x\right)}\left\\{y_{0}^{f}\mid x_{1}\right\\}\right\\}$ (4)

Note that in Equation (4), $E_{y_{1}\sim g_{1}\left(y\mid
x\right)}\left\\{y_{1}^{f}\mid x_{1}\right\\}-E_{y_{0}\sim g_{0}\left(y\mid
x\right)}\left\\{y_{0}^{f}\mid x_{1}\right\\}$ can be understand as the
conditional average treatment effect (CATE) under condition $x_{1}$. Besides,
the presumption of using Equation (4) is that the range of $x_{1}$ values is a
subset of the range of $x_{0}$ values, otherwise it cannot be guaranteed that
any condition $x_{1}$ can be found in the control group for $y_{0}^{f}\mid
x_{1}$, which means Asumption 2: Overlap cannot hold.

Classic matching-based ATT estimator such as PSM-DID and CEM-DID are all based
on the idea of estimating $y_{1}^{cf}\mid x_{1}$ via $y_{0}^{f}\mid x_{1}$.
However, no matter which matching method is used, samples without common
support domains and samples without matched pairs will be dropped, leading to
an inevitable trade-off between estimation accuracy and information loss: the
less samples are dropped, the greater the selection bias; the more samples are
dropped, the more data information is lost. So, it is impossible for matching
approaches to accurately estimate $y_{1}^{cf}\mid x_{1}$ without information
loss. To bypass this unsolvable problem, in the following section we propose a
GAN-ATT estimator, which is able to estimate CATE in Equation (4) via GAN
machine learning so that no matching process is required. For notation
simplicity, in the following sections $y_{0}^{f}$ and $y_{1}^{f}$ are
simplified to $y_{0}$ and $y_{1}$.

##  3 GAN-Counterfactual Inference Framework

###  3.1 Principles of GAN

Generative Adversarial Network (GAN) is first systematically proposed by
Goodfellow et al. (2014). Generally, the training procedure of GAN can be
summarized as the mutual iteration of two neural networks: generator
$\mathscr{G}$ and discriminator $\mathscr{D}$. First, the generator
$\mathscr{G}$ generates a set of data based on a white noise input. Next, with
the comparison of the real data set, the discriminator $\mathscr{D}$
determines whether the synthesized data is real or fake (generated). As long
as the value returned by discriminator $\mathscr{D}$ is fake, the generator
$\mathscr{G}$ optimizes the synthesized data through a cost function, making
it closer to the real. When the discriminator $\mathscr{G}$ cannot distinguish
between real and fake, it can be considered that the training has converged.
Based on this idea, Mirza & Osindero (2014) extend GAN into a conditional
version, making inputs with conditions available to be trained. Referring to
their work, conditional GAN will be used in this paper for CATE estimations.
The specific algorithm is shown in the next section.

###  3.2 Algorithm of GAN Training

Figure 1(a) illustrates the detailed GAN training procedure. The two GAN
training structures shown in Figure 1(a) are identical. Each training is
consists of two adversarial neural networks: generator $\mathscr{G}_{d}$ and
discriminator $\mathscr{D}_{d},\ d=0,1$, where $d=1$ represents the GAN
training for the treatment group and $d=0$ represents GAN training for the
control group.
$\left\\{y_{d}\left(i\right),x_{d}\left(i\right)\right\\}\,,i=1,\cdots,N_{d}$
is the real data set that is collected from sample pool.
$\left\\{\tilde{y}_{d}(i),\tilde{x}_{d}(i)\right\\}\,,i=1,\cdots,\tilde{N}_{d}$
is the synthesized data set that is generated from generator
$\mathscr{G}_{d}$.

According to Goodfellow et al. (2014) and Mirza & Osindero (2014), generator
$\mathscr{G}_{d}$ is a multi-layer feedforward neural network with adjustable
parameters. The input of $\mathscr{G}_{d}$ is an independent and identically
distributed Gaussian variable $z_{d}(i)$ together with a treatment dummy $d$.
The output node of $\mathscr{G}_{d}$ is linear. The output is a synthesized
data set $\left\\{\tilde{y}_{d}(i),\tilde{x}_{d}(i)\right\\}$ that will be
next judged by the discriminator $\mathscr{D}_{d}$.

Discriminator $\mathscr{D}_{d}$ is also a multi-layer feedforward neural
network. The input of $\mathscr{D}_{d}$ is
$\left\\{y_{d}(i),x_{d}(i)\right\\}$ and
$\left\\{\tilde{y}_{d}(i),\tilde{x}_{d}(i)\right\\}$. The output node of
$\mathscr{D}_{d}$ is a sigmoidal function. The output value is $D_{d}(i)$,
which gives a high score from input that is recognized from a real data set
$\left\\{y_{d}(i),x_{d}(i)\right\\}$, or gives a low score from input that is
recognized from a fake (synthesized) data set
$\left\\{\tilde{y}_{d}(i),\tilde{x}_{d}(i)\right\\}$.

Next, with the output from discriminator $\mathscr{D}_{d}$, generator
$\mathscr{G}_{d}$ reiterate the above procedure through a min-max game:

$\min_{\mathscr{D}_{d}}\max_{\mathscr{G}_{d}}V(\text{$\mathscr{G}_{d}$},\text{$\mathscr{D}_{d}$})=\sum_{\left\\{y_{d}(i),x_{d}(i)\right\\}\sim
g_{d}^{real}\left(y,x\right)}^{i\in\left[1,N_{d}\right]}\log\left[D_{d}(y_{d}(i),x_{d}(i))\right]+\sum_{z_{d}(i)\sim\mathscr{N}\left(z_{d}\right)}^{i\in\left[1,N_{d}\right]}\log\left[1-D_{d}(\tilde{y}_{d}(i),\tilde{x}_{d}(i))\right],$
(5)

where $\left\\{y_{d}(i),x_{d}(i)\right\\}\sim g_{d}^{real}\left(y,x\right)$
represents that $\left\\{y_{d}(i),x_{d}(i)\right\\}$ is collected from the
real data set with the joint PDF of $g_{d}^{real}\left(y,x\right)$;
$z_{d}(i)\sim\text{$\mathscr{N}$}\left(z_{d}\right)$ represents that
$z_{d}(i)$ is collected from the sampling of a Gaussian distribution. It can
be proved that when the real data set is sufficiently large, the output of the
generator $\mathscr{G}$
$\left\\{\tilde{y}_{d}(i),\tilde{x}_{d}(i)\right\\}\sim
g_{d}^{syn}\left(y,x\right)$ will approximate to the real data PDF
$g_{d}^{real}\left(y,x\right)$ in terms of the Kullback-Leibler (KL)
divergence (Goodfellow et al., 2014).

To sum up, GAN requires the input of $z_{d}(i)$ and $d$, and the output
generated is an arbitrarily large synthesized data set
$\left\\{\tilde{y}_{d}(i),\tilde{x}_{d}(i)\right\\}$ that approximately obeys
the PDFs of the real data set $g_{d}^{real}\left(y,x\right)$. The content of
Figure 1(b) will be explained in the next section.

###  3.3 Combining GAN with Counterfactual Inference

In order to integrate GAN into counterfactual inference framework, two
separate GAN training procedures for the two data sets
$\left\\{y_{d}\left(i\right),x_{d}\left(i\right)\right\\},\,d=0,1$ are
required to perform. After GAN training converges, the synthesized data sets
$\left\\{\tilde{y}_{d}(i),\tilde{x}_{d}(i)\right\\}\sim
g_{d}^{real}\left(y,x\right)$ can be used for CATE estimations. According to
Figure 1, the specific estimating procedure can divided into 4 steps:

#### 1) Joint probability learning

As it is shown in Figure 1(a), two conditional GANs are first used to learn
the joint PDFs of the treatment group $g_{1}^{syn}\left(y,x\right)\rightarrow
g_{1}^{real}\left(y,x\right)$ and the control group
$g_{0}^{syn}\left(y,x\right)\rightarrow g_{0}^{real}\left(y,x\right)$ based on
their corresponding real data sets $\left\\{y_{1}(i),x_{1}(i)\right\\}$ and
$\left\\{y_{0}(i),x_{0}(i)\right\\}$. Through the training process mentioned
in section 3.2, when reaching convergence, two trained models which can
reflect the corresponding PDFs of treatment group and control group can be
saved for further calculations.

#### 2) Synthesized data generating

As it is shown in Figure 1(b), based on the two obtained trained models.
$\text{$\mathscr{G}$}{}_{1}$ and $\text{$\mathscr{G}$}{}_{0}$ can generate two
synthesized data sets $\left\\{\tilde{y}_{d}(i),\tilde{x}_{d}(i)\right\\}\sim
g_{d}^{syn}\left(y,x\right)\rightarrow g_{d}^{real}\left(y,x\right),\>d=0,1$ .
With joint PDFs approximated to the real, the synthesized data sets follow the
conditional average $E\left\\{\tilde{y}_{d}\mid x_{d}\right\\}\rightarrow
E\left\\{y_{d}\mid x_{d}\right\\}$. Note that in this step, the samples size
generated $\tilde{N}_{d}$ can be arbitrarily large. The larger the sample size
generated, the less common support domains between
$\left\\{\tilde{y}_{1}(i),\tilde{x}_{1}(i)\right\\}$ and
$\left\\{\tilde{y}_{0}(i),\tilde{x}_{0}(i)\right\\}$ are missing.

#### 3) CATE approximating via histogram

After data sets are generated by $\text{$\mathscr{G}$}{}_{1}$ and
$\text{$\mathscr{G}$}{}_{0}$, histogram method is used on the synthesized
sample sets to approximate the conditional average $E\left\\{y_{d}\mid
x_{d}\right\\}$. Following the basic idea of Pearson (1894) and Iacus et al.
(2012), the conditional probability of each covariate will be fitted by a
combination of multiple bars so that a multidimensional distribution can be
approximated.

In details, define the dimension of $x_{d}$ in the data set be $q$. Coarsen
$x_{d}$ into multiple small $q$-dimensional equal-sized cubes. Let
$\varDelta\mid x_{d}$ denote such a cube that centered at $x_{d}$. All samples
of $\tilde{y}_{d}\left(i\right)$ with their covariates
$x_{d}\left(i\right)\in\varDelta\mid x_{d}$ will be averaged to obtain the
conditional average $\bar{y}_{d}|x_{d}\rightarrow E\left\\{y_{d}\mid
x_{d}\right\\}$. When the number of cubes is sufficient large, the target
distribution can be approximated. Note that the smaller the cube size
$\varDelta$ is chosen, the more accurate the approximation of
$E\left\\{y_{d}\mid x_{d}\right\\}$ can achieve. Yet with the decrease of cube
size $\varDelta$, the sample size of the synthesized data set must be
increased to ensure the existence of the common support domains.

It can be found that the $E\left\\{y_{d}\mid x_{d}\right\\}$ estimation
obtained is in fact a function of $x_{d}$. So, define a function as $\varDelta
y(x)=E\left\\{y_{1}\mid x\right\\}-E\left\\{y_{0}\mid x\right\\}$, making
$\varDelta y(x)$ be a function of conditional average treatment effect (CATE).

#### 4) ATT estimating

At last, bring $\varDelta y(x)$ into Equation (4), the estimation of ATT can
be obtained by averaging over the treatment group data set111In fact, any data
set that shares common support domains can be brought back. If there are no
privacy or other concerns, it is an easiest way to use samples of the
treatment group in real data set as input because they have very close common
support domains with the synthesized data sets.:

$ATT=E_{x_{1}\sim g_{1}(y_{1}\mid x_{1})}\left\\{\varDelta
y(x_{1})\right\\}\approx\frac{1}{N_{1}}\sum_{j=1}^{N_{1}}\varDelta
y(x_{1}(j))$ (6)

It can be found that by replacing traditional matching procedure, this
estimate is theoretically proved to be able to accurately estimate ATT
provided that the GAN is perfectly trained. In section 4.1, a standard linear
benchmark with ground truth will be taken as an example to show that the
proposed approach can exactly recover the preset ATT ground truth.

###  3.4 Extension to Conditional Tabular GAN

In practice, it is possible for covariates to be discontinuous, leading to
failures that the generator of traditional GAN cannot deal with. To make our
estimator more applicable to tabular economic data sets, in this paper we
proceed the training through the conditional tabular generative adversarial
network (CTGAN) proposed by Xu et al. (2019). By introducing conditional
generator joint with training-by-sampling algorithm, CTGAN can effectively
identify and learn distributions of data with both continuous and discrete
inputs.

Referring to the Python code222Detailed code introduction see
https://sdv.dev/SDV/user_guides/single_table/ctgan.html provided by CTGAN (Xu
et al., 2019), the joint probability density functions of the real data sets
in this paper are learned through a method equivalent to the algorithm
described in section 3.2. Figure 2 shows the detailed training process. It can
be found in the figure that the only difference from the original method is
that two separate GANs are merged into one and are trained separately by
distinguishing the one-hot input vector $\bar{d}$ ($\bar{d}=$’10’ for
treatment group and $\bar{d}=$’01’ for control group). The output is same to
the previously-mentioned methods. The major reason of using this equivalent
algorithm is to save computing resources. Detailed algorithm and neural
network structure see Xu et al. (2019, section 4).

With the trained model, any number of synthesized tabular sample with the
joint PDFs same to the real can be generated.

##  4 Standard Benchmark Establishment and Estimation Accuracy Tests

###  4.1 Linear Benchmark Establishment

In order to verify the accuracy of GAN-ATT estimator, a standard linear
benchmark with preset ATT ground truth is established in this section.

First, assume that the data of both treatment group ($d=1$) and control group
($d=0$) satisfy the following linear model:

$y_{d}=\alpha+\beta x_{d}+\gamma\cdot d+\varepsilon_{d},\>d=0,1$ (7)

In this model, $\alpha$ is the fixed effect, $\beta$ is the parameter of
covariates $x_{d}$, $\gamma$ is the treatment effect that need to be
estimated, $\varepsilon_{d}$ is an Gaussian white noise.

In order to make the benchmark closer to the real, we artificially set the
selection bias as follows:

$x_{d}\sim\mathscr{N}\left(\mu_{d},\>\sigma_{xd}\right),\>d=0,1,$ (8)

which means the covariates $x_{0}$, $x_{1}$ follows two different Gaussian
distributions with mean $\mu_{d}$ and variance $\sigma_{xd}$.

Subsequently, when $\varepsilon_{d}$ is set to be additive white Gaussian
noise, combining with Equation (7), there is:

$y_{d}\mid x_{d}\sim\text{$\mathscr{N}$}\left(\alpha+\beta x_{d}+\gamma
d,\>\sigma_{\varepsilon d}\right),\>d=0,1$ (9)

Next, follow the idea in the section 2.2, combine Equation (8)-(9) with
Equation (2), there is:

$E_{y_{1}\sim g_{1}\left(y\mid x\right)}\left\\{y_{1}^{f}\mid
x_{1}\right\\}=\alpha+\beta x_{1}+\gamma$ (10)

$E_{y_{0}\sim g_{0}\left(y\mid x\right)}\left\\{y_{0}^{f}\mid
x_{1}\right\\}=\alpha+\beta x_{1}$ (11)

Bring Equation (10) and (11) back in Equation (4), there is $ATT=\gamma$.

The above derivation proves that no matter how much selection bias exists
between $x_{0}$ and $x_{1}$, as long as the left sides of Equation (10) and
Equation (11) can be calculated from sample sets, the ATT estimated by
Equation (4) is theoretically the preset ground truth $\gamma$. Note that the
ATT estimate of Equation (6) is an approximation of ATT estimate in Equation
(4). That is to say, when the size of sample used for calculation is
sufficiently large, the estimate of GAN-ATT is the preset ATT ground truth
$\gamma$.

With the ATT ground truth preset, a linear benchmark is established. Referring
to Equation (7), the model is set with
$x_{0}\sim\text{$\mathscr{N}$}\left(0,\>1\right)$,
$x_{1}\sim\text{$\mathscr{N}$}\left(1,\>2\right)$;
$\varepsilon_{0},\varepsilon_{1}\sim\text{$\mathscr{N}$}\left(0,\>0.1\right)$;
$\alpha=0$;$\beta=1.5$; $\gamma=1$ is the ATT preset ground truth.

###  4.2 Estimator Performance Evaluation

After the linear benchmark is established, two toy data sets
$\left\\{y_{d}(i),x_{d}(i)\right\\}$ (50,000 samples in each) are randomly
generated from the given model. In order to avoid confusion, these two data
sets will be called ’real’ data sets in this section. Subsequently, two
synthesized data sets (200,000 observations in each333These samples are
randomly generated by the trained model of GAN. These two synthesized data
sets with arbitrarily large sample size will only be used for intermediate
calculations of CATEs and does not affect the sample size for the final
calculation of ATT.) with ideally the same distribution to the real data set
is generated from GAN machine learning. Next, ATT will be estimated via CATE
calculations, and the result will be compared to the preset ground truth
$\gamma=1$.

Figure 3 illustrates the $\varDelta y\left(x\right)$ values (CATE) in all bar
$\varDelta$ in the real data set with common support domain; Figure 4
illustrates the $\varDelta y\left(x\right)$ values in all bar $\varDelta$ in
the synthesized data set with common support domain. It can be found that
there do exist some differences between the two figures, especially in the
fringes of the bar axis. This is mainly caused by insufficient samples for
CATE calculations on the fringe. That is, very few synthetic samples fall in
these bars, making the estimation of $\varDelta y\left(x\right)$ biased.
However, in the case of estimaing ATT, this difference has no major effect on
the estimation accuracy. This is because similarly, when bringing the real
treated samples back into $\varDelta y\left(x\right)$, very few samples will
fall into these biased bars, so the biased estimation will be few.
Furthermore, by increasing the sample size of the synthesized data sets, the
bias can be significantly reduced.

Figure 3 CATE values of all bars of real data set

Figure 4 CATE estimates of all bars of synthesized data set

Table 1 is the ATT estimation statistics of this benchmark, including real
data sets and synthesized data sets of both treatment group and control group.
In the last column, the mean of $Estimated\\_ATT=0.9949$ is the estimated
average treatment effect of the treated (ATT) according to Equation (6). It
can be observed that the estimated ATT is 0.51% away from the preset ATT
ground truth $\gamma=1$. So, it can be concluded that the GAN-ATT estimator
has a good performance over this toy data set. Note that the standard error of
$Estimated\\_ATT$ can also be given by our estimator. If required, the
confidence interval of ATT estimate can be derived by the obtained mean and
variance under Gaussian distributions.

Table 1 ATT estimate - Benchmark 1

| $Real\\_y_{0}$ | $Real\\_y_{1}$ | $Synthetic\\_y_{0}$ | $Synthetic\\_y_{1}$ | $Estimated\\_ATT$
---|---|---|---|---|---
No. Obs | 50,000 | 50,000 | 200,000 | 200,000 | 49,280
Mean | -0.0090 | 2.5152 | 0.0813 | 2.4337 | 0.9949
Std. Err. | 1.5012 | 2.9971 | 1.7132 | 2.4812 | 5.55e-4
Inverted Kolmogorov-Smirnov D Statistic | - | - | 0.9796 | 0.9651 | -
Continuous Kullback–Leibler Divergence | - | - | 0.6867 | 0.6247 | -

###  4.3 Advanced Non-linear Benchmark Establishment

To further check the performance of the estimator proposed in this paper, in
this section the standard linear benchmark is extended into a 2-dimensional
non-linear data set with the treatment effect dependent on the covariates.

The benchmark is established as follows:

$y_{0}^{k}=\alpha\cdot\frac{1-e^{-\mathbf{W}\cdot\mathbf{X}_{0}}}{1+e^{-\mathbf{W}\cdot\mathbf{X}_{0}}}+\varepsilon_{0}^{k},$

$y_{1}^{k}=\beta\cdot\frac{1-e^{-\mathbf{W}\cdot\mathbf{X}_{1}}}{1+e^{-\mathbf{W}\cdot\mathbf{X}_{1}}}+t+\varepsilon_{1}^{k},$

$t=\gamma\cdot e^{\frac{-\parallel X_{1}\parallel^{2}}{2\sigma}}$ (12)

In this model, $t$ is the preset CATE. $\mathbf{W}$ is a $2\times 1$ random
vector that satisfies a uniform distribution:
$\mathbf{W\mathrm{\sim\mathit{\mathscr{U}}\left(-1,\,1\right)^{2\times 1}}}$;
$\boldsymbol{\mathbf{X}}_{0}$ and $\mathbf{X}_{1}$are the vectors of
covariates in control group and treatment group that
satisfy$\boldsymbol{\mathbf{X}}_{0}^{k}\sim\text{$\mathscr{N}$}\left(\mu_{0},\>\mathbf{\mathbf{\mathbf{\Sigma}}}\cdot\mathbf{\Sigma}^{T}\right)^{2\times
1}$,
$\mathbf{X}_{1}^{k}\sim\text{$\mathscr{N}$}\left(\mu_{1},\>\mathbf{\Sigma}\cdot\mathbf{\Sigma}^{T}\right)^{2\times
1},$ $\mathbf{\Sigma\sim\mathrm{\mathscr{U}\left(-1,\>1\right)}}^{2\times 2}$,
$\mu_{0}=0^{2\times 1}$,
$\mu_{1}\sim\text{$\mathscr{U}$}\left(-1,\,1\right)^{2\times 1}$;
$\alpha,\beta,\gamma,\sigma$ are parameters to control the value ranges of $y$
and $t$, here $\alpha=\beta=5,\gamma=1.5,\sigma=4$; $\varepsilon_{0}$ and
$\varepsilon_{1}$ are white noises that satisfy $\varepsilon_{0}$,
$\varepsilon_{1}\sim\text{$\mathscr{N}$}\left(0,\>0.1\right)$. Note that in
order to keep selection bias exist, $\mu_{0}\neq\mu_{1}$. With these settings,
the ATT preset ground truth is 0.6836444Here, the value of the preset ATT
ground truth is approximated by repeating Monte Carlo simulation since the
integral calculation of $t$ is over-complicated.. Similarly, it can be proved
in the same way in section 4.1 that the ATT estimated by the GAN-ATT estimator
is theoretically equal to the preset ATT ground truth.

###  4.4 Advanced Estimator Performance Evaluation

After the advanced benchmark is established, two toy data sets
$\left\\{y_{d}(i),x_{d}(i)\right\\}$ (100,000 samples in each) are randomly
generated from the given model. To avoid losing common support domains, the
sample size is also increased as the covariate dimension increases. Same to
section 3.1, two synthesized data sets (400,000 observations in each) with
ideally the PDF same to the real data set is generated from trained GAN model,
and the ATT is estimated and compared to the preset ground truth. Note that
here since the data dimension is upgraded to two-dimensional, the bars used
for CATE estimations should be upgraded into two-dimensional cubes
$\varDelta$.

In Figure 5, the CATE values in all cube $\varDelta$ over the real data set
with common support domains are presented, while the CATE values in all cube
$\varDelta$ over the synthesized data set with common support domain are
presented in Figure 6.

Figure 5 CATE values of all cubes in real data set

Figure 6 CATE estimates of all cubes in synthesized data set

Table 2 is the ATT estimation statistics of this benchmark. It can be observed
that the mean of $Estimated\\_ATT$=0.6720 is the estimated average treatment
effect of the treated (ATT) according to Equation (6), which is 1.70% away
from the preset ATT ground truth 0.6836. It can be concluded that the
estimator has an acceptable performance over this toy data set.

Table 2 ATT estimate - Benchmark 2

| $Real\\_y_{0}$ | $Real\\_y_{1}$ | $Synthetic\\_y_{0}$ | $Synthetic\\_y_{1}$ | $Estimated\\_ATT$
---|---|---|---|---|---
Mean | 0.0009 | 0.8094 | -0.0173 | 0.7905 | 0.6720
No. Obs | 100,000 | 100,000 | 400,000 | 400,000 | 99,166
Std. Err. | 2.0300 | 2.0472 | 2.2268 | 2.0317 | 0.0021
Inverted Kolmogorov-Smirnov D Statistic | - | - | 0.9389 | 0.9570 | -
Continuous Kullback–Leibler Divergence | - | - | 0.7483 | 0.7139 | -

Furthermore, in order to make our estimator better adapt to the practical
cases of insufficient samples, we randomly select 1,000 treated samples in our
benchmark 2 as a new treatment group, so without changing the preset ATT
ground truth, the sample size of $Real\\_y_{1}$ is reduced from 100,000 to
1,000.

In the case when the training data is insufficient, problems of overfitting
may exist. To solve problems of this kind, methods of data augmentations,
which add noise to the inputs, are widely used in the field of machine
learning. Referencing to Bishop (1995), samples of the treatment group are
augmented to 100,000 with input white noise. From Table 3 it can be found that
the ATT estimation obtained from our GAN-ATT estimator is 0.6796, which is
0.59% away from the preset ground truth. It can be concluded in the case of
insufficient treated samples for machine learning, GAN-ATT can still provide
relatively accurate estimations.

Table 3 ATT estimates comparing to matching approaches- Benchmark 2

| GAN-ATT | PSM-NN | PSM-Kernel | CEM
---|---|---|---|---
Eistimated_ATT | 0.6796 | 0.7038 | 0.7706 | 0.7064
No. Obs (Treatment Group) | 996 | 999 | 999 | 971
No. Obs (Control Group) | - | 2,918 | 99,583 | 86,722

Additionally, for comparison propose, two traditional matching approaches: PSM
and CEM are also used to estimate ATT over this benchmark. From Table 3 it can
be observed that the ATT estimation via PSM-Nearest Neighbour 1:3 is 0.7038,
which is 2.95% away from the ground truth; estimation via PSM-Kernel is
0.7706, which is 12.73% away from the ground truth; estimation via CEM is
0.7064, which is 3.34% away from the ground truth.

From the table it can be observed that comparing to the other three matching
methods, the estimate of GAN-ATT is more accurate. In details, the ATT
estimate of PSM-NN is relatively accurate, yet loses too many samples in the
control group. PSM-Kernel retains nearly full samples in both two groups, but
the estimation is not accurate. This might be attributed to the inaccurate
weightings, because the distance between propensity scores cannot precisely
reflect the real distance before dimensionality reduction. CEM performs a
better trade-off between information loss and estimation accuracy, yet still
cannot get rid of the problem of sample dropping due to insufficient common
support domains. Through the performance test of the benchmark in this
section, it can be concluded that the GAN-ATT estimator has obvious advantages
over traditional matching approaches.

###  4.5 Estimator Performance Evaluation via Firm-level Data

Referring to our previous study (You & Papps, 2022), we use the same data set
to further test the performance of our estimator in a more complex case. The
obtained results will also be compared to previous results estimated by PSM
and CEM.

Table 4 Descriptive statistics - Firm-level data

| Profit | Size | Age | Asset | Output
---|---|---|---|---|---
No. obs | 1,401,594 | 1,411,470 | 1,411,472 | 1,411,467 | 1,411,472
Mean | 10,549.64 | 260.32 | 9.88 | 167,026.2 | 200,378
Std. Err. | 32,685.11 | 430.92 | 9.25 | 2,162,198 | 4,208,789

Table 4 is the descriptive statistics of the original data set used in our
previous study, which is a panel firm-level data set 2007-2013 extracted from
the Chinese Industrial Enterprise Database. According to the previous
discussions, 2007-2009 is the time period before the treatment, while
2011-2013 is the time period after the treatment. Samples are previously
matched based on firm features of each year before the treatment to eliminate
selection bias. Two ATT estimations are obtained via difference-in-difference
(DID) after matching. The DID calculation used for ATT estimation is: 1)
Calculate the subtractions of the three-year means of the outcome variables
before and after the treatment; 2) Calculate the subtractions of samples
matched between the treatment group and the control group.

In order to incorporate the GAN-ATT estimator into an equivalent estimation to
the previous, step 1 remains unchanged, while step 2 is replaced by the
calculation of the CATEs. In our previous study, each firm feature is divided
into three covariates for matching purpose (e.g. firm profit between 2007-2009
is divided into profit2007, profit2008, profit2009). Firm size, which takes
the average of 2011-2013, is selected as the outcome variable, and the rest
four firm features are divided into covariates with 10 dimensions. Note that
firm age is constant over time, for calculation simplicity, only one dimension
is assigned to firm age. With this, the original data set is transformed into
an equivalent data set with 10-dimensional covariates and 1-dimensional
outcome variable. Besides, since the GAN training requires the integrity of
data input, we eliminate samples with missing input and fill a small amount of
data (less than 4%) through the mean over time. The final sample size is 1,250
for the treatment group and 241,289 for the control group. It can be observed
that the sample size of the treatment group is too small for machine learning.
In order to prevent overfitting, white noise is similarly added to the input
according to Bishop (1995). By this method, the number of input sample for GAN
training is extended into 125,000 (treatment group) and 241,289 (control
group).

Table 5 ATT estimates comparing to matching approaches - Firm-level data

| GAN-ATT | PSM-NN | PSM-Kernel | CEM
---|---|---|---|---
Eistimated_ATT - Firm Size | 6.258 | 7.269 | 2.029 | 10.009
No. Obs (Treatment Group/Control Group) | 1,216/- | 1,249/3,746 | 1,249/240,956 | 1,182/303,699
Inverted Kolmogorov-Smirnov D Statistic (T/C) | 0.9795/0.9417 | - | - | -
Continuous Kullback–Leibler Divergence (T/C) | 0.9068/0.9733 | - | - | -

Table 5 is the results of ATT estimates via our GAN-ATT estimator together
with PSM (Nearest Neighbour 1:3), PSM (Kernel) and CEM. It can be observed
that the estimation of GAN-ATT estimator is 6.258, in which CATEs are
estimated by 2,000,000 synthesized samples in both treatment group and control
group. It can be found that GAN-ATT still suffers a loss of 34 treated
samples, which is cause by the limited personal computing power. To improve
the results, simply increase the number of synthesized samples generated by
GAN for CATE estimations. Furthermore, the results obtained previously by the
three matching approaches are also presented for comparison purpose. It can be
found that the GAN-ATT estimate is not far from the estimates of the other
three methods. Short of ground truth, it is hard to say which method is better
in this section, but it can be concluded that our estimator is competent to
estimate ATT in the case of high-dimensional covariate inputs and insufficient
treated samples, while providing theoretically more accurate estimations with
less information loss.

##  5 Discussion

Based on the preliminary performance tests for the GAN-ATT estimator in
section 4, there are several points we need to further discuss:

1) The aim of section 4.1-4.2 is to validate the usability of our estimator
via the simplest benchmarked data set. In this regard, a standard linear
benchmark with one-dimensional covariate and with selection bias preset is
established. By estimating a preset constant ATT ground truth, the estimation
accuracy of our estimator can be tentatively revealed. Compared to commonly-
used semi-benchmarked data sets, such as TWINS, JOBS, and IHDP (Yao et al.,
2021), tests based on our benchmark can more intuitively reflect the absolute
accuracy of ATT estimates instead of relative accuracy of various estimation
methods, especially methods based on linear regressions. Our evidence shows
that the estimator proposed are within an accuracy loss of 1%. So it can be
concluded that the GAN-ATT estimator is effective and accurate in estimating
this benchmark.

2) In section 4.3-4.4, to make our estimators more broadly applicable to
economic data sets, the standard linear benchmark is extended into an advanced
non-linear benchmark, which has not been used in the existing literature. The
selection bias is same preset, yet the treatment effect is set dependent to
the covariates with a non-linear relationship. Furthermore, the dimension of
the covariates input is increased to two. The goal of establishing this
benchmark is to test the learning ability of neural networks towards the non-
linearities implicit in the data set, so as to check the accuracy of
estimating conditional average treatment effect (CATE) and ATT. Our evidence
shows that the accuracy loss of our estimator is 1.70%. Compared to
traditional matching approaches: propensity score matching (PSM) and coarsened
exact matching (CEM), the GAN-ATT estimate is proved more accurate while
having less information loss. Thus, it can be concluded that the estimator has
a good performance in this benchmark.

3) In section 4.5, we implement our GAN-ATT estimator to a real firm-level
data set, which is previous used in our study (You, Papps, 2022) for a
matching-based counterfactual inference estimation. With proper data cleaning
and augmentation, we find that the estimator is fully competent for ATT
estimation in the case of high-dimensional covariate inputs and insufficient
treated samples. Unfortunately, due to the need to privacy protections, the
database stopped updating after 2013. However, if our GAN-ATT estimator could
be used for casual inference proposes, privacy of the data providers would be
secured. By officially providing the pre-trained GAN models, the usage of real
data set with privacy concerns can be replaced by a synthesized data set under
same PDF. Joint with the ATT estimation procedure proposed in this paper,
sample privacy issues can be effectively resolved.

4) GAN is a relativity new deep learning technique, especially for tabular
data sets. It must be admitted that there are still shortages of GAN
performance in practice, such as the training validity problem existing in our
estimator. Ideally, the accuracy of our estimator would approach 100% if GAN
were able to learn the PDF of a data set with complete accuracy. However in
practice, there is currently no systematic way to verify the performance of
GANs in the case with multidimensional inputs. We found that in the cases when
dimension of covariates is high, performance evaluation criteria provided by
CTGAN (Xu et al., 2019), including KL distance, KS statistic, etc. may fail.
In this paper we try to avoid this failure by repeating training process with
adjusted hyperparameters and layer scales. The results with closest mean and
variance of the real data set are selected for further calculations.
Nevertheless, this is not sufficient to prove that the two sets of data are
equally distributed. We believe that with the development of GAN training
techniques, better testing methods can be found in our future research.

5) As it is mentioned in section 2.1, to use our estimator, the
unconfoundedness premise must be satisfied. Yet this premise is usually
difficult to check in the field of economics. In order to more effectively
eliminate confounding factors, several methods have been developed in the
early stage, such as learning representation (Bengio et al., 2013), T-Leaner,
R-Learner (KÃŒnzel et al. 2019), etc.. To better solve this problem, we are
considering a promising approach to fit all possible covariates using neural
networks. If the R-squared is sufficiently close to 1, it can be considered
that there is no major unobserved covariates. In order to better use our GAN-
ATT estimator, this issue awaits our further research.

##  6 Conclusion

In this paper we propose the GAN-ATT estimator that can better estimate
average treatment effect on the treated (ATT) without matching. With the
support of the latest mchine learning technique: Generative Adversarial
Network (GAN), the conditional probability density functions (PDFs) of samples
in both treatment group and control group can be obtained. By differentiating
conditional PDFs of the two groups under identical input condition, the
conditional average treatment effect (CATE) can be estimated, and the ensemble
average of corresponding CATE over treatment group samples is the estimate of
ATT. In particular, when GAN could perfectly learn the conditional PDFs
through training, the ATT estimate can be proved completely accurate.

Furthermore, we generate two benchmarks with preset ATT ground truth. Together
with a real firm-level data set, the performance of GAN-ATT estimator is
directly tested. Evidence shows that: 1) For the two benchmarks, whether the
model is set linear or non-linear, and whether the treatment effect is set
dependent to covariates or constant, the estimator can closely approximate the
ATT ground truths. 2) A real firm-level data set with high-dimensional
covariates and insufficient number of treated samples is further used for the
test. With proper data augmentations, the estimator is considered competent
for estimating ATT in such case.

Compared to matching approaches, the evidence shows that the GAN-ATT estimator
does have advantages in estimating ATT. Without matching, there is no
compromise between selection bias and information loss need to take into
considerations in our estimator; With GAN-based sample augmentations, infinite
samples can be generated based on the trained model, so that the problems of
insufficient samples or common support domains are solved; With good GAN
learning performance, ATT estimations can be very close to the fact; With
machine learning technique, the estimates are less influenced by human
factors; With providing per-trained GAN models instead of real samples, fewer
privacy concerns will be taken into consideration.

However, we also found that the performance of GAN deep learning is away from
our expectations. Same to other machine learning approaches, training results
are influenced by the hyper-parameters. Yet in the case of GAN, with high-
dimensional variable inputs, there is no systematic way so far we know to
check training performance. Nevertheless, this paper has given a theoretical
proof that the GAN-ATT estimate is exact under the premise that GAN performs
perfectly. With future advancement of GAN techniques, we believe our estimator
can perform completely accurate estimate of ATT.

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