Using small businesses for individual tax planning: evidence from special tax regimes in Chile

Abstract

Many countries have special tax regimes (STRs) for small businesses. Even though these regimes may reduce compliance costs, they increase the complexity of the tax system and can be used by high-income individuals to avoid taxes. This paper uses administrative data from Chile to analyze whether the use of STRs is associated with strategic tax planning at the individual level. A descriptive analysis of the data finds three stylized facts that, taken together, are consistent with strategic behavior: STRs are used frequently, they are used mainly by high-income taxpayers, and high-income taxpayers are more likely to hold a portfolio of businesses filing taxes under STRs. We rationalize these facts with a simple model of small business creation and tax planning and test the model’s predictions. We find that following a reform that made a particular STR more restrictive, reported individual incomes from businesses filing under that STR decreased between 10 and 15%, while income reported from alternative sources increased. Overall Taxable Income increased between 4 and 7%. This increase is explained by the more restrictive scenario for avoiding taxes through STRs, consistent with individuals using these regimes for tax planning.

Appendices

Appendix A: Estimation of financial profits of businesses subscribed to STRs

As businesses subscribed to some STRs are not forced to carry detailed internal accounting, financial profits are not observed. Nevertheless, the Internal Revenue Service of Chile carried out a procedure for estimating financial profits for these businesses in year 2013. In this appendix, we briefly describe the procedure carried out by the Chilean tax authority.

The central assumption is that financial profits are proportional to cash flow. Then, from other forms filed by the businesses, it is possible to compute a cash flow measure for every business i, $CF_i$, defined by

$$\begin{aligned} CF_i= & {} S_i-E_i-R_i, \end{aligned}$$

where $S_i$ are the sales, $E_i$ are the expenses, and $R_i$ are all wages and salaries paid. This is calculated for all businesses, regardless the tax regime associated, i.e., for the ones taxed by the general scheme and the ones subscribed to STRs.

Consider a set of businesses, A, that do not report profits given they are registered as STR firms. This set is defined by observables (for example, size or economic sector). Then, consider a set of businesses similar in observables, ${\hat{A}}$, that are taxed under the general regime and, therefore, report information about profits. For those businesses, it is possible to calculate a factor, $F_{{\hat{A}}}$, from the following relation

$$\begin{aligned} F_{{\hat{A}}}= & {} \frac{\sum _{i\in {\hat{A}}}P_i}{\sum _{i\in {\hat{A}}}CF_i}, \end{aligned}$$

where $P_i$ are the profits of firm i in ${\hat{A}}$. Then, for businesses in A it is possible to estimate profits, $P_i$, from the following relation

$$\begin{aligned} P_i= & {} F_{{\hat{A}}}CF_i, \quad \forall i\in A, \end{aligned}$$

i.e., by assuming a proportional relation between profits and cash flow. The groups of businesses taxed by the general regime considered for calculating the factors for the different regimes are

• 14B regime: Businesses with sales under 318,000 USD.

• 14T regime: Businesses with sole proprietorship legal status and sales under 318,000 USD.

• Agricultural PT regime: Businesses of the agricultural sector.

• Mining PT regime: Businesses of the mining sector.

• Freight transportation PT regime: Businesses of the freight transportation sector.

• Passengers transportation PT regime: Businesses of the passengers transportation sector.

Appendix B: Additional tables for stylized fact 3

See Tables 15, 16 and 17.

Table 15 Disaggregation of Table 6

Table 16 Disaggregation of Panel A of Table 7

Table 17 Disaggregation of Panel B of Table 7

Appendix C: A model of optimal business creation under STRs: detailed version

In this appendix we develop a simple model of individual tax planning in the presence of preferential tax regimes for small businesses. This model leads to the results reported in Sect. 5.

The agent chooses the number of enterprises that benefit from various preferential tax regimes so as to maximize after-tax income. The trade-off she faces when creating a new business is between lowering her tax burden and the cost of setting up and managing the additional enterprise.

We first consider the case with only one tax regime and then extend the model to incorporate two regimes, as in the case considered in Sect. 5. We show that the model’s implications are consistent with the stylized facts described in Sect. 4 and derive the implications that are tested in Sect. 6.

1.1 Appendix C.1: Model

The model is static. The agent receives exogenous income Y that is strategically broken up into two components, a component that is sheltered from the income tax, $Y_s$, and an unsheltered component, $Y_u$, with $Y=Y_s+Y_u$. The unsheltered component pays income tax at marginal rate $\tau _m(Y_u)$; hence, the average income tax rate, $\tau (Y_u)$, satisfies

$$\begin{aligned} \tau (Y_u) + Y_u\tau '(Y_u) = \tau _m(Y_u). \end{aligned}$$

(3)

In line with Chile’s (and most country’s) income tax schedule, we assume $\tau _m(0)=0$ and $\tau _m'\ge 0$.¹⁸

The component $Y_s$ is sheltered in small businesses created for the sole purpose of lowering the agent’s tax burden. Income reported by each of these businesses up to L is taxed at a constant rate t. Firms have no incentive to report income above L since they loose eligibility for the PT regime should they do so. Consistent with the application in the main text, the special tax regime considered here is referred to as the PT regime.

Creating businesses comes at a cost captured by the function c(S), where S denotes the number of businesses created. These costs can be interpreted as set up costs or the cost of managing the businesses. We assume $c(0)=0$, $c'>0$ and $c''>0$.¹⁹

We also assume that these costs cannot be subtracted from the tax base. For simplicity and without loss of generality we ignore integer constraints on S. It then follows that

$$\begin{aligned} S = \frac{Y_s}{L}, \end{aligned}$$

(4)

since it is optimal to shelter income in each business to the maximum, L, that benefits from a lower tax rate. It also follows that S is the sum of the agent’s participations in all businesses and whether the agent holds entire businesses or only a fraction thereof is irrelevant.

The agent maximizes after-tax income, Z. Given the above assumptions, her problem is

$$\begin{aligned} \max _{0\le Y_s\le Y}Z = [1-\tau (Y-Y_s)](Y-Y_s) + (1-t)Y_s - c(Y_s/L). \end{aligned}$$

(5)

As shown in the expression above, after-tax income, Z, has two components. The first component is unsheltered income net of income taxes, $(1-\tau )Y_u$. The second component is sheltered income net of taxes and setup costs, $(1-t)Y_s - c(S)$.

The above setup captures, albeit in a simplified way, one of the main features of preferential tax regimes for small enterprises described in Sect. 2, namely that their benefit expires beyond a certain size-related threshold. This characteristic provides incentives for high-income individuals to create many such businesses.

Next we solve the agent’s problem. Differentiating (5) w.r.t. $Y_s$ and using (3) yields

$$\begin{aligned} Z'(Y_s) = [\tau _m(Y-Y_s)-t] - \frac{c'(Y_s/L)}{L}. \end{aligned}$$

(6)

The marginal benefit from creating an additional enterprise is equal to the difference between the gap between both tax rates and the marginal cost of setting up and managing the additional business (where the latter is normalized by the maximum income that benefits from the preferential regime). The first term on the r.h.s. of (6), the difference between both marginal rates, is increasing in sheltered income because the marginal tax rate increases with Taxable Income. The second term, $c'(Y_s/L)/L$, also increases with sheltered income because the marginal cost of creating businesses is increasing. If follows that marginal after-tax income is decreasing in sheltered income:

$$\begin{aligned} Z''(Y_s) = -\tau _m'(Y-Y_s) - \frac{c''(Y_s/L)}{L^2} < 0. \end{aligned}$$

(7)

A first implication of (6) and (7) is that the agent will not set up any business if the cost of setting up the first business is larger than the benefit, that is, if $Z'(0)\le 0$. This leads to

Result C1

The agent will use the special tax regime only if

$$\begin{aligned}{}[\tau _m(Y)-t]L> c'(0). \end{aligned}$$

(8)

It follows that there exists a strictly positive income threshold ${\overline{Y}}$ characterized as the largest value of Y that satisfies²⁰

$$\begin{aligned}{}[\tau _m({\overline{Y}})-t]L= c'(0), \end{aligned}$$

(9)

such that the agent uses the special tax regime only if $Y>{\overline{Y}}$. Also, the threshold ${\overline{Y}}$ is increasing in the preferential tax rate t.

Proof

Expression (8) follows from $Z'(Y_s=0)>0$. The other statements follow from the assumption that $\tau _m'\ge 0$. $\square$

Agent’s problem (5) will have an interior solution if (8) holds and if $Z'(Y_s=Y)<0$. The latter is equivalent to:

$$\begin{aligned}{}[\tau _m(0)-t]L<c'(Y/L) \end{aligned}$$

which holds always given the assumption that $\tau _m(0)=0$. We are ready to characterize the optimal values of sheltered income and the number of businesses:

Result C2

Consider ${\overline{Y}}$ defined in (9) and denote by $Y^*_s$ and $S^*$ the optimal choices of $Y_s$ and S, respectively.

If $Y\le {\overline{Y}}$, we have $Y_s^*=S^*=0$. By contrast, if $Y>{\overline{Y}}$, $Y^*_s$ and $S^*$, are characterized by²¹

$$\begin{aligned}{}[\tau _m(Y-Y^*_s)-t]L = c'(Y_s/L) \end{aligned}$$

(10)

and

$$\begin{aligned}{}[\tau _m(Y-SL)-t]L = c'(S). \end{aligned}$$

(11)

Proof

Follow from (6), (7) and Result C1. $\square$

Result C2 is consistent with stylized facts 1 and 2 in Sect. 4. Special tax regimes will be used by all individuals with income above ${\overline{Y}}$, with ${\overline{Y}}$ at least as large as the highest income with an average tax rate of t.

The following result shows that, among those agents that create businesses, the number of businesses held increases with income.

Result C3

Assume the agent’s income is larger than ${\overline{Y}}$ defined in (9). Then $Y^*_s$ and $S^*$ are strictly increasing in Y, with

$$\begin{aligned} \frac{\partial Y^*_s}{\partial Y}= & {} \frac{ \tau _m'(Y_u^*)L^2 }{ \tau _m'(Y_u^*)L^2 + c''(S^*) } \in \left[ 0,1\right) , \end{aligned}$$

(12)

$$\begin{aligned} \frac{\partial S^*}{\partial Y}= & {} \frac{ \tau _m'(Y_u^*)L }{ \tau _m'(Y_u^*)L^2 + c''(S^*) }\in \left[ 0,\frac{1}{L}\right) , \end{aligned}$$

(13)

where $Y^*_u=Y-Y^*_s$.

Proof

Follows from implicit differentiation of (10) w.r.t. Y. $\square$

The intuition for (12) is the following: as the agent’s income increases so does the marginal income tax rate she must pay. For this reason, the agent is prepared to pay higher setup costs when her income is higher. Equation (12) also shows that the marginal propensity to shelter income will lie between zero and one and will be smaller if setup costs grow faster (larger value of $c''$).

Equation (13) shows that the number of enterprises created will increase with income, as noted in stylized fact 3 in Sect. 4.

The following result complements Result C3 by providing comparative statics w.r.t. variables other than income.

Result C4

Under the assumptions of Result C3:

$$\begin{aligned} \frac{\partial Y_s^*}{\partial t}= & {} -\frac{L^2}{\tau _m'(Y^*_u)L^2 + c''(S^*)}<0,\\ \frac{\partial Y_s^*}{\partial L}= & {} \frac{c'(S^*) + c''(S^*)S^*}{\tau _m'(Y_u^*)L^2 + c''(S^*)}>0, \end{aligned}$$

where $S^*=Y^*_s/L$ and $Y^*_u=Y-Y^*_s$. We also have:

$$\begin{aligned} \frac{\partial S^*}{\partial t}= & {} -\frac{L}{\tau _m'(Y^*_u)L^2 + c''(S^*)}<0,\\ \frac{\partial S^*}{\partial L}= & {} \frac{c'(S^*) - \tau _m'(Y^*_u)}{\tau _m'(Y_u^*)L^2 + c''(S^*)}. \end{aligned}$$

Finally, to capture changes in the cost of creating businesses, we replace c(S) by ac(S), where $a>0$ is a scale parameter that captures how fast marginal costs increase with the number of firms. We then have:

$$\begin{aligned} \frac{\partial Y_s^*}{\partial a}= & {} -\frac{\tau _m'(Y^*_u)L^2 + ac''(S^*)}{c'(S^*)L}<0 , \end{aligned}$$

(14)

$$\begin{aligned} \frac{\partial S^*}{\partial a}= & {} -\frac{\tau _m'(Y^*_u)L^2 + ac''(S^*)}{c'(S^*)L^2}<0 . \end{aligned}$$

(15)

Proof

The expressions follow from implicit differentiation of (10) and (11) w.r.t. t, L and a after replacing c(S) by ac(S). Also, to obtain the expressions for partial derivatives w.r.t. L we use (10) to get rid of an expression involving $\tau _m-t$. $\square$

The intuition underlying the first three expressions in Result C4 is straightforward. If the benefits associated with the special tax regime decrease, because t increases or L decreases, sheltered income decreases as well.

In general, the identity $S^*=Y^*_s/L$ implies that the partial derivatives for $S^*$ are obtained dividing partial derivatives for $Y^*_s$ by L. The case of the partial derivative w.r.t. L is different since the denominator in $Y^*_s/L$ also varies with the variable of interest in this case. It is therefore not surprising that the sign of the expression we obtained for $\partial S^*/\partial L$ is indeterminate. If $c'(S^*)>\tau _m'(Y^*_u)$, the optimal number of firms increases with L, while the opposite happens if $c'(S^*)<\tau _m'(Y^*_u)$.

Finally, the intuition for the impact of changes in the cost function is straightforward: a shift upward of this function makes business creation more costly and therefore lowers income sheltered and the optimal number of businesses.

1.2 Appendix C.2: The case with two STRs

We extend the above model to the case with two special tax regimes and denote by $Y_{is}$ income sheltered in regime i with $i=1,2$ so that now unsheltered income is given by

$$\begin{aligned} Y_u = Y-Y_{1s}-Y_{2s}. \end{aligned}$$

The preferential tax rate of regime i is $t_i$, valid for reported business income less than $L_i$, and the number of businesses that benefits from tax regime i is $S_i$, with

$$\begin{aligned} S_i = \frac{Y_{is}}{L_i}, \qquad i=1,2. \end{aligned}$$

As we did in the main text, regime 1 is the PT regime and we will consider the impact of parameter changes in this regime on the agent’s choice variables.

We assume two separate cost functions for setting up businesses, one for those of type 1, the other for those of type 2. That is, the cost of setting up and managing $S_1$ firms of type 1 and $S_2$ firms of type 2 is

$$\begin{aligned} c(S_1,S_2) = c_1(S_1) + c_2(S_2) + c_3(S_1+S_2), \end{aligned}$$

with $c_i(0)=0$, $c_i'>0$ and $c_i''>0$. The first two components are important for the results that follow, that is, both tax regimes involve separate cost components. This will be the case, for example, if both regimes apply to different economic sectors and sector-related sunk investments are needed to be eligible for each regime (see Sect. 2 for examples). The third component captures economies of scope between all businesses where the agent has participation; since its role is not essential to derive the results that follow, we assume it is equal to zero.

Given $Y>0$, the agent’s problem now is:

$$\begin{aligned} \max \nolimits _{Y_{1s},Y_{2s},Y_u}\quad Z= & {} [1-\tau (Y_u) ]Y_u + (1-t_1)Y_{1s} +(1-t_2)Y_{2s} - c_1(S_1) - c_2(S_2),\nonumber \\ \text{ s.t. }\quad Y= & {} Y_u+ Y_{1s}+Y_{2s},\nonumber \\ S_1= & {} \frac{Y_{1s}}{L_1}, S_2 = \frac{Y_{2s}}{L_2},\nonumber \\ Y_u\ge & {} 0, Y_{1s}\ge 0, Y_{2s}\ge 0. \end{aligned}$$

(16)

Result C5

Assume the solution to (16) is interior and replace $c_1(S)$ by $a_1c_1(S)$ in the results with comparative statics involving $a_1$. Denote the optimal values of $Y_{1s}$, $Y_{2s}$, $Y_{s}$ and $Y_{u}$ by $Y_{1s}^*$, $Y_{2s}^*$, $Y_{s}^*$ and $Y_{u}^*$, respectively. And denote the value of the problem, that is taxes paid under the optimal strategy, by $Z^*$.

Then

$$\begin{aligned} \frac{\partial Y^*_{2s}}{\partial L_1}= & {} - \frac{[c_1''(S_1^*)S_1^* + c_1'(S_1^*)]L_2^2\tau _m'(Y^*_u)}{c_1(S^*_1)c_2(S^*_2) + [c_1''(S_1^*)L_2^2 + c_2''(S^*_2)L_1^2]\tau _m'(Y^*_u)} < 0, \end{aligned}$$

(17)

$$\begin{aligned} \frac{\partial Y^*_{s}}{\partial L_1}= & {} -\frac{c_2''(S^*_2)}{\tau _m'(Y^*_u)L_2^2} \frac{\partial Y^*_{2s}}{\partial L_1}>0,\end{aligned}$$

(18)

$$\begin{aligned} \frac{\partial Y^*_{2s}}{\partial a_1}= & {} \frac{{c_1'(S^*_1)}{L_2^2}\tau _m'(Y^*_u)}{{c_1''(S_1^*)c_2''(S_2^*)} +\left[ {c_1''(S^*_1)}{L_2^2} + {c_2''(S^*_2)}{L_1^2}\right] \tau _m'(Y^*_u)} >0 ,\end{aligned}$$

(19)

$$\begin{aligned} \frac{\partial Y^*_{s}}{\partial a_1}= & {} -\frac{c_2''(S^*_2)}{L_2^2\tau _m'(Y^*_u)}\frac{\partial Y^*_{2s}}{\partial a_1}<0, \end{aligned}$$

(20)

where $Y^*_s=Y^*_{1s}+Y^*_{2s}$.

Also, using the identities $Y^*_{1s} = Y^*_s - Y^*_{2s}$ and $Y^*_u=Y-Y^*_s$, the above expressions lead to explicit expressions for $Y^*_{1s}$ and $Y^*_u$ that satisfy $\partial Y^*_{1s}/\partial L_1>0$ and $\partial Y^*_u/\partial L_1{<}0$.

Finally,

$$\begin{aligned} \frac{\mathop {}\!\mathrm {d}Z^*}{\mathop {}\!\mathrm {d}L_1}= & {} - c_1'(S_1^*)\frac{S_1^*}{L_1}<0, \end{aligned}$$

(21)

$$\begin{aligned} \frac{\mathop {}\!\mathrm {d}Z^*}{\mathop {}\!\mathrm {d}a_1}= & {} c_1(S_1^*)>0, \end{aligned}$$

(22)

where $S^*_1=Y^*_{1s}/L_1$.

Proof

We derive the partial derivatives w.r.t. $a_1$; the derivation of partial derivatives w.r.t. $L_1$ is similar. We also omit the subindex s in $Y_{1s}$ and $Y_{2s}$ in what follows.

From (16) we have that the first-order conditions w.r.t. $Y_{1}$ and $Y_{2}$ are:

$$\begin{aligned} \tau _m(Y-Y_1-Y_2) - t_1= & {} \frac{a_1}{L_1}c_1'(Y_1/L_1), \end{aligned}$$

(23)

$$\begin{aligned} \tau _m(Y-Y_1-Y_2) - t_2= & {} \frac{1}{L_2}c_2'(Y_2/L_2). \end{aligned}$$

(24)

Implicit differentiation of both expressions above w.r.t. $a_1$, and omitting arguments whenever this does not lead to confusion, leads to:

$$\begin{aligned} -\tau _m'\left( \frac{\partial Y_1}{\partial a_1} + \frac{\partial Y_2}{\partial a_1}\right)= & {} \frac{c_1'}{L_1} + \frac{c_1''}{L_1^2} \frac{\partial Y_1}{\partial a_1}, \end{aligned}$$

(25)

$$\begin{aligned} -\tau _m'\left( \frac{\partial Y_1}{\partial a_1} + \frac{\partial Y_2}{\partial a_1}\right)= & {} \frac{c_2''}{L_2^2} \frac{\partial Y_2}{\partial a_1}. \end{aligned}$$

(26)

Subtracting (26) from (25) yields:

$$\begin{aligned} \frac{\partial Y_1}{\partial a_1} = \frac{c_2''L_1^2}{c_1''L_2^2}\frac{\partial Y_2}{\partial a_1} - \frac{c_1'L_1}{c_1''}. \end{aligned}$$

Substituting this expression for $\partial Y_1/\partial a_1$ in (26) and solving for $\partial Y_2/\partial a_1$ leads to (19). And substituting the expression for $\partial Y_2/\partial a_1$ from (19) in (24) yields (20).

Finally, (21) and (22) follow from the envelope theorem. To see this, we write the taxpayer’s problem substituting the constraints into the objective function:

$$\begin{aligned}&\max \nolimits _{Y_{1s},Y_{2s}} [1-\tau (Y-Y_{1s}-Y_{2s}) ](Y-Y_{1s}-Y_{2s}) + (1-t_1)Y_{1s}\\&\quad +\,(1-t_2)Y_{2s} - a_1c_1(Y_{1s}/L_1) - c_2(Y_{2s}/L_2). \end{aligned}$$

This concludes the proof. $\square$

As stressed in Sect. 5, Result C5 establishes what may be viewed as “income” and “substitution” effects when a preferential tax regime—the PT regime in the case of this paper—becomes less attractive, either because $L_1$ decreases or because $a_1$ increases. In both cases total sheltered income decreases, which follows from (18) and (20). This is the income effect and is closely related to the fact that the agent is poorer. At the same time, the individual switches sheltered income from the regime that became less attractive to the one unaffected by the reform, as shown by (17) and (19). This is the substitution effect.

1.3 Appendix C.3: Proof of results in Sect. 5

Result 1 in Sect. 5 is a straightforward extension of Results C1, C2 and C3 to the case of two STRs. Result 2 follows from Result C5.

Appendix D: Statistics for assessing balance

In this section, details about the statistics proposed by Imbens and Rubin (2015) for assessing balance in covariates are discussed. The first one, normalized differences, is a scale-free way for measuring the difference in locations of the distributions. It is defined by

$$\begin{aligned} \mathrm{ND}_{tc}=\frac{\mu _t-\mu _c}{\sqrt{\left( \sigma ^2_t+\sigma ^2_c\right) /2}}, \end{aligned}$$

where t and c denote treatment and control groups, respectively, and $(\mu _i,\sigma ^2_i)$ are the population mean and variance of group i, for $i=t,c$, of a given variable X. This measure can be empirically implemented by

$$\begin{aligned} \widehat{\mathrm{ND}}_{tc}=\frac{\bar{X_t}-\bar{X_c}}{\sqrt{\left( s^2_t+s^2_c\right) /2}}, \end{aligned}$$

where $\bar{X_i}=\frac{1}{N_i}\sum _{j\in i}X_j$ and $s^2_i=\frac{1}{N_i-1}\sum _{j\in i}\left( X_j-\bar{X_i}\right) ^2$, with $N_i$ denoting the number of observations belonging to group i, for $i=t,c$. Imbens and Rubin (2015) suggest that $\widehat{\mathrm {ND_{tc}}}$ is better than the t-statistic for assessing differences in distributions. The central idea behind assessing balance is not to determine whether there is enough information about differences in covariate means, but to analyze whether or not differences are large enough to invalidate a posterior econometric application. The scale-free nature of the statistic is beneficial for that purposes.

For assessing differences in distributions’ dispersion, the authors propose the use of the logarithm of the ratio of standard deviations,

$$\begin{aligned} \Gamma _{tc}= & {} \ln \left( \frac{\sigma _t}{\sigma _c}\right) = \ln (\sigma _t)-\ln (\sigma _c), \end{aligned}$$

which can be empirically implemented by

$$\begin{aligned} {\hat{\Gamma }}_{tc}= & {} \ln (s_t)-\ln (s_c). \end{aligned}$$

Finally, the analysis can be complemented by calculating the fraction of treated and control observations whose covariate values are in the tails of the other group’s distribution. The idea is to determine whether the comparison between units of the different groups will rely too much on extrapolation. Fixing a confidence value $\alpha$, the probability mass that is outside the tails of the other group’s distribution is

$$\begin{aligned} \pi _i^\alpha= & {} \left( 1-F_i\left( F^{-1}_j(1-\alpha /2)\right) \right) +F_i\left( F^{-1}_j(\alpha /2)\right) , \end{aligned}$$

where $F_i$ is the cumulative distribution function for $i=t,c$ and j is the other group. With F unknown, this statistic can be empirically implemented using the empirical distribution functions

$$\begin{aligned} \hat{F_i}(x)= & {} \frac{1}{N_i}\sum _{j\in i}1_{X_j\le x}, \end{aligned}$$

where $1_{X_j\le x}$ is an indicator variable that takes value 1 if $X_j\le x$, and

$$\begin{aligned} {\hat{F}}^{-1}_i(q)= & {} \min _{-\infty<x<\infty }\left\{ x:{\hat{F}}_i(x)\ge q\right\} , \end{aligned}$$

for $i=t,c$. Then, fixing $\alpha =0.05$, statistics can be empirically implemented by

$$\begin{aligned} {\hat{\pi }}_i^{0.05}= & {} \left( 1-{\hat{F}}_i\left( {\hat{F}}^{-1}_j(0.975)\right) \right) +{\hat{F}}_i\left( {\hat{F}} ^{-1}_j(0.025)\right) . \end{aligned}$$

Appendix E: Previous trends: other variables

Appendix F: Robustness checks

See Tables 18, 19 and 20.

Table 18 Fixed-effects estimations

Table 19 Model in differences

Table 20 Difference-in-differences with longer pre-reform period: 2010–2013