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About below the text content, author: Wang Yu Zhou, econometrics University of Wisconsin-Madison, communication mail: [email protected]
Today, we mainly introduce regression Kink design (inflection point regression design), which is closely related to regression discontinuity design (breakpoint regression design).
Regarding the design of breakpoint regression, you can refer to: 1. Breakpoint regression design RDD classification and operation cases, 2. RDD breakpoint regression, an encyclopedia of Stata programming, 3. Current status of cutting-edge research on breakpoint regression design, RDD, 4. What the hell is the breakpoint regression design? Let's listen to the analysis of Harvard guest, 5. Breakpoint regression and readers' questions and answers, 6. Comprehensive explanation of breakpoint regression design RDD, there are many users in the education field, 7. There are no instrumental variables, breakpoints and random impacts, and attribution can be inferred. , 8. What should I do if I can’t find IV, RD and DID? This is an alternative method, Volume 9.2 RDD Breakpoint Regression Manual, including Stata and R software operating procedures, 10. DID, synthesis control, matching, RDD four Comparison of methods, scope and characteristics, 11. Anshen + Clark Prize winner’s RDD paper, breakpoint regression design, 12. Is the Islamic government unfriendly to women friends? RDD classic literature, 13. PSM, RDD, Heckman, Panel model operating procedures, 14. RDD classic literature, RDD model validity and robustness test, 15. Interesting articles published on JDE in 2019, the latest trends in measurement methods
1. Empirical analysis Tools: Overview of inflection point regression design The
breakpoint regression model tool RDD has become a supplement to the current popular impact assessment tools. It can show the impact of the research object near the critical value in a way that has a prominent visual effect. As an extension of this method, regression to inflection point design (RKD) has become more widely used. I have never used these methods for estimation, nor am I an expert in this area, but I think that introducing this method and providing some links to articles for subsequent reference may be very helpful to those who need to use this method.
basic concept
Breakpoint regression design uses jumps or discontinuities at the critical point of the possibility of intervention. In the inflection point regression design, the slope of the possibility of being interfered changes at the inflection point, which leads to the discontinuity of the first derivative of the assignment function. This kind of turning point is not uncommon in many policies. For example, Simonsen et al. (2015) used the inflection point method in the Danish government's prescription drug reimbursement schedule: the subsidy is based on the total prescription cost paid by the individual in the year; the first 500 Danish kroner is 0%, and the payment is 500 kronor to 1200 kronor. A subsidy of 50% in between, followed by a subsidy of 75%, and an 80% subsidy when the final payment exceeds DKK 2,800. The result is that the price share paid by pocket money is shown in Figure 1:
Figure 1: The Y-axis is the actual price share paid. The decrease is close to 500 Danish kronor, because if you spend 480 and buy something for 50 kronor, the 20 kroner part of the subsidy you pay is 0%, and the 30 kroner part that exceeds the threshold is calculated as 50%.
Second An example is the government's unemployment insurance payment, which can usually cover a certain percentage of the previous salary under the maximum limit, and sometimes has a lower limit. For example, in a recent paper by Landais, Figure 2 plots the unemployment benefits in Louisiana as a function of the highest quarterly income:
Afterwards, it is studied whether the kink/slope change of the exploration result variable with respect to the operating variable occurs at the same point. For example, Landais is interested in the relationship between the amount of time people spend in unemployment and the amount of benefits they receive. Figure 3 below shows that at this inflection point, the slope of the function of unemployment time with respect to the highest quarterly income has changed a lot at the inflection point.
Figure 3: The inflection point of the outcome variable and the intervention variable are at the same place.
Then the causal effect can be obtained by dividing the slope change of the outcome variable by the slope change of the intervention variable. When there is an error in the implementation of policy rules, fuzzy RKD design can be used (see Card et al. (2015)). The
estimation
can be estimated by using polynomial regression. Let x be the operating variable (for example, the highest income in the previous quarter), Y be the outcome variable (for example, the time of unemployment); k is the turning point, and D is a dummy variable located on the right side of the turning point. Then estimate:
Y = a0 + a1 (xk) + b1 D (xk) + a2 (xk)^ 2 + a3 D (xk)^ 2 +...
where b1 shows the change in the slope of the result variable at the inflection point with respect to the operating variable.
Run the same regression to obtain the slope change of the intervention variable with respect to the operating variable at the inflection point, and take the ratio of the two coefficients.
The standard deviation can be obtained by the Delta method. Estimation can also be done by non-parametric local polynomials.
The key hypothesis and the hypothesis for testing the
inflection point regression are similar to the breakpoint regression and are tested in a similar manner:
1. The slopes of other covariates at the inflection point should not change (for example, Landais shows the age, education level, capital at the inflection point) There is no slope change in income or the number of dependants at the turning point)
2. There is no manipulation of the assignment variable at the turning point-for example, in the case of unemployment, people will not strategically conceal past income because of their impact on future unemployment benefits. This can be tested by the McCrary-type test, which shows that the distribution of the assigned variable is continuous at the inflection point, and the first derivative of the probability density distribution function is also very smooth. Card et al. (2015) showed that as long as the error or error in the optimization of the sample object is so small that the sorting is not completely certain, you can allow some non-random selection of the object around the inflection point.
3. This article suggests that you can perform some placebo tests at non-inflection points.
Technical details can be found in Econometrica paper by Card et al. (2015).
Matters needing attention
RKD design may perform poorly in small samples, and usually requires larger bandwidth or larger sample size than RDD.
There are some methods to effectively select the best bandwidth, but papers that cannot demonstrate that this option is the best compared to other potential options or cannot verify its robustness will face criticism. For examples, see the comments on this AEJ Economic Policy Paper.
2. An empirical example using inflection point regression design: Evaluating the welfare impact of unemployment
is similar to breakpoint regression (RDD). Inflection point regression (RKD) has attracted the attention of today's scholars for its outstanding visual effects and controllability of changes in market factors. , Especially suitable for the study of the causal effects related to welfare policies. But at the same time, the model also has certain limitations, and has certain requirements for the relationship between the assignment variable and the covariate, and the sample:
-
First, you need to find suitable assignment variables (for example, the insurance amount and income are linearly related and have upper and lower limits), so as to be able to define the inflection point;
-
Except for the target variable, the slope of the covariate near the inflection point should not change significantly (such as age, education level, etc.);
- The distribution of the assigned variable at the inflection point should be continuous and first-order smooth, which means that the sample will not manipulate the assignment change at the inflection point.
(E.g. people will not under-report their income just because they have reached the maximum limit);
-
RKD is greatly affected by the limitations of small samples and requires a larger bandwidth than RDD;
- RKD can only estimate the causal effect near the inflection point, and cannot simply be generalized to the overall.
Landais (2015) uses a clear inflection point regression model to study the impact of changes in the amount and duration of unemployment insurance policies on labor supply. The author believes that most of the current data on unemployment insurance research comes from the changes in laws and regulations in various states based on time changes, so it is difficult to solve the endogenous problem of labor market conditions, and it is difficult to control the changes in market fluctuations over time. The method used in this article can effectively circumvent this problem, that is, the market factors can be considered almost unchanged in the cell at the turning point of RKD design, and there is no need for the assumptions of traditional regression models. Researchers can use the RK method to identify the behavioral response of the same worker to changes in the amount and duration of insurance premiums under a constant market environment. The well-known RDD model uses the discontinuity of the assignment rule at the breakpoint, while RKD focuses on the change in the relationship between the policy-related variable and the assignment variable near the inflection point, that is to say, it is no longer the intercept but the slope change. estimate.
Landais (2015) data comes from the Continuous Wage and Benefit History Project (CWBH), which contains unemployment records in five states in the United States from the late 1970s to 1984. The RKD design in this article is based on the issuance characteristics of the unemployment insurance UI. First of all, the UI collection time must not exceed its upper limit, which is generally 26 weeks. At the same time, the total amount must not exceed the upper limit amount determined by the highest quarterly income. Figure 2 in the text shows the weekly unemployment benefit in Louisiana as a function of the highest quarterly income:
Figure 3 shows the potential duration as a function of the highest quarterly income: in the
above two figures, we can see the obvious inflection point, and the inflection point varies with the policies of different years or different projects. Next, Landais observed that the variables he was interested in, such as the duration of unemployment, showed a slope change at both ends of the same inflection point, as shown in the following figure:
The author establishes the RKD model based on this: the short-term insurance amount is equal to the highest quarterly income multiplied by a constant (linear change) when the short-term insurance amount is lower than the upper limit Bmax, and the analysis is performed based on whether it is higher than the upper limit; similarly, in the income time The author also analyzes whether it reaches the upper limit Dmax as an inflection point. After linear regression is performed on both sides of the inflection point, the slopes at both ends are divided to obtain the causal effect ATET of each state.
Landais found through descriptive statistics that the degree of tolerance of UI varies greatly from state to state, but the overall structure has hardly changed in the 30 years of the data set. This means that the inflection point of the current income distribution is close to the data coverage period, which brings convenience to the study of the current UI optimal policy. However, if you want to use the empirical model as a reference standard for UI policy formulation, you still need to analyze and test the effectiveness of the model.
The design of sharp RK in this article is based on two basic assumptions: First, the direct marginal effect of the assigned variable on the outcome variable should be smooth; second, the density of heterogeneity at the inflection point should also be changed with the assigned variable. Smooth. The author believes that because few people pay attention to the UI schedule when in office, the hypothesis is credible. If the UI occupancy rate leads to non-random allocation, then the effective type of RKD will also be affected. CWBH provided evidence that the lack of occupancy rate is mainly caused by insufficient information, so the second hypothesis can also be supported. However, in addition to this, we must be able to keep abreast of the UI situation of the respondent before and after being dismissed and keep abreast of his resignation date in time.
The author uses a variety of methods to carry out auxiliary analysis to verify the effectiveness of RKD. First, it uses the McCrary method to test the discontinuity of the probability density function and test the continuity of the first derivative. Then the image shows the smoothness of the covariate at the inflection point.
The sample size has no breakpoint at the dividing point of the assignment variable, while the covariates such as age, education years, and the number of dependent variables are smoothed at the inflection point, which conforms to the basic hypothesis of RKD. Subsequently, the author conducted multiple tests on the robustness of RKD. First, the estimated values are not significantly different under the assumptions of linear, quadratic and cubic polynomials. In response to the correlation between the outcome variable and the forcing variable mentioned above, that is, the RKD estimate may only be caused by the correlation between the highest quarterly income and the outcome variable and has nothing to do with the improvement of UI tolerance, Landais Double difference RKD analysis and placebo analysis were performed. Double-difference analysis uses the difference in the position of the inflection point in different periods. As can be seen from the figure below, the sample data in 1979 did not change significantly at the inflection point in 1982, and the data in 1982 did not change significantly at the inflection point in 1979, while the slope of the fit between the two inflection points changed significantly in two years. .
In the placebo analysis, after the author replaced the outcome variable in the main analysis with a "false result" that would not be affected by treatment, no significant results in the main analysis were observed. Both test methods verify the robustness of RKD in this study.
In terms of unemployment insurance payment period, the author mainly focuses on three aspects: payment period, claimed unemployment period, and initial unemployment period. The following figure shows the estimation results of the impact of the UI amount in Louisiana on the three unemployment periods in different periods using the RKD method. Estimates show that Ɛb is an estimate of the elasticity of the allowance amount during the unemployment period, and its estimated value depends on different periods in the range of 0.2 to 0.7. The estimation of the regression coefficient shows that a 10% increase in the weekly allowance will lead to a 2-7% increase in the duration of unemployment. This value is consistent for the three different term calculation methods.
After performing RKD estimates for five states and different periods, the author found that the estimated elasticity of the unemployment period relative to the level of welfare is always between 0.1 and 0.7, while the average initial duration of a total of 26 estimates in all 5 states and different periods is between 0.1 and 0.7. The time flexibility is 0.32. In addition, estimates for Louisiana also show that unemployment will increase by 0.2 to 0.4 weeks for every additional week of UI duration, which is consistent with previous estimates of other academic studies. Finally, the author made a separate RKD estimate for Washington (because only Washington can obtain non-employment time information by matching UI and salary records) calculating the proportion of liquidity and moral impact is 88%, which is smaller than the result of Chetty (2008).
In general, RKD has great advantages in UI policy research. Since this method only relies on the time change of returns and the individual's first-order conditions, it is also generally applicable to other projects whose returns change over time. In addition, the method used in this article helps government agencies to find the optimal strategy more quickly in the face of minor insurance policy adjustments. However, in the equilibrium search and matching model of the labor market, only the response of labor supply to UI is not enough to calculate the best trade-off between insurance and moral hazard, and the change in employment equilibrium needs to be estimated. Therefore, a more accurate and clear estimation of the two different effects and weighing them is one of the valuable research directions in the future.
Long press the above QR code to read the original English PDF
Reference：Landais, Camille. 2015. "Assessing the Welfare Effects of Unemployment Benefits Using the Regression Kink Design." American Economic Journal: Economic Policy, 7 (4): 243-78.**
Source: https://blogs.worldbank.org/impactevaluations/tools-trade-regression-kink-design
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