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Research articles · Aug 31, 2020

Estimating Confidence Intervals in a Tax Microsimulation Model

Robert McClelland, Surachai Khitatrakun, Chenxi Lu

The Tax Policy Center, Washington, USA

Volume 13(2) Summer 2020

Cite as: R. McClelland, S. Khitatrakun, C. Lu; 2020; Estimating Confidence Intervals in a Tax Microsimulation Model; International Journal of Microsimulation; 13(2); 2–20. doi: 10.34196/ijm.00216

Abstract

Since the creation of microsimulation models, the need to measure uncertainty has been recognized, but little research has been conducted in spite of the widespread use of these models. In this article, we calculate confidence intervals for a large tax microsimulation model, comparing a normal approximation to a bootstrap estimator. We estimate confidence intervals for five proposed changes to tax law. We explore the relationship between the size of proposals’ point-estimated impacts and their confidence intervals by considering high, central, and low scenarios for three proposals. To explore uncertainty around small-magnitude estimates, we consider a proposal designed to keep tax revenue unchanged. Finally, we consider a proposal that affects a small number of taxpayers but has a large heterogeneous effect. We find that, overall, confidence intervals in the model fit tightly around the point estimates, but there are exceptions. We also find that in many cases the normal approximation is close to the bootstrap estimator but may differ for policy changes that affect a small number of taxpayers.

This article contains 8 tables.

Tables

Table 1. Population and Sample Size of TPC Microsimulation Model, 2019
Expanded Cash Income Level (thousands of 2019 dollars) Population (Tax Unit) Sample Size
Number (thousands) Percent of Total Number (thousands) Percent of Total
Less than 10 12,323 7.1 8 3.3
10–20 21,651 12.4 16 6.3
20–30 19,523 11.2 16 6.3
30–40 15,976 9.2 14 5.5
40–50 13,399 7.7 12 4.8
50–75 25,288 14.5 27 10.4
75–100 16,983 9.7 20 7.8
100–200 31,667 18.1 43 16.8
200–500 14,026 8.0 35 13.8
500–1,000 1,826 1.1 16 6.3
More than 1,000 837 0.5 40 15.7
All 174,690 100.0 256 100.0

Source: Urban-Brookings Tax Policy Center Microsimulation Model (version 0319-1).

Note: The sample exclude tax units with negative AGI or ECI.

Table 2. Partially Restore the Personal Exemption to $4,000, Selected Income Groups, 2019
Expanded Cash Income Level (thousands of 2019 dollars) Tax Units with Tax Cut Percent Change in After-Tax Income Average Federal Tax Change ($)
Percent of Tax Units Average Tax Change ($)
75–100   94.239 (0.314) −1,085.1 (4.5) 1.370 (0.008) −1,022.6 (5.7)
  [93.599, 94.843] [−1,094.0, −1,076.8] [1.351, 1.383] [−1,032.4, −1,009.9]
200–500   98.470 (0.080) −2,496.9 (10.7) 1.077 (0.006) −2,458.6 (10.9)
  [98.296, 98.615] [−2,515.1, −2,475.3] [1.065, 1.088] [−2,479.3, −2,435.9]
More than 1,000   0.620 (0.176) −1,214.3 (242.8) 0.000 (0.000) −7.5 (2.6)
  [0.334, 1.054] [−1,680.9, −789.1] [0.000, 0.001] [−13.2, −2.9]
All   62.332 (0.131) −1,216.4 (2.8) 0.933 (0.004) −758.2 (1.7)
  [62.033, 62.542] [−1,220.9, −1,210.7] [0.927, 0.943] [−761.6, −754.6]
Addendum: Confidence Intervals Calculated Using Bootstrap and Using Normal Approximation of BSE
75–100 Bootstrap [93.599, 94.843] [−1,094.0, −1,076.8] [1.351, 1.383] [−1,032.4, −1,009.9]
Normal with BSE [93.624, 94.854] [−1,093.9, −1,076.2] [1.354, 1.386] [−1,033.7, −1,011.4]
200–500 Bootstrap [98.296, 98.615] [−2,515.1, −2,475.3] [1.065, 1.088] [−2,479.3, −2,435.9]
Normal with BSE [98.313, 98.628] [−2,517.9, −2,475.9] [1.065, 1.089] [−2,479.9, −2,437.4]
More than 1,000 Bootstrap [0.334, 1.054] [−1,680.9, −789.1] [0.000, 0.001] [−13.2, −2.9]
Normal with BSE [0.275, 0.966] [−1,690.2, −738.4] [0.000, 0.001] [−12.6, −2.4]
All Bootstrap [62.033, 62.542] [−1,220.9, −1,210.7] [0.927, 0.943] [−761.6, −754.6]
Normal with BSE [62.075, 62.589] [−1,221.8, −1,210.9] [0.926, 0.941] [−761.6, −754.8]

Source: Urban-Brookings Tax Policy Center Microsimulation Model (version 0319-1).

Notes: Proposal would partially restore the personal exemption of $4,000 in 2019.

Numbers in parentheses are bootstrapped standard errors (BSE). Numbers in brackets are confidence intervals.

Table 3. Comparison of Population and Sample Size Affected by Partially Restoring the Personal Exemption to $4,000, Selected Income Groups, 2019
Expanded Cash Income Level (thousands of 2019 dollars) All Tax Units Tax Units with Tax Change Tax Change ($)
Population Number Sample Number Population Sample Minimum Maximum
Number Share within Income Group Number Share within Income Group
200–500 14,026,200 35,400 13,815,800 98.50% 32,220 91.00% 14 11,000
More than 1,000 836,500 40,100 5,190 0.62% 71 0.18% 15 3,840

Source: Urban-Brookings Tax Policy Center Microsimulation Model (version 0319-1).

Table 4. Increase the Standard Deduction to $4,000, Selected Income Groups, 2019
Expanded Cash Income Level (thousands of 2019 dollars) Tax Units with Tax Cut Percent Change in After-Tax Income Average Federal Tax Change ($)
Percent of Tax Units Average Tax Change ($)
50–75   79.993 (0.399) −625.8 (2.4) 0.921 (0.006) −500.6 (2.9)
  [79.140, 80.676] [−629.7, −620.9] [0.909, 0.931] [−505.9, −494.4]
200–500   79.289 (0.329) −1,621.6 (5.3) 0.563 (0.003) −1,285.7 (7.0)
  [78.622, 79.919] [−1,631.7, −1,612.3] [0.556, 0.569] [−1,296.7, −1,268.5]
More than 1,000   23.048 (0.548) −2,130.7 (26.4) 0.023 (0.001) −491.1 (13.4)
  [22.019, 24.250] [−2,180.3, −2,077.4] [0.021, 0.025] [−521.0, −466.6]
All   58.636 (0.141) −889.3 (1.8) 0.642 (0.003) −521.4 (1.2)
  [58.391, 58.910] [−893.3, −886.0] [0.638, 0.648] [−523.8, −519.0]
Addendum: Confidence Intervals Calculated Using Bootstrap and Using Normal Approximation of BSE
50–75 Bootstrap [79.140, 80.676] [−629.7, −620.9] [0.909, 0.931] [−505.9, −494.4]
Normal with BSE [79.211, 80.775] [−630.4, −621.2] [0.910, 0.932] [−506.3, −494.9]
200–500 Bootstrap [78.622, 79.919] [−1,631.7, −1,612.3] [0.556, 0.569] [−1,296.7, −1,268.5]
Normal with BSE [78.645, 79.934] [−1,632.0, −1,611.1] [0.556, 0.570] [−1,299.5, −1,271.9]
More than 1,000 Bootstrap [22.019, 24.250] [−2,180.3, −2,077.4] [0.021, 0.025] [−521.0, −466.6]
Normal with BSE [21.974, 24.122] [−2,182.5, −2,079.0] [0.021, 0.024] [−517.3, −464.9]
All Bootstrap [58.391, 58.910] [−893.3, −886.0] [0.638, 0.648] [−523.8, −519.0]
Normal with BSE [58.361, 58.912] [−892.7, −885.8] [0.637, 0.647] [−523.8, −519.1]

Source: Urban-Brookings Tax Policy Center Microsimulation Model (version 0319-1).

Notes: Proposal would increase the standard deduction by $4,000 for single filers and married couples filing separately, $8,000 for married couples filing jointly, and $6,000 for heads of household in 2019.

Numbers in parentheses are bootstrapped standard errors (BSE). Numbers in brackets are confidence intervals.

Table 5. Increase the Top Income Tax Rate to 41 Percent, Selected Income Groups, 2019
Expanded Cash Income Level (thousands of 2019 dollars) Tax Units with Tax Increase Percent Change in After-Tax Income Average Federal Tax Change ($)
Percent of Tax Units Average Tax Change ($)
200–500   0.062 (0.014) 2,044.6 (444.8) −0.001 (0.000) 1.3 (0.3)
  [0.038, 0.091] [1,349.4, 2,956.2] [−0.001, 0.000] [0.8, 1.9]
500–1,000   18.225 (0.530) 4,127.8 (111.2) −0.147 (0.006) 752.3 (29.8)
  [17.236, 19.150] [3,909.4, 4,363.9] [−0.156, −0.135] [691.6, 800.8]
More than 1,000   77.965 (0.571) 52,878.8 (1,006.8) −1.917 (0.049) 41,226.8 (808.0)
  [76.552, 78.925] [51,126.9, 54,963.1] [−1.996, −1.827] [39,944.2, 43,177.1]
All   0.569 (0.006) 36,127.7 (759.5) −0.253 (0.005) 205.7 (3.9)
  [0.555, 0.581] [34,865.4, 37,852.8] [−0.264, −0.246] [199.5, 214.6]
Addendum: Confidence Intervals Calculated Using Bootstrap and Using Normal Approximation of BSE
200–500 Bootstrap [0.038, 0.091] [1,349.4, 2,956.2] [−0.001, 0.000] [0.8, 1.9]
Normal with BSE [0.034, 0.089] [1,172.8, 2,916.3] [−0.001, 0.000] [0.7, 1.9]
500–1,000 Bootstrap [17.236, 19.150] [3,909.4, 4,363.9] [−0.156, −0.135] [691.6, 800.8]
Normal with BSE [17.186, 19.264] [3,909.9, 4,345.6] [−0.158, −0.135] [693.9, 810.6]
More than 1,000 Bootstrap [76.552, 78.925] [51,126.9, 54,963.1] [−1.996, −1.827] [39,944.2, 43,177.1]
Normal with BSE [76.846, 79.084] [50,905.5, 54,852.0] [−2.013, −1.822] [39,643.2, 42,810.5]
All Bootstrap [0.555, 0.581] [34,865.4, 37,852.8] [−0.264, −0.246] [199.5, 214.6]
Normal with BSE [0.557, 0.582] [34,639.1, 37,616.4] [−0.262, −0.244] [198.1, 213.2]

Source: Urban-Brookings Tax Policy Center Microsimulation Model (version 0319-1).

Notes: Proposal would increase the top income tax rate to 41 percent in 2019.

Numbers in parentheses are bootstrapped standard errors (BSE). Numbers in brackets are confidence intervals.

Table 6. Replace Farm Loss Deduction with a Revenue-Neutral Refundable Tax Credit, Selected Income Groups, 2019
Expanded Cash Income Level (thousands of 2019 dollars) Tax Units with Tax Cut Tax Units with Tax Increase Average Federal Tax Change ($)
Percent of Tax Units Average Tax Change ($) Percent of Tax Units Average Tax Change ($)
40–50   0.109 (0.029) −1,368.9 (537.1) 0.267 (0.065) 404.1 (101.8) −0.4 (0.7)
  [0.062, 0.180] [−2,743.8, −685.0] [0.166, 0.462] [234.1, 614.2] [−2.0, 1.0]
500–1,000   0.531 (0.109) −3,465.7 (1,568.9) 1.338 (0.170) 2,779.1 (400.8) 18.8 (9.1)
  [0.386, 0.812] [−8,022.0, −1,416.6] [0.990, 1.652] [2,082.2, 3,748.6] [−2.7, 32.6]
More than 1,000   0.994 (0.077) −15,499.8 (4,948.4) 0.922 (0.078) 7,473.2 (787.6) −85.1 (50.9)
  [0.852, 1.171] [−26,531.3, −7,571.9] [0.781, 1.073] [6,012.7, 9,182.7] [−176.3, −3.0]
All   0.207 (0.013) −1,909.9 (221.3) 0.485 (0.020) 770.2 (47.8) −0.2 (0.5)
  [0.186, 0.230] [−2,420.9, −1,566.6] [0.450, 0.522] [700.7, 884.2] [−1.3, 0.8]
Addendum: Confidence Intervals Calculated Using Bootstrap and Using Normal Approximation of BSE
40–50 Bootstrap [0.062, 0.180] [−2,743.8, −685.0] [0.166, 0.462] [234.1, 614.2] [−2.0, 1.0]
Normal with BSE [0.052, 0.166] [−2,421.6, −316.2] [0.140, 0.394] [204.6, 603.6] [−1.8, 1.0]
500–1,000 Bootstrap [0.386, 0.812] [−8,022.0, −1,416.6] [0.990, 1.652] [2,082.2, 3,748.6] [−2.7, 32.6]
Normal with BSE [0.318, 0.744] [−6,540.8, −390.7] [1.005, 1.670] [1,993.5, 3,564.6] [1.0, 36.5]
More than 1,000 Bootstrap [0.852, 1.171] [−26,531.3, −7,571.9] [0.781, 1.073] [6,012.7, 9,182.7] [−176.3, −3.0]
Normal with BSE [0.843, 1.145] [−25,198.5, −5,801.0] [0.769, 1.076] [5,929.6, 9,016.9] [−184.8, 14.6]
All Bootstrap [0.186, 0.230] [−2,420.9, −1,566.6] [0.450, 0.522] [700.7, 884.2] [−1.3, 0.8]
Normal with BSE [0.182, 0.232] [−2,343.6, −1,476.2] [0.445, 0.524] [676.5, 863.9] [−1.3, 0.8]

Source: Urban-Brookings Tax Policy Center Microsimulation Model (version 0319-1).

Notes: Proposal would replace current-law farm loss deduction with a refundable tax credit equal to 82.5% of total farm loss times the statutory income tax rate in 2019.

Numbers in parentheses are bootstrapped standard errors (BSE). Numbers in brackets are confidence intervals.

Table 7. Rescind the Tax-free Treatment of Interest Accruing from Certain Municipal Bonds, Selected Income Groups, 2019
Expanded Cash Income Level (thousands of 2019 dollars) Tax Units with Tax Increase Percent Change in After-Tax Income Average Federal Tax Change ($)
Percent of Tax Units Average Tax Change ($)
40–50   0.598 (0.105) 382.8 (142.4) −0.006 (0.002) 2.3 (0.9)
  [0.412, 0.830] [174.3, 815.8] [−0.011, −0.002] [1.0, 4.5]
200–500   10.871 (0.289) 2,693.2 (124.2) −0.128 (0.006) 292.8 (14.4)
  [10.348, 11.518] [2,442.2, 2,965.2] [−0.141, −0.116] [263.7, 323.0]
More than 1,000   50.162 (0.535) 27,001.5 (1,425.9) −0.630 (0.033) 13,544.6 (729.8)
  [49.325, 51.238] [24,610.9, 30,036.2] [−0.697, −0.576] [12,302.0, 15,143.7]
All   2.924 (0.050) 4,272.1 (140.4) −0.154 (0.005) 124.9 (3.9)
  [2.826, 3.014] [4,021.8, 4,613.3] [−0.164, −0.145] [118.0, 133.8]
Addendum: Confidence Intervals Calculated Using Bootstrap and Using Normal Approximation of BSE
40–50 Bootstrap [0.412, 0.830] [174.3, 815.8] [−0.011, −0.002] [1.0, 4.5]
Normal with BSE [0.393, 0.803] [103.8, 661.9] [−0.010, −0.001] [0.6, 4.0]
200–500 Bootstrap [10.348, 11.518] [2,442.2, 2,965.2] [−0.141, −0.116] [263.7, 323.0]
Normal with BSE [10.305, 11.437] [2,449.8, 2,936.6] [−0.141, −0.116] [264.6, 321.1]
More than 1,000 Bootstrap [49.325, 51.238] [24,610.9, 30,036.2] [−0.697, −0.576] [12,302.0, 15,143.7]
Normal with BSE [49.113, 51.212] [24,206.9, 29,796.1] [−0.695, −0.565] [12,114.2, 14,975.1]
All Bootstrap [2.826, 3.014] [4,021.8, 4,613.3] [−0.164, −0.145] [118.0, 133.8]
Normal with BSE [2.825, 3.022] [3,996.9, 4,547.4] [−0.163, −0.144] [117.3, 132.5]

Source: Urban-Brookings Tax Policy Center Microsimulation Model (version 0319-1).

Notes: Proposal would rescind the tax-free treatment of interest accruing from certain municipal bonds in 2019.

Numbers in parentheses are bootstrapped standard errors (BSE). Numbers in brackets are confidence intervals.

Table 8. Comparison of Confidence Intervals with Standard Errors Calculated from Bootstrapping and the Gatz and Smith Formula
Proposal Average Tax Change for All Tax Units ($)
Mean Standard Error Confidence Interval
BSE GSSE Bootstrap Normal Approximation with GSSE
Restore Personal Exemption −758.2 1.7 1.9 [−761.6, −754.6] [−761.9, −754.4]
Increase Standard Deduction −521.4 1.2 1.2 [−523.8, −519.0] [−523.8, −519.1]
Increase Tax Rate 205.7 3.9 19.3 [199.5, 214.6] [167.9, 243.4]
Rescind Tax-free Treatment of Certain Bonds 124.9 3.9 11.2 [118.0, 133.8] [102.9, 146.9]
Farm Loss Deduction Replaced with Credit −0.2 0.5 2.2 [−1.3, 0.8] [−4.5, 4.1]

Source: Urban-Brookings Tax Policy Center Microsimulation Model (version 0319-1).

Notes: The BSE column shows bootstrapped standard errors. The GSSE columns shows the Gatz and Smith formula for calculating standard errors. The Bootstrap Confidence Interval column shows the bias-corrected bootstrap confidence intervals. Finally, the Normal Approximation with GSSE Confidence Interval column shows confidence intervals derived as the means plus or minus the GSSE multiplied by 1.96. Due to rounding, that interval differs slightly from an interval calculated using the values in the table.