Office of Tax Analysis

Technical Paper 6

June 2015

Re-weighting to Produce State-Level Tax 

Microsimulation Estimates

Robin Fisher and Emily Y. Lin

OTA Technical Papers is an occasional series of reports on the research, models and datasets 

developed to inform and improve Treasury’s tax policy analysis. The papers are works in 

progress and subject to revision. Views and opinions expressed are those of the authors and do 

not necessarily represent official Treasury positions or policy. OTA Technical Papers are 

distributed in order to document OTA analytic methods and data and invite discussion and 

suggestions for revision and improvement. Comments are welcome and should be directed to the 

authors. OTA Papers may be quoted without additional permission.
1

RE-WEIGHTING TO PRODUCE STATE-LEVEL TAX 

MICROSIMULATION ESTIMATES

June 3, 2015

Robin Fisher and Emily Y. Lin

Revenue and distributional effects of a Federal tax law change can vary widely across 

states and the results are of policy relevance. A number of data files might be used for 

state-level tax simulation but each file has its deficiencies. For example, the Treasury’s 

Individual Tax Model (ITM) is used to simulate the revenue and distributional impact but 

its sample is not representative at the state level. Hence, while the file includes a variable 

for state of residency, the sampling errors of state-level statistics based on this variable are 

likely to be unacceptably large. The Internal Revenue Service’s Individual Returns 

Transaction File (IRTF) contains the population of individual income tax returns filed for a 

tax year so its state-level statistics are free of sampling errors. However, the file is 

unedited, contains random data errors, and more importantly, without substantial edits and 

modeling, lacks any capacity to project a tax law’s effect over the budget window.

In this paper, we use a statistical method to impute the information about each state’s tax 

return filers (e.g., number of filers and the distribution of their income, filing status, and 

number of dependents) from the population IRTF to the ITM. To the extent that the effect of 

a tax law change differs across states due to states’ differences in the size of the filing 

population and other tax characteristics, imputing this information to the ITM provides key 

identification for state-level tax effects. Specifically, we calculate the distributions of filers 

across states—unconditional and conditional on a set of tax variables—from the IRTF, and 

impute 52 state weights to each ITM record to generate filer distributions that resemble 

those calculated from the population file. 

By re-weighting ITM records, our method facilitates a straightforward way to estimate 

state-level revenue and distributional effects of Federal tax law changes because the same 

ITM calculator developed for analyzing the national effect can be used for state-level 

microsimulation. We also propose a method to evaluate the model fit of the state weight 

estimation and apply the state weights to an ITM run to produce state-by-state effects of 

repealing the alternative minimum tax (AMT). Lastly, we note that the re-weighting 

technique can be used to address other instances in which the sample in the ITM is not 

sufficiently representative of a specific population of interest.

Robin Fisher: Office of Tax Analysis, U.S. Department of the Treasury, Washington, DC 

(Robin.Fisher@treasury.gov)

Emily Y. Lin: Office of Tax Analysis, U.S. Department of the Treasury, Washington, DC 

(Emily.Lin@treasury.gov)
2

1. Introduction 

Revenue and distributional effects of a proposed or enacted change in Federal tax 

law can vary widely across states. If every state had the same distribution of taxpayer 

characteristics that impact tax liabilities (such as income, filing status, and number of 

dependents) and all other factors for the Federal tax effect were identical across states, a 

Federal tax law change would result in a proportionally larger change in the Federal tax 

liability for larger states than for smaller states. To the extent that taxpayer characteristics 

and other factors differ across states, changes in Federal tax liabilities resulting from a 

Federal tax change will vary by state for reasons other than differences in state population 

size. This geographic distribution of Federal tax liability changes can play an important 

role in lawmakers’ decision-making. Despite the policy relevance, due mainly to lack of 

appropriate data, analyses that show effects of Federal tax law changes by state are scarcely 

available. Using tax information derived from the population file of Federal individual 

income tax returns, we impute state weights to the Treasury’s Individual Tax Model (ITM) 

to facilitate state-by-state tax microsimulation analysis. In particular, we re-weight the ITM 

records to create 52 (for 50 states, the District of Columbia, and other areas) weighted 

samples that resemble the filing population of each state along certain important tax 

characteristics. 

A number of existing data can potentially be used to provide state-level tax analysis 

but each data has its own deficiencies. The Treasury Department uses the Individual Tax 

Model (ITM) to simulate the revenue and distributional impact of Federal tax law changes.

The sample of tax return filers in the ITM is based on the Individual and Sole 

Proprietorship (INSOLE) returns drawn by the Statistics of Income (SOI) Division of the 

Internal Revenue Service (IRS), and the data are extrapolated and modeled by the Treasury 

Department’s Office of Tax Analysis (OTA) to provide microsimulation estimates of the 

effects of tax changes over the 10-year period of the budget window. Although tax return 

data include a variable indicating a taxpayer’s state of residency, the INSOLE sample is not 

randomly drawn across states and is not intended to be representative at the state level. 

Hence, while state-level tax statistics can be made available on the ITM based on the 

taxpayer’s state of residency shown on the return, the sampling errors of some of these 
3

statistics are likely to be unacceptably large, substantially reducing the precision of the 

point estimates. 

Alternatively, state-level tax statistics can be directly obtained from the IRS’ 

Individual Returns Transaction File (IRTF), which contains the population of Federal 

individual income tax returns processed by the IRS each year. The advantage of using the 

population file is elimination of sampling errors because it includes all returns, rather than a 

sample of returns, filed for a tax year. However, unlike the extensively edited INSOLE file, 

this transaction file is unedited and is known to include random data errors. More 

importantly, without any of the extrapolation, imputation, and modeling of the data, as in 

the ITM, the file provides historical statistics by state but has a limited capacity to project 

the state-by-state effect of a Federal tax law change that will occur during the budget 

window.

Another method in common use is to treat the states as the unit of measurement and 

use a model together with auxiliary information to find estimators with improved 

characteristics. Well-known examples include the Small Area Income and Poverty 

Estimates (SAIPE) and Small Area Health Insurance Estimates (SAHIE) by the Bureau of 

the Census.

1 Henry et al. (2007) applied the same method to the current situation. While 

these methods can produce state-level estimates with good properties, they are laborintensive and can produce estimates only for a small set of variables, since each variable 

for which small-area estimates are desired must be explicitly included in the modeling. It is 

also not easy to see how to use these estimates for microsimulation problems like those 

encountered in the OTA; in general, the microsimulation would necessarily be run first, and 

then the small area model parameters would be estimated. 

One way to form state-level microsimulation estimates is to use the ITM for its 

edited data, extrapolation, imputation and modeling advantage, coupled with the 

information obtained from the IRTF about the filing population for each state. From the 

population IRTF, we know the unconditional distribution of filers across states (e.g., X 

 1 Information about SAIPE and SAHIE estimates can be found at http://www.census.gov/did/www/saipe/

and http://www.census.gov/did/www/sahie/. 
4

percent of all filers live in state Y) and the distribution of filers conditional on observable 

tax characteristics, such as filing status, income classes, or number of dependents (e.g., Z 

percent of the nation’s married-file-jointly couples who have income below $50,000 and 

two children live in state Y). To the extent that the effect of a proposal or law change on 

tax liability differs by the tax characteristics but is expected to be uniform among families 

with the same set of characteristics, we could use the ITM to estimate the projected total 

liability effect for each “cell” of family type partitioned by the selected characteristics, and 

then allocate the cell’s aggregate effect across states based on the conditional distribution of 

filers we tabulate from the population file. By modeling on the ITM while utilizing 

geographic distributions of filers and their tax characteristics from the population file, this 

method addresses the issues that the ITM sub-samples are not representative of state 

populations and that the population file lacks certain modeling capacity. 

Consider a tax cut for which the generosity increases with the tax unit’s number of 

exemptions and decreases with the tax unit’s adjusted gross income (AGI). We can model 

the tax cut on the ITM to calculate the amount of tax change for each family type, defined 

by the number of exemptions and AGI category. Note that the amount of tax cut varies by 

family type but is the same magnitude for tax units of the same type. From the ITM, we 

know the weighted count of tax units for each family type. From the IRTF, we know the 

number of tax units of each family type by state. With this information, we use the ITM to 

calculate the aggregate tax liability change for a family type and allocate the amount across 

states based on the conditional distribution of tax units across states we obtained from the 

IRTF tabulation. We then sum up the results across family types within a state to arrive at 

the state’s total liability change.

In this paper, we use a statistical method to bring over the information about the 

distribution of tax units across states—unconditional as well as conditional on observed tax 

characteristics—from the IRTF to the ITM by splitting the weight of each ITM record into 

52 state weights. The goal is to make the 52 re-weighted ITM samples to generate the 

conditional and unconditional distributions of tax units that resemble those calculated from 

the population file. The statistical method employed in this paper is closely related to that 
5

described in Schirm et al. (1999) and Schirm and Zaslavsky (1998), which is an elaboration 

of the informal tabulation described in the previous paragraph. These two papers describe 

the utility of sample weight adjustments, especially in the microsimulation setting.

The new state weights generate 52 state samples in the ITM. Because we expect the 

liability effect of a tax change to vary by family type (defined by the selected tax 

characteristics), state-level microsimulation can be performed on the ITM with reduced 

sampling errors once the ITM’s state samples have the population-like distributions of tax 

characteristics across states. Consequently, we use the same ITM tax calculator developed 

for analyzing the national effect and weight the sample with the newly imputed 52 state 

weights to simulate the effect of Federal tax changes state by state. 

This paper is organized as follows. Section 2 describes the data files, the tax 

characteristics included in estimation, and the properties we expect the estimates to have. 

Section 3 describes the estimation model, with details provided in the Appendix. Section 4 

presents the estimation results and verifies that the estimates generally have the desired 

properties. Section 5 shows state-by-state tax liability estimates (without behavioral 

change) resulting from repeal of the alternative minimum tax (AMT) as simulated by the 

re-weighted scheme on the ITM. Section 6 provides concluding remarks. 

2. Data 

The Individual Returns Transaction File (IRTF) contains records of all individual 

income tax returns filed for a given tax year. There is a variable for most (but not all) of the 

lines on the form 1040 and any schedules that may be attached. We use a set of X-variables 

(listed below) from the IRTF for tax year 2007 to estimate the probability a tax return was 

filed in a state st given the observed values of a set of those variables on that return. We 

denote this probability p(SS = ss|X). There are 52 “states” in our model: 50 states, the 

District of Columbia, and other areas. We rely on this population file to form the target 

distributions, including the distribution of returns across states and the distributions 

conditional on the selected tax variables. The variables we chose at this stage are 
6

AGI: Adjusted Gross Income, categorized as in Table 1 

EX: Number of exemptions, 1, 2, or 3 or more 

AGE: Primary filer’s age, grouped as in Table 2 

FamType: Family type, recoded from filing status; see Table 3 

SCHA: An indicator for filing Schedule A 

SCHB: An indicator for filing Schedule B 

STLCINTX: State and local income tax deduction, grouped as in Table 4 

realEstTax: Real estate property tax deduction, grouped as in Table 5 

Mort: Home mortgage interest deduction, grouped as in Table 6 

The AGE variable is obtained from Social Security records, which include date of birth. 

We merge that variable onto the IRTF. 

Each of the continuous variables in this list is converted to a discrete variable by 

grouping. We assume that the grouped variables are sufficient in the model. If this 

assumption is violated, information about the variables is lost and the estimates lose 

precision. The alternative is to treat some of the variables as continuous, but modeling the 

continuous distribution is a more complicated task. 

For example, it may seem reasonable to use the continuous version of the Xvariables as predictors in a multinomial logit model, for example, and estimate 

lll( P(SS=ss|X)

P(SS=AAAAAAA|X)

) = f(X, θ) + ε, or some variation of that. In this case it is necessary 

to model the function f as well as the distribution of ε. We know nothing about f, however, 

and by representing the variables as continuous, we introduce more computational 

complexity. The presence of large outliers in the IRTF also makes statistical robustness a 

consideration. Since the variables are grouped into categories, even a large data error only 

changes an observation's category and thereby contributes to a relatively small model error.
7

Let Y denote other variables for which we wish to have state-level estimates but that 

are not used to calculate state conditional probabilities. We assume that the relationship 

between Y and the state probability is fully explained by variables X included in the model. 

Examples of Y include the child tax credit (CTC) or earned income tax credit (EITC) 

claimants or amounts. Violation of this assumption leads to a loss of precision.

In our application, we take the X-variables described above and tabulate them by 

the generating classes described in the Appendix. Once these tables are produced, we are 

finished with the IRTF for estimation purposes.
8

Table 1 Grouping Categories for AGI

Table 2 Grouping Categories for Age

Category label Range 

1 

2 

3 

4 

(0,32) 

(32,50] 

(50,65] 

(65,∞] 

Table 3 Recode of Filing Status for Family Type

Category Label Filing Status

1

2

3

4

Single 

Head of Household 

Married Filing Jointly 

Married Filing Separately

Category label Range 

1 (−∞, 0] 

2 (0,10000) 

3 (10000,20000] 

4 (20000,30000] 

5 (30000,50000] 

6 (50000,75000] 

7 (75000,100000] 

8 (100000,150000] 

9 (150000,200000] 

10 (200000,500000] 

11 (500000,1000000] 

12 (1000000,2000000] 

13 (2000000,5000000] 

14 (5000000,∞] 
9

Table 4 Recode of State and Local Income Taxes

Table 5 Recode of Real Estate Property Taxes

Category Label Range

1

2

3

4

0

(0,1232] 

(1232,9407] 

(9407,∞]

Table 6 Recode of Mortgage Deduction

Category Label Range

1 

2 

3 

4 

5 

0

(0,6382] 

(6382,9150] 

(9150,13316] 

(13316,∞] 

The Individual and Sole Proprietorship (INSOLE) file is a stratified Bernoulli 

sample of individual income tax returns drawn from the IRTF. The IRS produces the 

INSOLE file for each tax year. Ratio adjustments are applied to the file to enforce 

consistency of stratum totals with known population totals. The stratification and sampling 

rates in the INSOLE are designed so that the highest income (and other ’high-interest’) 

returns are sampled at the highest rate; some high-interest returns are sampled with 

Category Label Range

1

2

3

4

5

0

(0,1522] 

(1522,2672] 

(2672,4402] 

(4402,∞]
10

certainty. Although the file contains a variable indicating a taxpayer’s state of residence, 

the sample is not randomly drawn across states and its sub-samples are not intended to be 

representative of state populations. 

Substantial effort has been made by the IRS to edit the INSOLE files so each record

has some internal consistency and obviously erroneous fields are corrected. The editing 

eliminates occasionally large errors found in the IRTF that can affect statistical estimates. 

In our application we use the 2007 INSOLE but exclude the subset of records identified by 

the IRS as being filed only to receive economic stimulus payments.

The X-variables on the INSOLE are grouped into categories in the same manner as 

described above for the IRTF. Our goal is to estimate 52 state weights (for 50 states, DC 

and other areas) for each INSOLE record so that, when applying these weights to the 

INSOLE file, the resulting distributions of tax returns across states, unconditional or 

conditional on certain tax variables, resemble the target distributions we calculated from 

the IRTF. Because state-level statistics should add up to national totals in the INSOLE, our 

method, in practice, splits the INSOLE weight for each record into 52 state weights. 

The Treasury Department’s Individual Tax Model (ITM) is used to simulate the 

revenue and distributional impact of tax law changes. The ITM is based on the IRS’ 

INSOLE file and is extrapolated by the Treasury’s Office of Tax Analysis (OTA) to meet 

targeted population and income growth, among other trends, over the 10-year period in the 

budget window. The ITM was based on the 2007 INSOLE file when the current project 

started. The ITM sub-samples, just like the INSOLE, are not intended to be representative 

of state populations. Furthermore, the ITM contains non-filing tax units outside of the 

INSOLE sample. Some X-variables for non-filing tax units are based on information 

returns filed by third-parties to the IRS. 

We merge taxpayers’ state weights estimated on the INSOLE onto the ITM. As 

mentioned, the filing units in the ITM are based on the INSOLE sample so we bring over 

state weights estimated for each INSOLE record to the same tax unit in the ITM. We also 

impute state weights to non-filing tax units in the ITM (see the Model section). To estimate 
11

state-by-state effects of tax law changes, we run the ITM as we do for national-level effects 

but use the imputed state weights (52 loops), instead of the file’s original weights, to 

produce 52 state-level outcomes. 

Once state weights are merged on to the ITM, we allow the weights to grow 

proportionally with the extrapolation of ITM weights for each tax unit over the budget 

window relative to the tax unit’s weight in the base year 2007. The evaluations below 

involve comparisons between state-level estimates based on the re-weighting method on 

the 2007 INSOLE and the corresponding state statistics calculated from the 2007 IRTF by 

the IRS and published in Statistics of Income (SOI) Tax Stats (2008). Alternatively, one 

can compare the re-weighted state-level estimates from the ITM with the published IRTF 

statistics for tax year 2012, the most recent year for which the publication is available. To 

prevent confounding the effect of ITM edits and extrapolation on state-level estimates with 

model errors in the re-weighting estimation, we chose to evaluate state-level estimates 

produced by the re-weighted INSOLE, rather than by the re-weighted ITM, against 

published IRTF statistics.

3. Model 

Let SS represent the random variable state, X a set of variables present in both the 

IRTF and the INSOLE, and Y a set of random variables in the INSOLE. Lower case letters 

denote specific values of the random variables. We assume SS ⊥ Y | X (which we read as

“ST is independent of Y, conditioned on X” and which implies 

p(ss, y|x) = p(ss|x)p(y|x) ); we further assume that p(x) > 0, so

 p(ss, x, y) = p(x,y)p(ss,x)

p(x) (1)

 = p(ss|x)p(x, y). (2)

Since the conditional probability of returns across states, p(ss|x), can be written as

 p(ss|x) = p(x|ss)p(ss)

p(x) (3)
12

and p(x, y) is given by the INSOLE weights and not subject to manipulation, by 

assumption, our estimation task is equivalent to estimating p(x|ss), which indicates the 

distribution of x in a given state, and p(ss), which indicates the distribution of returns 

across states. The latter is easy; we calculate it directly from the IRTF. A description of the 

methodology for estimating p(x|ss) is given in the Appendix.

Combining (3) with INSOLE weights, 

Np̂(ss, x, y) = p(x|ss)p(ss)wi

∑ss p(x|ss)p(ss)

 = p̂(ss|x)wi.

where N is the population count and wi is the INSOLE weight for observation i. Thus the 

new weight for observation i in state st is 

 wi,ss = p̂(ss|x)wi.

The new weight wi,ss is just the INSOLE weight, split into smaller shares. Those shares are 

proportional to the estimated proportion of units that are in state ss among all returns that

are similar to the i h return, in the sense that X = xi for those returns. 

3.1 Non-filing Tax Units 

We further assume that p(ss|x)is the same for non-filing units as it is for filing 

units. Evaluating this assumption and the quality of the estimator for non-filers is the 

subject of ongoing research. 

4. Evaluation on State-Level Estimates from the INSOLE 

Using the state weights, we produce state-level estimates for several tax variables 

from the INSOLE and compare them to the corresponding set of estimates calculated by 

the IRS based on the IRTF and published in SOI Tax Stats (2008). Figures 1 through 9 

present scatterplots of estimates based on the method in this paper (the re-weighting 
13

method) versus the corresponding statistics from the IRTF published by the SOI. Recall 

that our estimation goal is to let the state weight estimators, when applied to the INSOLE 

records, produce similar cross-state distributions (unconditional or conditional) to those 

observed from the IRTF population file. Each dot on the graph represents a state, with X 

denoting the value calculated in the IRTF and Y denoting the value calculated from the 

INSOLE using our state weight estimators. Under a perfect fit, all dots would lie on the 45-

degree line.

In addition to using scatterplots for visual evaluation, we also measure the model 

performance by calculating the correlation, as well as relative differences, between the two 

statistics in each scatterplot. Figure 10 contains a table of the correlation coefficients, 

estimated Coefficients of Variation (CV’s), and Mean Absolute Relative Difference 

(MARD) for the same set of variables. 

The correlation coefficients, c, is the usual one, defined on the linear scale, though 

the plots are on the ablog scale, where 

aa (c) = s s(c) log10(|c| + 1).

This transformation has many of the properties of the log10 function, but it is continuous at 

zero and defined for (and preserves the sign of) negative numbers. This transformation is 

useful for visualizations of data from distributions with a long tail while supporting those 

that are not subsets of the positive real numbers. The CV’s are estimated with 

 cc� = �

1

n ∑ �

y�i−yi

yi

�

2 n

i=1 �

1

2

, 

where n is the number of cells in the relevant table, yi is the population proportion in that 

cell, and y�i is its estimate formed from the re-weighting method. The MARD is 
14

MMMM = 1�|y�i − yi|

yi

.

n

i=1

The results presented in Figures 1 through 9 appear acceptable with the possible 

exception of the results for other areas and the Real Estate Tax variable. It is confirmed in 

Figure 10 that the correlation on the linear scale is low and the MARD is high for the Real 

Estate Tax, relative to the other variables. There are a number of possible reasons for these 

results. On the Real Estate Tax variable, it is possible that the categories are too wide when 

we group the variable and therefore the within-category variance is large, causing the 

excess variance in the estimates. For both other areas and Real Estate Tax, it is possible 

that the measurement differences between the INSOLE and the IRTF (particularly edits to 

the INSOLE) are large enough to matter. 

5. State-Level Microsimulation from the ITM

In this section, we apply state weights to an ITM run to produce state-by-state tax 

liability effects of repealing the AMT. Note that neither the AMT status nor the amount is 

included in the X variables in the re-weighting estimation. However, because certain 

deductions and credits are preferences in the AMT, we expect that the AMT effect is 

explained in the model by taxpayer income and by itemized deductions for state and local 

taxes, the deductions of which are not allowed in the AMT. 

The ITM predicts that, without behavioral changes, 4.35 million non-dependent 

taxpayers would be affected by a repeal of the AMT under 2016 law and income for a total 

decrease in the Federal individual income tax liability by $29.06 billion.2 This liability 

effect takes place either through an elimination of AMT payments or increases in the tax 

credits lost under the AMT. The effect is unequally distributed across states. About 17 

percent of the total liability effect resulting from the AMT repeal is attributable to 

California while another 11 percent and 7 percent are attributable to New York and New 

Jersey, respectively. Other states that are largely affected by the AMT include Illinois, 

 2 This model run is performed on the version of the ITM based on the 2010 INSOLE file and FY2016 winter budget 

extrapolation. 
15

Texas, Florida, Pennsylvania and Massachusetts, each of which contributes to at least 4 

percent of the total liability effect. In contrast, the bottom 25 states (including the District 

of Columbia and Other Areas) combined make up just under 10 percent of the total AMT 

effect. 

We divide each state’s AMT liability effect by its total Federal individual income 

tax liability after refundable tax credits to account for the state’s percentage change in tax 

liability from eliminating the AMT. According to this percent change, states are classified 

into four roughly equal sized groups. Figure 11 depicts the results. Within the top and 

bottom groups of states, 5 states are predicted to have their Federal individual income tax 

liability declined by more than 2.2 percent in 2016 with an AMT repeal, including New 

Jersey (with the greatest effect, 3.0 percent), California, Vermont, Maryland and Illinois, 

and 6 states below 1.2 percent, including Alaska (with the smallest effect, 0.7 percent), 

Wyoming, Nevada, Tennessee, South Dakota and Texas.

6. Conclusion 

With the re-weighting method described in this paper, micro-simulations for tax 

proposals are conducted on the ITM in a straightforward way to produce state-by-state 

estimates. Based on this application, OTA estimated the number of workers by state who 

would benefit from the proposed EITC expansion for childless workers in the President’s 

FY2015 Budget. The estimation results are released in a joint paper by the Executive 

Office of the President and U.S. Treasury (2014). 

It is worth noting that, while we implement the re-weighting method to perform 

state-by-state analysis, the same method can be applied for other small domains whether 

geographic or not. For example, same-sex married couples were not allowed to file as 

married filing jointly in 2007. To project the tax consequence of joint filing for same-sex 

married couples, we re-weighted married-filing-jointly couples in the 2007 INSOLE-based 

ITM to make the key tax characteristics of these joint filers distributed similarly to those of 

same-sex married couples observed in the survey data in 2007. As a result, the tax liability 

of same-sex married couples had they filed jointly in 2007 can be inferred from the re-
16

weighted tax liability of married couples in the 2007 ITM. 

The key assumption in this method is the conditional independence assumption. 

The results will still be valid if most, if not all, of the dependence between Y and ST is 

explained by X. One counter example, which gives rise to limitations in using the state 

weights, is when Y depends specifically on state policies, state-specific characteristics or 

state economic conditions whose effects are not captured by X. When using the state 

weights to produce state-level simulation estimates, one should exercise judgment on 

whether the Y variable(s) of interest, or the tax law’s effect, can be reasonably explained by 

the X variables included in the estimation. 

It is natural to consider extending the model by adding variables to the X-vector. 

There is trade-off, however. Specifically, P(Y, X) is estimated from the INSOLE sample, 

which has a limited sample size. As an illustration, consider the extreme case where X has 

ST as an element. Then P(Y|X) becomes P(Y|ST). As the INSOLE sample is not designed 

for state-level estimation, estimates of this density will be based on unsuitably small 

samples and it follows that estimates based on this density are likely to have unacceptably 

large variances. The problem persists for any X where the cells of the table defined by X are 

too small or, more generally, where the degrees of freedom in some sub-table are too small. 

Given the trade-off between model fit and variance, upon evaluating the re-weighting 

model, one should consider which Y variables are relevant and what level of error renders 

an estimate unacceptable.
17

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Executive Office of the President and U.S. Treasury Department (2014). The President’s 

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https://www.whitehouse.gov/sites/default/files/docs/eitc_report.pdf .

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(FCSM) Research Conference. 1999. www.fcsm.gov/99papers/schirm.pdf, downloaded 

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http://www.irs.gov/uac/SOI-Tax-Stats-Historic-Table-2, downloaded November 22, 

2010. 
18

Figure 1: Total numbers of returns from reweighting method versus tabulations 

in the IRS publications. Log10 scale.
19

Figure 2: Numbers of Returns with EITC from reweighting method versus 

tabulations in the IRS publications. Log10 scale.
20

Figure 3: EITC payments from reweighting method versus tabulations in the IRS 

publications. Log10 scale.
21

Figure 4: Numbers of returns with Child Tax Credits from reweighting method 

and from published tabulations from the IRTF. Log10 scale.
22

Figure 5: CTC from reweighting method versus tabulations in the IRS 

publications. Log10 scale.
23

Figure 6: Numbers of returns with Additional Child Tax Credits from rewighting 

method and from published tabulations from the IRTF. Log10 scale.
24

Figure 7: Additional CTC from reweighting method versus tabulations in the IRS 

publications. Log10 scale.
25

Figure 8: Numbers of returns with real estate tax property tax deductions from 

reweighting method versus tabulations in the IRS publications. Log10 scale.
26

Figure 9: Amounts of real estate tax property tax deductions from reweighting 

method versus tabulations in the IRS publications. Log10 scale
27

Variable Correlation, 

Linear Scale 

Estimated 

CV 

Mean 

Absolute 

Relative 

Difference 

All Returns 

EITC Number of Returns 

EITC Amount 

Mortgage Interest Deduction 

Amount CTC Number of Returns 

CTC Amount 

Additional CTC Number of Returns 

Additional CTC Amount 

Number of Returns with Real Estate 

Tax Amount of Real Estate Tax 

0.999 

0.998 

0.996 

0.999 

0.999 

0.998 

0.999 

0.998 

0.997 

0.986 

0.026 

0.051 

0.076 

0.093 

0.038 

0.042 

0.082 

0.120 

0.047 

0.26 

0.020 

0.039 

0.057 

0.076 

0.026 

0.028 

0.065 

0.094 

0.038 

0.22 

Figure 10: Correlation and Mean Absolute Relative Differences between Re-weighting Estimates 

and Published Estimates without ”Other Areas.”
28

Figure 11: Liability Change from Repeal of the AMT
29

Appendix

Since the variables in the model are discrete, either naturally or by grouping, we 

represent the joint conditional distribution of the variables as a table. The table describing 

the joint distributions of the variables and state is potentially very large so our model 

potentially has a large number of parameters. To decrease this number, we search for 

parsimonious models for the relationships between the variables. In particular, we model 

independence relationships which take the form p(X, Y|Z) = p(X|Z)p(Y|Z) for some 

variables {X, Y, Z}; we denote this relationship X ⊥ Y | Z. There is an intuitive and rigorous 

method to map the independence assumptions in a model, representing the model as an 

undirected graph G=(V,E), where V is the set of vertices (or nodes) in G, and E is the set of 

edges connecting pairs of vertices. Each node in the graph represents a variable in the 

model and the edges are defined so that the absence of an edge between nodes, say A and 

B, represents the assumption that A is independent of B given all of the other variables, V\

{ , B}. That is, p( , B|V\{ , B}) = p( |V\{ , B})p(B|V\{ , B}) or ⊥ B|V\{ , B}.

Such models are called Markov Random Fields or, alternately Undirected Graphical 

Models, and there is a substantial literature on their theoretical properties, and many 

efficient algorithms have been designed to take advantage of those properties (for example, 

see Cowell et al. (2003)). We are particularly interested in tools to search for simple models 

for the dependencies in these models and, generally, to reason about the joint distribution 

of the tax variables in our datasets. . 

For an example, consider the graph in Figure A1. Here X has four components, and 

the graph corresponds to the model where 

p�x1, x2, x3,x4� = p�x1,x2,x3�p(x1,x2,x4)

p(x1x2) .

Under this model, the original four dimensional table has been replaced with two smaller 

tables. Each table corresponds to a fully connected subgraph in the graph. Reductions like 

this can reduce the total number of parameters dramatically. Formally, the number of 

parameters increases asymptotically exponentially in the dimension of the largest table, so 

reducing the size of the largest table reduces the complexity, even if there are many such 

tables as a result. The generating class of this model is

(A1)
30

[[X1,X2,X3],[X1,X2,X4]],

using notation like that in Bishop et al. (2007) (also see the reference for a detailed 

discussion of the interpretation of generating classes). The variables S = (X1,X2) separate 

X3 from X4 in the graph and in the sense that X3 ⊥ X4 | S; S is therefore sometimes called 

a separator. The members of each set in the generating class are all connected to each 

other in the graph; in the graph theory literature, these sets are called cliques. They 

correspond to the largest tables we need to estimate. Equation (A1) can be generalized: for 

a set C of cliques with a set S of seperators, 

p�x� = ∑ccc p�xc�

∑s s p(xs)

,

where we abuse notation a little to overload p(. ) so it is the density of its arguments, 

whatever they are. The maximum likelihood (ML) estimate of p(x) is just 

p̂�x� = ∑ccc p��xc�

∑s s p�(xs)

,

where the hat indicates the ML estimate, and, recall, the ML estimator for a saturated table 

is just the table of empirical proportions. This simplifies the task enormously, and 

eliminates the need for iterative methods like Iterative Proportional Fitting (see Cowell et 

al. (2003) for details).

(A2)
31

Figure A1. A graph corresponding to a model where p�x1, x2, x3,x4� =

p�x1,x2,x3�p(x1,x2,x4)

p(x1x2)

In our application we partition the set of variables associated with each tax unit into 

three parts. The first part contains the state (or DC or Other Areas) of residence, which we 

denote ST , ST ∈{1, ..., 52}. The second and third parts contain set of variables X and Y for 

which we make the assumption that SS ⊥ Y | X, where Y are the remaining variables of 

interest in the INSOLE. That is, ST is independent of Y, conditioned on X. Denote the 

density (relative to the appropriate measure) of that distribution by p(ss, x, y) = p(SS =

ss, X = x, Y = y). To set the context, ST and X are variables available from the IRTF while 

X and Y are present on the INSOLE/ITM and we wish to construct weights on the 

INSOLE/ITM so that we can estimate state-level summaries of functions of (X, Y). For 

example, X may be AGI, total exemptions, and state and local income tax, and Y may be 

taxes paid. Then the assumption is that taxes paid is independent of ST given AGI, total 

exemptions, and state and local income tax. Put another way, ST adds no information 

about taxes paid, once AGI, total exemptions, and state and local income tax are observed. 

This illustrative example is, of course, unrealistic; the modeling exercise involves making 

good choices for X so the model holds well for a useful set of choices of Y.

These assumptions imply, assuming p(x) > 0,

 p(ss, x, y) = p(x,y)p(ss,x)

p(x) (A3)

X

X X

X
32

 = p(ss|x)p(x, y). (A4)

Since the conditional probability of returns across states, p(ss|x), can be written as

p(ss|x) = p(x|ss)p(ss)

p(x) ,

and p(x, y) is given by the INSOLE weights and is not subject to manipulation, our 

estimation task is equivalent to estimating p(x|ss), which indicates the distribution of x in a 

given state, and p(ss), which indicates the distribution of returns across states. The latter is 

easy; we calculate it directly from the IRTF data. 

Note in equation (A3) that we group the denominator with the rightmost factor in 

the numerator, so information from the INSOLE informs the inference about ST, and the 

joint distribution preserves the marginal distribution P(X, Y), which is the empirical 

distribution in the INSOLE. If we had grouped the denominator with the leftmost factor in 

the numerator, information would have propagated the other way, and the method would 

have been a post-stratification on states with raking, controlling to population estimates of 

SS ∗ X totals. 

We assume the dependence structure of X|SS can be represented by a graphical 

models as we have described. This is a weak assumption; it is implied by the generalized 

linear model (GLM). We also assume it is decomposable, which is a stronger assumption. 

See Bishop et al. (2007) and Cowell et al. (2003) for the implications. These models are all 

members of the class of log linear models. Finally, we also assume the Markov graph of 

X|SS is the same for every value of st and that we can estimate that graph from a sample 

from the nation. This implies that, for the purpose of fitting the model structure, there are 

no aggregation issues.

We estimate the structure of the graph using a sequence of the forward and 

backward search algorithms implemented in R (R Core Team (2012)) in the package gRim 

(Hojsgaard (2012)). The transaction file has nearly 150 million records, so it will be 
33

sensitive to even very small effects in the model. The resulting fitted model is likely to be 

over-fit for use with the INSOLE, which has a fraction of the sample. We form a sample by 

resampling without replacement from the INSOLE with probabilities proportional to the 

INSOLE sample weights, resulting in an approximately simple random sample without 

replacement. We do this because the theory is poorly developed for weighted data; we use 

the INSOLE in the first place to preserve the effect of the edits. This sample has 100,000 

records in it. This sample size should be appropriate to detect dependencies useful for the 

INSOLE. After running the forward-backward procedure until the model stopped changing, 

which only took two iterations, the result was a log linear model with the following 

generating classes

[[SCHB, STLCINTX, FamType, SCHA, AGE, AGI], 

[EX, SCHB, FamT ype, STLCINTX, AGE], 

[FamT ype, SCHA, SCHB, AGE, Mort, AGI], 

[AGE, SCHA, realEstTax, SCHB, Mort, AGI]].

Recall the only interactions in a log linear model are between variables within one member 

set of the generating class. In this model, for example, interactions are allowed between 

SCHB and SCHA, since they are common to the first set, but not STLCINTX and Mort, 

since they are not in any of the same sets in the generating class. The Markov Random 

Field associated with this model is shown in two forms in Figures A2-1 and A2-2. Figure 

A2-1 shows the whole graph, including state. Figure A2-1 shows the graph with state

excluded; this is the graph for the model estimated within each state.
34

Figure A2-1. Markov Random Field for the model with state.
35

Figure A2-2. Markov Random Field for variables within state

It follows that we can summarize the data from the IRTF with the set of four tables with the 

variables listed in the generating class for each state. 

We estimate the joint probabilities of the tables directly by forming the crosstabulations from the IRTF separately for each state, 

 p̂(Xc = x_c|SS = ss) = ∑ I(Xc,i=xc,S i=ss) i

∑ I(S i=ss) i

,

where i indexes the records in the IRTF. The distribution of x and y, normalized so the 
36

total is the population size, evaluated at (xi, yi), is N (x, y), and is represented by the 

weight wi.

Putting it all together, 

Np̂(ss, x, y) = p(x|ss)p(ss)wi

∑ss p(x|ss)p(ss)

 = p̂(SS|x)wi.

Thus our new weight for observation i in state st is 

 wi,ss = p̂(ss|xi)wi.

The new weight wi,ss is just the INSOLE weight, split into smaller shares. Those shares are 

proportional to the estimated proportion of units that are in state ss among all returns that 

are similar to the i h return, in the sense that X = xi for those returns. Note 

∑ss ∑ii wi,ss = N. This fits the framework in Schirm et al. (2010), but we use more 

sources of data; in particular, we use the external tabulations and the model based on 

external data, rather than relying on data internal to the data set of interest to form the 

weight adjustment. We also use the equivalence of the decomposition of the Markov graph 

with the arrangement of the data into two datasets. In the decomposition of the graph into 

two graphs G1 and G2, G1 corresponds to the first data set (here, the INSOLE) while 

G2 corresponds to the second (here, the IRTF). The variables common to the two data sets 

(here, X) correspond to a separator S. This method should generalize to larger collections of 

data sets, each represented by a graph Gi, i indexing data sets, and a collection of models , 

Mi.