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GEP 4 — A DAG—based Computational Backend#

Author

Max Blesch, Janos Gabler, Hans-Martin von Gaudecker, Tobias Raabe, Christian Zimpelmann

Status

Provisional

Type

Standards Track

Created

2022-03-28

Resolution

Accepted

Abstract#

This GEP explains the directed acyclic graph (DAG)-based computational backend for GETTSIM.

The graph operates on columns of data. Stringified function names and their inputs correspond to columns in the data, i.e., nodes in the graph.

Unless functions perform aggregations, they are written in terms of scalars and vectorized during DAG setup.

Motivation#

The implementation choice to use a DAG to represent the taxes and transfers system is motivated by two main reasons.

  1. The taxes and transfers system is constantly evolving in many dimensions, flexibility is thus needed internally. Additionally, it is not enough to represent the state of the system at any given point in time, but users need to be able to introduce their own changes. Being able to change or replace any part of the taxes and transfers system is crucial. Put differently, there is no meaningful distinction between parts of the system only ever touched by developers and parts that are modifiable by users. A DAG implementation allows to eliminate this usual boundary for almost all use cases.

  1. The DAG allows a user to limit computations to generate a set of target variables, which she is ultimately interested in. Doing so allows cutting down on the number of input variables, it prevents unnecessary calculations, and it increases computation speed.

Basic idea#

Based on the two requirements above we split the taxes and transfers system into a set of small functions. Each function calculates one clearly defined variable (identical to the function’s stringified name) and returns a 1d-array.

Note

The function code itself will typically work on scalars and is vectorized by GETTSIM; this is irrelevant for the DAG.

Function arguments can be of three kinds:

  • User-provided input variables (e.g., bruttolohn_m).

  • Outputs of other functions in the taxes and transfers system (e.g., eink_st_y_sn).

  • Parameters of the taxes and transfers system, which are pre-defined and always end in _params (e.g., ges_rentenv_params).

GETTSIM will calculate the variables a researcher is interested in by starting with the input variables and calling the required functions in a correct order. This is accomplished via a DAG (see below).

Splitting complex calculations into smaller pieces has a lot of the usual advantages of why we use functions when programming: readability, simplicity, lower maintenance costs (single-responsibility principle). Another advantage is that each function is a potential entry point for a researcher to change the taxes and transfers system if she is able to replace this function with her own version.

See the following example for capital income taxes.

def abgelt_st_y_sn(zu_verst_kapitaleink_y_sn: float, abgelt_st_params: dict) -> float:
    """Calculate Abgeltungssteuer on Steuernummer-level.

    Parameters
    ----------
    zu_verst_kapitaleink_y_sn
        See :func:`zu_verst_kapitaleink_y_sn`.
    abgelt_st_params
        See params documentation :ref:`abgelt_st_params <abgelt_st_params>`.

    Returns
    -------

    """
    return abgelt_st_params["satz"] * zu_verst_kapitaleink_y_sn

The function abgelt_st_y_sn() requires the variable zu_verst_kapital_eink_y_sn, which is the amount of taxable capital income on the Steuernummer-level (the latter is implied by the _sn suffix, see GEP 1 — Naming Conventions). zu_verst_kapital_eink_y_sn must be provided by the user as a column of the input data or it has to be the name of another function. It is also possible to specify zu_verst_kapital_eink_y and aggregation to the sn-level will happen automatically. abgelt_st_params is a dictionary of parameters related to the calculation of abgelt_st_y_sn.

Another function, say

def soli_st_y_sn(
    eink_st_mit_kinderfreib_y_sn: float,
    anz_personen_sn: int,
    abgelt_st_y_sn: float,
    soli_st_params: dict,
) -> float: ...

may use abgelt_st_y_sn as an input argument. The DAG backend ensures that the function abgelt_st_y_sn will be executed first.

Note that the type annotations (e.g. float) indicate the expected type of each input and the output of a function, see GEP 2 — Internal Representation of Data on Individuals.

Directed Acyclic Graph#

The relationship between functions and their input variables is a graph where nodes represent columns in the data. These columns must either be present in the data supplied to GETTSIM or they are computed by functions. Edges are pointing from input columns to variables, which require them to be computed.

Note

GETTSIM allows to visualize the graph, see this guide.

The resulting structure is a special kind of graph, called a directed acyclic graph (DAG). It is directed because there are clearly inputs and outputs, i.e., there is a sense of direction. Acyclic means that there exist no path along the direction of the edges, where you start at some node and end up at the same node. Equivalently, a DAG has a topological ordering which is a sequence of nodes ordered from earlier to later in the sequence. The topological ordering is what defines the sequence in which the functions in the taxes and transfers system are evaluated. This ensures that the inputs are already computed before a function that requires them is called.

In order to calculate a set of taxes and transfers, GETTSIM builds a DAG based on three inputs provided by the user:

  • Input data.

  • A set of functions representing the taxes and transfers system, which consist of the ones pre-implemented in GETTSIM and potentially user-written additional functions.

    Parameters of the taxes and transfers system can be ignored in the following (they amount to collections of constants; in practice they will already be partialled into these functions). These functions need to be written for scalars; they will be vectorised during the set up of the DAG.

  • A set of dictionaries specifying aggregation functions, calculating, for example, household-level averages.

  • The target columns of interest.

The DAG is then used to call all required functions in the right order and to calculate the requested targets.

Level of the DAG and limitations#

In principle, GETTSIM will import all functions defined in the modules describing the taxes and transfers system. In principle, these functions refer to all years in GETTSIM’s scope. There has to be some discretion in order to allow for the interface of functions to change over time, new functions to appear, or old ones to disappear.

Some examples include:

  1. arbeitsl_hilfe being replaced by arbeitsl_geld_2.

  2. kinderbonus being active only in a few years.

  3. The introduction of kinderzuschl.

  4. Capital income entering sum_brutto_eink or not.

The goal is that the graph for any particular point in time is minimal in the sense that arbeitsl_geld_2 does not appear before it was conceived, it is apparent from the interface of sum_brutto_eink whether it includes capital income or not, etc..

In the yaml-files corresponding to a particular tax / transfer, functions not present in all years will need to be listed with along with the dates for when they are active. See :gep-3-keys-referring-to-functions: for the precise syntax. That mechanism should be used for:

  1. Functions that are newly introduced.

  2. Functions that cease to be relevant.

  3. Functions whose interface changes over time.

  4. Functions whose body changes so much that

    • it is useful to signal that things have changed and/or

    • it would be awkward to program the different behaviors in one block with case distinctions.

Needless to say, the different reasons may appear at different points in time for the same function.

Additional functionalities#

We implemented a small set of additional features that simplify the specification of certain types of functions of the taxes and transfers system.

Group summation and other aggregation functions#

Many taxes or transfers require group-level variables. <GEP-2 describes gep-2-aggregation-functions> how reductions are handled in terms of the underlying data. This section describes how to specify them.

In order to inject aggregation functions at the group level into the graph, scripts with functions of the taxes and transfer system should define a dictionary aggregate_by_group_[script_name] at the module level. This dictionary must specify the aggregated columns as keys and a dictionary with keys source_col and aggr as values. If aggr is count, source_col is not needed.

For example, in demographic_vars.py, we could have:

aggregate_by_group_demographic_vars = {
    "anz_kinder_hh": {"source_col": "kind", "aggr": "sum"},
    "anz_personen_hh": {"aggr": "count"},
}

The group identifier (hh_id, sn_id, fg_id, bg_id, eg_id, ehe_id) will be automatically included as an argument; for count nothing else is necessary.

The output type will be the same as the input type. Exceptions:

  • Input type bool and aggregation sum leads to output type int.

  • Input type int and aggregation \(\in \{\) any, all \(\}\) leads to output type bool

  • Aggregation count will always result in an int.

The most common operation are sums of individual measures. GETTSIM adds the following syntactic sugar: In case an individual-level column my_col exists, the graph will be augmented with a node including a group sum like my_col_hh should that be requested. Requests can be either inputs in a downstream function or explicit targets of the calculation.

Automatic summation will only happen in case no column my_col_hh is explicitly set. Using a different reduction function than the sum is as easy as explicitly specifying my_col_hh.

Consider the following example: the function kindergeld_m calculates the individual-level child benefit payment. arbeitsl_geld_2_m_bg calculates Arbeitslosengeld 2 on the Bedarfsgemeinschaft (bg) level (as indicated by the suffix). One necessary input of this function is the sum of all child benefits on the Bedarfsgemeinschaft level. There is no function or input column kindergeld_m_bg.

By including kindergeld_m_bg as an argument in the definition of arbeitsl_geld_2_m_bg as follows:

def arbeitsl_geld_2_m_bg(kindergeld_m_bg, other_arguments): ...

a node kindergeld_m_bg containing the Bedarfsgemeinschaft-level sum of kindergeld_m will be automatically added to the graph. Its parents in the graph will be kindergeld_m and bg_id. This is the same as specifying:

aggregate_by_group_kindergeld =  = {
    "kindergeld_m_bg": {
        "source_col": "kindergeld_m",
        "aggr": "sum"
    }
}

Aggregation based on person-to-person pointers#

For some taxes and transfers, one person may establish a claim for another person. A parent, for example, has a claim on the basic child allowance (Kindergeld) because their child is eligible for it. Similarly, parents receive a tax allowance because their child satisfies the criteria for it. These aggregation operations are based on the p_id column. This section describes how to specify such taxes and transfers.

The implementation is similar to aggregations to the level of groupings: In order to specify new aggregation functions, scripts with functions of the taxes and transfer system should define a dictionary aggregate_by_p_id_[script_name] at the module level. This dictionary must specify the aggregated columns as keys and a dictionary with keys source_col, p_id_to_aggregate_by and aggr as values. If aggr is count, source_col is not needed.

The key source_col specifies which column is the source of the aggregation operation. The key p_id_to_aggregate_by specifies the column that indicates to which p_id the values in source_col should be ascribed to. The key aggr gives the aggregation method.

For example, in kindergeld.py, we could have:

aggregate_by_p_id_kindergeld = {
    "kindergeld_anz_ansprüche": {
        "p_id_to_aggregate_by": "p_id_kindergeld_empf",
        "source_col": "kindergeld_anspruch",
        "aggr": "sum",
    },
}

This dict creates a target function kindergeld_anz_ansprüche which gives the amount of claims that a person has on Kindergeld, based on the kindergeld_anspruch function which returns Booleans, which show whether a child is a reason for a Kindergeld claim.

The output type will be the same as the input type. Exceptions:

  • Input type bool and aggregation sum leads to output type int.

  • Input type int or float and aggregation \(\in \{\) any, all \(\}\) leads to output type bool

  • Aggregation count will always result in an int.

Conversion between reference periods#

Similarly to summations to the group level, GETTSIM will automatically convert values referring to different reference periods defined in GEP 1 — Naming Conventions (years (default, no suffix), months _m, weeks _w, and days _d).

In case a column with annual values [column] exists, the graph will be augmented with a node including monthly values like [column]_m should that be requested. Requests can be either inputs in a downstream function or explicit targets of the calculation. In case the column refers to a different level of aggregation, say [column]_hh, the same applies to [column]_m_hh.

Automatic summation will only happen in case no column [column]_m is explicitly set. Using a different conversion function than the sum is as easy as explicitly specifying [column]_m.

Conversion goes both ways and uses the following formulas:

time unit | suffix | factor |
Year | | 1 |
Month | _m | 12 |
Week | _w | 365.25 / 7 |
Day | _d | 365.25 |

These values average over leap years. They ensure that conversion is always possible both ways without changing quantities. In case more complex conversions are needed (for example to account for irregular days per month, leap years, or the like), explicit functions for, say, [column]_w need to be set.

Alternatives#

We have not found any alternatives which offer the same amount of flexibility and computational advantages.

Discussion#

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