1.1 Marketable vs. non-marketable debt

Figure 1 can be disaggregated to compare the marketable debt (most of which is publicly held debt) vs. the non-marketable debt (most of which is internally held debt). This breakdown is shown in Figure 2 where we plot total marketable and non-marketable debt as well as total marketable and non-marketable redemptions and reissuances. Also reported in the figure is debt subject to limit. Figure 2 clearly shows that non-marketable redemptions/reissuance has been growing rapidly and has experienced notable fluctuations from year to year. A significant portion of non-marketable debt is held internally as 1-day securities that are turned over every business day. Figure 3 shows that most of the non-marketable redeemed and issued debt comes from Government Account Series debt. Notice that non-marketable redemptions/reissuance series are very similar to government account redemptions/reissuance series as Figures 2 and 3 indicates. Therefore, we also provide Tables A2 and A3 in the Appendix to highlight the subtle differences between both series.

Figure 2.

Marketable and non-marketable redemptions and reissuance, and marketable and non-marketable debt subject to limit, 2000–2020.

Source: Author's calculations based on U.S. Department of the Treasury information.

Note: The data used to generate this graph correspond to Table A.2 in the Appendix.

Figure 3.

Marketable and non-marketable redemptions and reissuance, and marketable and non-marketable debt subject to limit, 2000–2020.

Source: Author's calculations based on U.S. Department of the Treasury information.

Note: The data used to generate this graph correspond to Table A.3 in the Appendix.

This broad summary of total federal debt, redemptions, and reissuance shows that most of the redemptions and reissuances are occurring in the non-marketable (internally held) debt category. We now turn our attention to the fund that is generating the largest portion of internal redemptions and reissuances – the G-Fund.

5.1 Data

In the previous sections, we introduced the time series of data that we will use in the empirical analysis. We compiled daily data the G-Fund, total public debt, and the debt ceiling. This information is used to create the variables we use to understand the relationship between the public debt, debt ceilings, and the G-Fund balance. Before presenting our formal analysis, we provide an overview of the G-Fund series and its relationship to historical events arising from debt-ceiling constraints, debt-ceiling expansions, and debt-ceiling suspensions.

Figure 6 presents G-Fund trends over time where we have identified all the major balance reductions that occurred between 2001 and 2020. In total, there are 11 time intervals where the G-Fund experienced significant variation in daily fund balances. This figure is central for the analysis in that it provides the broad perspective of each of the events that we analyze separately below. The beginning and end of each of these periods is marked by one of the following events.

• • FDL (Federal Debt Limit): This variable marks the day when the Total Federal Debt (GFDEBTN) is constrained by the debt ceiling.Footnote ⁸

• • DCE (Debt-Ceiling Expansion): This variable marks the day on which a debt-ceiling expansion occurs (recall that the dates of these events are given in Table 1).

• • SUSP (Suspension of the Debt Ceiling): This variable indicates when the debt ceiling was suspended.

Taking into consideration Figure 6, we isolate each of the decreases into separate events. Figure 7 shows each debt-ceiling constraint period overlaid with G-Fund balances. For each event, we identify the dates related to the large decreases in the G-Fund. Each event corresponds to the key days marked in the figure using colored vertical lines: (1) a red line for FDL days, (2) a green line for DCE days, and (3) a purple line for SUSP days.

Figure 7.

(a) Large G-Fund balance decreases and fund recovery. (b) Large G-Fund decreases and fund recovery.

Source: Author's calculations.

Note: Federal debt limit (FDL) marks the day when the total federal debt is constrained by the debt ceiling. Debt-ceiling expansion (DCE) marks the day on which a debt-ceiling expansion occurs (see Table 1). Suspension of the debt ceiling (SUSP) indicates the date when the debt ceiling was suspended.

Two patterns emerge in Figure 7. First, we see that the beginning of each graph is marked by an FDL day. Within days of an FDL day there is a significant decrease in the G-Fund; in a number of cases the balance decreases to nearly 0, and then recovers completely once a DCE occurs. In other words, when a budget constraint is imposed, the G-Fund balance diminishes. During these periods, the funds are shifted into another account/location and may be used for other purposes. Once the debt ceiling is expanded or suspended, the balance is restored. This pattern is observed in the first four events. The SUSP becomes an instrument of budgetary relief. The fifth event begins with an FDL day, the G-Fund balance decreases by nearly half and then recovers after an SUSP day. This pattern is repeated for all subsequent events, with the difference that each fund decline begins with an FDL and ends with either a DCL or SUSP day.

In Table 2 we present detailed information on each of these events. Columns (2) and (3) indicate the exact start and end date of each decline in the G-Fund, where one of the following three cases occurs: FDL, DCE, or SUSP. Columns (7) and (8) show precisely what type of day marks the beginning and end of each event. Additionally, column (4) indicates the number of days that elapse during each of the events. The average number of days is approximately 76 (business days). Column (5) indicates the number of days that occur before the G-fund recovers the full balance. Notice that the number of days in column (5) will be less than the number of days in column (6), because we are not counting that small window of time where the balance recovers to the levels it had prior to when the G-Fund declined. Finally, column (6) indicates the number of days of the most binding period; that is, it includes only those days where the G-Fund actually declined. Column (4) shows the number of days between the two vertical lines that appear in Figure 7. Likewise, for the interested reader we present Figures 4A and 5A in the Appendix that correspond with columns (5) and (6).

Table 2.

Descriptions of each G-Fund decrease

Source: Author's calculations.

Note: Column (1) indicates the event number that appears in Figure 7. Each event corresponds to a noticeable decrease in the G-Fund balance. Columns (2) and (3) indicate the month, day, and year in which each event begins and ends, respectively. Column (4) indicates the total number of days the events last. Column (5) indicates the total number of days since an event began and the G-Fund decreases to its minimum amount, that is, we do not count the day or days of recovery of the balance. Column (6) counts only the days in which the TSF declines. The start dates of each event are marked by an FDL or DCE day. However, when an FDL or DCE day occurs, several days may pass before the G-Fund drops. Therefore, this column does not consider initial days without movement in the G-Fund balance. Finally, columns (7) and (8) indicate what type of day (FDL, DCE, SUSP) marks the beginning and end of each of the events.

Note that we named columns (4), (5), and (6) as Binding Period 0 (BP0), Binding Period 1 (BP1), and Binding Period 2 (BP2), respectively. The information presented in these columns is the key to generating the independent variables that we will use in the next section. Specifically, each of these variables corresponds to an indicator variable that takes on a value of 1 during the days that we have defined as a binding period in Table 2, and 0 for the other days in our sample. The main idea is that these variables signal the periods when the public debt is constrained by a debt ceiling, and then use this information to explain the decreases in the G-Fund balance. In the following section, we provide a formal time series analysis of this relationship.

5.2 Empirical strategy

In this section we provide the empirical strategy we use to estimate the effect of public debt behavior, debt ceiling, and debt suspension on the G-Fund based on a time series econometric specification as outlined in Wooldridge (Reference Wooldridge2018). We rely on the three debt-ceiling measures described above as independent variables. However, by construction BP1 and BP2 correspond to subsets of BP0, just as BP2 is also a subset of BP1. Mathematically, we define $t_{FDL, DCE}^0$ as the day that marks the start of a binding period which can be either an FDL, a DCE, or both as indicated in Table 2. Similarly, we define $t_{DCE, SUSP}^1$ as the day that marks the end of a binding period, which can be either a DCE or an SUSP day. Using these definitions, equations (1)–(3) define the rules associated with each variable:

(1)$$BP0_t = \left\{{\matrix{ {1\;\quad if\;t\in [ {\;t_{FDL, DCE}^0 , \;\;t_{DCE, SUSP}^1 } ] } \hfill \cr {0\;\quad Otherwise} \hfill \cr } } \right.$$

(2)$$BP1_t = \left\{{\matrix{ {1\quad if\{ {\;t\in [ {t_{FDL, DCE}^0 , \;\;t_{DCE, SUSP}^1 } ] } \} \;\wedge \{ {\;TSF_t < TSF_{t_{DCE, SUSP}^1 }} \} \wedge \;\{ {TSF_t-\;TSF_{t-1}{\kern 1pt} {\rm \leqslant }{\kern 1pt} 0} \} } \hfill \cr {0\quad Otherwise} \hfill \cr } } \right.$$

(3)$$BP2_t = \left\{{\matrix{ 1&{if\{ {\;t\in ( {t_{FDL, DCE}^0 , \;\;t_{DCE, SUSP}^1 } ) } \} \wedge \{ {\;TSF_t < TSF_{t_{FDL, DCE}^0 }} \} \;}\cr&\wedge \{ {\;TSF_t < TSF_{t_{DCE, SUSP}^1 }} \} \wedge \;\{ {TSF_t-\;TSF_{t-1}{\kern 1pt} {\rm \leqslant }{\kern 1pt} 0} \}\cr 0&{\it Otherwise}\hfill}}\right.$$

Each variable imposes additional restrictions on the set that is included in the binding period. BP0_t takes a value of 1 for the days that are in between the binding period including the first and last day marked by an FDL, DCE, or SUSP, and 0 otherwise. BP1_t takes the value of 1 for the days that are included in the binding period but without considering the last day, because the G-Fund balance had already recovered by that time. Hence, the two additional restrictions imposed with BP1_t are such that this variable takes on value of 1 only when the balance is decreasing and until it reaches the lowest point. Finally, BP2_t takes the value of 1 only for the period where the G-Fund decreases, which is a subset of the whole period $[ {\;t_{FDL, DCE}^0 , \;\;t_{DCE, SUSP}^1 } ]$. Because the three variables are similar and the interpretation is the same, for ease of notation from this point forward we collapse the notation to BP _t to refer to all three variables. The main purpose of these variables is to define the time interval for when the large decrease occurs in three different ways, in case the minimum and maximum of the interval interfere with the average decrease estimate. Due to the stylized facts emerging from Figure 7, we expect a negative effect from BP _t to TSF _t and hence, negative regression coefficients associated with these variables.

The first time series model we use is the geometric distributed lag (GDL) model, which is a simplification of the more generalized infinite distributed lag (IDL). The IDL is specified by equation (4), where TSF _t is the balance of the G-Fund in time t, BP _t is a dummy variable that takes the value of 1 if we are in the interval of the binding period from Table 2 and 0 otherwise, α is the intercept, and u _t is the error term:

(4)$$TSF_t = \alpha + \delta _0BP_t + \delta _1BP_{t-1} + \delta _2BP_{t-2} + \cdots + u_t$$

The binding period begins when the debt is constrained by the debt ceiling and ends after a DCE or suspension. The G-Fund decreases in response to being constrained by the debt ceiling and then increases once the debt ceiling is either expanded or suspended. Therefore, the G-Fund balance depends on BP and potentially its lagged values. Hence, the next question is how many lags are needed to accurately model this relationship. The IDL model proposes infinite lags which is empirically impossible to implement; however, by making assumptions we simplify the model to create an estimable equation. Consequently, we assume that δ _j depends on two parameters, γ and ρ, such that δ _j = γρ ^j for all j = 0, 1, 2, …, with the restriction that |ρ| < 1. This restriction ensures that δ _j → 0 as j → ∞, which means that the impact of BP _t−j on TSF _t will decrease as j becomes larger. The intuition behind this assumption is that time lags of BP will at some point end in terms of identifying days outside the binding period, and therefore the days where the G-Fund balance is no longer affected. Applying this assumption to equation (4) results in equation (5). If we lag equation (5) over one period and then multiply by ρ, we obtain equation (6). Finally, subtracting equation (6) from equation (5) results in the equation to be estimated (7), where α ₀ = α(1 − ρ) and v _t = u _t + ρu _t−1:

(5)$$TSF_t = \alpha + \gamma BP_t + \gamma \rho BP_{t-1} + \gamma \rho ^2BP_{t-2} + \cdots + u_t$$

(6)$$\rho TSF_{t-1} = \rho \alpha + \gamma \rho BP_{t-1} + \gamma \rho ^2BP_{t-2} + \gamma \rho ^3BP_{t-3} + \cdots + \rho u_{t-1}$$

(7)$$TSF_t = \alpha _0 + \rho TSF_{t-1} + \gamma BP_t + v_t\quad for\,\,\;t = 1, \;2, \;\ldots $$

There are two important features about equation (7) that require some discussion. First, one of the assumptions to obtain consistent estimates from an ordinary least squares (OLS) estimator is the zero-correlation assumption, $E( {x_t^{\prime}u_t} ) = 0$. However, by construction we have E(TSF _t−1u _t) ≠ 0, even assuming independence between u _t with respect to BP _t and all past values of TSF _t and BP _t. Therefore, assuming BP _t is exogenous, we need an instrumentalized TSF _t−1 in equation (7). The most suitable instruments to estimate (7) are BP _t and BP _t−1. Note that BP _t−1 is a valid instrument for TSF _t−1 because it meets the requirements of an instrumental variable (IV). This IV complies with the exclusion restriction since u _t and u _t−1 are uncorrelated with BP _t−1, which implies that v _t is uncorrelated with BP _t−1. This suggests that the previous binding period follows the same assumptions as the current binding period, that is, they affect the TSF directly and in the period in which they occurred. In addition, the instrument also complies with the relevance assumption because BP _t−1 and TSF _t−1 are correlated, according to our model. The binding periods will (and do) affect the balance of the TSF.

In this case, we estimate equation (7) with IVs and adjust by serial correlation in v _t. Second, we exploit the fact that {u _t} may contain a specific kind of serial correlation such as assuming that {u _t} follows an AR(1) model, as indicated by equation (8), with E(e _t) = 0 and $Var( {e_t} ) = \sigma _e^2$:

(8)$$u_t = \rho u_{t-1} + e_t$$

Using the assumption from equation (8) we restate equation (7) as a dynamically complete model as shown in equation (9). This model can be estimated by OLS, providing consistent estimates. Nevertheless, obtaining consistent estimates depends greatly on the assumption that {u _t} follows an AR(1) model, which implies that ρ is the same parameter in equations (8) and (9). We use the approach suggested by McClain and Wooldridge (Reference McClain and Wooldridge1995) to test this assumption. Finally, it is worth noting that due to the nature of the underlying geometric series in this model, the long-run propensity (LRP) impact is γ/(1 − ρ), and can be generated from the regression coefficients:

(9)$$TSF_t = \alpha _0 + \rho TSF_{t-1} + \gamma BP_t + e_t$$

In addition to the GDL model, we also use a more general model called rational distributed lag (RDL). Following the same notation as in the previous equations, the RDL model can be specified as shown in equation (10), which now includes a lag of BP _t as an additional independent variable. The LRP in this case is (γ ₀ + γ ₁)/(1 − ρ):

(10)$$TSF_t = \alpha _0 + \rho TSF_{t-1} + \gamma _0BP_t + \gamma _1BP_{t-1} + v_t\;\quad \;for\;t = 1, \;2, \;\ldots $$

Below, we present the findings from both models with their LRP specifications to estimate the effect of BP _t on TSF _t.