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# July, 2014

## THE PRICE LEVEL: TESTING FOR FISCAL DOMINANCE 5

As Campbell and Shiller (1987) point out, if the discount factor is constant, the bivariate VAR includes all of the relevant information (because Liabilities/GDP is the expected present value of future Surplus/GDP). Of course, discount factors need not be constant so we also include the discount factor and consider a three-variable system. This allows us to determine whether the effect of an innovation to st on wt+1 is robust to conditioning on a, the discount factor, and to investigate the effect of innovations to st on future discount factors. Figure 4 shows impulse response functions from a VAR with a constant and two lags.

Once again the results are quite robust: adding a deterministic time trend, or using lag lengths of one, three or four, we get similar pictures. The VAR’s residuals are highly correlated; we show what would seem to be the two relevant orderings. In the top panel, Surplus/GDP comes first (as may make more sense in a FD regime), followed by Liabilities/GDP and a; in the bottom panel, Liabilities/GDP comes first (as may make more sense in a MD regime), followed by Surplus/GDP and a. The dashed lines represent the two standard deviation bands obtained by a Monte Carlo simulation with 500 repetitions.

## THE PRICE LEVEL: TESTING FOR FISCAL DOMINANCE 4

Figure 3 shows that the response of Liabilities/GDP in period 2 to an innovation in Surplus/ GDP in period 1 is negative and significant, regardless of the ordering used.20 The response of Liabilities/GDP in period 3 is also negative and significant. This can be readily explained in terms of a MD regime. But it is also consistent with a FD regime in which the primary surplus is exogenous (or at least unrelated to liabilities) but exhibits negative autocorrelation.

Does the primary surplus respond to liabilities thereby yielding a MD regime or is Surplus/GDP exogenous but negatively autocorrelated? We begin answering this question by examining the autocorrelation function of Surplus/GDP, which is found in Table 1. The autocorrelations and the corresponding Q-statistics clearly indicate that there is significant positive autocorrelation in Surplus/GDP, at least at lags of up to 9 years. Next, we return to the joint dynamics of Surplus/GDP and Liabilities/GDP and look at the impulse response functions from the VAR.

## THE PRICE LEVEL: TESTING FOR FISCAL DOMINANCE 3

Next we estimate a VAR with two lags of Surplus/GDP and Liabilities/GDP (and a constant).19 We tried other specifications: we added a deterministic trend; we used lags of one, three and four years; and we estimated the VAR in first differences. All of these VARs produced very similar results. Figure 3 shows the impulse response functions for an innovation in Surplus/GDP. Since the residuals in this VAR are highly (negatively) correlated, the ordering in the Cholesky decomposition may matter.

For this reason, we show the results for both orderings. In the top panel, Surplus/GDP comes first. This ordering allows for a contemporaneous affect on Liabilities/GDP as is consistent with a FD regime (where nominal GDP has to jump to make the value of the existing debt equal to the expected present value of surpluses). In the bottom panel, Liabilities/GDP comes first. This ordering may make more sense in a MD regime (where GDP can be determined elsewhere in the model) because it does not allow for a contemporaneous affect on Liabilities/GDP. The dashed lines represent the two standard deviation bands obtained by a Monte Carlo simulation with 500 repetitions.

## THE PRICE LEVEL: TESTING FOR FISCAL DOMINANCE 2

Before estimating the VAR, we test for stationarity. Both Phillips-Perron and augmented Dickey-Fuller tests reject a unit root in Surplus/GDP at the 1% significance level. The evidence on the stationarity of Liabilities/GDP is somewhat mixed. The Phillips-Perron test rejects a unit root at the 5% level. The augmented Dickey-Fuller test, on the other hand, does not reject a unit root at conventional levels.

These results are consistent with those found in the literature. Kremers (1989), using data from the inter and post war periods (1923-1940, 1951-1985), found Augmented Dickey-Fuller tests reject a unit root in the debt to GDP ratio at about the 10% level. Bohn (1995), using a longer data set (1916-1989), found that ADF tests reject unit roots in the surplus to GNP ratio at the 1% level and in the debt to GNP ratio at about the 5% level. The general consensus from the literature seems to be that there is strong evidence that the surplus to GDP ratio is stationary while the evidence on the stationarity of the debt to GDP ratio is somewhat weaker.

## THE PRICE LEVEL: TESTING FOR FISCAL DOMINANCE

This discussion does however suggest a way to differentiate between MD and FD regimes. Consider how a positive innovation in st passes to wt+1. In a MD regime, the surplus pays off some of the debt, and wt+1 falls. In a FD regime, there are several possibilities. Consider first the case in which an innovation in st is not correlated with the surpluses and discount factors that follow st on the RHS of (4). In a FD regime, the value of wt+1 can be found by shifting (4) forward one period.

In the case we are considering, w,+1 should not be affected by the innovation in st. Consider next the case in which an innovation in s, is positively correlated with future surpluses and discount factors. In this case, wt+1 should rise. In either of these cases, it should in principle be possible to differentiate between MD and FD regimes. For example, the impulse response function from a VAR in s, and wt would tell us how wt+1 responds to an innovation in st. If wt+l falls, we have an MD regime; if it does not, we have a FD regime.

## THE PRICE LEVEL: THEORETICAL CONSIDERATIONS 5

We don’t consider the above Proposition’s assumptions about fiscal policy to be overly restrictive. It seems natural to assume that the ratio of the primary surplus to GDP, and thus {e^}, is bounded and that an increase in debt does not cause a decrease in the primary surplus (so that Cj г 0 holds). The lower bound a. on aj and the upper bound c* on Cj are plausible (and analytically convenient) assumptions. The growth rate of the economy can exceed the interest rate for finitely many periods without violating (Cl). However, the condition lim sup otj < 1 is necessary for the government’s present value budget to be well defined; without it the government could roll over debt indefinitely regardless of the value of wt. The only substantive assumption is that Cj is bounded away from zero infinitely often. This will be the case unless the fiscal authority tries to rollover the interest due on debt indefinitely. In periods in which Cj > 0, Sj is indeed moving to stabilize Wj. However, it need not do so each and every period. A stabilizing policy could be in effect every other year, or every third year, or every decade. Indeed, the required fiscal retrenchment need not occur in the next 100 years! All that is necessary is that the private sector expects that there will sooner or later be a retrenchment. In the meantime, fiscal policy can respond to economic or political conditions (as represented here by the random variable €j).

## THE PRICE LEVEL: THEORETICAL CONSIDERATIONS 4

The MD regime might seem like a rather special case, but it turns out that many different fiscal policy rules lead to MD regimes. Moreover, these rules can be quite lax. The MD regime is not as implausible as it may at first appear. To illustrate this point, suppose that the sequence {Sj} is expected to follow the rule

where Cj is a time varying response parameter and ej is a random variable. €j could represent political factors or economic conditions, such as unemployment. We have the following proposition:
Proposition: Assume that {Cj}, {ctj} and {Gj} are deterministic sequences, that {€j} is bounded, and that the following conditions hold:

(5) imply that the present value constraint (4) holds for any arbitrary value of wt. In other words, the fiscal rule (5) results in a MD regime if {€j} is bounded and conditions (Cl), (C2) and (C3) hold. We prove this proposition in an appendix, and we extend it to a stochastic environment.

## THE PRICE LEVEL: THEORETICAL CONSIDERATIONS 3

The MD and FD regimes can now be defined in terms of the present value constraint (4). If primary surpluses (or more precisely the surplus to GDP ratios) are determined independent of the level of the debt, then nominal income and/or discount factors must “jump” in equilibrium to satisfy (4). We call this a FD regime. If on the other hand primary surpluses are determined in such a way that (4) is always satisfied no matter what nominal income and discount factors are fed into it, then nominal income and the discount factors can be determined elsewhere in the model. We call this a MD regime. In summary, nominal income is determined by the needs of fiscal solvency in a FD regime; it can be determined in more conventional ways in a MD regime. Once we specify the way in which changes in nominal income are split between price and output, we have a theory of price determination. This last step is obviously model specific, and controversial. Fortunately, the restrictions we need for Section III do not require us to take a stand on these issues.

## THE PRICE LEVEL: THEORETICAL CONSIDERATIONS 2

We want to express the budget constraint in terms of total government liabilities, M + B, and to scale the fiscal variables on GDP. We do this to facilitate policy discussions and empirical applications. After some tedious algebra, the budget constraint becomes

(2) says that the ratio of total government liabilities to GDP has to be equal to the ratio of the primary surplus (now inclusive of central bank transfers) to GDP plus the discounted value of the ratio of next period’s liabilities to GDP; the discount factor is the ratio of the real growth in GDP to the real interest rate. Finally, we want to simplify our notation by replacing (2) with

Wj is the liabilities to GDP ratio, Sj is the surplus to GDP ratio, and a} is the discount factor. It should be kept in mind that Sj includes central bank transfers (or seignorage).