# Debt–liquidity shock risk: intertemporal effects and probability measures

## Bernardo Maggi

#### Need to know

• The authors study the liquidity of the Italian Government bonds market.
• The intertemporal effects of a debt liquidity shock are estimated.
• The opportunity of an issuance procedure that allows to reduce issuance costs is assessed.
• The authors evaluate the relevance of the average liquidity cost indicator by using probability measures of liquidity shock risk.

#### Abstract

ABSTRACT

We analyze how the yield of government securities may be managed in order to save costs in the face of the risk of a liquidity shock. This issue is especially relevant for highly indebted countries that are under the threat of positive changes in the yields of public debt bonds. We study the liquidity features of the Italian government bond market with the twin aims of formalizing and estimating the intertemporal effects of a debt-liquidity shock on yields, with a view to understanding how to reduce costs when issuing new bonds, and evaluating the effectiveness of the average liquidity cost index by finding its conditional probability.

#### Introduction

We study how the yield of government securities may be managed in order to save costs in the event of a debt–liquidity shock risk. This issue is especially relevant for officers in charge of government debt management in highly indebted countries, which are under the threat of increasing yields, since a more liquid security usually has a higher price or a lower yield (and vice versa). As is known, a liquidity shock may occur following the auction of a new security, as a result of the pressure exerted by primary dealers selling analogous securities in the secondary market before purchasing the new securities at the treasury auctions. As a consequence, a liquidity premium arises, consisting in the increase in the yield to compensate for the lower liquidity of the security being sold. This fact, together with the imperfect capital mobility due to few end-investors and arbitrageurs in the liquidity market, supports our data; we find reductions in the yield after the auction, once the pressure by primary dealers in the secondary market has vanished. Therefore, the yield (price) quoted at the auction is often higher (lower) than that of the following period in the secondary market (underpricing). In this regard two problems are relevant for highly indebted countries. On the one hand, it is important to formalize the time profile of the effects on the yield due to the risk of a liquidity shock in order to determine a more appropriate issuance schedule that allows the government agency to reduce the costs associated with these liquidity shocks. On the other hand, there is a need for more information on the data-generating process regarding the liquidity shocks in order to evaluate, in terms of conditional probability, the related risk.

Regarding the policy for new issuances, Lou et al (2013) look at the increasing price dynamics in the US secondary market during the auction period, in order to discern the amount saved by the US Treasury when fixing the auction price with an efficient analysis of the price changes in the secondary market. However, they also highlight the difficulties in anticipating the effects of a liquidity shock, which would help to improve the design of the treasury auction mechanism. In line with this result, we find that for the Italian treasury bonds secondary market there is evidence of underpricing, namely an increasing return for the most recently issued (ie, on-the-run) note before the auction and a decreasing return during the post-auction period. Hence, we argue that an appropriate issuance schedule would reasonably reduce costs. We perform a dynamic analysis that allows us to evaluate the persistence of a liquidity shock driven by the issuance of a new on-the-run security. We show that in a single year €6.06 million in costs could be cut, against issuances of €7 billion, with a yield reduction ranging from 2.4 to 18.6 basis points (bps).

Regarding the evaluation of the liquidity cost indicators that may be used in the analysis, Goldreich et al (2005) suggest deriving these indicators from a regression of the yield difference between two securities, identical but for the degree of liquidity, on the difference between the liquidity cost indicators. We first show that, though it is empirically useful, the average liquidity cost indicator does not derive directly from the equations of the theory underlying the pricing of an illiquid bond. Further, after having performed the empirical analysis on the basis of the average liquidity cost indicator, we compare it with the one that is actually implicit in the theory. We are able to derive both the joint probability of all costs due to the resulting liquidity shocks and their probability conditioned on the traditionally used average liquidity cost indicator, thus allowing us to evaluate its effectiveness.

As a representative case of a high-debt country, we consider Italy for both its large debt/GDP ratio (132.1% in 2014) and the high share of debt held abroad, which was around 35% of the total debt in 2014 (Banca D’Italia 2015). We use liquidity indicators that have been proved to better represent the liquidity of the Italian bond market, and distinguish between the most recent, ie, on-the-run, auctioned government securities in each maturity sector and those issued previously (ie, off-the-run), which differ with respect to the degree of liquidity.11Another approach, applied to the Spanish treasury bonds market, is given in Alonso et al (2000). It takes into consideration the yield curve for a wide set of securities over two years by means of the Nelson and Siegel (1987) model. They gauge the liquidity effect by introducing dummies according to the liquidity status of the security considered: benchmark, pre- and post-benchmark, strippable and nonstrippable. They find evidence in favor of the existence of a small liquidity premium for post-benchmark bonds. A similar analysis on the term structure for the Italian government bonds with the Cox–Ingersoll–Ross model may be found in Maggi and Infortuna (2008). In particular, we examine the relationship between several proxies of current and future liquidity and the yield difference between off-the-run and on-the-run bonds over the on/off cycle.22While theoretical studies suggest that the bond price is affected by both current and expected future liquidity, the empirical literature has very often focused only on the former. So far, similar studies have been conducted only on US data. Nonetheless, although the Italian government bonds market differs greatly from the US both in the issuance modality and in other aspects affecting the liquidity of securities, we are able to capture and estimate a liquidity effect on the yield difference between off-the-run and on-the-run bonds. The estimated liquidity premium for an increase in the illiquidity is quantitatively relevant, and amounts to about 0.42bps for a change in the liquidity cost of 0.01bps.

The paper is organized as follows: Section 2 presents the data set; Section 3 shows the model and the empirical methodology; Section 4 presents the empirical analysis for the liquidity cost; Section 5 deals with the intertemporal effects of a liquidity shock on the yield and with the possibility of reducing costs through more efficient issuance scheduling; Section 6 evaluates different probabilities in the event of liquidity shock and assesses the effectiveness of the liquidity cost indicators considered; Section 7 concludes. (Appendix A online reports the formulas for the liquidity indexes adopted (price and quantity), and Appendix B online shows the covariance matrix specific to the estimated model.)

## 2 Bond market data

We collected intraday data of seven two-year zero-coupon bonds (Certificati del Tesoro Zero coupons, or CTZs) ranging from January 2004 to December 2006 using the MTS market platform provided by the Italian Ministry of Economy and Finance. MTS is a leading wholesale market for the electronic trading of fixed-income securities in Europe, where electronic transactions make up about 80% of the total trade volume of government securities. We constructed a data set that comprised five-minute snapshots of the entire order book.33We implemented a filtering methodology to deal with the huge original database. Because of the size of methodology needed to treat outliers, missing records, etc, appropriately, it is not reproduced here; the data is available upon request from the author. The empirical analysis and the filtering program were developed with Stata 12 software. Thus, one main feature of this data set is that it includes all intraday quotes and transactions, rather than trades that go through the major interdealer brokers as in typical previous studies. Further, while market participants usually have access only to the five best bid and ask quotes and related quantities, we had at our disposal the entire book, which helped us to investigate the market in depth. We also used daily yields drawn from the Bloomberg database on market detection at the end of each trading day.

Table 1 summarizes the on- and off-the-run status and the periods during which data was collected for the seven CTZs used in our study. When the first CTZ is auctioned, it is on-the-run, or benchmark, until the second CTZ is issued. From that moment on, the first CTZ becomes first off-the-run. After the third CTZ is issued, it then becomes second off-the-run, and so on. To simplify the reading, we numbered our securities progressively from one to seven; these correspond to the International Securities Identification Number (ISIN) codes 347137, 353172, 364676, 369706, 383119, 392699 and 405105, respectively.

There are two main reasons why we adopted CTZs in our analysis. First, the yield differential cannot depend on differences in coupons. Second, the relative short-term maturity of these securities allowed us to compute and analyze liquidity over the entire on/off cycle of each note.

As the auctions do not follow a regular schedule, but instead vary between four and eight months, the period during which a specific note is considered on-the-run or benchmark turns out to be irregular.44This is a peculiarity of Italian government bonds. Actually, CTZs are auctioned monthly, but most of these monthly auctions are reopenings of “old” securities. As a result, if we consider the six pairs of on-the-run and first off-the-run CTZs available progressively in the market, the time periods for each pair are also irregular and roughly amount to three, four, eight, six, seven and eight months, respectively.

Our next step is to relate the yield differential of two securities to specific indicators suitable for measuring their different degrees of liquidity, and thus capable of extrapolating the liquidity premium, ie, the cost of an increase in the illiquidity embedded in the differential. Toward this aim, we adopt some liquidity indicators suggested in the literature (see, for example, Goldreich et al 2005) and constructed from the order book. Moreover, the liquidity indicators we used proved to be very suitable for our specific study, as they are extremely relevant for describing the market liquidity of the Italian government bonds.55See, in particular, Coluzzi et al (2008), who test these measures and find them to be representative of the liquidity of the Italian public debt traded in the secondary market. They are

1. (1)

2. (2)

3. (3)

the slope (slope),

4. (4)

5. (5)

the market quality index (mqi).

The first three are price indexes; the remaining two are quantity indexes.66For a detailed descriptive analysis of the securities considered and the liquidity indicators adopted, see Delle Chiaie and Maggi (2014). Importantly, bs and ws, referring to the bid–ask spread of the bond market, respectively represent the “exact” cost and the “weighted” cost that the ask side must pay above the bid price for a desired quantity of security, ie, the trading cost due to the risk of a liquidity shock. Therefore, price indexes are related negatively to the degree of liquidity and positively to the yield. A similar reasoning may be applied to the slope, which resembles the angular coefficient between prices and quantities, as may be checked in Appendix A online; the opposite reasoning may be applied for the quantity indexes, because an increase in the quantity of the traded security reduces the gap between the ask and bid sides of the market.77Of course, we are assuming that for the quantity traded the ask price is higher than the bid price, which is the condition that makes new auctions possible, and opens up the risk of liquidity shock.

As mentioned above, we choose to compare the most liquid security (the first off-the-run) with the less liquid on-the-run security.

## 3 Model and methodology

### 3.1 Model and strategy analysis

We group the seven two-year treasury notes issued in our sample period into six pairs of consecutive notes. From the issuance date until the on-the-run note goes off-the-run, we compute the daily yield difference between the two off/on-the-run securities.

A potentially serious problem is that, although they are very close in maturity, the two notes we compare are not at exactly the same point on the yield curve. Hence, if the yield curve is not flat we would expect them to have different yields even in the absence of any liquidity effect. In order to cope with such a problem, we use the slope of the euro swap curve. From a computational point of view, we subtract from the yield differential of the two CTZs the difference of their asset swap spreads.88The asset swap spread is a proxy for the difference between the bond yield and the swap rate for the same maturity. After this correction, we can study the difference in liquidity for each off/on-the-run pair, since the yield differentials do not depend on difference in coupons for a CTZ. As mentioned earlier, the less liquid security is expected to have a higher yield to compensate for the illiquidity, which in our case means that the on-the-run security, being traded over a shorter time and thus being less liquid, has a higher yield in the pairs considered.

Figure 1 shows the average yield differential for each day of the on/off cycle.99The average is computed over the cross-section of the four pairs of securities with a longer life span. Overall, such a differential is negative, ie, the on-the-run shows a higher yield than the off-the-run once the curve effect has been taken into account.1010Another difference that does vary systemically over the on/off cycle and merits further research is the specialness in the repo market. This affects the liquidity premium associated with on-the-run issues as predicted by the theoretical model of Duffie (1996), in line with the empirical findings in Jordan and Jordan (1997). The repo specialness is particularly pronounced in Italy (Corradin and Maddaloni 2015). Sundaresan (1994) documents that newly issued treasury securities tend to trade as special repo contracts in repo markets, with the extent of specialness reflecting auction cycles. However, the differential is not stable over the period: while within the first sixty trading days it moves by around 10bps, after this period the differential falls significantly to around 20bps and remains at this level.

First, we note that the negative difference in the yields tends toward zero around the second monthly reopening auction. Second, the fall in the differential in the last part of our observation period may be explained by the fact that the residual maturity of the off-the-run security in each pair approaches one year. In fact, when the CTZ’s maturity gets close to one year, the security becomes a short-term instrument, so that the trading activity jumps even further in favor of the more liquid off-the-run security. Moreover, after (before) each sell in the primary market (ie, the auction at time zero and the two following monthly reopening auctions), there is a rise (fall) in the yield of the on-the-run security with respect to the off-the-run one. This highlights the higher liquidity of the off-the-run security, especially once the effect of the new issuances in the primary market has vanished, and provides evidence of the way the underpricing operates. Hence, we underline the opportunity to manage the new issuances schedule appropriately with such a price swing, in order to reduce costs by taking advantage of the higher prices, ie, lower yields, at which the new securities might be issued.

Despite the peculiar behavior of the yield differential in the Italian government bonds market, we are able to detect a liquidity effect on the yield differentials.1111As stated earlier, the on-the-run security reaches its outstanding value, on average, around the second monthly reopening auction, so that all this period is characterized by the underpricing phenomenon. This is another reason (besides the liquidity difference explained in the theoretical section) for the negative difference in the yields off/on-the-run. In fact, once the issuance period is concluded, the difference becomes positive. Moreover, this positive trend may be explained by the possible over-bidding at the auction by the primary dealers, who aim to reach the highest ranks of the dealers favored by the treasury. These primary dealers in Italy are the so-called specialists, who, before auctions, sell in the secondary market in order to have the largest proportion of new auctioned securities in their portfolio. The reason the positive yield differential in the United States is more persistent than in Italy, despite decreasing (see Goldreich et al 2005), probably lies in the more concentrated (monthly) auctions without reopenings. Further, as outlined by Pelizzon et al (2016), the frequent reopenings in the Italian bond market attenuate the liquidity differences between off- and on-the-run securities.

Now, we briefly present the theoretical foundation of our econometric analysis. Theoretical studies show that the yield of a bond is equal to the yield of a perfectly liquid bond plus a term that captures current and expected future trading costs. Accordingly, we assume that the yield differential of two bonds with different degrees of liquidity may be given by

 $\mathrm{YD}_{it}=\beta_{i}(I_{\mathrm{off},it}-I_{\mathrm{on},it}),$ (3.1)

where $\mathrm{YD}_{it}$ is the yield spread (the difference between $y_{\mathrm{off}}$ and $y_{\mathrm{on}}$) and $I_{j,it}$ is the index of liquidity adopted for the security in the $j$th (off, on) state and belonging to the $i$th pair at time $t$. Indeed, provided that the index $I_{j,it}$ represents well the cost of liquidity, (3.1) may be proved to derive from the expression of the price of an illiquid bond, once the forward rate and the probability that a liquidity shock hits the bond are taken into account. In fact, by defining the instantaneous forward rate $f_{\tau}$, the price of an illiquid bond $P_{t}^{\mathrm{I}}$, the price for a liquid bond $P_{t}^{\mathrm{L}}$ and the instantaneous cost rate for the risk of a liquidity shock, $c_{\tau}$, to which the probability, $\beta$, is related, it is true that1212We assume, for simplicity, a constant probability of liquidity shock to which is associated a certain cost rate $c_{t}$. If the probability of a liquidity shock and the associated cost rate are uncertain, then an expectation operator and covariance terms may be considered without altering the conclusions of the analysis.

 $\displaystyle P_{t}^{\mathrm{I}}=\exp\bigg({-}\int_{t}^{T}(f_{\tau}+\beta c_{% \tau})\,\mathrm{d}\tau\bigg)=\mathrm{e}^{-\beta\bar{c}_{t}(T-t)}P_{t}^{\mathrm% {L}},$ $\displaystyle C_{t}=\int_{t}^{T}c_{\tau}\,\mathrm{d}\tau,\quad\bar{c}_{t}=% \frac{C_{t}}{T-t},$ (3.2)

which, in terms of yields, is

 $\displaystyle P_{t}^{\mathrm{I}}=\exp(-\bar{y}_{t}^{\mathrm{I}}(T-t))=\exp(-(% \beta c+\bar{y}_{t}^{\mathrm{L}})(T-t)),$ $\displaystyle Y_{t}^{\mathrm{I}}=\int_{t}^{T}(f_{\tau}+\beta c_{\tau})\,% \mathrm{d}\tau,\quad\bar{y}_{t}^{\mathrm{I}}=\frac{Y_{t}^{\mathrm{I}}}{T-t},% \quad Y_{t}^{\mathrm{L}}=\int_{t}^{T}f_{\tau}\,\mathrm{d}\tau,\quad\bar{y}_{t}% ^{\mathrm{L}}=\frac{Y_{t}^{\mathrm{L}}}{T-t},$ (3.3)

where, $\bar{y}_{t}^{\mathrm{L}}$, $\bar{y}_{t}^{\mathrm{I}}$ and $\bar{c}_{t}$ represent, for the maturity considered, the average of the current and future interest rates and the cost rate associated with the yields of a liquid ($Y_{t}^{\mathrm{L}}$) or illiquid ($Y_{t}^{\mathrm{I}}$) bond and with the cost of a liquidity shock ($C_{t}$), respectively. From (3.3), it follows that

 $\bar{y}_{t}^{\mathrm{I}}=\beta\bar{c}_{t}+\bar{y}_{t}^{\mathrm{L}}.$ (3.4)

Expression (3.4) implies that, if the cost rate due to a liquidity shock increases, a liquidity premium consisting in the rise of the yield is to be paid by the more illiquid security. Moreover, if we consider two securities, off- and on-the-run, the latter expression can be formulated as

 $\bar{y}_{\mathrm{off},t}-\bar{y}_{\mathrm{on},t}=\beta(\bar{c}_{\mathrm{off},t% }-\bar{c}_{\mathrm{on},t}).$ (3.5)

Now, since $\smash{\bar{c}_{t}^{j}}$ is an index representing the cost of the risk of a liquidity shock, (3.5) is the analog of (3.1) apart from the use of the yields instead of the interest rates as the dependent variable. It is worth stressing that (3.5) allows us to estimate $\beta$, as opposed to (3.4), since it is difficult to find data for a perfectly liquid bond ($\bar{y}_{t}^{\mathrm{L}}$).

### 3.2 Empirical methodology

Then, according to (3.5), if we want $\beta$ to be interpreted as the probability associated with a liquidity shock, the index $I_{j,it}$ of (3.1) must necessarily be based on the best spread of the possible measures (1)–(5). In fact, as stated above, bs represents, by definition, the trading cost between the ask and bid sides of the order book in the case of a liquidity shock, which is just $c_{t}$ in the theoretical model. More specifically, in such a case, $I_{j,it}$ would coincide with the average cost of liquidity ($\bar{c}_{t}$). However, in the rest of the analysis and in the following estimations, although price measures, and the best spread in particular, will be examined for the reasons given above, we will also deal with quantity indexes in order to test the robustness of our model.

As argued above, the smaller the price indexes, the larger the quantity indexes, and thus the more liquid the order book. Then, a fall in the differentials of the spreads or slope indexes and an increase in the differentials of the quantity indicators are associated with a fall in the off-minus-on yield. Hence, we expect a positive (respectively, negative) coefficient $\beta$ for the price (respectively, quantity) regressor.

We present two sets of estimations. The first provides a comparison (and is in line) with the literature cited earlier. It is based on the definition of the index $I_{j,it}$ of (3.1) as the average over the time to maturity $(T-t)$ of the current and future liquidity ($\bar{l}_{j,it}$) measured by the indicators (1)–(5) of Section 2, here represented by $l_{j,it}$:

 $\bar{l}_{j,it}=\frac{1}{T-t}\sum_{\tau=t}^{T}l_{j,i\tau},\qquad L_{j,it}=\sum_% {\tau=t}^{T}l_{j,i\tau}.$ (3.6)

Therefore, from (3.6), $\beta$ accounts for the time to maturity $(T-t)$ and refers, with daily data, to a CTZ of 720 days. As a consequence, if the index $\bar{l}_{j,it}$ refers to the average cost rate of liquidity, ie, the average of bs, we need to assess the estimation of $\beta$ over the above period in order to evaluate the daily probability of a liquidity shock.

A second estimation will be carried out by defining the index $I_{j,it}$ of (3.1) as the summation, $L_{j,it}$, of the present and future liquidity cost indicators, corresponding to the theoretical background (3.2)–(3.5), which relates that summation to the yield, or the average indexes to the interest rate. In effect, in the literature (see Goldreich et al 2005) this aspect is not analyzed in depth, and the concepts of yield and interest rate are interchanged for continuous time. Indeed, analyzing this point in detail is crucial for the calculation of a reliable probability of the liquidity shock risk.

Our empirical methodology consists in the econometric estimation of (3.1) with autoregressive AR(1) residuals and random effects

 $\displaystyle\mathrm{YD}_{it}=\beta_{i}(I_{\mathrm{off},it}-I_{\mathrm{on},it}% )+\varepsilon_{it},$ $\displaystyle\varepsilon_{it}=\rho\varepsilon_{it-1}+\mu_{i}+v_{it},\quad v_{% it}\sim\mathrm{iid}(0,\sigma_{v}^{2}),\quad\mu_{i}\sim\mathrm{iid}(0,\sigma_{% \mu}^{2}),$ (3.1′)

where $\varepsilon_{it}$ is the error term, $\rho$ is the coefficient of the lagged errors, $\nu_{it}$ and $\mu_{i}$ are the time and random residuals effects, respectively, and iid denotes independent and identical distribution.

The above formulation is particularly suitable for our analysis, and for models having independent forward variables in general, because of the autocorrelation generated in the residuals. More specifically, we follow Hamilton (1994) and exploit the asymptotic proprieties, implied by our high-frequency data, of the generalized-least-squares (GLS)–Cochrane–Orcutt iterated procedure, which is consistent with spurious regressions and allows us to account for the structure of the error terms. Therefore, at each iteration step we calculate $\beta$ and then apply the robust GLS estimator to find $\beta$ according to the new error-covariance matrix. We use this procedure for both time series (single pairs) and panel data series (panel of pairs) analyses.1313The estimation method mentioned above is consistent. Nonetheless, we calculate robust standard errors as in White (1980, 1982) to control for possible misspecifications. Actually, the panel data models are estimated with fixed effects. Still, in order to obtain robust estimations, we allow for a full error-covariance matrix, which implies, for each single security, both autocorrelations dependent on lags and possible fixed autocorrelations, ie, random effects, as described in the online appendix. From the theory, we also deduce a more appropriate random effect among securities rather than a fixed one, the yield differentials in (3.5) being dependent only on the different degrees of liquidity (see Appendix B online for the derivation of the error-covariance matrix). Moreover, for the panel case a different methodology will be introduced in Section 5 by considering, through the robust generalized-method-of-moments (GMM)–Arellano–Bond estimator, the effect of the lagged dependent variable, in order to appositely deal with the intertemporal effects of a liquidity shock and to check the validity of our results under a different estimation method.1414In the following estimations we omit the constant terms, for simplicity, because our prime interest is in the effect of a change in the liquidity cost on the yield. Moreover, such terms are very small compared with the coefficient $\beta$ of interest, and in many cases not significantly different from 0, in accordance with the expounded theory.

## 4 Liquidity shocks

### 4.1 Overall estimates

We first estimate (3.1) by performing single equation regressions for each of the five measures constructed from proposals of the order book, as outlined in Section 2, and considering the case in which our regressor is based on the index $I_{j,i\tau}=\bar{l}_{j,it}$.

We find quite a consistent yield premium, amounting to about 0.31–0.49bps, per percentage basis point rise in the spread, if we consider the best spread as the cost indicator.

### 4.2 Liquidity cost

We observe that in the previous estimations the autocorrelation coefficients of residuals ($\rho$) are almost identical for the same pair via several liquidity indexes. Such evidence means that the liquidity indexes presented are capable of interpreting the dynamics of the dependent variable in a similar way, and hence have the same explicative economic meaning in terms of liquidity, even if they are constructed on different bases (price and quantity).1515Using the same methodology, we also carried out rolling regressions, obtaining similar paths for the coefficients of the several liquidity indicators, ie, again corresponding to the common meaning of liquidity. However, since the values of $\beta$ are different between pairs, we need to perform a panel analysis in order to check whether a unique and reliable coefficient of the cost rate of liquidity is retrievable, especially with regard to price indicators, and the best spread in particular. In Table 3, panel regressions on price indexes1616We also carried out a panel regression for the quantity indicators, but in this case we found a greater variability across securities, and a specific security effect on the coefficients for each measure was found to be necessary in order to obtain good results, which rendered the panel estimation useless. are carried out with the same GLS–Cochrane–Orcutt procedure as before. We also control for random effects, heteroscedasticity and correlation across different pairs of securities.

Looking at the best spread, the yield premium is significant and is estimated to be about 0.42bps for a change in liquidity cost of 0.01bps. Yet the other two measures perform well and, in the case of ws, even better than we saw in Table 2. Importantly, the $\rho$ terms, all with the same order of magnitude, are a first indication of the intertemporal effect of the liquidity cost on the yield. In fact, as is known, they may be associated with the lagged yield differential once (3.1) is reformulated in terms of quasi first difference according to the Cochrane–Orcutt procedure. However, the $\beta$ coefficients presented in both Tables 2 and 3 refer to the completion of the adjustment process, in that the autoregressive scheme of the errors specified in (3.1) is relevant only for the estimation in the aforementioned procedure and does not account for lagged variables in the structure of the model.

## 5 Intertemporal effects of a liquidity shock and efficient auction design

In this section, we quantify the future effects of a current liquidity shock in more detail in order to know if it is possible to improve the efficiency of the schedule of the new issuances and manage cost savings. In fact, once the liquidity shock occurs, the yield of the on-the-run security declines after the auction. As stated above, primary dealers sell on-the-run notes before a new issuance in order to buy new analogous securities in the auction. This causes a rise in the yield for the increasing illiquidity of the on-the-run security sold in the secondary market, and a consequent high yield is quoted at the auction. After the new issuance, once the pressure from primary dealers disappears, we find a decline in the yield, as shown later in Table 5, due to imperfect capital mobility and the few end-investors and arbitrageurs in the Italian liquidity market.1717Actually, Lou et al (2013) observe the same problem for the US bond market, which underlines the puzzling question of large markets and “persistence” of low prices, ie, the inability of the arbitragers to restore the price before auction. Therefore, we perform here a dynamic panel estimation based on the lagged effect of the dependent variable, which, besides having general relevance to the study of the resilience of a liquidity shock over time, may help improve the efficiency of the auction design. Of course, we conduct such a dynamic analysis on the basis of the average liquidity cost indicator ($\smash{\bar{l}_{j,it}}$), rather than on $\smash{L_{j,it}}$, since in this section we are interested in the intertemporal effects of a single liquidity shock due to auction.

Given that we need to cope with a dynamic panel data problem, we choose to adopt the GMM–Arellano–Bond estimator. In particular, we use the robust error-covariance version of this estimator. According to this method, strictly predetermined variables, ie, the lagged dependent and independent variables of (3.1), can be used as instruments. We then consider

 $\displaystyle\mathrm{YD}_{it}=\beta_{i}(I_{\mathrm{off},it}-I_{\mathrm{on},it}% )+\gamma_{\mathrm{YD}_{it-1}}\mathrm{YD}_{it-1}+\varepsilon_{it},$ $\displaystyle\varepsilon_{it}=\mu_{i}+v_{it},\quad v_{it}\sim\mathrm{iid}(0,% \sigma_{v}^{2}),\quad\mu_{i}\sim\mathrm{iid}(0,\sigma_{\mu}^{2}),$ (3.1″)

where the autoregressive structure of the error term has been replaced by lagged dependent variables. Therefore, we estimate

 $\bm{Z}^{\prime}\Delta\mathbf{YD}_{t}=\bm{Z}^{\prime}\Delta\mathbf{YD}_{t-1}% \gamma_{\mathrm{YD}_{t-1}}+\bm{Z}^{\prime}\Delta\bm{X}_{t}\beta+\bm{Z}^{\prime% }\Delta\bm{v}_{t},$ (5.1)

where $\bm{X}_{t}$ refers to $I_{\mathrm{off},it}-I_{\mathrm{on},it}$, with $i=1,\dots,N$, $t=1,\dots,T$,1818Note that “$T$” here is the final observation period of the considered pair, which is different from $T$ in (3.6) and in the previous formulas, which is the reference maturity time of a security. and $\bm{Z}$ is the instrument matrix for the whole panel of security pairs of order $N(T-2)P$, with $\smash{P=2\sum_{l=1}^{T-2}p}$. From the error structure of (3.1″), the first time difference of $v_{it}$ will have the following properties:

 $\left.\begin{gathered}\displaystyle\Delta\varepsilon_{it}=\Delta v_{it}\sim% \text{MA}(1),\\ \displaystyle\bm{V}_{i}=E(\Delta\bm{v}_{it}\Delta\bm{v}^{\prime}_{it-k})=% \begin{cases}2\sigma_{v}^{2},&k=0,\\ -\sigma_{v}^{2},&k=1,\\ 0,&k>1,~{}k=1,\dots,T-3,\end{cases}\end{gathered}\right\}$ (5.2)

of order $T-2$, where the error-covariance matrix for the whole panel, $\bm{V}$, is of order $N(T-2)$. Finally, the GMM-consistent estimator, $\bm{\varTheta}^{\prime}=(\gamma_{\mathrm{YD}_{t-1}},\beta)$, can be derived by application of the GLS algorithm:

 $\bm{\varTheta}=\{\Delta\bm{H}^{\prime}_{t}\bm{Z}[\bm{Z}^{\prime}\bm{V}\bm{Z}]^% {-1}\bm{Z}^{\prime}\Delta\bm{H}_{t}\}^{-1}\Delta\bm{H}^{\prime}_{t}\bm{Z}{[\bm% {Z}^{\prime}\bm{V}\bm{Z}]^{-1}}\bm{Z}^{\prime}\Delta\mathbf{YD}_{t}$ (5.3)

with variance regression $E(\bm{Z}^{\prime}\Delta\bm{v}\Delta\bm{v}^{\prime}\bm{Z})=\sigma_{v}^{2}\bm{Z}% ^{\prime}\bm{V}\bm{Z}$ and $\bm{H}^{\prime}_{t}=(\mathbf{YD}_{t-1},\bm{X}_{t})$.

In Table 4, $\beta$ is the coefficient of liquidity, now estimated in the short run, and $\gamma_{\mathrm{YD}_{t-1}}$ is the analog of $\beta$, ie, the coefficient of the lagged dependent variable. Again, all the coefficients are highly significant and with correct signs.1919The default number of instruments used for the estimation is consistent with (5.1). However, in order to disprove its occurrence for the coefficients obtained, we also tried regressions with a lower number of instruments and obtained similar results, which, as observed in the text, are still confirmed by the GLS robust regressions of Table 3. Above all, $\gamma_{\mathrm{YD}_{t-1}}$ allows us to derive a long-run liquidity shock effect, $\beta_{\text{long~{}run}}$, with the same order of magnitude as the earlier estimations of $\beta$, which amounts to about 0.4bps, for bs, for a change in liquidity cost of 0.01bps. In the short run, such an effect is about 0.22bps. The dynamic effect $\gamma_{\mathrm{YD}_{t-1}}$ is also consistent with that of $\beta$ obtained in Section 4.2, especially for bs, which is our primary concern. However, since the latter estimation method provides the significance test on the lagged term, we use this result to derive the speed of adjustment and the mean time lag (reported in Table 4), which characterize the intertemporal effects of a liquidity shock. The former measures the part of the gap between the actual value of the dependent variable and its “prescribed” value (usually an equilibrium value as a function of the independent variables) that is covered within the time unit. The latter is the amount of time, on average, it takes for a representative effect on the dependent variable to occur after a variation in the independent variable. In the present context, in order to show the dynamical quantitative implications for the yield of these two terms, it is useful to interpret our high-frequency data in continuous time. In fact, by considering as a prescribed value the long-run equilibrium, $\smash{\beta(I_{\mathrm{off},it}-I_{\mathrm{on},it})/(1-\gamma_{\mathrm{YD}_{t% -1}})}$, obtained using (3.1″) and omitting the residual term for simplicity (and with no loss of generality), the Arellano–Bond procedure implies the following adjustment relationship:

 $\Delta\mathrm{YD}_{it}=(1-\gamma_{\mathrm{YD}_{t-1}})\bigg[\frac{\beta}{1-% \gamma_{\mathrm{YD}_{t-1}}}(I_{\mathrm{off},it}-I_{\mathrm{on},it})-\mathrm{YD% }_{it-1}\bigg].$ (5.4)

By defining $\beta=(1-\gamma_{\mathrm{YD}_{t-1}})$, (5.4) can be rewritten in terms of continuous-time exponential lag distribution:2020The exponential lag distribution (5.5a), with probability function $\delta\mathrm{e}^{-\delta\tau}$, is the continuous counterpart of the discrete development of (5.4) in terms of geometric lag distribution, ie, the Koyck distributed lag equation (see Kenkel 1974).

 $\displaystyle\mathrm{YD}_{it}$ $\displaystyle=\int_{0}^{+\infty}\delta\mathrm{e}^{-\delta\tau}\frac{\beta}{1-% \gamma_{\mathrm{YD}_{t-1}}}(I_{\mathrm{off},it-\tau}-I_{\mathrm{on},it-\tau})% \,\mathrm{d}\tau$ (5.5a) or $\displaystyle\mathrm{YD}_{it}$ $\displaystyle=\int_{-\infty}^{t}\delta\mathrm{e}^{-\delta(t-s)}\frac{\beta}{1-% \gamma_{\mathrm{YD}_{t-1}}}(I_{\mathrm{off},is}-I_{\mathrm{on},is})\,\mathrm{d% }s,$ (5.5b)

where $s=t-\tau$.

According to (5.4) and (5.5b), $\delta$ is the speed of adjustment and

 $\int_{0}^{+\infty}\delta\mathrm{e}^{-\delta(t-s)}\tau\,\mathrm{d}\tau=\frac{1}% {\delta}$

is the expected amount of time taken to observe a representative effect on the dependent variable after a change in the independent variable, ie, the mean time lag. Therefore, with reference to the best spread, such measures are 0.45 and about 2.22 days, respectively, as indicated in Table 4.

Moreover, from (5.5b), defining $\Delta I_{\mathrm{off}-\mathrm{on}}$ as the constant change in the prescribed value coming from a change in the degree of liquidity between the two securities, we can write

 $\displaystyle\Delta\mathrm{YD}_{it}$ $\displaystyle=\int_{-\infty}^{t-\theta}\delta\mathrm{e}^{-\delta(t-s)}\frac{% \beta}{1-\gamma_{\mathrm{YD}_{t-1}}}(I_{\mathrm{off},is}-I_{\mathrm{on},is})\,% \mathrm{d}s$ $\displaystyle\qquad+\int_{t-\theta}^{t}\delta\mathrm{e}^{-\delta(t-s)}\bigg[% \frac{\beta}{1-\gamma_{\mathrm{YD}_{t-1}}}(I_{\mathrm{off},is}-I_{\mathrm{on},% is}+\Delta I_{\mathrm{off}-\mathrm{on}})\bigg]\,\mathrm{d}s$ $\displaystyle\qquad-\int_{-\infty}^{t-\theta}\delta\mathrm{e}^{-\delta(t-s)}% \frac{\beta}{1-\gamma_{\mathrm{YD}_{t-1}}}(I_{\mathrm{off},is}-I_{\mathrm{on},% is})\,\mathrm{d}s$ $\displaystyle\qquad+\int_{t-\theta}^{t}\delta\mathrm{e}^{-\delta(t-s)}\bigg[% \frac{\beta}{1-\gamma_{\mathrm{YD}_{t-1}}}(I_{\mathrm{off},is}-I_{\mathrm{on},% is})\bigg]\,\mathrm{d}s,$ (5.6)

and, finally,

 $\displaystyle\Delta\mathrm{YD}_{it}$ $\displaystyle=\frac{\beta}{1-\gamma_{\mathrm{YD}_{t-1}}}\Delta I_{\mathrm{off}% -\mathrm{on}}\int_{t-\theta}^{t}\delta\mathrm{e}^{-\delta(t-s)}\,\mathrm{d}s$ $\displaystyle=\frac{\beta}{1-\gamma_{\mathrm{YD}_{t-1}}}\Delta I_{\mathrm{off}% -\mathrm{on}}\int_{0}^{\theta}\delta\mathrm{e}^{-\delta\tau}\,\mathrm{d}\tau$ $\displaystyle=\frac{\beta}{1-\gamma_{\mathrm{YD}_{t-1}}}\Delta I_{\mathrm{off}% -\mathrm{on}}[-\mathrm{e}^{-\delta\tau}]_{0}^{\theta}$ $\displaystyle\cong 0.632\frac{\beta}{1-\gamma_{\mathrm{YD}_{t-1}}}\Delta I_{% \mathrm{off}-\mathrm{on}}.$ (5.7)

In (5) we return to the original variable $\beta$ and impose $\theta=1/\delta$ in order to understand the quantitative importance of the effect of an impulse from a change in the prescribed value, ie, the long-run equilibrium, within the mean time lag. We may conclude that the mean time lag represents the time required to close around (a quantitatively relevant) 63% of the gap between the actual value and the prescribed value. Therefore, the mean time lag in our case (being an average of the lags) clearly indicates the intertemporal effect of a liquidity shock.2121See Gandolfo (1981) for an extended exposition of these problems in a more general context, where the aim is to find the correspondence between stochastic difference and differential equations.

According to this finding, we now know that an increase (decrease) in the degree of liquidity due to a liquidity shock will produce a large reduction (increase) in the yield differential within the first two to three days after the liquidity shock.

As mentioned above, the conclusions we have reached for model (3.5) in terms of the yield differential are also valid for the yield of a single security, as indicated in (3.4). With specific reference to the on-the-run yield, the results obtained have important implications in understanding how much of the initial upward swing in the price immediately after the auction might be exploited by the treasury or, vice versa, how much of the lower cost of liquidity, resulting in a lower yield, might be saved. We may state that, on average, 63% of the liquidity shock, induced by the primary dealers in the secondary market before the auction, is absorbed by the price of the new on-the-run security in a span of $1/\delta=2.\bar{2}$ time units. Therefore, according to the issuance volume, appropriate auctions should be scheduled by taking into account such a frequency.

In Table 5, we report, by year and security, the observed lag, consisting in the number of days necessary for the largest reduction in the yield to occur since the new security issuance date. We also report the issuance amount and, as a consequence, the possible cost savings. Hence, we may verify whether our model accurately interprets the intertemporal effects on the yield of a liquidity shock due to auctions, and may assess the consequent implications for a more efficient auction design.

We underline that the exponential decline of the effect of a liquidity shock on the yield, implied by (5.5b), is confirmed empirically by the lag results, which are consistently equal to 2 (except for security 6), as prescribed by (5).2222Note that the lag presented in Table 3 is slightly different from the estimated mean time lag, since the latter was obtained by considering all the liquidity shocks in our sample period, not just those referring to the issuance of a new bond. Moreover, we note that the new issuances would have admitted cost savings of up to €6.06 million in 2004, and reductions in yield range from 2.4bps to 18.6bps.

In practice, as suggested by Lou et al (2013), the treasury might minimize the issuance cost by increasing the auction frequency so as to reduce the amount per auction. We leave the cost–benefit analysis of such a choice for future research. Nonetheless, we stress how important it is to apply similar analysis to the whole set of securities of public debt and to account for an appropriate schedule of issuances in order to manage the risk of a liquidity shock efficiently.

We now need to find a criterion for assessing the effectiveness of the liquidity cost index. Since in our estimations and in the literature the indicator used is the average cost of liquidity, in the next section we focus on this specifically.

## 6 Probability of liquidity shock risk

We now turn to the question raised in Section 3, of the probability of a liquidity shock risk. We study this problem in more depth by formalizing the link between the empirical and the theoretical setups of such a probability, and by assessing the importance of the average cost as a predictor of the cost of a liquidity shock. In particular, we want to distinguish the probability of the average cost of the liquidity shocks from the joint probability associated with all the liquidity shocks, so as to derive a conditioned probability that measures the relevance of the mean on the distribution of the liquidity shocks.

In Section 3.1, we mentioned that the cost linked to a liquidity shock has the same dimension as the interest rate. This comes from the definition of the forward rate, $f_{t}$, referring to the $n$th financial operation occurring in the time interval $T-t$, to which the cost, $c_{t}$, is associated.

As stated in Section 2, in our analysis we use daily data drawn from a larger database that collects minute-by-minute data, and therefore we may reliably assume the number of financial operations traded tends to infinity, which implies that the capitalized value relative to the yield is

 $\prod_{n=1}^{N}\bigg(1+(f_{\tau}+\beta c_{\tau})\frac{T-t}{n}\bigg)\to\exp% \bigg(\int_{t}^{T}(f_{\tau}+\beta c_{\tau})\,\mathrm{d}\tau\bigg),\quad\text{% with~{}}N\to+\infty,~{}\tau\in[t,T].$ (6.1)

From (6.1) we know that, at each $n$th instantaneous operation, the rate $(f_{\tau}+\beta c_{\tau})$ is applied relative to the period $T-t$, which, as explained in Section 3, in our case of a CTZ note, may be 720 days from the issuance date. Therefore, if the dependent variable of our regressions is the yield to maturity $T$, and the independent variable is the average of the liquidity cost rates conditioned on time $t$ (as in (3.6)), the resulting coefficient estimate is the total probability that, in the period $T-t$, only one liquidity shock occurs at the $n$th operation. According to our notation, and with reference to the estimation of $\beta$ in Table 2, the total probability of $\bar{l}_{j,it}$ (given by the union of the possible liquidity shocks between $t$ and $T$, referring to bs and conditioned on time $t$) is

 $P\bigg(\bigcup_{\tau=t}^{T}l_{j,i,\tau}=\bar{l}_{j,i,t}\bigg|_{t}\bigg)=41.96\%.$

Dividing such a probability by the above term of reference, we obtain $P(\bar{l}_{j,i,t}|_{t})=5.83$%, which is the probability that a daily liquidity shock is equal to the sample mean at time $t$.

Alternatively, if we were to comply more closely with the expressions of the yield (3.3) and (6.1), we would have to regress, using the method employed in Section 4.1, the yield differential on the differential of the summation of the current and future liquidity indicators, which means considering the case $\smash{I_{j,i\tau}=L_{j,it}}$ in (3.1). In such a case, $\beta$ should be interpreted as the joint probability ($\cap$) of all the daily liquidity shocks occurrences up to maturity.

The results, reported in Table 6, are relative to all the indicators and securities and confirm the underlying theory. In fact, the pair 6–7 provides good results for price measures when regressions are based on the average liquidity cost indicators (Table 2), but not in this case, where the wrong signs appear, except for the slope. Actually, such a pair has the longest on/off cycle but the shortest observation period, which does not cover the terms of the two securities. Therefore, in this case, the calculation of the joint probability is unsatisfactory, since the indicators used are not representative of the summation of the liquidity shocks throughout the time to maturity.

The estimations show that all the indicators apart from the slope for the pair 5–6 and the two spreads for the pair 6–7, including the quantity ones, are significant and with correct sign (which is better than what we obtained in Table 2), and provide the empirical validation of the theory developed in Section 3. In Table 6, we see that the joint probability for the risk of a liquidity shock associated with the cost of the best spread goes from 0.013% to 0.24%, depending on the security considered. To gain more precision, in this case we also perform a panel regression, reported in Table 7, using the procedure in Section 4.2. Again, we obtain significant results at the 99% confidence level for the best and weighted spread, while for the slope the sign of the coefficient is incorrect but much smaller and less significant than the others.2323The smaller coefficient for the slope than for the others in this panel estimation is as we would expect. In fact, the slope is a more unstable measure than the others (being given by the ratio between small values). This is also confirmed by the highest autoregressive coefficient for residuals. In other words, in the panel regression of Table 3, where the slope index is averaged over time (so that it is more stable), it is possible to identify the correct result across all pairs of securities.

According to such an estimation, the joint probability conditional on time $t$ and associated with the cost of the best spread amounts to $\smash{P(\bigcap_{\tau=t}^{T}l_{j,i,\tau}|_{t})=0.025}$%. As expected, given the meaning of the joint probability, such a value is definitely smaller than that of the daily total probability. This is a clear indication of how sparsely the occurrences of liquidity shocks are distributed throughout the term of reference, and points to the exigency of having more information on the corresponding density function. More specifically, we need to account for the degree of dependence between the occurrence of the mean at time $t$ and the other occurrences of the distribution of the liquidity shocks at the other times. In fact, in managing the risk of a liquidity shock, the most commonly used indicator is the sample mean, whose small probability of occurrence at time $t$, ie, $P(\bar{l}_{j,i,t}|_{t})=5.83$%, is a poor indication of its effectiveness given the wide time interval over which the occurrences may take place. Therefore, by dividing these two probability measures, it is possible to obtain a more reliable evaluation of the indicator consisting in the probability of all-but-the-mean liquidity shocks occurrences, at times different from $t$, conditional on the occurrence of the mean at time $t$, which reflects the linkage of the mean with the rest of the distribution,

 $P\bigg(\frac{\bigcap_{\tau=t}^{T}l_{j,i,\tau}\neq\bar{l}_{j,i,t}}{\bar{l}_{j,i% ,t}}\bigg|_{t}\bigg)=0.43\%.$

Hence, the latter probability is a measure of the risk of a liquidity shock in terms of implied costs. In particular, it shows the quite evident increase from the joint probability, thus confirming the appropriateness of the average cost of liquidity in the evaluation of the trading costs.

## 7 Conclusions

This research focuses on the liquidity premium of Italian government bonds, with the aim of assessing whether, for a high-debt country, a cost saving is possible in the presence of a liquidity shock risk produced by the auction of new securities. Such a problem is particularly relevant in Europe, given the efforts of the European Central Bank to finance auctions indirectly via a quantitative easing policy. Toward this aim we build and examine indexes of liquidity for the lifetime of the securities considered, on which the yield differential off- and on-the-run is regressed. We perform both single and panel robust dynamic regressions for each indicator and emphasize the need for a specific analysis to better interpret the characteristics of the Italian liquidity market. We find an important liquidity premium and a significant intertemporal effect, which allows us to better characterize the underpricing phenomenon, with the opportunity of managing the risk of liquidity shock costs, in the case of new issuances, by improving the auction design. We find several measures of the probability associated with the cost of illiquidity in order to assess the appropriateness of the empirical indicators used. Finally, given that the estimated liquidity premium is quantitatively relevant, we underline the need to extend our analysis to securities not considered here and to other countries suffering from high public debt.

## Declaration of interest

The author reports no conflicts of interest. The author alone is responsible for the content and writing of the paper.

## Acknowledgments

The author is grateful to the Italian Public Debt Department at the Ministry for Economy and Finance for its support. He also thanks the University of Rome “La Sapienza” and the Ministero dell’Istruzione, dell’Università e della Ricera (MIUR) for having provided research funds. The author thanks Chiara Coluzzi, Tiziana Capponi, Davide Iacovoni, Paola Fabbri, Paola De Rita, Sergio Ginebri, Manuel Turco, Stefano Fachin, Christine Parlour and Antonio Scalia for their contributions to a preliminary version of this paper presented at the Italian Ministry of Economy and Finance. He is grateful to the Italian Public Debt Department at the Ministry for Economy and Finance for the extremely helpful suggestions and comments. Particular thanks go to Simona Delle Chiaie, who was fundamental in preparing the data set, and Eleonora Cavallaro, who gave much useful advice on the preparation of the print version. The author is also particularly indebted to an anonymous referee for suggesting valuable improvements to the paper.

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