## _wp03115

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---

### Bayesian updating and elapsed inter-transaction time
- Equation (12) implies that as elapsed inter-transaction time increases (τ∆→∞) the dealer places the majority of the weight on the incoming order’s information, ts.
- The precision of the quantity shock signal 1()tκγ− is assumed to decrease (variance increases) as clock-time elapsed from event t−1 to t increases; hence 1()tκγ− becomes noisier with longer τ∆.
- The appendix shows the estimate of the full-information asset value, tμ, generates an unbiased estimate of the liquidity trade, Xt: [|]tt t EXxΩ=.

### The dealer’s optimization problem
- Dealer solves stochastic dynamic programming problem summarized in equation (13), subject to evolution constraints:
  - Inventory evolution (14): constrains inventory and links inventory to trades and shocks.
  - Noise trading (15): constrains liquidity trades to be zero in expectation.
  - Information (16): constrains the asset to a random walk.
  - Capital evolution (17): constrains capital and specifies expected price when trading out, including a price impact term (out tt qμα+); α captures the price impact of a marginal increase in outgoing quantity.
- Interpretation:
  - Dealer faces a downward sloping demand curve when trading out (α>0).
  - Dealer spreads inventory-management costs across available instruments.

### Model solution and pricing equations (key relations)
- Closed-form solutions (appendix) include:
  - Price function (18): price as function of estimated asset value tμ, deviation from optimal inventory (d t II−), and liquidity shocks xt.
  - Outgoing quantity (19): outt q depends on price impact α; as 0α→ outgoing trades fully adjust inventories to the optimal level.
  - Inventory evolution parameter (20): links inventories and shocks via β.
  - Incremental dealer price change (21): contains information-driven components from current incoming order (stη), previous inventory shock (1(1)ttηγ−−), inventory-change term (111out ttt qqγ−−− ++), and noise-trading component (tx∆); weights involve the Bayesian updating term ψ and elapsed time measure 1 τ τ η ∆ +∆ = .
- Economic interpretation:
  - Dealer balances price-impact costs from trading out (α) and expected loss to informed dealers via (pt δ μ−).
  - Price is centered on best guess of vt derived from st and 1()tκγ−, weighted by elapsed clock-time.

### Comparison with existing (canonical) models
- Restricting the model to no outgoing trades yields Madhavan and Smidt (1993) pricing behavior (22).
- Canonical pricing assumption (23) yields price change form (24); paper shows canonical specifications are misspecified because τ∆ systematically affects precision of informative variables (jtt qγ , ).
- Omitted terms in canonical models cause confounding: lagged inventory and information effects can be conflated; inter-transaction time alters relative informativeness, creating breaks in canonical estimates.
- Predicted behavior as τ∆ varies:
  - Long inter-transaction times (τ∆→∞; 1 τ τ η ∆ +∆ ≡→): incoming order flow coefficient (1 φ) more significant; 1t γ − mostly noise.
  - Short inter-transaction times (0,τ∆→ and 0η→): order flow coefficient (1 φ) less significant; inventory coefficients more significant.

### Data considerations (sample and descriptive facts)
- Data: one week of a New York based FX dealer’s intraday recordings (prices pt, incoming order flow qjt, inventories It, transaction clock times τ∆ and η) from a Reuters Dealing trading system.
- Sample counts and adjustments:
  - 843 transactions in the week; four overnight price changes discarded (intraday model).
  - A few transaction clock-time measurement errors treated with a dummy variable.
- Key descriptive statistics:
  - Dealer keeps average inventory at $2.1 million, but inventory deviates as much as ±$50 million.
  - Median incoming order roughly $3 million; reversing a one standard deviation swing in inventory requires about five sequential incoming trades.
  - Table 3: on average 20 outgoing trades per day initiated by the dealer.
  - Observed spread on both incoming and outgoing trades is tightly maintained at the market convention of 3 pips.
- Visual evidence:
  - Figures (2 and 3): cumulative daily incoming order flow and cumulative daily inventory shocks show varying correlations with price across days (e.g., Monday and Wednesday: incoming order flow more precise; Friday: inventory shocks more precise).
  - Table 4 reports daily correlations and average inter-transaction clock-time illustrating these patterns over the five-day sample.

### Estimation strategy and system identification
- System estimated almost directly from model solution; added elements: leveling constants, AR(1) error on inventory equation, bid-ask bounce dummies on pricing equation.
- Identification:
  - Simplified outgoing-order identification (25) and transformations (26)–(28) identify 1ˆ t γ − as outgoing trade shock and allow formation of regressors for pricing without multicollinearity.
  - Observed inter-transaction times (η) used to distinguish informativeness of incoming vs outgoing order flow; model constrains components of 1t γ − to have same coefficient as incoming order flow after accounting for η.
- Estimation method: Seemingly Unrelated non-linear least squares; system estimated simultaneously.
- Robustness:
  - Estimates robust to using order flow or its unexpected component.
  - Elapsed time measured in minutes; estimates robust to monotonic transformations of η.
  - Measurement-error dummy for some timestamps included (insignificant).

### Main estimation results and fit
- System estimates (Table 6) indicate very significant and properly signed coefficients on information and inventory effects: c11 and c12.
- Point estimates:
  - Dealer plans to trade out roughly one-third of each incoming trade: 3ˆ c =0.34.
  - Estimated dealer target inventory: ˆ d I =2.09.
  - Observed average inventory from Table 2: 2.16; a Wald test fails to reject equality of the mean to the target with a p-value of 0.94.
- Canonical model estimates (Table 7) for comparison:
  - Canonical estimates not robust to subsample estimation; canonical predictions of inventory effects rejected in first half and information effects rejected in second half.
  - Presented model is robust to subsample estimation (though first sub-sample p-values are lower).

### Asymmetric information (empirical magnitude and interpretation)
- Asymmetric information coefficient c11 is significant and larger than canonical estimate 1 β in Table 7 (pricing coefficients scaled by 10^5).
- Interpretation in spread terms:
  - Dealer widens her spread by 3.5 pips per $10 million of incoming order flow or inventory shocks (twice c11 since orders are quoted on absolute size).
- Economic significance:
  - Marginal $1 million order pushes the dealer’s price by about 2 basis points given sample average exchange rate roughly DM 1.5 per US$1 (equivalently 2 percentage points per excess US$1 billion traded).
  - Market-wide estimates around half a percent per US$1 billion; direct comparisons are problematic.
- Explanation for prior underestimation:
  - Previous work assigns all information-driven price changes to incoming order flow; this model shows information-driven inventory changes also transmit information (via 1t γ −) and vary with τ∆.

### Inventory effects (structural comparison)
- Canonical inventory specification (23) leads to canonical inventory-related price-impact parameter: 3ˆ 0.72 β = (from (29) and Table 7).
- In this model:
  - Structural inventory effect on prices is (1)( ) α δα β + (from equation (18) first-order conditions).
  - Estimated β via system relations: 2ˆ ˆ (1)c β = − yields ˆ β = −0.34.
  - For 0α> and 0δ> consistent with model, (1)( ) α δα + <1; total inventory effect is β multiplied by factor approaching unity from below.
- Comparison implication:
  - Canonical model estimates of inventory costs are at least two to three times larger than estimates here.
  - Ignoring multiple instruments overweights inventory component because price changes are empirically assigned too large a role.
  - Conclusion: price changes are only one of multiple instruments used to control inventory; inventory accumulation matters less for price changes than canonical models imply.

### Expected cost of outgoing trades and base spread
- Dealer’s expected marginal cost of trading out an extra million dollars at time of price setting estimated at ˆ α =0.35 pips.
- This reflects dealer’s opportunity cost of changing the spread in response to a $1 million incoming trade.
- Estimated proportion of excess inventory traded out, C3 = 0.33.
- Baseline effective spread for qjt close to zero implies baseline spread roughly 2.5-2.8 pips (twice c4 times 10^-5), approximately equal to median interdealer spread observed of 3 pips.

### Fed intervention (August 7, 1992) — empirical impact
- Intervention size: dollar purchases totaling $300 million after the close of European markets.
- Intervention moved market price between 20 and 32 pips before falling back.
- Wald tests fail to reject equality between model estimates with and without intervention period (i.e., 95 percent of the sample, versus 100 percent).
- Table 8 / parameter impact:
  - Intervention increases asymmetric information effect of incoming order flow (c11) by over 8 percent.
  - Changes in estimated inventory effect (c12) and other model parameters negligible.
- Comparison with prior studies:
  - At lower end of 20–32 pips range, concords with estimates of between 5 and 8 pips per 100 million from Evans and Lyons and Dominguez and Frankel.
  - Higher end implies 12 pips per $100 million and market-wide elasticity closer to dealer cost estimates in this study.
- Timing notes:
  - Wall Street Journal quote documents intervention start corresponding to 12:32 pm (dollar traded at 1.4720).
  - Alternative possible start points considered: 12:02 pm and 12:26 pm.

### Model contributions and conclusions
- Model incorporates realistic options available to market makers for absorbing portfolio flows, allowing multiple instruments for inventory control (price changes, planned outgoing trades, learning through shocks between events).
- Outgoing calls modeled with price impact (increasing marginal costs); dealers equate marginal loss from trading unwanted inventory via outgoing calls with marginal loss of inventory imbalance.
- Dealers learn during outgoing call periods; updates from trading at those times cause price changes not explained solely by inventory costs or incoming order flow.
- Multiple instruments and signal precision:
  - Canonical models that omit alternate instruments overemphasize role of price changes in inventory management, biasing downward the role of information in price changes, making inventory effects appear insignificant, and tightening the bid-ask spread.
  - Data-generating process here suggests information effects are biased downward in canonical estimations because dealers infer asset values from multiple signals with varying precision; canonical estimates overweigh signals at uninformative times.
- Permanent vs. transitory price components:
  - Presence of inventory effects implies part of observed price changes is transitory.
  - With multiple instruments, dealers exhaust gains from sharing a large inventory position with less price impact, so transitory component is less important than information components.
  - Model favors a permanent impact of portfolio flows on prices, though both transitory and permanent effects are present.
- Liquidity/cost implications of intervention:
  - Fed intervention increases informativeness of order flow, lowers cost of liquidity for dealer, lowers inventory costs, and tightens the spread.

### Timing and mechanics of the model (event sequence)
- At every event (if t ≠ T):
  1. Dealer knows current inventory It and an incoming trade qjt occurs (one information source).
  2. Dealer decides price Pt and plans outgoing trade qout_t.
  3. Between events, planned outgoing trade qout_t executes and dealer faces quantity shock γt.
  4. Dealer observes elapsed time between trades τ∆.
  5. At t+1, dealer updates priors using qjt+1, the between-trade quantity shock, and elapsed time, then sets prices.
- Pricing and execution notation in timing figure and equations preserved in model derivation.

### Descriptive statistics and observed trading (selected exact figures)
- Lyons (1995) dataset: NY based dollar/DM dealer, August 3–7, 1992.
- Table 2 — descriptive statistics (Inventory / Order flow / Order flow (absolute value)):
  - Mean: 2.16 / -0.43 / 3.8
  - Median: 0.70 / 0.5 / 2.5
  - Maximum: 56.8 / 20.0 / 28.0
  - Minimum: -42.7 / -28.0 / 0.0
  - Std. Dev.: 15.4 / 5.2 / 3.6
  - Observations: 838 / 838 / 838
- Table 3 — observed trades (dealer-initiated, not including a small amount of brokered trades):
  - Observed Incoming trades: 170
  - Observed Outgoing trades: 205
- Table 4 — daily correlations (price and order flow variables, August 3–7, 1992):
  - Monday: Order Flow 0.83 / Unexpected Shocks 0.83 / Inventory Time* -0.54 / Mean Elapsed Observation 1.77 / End of Day 181
  - Tuesday: 0.69 / 0.71 / 0.58 / 1.86 / 330
  - Wednesday: 0.53 / 0.48 / 0.03 / 2.44 / 440
  - Thursday: 0.82 / 0.81 / 0.66 / 2.01 / 592
  - Friday: -0.03 / -0.02 / 0.71 / 1.31 / 843
  - *Reporting errors imply mean absolute value transaction time.
- Table 6 — selected parameter interpretations and sample estimates:
  - α measures expected price impact of augmenting planned outgoing trade by $1 million in pips.
  - Id measures implicit optimal inventory level used by dealer.
  - Full sample α (reported in table) = 2.09 (pips); adj R^2 and sample subsplit estimates reported in Table 6 (all estimates multiplied by 10^5; p-values in italics in original table).

### Model solution highlights (appendix material preserved)
- Inventory carrying cost and dealer belief updating derived with Kalman-filter-style recursive updating.
- Two signals used to update beliefs on asset value vt at price-setting: incoming-order innovation st and signal from between-trade inventory shock γt (weighting depends on elapsed time τ∆).
- Signal-to-noise ratio and steady-state variance relationships preserved (notation and equations in appendix).
- Optimal policy functions:
  - Optimal outgoing quantity: qout = A_d^(')^{-1} I (eq. (63): 1(') d out A I I q α − = .)
  - Pricing equation (policy): p = (1/(1'))( ... ) with full derivation in appendix (equations (64)–(69)).
- Value function conjecture and verification: quadratic form with coefficients solved via envelope and first-order conditions; economic parameter restrictions discussed (e.g., requirement A1 < 0 leading to condition (71)).
- Informed trader’s problem and optimality: deviation analysis shows suboptimality of nonzero deviations under model conditions; derivation parallels Madhavan and Smidt (1993) and imposes equilibrium conditions (equations (72)–(84)).

*Source: _wp03115*

### appendix shows, this gives an updating as a function of:

### _wp03115 - appendix shows, this gives an updating as a function of:

### Bayesian updating and elapsed inter-transaction time
- Equation (12) implies that as elapsed inter-transaction time increases (τ∆→∞) the dealer places the majority of the weight on the incoming order’s information, ts.
- The precision of the quantity shock signal 1()tκγ− is assumed to decrease (variance increases) as clock-time elapsed from event t−1 to t increases; hence 1()tκγ− becomes noisier with longer τ∆.
- The appendix shows the estimate of the full-information asset value, tμ, generates an unbiased estimate of the liquidity trade, Xt: [|]tt t EXxΩ=.

### The dealer’s optimization problem
- The dealer solves a stochastic dynamic programming problem summarized in equation (13), subject to evolution constraints:
  - Inventory evolution (14): constrains inventory and links inventory to trades and shocks.
  - Noise trading (15): constrains liquidity trades to be zero in expectation.
  - Information (16): constrains the asset to a random walk.
  - Capital evolution (17): constrains capital and specifies expected price when trading out, including a price impact term (out tt qμα+); α captures the price impact of a marginal increase in outgoing quantity.
- Interpretation: the dealer faces a downward sloping demand curve when trading out (α>0) and spreads inventory-management costs across available instruments.

### Model solution and pricing equations (key relations)
- Closed-form solutions shown in appendix include:
  - Price function (18): price as function of estimated asset value tμ, deviation from optimal inventory (d t II−), and liquidity shocks xt.
  - Outgoing quantity (19): outt q depends on price impact α; as 0α→ outgoing trades fully adjust inventories to the optimal level.
  - Inventory evolution parameter (20): links inventories and shocks via β.
  - Incremental dealer price change (21): contains information-driven components from current incoming order (stη), previous inventory shock (1(1)ttηγ−−), inventory-change term (111out ttt qqγ−−− ++), and noise-trading component (tx∆); weights involve the Bayesian updating term ψ and elapsed time measure 1 τ τ η ∆ +∆ = .
- Economic interpretation:
  - Dealer balances two competing losses when restoring inventory: price-impact costs from trading out (α) and expected loss to informed dealers via (pt δ μ−).
  - Price is centered on the best guess of vt derived from two information sources, st and 1()tκγ−, weighted by elapsed clock-time.

### Comparison with existing (canonical) models
- Restricting the model to no outgoing trades yields the Madhavan and Smidt (1993) pricing behavior (22).
- Canonical pricing assumption (23) yields price change form (24). The paper shows such canonical specifications are misspecified for these data because inter-transaction time (τ∆) systematically affects the precision of informative variables (jtt qγ , ).
- Omitted terms in canonical models cause confounding: lagged inventory and information effects can be conflated; inter-transaction time alters relative informativeness, creating breaks in canonical estimates.
- Predicted behavior as τ∆ varies:
  - Long inter-transaction times (τ∆→∞; 1 τ τ η ∆ +∆ ≡→): incoming order flow coefficient (1 φ) more significant; 1t γ − mostly noise.
  - Short inter-transaction times (0,τ∆→ and 0η→): order flow coefficient (1 φ) less significant; inventory coefficients more significant.

### Data considerations (sample and descriptive facts)
- Data: one week of a New York based FX dealer’s intraday recordings (prices pt, incoming order flow qjt, inventories It, transaction clock times τ∆ and η) from a Reuters Dealing trading system.
- Sample counts and adjustments:
  - 843 transactions in the week; four overnight price changes discarded (intraday model).
  - A few transaction clock-time measurement errors treated with a dummy variable.
- Key descriptive statistics:
  - Dealer keeps average inventory at $2.1 million, but inventory deviates as much as ±$50 million.
  - Median incoming order roughly $3 million; reversing a one standard deviation swing in inventory requires about five sequential incoming trades.
  - Table 3: on average 20 outgoing trades per day initiated by the dealer.
  - Observed spread on both incoming and outgoing trades is tightly maintained at the market convention of 3 pips (market convention noted).
- Visual evidence:
  - Figures (2 and 3): cumulative daily incoming order flow and cumulative daily inventory shocks show varying correlations with price across days (e.g., Monday and Wednesday: incoming order flow more precise; Friday: inventory shocks more precise).
  - Table 4 reports daily correlations and average inter-transaction clock-time illustrating these patterns over the five-day sample.

### Estimation strategy and system identification
- The system is estimated almost directly from the model solution; added elements: leveling constants, AR(1) error on the inventory equation, bid-ask bounce dummies on pricing equation.
- Equations used to identify key objects:
  - Simplified outgoing-order identification (25) and transformations (26) through (28) identify 1ˆ t γ − as the outgoing trade shock and allow formation of regressors for pricing without multicollinearity.
  - Inter-transaction times (η) are observed and used to distinguish informativeness of incoming vs outgoing order flow; the model constrains components of 1t γ − to have same coefficient as incoming order flow after accounting for η.
- Estimation method: Seemingly Unrelated non-linear least squares; system estimated simultaneously.
- Robustness notes:
  - Estimates robust to using order flow or its unexpected component.
  - Elapsed time measured in minutes; estimates robust to monotonic transformations of η.
  - Measurement-error dummy for some timestamps included (insignificant).

### Main estimation results and fit
- System estimates (Table 6) indicate:
  - Very significant and properly signed coefficients on information and inventory effects: c11 and c12.
  - Predicted inventory evolution and outgoing trade estimates: c1, c2, c3 identified.
- Point estimates highlighted:
  - Dealer plans to trade out roughly one-third of each incoming trade: 3ˆ c =0.34.
  - Estimated dealer target inventory: ˆ d I =2.09.
  - Observed average inventory from Table 2: 2.16; a Wald test fails to reject equality of the mean to the target with a p-value of 0.94.
- Canonical model estimates (Table 7) are shown for comparison:
  - Canonical estimates are not robust to subsample estimation; canonical predictions of inventory effects rejected in first half of sample and information effects rejected in second half.
  - The presented model is robust to subsample estimation (though first sub-sample p-values are lower).

### Asymmetric information (empirical magnitude and interpretation)
- Asymmetric information coefficient c11 is significant and larger than canonical estimate 1 β in Table 7 (pricing coefficients scaled by 10^5).
- Interpretation in spread terms:
  - Dealer widens her spread by 3.5 pips per $10 million of incoming order flow or inventory shocks (twice c11 since orders are quoted on absolute size).
- Economic significance:
  - Marginal $1 million order pushes the dealer’s price by about 2 basis points given sample average exchange rate roughly DM 1.5 per US$1 (equivalently 2 percentage points per excess US$1 billion traded).
  - This is higher than market-wide estimates (around half a percent per US$1 billion), but direct comparisons are problematic.
- Explanation for prior underestimation:
  - Previous work assigns all information-driven price changes to incoming order flow; this model shows information-driven inventory changes also transmit information (via 1t γ −) and vary with τ∆, so canonical estimates can understate information effects.

### Inventory effects (structural comparison)
- Canonical inventory specification (23) leads to inventory-related price-impact parameter estimates; canonical estimate reported: 3ˆ 0.72 β = (from (29) and Table 7).
- In this model:
  - Structural inventory effect on prices is (1)( ) α δα β + (from equation (18) first-order conditions).
  - Estimated β via system relations: 2ˆ ˆ (1)c β = − yields ˆ β = −0.34.
  - For 0α> and 0δ> consistent with model, (1)( ) α δα + <1; thus total inventory effect is β multiplied by a factor approaching unity from below.
- Comparison implication:
  - Canonical model estimates of inventory costs are at least two to three times larger than the estimates here.
  - Ignoring multiple instruments (incoming and planned outgoing trades) will overweigh the inventory component because price changes are empirically assigned too large a role in inventory management.
  - Conclusion: price changes are only one of multiple instruments used to control inventory; inventory accumulation matters less for price changes than canonical models imply.

### Expected cost of outgoing trades and base spread
- The dealer’s expected marginal cost of trading out an extra million dollars at the time of price setting is estimated at ˆ α =0.35 pips.
- This reflects the dealer’s opportunity cost of changing the spread in response to a $1 million incoming trade.

### 0.34 pips per million, as discussed above.  Hence, the estimates suggest that trading out

### _wp03115 - 0.34 pips per million, as discussed above.  Hence, the estimates suggest that trading out

### Key empirical estimates and market microstructure findings
- Estimated proportion of excess inventory that is traded out, C3 = 0.33 (meaning for each incoming dollar, the dealer expects to trade out one third).
- Baseline effective spread for qjt close to zero implies a baseline spread roughly 2.5-2.8 pips (twice c4 times 10^-5). This is approximately equal to the median interdealer spread observed in the FX market of 3 pips.
- Estimated marginal cost implication: trading out excess inventory has a higher marginal cost for the dealer than accepting incoming trades.
- At the time of price setting, planned outgoing trades are one-third of the difference between dealer’s current and optimal inventory positions (restating the C3 implication).

### Fed intervention (August 7, 1992) — empirical impact
- Intervention size: dollar purchases totaling $300 million after the close of European markets.
- The intervention moved the market price between 20 and 32 pips before falling back.
- Wald tests fail to reject equality between model estimates with and without the intervention period (i.e., 95 percent of the sample, versus 100 percent).
- Table 8 / parameter impact: the intervention increases the asymmetric information effect of incoming order flow (c11) by over 8 percent; changes in the estimated inventory effect (c12) and other model parameters are negligible.
- Comparison with prior studies: at the lower end of the 20–32 pips range, this concords with estimates of between 5 and 8 pips per 100 million from Evans and Lyons (5 pips per $100 million), and Dominguez and Frankel (8 pips per $100 million). At the higher end, 12 pips per $100 million implies a market-wide elasticity closer to the estimates of dealer costs in this study.
- Note on timing: the Wall Street Journal quote (August 10, 1992) documents the most precise available intervention start corresponding to 12:32 pm (dollar traded at 1.4720). Alternative possible start points considered: 12:02 pm (reports of a mid-day start) and 12:26 pm (price jumps 36 pips suggesting a possible intervention start).

### Model contributions and conclusions
- The model incorporates realistic options available to market makers for absorbing portfolio flows, extending canonical models by allowing multiple instruments for inventory control (price changes, planned outgoing trades, and learning through shocks between events).
- Outgoing calls (planned outgoing trades) are modeled with price impact (increasing marginal costs). Dealers equate marginal loss from trading unwanted inventory via outgoing calls with marginal loss of inventory imbalance (inventory carrying cost).
- Dealers learn during outgoing call periods; updates from trading at those times cause price changes not explained solely by inventory costs or incoming order flow.
- Multiple instruments and signal precision:
  - Canonical models that omit alternate instruments overemphasize the role of price changes in inventory management, biasing downward the role of information in price changes, making inventory effects appear insignificant, and tightening the bid-ask spread.
  - The data generating process modeled here suggests information effects are biased downward in canonical estimations because dealers infer asset values from multiple signals with varying precision; canonical estimates overweigh signals at uninformative times.
- Impact on permanent vs. transitory price components:
  - Presence of inventory effects implies part of observed price changes is transitory.
  - With multiple instruments, dealers exhaust gains from sharing a large inventory position with less price impact, so the transitory component is less important than information components.
  - The model favors a permanent impact of portfolio flows on prices, though both transitory and permanent effects are present in the data.
- Liquidity/cost implications:
  - A Fed intervention increases the informativeness of order flow, lowers the cost of liquidity for the dealer, lowers inventory costs, and tightens the spread.

### Timing and mechanics of the model (event sequence)
- At every event (if t ≠ T):
  1. Dealer knows current inventory It and an incoming trade qjt occurs (one information source).
  2. Dealer decides price Pt and plans outgoing trade qout_t (alternatives to shading incoming price).
  3. Between events, the planned outgoing trade qout_t executes and the dealer faces a quantity shock γt (another information source).
  4. Dealer observes elapsed time between trades τ∆.
  5. At t+1, dealer updates priors using qjt+1, the between-trade quantity shock, and elapsed time, then sets prices.
- Pricing and execution notation in the timing figure and equations were preserved in the model derivation.

### Descriptive statistics and observed trading (selected exact figures from dataset summaries)
- Lyons (1995) dataset: NY based dollar/DM dealer, August 3–7, 1992.
- Table 2 — descriptive statistics (Inventory / Order flow / Order flow (absolute value)):
  - Mean: 2.16 / -0.43 / 3.8
  - Median: 0.70 / 0.5 / 2.5
  - Maximum: 56.8 / 20.0 / 28.0
  - Minimum: -42.7 / -28.0 / 0.0
  - Std. Dev.: 15.4 / 5.2 / 3.6
  - Observations: 838 / 838 / 838
- Table 3 — observed trades (dealer-initiated, not including a small amount of brokered trades):
  - Observed Incoming trades: 170
  - Observed Outgoing trades: 205
  - Daily No. (mean)(median)(median) and Size and Spread fields are reported in the table (preserve original table formatting for full detail).
- Table 4 — daily correlations (price and order flow variables, August 3–7, 1992):
  - Monday: Order Flow 0.83 / Unexpected Shocks 0.83 / Inventory Time* -0.54 / Mean Elapsed Observation 1.77 / End of Day 181
  - Tuesday: 0.69 / 0.71 / 0.58 / 1.86 / 330
  - Wednesday: 0.53 / 0.48 / 0.03 / 2.44 / 440
  - Thursday: 0.82 / 0.81 / 0.66 / 2.01 / 592
  - Friday: -0.03 / -0.02 / 0.71 / 1.31 / 843
  - *Reporting errors imply mean absolute value transaction time.
- Table 6 — selected parameter interpretations and sample estimates:
  - α measures the expected price impact of augmenting the planned outgoing trade by $1 million in pips.
  - Id measures the implicit optimal inventory level used by the dealer.
  - Full sample α (reported in table) = 2.09 (pips), adj R^2 and sample subsplit estimates reported in Table 6 (all estimates multiplied by 10^5; p-values in italics in original table).

### Structural-model implications and estimations
- The system is estimated simultaneously using seemingly unrelated non-linear least squares.
- Table 5 summarizes the mapping from model algebraic solution to estimable equations and lists testable restrictions. The structural parameter measuring expected cost of liquidity at time of price setting is ˆα.
- Canonical model reproduction (Table 7) shows canonical microstructure estimates (all estimates multiplied by 10^5). Estimating over two halves of the sample reveals canonical simultaneous presence of inventory and information effects are significantly not different from zero in those subsamples (see discussion and referenced robustness checks).

### Model solution highlights (appendix material preserved)
- Inventory carrying cost and dealer belief updating are derived with Kalman-filter-style recursive updating.
- Two signals used to update beliefs on asset value vt at price-setting: incoming-order innovation st (unexpected order flow) and a signal from the between-trade inventory shock γt (weighting depends on elapsed time τ∆).
- Signal-to-noise ratio and steady-state variance relationships preserved (notation and equations in the appendix).
- Optimal policy functions:
  - Optimal outgoing quantity: qout = A_d^(')^{-1} I (eq. (63) in text: 1(') d out A I I q α − = .)
  - Pricing equation (policy): p = (1/(1'))( ... ) with full derivation in the appendix (equations (64)–(69)).
- Value function conjecture and verification: quadratic form with coefficients solved via envelope conditions and first-order conditions; economically sensible parameter restrictions discussed (e.g., requirement A1 < 0 leading to condition (71)).
- Informed trader’s problem and optimality: deviation analysis shows suboptimality of nonzero deviations under model conditions; derivation parallels Madhavan and Smidt (1993) and imposes equilibrium conditions (equations (72)–(84)).

*Italicized original PDF: _wp03115 - 0.34 pips per million, as discussed above.  Hence, the estimates suggest that trading out*

### REFERENCES

### _wp03115 - REFERENCES

### Market microstructure theory and dealer markets
- Admati, A., and P. Pfleiderer, 1988, “A Theory of Intraday Patterns: Volume and Price Variability,” Review of Financial Studies, Vol. 1 (Spring), pp. 3–40.
- Amihud, Y., and H. Mendelson, 1980, “Dealership Market: Marketmaking with Inventory,” Journal of Financial Economics, Vol. 8 (March), pp. 31–53.
- Glosten, L., and P. Milgrom, 1985, “Bid, Ask, and Transaction Prices in a Specialist Market with Heterogeneously Informed Agents,” Journal of Financial Economics, Vol. 14 (March), pp. 71–100.
- Ho, T., and H. Stoll, 1981, “Optimal Dealer Pricing Under Transactions and Return Uncertainty,” Journal of Financial Economics, Vol. 9 (March), pp.47–73.
- Ho, T., and H. Stoll, 1983, “The Dynamics of Dealer Markets under Competition,“ Journal of Finance Vol. 38 (September), pp. 1053–74.
- Stoll, Hans R., 1978, “The Supply of Dealer Services in Securities Markets,“ Journal of Finance Vol. 33 (September), pp. 1133–51.
- O'Hara, M.; Oldfield, G. S., 1986 “The Microeconomics of Market Making,” Journal of Financial and Quantitative Analysis, Vol. 21 (December), pp. 361–76.
- Manaster, S. and S. C. Mann, 1996, “Life in the Pits: Competitive Market Making and Inventory Control,” Review of Financial Studies Vol. 9 (Fall), pp. 953–75.
- Madhavan, A., and G. Sofianos, 1998, “An Empirical Analysis of NYSE Specialist Trading,” Journal of Financial Economics Vol. 48 (May), pp. 189–210.
- Madhavan, A., and S. Smidt, 1991, “A Bayesian Model of Intraday Specialist Pricing,” Journal of Financial Economics, Vol. 30 (November), pp. 99–134.
- Madhavan, A., and S. Smidt, 1993, “An Analysis of Daily Changes in Specialist Inventories and Quotations.” Journal of Finance, Vol. 48 (December), pp.1595–1648.
- Manaster, S. and S. C. Mann, 1996, “Life in the Pits: Competitive Market Making and Inventory Control,” Review of Financial Studies Vol. 9 (Fall), pp. 953–75.

### Information, inventories, and trade informativeness
- Easley, D., and M. O’Hara, 1987, “Price, Trade Size, and Information in Securities Markets,” Journal of Financial Economics, Vol. 19 (September), pp. 69–90.
- Easley, D., and M. O’Hara, 1992, “Time and the Process of Security Price Adjustment,” Journal of Finance, Vol. 47 (June), pp. 577–605.
- Hasbrouck, J., 1988, “Trades, Quotes, Inventories, and Information,” Journal of Financial Economics Vol. 22 (December), pp. 229–252.
- Hasbrouck, J., 1991a, “Measuring the Information Content of Stock Trades,” Journal of Finance, Vol. 46 (March), pp.179–207.
- Hasbrouck, J., 1991b, “The Informativeness of Stock Trades: An Econometric Analysis,” Review of Financial Studies, Vol.4 (No. 3), pp. 571–595.
- Kyle, A., 1985, “Continuous Auctions and Insider Trading,” Econometrica, Vol. 53 (November), pp. 1315–35.
- Cao, H. H., M. D. D. Evans, and R. K. Lyons, 2006, “Inventory Information,” The Journal of Business, Vol. 79, pp. 325–364.
- Ito, T., R. Lyons, and M. Melvin, 1998, “Is There Private Information in the FX Market? The Tokyo experiment,” Journal of Finance, Vol. 53 (June), pp. 1111–30.
- Lyons, R., 1995, “Tests of Microstructural Hypotheses in the Foreign Exchange Market,” Journal of Financial Economics, Vol. 39 (October), pp. 321–51.
- Lyons, R., 1996, “Optimal Transparency in a Dealer Market with an Application to Foreign Exchange,” Journal of Financial Intermediation, Vol. 5 (July) pp. 225–54.
- Lyons, R., 1997, “A Simultaneous Trade Model of the Foreign Exchange Hot Potato,” Journal of International Economics, Vol. 42 (May), pp. 275–98.
- Lyons, R., 1998, “Profits and Position Control: A Week of FX Dealing,” Journal of International Money and Finance, Vol. 17 (February), pp. 97–115.
- Lyons, R., 2001, The Microstructure Approach to Exchange Rates, (Cambridge and London: MIT Press).
- Massa, M. and A. Simonov, 2003, “Reputation and Interdealer Trading. A Microstructure Analysis of the Treasury Bond Market.” Journal of Financial Markets, Vol. 6, Number 2, (April), pp. 99-141.
- Yao, J., 1998, “Market Making in the Interbank Foreign Exchange Market” (unpublished; New York: Salomon Center Working Paper No. S-98-3, New York University).

### Foreign exchange market structure, intervention, and portfolio flows
- Dominguez, K., and J. Frankel, 1993a, “Does Foreign-Exchange Intervention Matter? The Portfolio Effect,” American Economic Review, Vol. 83 (December), pp.1356–69.
- Dominguez, K., and J. Frankel, 1993b, “Does Foreign-Exchange Intervention Work?” (Washington: Institute for International Economics).
- Evans, M. D. D. and R. K. Lyons, 2002. “Order Flow and Exchange Rates Dynamics.” Journal of Political Economy, Vol. 110 (February), pp. 170–180.
- Froot, K. A., P. O’Connel, and M. Seasholes, 2001, “The Portfolio Flows of International Investors,” Journal of Financial Economics, Vol. 59 (February). pp.151–93.
- Froot K. A. and T. Ramadorai, 2001a. “The Information Content of International Portfolio Flows,” NBER Working Papers 8472, National Bureau of Economic Research, Cambridge, MA.
- Froot, K., and T. Ramadorai, 2002. “Currency Returns, Institutional Investor Flows, and Exchange Rate Fundamentals.” NBER Working Paper 9101, National Bureau of Economic Research, Cambridge, MA.
- Rime, D., 2000, “Private or public information in foreign exchange markets? An empirical analysis,” (unpublished; Oslo: Norwegian School of Management).
- Romeu, R. B., 2005, “Why Are Asset Markets Modeled Successfully, But Not Their Dealers?” IMF Staff Papers, Vol. 52, No. 3 (December), pp. 369–86.
- Ausubel, L. and R. Romeu, 2005, “Bidder Participation and Information in Currency Auctions.” (unpublished; IMF Working Paper No. 05/157, International Monetary Fund).

### Exchange rate modeling, macro approaches, and empirical failures
- Flood, R., and A. Rose, 1995, “Fixing Exchange Rates: A Virtual Quest for Fundamentals,” Journal of Monetary Economics, Vol. 36 (December), pp 3–37.
- Flood, R., and M. Taylor, 1996, “Exchange Rate Economics: What’s Wrong with the Conventional Macro Approach?” The Microstructure of Foreign Exchange Markets, ed. by J. Frankel, G. Galli, and A. Giovannini (Chicago: The University of Chicago Press), pp. 261–294.
- Frankel, J., G. Galli, and A. Giovannini, 1996, The Microstructure of Foreign Exchange Markets, (Chicago: University of Chicago Press).
- Frankel, J., and A. Rose, 1995, “Empirical Research on Nominal Exchange Rates,” Handbook of International Economics, ed. By G. Grossman and K. Rogoff (Amsterdam: Elsevier Science) pp.1689–1729.
- Meese, R., and K. Rogoff, 1983, “The Out-of-sample Failure of Empirical Exchange Rate Models,” Exchange Rate and International Macro-economics, ed. by J. Frenkel (Chicago: University of Chicago Press).
- Mello, A., 1996, Comment–Chapter 5. The Microstructure of Foreign Exchange Markets, ed. by J. Frankel, G. Galli, and A. Giovannini (Chicago: The University of Chicago Press).

### Econometric methods and structural change
- Andrews, D.W.K., 1993, “Testing for Parameter Instability and Structural Change with Unknown Change Point,” Econometrica, Vol. 61 (July), pp. 821–56.
- Bai, J. and P. Perron, 1998, “Estimating and Testing Linear Models with Multiple Structural Changes,” Econometrica, Vol. 66 (January), pp. 47–78.
- Hasbrouck, J., 1991a, “Measuring the Information Content of Stock Trades,” Journal of Finance, Vol. 46 (March), pp.179–207.
- Hasbrouck, J., 1991b, “The Informativeness of Stock Trades: An Econometric Analysis,” Review of Financial Studies, Vol.4 (No. 3), pp. 571–595.

### Press reports and contemporary accounts
- Blitz, J., 1992, “Dollar Below Intervention level,” Financial Times (London: August 11, 1992), p. 23.
- “US Fed Intervenes as Dollar Nears Low Against D-Mark,” Financial Times (London: August 8, 1992), p.1.
- Linton, C., “Dollar Likely to Fall Though Banks May Try to Slow Currency's Descent,” Wall Street Journal (New York: Aug 10, 1992), p. C13.

*Source: _wp03115 - REFERENCES*

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