Forward-looking information in VAR models and the price puzzle
📄 Summarized from the full manuscript · Human-reviewed for faithfulness before publication
In brief
Can the puzzle of prices rising after a rate increase be fixed by giving the statistical model the forward-looking information a central bank actually uses? Using monthly United States data from 1989 to 2004, this 2006 paper adds a composite leading indicator of activity and the market's expectation of the coming month's policy rate. Prices then edge up only slightly and insignificantly for two or three months before declining, significantly so after fourteen months, while industrial production falls and recovers within roughly two years. Why it matters: it locates the puzzle in what the model omits, not in how the economy works.
What this paper finds — and why it matters
This 2006 Journal of Monetary Economics paper by Sophocles Brissimis and Nicholas Magginas asks whether augmenting a standard monetary VAR with forward-looking information can resolve the “price puzzle” — the counter-theoretical finding that prices rise, rather than fall, after a contractionary monetary policy shock — in a modern US sample dominated by the Greenspan Federal Reserve. Using monthly US data from January 1989 to June 2004 (the period for which federal funds futures data are available, and one the authors characterize as a homogeneous policy regime), they first show that a standard four-variable recursive (Cholesky) VAR — industrial production, CPI, a commodity price index, and the federal funds rate, with 6 lags chosen by AIC and SBC — produces “a strong counter-theoretical response of CPI to the monetary policy shock,” with CPI rising throughout all 48 months of the impulse response. They then replace this baseline with a system ordered {composite leading indicator (LCOM), industrial production, CPI, federal funds rate}, in which the one-month-ahead federal funds futures rate (FFF) — the market’s expectation of the current month’s funds rate — enters as an exogenous variable rather than as part of the endogenous ordering, so that the monetary policy shock is identified as the orthogonalized residual of the funds-rate equation. In this augmented system “the price puzzle is solved”: CPI shows only a small, statistically insignificant positive response for the first 2-3 months, then declines gradually to a trough 30 months after the shock, with the decline becoming statistically significant after 14 months; industrial production declines steadily to a trough 5-6 months out and returns to its pre-shock level roughly two years (24 months) after the shock, tracking LCOM’s response with a 1-2 month lag; and the funds-rate shock itself dies out quickly (4-5 months), so there is “no policy innovation paradox.” The result is robust to replacing FFF with an implied one-month forward rate constructed from two-month Libor and the effective funds rate, both over the same 1989-2004 window and over a longer 1986-2004 sample (the earliest for which Libor data exist), and the authors note that further extending the system to seven variables in a Christiano-Eichenbaum-Evans/Kim-style specification with total and non-borrowed reserves and M2 yields “no significant gains” in resolving the puzzle. The paper’s overall diagnosis is that the price puzzle reflects the VAR’s omission of the richer, forward-looking information set the central bank actually uses to forecast inflation and output; the federal funds futures rate and the composite leading indicator proxy for that omitted information without expanding the VAR’s dimensionality.
Summary of a classic paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.
Questions & answers
Q1. What is the “price puzzle,” and does it show up in this paper’s own baseline VAR?
Yes, strongly. The price puzzle is the empirical finding that CPI rises, rather than falls, following a contractionary monetary policy shock in a standard recursive VAR — the opposite of what conventional theory predicts. The authors first estimate a standard four-variable recursive (Cholesky) VAR — industrial production (INDP), CPI, a commodity price index (COMP), and the federal funds rate (FFR) — over January 1989-June 2004 using monthly US data with 6 lags (chosen by both AIC and SBC). This baseline system produces “a strong counter-theoretical response of CPI to the monetary policy shock,” with CPI rising for the entire 48-month impulse-response horizon.
Q2. How is the augmented system set up, and how is the monetary policy shock identified?
The main system replaces COMP with a composite leading indicator (LCOM) and adds the one-month-ahead federal funds futures rate (FFF) as an exogenous variable, keeping a recursive (Cholesky) identification with Wold ordering {LCOM, INDP, CPI, FFR}. FFF is the market-implied expected level of the current month’s FFR, taken from the futures contract price on the last business day of the previous month; it enters as exogenous rather than as an ordered endogenous variable, so the monetary policy shock is identified as the orthogonalized residual of the FFR equation. LCOM is the Conference Board’s composite leading index (manufacturing employment, confidence indicators, new orders, a monetary aggregate, the term spread), used with a one-month lag because much of its underlying data are not available until more than 15 days after month-end. The ordering places LCOM first (on the view that the Fed’s information advantage is compressed into it), then INDP and CPI (reflecting the Fed’s contemporaneous access to output data and its emphasis on forecasting prices), then FFR.
Q3. Does adding FFF and LCOM actually resolve the price puzzle, and by how much?
Yes — the authors state plainly that “the price puzzle is solved.” In the augmented system, CPI shows only “a small, but not statistically significant positive response” for the first 2-3 months after a contractionary shock, then declines gradually, reaching a trough 30 months after the shock; the decline “becomes significant after 14 months.” This is a qualitative resolution (the puzzle disappears and the expected sign eventually dominates and becomes significant) rather than an instantaneous one — the initial months still show a positive, if insignificant, CPI response.
Q4. What do output and the policy rate do in the resolved system, and is there a “policy innovation paradox”?
Industrial production falls steadily to a trough 5-6 months after the shock and returns to its pre-shock level about two years (24 months) later, tracking the response of LCOM with a 1-2 month lag; the FFR shock itself dies out quickly, 4-5 months after impact. The authors read the quick decay of the policy shock together with the plausible, hump-shaped output response as evidence there is “no policy innovation paradox” in this specification — i.e., the model does not require an implausibly persistent policy shock to generate a sensible real response.
Q5. Is the result specific to the federal funds futures rate, or does it hold under an alternative measure of policy expectations?
The result is robust to replacing FFF with an implied one-month forward rate constructed from the two-month Libor rate and the effective FFR (ffr_{t,t+1} = 2 x Libor_{t-1} - ffr_t), which is available for a longer sample (Libor data begin January 1986, versus January 1989 for FFF). Using this implied forward rate, the price puzzle is resolved “qualitatively similar[ly]” in both the original 1989:1-2004:6 window and the extended 1986:1-2004:6 window: the initial positive CPI response is statistically insignificant, and the price level then declines gradually, remaining below its baseline path through the end of the 48-month horizon.
Q6. Does simply adding more monetary/financial variables to the VAR (without forward-looking information) achieve the same result?
No — the authors report that a seven-variable system in the style of Christiano-Eichenbaum-Evans (1999) / Kim (2001), which adds total reserves, non-borrowed reserves, and M2 to the baseline four variables, yields “no significant gains in terms of dealing with the price puzzle.” This is reported only as a footnote result without further detail, but it is used to argue that the resolution is specifically attributable to the forward-looking content of FFF/the implied forward rate and LCOM, not merely to enlarging the VAR’s information set with additional monetary aggregates.
Q7. What mechanism do the authors propose for why the price puzzle arises in standard VARs, and why forward-looking variables fix it?
The authors attribute the price puzzle to two shortcomings of a low-dimensional VAR’s implied policy reaction function: it omits much of the central bank’s actual (richer) information set, and it contains no forward-looking elements, so it cannot capture the pre-emptive character of modern monetary policy. Drawing on Romer and Romer (2000) and Brunner (2000), they argue that “a significant part of this information is omitted from the VAR, which provides [only] a simplified reduced-form expression of the central bank’s feedback rule.” Federal funds futures and the implied forward rate reflect market expectations of future policy, and so implicitly embed information about future output and price developments that the Fed itself uses; entering them as exogenous variables effectively conditions the identified shock on (an proxy for) the Fed’s own information set. The composite leading indicator plays an analogous role by compressing forward-looking information from several series into one variable without expanding the VAR’s dimension. The authors summarize: “the greater openness and transparency that characterize monetary policy-making during the last 15 years have increased the ability of the markets to anticipate policy actions,” making instruments like federal funds futures “effective means for incorporating in a parsimonious way a large amount of information in VARs.”
Q8. What are the paper’s stated caveats and limitations?
The authors flag several limits on their own results. LCOM is used with a one-month lag purely due to data-availability constraints, which “potentially understates” its forward-looking content. The FFF-based results are confined to 1989 onward because federal funds futures only began trading at the CBOT in 1988. Even in the resolved system, the initial 2-3 month CPI response remains positive (just statistically insignificant), so the puzzle is not eliminated instantaneously, only over the medium run. All reported impulse-response magnitudes (other than a handful of stated timing facts — the 14-month significance point, the 30-month CPI trough, the 5-6 month and 24-month INDP milestones, and the 4-5 month FFR decay) are shown only graphically in Figures 1-3 and are not given as numbers in the text. Finally, the paper treats the expected FFR (FFF) as fully exogenous to the VAR system and does not formally test that exogeneity assumption.
Key terms in this paper
Definitions below follow the paper's own usage.
- Price puzzle
- in this paper, the empirical finding — reproduced in the baseline four-variable recursive VAR (INDP, CPI, COMP, FFR) over 1989:1-2004:6 — that CPI responds positively, rather than negatively, to a contractionary federal-funds-rate shock, described by the authors as "a strong counter-theoretical response of CPI to the monetary policy shock" that persists across the full 48-month horizon in the baseline system.
- Recursive (Cholesky) VAR / Wold ordering
- the identification scheme used throughout, in which a specific causal ordering of the endogenous variables — here {LCOM, INDP, CPI, FFR} — is imposed via a Cholesky factorization of the reduced-form residual covariance matrix, so that the monetary policy shock is the orthogonalized innovation to the FFR equation, with FFR ordered last so it can respond contemporaneously to innovations in LCOM, INDP, and CPI.
- Federal funds futures rate (FFF)
- the market-implied expected level of the federal funds rate for the current month, taken from the price of the one-month-ahead federal funds futures contract on the last business day of the prior month; in this paper it enters the main VAR system as an exogenous variable (not part of the endogenous Cholesky ordering) so as to bring the market's forward-looking information about policy into the system without conditioning the expectation itself on the VAR's own information set.
- Composite leading indicator (LCOM)
- the Conference Board's US leading index, a weighted average of forward-looking and coincident series (manufacturing employment, confidence indicators, new orders in manufacturing, a monetary aggregate, and the term spread), used in this paper with a one-month lag (because several of its component series are not available until more than 15 days after month-end) as a compact way to bring forward-looking information about real activity into the VAR without adding many extra variables.
- Implied one-month forward rate
- an alternative to FFF, constructed as ffr_{t,t+1} = 2 x (two-month Libor rate at t-1) - (effective FFR at t), used because it is available over a longer sample (from January 1986) than the federal-funds-futures data (from January 1989); the paper uses it as a robustness check on the FFF-based results.