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What Monetary Policy Prevents Financial Chaos? James M. Haley1 The continuing debate about what caused the Great Recession challenges the traditional view of how rational expectations theory explains the evolution of financial markets. Monetary policy can cause individuals to chaotically hedge in ways that destabilize financial markets and the whole economy. It is not possible to make reliable economic forecasts without knowing how everyone, including the central bank, corrects their mistakes in a rational way. “If I had a simple answer, I would be spreading it around the world,” said Professor Sims after recently winning the Nobel Prize in Macroeconomics (Associated Press, October 11, 2011). This paper provides an answer. 1. Introduction There is a simple way to model financial chaos in three dimensions and continuous time. Specifically, a Sprott nonlinear model perturbed by noise can represent the evolution of forecast errors of real stock returns, inflation, and excess bond returns. A strange attractor emerges when monetary policy is consistent with the Taylor Rule, which targets interest rates to vary directly with inflation and real output. In order to promote stability the nominal interest rate and long run bond return should be targeted to equal the same fixed real interest rate expectation, such that the expectation of inflation is zero. This policy makes the forecast errors of stock returns behave more normally like a Langevin errorcorrecting, differential equation. The failure to predict the dynamics of how financial markets correct their mistakes explains why economists, including central banks, and investors cannot make reliable forecasts. Central banks can make matters worse by setting policy, based on inaccurate and skewed forecasts, that disrupts the economy even more. The resulting forecast and coordination errors can precipitate a financial crisis, causing chaotic business cycles to emerge. It is now possible to develop a more complete dynamic model of the evolution of economic errors, caused by central bank policy. This complexity is due in large part to the nonlinear feedback in the stock market, which creates speculative bubbles, when interest rates are set too low to stimulate the economy. But financial markets can suddenly panic, when interest rates are raised too high to curtail excessive speculation. This evolution of all possible errors, especially financial chaos, is more tractable to analyze in a macro framework. Predicting the collective action of interacting 1 H.J. Heinz Endowed Chair of Management, Ph.D., Economics, and MBA, Finance, the Business School of Point Park University, Pittsburgh, Pennsylvania, U.S.A., [email protected] 1 heterogeneous agents, as they search for consensus, is the only way to make more realistic predictions of how financial markets behave. So it should not be surprising that dynamic stochastic general equilibrium models, based on the behavior of an isolated representative agent, fail to address the most basic questions about the economy’s true dynamics (Colander, Howitt, Kirman, Leijonhufvud, & Mehrling, 2008). The real reason that new forecasting models should be developed is that standard financial models have failed to predict the financial crisis that led to the Great Recession. Furthermore, most macroeconomic forecasting models cannot predict what happened, because they ignore financial market dynamics altogether. For example Sargent (2006) and Sims (2006), who both recently won the 2011 Nobel Prize in Macroeconomics, do not include the behavior of the stock market and the long-term bond market in their analysis of the macroeconomy’s response to changes in policy. Interestingly, these same distinguished economists cannot agree about whether inflation persistence has declined. Sargent (2002) believes it has, while Sims (2002) believes it is unchanged. This paper proposes that inflation persists, when financial markets are chaotic that would be caused by the Taylor Rule being applied as central bank policy. When the central bank provides prudent guidance by “pegging” interest rates, to always equal the same real expectation, inflation no longer persists but mean-reverts to a zero expectation. This prudent policy should be implemented by central banks and investors, since it eliminates chaos and reduces risk, making forecasts more reliable. In fact this policy of pegging short and long interest rates to be the same, resembles a monetary policy specified by Woodford (2011). He claims that an equilibrium exists in a deterministic interpretation of a New Keynesian dynamic stochastic general equilibrium model. He assumes that the dynamics of this policy are consistent with the Taylor Rule. But this paper proves that only chaos will emerge in that case. Financial stability requires different policy dynamics. 2. Financial Market Dynamics Previously, I proved at the January 2007 AMS Conference in New Orleans that the financial markets can behave like a Rössler nonlinear system, which predicted the chaos of the Great Recession that started at the end of 2007 (Haley, 2010). There exists a simpler way to model financial chaos with a nonlinear dynamic system of forecast errors of the real stock return, R * , with respect to its expectation, Re* , inflation, pˆ , with respect to its expectation, pˆ e , and the coordination error of excess bond returns, x B . Specifically, a Sprott nonlinear model perturbed by noise can be derived in this system, by targeting excess demand for real money to be consistent with the Taylor Rule, a monetary policy that has generally failed to stabilize financial markets. 2 There is another bifurcation of policy, which more reliably guides the economy's search for a rational expectations equilibrium, such that the long-term bond return, rB , should be targeted to equal a fixed the short-term nominal interest rate, r , that equals its real expectation, re*, assuming expectation of inflation is zero. This policy peg, unlike the varying interest rate target of the Taylor Rule, makes the forecast errors of the real stock return, * , and inflation, y , to each differential equation. Consequently, the behave like a Langevin error-correcting, errors of forecasting real stock returns evolve smoothly and quickly converges to a normal density with a bounded variance and a mean error of zero. Whether the interest rate financial markets behave rationally or not, depends on setting M target, r, with respect to at any level of real money, , and real economic p output, Q , that is derived from the following assumptions about the macro search for correcting economic errors: * Ý* = R * = 1 x B2 - (A1) 2 + , such that * R* - Re* , x B = rB - r , Re* > re*, 1 ~ N ( 0 , 1), and 2 is sufficiently large to ensure fast convergence; xÝB = - a1 * + a 2 ; p̂ = b1 x B - b2 y - b3 , such that y = p̂ - p̂ e ; M = - l1 r + l 2 ln Q - ln , if r = r * + p̂ . p (A2) (A3) (A4) What do these financial market dynamics mean? Clearly, the real stock return, R * , which is the consensus forecast in (A1), mean-reverts to its expectation, Re* , which is greater the expectation of the real rate of interest, re*, as long as, Br = r . Inflation, pˆ , as a consensus forecast in (A3), mean-reverts as well with respect to random shocks of the excess demand for money, , as long as Br = r . Thus real stock returns and rates of inflation are self-correcting, when the short and long-term interest rates are the same. What distorts this normal convergence is when the real stock return exceeds its expectation, making returns on long-term bonds fall in (A2). Financial stability requires: * rB = r = re , if pˆ e = 0 . This stabilization policy means that the “real term structure” is flat, such that all * interest rates are equal to the same fixed real rate expectation, re , based on a zero inflation expectation, Interestingly, this deterministic equilibrium pˆ e . condition means there are no arbitrage profits in the bond market. Furthermore, 3 in the long run forecast errors in the stock market are normally distributed, such that the prices of risky assets asymptotically behave like a random walk. Put simply, financial markets can become a “fair game”. Then no one can beat the stock market, if real interest rates are pegged by the central bank. 3. Financial Chaos The evolution of a complex economy's forecasting errors, summarized in the dynamic system of economics errors, stated above, is best understood by analyzing its dynamics holistically. One way to see the system's complexity is to discover how different monetary policies affect the economy's process of learning from its mistakes. By experimenting with alternative policy rules it can be proven that a monetary policy exists that efficiently speeds up the convergence to an unbiased stochastic steady state of forecasting and coordination errors. One source of noise, which can not be reduced or eliminated, is random news, specified by 1 in (A1), since the variance of this noise is exogeneous and is not effected by monetary policy. But endogenous instability can be caused by the expectation of inflation or deflation that disrupts the economy, according to Meltzer (1986). This distortion becomes evident by analyzing the partial derivative of (A2) with respect to the expectation of inflation, which is positive and follows from: xÝB = a2l1 > 0. pˆ e Under these circumstances interest rates can not be easily pegged, which as inflation expectations change. increases instability in financial markets Many economists (Taylor, 1993) believe that maintaining long-term price stability reduces volatility in the economy. The benefits of this policy become evident by assuming that the standard deviation of inflation expectations, 2 , is zero, if and only if the expectation of inflation, p̂ e , is also zero, This assumption about when inflation expectations are fixed, reduces the uncertainty facing the economy as follows in assumption (A5): p̂ e = 0 if and only if 2 = 0. (A5) So far, the model specified by assumptions (A1) through (A5) is complete in terms of describing the evolution of economic errors. If the objective is to predict the behavior of real stock returns, interest rates, inflation, and money supply for any given level of output, a specific monetary policy is required to completely describe alternative monetary regimes of a complex learning economy. Knowing how the money supply behaves, based on an interest rate target, completes the 4 dynamic system at economy's evolution. Consequently, sufficient central bank intervention makes it possible to avoid indeterminancy in the model (Cass, 1995). It can be proven that there exists a policy, which approximates how many central banks currently set interest rates. It requires targeting the short-term nominal interest rate, r , to vary directly with its expectation, re , the forecast error for inflation, y , and the forecast error for the natural log of real economic output, . Furthermore, r varies inversely with excess real liquidity, L* , which is the M difference of the natural log of real money, , and its expectation, measured by p MT its trend, . This monetary policy is assumed to be mathematically specified pe as follows: r = re + w1 y + w2 - w3 L* , if w1 = l a1 1 , w 2 = 2 , w3 = l1a2 l1 l1 M M * such that = ln Q - ln Q - ln T . T , L = ln p p e (A6) This policy rule implies, that when inflation, pˆ , is greater (less) than its expectation or real output, Q, is greater (less) than its trend, the target interest rate should rise (fall), assuming its expectation, re , and excess real liquidity, L , are fixed. This policy is consistent with the Taylor Rule, which constrains the economy by raising interest rates when the economy is over-expanding, > 0, and inflation is too high, y 0 . Or the central bank, according to Taylor, should stimulate the economy when real output and inflation are too low, < 0 and y 0 , by lowering interest rates. Interestingly, when inflation, output, and real excess liquidity are normal, that is equal to their expectations, then the nominal interest rate target should equal its expectation as well. Furthermore this rule is equivalent to the following excess demand for money, : =- M a1 y , if ln T = - l1 re + l2 ln QT . pe a2 Applying this behavioral rule to a complex learning economy given by the differential system (A1) makes monetary regime emerge. a chaotic through (A6), (A2), it easily follows that: assumption By substituting in xÝB = - a1 ( * + y ), if = - 5 a1 y. a2 Clearly, after substituting again for in assumption (A3), it simply follows that: y = b1 x B + b y , if = - a1 a y , b = ( 1 b3 - b2 . a2 a2 Combining the above reduced form equations of (A2), and (A3) with (A1), it there exists a dimensional follows from (A6), three that nonlinear dynamic model of financial markets unperturbed by noise. For simplicity assume: 1 = 2 = a1 = b1 = 1, b = 0.5, x B = - x˜ B . (A7) Therefore, after making the necessary substitutions, the following theorems can be derived from assumptions (A1), (A2), (A3) and (A7): = x˜ B2 - * , (T1) ˜ B = ( * + y ). xÝ (T2) y = - x˜ B + .5 y . (T3) * This three dimensional nonlinear system represents one of the simplest ways to model chaos in continuous time and is consistent with a chaotic system first devised by Sprott (1994). A central bank that changes interest rates too much, makes it impossible to correct forecasting and coordination errors in the economy. This occurs, because there is confusion about what interest rates should be. Imprudent central bank guidance can cause interest rates to rise too high, increasing the risk of a bear market in stocks. If interest rates are targeted too low, a speculative bubble can emerge, as well. 4. A Better Way to Stabilize the Stock Market Whether expectations become rational, meaning that the forecast errors become unbiased and reliable, critically depends on the stock market being efficient, such that it quickly converges to a stochastic equilibrium. Then expected excess returns of a diversified portfolio of stocks equal the market risk premium, and the probability density of excess returns is asymptotically normal. The stock market avoids financial chaos, caused by nonlinear feedback, when real stock returns mean-revert, a necessary condition for rational expectations to emerge. Thus the variance, caused by the random shocks of news, is reduced asymptotically by reinforcing mean-reversion, as the financial markets learn to correct errors in a normal way. 6 In order to stabilize financial markets it is necessary to peg the short-term nominal interest rate, r , and the long-term bond return, rB , to equal their expectation, re . This new monetary policy assumption would replace the current monetary regime stated in (A6): r = rB = re . (A6)* Now it is easily follows that the interest rate peg should equal its real rate expectation, re* , if price stability is the other necessary condition for further reducing economic uncertainty: r = re* + p̂ e = re* , if and only if p̂ e = 0 . (T4) Furthermore, it follows from (A6)* that the excess demand for real money, , behaves as follows: = a1 * . a2 Then it can be shown that the complex learning economy reduces to a selfcorrecting monetary regime, after substituting the above value of into the appropriate error-correcting differential equations. So a different monetary regime emerges by replacing assumption (A6) with (A6)*, such that a new system of differential equations, perturbed by noise can be derived. Specifically, after substituting assumption (A6)* into (A1), the following error correcting, Langevin equation is implied for the forecast error, * , of a real stock return, R * , with respect to its expectation, Re* , shocked by noise, 1 : * = - 2 * + 1 , if x B = p̂ e = 0 , 1 ~ N ( 0 , 1 ) (T5) Furthermore, the long run stochastic equilibrium of this mean-reverting differential equation can be proven to be a normal density of * with a zero mean and standard deviation of 3 , which is given by: Asymptotically, * ~ N ( 0, 3 ), if rB = r , (T6) such that lim E ( * ) = 0 , 32 = 12 / 2 2 , * = R * - Re* . t Eventually the long-term variance of the unbiased forecast errors of real stock returns, 32 , is less than the variance of the continuous random shocks of news, 12 , assuming that 2 is relatively large (Lasota & Mackey, 1994). 7 Current research tends to model the density of stock returns as not being normally distributed, especially for the short run returns (Campbell, Lo, & MacKinley, 1997). In fact actual returns exhibit excess kurtosis or fat tails, as well as skewness. Thus positive and negative extreme returns are more likely to occur than standard finance theory would predict. This paper proves that standard theory is correct, if central banks properly target interest rates to equal the expectation of the real rate. Then according to theorem (T6) real stock returns will behave “normally” in the long run, such that extreme returns, resulting from skewness or excess kurtosis, are not likely. Finally under certain parameter specifications it can be proven that the whole economy becomes rational. After making the necessary substitutions it can be shown that expected forecast errors for real stock returns, * , real output, , and inflation, y , eventually converge to zero as follows (Haley, 2010): Asymptotically, E ( * ) = E ( ) = E ( y ) = 0, (T7) a b if x B = p̂ e = 0, 1 ~ N ( 0, 1 ), 1 > 2 . a2 b3 The complex evolution of the economy converges to a simple state, where stock returns and inflation mean-revert by evolving as independent Langevin equations. Clearly, how a central bank targets interest rates determines whether the whole economy behaves rationally or not. Guesnerie and Woodford (1995) state it simply, “... An interest-rate pegging regime provides an anchor for expectations.” Then it becomes easier to forecast and plan, which eventually stabilizes the stock market and the whole economy. In fact it speeds up the smooth convergence to a stochastic steady state, such that expectations become rational in the long run. 5. Conclusion This paper proves that the theory of monetary policy can be made more rational, if it exploits how a complex learning economy learns from its mistakes. This search for an error correcting stationary process can be disrupted in the short run, if a central bank distorts the normal relationship between the excess bond returns with real stock returns. To avoid this confusion the short-term, nominal interest rate should be pegged to eventually equal its real expectation of 1.8%, as estimated by Campbell, Lo and MacKinlay (1997), which equals the return of a long run bond in equilibrium, assuming a zero inflation expectation. 8 Therefore the Federal Reserve's current federal funds target rate of less than .25% or greater than zero is biased too low. This cheap interest rate policy to stimulate the economy raises the risks of higher inflation and the emergence of a chaotic bubble economy in the long run. 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