سیاست های پولی و ثبات در طول شش دوره در تاریخ اقتصادی آمریکا : 1959-2008: یک قانون سیاست های پولی غیر خطی جدید

We investigate the monetary policy of the Federal Reserve Board during six periods in US economic history 1959–2008. In particular, we examine the Fed's response to changes in three guiding variables: inflation, π, unemployment, U, and industrial production, y, during periods with low and high economic stability. We identify separate responses for the Fed's change in interest rate depending upon (i) the current rate, FF, and the guiding variables’ level below or above their average values and (ii) recent movements in inflation and unemployment. The change in rate, ΔFF, can then be calculated. We identify policies that both increased and decreased economic stability.

We explore the role of monetary policy on the stability of business cycles during six periods in US economic history from 1959 to 2008. We examine in particular the Federal Reserve's response to three candidate target, or guiding, variables: inflation, industrial production and unemployment. The first two correspond to the guiding variables in the Taylor rule (Taylor, 1993 and Doran and Hickey, 2009). We develop a nonlinear (compass-type) policy reaction function in which the reaction coefficients depend upon movements in both FF and the explanatory variable and how the movements relate to their average values. The method is based upon the assumption that the chosen candidate explanatory variables are sufficient to explain differences in economic volatility among periods. The Taylor rule is currently debated (2012) because of its role during the current recession both in the US and the EU (Cancelo et al., 2011, Fernandez et al., 2010, Fourcans and Vranceanu, 2007, Giammarioli and Valla, 2004 and Rudebusch, 2009). Recently, nonlinear Taylor rule functions have been discussed by Taylor and Davradakis (2006), Aksoy, Orphanides, Small, Wieland, and Wilcox (2006), Orphanides and Wieland (2008) and Hayat and Mishra (2010). Below, we give our working version of the Taylor rule, e.g. (Taylor, 2009), and then we discuss Okun's law that links unemployment to output (GDP). The latter rule helps explain why a response to output may be confounded with a response to unemployment. The Taylor rule in its original formulation states how much the central bank should change the nominal interest rate, in US, a federal funds rate, i , in response to the difference between actual inflation π t and the target inflation, π *, and to the actual output, GDP, from the potential output GDP. The output parameters y t and View the MathML sourceyt* are defined as the logarithm of the GDPs: equation(1) View the MathML sourceit=r*+π*+βπ(πt−π*)+βy100×yt−yt*yt* Turn MathJax on Here r* is the equilibrium real interest rate corresponding to the potential GDP. r* + π* is the long-term equilibrium interest rate. According to the rule, both βπ and βy should be positive. Taylor (1993) proposed the values βπ = 1.5 and βy = 0.5. Clarida, Gertler, and Galí (2000) showed that during the highly volatile pre-Volcker era 0 < βπ < 1, and in the much more stable post-Volcker era (from 1982) the slope βπ was ≫1 (pre- and post aggregate volatility indicators were 2.77 and 1.00 respectively). Rudebusch (2006) found that βπ = 1.39 and βy = 0.92 for the period 1988–2005 with least squares regression. Hayat and Mishra (2010) found that the βπ-coefficient would be zero at less than 6.5–8.5% changes in inflation, independent of period. Okun's law states that the gross domestic product, yt, is negatively related to unemployment, u: equation(2) View the MathML sourcey−y*y*=−ω(u−u*) Turn MathJax on Okun's law is reported to show a consistent negative correlation where ω is about 2, as summarized by Dornbusch, Fischer, and Stratz (2008). We construct phase plots for the variables (one variable on the x-axis and the other variable on the y-axis, Fig. 1 lower panels). For 6 variables there will be 15 such pairs. These phase plots describe graphically the relationship between paired variables. The relationship is formally quantified by calculating the slopes νi,i+1 for the trajectories between sequential states i and i + 1 and the x-axis. We also determine if the initial values of the trajectories are below or above the average value of the variables, and we determine from which of six historic periods the observations were taken. We thereafter calculate a measure of the stability of business cycles during the six periods. This allows us to examine which moves were characteristic during periods with low or high stability. The technical method is called the angle frequency method, AFM ( Sandvik, Jessup, Seip, & Bohannan, 2004), and to our knowledge it is the only method that allows detection of cyclical or spiraling movements in phase plots, e.g. Brunnermeier (2009, Fig. 9) for spiraling effects in an economy.The trajectory angles and their interpretation in economics. To give a rationale for the AFM, we show how the trajectory angles can be interpreted in economic terms. For illustration purposes, we assume that the business cycles are represented by perfect sines, Fig. 1a. Observed cycles can be looked upon as sine curves with added noise. The upper two panels show a target variable (a1 and b1) and an alternative series that is shifted in time relative to the target. The two lower panels show the phase plots corresponding to the time series in the upper panels. The alternative variable may relate to the target variable by being coincident (a), leading, counter cyclic, or lagging (b). It is seen that the phase plots in the lower two panels give signatures of the relationship between the paired variables. Variables that are coinciding, for example variables that belong to National Bureau of Economic Research's, NBER's, coinciding indicator, would show a pattern like the ones in (a) and (c). This means that the angle νi,i+1 in the phase plots for the trajectory from i to i + 1 will be around 45° or 225° whether the trajectory starts below or above average values of the two variables, and most of the trajectories will start in quadrant I or III. However, coinciding indices may be shifted within three months ( Kholodilin & Yao, 2005), so angles may deviate from the ideal values for this reason and because of noise in the data or because there is friction between variables. See Aksoy et al. (2006, Fig. 1) for graphs similar to our graphs (c) and (d), but without our interpretations. If the two sines are shifted 〈0,λ/2〉 relative to each other in time, rotational patterns will emerge. For a lagging variable (y-axis) that has its peak after the target peak (x-axis), the trajectories will rotate counter clock-wise (d) and the directions of the trajectory will be (for sines without noise) between 90° and 180° if the trajectories start with high levels of both variables, and between 270° and 360° if the trajectories start at low values of the two variables. The federal funds rate should be a lagging variable to output growth ( Herrera and Pesavento, 2009) and would rotate counter clockwise in phase plots when output growth is plotted on the x-axis. The sets of angles we obtain for each pair of variables within each period (15 × 6 period – pairs, about 9000 trajectories) can be regarded as fingerprints of the interaction between the two variables. The “fingerprints” are further described in Section 3. We developed an algorithm for setting the Feds rate as a function of (i) the current values of the Fed's rate, inflation and unemployment, and of (ii) observations of recent movements in the two latter variables. Our prescription distinguishes itself from the Taylor rule in that it is data driven and that it requires the additional assumption that successful moves by the Fed in the past can be used as prescriptions for successful moves in the future, e.g. (Doran and Hickey, 2009, Rudebusch, 2006 and Taylor, 2009). The method is easy to implement with standard computational packages. Our main result is a nonlinear calculating rule for the federal funds rate. The rule is formulated as compass directions rather than slopes, and thus different from the Taylor rule (1993). However, our AFM-rule supports to a certain degree the same decisions as the Taylor rule recommend. There are, however, important differences. It does not recommend changing the federal funds rate if monitoring variables are already moving toward a “natural” equilibrium. We also develop an algorithm for moves that counteract stability and thus gives clues to the sensitivity of the Fed's moves during “good” and “bad” economic periods (in terms of volatility in inflation and unemployment). The recommended changes in the federal funds rate should be interpreted and understood within a structural economic framework, probably more so than recommendations obtained from theoretical studies. The rest of the paper is organized as follows: We first present the material for our study. Thereafter we give an outline of methods employed: identification of variables to be paired and their pre-treatments, the construction of phase plots, the angle frequency method, AFM, and the partial least squares method, PLS. The latter method is required for strongly co-varying regressors and when there are more regressors than samples. Finally, we discuss the results relative to monetary policies and the Taylor rule

We first present results for the relationship between volatility during six periods in US economic history and the Fed's interest rate response to changes in 3 variables: inflation, unemployment and industrial production during these six periods. The mean value will be compared to the variables’ “natural” rates in the economy. 4.1. The Fed's moves during low and high stability The characteristic angles for moves that support or counteract stability are retrieved from the loading plots of the PLS and are shown in Table 1 and Table 2. The PC1 of the plot explains 24% of the variance in the stability and PC2 explains 19% of the variance (plots not shown). The characteristics are given in terms of guiding variable (FF versus inf, U, or IP respectively), quadrants (I–IV) and angles (0–360°).Fig. 3B and C shows the result for the Fed's rate versus inflation and unemployment as arrows in their respective phase plots. Note that the arrows are important, it is only movements in the direction of the arrows that will give ΔFF ≠ 0. The two dashed arrows a and b will be explained below. The results for IP were not significant and are not shown. Numbers at midpoint of axes are the average of the averages of variable values over the six periods A–G and their standard deviation. The graphs are similar to the policy response graphs in Aksoy et al. (2006, Fig. 1), but our responses are here represented by arrows rather than bi-directional lines. The results will be formulated as follows; first we identify the current states of FF and the guiding variables. Then we describe the stability supporting moves for FF depending on moves in the guiding variable. Lastly, we describe moves that would have counteracted stability. The arrows show the direction of the movements. The upper two panels, A, in Fig. 3 show a schematic version of the Taylor rule.4.1.1. Inflation as target Panel B, left square, shows moves that increase stability. We found that at a high value of FF and at lower than average inflation, stability supporting FF moves were a sharp decrease in FF with slowly decreasing inflation (quadrant II). However, from a high value of both FF and inflation, a slow increase in FF with rapidly increasing inflation supported stability. Panel C, lower left square shows moves that decreased stability. At low values of both FF and inflation, stability decreased if FF is increased with increasing inflation. At high values of both FF and inflation, stability decreased if FF was decreased sharply with a slowly decreasing inflation (quadrant I). 4.1.2. Unemployment as target Panel B, upper left square shows moves that increased stability (quadrant II). We found that at high levels of FF and low levels of unemployment slowly decreasing the Feds rate with increasing unemployment supported stability. However, the dashed arrow (a) – which is only weakly significant – shows that at low unemployment, increasing the Fed's rate with decreasing unemployment, may also contribute to increasing stability. At low levels of FF and high level of unemployment, rapidly decreasing FF with slowly increasing unemployment supports stability. The second dashed arrow (b) will be discussed below. Panel C, lower right square, shows moves that decrease stability. At low levels of FF, but at low as well as high level of unemployment, increasing FF sharply with decreasing unemployment decreased stability. The destabilizing effects appeared to be particularly strong at low values of both FF and unemployment. 4.1.3. Comparison with the Taylor rule We have indicated a comparison with the Taylor rule in the upper panel of Fig. 3 by recalculating the Taylor slope, 1.5 for inflation, to its standardized units, 1.1, and positioning it approximately at an inflation rate of 2%. Our prescription is overall consistent with the Taylor rule, but recommends a more rapid decrease at low inflation rates and a slower increase at higher inflation rates. The turning point for inflation is higher in our model than for the Taylor rule (4% versus 2%). The Taylor rule is linear and gives the same slopes for the Fed's response above and below the guiding variables’ “natural” rates, and it describes a response irrespectively of how the economy moves at the time of the decision if the economy is not at equilibrium. 4.2. Applications to the period 2000–2008 In this section we compare the Fed's actual funds rate, our AFM policy rule, and the Taylor rule for the period 2000 to 2008. We find that the AFM policy rule is fairly consistent with the Fed's policy during this period, prescribing a lowering of the Fed's rate when the rate was actually lowered, Fig. 4, but only prescribed an increase in the rate for a short period (2007). Our rule contrasts with the Taylor rule in that it (with one exception) does not give prescriptions when the economy is moving toward equilibrium.4.2.1. Backgound The period 2000–2008 corresponds to our last period, G. It is the period with the next lowest volatility out of our six periods. Thus, we would expect that our AFM-rule would reflect fairly well the Fed's interest rate movements during this period. We chose this period because it is discussed by Rudebusch (2009), Fernandez et al. (2010), Catte, Cova, Pagano, and Visco (2011) and others. For the sub period 2003–2007, Taylor (1993) asserted that the federal funds rate should have been higher than the rates the Fed actually set. 4.2.2. Conducting the test In the test we chose to smooth the inflation time series by calculating annual data and then interpolate to obtain monthly data.6 It will then be easier to see how our ΔFF values are calculated. At the end of each month we calculate the movements in inflation, ΔInf, and unemployment, ΔU (based on the smoothed series). We then identify if the Federal fund's rate, inflation, and unemployment are below or above their averages (as in Fig. 3). Depending upon their current values we calculate the response values ΔFFπ = tan(υ FF,π) × Δπ and ΔFFU = tan(vFF,UvFF,U) × ΔU with the angles from Table 1. Note that ΔFF is only given a non-zero value if the movements are in the direction shown by the arrows in Fig. 3. All calculations were made in Excel. The results are shown graphically for unemployment ( Fig. 4c) and inflation ( Fig. 4d) and with a weighted sum ( Fig. 4b). In the latter case we let the two components weight equally by multiplying each with the standard deviations of the series of moves 2000–2008. To compare with the Taylor rule, we also relaxed the restriction on the direction of the recent movements in inflation and unemployment, that is, there is no reference to whether the economy is moving away from equilibrium or toward it. The results are shown in Fig. 4e and f. Our AFM-rule basically describes what the Fed actually did during the 2000–08 period, Fig. 4(c) and (d). The largest contribution for changes came from observations of unemployment, but both variables tend to give the same recommendation. There is one exception; in 2007 there is a prescription to increase the FF-rate based on the observations of the inflation rate. The figure also shows contributions that came from low and high values of the two variables. When we relaxed the restriction on directions, the unemployment component of the rule recommends an increase in the Fed's rate from mid 2003 to 2007, Fig. 4(e). However, except for the small rise from mid 2006 to 2007, there was no support for this rise in the learning sets of the AFM-rule (six periods during 1959–2008).