Monetary Policy Dynamics in the United States

This work is licensed under the Creative Commons Attribution alone 4.0 License. Open Economics 2021; 4:14–30 Research Article Open Access Oladimeji T. Shodipe* and Olatunji Abdul Shobande Monetary Policy Dynamics in the United States https://doi.org/10.1515/openec-2020-0111 Received Feb 6, 2021; accepted Mar 16, 2021 Abstract: The recognised approach to designing an optimal monetary policy model is based on the central bank’s ability to mitigate losses using a quadratic criterion subject to the linear structure of the economy. This study examines the United States Federal Reserve’s (Fed) monetary policy in di erent economic environments. It provides an empirical solution to the central bank’s optimisation problem when preferences are asymmetric in both in ation and output gaps. The study tested for structural breaks and uncovered potential evidence of nonlinearities in the Fed’s reaction function, which provides more information on policy objective. The empirical evidence suggests that the Fed’s policy rate di ers in these periods. This strongly indicates the presence of asymmetry. Further evidence suggests that the predictive power of the estimated model increases when a smoothing process is allowed.


Introduction
The traditional approach to monetary policy is to use fund rates as a tool to manage irregular uctuations in the economy. During a recession, the reserve bank rate is the most e cient policy instrument to spur a recovery (Shobande & Shodipe, 2019a. However, caution must be exercised to ensure that the rates do not drop below the thresholds (Shobande, 2019a;Shobande & Shodipe, 2019b). For the United States Federal Reserve (Fed), unconventional monetary policy is often used in response to macroeconomic uncertainty. However, it is unclear whether this approach is e ective, especially during economic turmoil. In this study, we examine the complex events that characterise the actions of the Fed in terms of changing periodic asymmetry in policy responses. By addressing the optimal control problem, we examine how central banks mitigate losses using a quadratic criterion to determine target rules in response to macroeconomic uctuations and provide new information that has important policy implications for e ective and e cient monetary policy formation in the United States.
There are three reasons to investigate the dynamic nature of the Fed's monetary policy. First, major empirical e orts have been based on the quantitative measures of monetary policy, whereas little attention has been paid to the issue of structural breaks and policy rules for asymmetry, which are important for forming expectations about the future (Moosavi & Cao, 2020;Luo & Tsang, 2020;Hang Xue, 2020). For example, Jansen (2011) has suggested that information asymmetry often misinforms the public regarding expectations, while Cihak and Jansen (2013) have shown that nancial market volatility is a product of unclear communication by monetary authorities. Second, a proper understanding of the linear and nonlinear nature of monetary policy can provide e ective communication between the monetary authorities and the public that can help to mitigate volatility in the nancial market and improve the forecasting accuracy of in ationary expectations (Jarocinski & Karadi, 2020;Heider et al., 2019;Mankiw & Reis, 2018;Clarida, 2019;Daetz, 2018;Shobande & Enemona, 2021). Third, many studies have argued that understanding how policymakers adjust their preferences within the macroeconomic environment can serve as yardsticks for designing policy rules to capture the time-varying properties of threshold values (Shobande, 2019b;Taylor & Davradakis, 2006;Bunzel & Enders, 2010;Koustas & Lamache, 2012;Qin & Enders, 2008;Tan & Habibullah, 2007;Kin, 2004;Martin & Milas, 2004).
Empirical evidence on the dynamics of the Fed's policy is inconsistent and widely controversial. The Fed's use of optimal policy rules has been examined in both conventional and unconventional studies. For example, studies by Taylor (1993), Clarida et al. (1999), and Woodford (2007) have extensively used the conventional linear functional approach to analyse the Fed's behaviour on monetary policy responses, but a lack of agreement among these scholars has continued to stimulate a growing body of literature on the subject. In another example, Bernanke (2010) has rejected the claim by Taylor (2009) that low Fed rates have caused the nancial meltdown. Mankiw et al. (2003) investigated 50 years of in ation expectations and discovered that time factors account for variations in the in ation rate. In another study, Filardo and Guinigundo (2008) reported that disagreements about in ation outlook were due to a lack of transparency. Dovern et al. (2012) analysed monetary policy behaviour in a panel of G7 countries and observed that disagreement on future expectations arise from an increase in uncertainty. The second group has focused on the dynamic macroeconomic environment that drives the Fed's behaviour to encourage economic recovery using unconventional monetary policy. For example, a study by Ben-habib et al. (2001) has highlighted that the policy rule is subject to multiple equilibria under the zero-lower bound (ZLB). Some studies have recognised that optimum monetary policy relies on the exogenous, endogenous inelastic, or endogenous elastic attention of agents (Luo & Tsang, 2020). For example, Luo & Tsang (2020) have shown that under elastic consideration, optimal monetary policy produces balances that are not feasible under the other two conditions: no exposure to any shocks that cause unstable economic uctuations. A similar study by Ouerk et al. (2020) used the shadow rate to measure monetary policy stance and reported that the e ect of unconventional monetary shocks was weaker and less persistent than that of conventional monetary policy. Du y and Engle-Warnick (2006) have applied a non-parametric method to examine the multiple policy regimes of the Fed's behaviour and reported that a structural test approach requires the identi cation of a number of policy regimes prior to the test and might pose a serious threat to the test outcomes. Siklos (2013) has shown that central bank forecasts increase disagreements, while Zhu and Chen (2017) examined a forward-looking threshold Taylor rule for the United States and have reported that response to in ation and output gaps are asymmetric. In addition, Bunzel and Enders (2010) have suggested that the Fed is likely to be more aggressive when in ation is high than when it is low. Marin and Milas (2004) have noted that if monetary policy is asymmetric, policymakers are more likely to respond to upward deviations of in ation away from in ation targets; while Shobande and Shodipe (2019) have argued that in ation targeting often leads to a change in monetary policy formation.
This study extends the existing literature in the following way.
(1) It examined the asymmetric nature of monetary policy in the United States. (2) By allowing for the presence of asymmetric preferences, the study tested for the structural break and uncovered potential evidence of nonlinearities in the Fed's reaction function and thereby provides more information on asymmetry in the policy objective. Our study used quarterly data and implemented both traditional and modernised models. Empirical evidence showed that monetary policy behaviour in the United States can be e ectively and e ciently characterised as a partial nonlinearity policy rule. Further ndings showed that the predictive power of the policy model increases when smoothing is allowed, both in the linear and asymmetric models. This new information can help policymakers conduct their monetary policy at a zero lower rate.
The remainder of this paper is organised as follows. Section 2 presents the empirical model, while Section 3 presents the data, sources, and descriptions. Section 4 presents the empirical results, and Section 5 provides concluding observations.

The Model
This section discusses the buildup for the policy rule. The traditional monetary policy framework links changes in nominal interest to macroeconomic fundamentals. Since policy credibility relies on consistency, an unwarranted reversal in the policy stance is usually avoided. Thus, the Fed depends on rules for policy guidance. Taylor (1993) de nes short-term nominal interest rates as a function of long-run equilibrium real interest rate, actual in ation, in ation and the output gaps. Historically, Taylor's policy setting was rst to compensate for the liquidity e ect and the sher e ect as it allows the central bank to pursue short-term expansion through a low nominal interest rate or a high nominal interest rate for long-term growth. Apparently, this policy rule o ers a guide for simultaneous in ation control and dampening the cyclical business uctuation.
i t is the reactive short-term nominal interest rate desired by the Fed,r and π * are the long-run equilibrium nominal interest rate and in ation target. π t and y t represent the in ation rate and output gap which constantly steers the policy strategy of the central bank. This strategy is determined by the parameters β and γ. The Fed pursues an active and stabilizing strategy if β , γ > 0. Otherwise, if policy responses lag, γ < 0 and β < 0, the Fed is merely accommodating the changes in the in ation; this policy stance leads to self-ful lling burst of in ation (Bernanke & Woodford, 1997;Clarida et al., 1999). The model follows some random walk, ξ t , arising from the unanticipated exogenous shock to the interest rates. That is, ξ t is independently and identically distributed (iid), E [ξt] ∼ (0, σ ). In the other speci cation we impose the assumption of the zero conditional mean of error terms E [ξt|πt , y t] = 0 as it is plausible to make a case for endogeneity in the model.

The Data
This study used the annualized fed funds rate (R FED ), in ation rate (INF) and output gap (CBO-Est) data from 1980:Q1 to 2019:Q4 for the model estimations. The fed funds rate is the quarterly short-term nominal interest rate. That is, the average rate the banks charge each other on the short-term basis -overnight lending rate. The output gap is constructed by GDP ltering. The lter shows the real GDP percentage deviation from potential. A negative deviation indicates a recession while positive represents an expansion. In ation is measured with the GDP implicit de ator data measured by year-over-year changes in the GDP de ator index. The GDP de ator is the nominal to real GDP ratio.
The dataset we used account for the recent dynamic changes to the macroeconomic environment. For instance, the period 2009:Q1 and 2017:Q2 correspond to the ZLB time therefore the fed funds rate is replaced with the Krippner's (2019) shadow interest rate for the period and we used the updated Holston et al. (2017) estimated time-varying natural rate in the modi ed model. Besides, we consider the sensitivity of the structural parameters to methods used in measuring output gap. Preliminary analyses reveal that the gap measure provided by the Congressional Budget O ce (CBO) predicts the business cycle better than other measures. As depicted in Fig.2, the CBO's gap prediction is clear, discovers recession dates earlier and these dates signicantly correlate with major recession dates reported by National Bureau of Economic Research (NBER). The studied data are sourced as follows.
The fed funds rate, in ation and real GDP data are from US Federal Reserve Bank of St-Louis, the real GDP potential is from the US Congressional Budget O ce (CBO), the Krippner's (2019) shadow rate from Reserve Bank of New Zealand and the Holston et al. (2017) natural rate of interest estimate is from US Federal Reserve Bank of Atlanta. Event-based historical relationships between the data are provided below.
The 1980 Paul Volker's aggressive policy stance came as a result of too much in ation in the late 1970s. The interest rate rose up to 17% in the 1982 and relaxed at 10% by 1985, Fig.1. The Fed defended this policy while understanding the need to control the excess of in ation. The commitment to this strategy continued till 1984. Although Volker's policy tightening has its minor adverse economic impact, in ation fell approximately 300% within 5 years, see Fig.1.
John B.Taylor is the most prominent in the Fed's policy critique of the 2008 nancial stress. Taylor (2009) characterized the Fed policy as loose and weak before the crisis. Many critics essentially linked the housing bubble to an overly low interest rate. The functional policy rule proposed is of the form: Equation 2, according to Taylor (1993), is an implied policy guide the Fed would coherently pursue in a stable environment. Simply, an active Fed policy responds by 1.5% to 1% changes in in ation and 0.5% to 1% output gap dynamics. Observers felt this simple policy requirement was lacking before the 2008 recession. Fig.1, from 2002 to the start of the crisis, the implied Taylor rate (R IM ) rose signi cantly above the Fed rate. For instance, policy-based rule peaked around 6% relative to the fed rate which hovered around 2% in the rst quarter of 2005. As a consequence, the wave of housing burst cut across all nancial departments, mortgage default surged and stock market crashed, by December 2007, the economy plunged into a deep recession and in the fourth quarter of 2009, unemployment surged to a 25 year high at about 10%. As the crisis deepened far into depression, the Fed resorted to unconventional policy measures.
Apparently, the relationship between the fed funds rate, in ation and output gap exist. However, what was not clear is how the Fed's behavior interacts with these macroeconomic fundamentals, Shodipe (2019). We explored this analysis under linearity and non-linearity in the Fed's responses. Fig.2 reports output gap measured by Hodrick Prescott (HP) , linear, quadratic data ltering method and the estimate from CBO. All the methods are virtually similar as they exhibit co-movement, but they exhibit slight dissimilar characteristics. The linear ltering over-exaggerates the size and length of the uctuation over the entire sample. Both HP and quadratic closely moved together from 1980 to 1990 but departed thereafter. More so, the HP measure underestimates output gap and shows a rapid recovery during the period of severe recession. During the heat of the great recession, HP's predicted gap barely went down to -2.78% in 2009:Q2 and in 2012, it gravitated around 0%. The quadratic ltering completely missed the 2001 recession. The CBO -based estimates show close to perfect predictions and align with the common knowledge about US recessions and its gradual recoveries, see Fig. 2. Therefore, we proceed with the CBO estimates in the analysis.

Empirical Strategy . Structural Break Test
At the foremost, the study tests for structural breaks in the fed funds rate. The rationale leading to the break test arises from the presupposition that policy changes can inadvertently cause a shift in the fed funds rate. For example, the Volker's policy, Alan Greenspan's -Great Moderation and policy challenge from the Year 2000s glitch are credible to cause multiple breaks in the fed funds data.
We follow two steps in determining the structural breaks (see Nikolsko-Rzhevskyy et al., 2019). First, the rate deviation ( i t ) from the implied policy rules¹ is derived and second, the break dates are serially determined using the deviation data. Because the result becomes skeptical using an intuition for obtaining a number of breaks in a given time-series data, we adopt Bai and Perron (1998) Sequential test with an Unknown Number of Breaks for multiple structural breaks. Equation 3 considers m + 1 policy regimes with m possible structural break points, where m is unknown.
represents the deviation of fed funds rate from the implied Taylor rule. η is a constant term. z t is a vector of covariates. τ j is the corresponding vector coe cient. ν t is the random term. Also, z t ∼ i.i.d N(1, 1) and ν t ∼ i.i.d N(0, 1) and both are uncorrelated. The t = T , ...,Tm captures the break points. The focus is to estimate the regression and identify the number of the breaks and the breakpoints.
The procedure sequentially tests the null of l versus l+1 breaks. The test starts with l number of break point, l ∈ (0, 1, . . .). Then, the F-statistics, F T (l+1|l) is tested against the null. If null fails to reject, the test is done, no structural breaks exist. By rejecting the null, the procedure is repeated by sequentially increasing l until the null cannot be rejected. Andrew (1993) procedure also yields parallel results. That is, allowing the rst l obtained from the regression to be the global minimizer of the residual sum of the square, tested by Sup F T (l+1|l). The Bai and Perron (2003) trimming parameter, ϵ = 0.15 and the maximum number of breaks, m = 5 are used for obtaining the break dates.

. The Baseline
Equation (5) is the reduced version of (1) and summarizes the simple optimal linear rule that minimizes the symmetric quadratic loss function of the central bank (Castrol, 2011). According to Taylor principle, both 1 + β and γ are the key response parameters. By simpli cation, equation (1) can be rearranged as: ψ ≡ + β is the interest rate response to in ation. θ ≡r − βπ * is the time-invariant intercept and composed of a natural rate of interest and fraction of in ation target. This type of rule implies both long-run parameters are constant. From (5), it is possible to derive the implicit in ation target, π * , practically pursued by the Fed and the implicit policy response to in ation, ψ * . The implicit value depicts the model-based long-run response to in ation at the chosen in ation target and nominal natural rate of interest. These parameters are constructed using the long-run averages of nominal interest rates and the in ation target.
The traditional Taylor rule has received a mixed review from critics. Svensson (1999) underscores the importance of traditional policy rule of in ation targeting in a backward looking model. On the other hand, Woodford (2003) draws attention to time-varying equilibrium interest rate in policy function. Laubach & Williams (2016) and the update in Holston et al. (2017) provides the US varying natural rate estimates. Clarida et al.(1999) speci es policy rules di erently based on two facts. The rst is the argument that the central banks react to the expected rather than the lagged in ation. The second is that the central banks do not have complete information available at the time of making policy. Fundamentally, it is instructive to think that the central banks respond to the macroeconomic aggregates using broad imperfect information.
i * t is the short-term (nominal) policy rate target. I is the information set available to the central bank. E is the expectation operator.r is the underlying equilibrium nominal interest rate/natural nominal rate which comprises the natural real rate and in ation expected in future date, E π t+n| I . Y t is the real GDP and Y * t is the trend of real GDP. By expansion, it is easy to arrive at equation (9) and to ease the model estimation, let y t ≡ 100 (Y t − Y * t )/Y * t and θ ≡r − ψπ * in equation (10).
To provide an adaptable monetary policy the central banks review past interest rate before deciding on new policy². Therefore, equation (10) requires a smoothing treatment to prevent sharp policy reversal between two periods.
Where i t is the actual policy rate, i * t is the target nominal interest rate, ρ ∈ [0,1] is the degree of interest rate smoothing (weight) and µ t is the i.i.d exogenous random shock. Substituting equation (10) into (11) results in the following constructs.
For ease of estimation, equation (13) is simpli ed as: Equation (14) is popularly regarded as the modernized Taylor Rule.
)] + µ t is the error term and comprises a stochastic exogenous term and the forecast error of in ation and output. The derivation of the implicit in ation target and response to in ation is similar to (6) and (7). Nonetheless, OLS econometric technique (14) would be impossible as it violates the Gauss-Markov assumption of zero conditional mean because both the output gap and in ation are components of ξ t , such that E [ξt|πt+n , y t] ≠ 0 provides incorrect moment conditions for the population. Hence, to consolidate the complication in the estimation, equation (14) requires vector of instrument variables z t that are correlated with π t+n , y t but uncorrelated with ξ t , µ t , that is, E [πt+n , y t |z t] ≠ and E [ξt|zt] = E [µt|zt] = 0 that demonstrate the correct moment conditions for the population. This vector represents the information set available to the central bank at the time of deciding policy rate. As a norm, this study resolves to the Generalized Method of Moment (GMM) estimation technique. Although equation (14) is adaptable to any instrument-based estimation mechanisms, GMM estimation provides the advantages of arriving at the most e cient parameters. As the case warranted in each sample, this study used up to four lags of in ation, output gap and growth in price index as the instrument variables.
Single equation linear GMM rationalizes the unbiasedness and consistency of the results in equation (14) given the following conditions: Consider the vector form of linear equation analogous to equation (14) v t is a K x 1 vector of explanatory variables, λ is the vector of unknown parameters and ξ t is the stochastic error term. Since λ is inconsistent and biased, it thus requires H x 1 vector of instrumental variables z t . Given 2 Central banks smooth the policy rate curve to avoid loss of credibility that can result from sharp policy reversal. Other theoretical reasons are: presence of ZLB, unquanti able economic shocks with some exogenous probabilities, presence of transaction frictions or fear of nancial market disruption that w t is vector of all variables including the instrumental variables in equation (15) (i t , v t , z t ) and follows a stationary stochastic process, then z t satisfy H orthogonality condition if: Where g t (w t , λ) = z t ξ t = z t (i t − v t λ). Simplifying equation (16) rank K which is necessary for λ to be unique solution of equation (17). Similarly, identi cation of λ requires the order conditions H ≥ K. That is, for λ to be identi ed in the equation (15) the number of instruments z t must be greater than or equal to the number of the set of endogenous regressors v t . Otherwise, the model is under-identi ed, hence the vector of parameters λ in the structural equation (15) is impossible.
Since it is most likely equation (15) will be over-identi ed considering the number of instruments available the Hansen (1982)'s J -statistics test of overidentifying restriction is examined for model misspeci cation. Where J ∼ χ (H -K). If some moment conditions are not satis ed, for some k's in (16) then the reported Jstatistics will be larger relative to χ (df= H -K). Therefore, there exists some redundant exogenous instrumental variables. Otherwise, the over-identifying restrictions cannot be rejected. The GMM procedure shows that by satisfying the above conditions the vector λ is consistent and asymptotic normal. Hence, equation (14) can be estimated.

Non-Linear Model
This study further examines whether the Fed policy preferences exhibit asymmetry, determined by the magnitude of the threshold variable such as current or lag value of interest rate, in ation, output, unemployment. Castro (2011) emphasizes that policy responses are best characterized as non-linear if the central banks attach di erent weights to negative and positive output gap and in ation in its loss function. We explored this analysis with threshold regression.
The widely used threshold autoregressive models in the literature are those proposed by Tong (1990) and Hansen (1999) where lag dependent is the threshold variable used for demarcating the regression into the regions. Whereas, this study used the output gap as a threshold variable because it provides coherent results compared to other choices of threshold variables.
By extension, equation (18) combines threshold variables with the equation (5)³ to form two regions endogenously chosen by the threshold χ. Region 1 de nes the subset of the total sample where the threshold variable y t is less than or equals the threshold χ and region 2 where the χ is greater than y t . This approach is e ective for identifying possible changes as it o ers a smooth endogenous regime shifts.
θ , ψ , γ and θ , ψ , γ are the policy parameters in the region 1 and 2 respectively. i t , π t and y t are the subset of data series for each region. Our focus is to investigate whether the Fed's behavior exhibits asymmetry given a low or a high output gap y t .

Baseline Results
The results o er a new perspective to the Fed's policy actions between 1980 and 2019. The presence of structural breaks in the Fed rate data motivates separate regression analyses in this study. Table 2 The table also presents the estimates' standard errors, the threshold regression results, the implicit in ation target (π * ) and the implicit policy response to in ation (ψ * ). Apparently, policy response to in ation diminished in the samples; 2002:Q2 -2010:Q2, 2010:Q3 -2019:Q4. One explanation we arrive at is that variation in in ation data is insu cient in the two periods. We also infer that these outcomes may have resulted from policy inconsistency. As a result, Fed's policy is insensitive to in ation during these periods. In spite of this, the response to output gaps improved generally. We observe that this result contradicts the trade-o between in ation and output. For the full sample, the Taylor's principles; ψ >1 , γ > 0 abide. In reaction to the in ation gap, the Fed raises the nominal interest rate signi cantly enough to stabilize the price level. Comparing the full sample results to the sample results show that econometric analysis is sensitive to the breaks in the data. The estimations reveal a slow down in the natural real interest rate, θ. The results also show that smoothing interest rate is relevant for policy consistency. More so, the result reveals that the Fed's policy exhibits a non-linear preference under di erent macroeconomic situations. Nevertheless, the preference can be regarded as being partial because the output gap is insignicant in the second regime. It is interesting that all the estimated parameters are statistically signi cant. More importantly, Hansen's J-tests of overidentifying restrictions fail to reject all the results. Thus, the structural parameters are identi ed, consistent and asymptotic normal. Our ndings connect with the literature as follows.
The study supports the modernized policy rule of Clarida, et el. 1999 buts reject the traditional rule of Taylor, 1993. The study lends support to the declining natural rate of interest asserted in Bullard, 2018;Laubach & Williams, 2016 but refutes the ndings of Taylor & Wieland (2016). The study partially supports the ndings of Zhu and Chen, 2017;Beckmann et al., 2017;Gogas et al., 2018 on policy asymmetry but refutes the ndings of Castro, 2011. The study supports the Nikolsko-Rzhevskyy et al. (2019) on multiple structural breaks in the fed rate. Our analysis contradicts the second-order partial smoothing treatment demonstrated in Clarida et al. (1999).
To have a feel of the result predictions in each sample and to provide insights to the disparity in the estimates, Fig.3a depicts the fed funds rate (R FED ) and the baseline predictions (R C , R T ), and Fig.3b shows the deviations from policy rule (R FED -R C , R FED -R T ). R T is the traditional rule prediction and R C is the modernized rule prediction. From the results, it is obvious the outcomes of the traditional (5) and the modernized (14) rule o er parallel ndings about the Fed's policy stance, but the predictions remarkably di er in size.
The estimates of the sample, 1980:Q1 -1986:Q4, show that interest rate moved to stabilize the in ation and output gap. However, the traditional policy rule (5) demonstrates a downward bias , ψ = 1.65% , γ = 0.07% relative to the modernized rule estimates; ψ = 1.84% , γ = 0.22%. In both cases, the estimates show that the Fed lived up to the expectation regarding the price stabilization objective. In Fig. 3b ( rst column), although the traditional rule depicts a gradual adjustment of the fed rate to one prescribed by rule before 1982, the result insigni cantly re ects the ne-tuning policy strategy adopted during the period. The deviation from modernized rule shows an alternating spike shortly before 1982. An explanation for these spikes is possible policy glitches⁴ negligently motivated by the Fed. In the modernized rule, the smoothing parameter ρ = 0.43, somewhat low for the sample, keeps the natural real rate in a more stable distance. We infer that the lower ρ, relative to estimates in the later samples, could be the cause of the spikes in the policy rate. The takeaway from this result is that interest rate smoothing has improved the Fed's policy over the year. Given the high level of in ation started within the sample the estimated in ation target (π * ) is fairly large.
The sample 1987:Q1 and 2002:Q1 marks the period of Alan Greenspan's policy as widely remarked as the "Great Moderation". The modernized rule estimation reveals that the Fed's policy exhibits minimum deviation. In fact, our result shows that the Fed strongly adhered to the rule. Fig.3a demonstrates that the Fed rate and the prediction closely moved together. We rationalize this policy improvement on the weight (ρ) increment. ρ remarkably increased to 0.83, depicting a strong smoothing process. The traditional model continued to over-exaggerate the prescribed target. The absence of smoothing treatment also plays out signi cantly in this sample. To anchor the claim that the smoothing treatment o ers a central role in the policy setting, the traditional Taylor rule prescribes rates overly high above the policy requirement as against the modernized rule prediction. Therefore, the major disparity in this class of baseline model arises from smoothing treatment. Generally, the estimates demonstrate that Fed's policy was well responsive to in ation and output gap; ψ = 1.74% , γ = 0.51% and ψ = 2.30% , γ = 1.12% for the traditional and the modernized models respectively.
The sample results of 2002:Q2 -2010:Q2 and 2010:Q3 -2019:Q4 demonstrate negative responses to in ation. These outcomes are the same for both baseline models; actually these do not characterize the principles expected in the standard policy rules. As indicated earlier, we make the case for two rationales for a negative relationship. First, to identify estimates in the policy function the in ation and output gap must possess su cient variations, and the sample period must be long enough. Obviously, the sizes of these samples are relatively small and in ation has less variation. According to Stock and Watson (2012), there was a "Missing In ation". More so, Fig.1b shows that in ation struggles to maintain the 2% target. Second, the US monetary policy behavior during these periods has earlier been characterized as being unconventional, erratic and unstable (see Taylor, 2009Taylor, & 2014Kroeger et al., 2018). Farmer (2012) described the unconventional Fed's behavior -due to the e ect of ZLB -as being e ective in averting de ation. Nonetheless, we observe that, even though there is less variation in in ation, the output gap displays a signi cant level of variation. Although we were tempted to rely on the full sample, since it provides a su cient long sample and the desired variation in the data, we realized that doing so provides an incoherent explanation to contradicting results in the earlier literature as compared to when we acknowledged the evidence of structural breaks in the data. For the purpose of analysis, this study explores the implication of the sample results with the notion that there was a dramatic change in the Fed's policies during those periods. Given that this is true, the estimates from these regressions can be rationalized as consistent and e cient as the Generalized Method of Moment (GMM) technique is well suited for a relatively small sample.
Pursuing the sample results from lack of su cient variation and policy shift stand-point, we interpret the dynamics in fed rate for the periods 2002:Q2 -2010:Q2 and 2010:Q3 -2019:Q4. Due to the insu cient variation in the in ation, the Fed redirected focus to the supply side in actualizing its (dovish) policy objective. Response to business cycles increased above one percentage point in the two samples. Obviously, this re-sponse shift was motivated to incentivize the factor and goods markets during the recession and exercise policy tightening during a heated economy. Implication derived from this policy pursuit is that when in ation is below the target the Fed pursues policies that in uence the supply side in a way that is synonymous to price stabilization. The smoothing parameter, ρ = 0.77; 0.83, signi cantly reveals that the Fed follows a pattern in deciding policy. The sample regressions generate interesting implicit in ation targets close to the desired 2%. The implicit target for the two sample regressions in the traditional model are: π * = 2.10% and 2.14%; and implicit targets for the modernized rules: π * = 1.80% and 3.13%. As the traditional rule does not allow for the smoothing treatment and forward-looking behavior of the in ation, the outcome of the predictions (Fig.3) enormously di er from the fed rate and the prediction from the modernized model.
The non-linearity test in policy response reveals that over the entire sample the Fed's policy preference can be described to be partially non-linear. The results of this test are reported at the bottom-left of Table  2. The threshold is estimated at χ = -2.37%. This indicates the Fed switches regime when the output gap is greater than χ. Below this threshold level, the results show that the Fed is more aggressive on both in ation (ψ = . %) and output gap (γ = . %) . Above it, the Fed still actively pursues in ation control (ψ = . %) but with a negative reaction to the business cycle. These results show partial asymmetry but the estimated parameters seem to possess some limitations. First, this test is conducted on the full sample without consideration to structural breaks in the data results⁵. Second, the test is conducted with traditional Taylor rule. We understand that these limitations can a ect the consistency of the outcome of the results. Therefore, this study can be extended. Woodford (2003) spurred research on time-variation in the US natural rates of interest. Over the years, significant bodies of studies have provided estimates for the underlying natural rates; for surveys, see Woodford, 2003;Barsky, et al., 2014;Kasyanenko & Papell, 2019. Recently, Laubach & Williams, 2016Cúrdia et al., 2015;Holston et al., 2017; show evident decline in natural rate. To accommodate this piece of evidence in the policy frameworks, the baseline model is adjusted for the time-varying natural rate of interest. We use the Holston et al. (2017) estimates of natural rate of interest because it is most acknowledged in the literature.

. Time-Varying Natural rate Treatment
The traditional Taylor is modi ed by replacing the invariant nominal interest (r) in equation (5) with the Holston et al. (2017) estimates of natural rate of interest (r t ). Where i † t ≡ i t −r t and θ ≡ −βπ * Also, the modernized Taylor rule (14) is modi ed for varying nominal natural rate of interest as follows.  Fig.4. R TV is the predicted rates from the modi ed traditional rule and R CV is the predicted rate from the modi ed modernized rule. Overall, the result outcomes slightly parallel the baseline results but the major di erences are underscored as follows.

Results for the Time-Varying Treatment
The result of the sample (1980:Q1 -2002:Q1) replicates the combined outcomes of the baseline model. Fed's policy was active at stabilizing price and minimizing the e ect of the business cycle. However, the response to the business cycle in the modi ed traditional Taylor rule is insigni cant and is averagely little. The modi ed modernized rule shows that the Fed attached larger weight, ρ = . , on the average, to past interest rate when deciding on policy; similar to the higher weight observed in the baseline estimations. The estimated implicit response to in ation, ψ * , also closely aligns with the Fed's practices; 1.38%, 2.59% for the modi ed traditional and the modernized rules respectively.
The estimations of the samples (2010:Q3 -2019:Q4, 1980:Q1 0 2019:Q4) provide concrete result comparison between the baseline and the modi ed policy rule (Table 2 and Table 3). Although the policy responses reported in the two tables have slightly di erent parameters, the generalized ndings are analogous. Similar to the previous discussion, we show that response to in ation remains absurd but there was improvement to responses to the business cycle. The responses to in ation were negative in the modi ed traditional model, ψ < , and ψ < in the modi ed-modernized models. In these two cases, in ation is frail in the Fed's policy equation. It appears apparently that such a policy stance can lead to a self-ful lling burst in in ation but policy response to the business cycle is large enough to dampen an unanticipated surge in the in ation.
The full sample (1980:Q1 -2019:Q4) averages demonstrate responses to in ation, ψ = . % and business cycle, γ = . % conform to the Taylor principle. The smoothing parameter is also sizable, ρ = . . Over the entire sample, US monetary policy behavior can be said to be price stabilizing. The implicit response to in ation, ψ * = . %, is close to the implied policy response, 1.5%, suggested by Taylor (1993). The full sample results support the non-linearity in the US policy behavior with the threshold estimate, χ = − . %. Similar to the baseline, below the threshold, the Fed's reaction to in ation and output gap are aggressive while it is less aggressive to in ation above the threshold.

. Second-Order Partial Treatment
Alternative policy mechanism studied in this paper is second-order partial adjustment mode. We allowed standard modernized Taylor to follow two lags weight, ρ , ρ . Such policy setting is justi ed when the central banks use both the rst and second lags for smoothing processes. Usually, the rst lag treatment is su cient to provide a smooth interest rate curve but a short-term disruption in the interest rate may occur if there is an unanticipated exogenous shock. Our survey of literature indicates no signi cant amount of recent studies have corroborated the piece of evidence reported in Clarida et al.(1999). We extended the author empirical piece using both baseline and modi ed. Equation (26) represents the model with constant natural rate of interest and (27) represents model with time-varying natural rate of interest.
Here, the sum of ρ j is tested for signi cance. The derivation of the implicit in ation target, π * and response to in ation, ψ * , follow equations (6), (7) & (25). Table 4 reports the estimation from second-order partial adjustment models. We observe ndings consistent with previous result comparisons. The model with constant natural rate of interest is upward bias of the time-variant models. The results demonstrate the same abnormal response between periods 2002:Q2 -2019:Q4; ψ < and γ > . On the average, over the years the Fed increasingly accords preference for smoothing processes, ρ + ρ .

Results for the Second-Order Partial Treatment
The in ation target gradually falls as in ation becomes more less variant. Over the entire sample (1980:Q1 -2019:Q4), the Fed's policy can be described as stabilizing, ψ > and γ > . The Fed su ciently adjusts the nominal interest rate in response to the dynamics of the business cycle. Generally, the study nds little or no support for improvement in second-order partial adjustment models relative to rst-partial adjustment. The predicted rates from these models are depicted in Fig. 5.

Conclusion and Policy Implications
This study is motivated by the periodic policy shifts and asymmetry observed in the Fed's behaviour. It discusses relevant issues surrounding policy setting and makes the following contributions First, it probes the presence of structural breaks in the Fed rate. Second, it provides estimations to test asymmetry over cyclical drift in the output. Third, it analyses the usefulness of interest rate smoothing and reports on disparities between alternative policy rules. These estimations were carried out using US data, 1980 Q1-2019 Q4.
The test results show that there were multiple breaks in the Fed's rate and the identi ed dates correlate with the Fed's reported dates of policy changes. Overall, the results of the full sample reveal that the Fed's policy is one of stabilising prices, also in the samples 1980 Q1-1986 Q4 and 1987 Q1-2002 Q1. We found that, even though the results of the samples 2002 Q2-2010 Q2 and 2010 Q3-2019 Q4 appear to fail Taylor principles, the Fed's policy stance was not weak, as criticised in the literature. Our results show that during the monetary crisis the Fed actively performed policy shifts towards the real sector.
The major ndings are as follows. Fed's policy preference is partially asymmetric. The estimation shows better policy improvement with a higher interest rate smoothing treatment. The results show that forwardlooking policy settings provide better predictions. Importantly, we found that the baseline policy rule has an upward bias compared to the modi ed policy rule. This study rejects second partial interest rate smoothing.
Our study makes two important contributions to the literature. At a theoretical level, it derived, tested, and provided solutions to the central bank optimisation problem when policy preference was asymmetric. We translated the quadratic form to a potential nonlinear monetary policy rule that engineers evidence in asymmetries in the objective. At an empirical level, our study indicates that the Fed's monetary policy behaviour follows a nonlinear policy rule. Our study shows that modelling in ation requires considering time variations within regimes. Similarly, evidence of a random path with a long cycle was observed.
Finally, the policy implication of our ndings is that smoothing treatment is signi cantly bene cial during an economic crisis if the Fed adheres to its policy rule. Our results on structural breaks extends the literature. By integrating monetary and scal policy, future studies should consider testing structural breaks in time-series analyses, which may provide additional value to the ndings of this study.
Financial Support: This research received no speci c grant from any funding agency, commercial or nonpro t sectors.

Con ict of Interests Statement:
The authors have no con icts of interest to disclose.