Abstract
Linear Taylor rule prescribes symmetric response to inflation rate and output gap in good and bad times. Central banks in the world, however, are more concerned about inflation when economy is in high inflationary regime. Similarly they are more reactionary to output fluctuations when economy is experiencing slowdown in the economic activity. Thus, most of the researchers in the area of monetary policy construct a nonlinear monetary policy reaction function. In the literature related to monetary policy of Pakistan, this reaction function has been modeled as threshold regression (TR), Markov regime switching regression, and Logistic smooth transition regression (LSTR). This study compares these three choices for the case of Pakistan and tries to find out which functional form of nonlinear Taylor rule fits the Pakistani data well.
Using quarterly data for the period 1993:12011:4, we find strong evidence that the monetary policy followed by the State Bank of Pakistan (SBP) exhibits nonlinearity. The results of this study show that threshold level of inflation rate is 6.37% and that of output gap is 2.5%. Moreover, threshold regression, with inflation rate as threshold variable, is found the best among the three specifications as it satisfies maximum number of criteria for comparison. However, LSTR model performs well if forecasting performance of the models is compared.
Keywords: Nonlinear Taylor Rule, Markov Switching Model, Logistic Smooth Transition Regression, State Bank of Pakistan

INTRODUCTION
Monetary policy objective is to maximize society's welfare by maintaining price stability along with keeping unemployment at its natural rate. A great deal of research has been done, since the early 1990's, on monetary policy reaction functions of central banks. In particular, Taylor rule (Taylor, 1993) has received considerable attention. The rule specifies relationship between policy instrument (short term interest rate) and the target variables (inflation rate and output gap). According to this rule central banks increases the interest rate in times of high inflation, or when output is above its potential level (unemployment is below the natural rate of unemployment), and vice versa. Therefore, the rule prescribes symmetric policy action in high and low inflationary regimes.
The theoretical basis of linear Taylor rule rests on two key assumptions, namely that central banks have quadratic loss function and that the Phillips curve is linear. Recently however, both of these assumptions have been criticized. For instance, Bec et al. (2002), Kim, Osborn and Sensier (2002), Martin and Milas (2004), Bruggemann and Riedel (2011), Cukierman and Muscatelli (2008), Castro (2008), and Ncube and Tshuma (2010) highlight asymmetric preferences of central banks regarding inflation and the output gap, which in turn lead to nonlinear policy reaction function. Moreover, Dolado et al. (2004), Corrado and Holly (2003) and Nobay and Peel (2003) specify the Phillips curve relationship as nonlinear which again lead to the nonlinear policy reaction function. Dolado et al. (2000) relax both the assumptions and have constructed a general model which departs from linearquadratic framework.
Hence, there are good theoretical reasons to hypothesize that central banks may not be following a linear Taylor rule; empirical evidence validates this hypothesis. The nonlinear Taylor rule spells out that weights assigned to negative vs. positive output gap and low vs. high inflation rate could be different. However, we do not directly observe non quadratic loss function or nonlinear Phillips curve so there exists unbounded universe of possible alternative nonlinear specifications of the Taylor rule.
There is limited literature and empirical work available on the monetary policy reaction function of Pakistan. In this regard, the pioneering study estimating linear Taylor rule for Pakistan is of Malik and Ahmed (2010).
The study finds, using threshold regression, that State Bank of Pakistan (SBP) has never followed Taylor rule during the period 19912005. Ahmed and Malik (2011) find nonlinearity in the reaction function; SBP has asymmetry in the degree of leaning against the wind in high and low inflationary regimes. Saghir (2014) and Satti (2014) find instability of parameters in the monetary policy reaction function of SBP. Moreover, asymmetry is found in the response to high vs. low inflation and positive vs. negative output gap. Sattar (2014), using Markov Regime Switching framework, also depicts nonlinearity in the policy reaction function of SBP. Finally, Alam (2015) reaches the same conclusion using Logistic Smooth Transition model.
The nonlinearity in the policy reaction function, once established, becomes part of the macroeconomic models analyzing monetary policy issues. But question remains how nonlinearity should be modeled; threshold regression, Markov regime switching framework, or smooth transition regression model. We believe that it is important to investigate the type and nature of nonlinearity in SBP's reaction function while avoiding specific parametric assumptions. Once we are out of the realm of linear framework, the specification problem has to be addressed. Adopting an incorrect nonlinear specification is more problematic than simply ignoring the nonlinearity altogether.
Our main contribution is to determine the most appropriate form of nonlinearity in the policy reaction function of SBP. In this regard, we compare the results of three models; Threshold regression model, smooth transition regression model and Markov regimeswitching model. These three models differ on the basis of their mode of transition from one state to another. In threshold regression model, parameters abruptly change from one regime to another regime implying sharp threshold while smooth transition regression allows for the smooth and gradual transition of the parameters from one state to the other. In Markov regime switching model there is exogenous regime switching having fixed probabilities.
In this study, we have first estimated the simple linear Taylor rule which did not fit the data well. Therefore, we have estimated nonlinear Taylor rule with the three potential nonlinear techniques, i.e. Threshold regression model, Smooth Transition regression model and Markov regime switching model. The objective of our study is to compare these models and to find the best fitted model for our data. The three models are compared on the basis of six criteria, i.e. Akaike information criterion, Shwartz information criterion, coefficient of determination, coefficient of correlation, root mean square error and mean absolute error.
The structure of rest of the study is as follows. The following section presents a review of the relevant literature. Section III describes in detail the nonlinear econometric models and their comparison techniques. Moreover, details of data and variables are also given in this section. Estimation results are then presented and discussed in section IV, and section V concludes the study.

LITERATURE REVIEW
Monetary policy is the demand side macroeconomic plan of action or stra tegy set by the nation's central bank in order to achieve the macroeconomic goals which are achieved by manipulating money and credit supplies or by changing interest rates. The idea of monetary policy originated in 1699 when the Bank of England printed notes backed by gold. Later on during 18701920, the developed or industrialized nations set up central body known as central bank for laying the monetary policy. The objectives of monetary policy may vary across countries but the main objectives do not change which are controlling inflation rate, exchange rate stability and stabilization of economic activity. For the monetary stability, there are policy tools like open market operations, discount window borrowing and reserve requirements. The operating target of monetary policy is set either by a pre specified rule or it remains discretionary choice of the central bankers.
Simons (1936) was the first to raise the issue regarding the rules vs. discretion of monetary policy and favoured policy rule for the economic stability. Discretion is authorization to enhance economic performance whereby actions are done solely on the basis of judgment whereas rule is considered a constraint. Monetary policy rule has been advocated against discretion by the prominent economists including Kydland and Prescott (1977), Fischer (1980), Barro and Gordon (1983), McCallum (1988) and Taylor (1993).
The idea of rule as a practical guide for monetary policy was popularized by McCallum (1988) and Taylor (1993). McCallum proposed changes in money growth rate in response to changes in inflation rate and GDP growth rate. Taylor rule prescribes changes in short term interest rate in response to changes in inflation rate and output gap; the relationship is assumed to be linear. The assumptions of quadratic loss function and linear Phillips curve lead to linear and symmetric response of central bank to inflation deviation from the target and output deviations from potential level.
Linear Taylor rule has been criticized by many intellectuals on the basis of its assumptions and once any of these assumptions is relaxed monetary policy response function becomes nonlinear. Central bankers' preferences regarding stabilization of economic activity and inflation rate are modeled symmetric, perhaps due to mathematical convenience, but actually these preferences might be asymmetric either due to the bankers' own choice or due to political pressure (Blinder, 1999). Policy makers tend to take more serious actions when output is below its potential (unemployment is higher) and/or inflation rate is above its target. The response to deviations is less severe when output is above its potential (unemployment is lower) and/or inflation rate is below its target. This kind of behaviour is quite close to the human psychology...