Specifications of the FL and BL MP Reaction Functions

This section specifies the rules as backward and forward looking reaction functions. In the spirit of CGG (1999), Taylor (2001), McCallum (2000), Clarida (2001; 2012) and Mehrotra and Sanchez-Fung (2011), the study specifies 10 empirical models for five monetary policy reaction functions for the UK data and estimates the coefficients and other statistics. The MP rules are the Taylor rule, the McCallum rule, the Hybrid of MaCallum-Taylor rule, the Hybrid McCallum-Hall-Mankiw rule and the Nominal Fixed Rate (NFR) – McCallum-Dueker-Fisher rule, also known as a Nominal Feedback Mechanism (NFM). The model specifications represent targets and instruments from the prevalent framework to analyse the UK monetary policy framework. The specifications are:

The Taylor’s rule BLRF/FLRF

𝑅_𝑡 = 𝛼_𝑇𝑅+ 𝜑_𝑇𝑅𝑅_𝑡−1+ 𝛽(𝜋_𝑡− 𝜋^∗) + 𝜆(𝑦_𝑡− 𝑦̃) + 𝛿_𝑇𝑅Δ𝑒_𝑡+ 𝑎₄𝑖_𝑡−1 𝑅_𝑡 = 𝑎₀+ 𝑎₁(𝐸_𝑡𝜋_𝑡+1− 𝜋^∗) + 𝑎₂∗ 𝐸_𝑡𝑦_𝑡+𝑖+ 𝑎₃Δ𝑒_𝑡−1+ 𝑎₄𝑖_𝑡−1 (2.20) The McCallum’s rule BLRF/FLRF

Δ𝑏_𝑡 = 𝛼_𝑀𝑅+ 𝜇_𝑀𝑅Δ𝑏_𝑡−1+ 𝜃(Δ𝑥_𝑡^∗− Δ𝑥_𝑡−1) + 𝛿_𝑀𝑇Δ𝑒_𝑡 Δ𝑏_𝑡 = 𝑏₀+ 𝑏₁(Δ𝑥_𝑡^∗− 𝐸_𝑡Δ𝑥_𝑡+1) + 𝑏₂Δ𝑒_𝑡−1+ 𝑏₃𝛿𝑏_𝑡−1 (2.21) The Hybrid McCallum-Taylor rule BLRF/FLRF

𝑅_𝑡 = 𝛼_𝑀𝑇+ 𝛾_𝑀𝑇𝑅_𝑡−1+ 𝜌(Δ𝑥_𝑡^∗− Δ𝑥_𝑡−1) + 𝛿_𝑀𝑇Δ𝑒_𝑡

𝑅_𝑡 = 𝑐₀+ 𝑐₁(Δ𝑥_𝑡^∗− 𝐸Δ𝑥_𝑡+1) + 𝑐₂Δ𝑒_𝑡−1+ 𝑐₃Δ𝑖_𝑡−1 (2.22) The Hybrid - McCallum-Hall-Mankiw rule BLRF/FLRF

Δ𝑏_𝑡 = 𝛼_𝑀𝐻𝑀+ 𝜇_𝐻𝑀Δ𝑏_𝑡−1+ 𝜒((𝜋_𝑡− α𝑝̅ + 𝑦_𝑡 ̅ )) + 𝛿_{𝑡 𝐻𝑀}Δ𝑒_𝑡

∆𝑏_𝑡 = 𝑑₀+ 𝑑₁[(𝐸𝜋_𝑡+1− 𝜋^∗) + 𝐸_𝑡𝑦_𝑡+1)] + 𝑑₂𝛿𝑒_𝑡−1+ 𝑑₃𝛿𝑏_𝑡−1 (2.23) The NFR - McCallum-Dueker-Fisher rule BLRF/FLRF

∆𝑚_𝑡− ∆(𝑚 − 𝑝)_{(𝑡|𝑡−1)} = 𝛼_𝑀𝐷𝐹+ 𝜔(∆𝑚_𝑡− ∆(𝑚 − 𝑝)_{(𝑡|𝑡−1)𝑡−1}+ 𝛽_𝐷𝐹(𝜋_𝑡− 𝜋̅_𝑡^∗) + 𝛿_𝐷𝐹𝛥𝑒_𝑡

∆𝑚_𝑡− ∆(𝑚 − 𝑝)_{(𝑡|𝑡+1)}

= 𝛼_𝑀𝐷𝐹+ 𝜔(∆𝑚_𝑡− 𝐸∆(𝑚 − 𝑝)_{(𝑡|𝑡+1)𝑡+1}+ 𝛽_𝐷𝐹(𝐸𝜋_𝑡+1− 𝜋̅_𝑡^∗) + 𝛿_𝐷𝐹𝛥𝑒_𝑡 (2.24)

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The lagged policy instrument is an important feature in Equations (2.20) to (2.24). The specification is intended to account for smoothing by the monetary authorities through the coefficients 𝜑_𝑇𝑅, 𝛾_𝑀𝑇, 𝜇_𝑀𝑅, 𝜇_𝐻𝑀 and 𝜔 (as in English, 2003). All 𝛼 terms are equation-specific intercepts. Equation (2.20) is the benchmark Taylor-type monetary policy reaction function. As in Taylor (2001), Svensson (2000) and Moron and Winkelried (2005), Equation (2.20) and the other specifications allow for feedback from the exchange rate. The exchange rate variable is the annual depreciation of the exchange rate expressed in percentage points, and an increase in 𝑒_𝑡 is a depreciation. In the same equation, an increase in the exchange rate is expected to produce an increase in the interest rate (𝛿_𝑇𝑅 > 0) if the monetary authorities lean against the wind. The output gap is based on HP filtered 𝐺𝐷𝑃 data. The coefficient on the output gap is expected to be positive (𝜆 > 0), indicating that the central bank increases the interest rate if actual output is above the potential output. According to the Taylor’s principle, the nominal policy interest should move one for one with average inflation’s deviations from target (𝛽 > 0). Using average of observed inflation in the specifications avoids overreacting to temporary movements in the variable. Equation (2.22) is a hybrid rule mixing Taylor’s interest rate instrument with a McCallum nominal income gap target and an exchange rate variables. An important variable in the rule is the nominal GDP target.

An increase in the nominal income gap²⁶ should lead to a reduction in the interest rate, that is 𝜌 <

0; in Equation (2.22) an increase in the exchange rate should lead the central bank to react by increasing the interest rate (𝛿_𝑀𝑇 > 0). Furthermore, the benchmark forward and backward looking models are specified from the Taylor Monetary Policy Rule and McCallum Monetary Policy Rule.

Equation (2.21) is McCallum’s benchmark feedback mechanism including an exchange rate variable (𝛿_𝑀𝑅< 0). Equation (2.23) is a hybrid mixing a monetary base instrument with a target following Hall and Mankiw (1994). The hybrid target is specified as the deviation of annual inflation from its moving average and an output gap. In Equations (2.21) and (2.22) an increase in the McCalum and in Hall-Mankiw targets should lead to a reduction in the monetary base, i.e. a tightening of the monetary policy stance; 𝜃 < 0 and 𝜒 < 0 are expected. In the reaction functions with a monetary base instrument, the coefficients on the exchange rate are expected to be negative (𝛿_𝑀𝑅 < 0 and 𝛿_𝐻𝑀 < 0) if the central bank tightens its policy stance following depreciation.

Equations (2.24 and 2.25) are a nominal feedback rule following Dueker and Fischer’s (1996) analysis of monetary policy. The analysis of interest estimates the variable Δ(𝑚 − 𝑝)_{(𝑡|𝑡−1),}. The estimation amounts to a technical approximation to the internal predictions a central bank is

26 The nominal income target is computed by applying the HP filter to the real GDP data and taking the growth rates of the resulting trend series, and adding this measure of real trend growth to the inflation target announced by the central bank.

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supposed to generate and use a technical approximation when designing its policy. In generating that variable, it estimates a structural time series model from which a data sequence is generated for all the points in the given sample using the Kalman filter. In producing the series, Δ(𝑚 − 𝑝)_{(𝑡|𝑡−1),}the analysis only uses information available up to the period 𝑡 − 1 (Harvery, 1989).

In Equation (2.24) the coefficients 𝛽_𝐷𝐹 and 𝛿_𝐷𝐹 are expected to be negative if the monetary authorities bring the implicit inflation target down following an increase in the inflation gap or a depreciation in the exchange rate. The data description states the items of the monetary policy instruments and identified targets. It also describes the type of data used to fit the reaction functions, the data generating process and the multiple sources used to obtain the data.

2.7.3 Data

As discussed in the above sections, to account for the post Second World War and the high inflation periods together with the frequently changing monetary policy instruments, the early 1960s period is a reasonable starting point for the empirical analysis. The monetary policy reaction functions are estimated over three policy regimes using quarterly data. To allow lags of monetary policy instruments and future expectation for the BL and FL settings, the actual sample series used for estimation are adjusted. The three policy regimes are identified as pre-inflation targeting policy regime (pre-1992), announcement/inflation targeting policy regime (1992 to 2007), and post-GFC policy regime (2007 to 2014). The Zivot and Andrew method confirmed that there is a significant structural (at 5%) shift in the stated policy regimes. The reaction functions simulate the policymakers’ behaviour based on the assumption that one of the five reaction functions might have been used in the given policy regime. The sample periods are identified based on the UK monetary policy structure and the onset of the GFC.

Following the data description, the empirical analysis allows investigating the UK monetary policy rules based on the Taylor, McCallum and the hybrid rules in three policy regimes using quarterly time series data from 1962 to 2014. The nominal interest rate is used to estimate the benchmark Taylor rule and the hybrid McCallum-Taylor counterpart are the official policy interest rates (controlled by the monetary authority) as shown in Table 2.3. The rate of inflation is the annual change in the consumer price index. The inflation gap is calculated as the difference between moving average of annual inflation and the inflation targets announced by the monetary authorities in IT economies. The exchange rate variable included in the reaction functions is the annual change in the price of domestic currency per U.S. dollar. The output gap is based on the GDP data calculated as a deviation of the log output from its trend using the Hodrick-Prescott (HP) filter. Table 2.3 presents the notation and illustrations of data transformations for the monetary policy instruments and policy targets.

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Table 2.3 Variables and Descriptions of the Monetary Policy Rules Reaction Functions

Variables Description Units Sources

For inflation targeting economies (the UK), is the sum of real output passed moving average and a measure of the real output gap Source: author’s data and variable overview in the spirit of MSF (2011).

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