# «MICHAEL W. MADSEN THESIS FOR THE DEGREE MASTER OF ECONOMIC THEORY AND ECONOMETRICS DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO MAY 2012 © Michael W. ...»

Siklos and Bohl (2008) include asset prices as instruments in monetary policy rules, and find that this improves the model fit. Their results suggest that asset prices are part of the information set used in the determination of policy responses to inflation and output gap.

This model is an expansion of the interest rate rule presented by Clarida et al. (1998), since I have included the exchange rate in my model. In that respect, the model I use is somewhat similar to the one utilized in de Andrade and Divino (2005). The exchange rate variable is included since my data set includes a time period when the central bank aimed to stabilize the exchange rate. Thus, one would expect that the exchange rate has had an impact on the central bank’s interest rate setting decision. Moreover, Norges Bank has often pointed out that they take the exchange rate into account when making their interest rate decision, even after the inflation targeting regime was implemented.

Goodfriend (1991) points to the tendency the central bank has in not adjusting its interest rate target immediately. Specifically, as argued by Clarida et al. (1998), a simple Taylor-type rule, e.g. Eq. (1), does not capture the central bank’s propensity to smooth interest rates over time.

That is, the central bank has a tendency to adjust its interest rate somewhat sluggishly to avoid highly volatile interest rates, which may act destabilizing on the real economy. Therefore, I

**assume that the interest rate adjusts gradually to the target rate in the following manner:**

(2), is a measure of the central bank’s degree of interest rate smoothing.

where Specifically, the parameter captures the tendency central banks have in adjusting the interest rate somewhat sluggishly toward their target.

6 Since one would expect the central bank to also adjust their policy rates in situations of financial turmoil, a variable measuring financial instability should be included. Insertion of (1) into (2) and including a term for financial instability,, yields:

Due to the inclusion of unobservable future dated variables in the former expression, the policy rule is rewritten in order to eliminate these and instead include observable future dated variables. Imposing this assumption, and defining, (3) can be

**rewritten in the following way:**

,

However, the fact that the central bank’s objective has changed through the years, in my sample from a stable exchange rate regime to the current inflation targeting regime, indicates that there are some difficulties by using the constant parameter rule described in (4). This can be overcome by dividing the sample into different sub-samples, as in Clarida et al. (2000).

Valente (2003) argues that one should apply a more general, nonlinear model rather than a linear time-invariant model. His argument in favour of a nonlinear model is strengthened as the test for parameter stability is rejected. Nevertheless, one can estimate the policy rule under different regimes in two alternative ways. That is, one may use a state-dependent Markovswitching model as Valente (2003), or a state-space model with time-varying parameters as in Kim (2001), Kim and Nelson (2003) and Trecroci and Vassalli (2009).

Since the shifts in the central bank’s regimes are more likely to follow a smooth transition rather than an abrupt and sudden change, I will employ a model where the parameters vary through time. In addition, the weights the central bank puts on its objectives are also likely to vary within each regime, which further strengthens the argument of using a time-varying 7 parameter model. That is, the central bank is likely to vary the emphasis it puts on its objectives, e.g. the inflation rate, both across regimes and within each regime. These shifts are likely to happen more smoothly rather than abruptly as in the Markov-switching model. In Norway, the transition from an exchange rate targeting regime to an inflation targeting regime was a gradual one. Furthermore, there might also be changes in the weights put on the variables in the objective function due to changes in the central bank management.

The inclusion of a measure for financial instability also points in the direction of having a model in which the parameters vary across time. One would expect that shocks stemming from financial instability would strike the economy with different power at different points in time. In that respect, a time-varying parameter model provides a viable alternative to the statedependent Markov-switching model.

As in Kim (2006) I will consider a model where the coefficients are allowed to vary across time, and where the regressors are endogenous. In contrast to Kim (2006), however, I will use time-invariant parameters in the equations for the endogenous regressors.

I will use a model framework as presented by Baxa et al. (2011), which is found by rewriting (4) along the lines of Kim (2006), resulting in the following model in a state-space

**framework:**

(5) (6) (7) (8) (9) (10) (11) (12) 8 (13) (14) (15) Equation (5) is the time-varying representation of the time-invariant policy rule, i.e. Eq. (4).

Eqs. (6) – (11) show how the parameters are assumed to vary across time. The movements of the parameters are represented by random-walk processes without a drift, where unexpected movements in the parameter values are picked up by the error terms. Eqs. (12) – (15) demonstrate the relationship between the endogenous right-hand side variables in (5),,, and, and their instrumental variables, represented by the vector. The instruments that will be used, are lagged values of the inflation rate, the output gap, the exchange rate, the interest rate and foreign interest rate, along with lagged values of oil price inflation, foreign interest rates, foreign output gap and foreign inflation.

The covariance between the error terms in (5) and the standardized errors in (12) – (15), are defined as,, and, where denotes are the correlation coefficients between residual and.

In the model framework above, the parameters in the equations for the endogenous regressors (equations 12 – 15) are assumed time-invariant. By contrast, Kim (2006) and Kim and Nelson (2006) assume time-varying parameters for the endogenous right-hand-side variable.

Consistent estimates of the coefficients in (5) are obtained by estimation in two steps. In the first step, (12) – (15) are estimated. In this step, standardized prediction errors of the residuals in the equations for the endogenous right-hand-side variables in the policy rule are found.

Furthermore, as is done in Kim and Nelson (2006), the error terms in (5) and (12) – (15) are decomposed utilizing the Cholesky method. Hence, the error term in (5) is allowed to be

**rewritten as follows:**

(16)

**Insertion of (16) into (5) yields the following equation to estimate:**

(17) In the second stage of the estimation, I estimate (17) along with (6) – (11) by employing the Varying Coefficients (VC) method proposed by Schlicht (1981, 2005) and Schlicht and Ludsteck (2006). In contrast to Kim and Nelson (2006), who use the maximum likelihood estimator through the Kalman filter to estimate the parameters in (17), the VC method suggested by Schlicht and Ludsteck (2006) is a generalization of the ordinary least squares method. Rather than minimizing the sum of squares,, their VC method minimizes the weighted sum of squares. The weights,, are defined as the inverse variance ratio of the residuals from eq. (5),, and the error terms of the time-varying parameters,, formally defined as.

Moreover, the time-average of the parameters in (17) coincides with the GLS estimate of the fixed parameters, i.e., where denotes the parameters in (17) that is to be estimated.

This method has several advantages when estimating the modified Taylor-rule. First, no initial values are needed, as the estimator uses an orthogonal parameterization (Schlicht & Ludsteck, 2006).

Schlicht and Ludsteck (2006) also compare the estimator generated by the VC-method with the corresponding estimator produced by estimation through utilization of the Kalman filter.

The performance of the two estimation methods is quite similar. However, the ease of estimation through applying the VC-method is greater than the Kalman filter, as no initial values are needed. They conclude that the VC-method is preferable for estimating linear models with parameters that are following a random walk.

10 4 Data All data are monthly observations, with the first observation being August 1998 and the last being December 2011. During this period, the central bank changed its objective from keeping a stable exchange rate to keeping a stable inflation rate. The era of the floating exchange rate regime was officially abandoned in 2001, from which the objective of the monetary authority has been to keep a stable inflation rate.

To represent the monetary policy interest rates, I use the three month Norwegian Interbank Offered Rate (NIBOR). In general, Norges Bank’s liquidity policy is able to ensure that the money market interest rate stays close to the policy rate, which is its overnight deposit rate.

However, during the recent financial crisis, there was occasionally relative large divergence between the policy rate and money market interest rates. Source: Norges Bank.

The inflation rate is measured as the 12-month percentage change in the consumer price index, excluding energy prices (CPI-AE). Starting in 2001, Norges Bank has had an explicit inflation target rate of 2.5 per cent, where the targeted inflation rate is the change in CPI adjusted for tax changes and excluding energy goods (CPI-ATE). Due to the limited number of observations based on this measure, I use CPI-AE as a proxy. Source: Statistics Norway.

The output gap is calculated as the log deviation of actual output from an estimated trend. The trend is calculated using an HP filter with the smoothing parameter set to 129 600. The output data consists of seasonally adjusted quarterly GDP in Norway4, and is measured in millions of Norwegian Kroner. As my data set consists of monthly data, I obtained monthly output data by linearly interpolating the quarterly data. Source: Statistics Norway.

The data from the last few quarters will typically be revised at a later point in time, and may thus not be alike for samples collected at different periods. To avoid this problem, the difference in the unemployment rate from its natural level could be used as a proxy for the output gap. There are some advantages from using the unemployment rate, such as the availability of monthly data and that the data will not be revised. That is, there are no uncertainties concerning the data gathered, as opposed to the use of the output gap. I will use 4 The data used are the Statistics Norway series called “Value added at basic prices. Current prices (NOK million), for Total Industry.”

As a measure of the exchange rate, I will use the monthly growth rate of the import weighted exchange rate, I44. It is the nominal effective exchange rate measured as a geometric average over Norway’s 44 most important trading partners. Source: Norges Bank.

Finally, the set of instruments consists of monthly data for oil price inflation, average 12month inflation rate for the G7 countries, the average nominal short term interest rate for the Euro economies and the output gap for the 27 European Union countries, measured as the deviation of the logged average gross production from its trend, where the trend is calculated using an HP filter with the smoothing parameter set to 129 600. Sources: FRED St. Louis Fed, OECD, EuroStat.

**Financial Stress Index (FSI)**

The variable measuring the degree of financial instability is based on the index created by Cardarelli et al. (2011). However, due to data availability the index presented in this thesis is somewhat different from the one constructed by them. All sub components of the index are demeaned and standardized. That is, I have subtracted the arithmetic mean and divided by the standard error. The financial stress index (FSI) is constructed as the sum of seven components, the beta for the banking sector, the spread between the NIBOR and the overnight lending rate, the inverted term spread, stock market return, stock market volatility and finally the exchange market volatility. The index itself is also demeaned and standardized.

The banking sector beta is a measure of the risk of the banking sector, and is defined as the covariance between the banking and market returns divided by the variance of the market.

This measure gives an indication of how risky the banking sector is – the higher the value of beta the more risky is the banking sector stocks. As a proxy for the banking market performance in the Norwegian stock market I use the stock price of Norway’s largest bank, DNB. The returns are measured as the twelve month growth in the DNB stock and the market index. Source: Oslo Børs.

As a proxy for the TED spread used by Cardarelli et al. (2011), I use the spread between the overnight lending rate and the three month NIBOR. Since the overnight lending rate is 12 considered to be safer than the interbank rate, this provides a measure of uncertainty in the interbank market as it captures the premium that banks charge over the overnight lending rate to lend out money in the interbank market. An increase in the spread corresponds to an

**increase in the premium and thus increased uncertainty in the interbank market. Source:**

Norges Bank.

The inverted term spread is calculated as the government overnight lending rate minus the NIBOR long term rate, the NIBOR 12-month rate. Hence, an increase in the term spread shows that there are increased difficulties in attaining short-term funding, indicating increased uncertainty in the banking sector. Source: Norges Bank.

The stock market returns are computed as the monthly change in the stock market index multiplied by minus one. By multiplying the returns with minus one, a drop in the stock prices corresponds to an increase in the index. Source: Oslo Børs.