Inflation and inflation uncertainty in Finland

Transcription

1 Mat Independent Research Projects in Applied Mathematics Inflation and inflation uncertainty in Finland Matti Ollila HELSINKI UNIVERSITY OF TECHNOLOGY Faculty of Information and Natural Sciences Department of Mathematics and Systems Analysis

2 Contents 1 Introduction 1 2 An overview of the inflation data Finland Sweden Stationarity of time series Empirical models and results Autoregressive model of inflation Impact of euro Volatility modeling An overview of GARCH-models Inflation and inflation uncertainty Discussion and conclusions 18 References 21 Appendices 24 i

3 1 Introduction Price stability is the primary objective of monetary policy in the European Monetary Union (EMU) (see e.g. Liikanen, 2007). There is a broad consensus that maintaining stable prices, i.e. controling the level of inflation, is the most effective means of monetary policy to support economic growth, job creation and social cohesion. The European Central Bank (ECB) aligns an inflation rate target of below but close to two per cent (ECB, 2004). One motivation for targeting a low inflation rate is the alleged relationship between inflation rate and inflation uncertainty. In 1977 Milton Friedman presented in his Nobel lecture that an increasing rate of inflation increases the uncertainty of inflation. This is because a burst of inflation tends to produce a strong pressure to counter it and consequently policy is altered, encouraging a deviation in the actual and anticipated rate of inflation. According to Friedman, increased uncertainty over the actual future level of inflation causes political unrest and economic efficiency is lowered. Ball formalized Friedman s hypothesis in 1992 and subsequently this causality between inflation rate and inflation uncertainty has been labeled the Friedman Ball hypothesis in literature. There has been a considerable volume of empirical research on the Friedman Ball hypothesis, but evidence regarding the impact of inflation rate on inflation uncertainty remains mixed. Engle (1983) compares the conditional variance and the inflation rate in the USA for different time periods, but finds no significant relationship between the two series. Later on, the ARCH 1 and GARCH 2 models presented by Engle in 1982 have been actively utilized in testing the Friedman Ball hypothesis. Evans (1991), Gried and Perry 1 autoregressive conditional heteroscedasticity 2 generalized ARCH 1

4 (1998) and Fountas (2001) present results in support of the Friedman Ball hypothesis. This study seeks to investigate the relationship between inflation and inflation uncertainty in Finland Particularly, the study aims at clarifying how the fixing of the Finnish markka to the euro has affected to this relationship. The Finnish markka was tied to euro Sweden that is not a full member of the EMU is used as a comparison. The rest of the study is structured as follows. Section 2 presents an overview of the inflation data. Section 3 identifies empirical models for inflation and inflation uncertainty and presents results. Section 4 concludes. 2 An overview of the inflation data The time period is from January 1985 to June Inflation is measured as the first difference of the seasonally adjusted 3 log consumer price index (CPI) π t = 100 (ln CP I t ln CP I t 1 ), (1) by using monthly observations. 3 Adjustment performed using the X-12-ARIMA routine (see more information at 2

5 2.1 Finland Finland s annual inflation rate and its volatility are presented in Figure 1. Inflation volatility is measured as the standard deviation 4 of the inflation rate. The volatilities are obtained such that first for each year a standard deviation is calculated from twelve monthly observations. These are then scaled to annual values by multiplying them with square root of twelve. The liberilization of the financial system in the late 1980s resulted in the Finnish economy overheating and consequently inflation rate soared. With the economy still in recession in 1993, the Bank of Finland (BoF) committed itself to stabilizing the inflation rate at around two per cent by 1995 (Akerholm and Brunila, 1994). In 1999 the Finnish markka was tied to the euro and the ECB took over the responsibility of conducting monetary policy. From Figure 1 it can be seen that BoF was successful in its inflation targeting as inflation rate dropped to around two per cent after the early 1990s. Figure 1 also gives some evidence of that periods of higher inflation correspond with periods of higher volatility. 2.2 Sweden Sweden experienced also an economic crisis at the break of 1990s. In 1993 the Swedish Central Bank (Riksbank) decided that flexible exchange rate would be combined with an explicit inflation rate target. Along with this policy a two per cent inflation rate was targeted with an interval of one percentage point from 1994 onwards (van Dijk et al., 2005). Sweden, unlike Finland, has not to this date become a full member of the EMU as it has maintained its own currency, the Swedish krona. Sweden s annual inflation 4 For the definition of standard deviation see Section 3.3 3

6 Figure 1: Inflation rate and inflation volatility in Finland rate and inflation volatility are presented in Figure 2. The shape of the inflation series is similar to Finland s. The average volatility seems to be approximately on the same level as Finland s and periods of higher inflation are associated with increases in inflation volatility. 2.3 Stationarity of time series To apply time series analysis, the series in consideration needs be stationary. A stochastic process y t t T is (weakly) stationary if it satisfies the following requirements (Greene, 2003) 1. E[y t ] = µ t T 2. V ar[y t ] = σ 2 t T 3. Cov[y t, y s ] = γ t s t T 4

7 Figure 2: Inflation rate and inflation volatility in Sweden The first requirement states that the expected value of y t is independent of t. The second requirement states that the variance of y t is a finite, positive constant, independent of t and the third that the covariance between observations in the series is a function of how far apart the two observations are in time, not the time at which they occur. Stationarity of time series can be tested with unit root tests. Augmented Dickey-Fuller and Phillips-Perron tests are used to investigate the stationarity of the time series. The tests are presented in Appendix A. The results of the tests are presented in Table 1. For the Augmented Dickey-Fuller test results are presented for lag length one, but other lag lengths were also tested with similar results. For both time series the null hypothesis of a unit root is rejected and thus time series can be treated as stationary. 5

8 Table 1: Unit root tests SWEDEN t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level Adj. t-stat Prob.* Phillips-Perron test statistic Test critical values: 1% level FINLAND t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level Adj. t-stat Phillips-Perron test statistic Test critical values: 1% level Empirical models and results Time series analysis is applied to identify suitable models for the Finnish and Swedish inflation time series, because no commonly accepted structural model for inflation is presented in literature. In Section 3.1 an autoregressive model for the conditional mean of inflation is identified. Section 3.2 investigates the impact of euro on inflation by adding a dummy variable to the models of Section 3.1. Section 3.3 goes over the basics of volatility modeling and Section 3.4 presents the theory of conditional variance which is used as a proxy for inflation uncertainty. Section 3.5 investigates the link between inflation and inflation uncertainty. 3.1 Autoregressive model of inflation Autoregressive specifications to model the conditional mean of inflation are popular in the literature (see e.g. Fountas, 2001). A general autoregressive 6

9 model is of the form (see e.g. Greene, 2003): y t = α 0 + α 1 y t 1 + α 2 y t α n y t n + ε t, (2) where ε t i.i.d.(0, σ 2 ). The model identification is started by including all lagged terms of up to one year in the model and then reducing non-significant terms while ensuring the whiteness of residuals. First, stepwise regression is implemented, but the model s residuals are strongly autocorrelated. Finally, Akaike information criterion (see e.g. Greene, 2003) together with residual tests is used. The following models are found to best to fit the data: π f t = γ 0 + γ 1 π f t 1 + γ 3 π f t 3 + γ 11 π f t 11 + u f 1t (3) πt s = γ 0 + γ 3 πt 3 s + γ 9 πt 9 s + u s 2t, (4) where the indexes f and s refer to Finland and Sweden, respectively. These models can be estimated with the ordinary least squares (OLS) method and the results of the significance of model parameters are presented in Table 2. The diagnostic tests for the residuals are presented in Appendix B. In Appendix C the results for the tests are presented. Results show that all model parameters in both models are significant with 95% confidence interval and residuals pass tests for serial correlation, normality and heteroscedasticity. It is somewhat surprising what lagged terms were found to best fit the data. A lagged term of 12 months, that is one year, would be more intuitive than the 11 months found in model (3). It is also reasonable to assume that current inflation depends on previous month s inflation, but in model (4) the nearest lagged term is three. If a one-month lagged term is included in model (4), the residuals are strongly autocorrelated and the term is therefore omitted. 7

10 Table 2: Results for the significance of model parameters of models (3) and (4) FINLAND Variable Coefficient Std. Error t-statistic Prob. gamma(0) gamma(1) gamma(3) gamma(11) R-squared F-statistic Adjusted R-squared Prob(F-statistic) SWEDEN Variable Coefficient Std. Error t-statistic Prob. gamma(0) gamma(3) gamma(9) R-squared F-statistic Adjusted R-squared Prob(F-statistic) Impact of euro In January the Finnish markka was tied to euro with a fixed exchange rate. The currency was formally introduced to consumers in January The major impact of Finland becoming a full member of the EMU was that the ECB became responsible for conducting monetary policy in the euro area. The primary objective of ECB s monetary policy is to control the inflation rate. Evidence of inflation rate convergence in the euro area remains, however, mixed (Mentz and Sebastian, 2003). This study seeks to investigate whether the average inflation rate in Finland has changed after the introduction of the euro. The same analysis is done 8

11 to Sweden which is not a full member of the EMU. A statistically significant drop in the average inflation rate of Finland would imply that Finland s entry to the EMU has had a stabilizing effect on the inflation rate. For Sweden no change in the average rate of inflation is expected. Whether the average inflation rate has changed in the two countries can be tested by introducing dummy variables to models (3) and (4). The dummy variable has a value of zero before the introduction of the euro and a value of one after the euro. In equation (3) a dummy variable will be used for lags one and eleven and in equation (4) for lag three. The new models are: π f t = γ 0 + (γ 1 + D t γ 1 )π f t 1 + γ 3 π f t 3 + (γ 11 + D t γ 11 )π f t 11 + u f 1t (5) π s t = γ 0 + (γ 3 + D t γ 3 )π s t 3 + γ 9 π s t 9 + u s 2t. (6) Results of parameter significance using OLS method are presented in Table 3. From the results it is seen that for Finland the coefficients of the dummy variables are not significant at 95% confidence interval. Therefore, the null hypothesis of the coefficients being zero cannot be rejected. This suggests that there has not been a change in the average inflation rate after the introduction of euro. Figure 1 supports this result as no visible change is detected in the inflation rate after For Sweden the coefficient of the dummy variable is also non-significant and therefore no change in average inflation rate has taken place in Sweden after Next, the uncertainty of inflation is modeled. Because the dummy variable models are not statistically significant, models (3) and (4) are used in the following analysis. 9

12 Table 3: Results for the significance of model parameters of models (5) and (6) FINLAND Variable Coefficient Std. Error t-statistic Prob. gamma(0) gamma(1) gamma(3) gamma(11) gamma(1*) gamma(11*) R-squared F-statistic Adjusted R-squared Prob(F-statistic) SWEDEN gamma(0) gamma(3) gamma(9) gamma(3*) R-squared F-statistic Adjusted R-squared Prob(F-statistic) Volatility modeling Volatility refers to a measure of uncertainty (see e.g. Hull, 2003). Especially in finance it is used to describe how much a time series (e.g. asset returns) has varied on average over some specific time span. The most common measure of volatility is the standard deviation σ of a set of observations ˆσ 2 = 1 N 1 N (π t π) 2, (7) t=1 10

13 where π is the mean of observations. In Section 2 this measure was used in Figures 1 and 2 to illustrate how the annual volatility of inflation has developed in Finland and in Sweden. Although simple to calculate, the problem with the specification of (7) is that it has limited value for forecasting as it does not account for past values. By this we mean that the standard deviation at time t does not depend on the values of standard deviation at time t 1,..., t N. For volatility forecasting, conditional volatility and also a conditional mean as modeled in Sections 3.1 and 3.2 are needed. Next section presents a class of such models. 3.4 An overview of GARCH-models In 1982 Engle provided an approach to model and forecast the conditional variance. If there exists a conditional density function f(π t π t 1 ) for inflation, the forecast for today s inflation is simply E(π t π t 1 ). In equations (3) and (4) this conditional mean for the inflation time series of Finland and Sweden is modeled. The variance of this forecast is V ar(π t π t 1 ), but for conventional econometric models the variance does not depend on π t 1. Engle proposed in his autoregressive conditional heteroscedasticity (ARCH) specification that the mean corrected inflation u t in (3) and (4) would be presented as: where h t = u t = h t ɛ t, (8) a 0 + a 1 u 2 t 1 + a 2 u 2 t a k u 2 t k and ɛ t i.i.d.n(0, 1). The parameters have conditions a 0 > 0 and a i 0. 11

14 Bollerslev (1986) presented an extension to Engle s ARCH model, namely generalized ARCH (GARCH). By using equation (8) for the mean corrected inflation, a GARCH(p,q) process, where p is the order of u 2 t terms and q is the order of the ARCH terms h 2 t, is defined as: h t = a 0 + p a i u 2 t i + i=1 q b j h 2 t j, (9) where a 0 > 0 and a i 0, b j 0 and ɛ t i.i.d.n(0, 1). One of the most j=1 popular specifications is the GARCH(1,1) model: h t = a 0 + a 1 u 2 t 1 + b 1 h 2 t 1. (10) These models are useful, because by knowing the parameter values, the variance at time t can be calculated from the observations at time t 1. The GARCH models are, thus, suited for forecasting variance. A statistical property these specifications have is that large past positive or negative errors are likely to be followed by another large error of either sign and similarly small errors are followed by a small error of either sign. In financial applications this phenomenon is often called volatility clustering. Another property that a GARCH-class model has is that the innovations are leptocurtic. A leptocurtic distribution has a higher peak and fatter tails than a normally distributed variable. Critics of GARCH models point out that as the models use squared error terms, they respond similarly to negative and positive shocks, which does not, however, reflect reality observed in empirical time series. Another weakness of these models is that although the innovations are leptocurtic, the tail behaviour is still too short (Tsay, 2001). 12

15 To be able to use GARCH models, the residuals of the conditional mean have to be tested for conditional heteroscedasticity. Heteroscedasticity means that the variance of the residuals is not constant over time. A standard test is the Lagrange multiplier test developed by Engle (1982) in which the squared residuals of the OLS estimation of the conditional mean are regressed against a constant and their lagged values: u 2 t = δ 0 + q δ i u 2 t i + θ t, (11) i=1 where the null hypothesis δ 1 = δ 2 =... = δ q = 0 implies a constant variance. Bollerslev (1986) shows that this test applied to a qth order ARCH is equivalent to a test for GARCH(i+j) where i+j=q. The results for Finland s and Sweden s inflation time series for lags one and two are presented in Table 4. From the results it can be seen that for Finland the null hypothesis of homoscedasticity cannot be rejected but for Sweden the null hypothesis is rejected. This implies further that inflation rates have behaved differently in the two countries. Table 4: ARCH LM tests for inflation time series lag F-statistic Prob. FINLAND SWEDEN Inflation and inflation uncertainty Conditional variance can be used as a proxy for inflation uncertainty. With help of the theory presented in Section 3.4 models to test the Friedman Ball 13

16 hypothesis can now be built. This is achieved by modeling the residuals of models (3) and (4) as having a time-varying variance. Because Finland s time series of inflation nearly passed the test of ARCH effects, for Finland a GARCH(1,0) model is used to model the residuals. Sweden s inflation implied strong effects of time-varying variance and consequently a GARCH(1,1) model is used to model the residuals. With these specifications the models (3) and (4) become: π f t = γ 0 + γ 1 π f t 1 + γ 3 π f t 3 + γ 11 π f t 11 + u f 1t u f 1t = (12) a 0 + a 1 u 2 1t 1ɛ 1t π s t = γ 0 + γ 3 π s t 3 + γ 9 π s t 9 + u s 2t u s 2t = h 2t ɛ 2t, (13) h 2t = a 0 + a 1 u 2 2t 1 + b 1 h 2 2t 1 where ɛ 1t = ɛ 2t i.i.d.n(0, 1). For the estimation of these models an OLS method is not applicable. A common method in the estimation of GARCH models is the maximum likelihood method (see more e.g. Engle, 1982). A likelifood function which is a function of the parameters to be estimated is constructed. As a density function is used, the sample is usually assumed to follow a normal distribution. The likelifood function is maximized with an iterative algorithm to find parameters that best fit the data. The results for the estimation of models 12 and 13 are presented in Table 5. The results for Finland show that although the model as a whole is significant as can be seen from the F-statistic and its p-value, the parameter of the conditional variance is not significant. This implies that using our model no clear relationship between inflation and inflation uncertainty is found in the case of Finland. The tests for residuals are presented in Appendix A. The tests indicate a correct model specification. The residuals are from normal 14

17 disribution as indicated by the Jarque-Bera test and there is no significant serial correlation in the residuals (Ljung-Box Q-statistics). For Sweden all the estimated model parameters are siginificant. This suggests that there is a relationship between inflation and inflation uncertainty. To investigate how an increase in past inflation affects inflation uncertainty, a scatter plot is drawn where on the x-axis there is inflation lagged by one month and on the y-axis the values of conditional variance. These values are used as a proxy for inflation uncertainty. The corresponding data for Finland is plotted as well, although the ARCH parameter is not significant. The scatter plots are presented in Figure 3. From the figures it can be seen that the shapes of the series are similar, but the scales differ. For Sweden the variability of inflation and its conditional variance are greater. The figures give no evidence in favor of the Friedman Ball hypothesis, as there appears to be no clear positive linear relationship between past inflation and current inflation uncertainty. Kontonikas (2004) uses a variant of the GARCH model used here, called the GARCH-in-mean (GARCH-M) to investigate the link between inflation and inflation uncertainty in the UK. In this specification the conditional variance is inserted directly into the equation of the conditional mean. GARCH-M model here, (3) and (4) become: π f t By using = γ 0 + γ 1 π f t 1 + γ 3 π f t 3 + γ 11 π f t 11 + δ h f t + v f 1t (14) π s t = γ 0 + γ 3 π s t 3 + γ 9 π s t 9 + δ h s t + v s 2t, (15) 15

18 Table 5: Results for the significance of model parameters of models (11) and (12) FINLAND Coefficient Std. Error z-statistic Prob. gamma(0) gamma(1) gamma(3) gamma(11) a(0) a(1) R-squared F-statistic Adjusted R-squared Prob(F-statistic) SWEDEN Coefficient Std. Error z-statistic Prob. gamma(0) gamma(3) gamma(9) a(0) a(1) b(1) R-squared F-statistic Adjusted R-squared Prob(F-statistic)

19 Figure 3: Scatter plots of inflation vs. inflation conditional variance where h t is the conditional variance of inflation defined: h t = ϕ + αe 2 t 1 + βh t 1 + λπ t 1. (16) The coefficient δ describes the effect of conditional variance on average inflation. Results for the model parameters significance for the models (14) and (15) are presented in Table 6. For Finland, most of the parameters are non-significant and thus it can be concluded that this model is not suitable for further analysis. For Sweden, the parameters γ 0 and γ 1 are not significant and therefore our model fails to fit the data properly. These results imply that an autoregressive specification with the residuals following a GARCH- 17

20 M process does not point to any evidence of a positive linear relationship between inflation and inflation uncertainty in the two countries. 4 Discussion and conclusions This study sought to investigate the relationship between inflation and inflation uncertainty in Finland from January 1985 to June A further goal was to find out whether Finland s inclusion to the EMU and the introduction of the euro has lowered the average inflation rate in Finland and on the other hand reduced inflation uncertainty. Friedman Ball hypothesis formalizes the relationship between inflation rate and inflation uncertainty. Autoregressive models were identified to model inflation time series, while GARCH-models popular in the literature were used in modeling inflation uncertainty. For Finland, no relationship between inflation and inflation uncertainty was found. Finland joining the EMU was found to have no impact in lowering the average inflation rate. Sweden that was used as a comparison in the study was not either found to have a statistically significant change in its average inflation rate after By using scatter plots, the existence of the Friedman Ball hypothesis was further examined to discover that a higher past inflation rate does not imply increase in current inflation uncertainty. Using similar models as presented here, Kontonikas (2004) and Caporale and McKiernan (1997) find evidence in favor of the Friedman Ball hypothesis for UK and US inflation data, respectively. There are several possible reasons why their results differ from the ones presented above. First, Kontonikas, Caporale and McKiernan use a longer time period, starting from 1970s and 1940s, respectively. Before 1990s inflation used to be more volatile, which 18

21 Table 6: Results for the significance of model parameters of models (13) and (14) FINLAND Coefficient Std. Error z-statistic Prob. gamma(0) gamma(1) gamma(3) gamma(11) delta phi alpha beta lambda R-squared F-statistic Adjusted R-squared Prob(F-statistic) SWEDEN Coefficient Std. Error z-statistic Prob. gamma(0) gamma(3) gamma(9) delta phi alpha beta lambda R-squared F-statistic 5.67 Adjusted R-squared Prob(F-statistic)

22 supports the use of GARCH models. Second, already before Finland joining the EMU, BoF used inflation targeting similar to that of the ECB with good success. It is, thus, not surprising that for the time period of 1985 to 2008 time-varying variance is not detected in Finland s inflation rate. Moreover, it would be more sensible to use 1993 as the breakpoint when investigating a change in the average inflation rate. Third, Finland joined the European Union in 1995 and after that it was likely that Finland would also join the EMU. It has been debated in the literature that inflation convergence was more significant before 1999 (e.g. Mentz and Sebastian, 2003), which further supports the use of an earlier date for studying a change in the average inflation rate. No clear conclusions can be drawn of the relationship between inflation rate and inflation uncertainty in Finland and in Sweden based on these results. A suitable line of further study would be to use a longer time period and to leave out seasonal adjustments of the inflation time series in order to find a model that better fits the data. 20

23 References [1] Akerholm, J. and A. Brunila (1994). Inflation Targeting: The Finnish Experience, paper presented in CEPR Workshop Inflation Targets, Milan, Italy. [2] Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroscedasticity, Journal of Econometrics, 31, [3] Caporale, G. M. and A. Kontonikas (2006). The Euro and Inflation Uncertainty in the European Monetary Union, CESifo working paper No [4] Caporale, T. and B. McKierman (1997). High and variable inflation: Further evidence on the Friedman hypothesis, Economic Letters, 54, [5] Dickey, D.A., Fuller W.A. (1979). Distribution of the Estimators for Autoregressive Time Series with a Unit Root. Journal of the American Statistical Association, 74, [6] van Dijk, D., H. Munander and C. M. Hafner (2005). The Euro Introduction and Non-Euro Currencies, Tinbergen Institute Discussion Paper, TI /4. [7] Engle, R. (1982). Autoregressive conditional heteroschedasticity with estimates of the variance of UK inflation, Econometrica, 50, [8] Engle, R. (1983). Estimates of the variance of US inflation based upon the ARCH model, Journal of Money, Credit and Banking, 15, [9] Evans, M. (1991). Discovering the Link Between Inflation Rates and Inflation Uncertainty, Journal of Money, Credit and Banking, 23, 2,

24 [10] European Central Bank (2004). The Monetary Policy of the ECB 2004, ISBN [11] Fountas, S. (2001). The relationship between inflation and inflation uncertainty in the UK: , Economic Letters, 74, [12] Fountas, S., A. Ioannidis and M. Karanasos (2004). Inflation, Inflation Uncertainty and a Common European Monetary Policy, The Manchester School, 72, 2, [13] Friedman, M. (1977). Inflation and Unemployment, Journal of Political Economy, 85, 3, [14] Greene, W.H. (2003). Ecometric Analysis, Fifth Edition, Prentice Hall, New Jersey. [15] Grier, K. B. and M. J. Perry (1998). On inflation and inflation uncertainty in the G7 countries, Journal of International Money and Finance, 17, [16] Judge G.G., W.E. Griffiths, R.C. Hill, H. Ltkepohl, T-C Lee (1985). The Theory and Practice of Econometrics, Second Edition, Wiley, New York. [17] Kontonikas, A. (2004). Inflation and inflation uncertainty in the United Kingdom, evidence from GARCH modelling, Economic Modelling, 21, [18] Liikanen, E. (2007). Monetary policy in theory and practice, presented in the Meeting of the Finnish Economic Association on 30 May [19] Mentz, M. and S. P. Sebastian (2003). Inflation convergence after the introduction of the Euro, CFS Working Paper 2003/30. [20] Muhleisen, M. (1995). Monetary Policy and Inflation Indicators for Finland, International Monetary Fund, WP/95/

25 [21] Tsay, R.S. (2001). Analysis of Financial Time Series, John Wiley & Sons, Inc. 23

26 Appendices Appendix A: Unit root tests Augmented Dickey-Fuller test The model of the test is y t = α + β t + γy t 1 + δ 1 y t δ p y t p + ɛ t (17) The null hypothesis is γ = 0 and the test statistic is compared to the relevant critical value of the Dickey-Fuller test that has a distribution known as the Dickey-Fuller table (Dikcey and Fuller, 1979). Phillips-Perron unit root test The model of the test is y t = α + β t + γy t 1 + ɛ t. (18) Here the ɛ t may be heteroscedastic. The test statistic is modified to correct for serial correlation and heteroscedasticity in the errors. The null hypothesis is γ = 0. The test statistic has the same asymptotic distribution as the Augmented Dickey-Fuller statistic. One advantage of Phillips-Perron test over the Augmented Dickey-Fuller is that there is no need to specify a lag length for the test regression. Appendix B: Tests for residuals Jarque-Bera normality test The Jarque-Bera test tests the normality of a given sample. The null hypothesis is that the data are from a normal distribution. The test statistic is 24

27 based on skewness and curtosis calculated from the sample and it is defined as JB = n 6 (S 2 + ) (K 3)2, (19) 4 where n is the sample size, S is the skewness and K the sample kurtosis, defined as S = µ 3 K = µ 4 σ 3 = 1 n n i=1 (x i x) 3 ( 1 n n i=1 (x i x) 2 ) 3/2 σ 4 = 1 n n i=1 (x i x) 4 ( 1 n, n i=1 (x i x) 2 ) 2 (20) where µ 3 and µ 4 are the third and fourth central moments, respectively and x is the sample mean. The test statistic follows a chi-square distribution with two degrees of freedom. Breusch-Godfrey Serial Correlation LM Test: The Breusch-Godfrey test can be used to test the serial correlation of residuals. Consider the model, y t = β 0 + β 1 x t + u t u t = ρ 1 u t 1 + ρ 2 u t ρ p u t p + ɛ t. (21) To test the serial correlation of residuals u t, the null hypothesis is H 0 : ρ 1 = ρ 2 =... = ρ p = 0 (22) The test statistic is LM = (n p)r 2, (23) where n is the sample size and R 2 is the model significance of (21). The test 25

28 statistic is chi-square distributed with p degrees of freedom. White s test for heteroscedasticity The null hypothesis in White s heteroscedasticity test is that the sample is homoscedastic. This means that the sample variance is constant in time. The test is based on an auxiliary regression with squared residuals as the dependent variable and regressors from the initial model, their squares and cross-products as independent variables. For example, with two independent variables x 1 and x 2 the auxiliary regression is: e 2 i = α 0 + α 1 x 1i + α 2 x 2i + α 3 x 2 1i + α 4 x 2 2i + α 5 x 1i x 2i (24) The test statistic nr 2, where n is the number of observations and R 2 is from (24), is chi-square distributed with q degrees of freedom, where q is the number of independent variables in the auxiliary regression. Appendix C: Tests for residuals - Results Table 7: Residual tests for Finland (model (3)) FINLAND Jarque-Bera (normality test) Probability Breusch-Godfrey Serial Correlation LM Test: F-statistic Prob. F(2,263) Obs*R-squared Prob. Chi-Square(2) White Heteroskedasticity Test: F-statistic Prob. F(6,262) Obs*R-squared Prob. Chi-Square(6)

29 Table 8: Residual tests for Sweden (model (4)) SWEDEN Jarque-Bera (normality test) Probability Breusch-Godfrey Serial Correlation LM Test: F-statistic Prob. F(2,267) Obs*R-squared Prob. Chi-Square(2) White Heteroskedasticity Test: F-statistic Prob. F(4,267) Obs*R-squared Prob. Chi-Square(6) Table 9: Tests of residuals for GARCH models (12) and (13) SWEDEN FINLAND Jarque-Bera Jarque-Bera 55.9 Probability Probability Ljung-Box Q-statistics Ljung-Box Q-statistics lag Q-stat Prob lag Q-stat Prob