Joao Sousa
email: jmsousa@bportugal.pt

Nuno Alves

Sandra Gomes

José Brandão Brito

Joao ; Sousa

José Brandão De Brito

João Sousa

The Transmission of Monetary Policy and Technology Shocks in the Euro Area

Keywords: E32 -Business Fluctuations|Cycles < E3 -Prices, Business Fluctuations, E32 -Business Fluctuations|Cycles < E3 -Prices, Business Fluctuations, and Cycles < JEL Classification: E32, E52 technology and monetary shocks, euro area, long-run restrictions

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Introduction

This paper analyzes the dynamic response of a set of euro area macroeconomic variables to monetary policy and technology shocks using structural VAR models. Our results build on previous work for the euro area, namely [START_REF] Peersman | The monetary transmission mechanism in the euro area: More evidence from VAR analysis[END_REF], who only identify monetary policy shocks, and [START_REF] Peersman | Technology shocks and robust sign restrictions in a euro area SVAR[END_REF], who estimate both monetary and technology shocks using model-based sign restrictions. In contrast to those papers, we identify both shocks simultaneously by imposing theoretically-consistent restrictions both in the short-run and the long-run, in line with [START_REF] Altig | Firm-specific capital, nominal rigidities and the business cycle[END_REF].

The identification of impulse response functions based on structural VAR models aims not only at assessing the properties of the data but also at understanding the set of shocks and frictions that should be incorporated in dynamic general equilibrium models. An important debate in the literature is whether Galí's conclusions, that positive technology shocks induce a short-run decline in hours worked, is robust to different specifications. Several arguments have been traded in this context, with [START_REF] Basu | Are technology improvements contractionary?[END_REF], [START_REF] Francis | Is the technology-driven real business cycle hypothesis dead? shocks and aggregate fluctuations revisited[END_REF], and [START_REF] Galí | On the role of technology shocks as a source of business cycles: Some new evidence[END_REF] broadly supporting Gali's (1999) original claim and [START_REF] Christiano | The response of hours to a technology shock: Evidence based on direct measures of technology[END_REF] and [START_REF] Uhlig | Do technology shocks lead to a fall in total hours worked?[END_REF] presenting a critical view of these conclusions 1 . We contribute to this debate by building a novel long-run series of hours worked per capita in the euro area and incorporating it in the VAR, which also includes measures of labour productivity, inflation, the consumption to output ratio, the investment to output ratio, capacity utilization, the productivity to real wage ratio, the interest rate and M1 velocity. The sample runs from 1970Q1 until 2006Q4.

Several of our main conclusions are worth anticipating. First, in line with the results for the US presented in [START_REF] Altig | Firm-specific capital, nominal rigidities and the business cycle[END_REF], we find that real variables in the euro area follow a hump-shaped pattern after a monetary policy shock and jump on impact after a technology shock. Second, we find that per capita hours worked fall in the euro area after a positive technology shock, in line with the conclusions of [START_REF] Galí | On the role of technology shocks as a source of business cycles: Some new evidence[END_REF] for the euro area using a measure of employment. It should be noted that we conclude from our analysis that per capita hours are non-stationary in the sample period. This conclusion is particularly important, given that, as in [START_REF] Christiano | The response of hours to a technology shock: Evidence based on direct measures of technology[END_REF], the response of hours worked to a neutral technology shock depends on the evidence regarding the stationarity of per capita hours. Third, both conclusions above are robust to different sample periods and to different specifications of the variables. Finally, we show that technology shocks explain a significant part of the fluctuations of output, hours worked and wages in the euro area, while monetary policy shocks have a substantial contribution to the variations in the short-term interest rate.

The remainder of the paper is organized as follows. Section 2 presents the strategy to identify the shocks. Section 3 describes the data and their main empirical properties. Section 4 discusses the main results and section 5 undertakes several robustness checks. Section 6 presents the historical contribution of the shocks to the variance of the variables in the VAR. Section 7 concludes.

Identification of shocks

In the identification of the technology and the monetary policy shocks, we follow the methodology of [START_REF] Altig | Firm-specific capital, nominal rigidities and the business cycle[END_REF]. The analysis is based on the following reduced-form VAR,

Y t = η + B (L) Y t-1 + u t , Eu t u 0 t = V (1)
where B(L) is a polynomial of order q in the lag operator, L, u t is the vector of the one-stepahead forecast errors to Y t . The Y t vector is defined as,

Y t |{z} 9×1 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ∆ ln (GDP t /Hours t ) ln (Hours t ) ∆ ln (GDP deflator t ) ln (C t /GDP t ) ln (I t /GDP t ) ln(Capacity Utilisation t ) ln (GDP t /Hours t ) -ln (W t /P t ) Interest Rate t ln (GDP deflator t ) + ln (GDP t ) -ln (M 1 t ) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
Starting with the monetary policy shocks, we identify monetary policy shocks as deviations from a policy rule that is assumed to be followed by policymakers (see [START_REF] Christiano | Monetary policy shocks: What have we learned, and to what end?[END_REF]):

R t = f (Ω t ) + ε Rt
where R t is the interest rate, f a linear monetary policy rule, which is a function of the information set Ω t available to the policymakers. ε Rt is the monetary policy shock. The identification of the monetary policy shocks is achieved by assuming that ε Rt is orthogonal to the information set Ω t . In addition, we require that the only date t variables in the information set are productivity, measures of economic activity (hours, capacity utilization), wages and inflation. The introduction of the monetary aggregates in the system allows the identification of money demand shocks. We assume that the central bank does not react contemporaneously to money velocity shocks.

In identifying the technology shock, we follow much of related literature by imposing the restriction that only technology shocks can affect labour productivity in the long run (this restriction on VARs has been originally proposed by [START_REF] Galí | Technology, employment, and the business cycle: Do technology shocks explain aggregate fluctuations?[END_REF]). In implementing it, we pursue the methodology advocated by [START_REF] Shapiro | Sources of business cycle fluctuations[END_REF]. The structural representation of the VAR models we use can be written as:

A 0 Y t = A(L)Y t-1 + e t , e t = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ e zt |{z} (1×1) e 1t |{z} (6×1) e Rt |{z} (1×1) e 2t |{z} (1×1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
where the structural shocks, e t , which are unobservable, are assumed to be mutually independent and related linearly to the one-step-ahead forecast errors, u t :

u t = Ce t , Ee t e 0 t = I.
The parameters of the structural form are therefore linked to those of the reduced form by:

C = A -1 0 , B (L) = A -1 0 A (L) (2)
where the first and eighth columns of C are the objects we need to uniquely identify in order to compute the impulse responses pertaining to a technology and monetary shocks, respectively. It turns out that we need to estimate A 0 in order to estimate the technological and monetary shocks as well as their dynamic effects on Y t . The identifying restrictions laid down above impose the following structure on the matrix A 0 :

A 0 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ A 1,1 0 |{z} (1×1) A 1,2 0 |{z} (1×1) A 1,3 0 |{z} (1×6) 0 |{z} (1×1) 0 |{z} (1×1) A 2,1 0 |{z} (6×1) A 2,2 0 |{z} (6×1) A 2,3 0 |{z} (6×6) 0 |{z} (6×1) 0 |{z} (6×1) A 3,1 0 |{z} (1×1) A 3,2 0 |{z} (1×1) A 3,3 0 |{z} (1×6) A 3,4 0 |{z} (1×1) 0 |{z} (1×1) A 4,1 0 |{z} (1×1) A 4,2 0 |{z} (1×1) A 4,3 0 |{z} (1×6) A 4,4 0 |{z} (1×1) A 4,5 0 |{z} (1×1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
As seen above, we assume that in the short-run real economic activity and prices do not react contemporaneously to monetary policy shocks or to shocks to money velocity, which explains the zero restrictions on the first seven rows of the two last columns of the A 0 matrix above. This restriction stems from the empirically grounded assumption that in the short-term real variables do not respond significantly to nominal shocks. We thus follow a long tradition in monetary economics (going back to [START_REF] Friedman | The lag in effect of monetary policy[END_REF]) in assuming the existence of lags in the transmission of monetary policy. As for the restriction that the interest rate does not to react contemporaneously to velocity shocks we follow Larry [START_REF] Larry Christiano | Nominal rigidities and the dynamic effects of a shock to monetary policy[END_REF]. The justification for this restriction is that it is necessary to make the VAR consistent with the general equillibrium model of Larry [START_REF] Larry Christiano | Nominal rigidities and the dynamic effects of a shock to monetary policy[END_REF] which we take as our reference. As monetary policy may still react to past velocity shocks this restriction is not very binding. Finally, the restriction that technology shocks (e zt ) are the only shocks having a non-zero effect on labour productivity in the long-run implies that the matrix A 0 -A(1) has all zeros in the first row except in the (1,1) element.

Data

Description

The data used in the paper refer to an aggregate of twelve euro area member states for the period from the first quarter of 1970 to the last quarter of 20062 . For periods prior to 1999, the data correspond to an aggregation of the available country series. The use of aggregated euro area data is now standard in the literature, even though it has the drawback of assuming that before 1999 the euro countries behaved as if they formed a monetary union, which was obviously not the case. Recent examples of the use of similar aggregated data in related studies are the VAR studies of [START_REF] Peersman | The monetary transmission mechanism in the euro area: More evidence from VAR analysis[END_REF] and [START_REF] Peersman | Technology shocks and robust sign restrictions in a euro area SVAR[END_REF] or the euro area-wide models of [START_REF] Fagan | An area wide (AWM) model for the euro area[END_REF], [START_REF] Smets | An estimated dynamic stochastic general equilibtium model of the euro area[END_REF] and Günter [START_REF] Günter Coenen | Tax reform and labour-market performance in the euro area: A simulation-based analysis using the new area-wide model[END_REF]. Note that, to some extent, the robustness tests in section 5, namely the analysis of the stability of the responses over time, should be able to detect potential problems related to the aggregation of the country series before 1999.

As far as possible we used official statistical sources, such as the Eurostat, the ECB, the European Commission and the OECD. However, euro area series at a quarterly frequency are often available only for a relatively short time-span and we had to backdate a number of series.

To do this we relied mostly on the database by [START_REF] Fagan | An area wide (AWM) model for the euro area[END_REF] (hereafter Area-Wide Model (AWM) database). This was the case of the Eurostat national account series in volume, which only start in 1991 and therefore had to be chained backwards. Therefore, some caution is needed when analyzing the data in the beginning of the sample, not only due to methodological considerations but also because country data availability becomes scarcer as we move back in time.

Regarding national accounts deflators, the Eurostat series were chain linked with ECB data, which corrects for exchange rate variations among member countries in the period prior to 1999. Data on compensation per employee are published by the ECB. The series starts in the first quarter of 1991, and was chain-linked with data from the AWM database. The euro area capacity utilization series is published by the European Commission and is available since 1985. For the earlier periods we constructed a proxy for the euro area aggregate based on available data for the member countries. The M1 monetary aggregate series is published by the ECB. The short-term interest rate series used is the three-month Euribor provided by Bloomberg and for periods before 1999 we used data from the AWM database.

One important novelty in this paper is the use of a quarterly series for hours worked in the euro area. As there is no official series of average hours worked in the euro area we had to construct a new series. To do this we used country data on average hours worked per person in employment published by the OECD. However, and as mentioned by the OECD, there are significant differences in the sources and coverage of national data, implying that comparisons of the level of average hours worked across countries are probably not suitable. Therefore, we aggregated the quarterly rates of change of country series based on the euro area structure of employment (across countries) to get an index of average hours worked in the euro area.3 Finally, we obtain per capita hours by multiplying average hours per persons in employment by total employment and then dividing by working age population.

All variables are in logarithms, except for the short-term interest rate. Inflation is measured as quarterly changes in the logarithm of the GDP deflator.

Low-Frequency Properties of the Variables

A proper identification of the monetary policy and technology shocks requires all variables in the VAR to be stationary. Looking at figure 1 it is apparent that while some variables seem clearly stationary, others are less so. Among the variables that do not look stationary, some seem to be driven by deterministic trends.

In order to ascertain as accurately as possible the low-frequency properties of the data, we carry out a series of formal tests in pursuit of sources of nonstationarity in the data. The strategy we put in place consists of three main steps. First, we apply a battery of unit root tests to distinguish the stationary from the nonstationary variables in the sample. Second, we test for the presence of linear or quadratic deterministic trends among the variables labeled as nonstationary. Lastly, whenever significant, the deterministic trends are removed and the unit root tests re-run4 . After the de-trending process, the variables that remain nonstationary according to the unit root tests enter the VAR in first-differences.

On a first passage, we subject each variable to three distinct unit root tests, namely the Augmented Dickey-Fuller (ADF), the Phillips-Perron (PP), the Elliott, Rothenberg, and Stock (ERS), and Ng and Perron (N-P) tests which have unit root as the null hypothesis, and the Kwiatkowski, Phillips, Schmidt and Shin (1992) test (KPSS), which has stationarity as the null hypothesis. The results of these tests are summarized in table 15 .

As seen in the table, on the basis of the results of the ERS and N-P tests all of the variables could be non-stationary. Looking at all the tests, there is strong evidence that inflation, the interest rate and money velocity are non-stationary variables. The results also point overwhelmingly to non-stationarity in the cases of hours per capita and investment to output ratio. The less clear cut cases pertain to productivity growth, consumption to output ratio, capacity utilization and productivity to real wage ratio. In the case of productivity growth, the results of both the ADF and the PP tests point to a rejection of the null of non-stationarity while the results of the KPSS test point to a rejection of the null of stationarity. Given that the ADF and PP tests have low power, we view these results as constituting strong evidence of the stationarity of this series. In this respect, it should also be noted that the KPSS test has severe size-distortions for highly persistent, albeit stationary series6 . Therefore we take productivity growth to be a stationary series. We also opt for stationarity in the case of the consumption to output ratio and of capital utilization as there is some evidence to reject the null of non-stationarity in the cases of the ADF and PP tests and no evidence to reject the null of stationarity in the KPSS test. The variable productivity to real wage ratio remains a borderline case.

We then gather the potentially nonstationary variables and search for the presence of significant deterministic components. Since none, among the group of potentially nonstationary variables, seems to have a linear trend along the whole sample, we resort to the procedure proposed by [START_REF] Zivot | Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis[END_REF] that enables to test the null of a unit root against the alternative of stationarity around a broken linear trend. In particular, this procedure estimates the (single) break date that maximizes the evidence against the null of nonstationarity, computes an ADF-type test-statistic and provides the relevant critical values7 . The outcomes of the application of the Zivot and Andrews test to the group of nonstationary variables are summarized in table 2.

Based on the results of table 2, we take the variables inflation, investment to output ratio, productivity to real wage ratio and the interest rate to be stationary around a broken linear trend. As regards hours and money velocity, it seems that a broken linear trend is not enough to induce the stationarity of these variables. However, in the case of money velocity, graphical inspection of the series suggests that a quadratic trend might be a better characterization of the deterministic components at play. In fact, the inclusion of a quadratic trend is found to be highly significant and to make detrended money velocity stationary 8 .

To sum up, to estimate the VAR we use productivity growth, consumption to output ratio and capital utilization as in their original series. We estimate and remove broken linear trends from inflation, investment to output ratio, productivity to real wage ratio and the interest rate and a quadratic trend from money velocity. As for per capita hours, we consider it to be a non-stationary variable and therefore include it into the VAR in first-differences.

Benchmark specification

In this section, we report the responses of our set of macroeconomic variables to a monetary policy shock and a neutral technology shock. In the following description of the results, we concentrate on the first 20 quarters after the occurrence of each shock.

Impulse responses to a monetary policy shock

The responses of the variables to the monetary policy shock are shown in figure 2. In this figure the solid lines depict point estimates and the grey areas their respective 95% confidence bands 9 . The responses of all variables are measured in percentages, except the interest rate, which is measured in basis points. There are several interesting features worth highlighting from this figure. First, the one-standard deviation monetary policy shock is estimated to be around 40 basis points which is somewhat higher than the estimate of 30 basis points obtained by [START_REF] Peersman | The monetary transmission mechanism in the euro area: More evidence from VAR analysis[END_REF]. The fall in interest rates is quite persistent, lasting over one year. Money growth and velocity react contemporaneously to the monetary policy shock, a feature that stems directly from our identifying assumptions. Not surprisingly, the expansionary monetary shock induces a liquidity effect that fades away in tandem with the nominal interest rate. Second, there is a hump-shaped response of output, consumption, investment, hours per capita and capacity utilization, with the peak effect occurring one and a half to two years after the shock. As expected, the response of investment is quantitatively stronger than that of consumption. Third, the response of inflation is characterized by a fall on impact followed by a rebound that takes approximately two years to reach its peak 10 . Surprisingly, the very short run reaction of real wages is to decline, in spite of inflation falling at the same time. That effect is reversed eventually so that real wages' response goes into positive territory.

Neutral technology shock

The responses of the variables to the neutral technology shock are shown in figure 3. Several conclusions are worth underlining from this figure. First, the impact of a one-standard deviation positive technology shock is to generate a steady increase in output that reaches roughly 0.8% after 20 quarters. There is also a positive permanent effect on real wages, consumption and investment, as expected. Second, the response of inflation to the technology shock is mostly 8 The ADF test-statistic for money velocity after removing the estimated quadratic trend is -3.75, thereby implying the rejection of the null of unit root at the 5% significance level. 9 These bands were computed by bootstrap simulation with 2.000 draws. 10 An initial negative response of inflation to an expansionary policy shock has been dubbed as the 'price puzzle'. This effect has also been identified in several empirical studies for the US [START_REF] Altig | Firm-specific capital, nominal rigidities and the business cycle[END_REF]. contemporaneous and the largest response occurs on impact. However, this response is not statistically different from zero. Third, per capita hours endure what seems to be a permanent fall, a result similar to the one reported by [START_REF] Galí | On the role of technology shocks as a source of business cycles: Some new evidence[END_REF] for the euro area11 . This outcome means that positive technology shocks generate a lasting drop in the amount of labour input used in the economy. Together with the empirical observation that hours worked are procyclical this result raises doubts regarding the validity of the RBC paradigm. In fact, if technology shocks were the main drivers of business cycles then our VAR findings would imply a negative relation between hours worked and output growth and not the positive one seen in the data. Our results using VARs with model based restrictions are in contrast with those of [START_REF] Peersman | Technology shocks and robust sign restrictions in a euro area SVAR[END_REF] who, using an identification procedure based on sign-restrictions, find that hours worked (or employment) rise following a positive technology shock. Nevertheless, they are very much in line with those of Fernald ( 2007) who shows that after properly accounting for the non-stationarity in the hours series for the US, hours fall after a positive technology shock. In this context, it is important to note that, in line with the conclusions reported in [START_REF] Christiano | The response of hours to a technology shock: Evidence based on direct measures of technology[END_REF], the response of hours to a technology shock would also change substantially if hours worked were found to be stationary.

Robustness tests

This section tests the robustness of the impulse responses obtained with the benchmark VAR specification with respect to different sample periods and also to different definitions of some variables.

The sample period tests consist of estimating the impulse responses recursively for different samples. Two exercises were conducted. In a first exercise, the start of the sample is progressively increased while maintaining the end of the sample unchanged. In a second exercise, the end of the sample is progressively decreased while the start of the sample is kept unaltered.

The impulse responses for the monetary policy shock and the technology shock estimated with a sample recursively starting in the first quarter of 1970 up to the last quarter of 1979 and ending in the last quarter of 2006 are shown in figures 4 and 5, respectively. Starting with the monetary policy shock, as seen in figure 4, the qualitative responses of the variables seem broadly to follow the patterns of the benchmark model. As for the technology shock, the results show that, at least in a qualitative sense, in most cases the impulse responses are robust to the different starting points. There seems to be some uncertainty regarding the magnitude of the responses of output, and consumption but not about their sign. In the case of investment, the very short-term response is negative in some sub-samples but becomes undoubtedly positive in the medium to long-term. Finally, the responses of money velocity and the interest rate appear to be somewhat erratic, which is consistent with the full-sample result that they are not significantly different from zero.

Figure 6 shows the impulse responses of the monetary policy shock estimated with a sample starting in the first quarter of 1970 and with a decreasing end of sample (backwards from the last quarter of 2006 to the fourth quarter of 1997). From figure 6, it can be concluded that the impulse responses of all of the variables are quite robust to changes in the end of sample as they appear to be rather stable. As for the technology shock (see figure 7), there are some signs of instability in the magnitude of the response of hours, but the responses always point towards a decline following a positive technology shock. Another variable which shows somewhat erratic responses is capital utilization. As with the case of the increasing start of the sample, the responses of money velocity and the interest rate to a technology shock are again those that As regard the sensitivity of the results to different variable specifications, we test the robustness of the results to the use of alternative measures of the labour input and monetary aggregates 12 .

As regards the labour measure, we redo our VAR analysis considering the first-difference of log per capita employment13 . It turns out that the qualitative pattern of the impulse responses to technology and monetary policy shocks using this alternative measure of the labour input remains broadly unchanged.

As for the monetary aggregate, replacing the measure of money M1 by the broader measure M3 does not impart any significant change to the pattern of the effects of the monetary policy shock on all variables, except money growth. In fact, the liquidity effect following a negative interest rate shock disappears when M3 is used instead of M1 (i.e. money growth drops at the time of the expansionary interest rate shock). Nevertheless, M3 growth in the following quarters increases. This different reaction of M3 compared to M1 may be related to the fact that the share of interest remunerated components in M3 is rather large when compared to M1. If these components are more reactive to interest rate movements than those in M1, then M3 growth is likely to temporarily fall when there is an unexpected decline in the short-term interest rate.

Variance Decomposition

We now assess the contribution of the monetary policy and technology shocks to the forecast error variance of the variables included in our VARs. Table 3 summarizes the results. Before describing the evidence obtained, it is useful to explain some patterns of the results that stem directly from the identification strategy. First, the contribution of the monetary policy shock to the fluctuations of the real variables placed before the interest rate in the VAR list in equation ( 1) is zero on impact (i.e. in period 1). That follows from our timing assumption that those variables are sticky in the short-term, reacting to monetary policy shocks only with a lag. Second, the contribution of the technology shock to the fluctuations in output increases as we move forward in the horizon, a feature that stems from our identifying assumption that these shocks are the sole source of the unit root in labour productivity.

From table 3 it is worth highlighting that technology shocks play a significant role in explaining the fluctuations of output, hours worked and real wages. This result, which is predicted by the RBC paradigm, contradicts that of [START_REF] Galí | Technology, employment, and the business cycle: Do technology shocks explain aggregate fluctuations?[END_REF] for the U.S., but is consistent with the evidenced gathered by [START_REF] Altig | Firm-specific capital, nominal rigidities and the business cycle[END_REF] for the U.S. and [START_REF] Peersman | Technology shocks and robust sign restrictions in a euro area SVAR[END_REF] for the euro area. Therefore, while the evidence presented in the previous section does not support some of the RBC propositions, we still find that technology shocks have quite an important role in explaining business cycle fluctuations.

As for the monetary policy shock, we find its contribution to be relative unimportant in explaining the fluctuations of all variables, except those of the interest rate, as expected. That is also the thrust of the results in [START_REF] Altig | Firm-specific capital, nominal rigidities and the business cycle[END_REF] and [START_REF] Peersman | Technology shocks and robust sign restrictions in a euro area SVAR[END_REF].

F

Conclusions

This paper characterized the responses of the main euro area macroeconomic aggregates to monetary policy shocks and neutral technology shocks. Most of our results are qualitatively in line with those found in the VAR literature for the euro area. Our characterization benefited from the introduction of a novel series for hours worked in the euro area thereby constituting a further contribution to the ongoing debate on the behaviour of hours after technology shocks. We argued that the main conclusions in [START_REF] Galí | Technology, employment, and the business cycle: Do technology shocks explain aggregate fluctuations?[END_REF][START_REF] Galí | On the role of technology shocks as a source of business cycles: Some new evidence[END_REF], that hours worked fall after a positive technology shock, also apply to the euro area. Further, we have found a substantial degree of robustness of the results, both to the sample period and to the definitions of the variables, Assessing whether changes to the VAR specification and the quality of the data substantively affect our conclusions remains an open issue. These qualifications and caveats call for more research on the quest for the set of shocks, frictions and policies that, embedded in a general equilibrium framework, explains the dynamics of real and nominal variables in the euro area business cycle.

F

  in the different sub-samples.

Figure

  Figure 1: Raw data

Figure 2 :

 2 Figure 2: Impulse responses to a negative monetary policy shock.

Figure 3 :

 3 Figure 3: Impulse responses to a positive technology shock.

Figure 4 :

 4 Figure 4: Impulse responses to a negative monetary policy shock with increasing start of the sample. Start: 1970Q2:1979Q4; end: 2006Q4.

Figure 5 :

 5 Figure 5: Impulse responses to a neutral technology shock with increasing start of the sample. Start: 1970Q2:1979Q4; end: 2006Q4.

Figure 6 :

 6 Figure 6: Impulse responses to a negative monetary policy shock with decreasing end of the sample. Start: 1970Q1; end: 1997Q1-2006Q4.

Figure 7 :

 7 Figure 7: Impulse responses to a neutral technology shock with decreasing end of the sample. Start: 1970Q1; end: 1997Q1-2006Q4.

Table 1 :

 1 Unit Root Tests

	Variable			Test-Statistic
		Null: Non-stationarity	Null: Stationarity
		ADF	PP	ERS N-P	KPSS
	Productivity growth	-3.38** -14.52*** 26.75 -0.91	1.22***
	Hours	-2.95*	-2.41	95.31 0.25	1.18***
	Inflation	-0.70	-1.27	21.65 -0.62	1.20***
	Consumption/GDP	-2.65*	-2.69*	6.73 -4.35	0.26
	Investment/GDP	-2.64*	-2.42	10.67 -2.81	0.61**
	Capacity Utilization	-2.78*	-2.89**	17.04 -1.81	0.15
	Productivity/Real Wage	-1.37	-3.23**	29.23 -1.05	0.31
	Interest Rate	-1.82	-1.84	3.65 -7.36	0.78***
	Money Velocity	-1.81	-1.81	70.13 -0.34	0.71**
	*,**,***, denote rejection of the null at the 10%, 5%, 1% significance level, respectively.
	F Variable o r ADF-statistic Break Date Zivot-Andrews Test
	Hours Inflation Investment/GDP Productivity/Real Wage	-3.76 -4.14* P -4.43** e -4.61**	1984Q4 1996Q2 1982Q3 1974Q1
	Interest Rate Money Velocity *,**,***, denote rejection of the null at the 10%, 5%, 1% significance level, respectively. -4.19* 1980Q2 e -3.14 1981Q4 r
			R
			e
				v
				i e w

Table 2 :

 2 Tests of Unit Root vs Broken Trend Stationarity

Table 3 :

 3 Variance decomposition

	o r
	P e
	e
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	R
	e
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Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

The data ser includes all the 12 countries participating in the euro area in 2006.

The behaviour of the constructed series seems reasonable, namely when compared with the one of the ECB's estimate of euro area average hours worked published in the October 2004 Monthly Bulletin. The ECB used annual data from the European Labor Force Survey, which is available only at an annual frequency and for a relatively short time span.

Since the presence of deterministic trends affects the power of the unit root tests, and the presence of unit roots tampers inference on the significance of deterministic trends by altering the sampling distribution of the relevant test-statistics, we also tried to apply the unit root tests after looking for the presence of deterministic trends. Although not reported here, inverting the testing sequence did not alter the outcome regarding the low-frequency properties of all our variables.

The reported tests include a constant, but no time trend.

This problem is well documented in[START_REF] Kwiatkowski | Testing the null hypothesis of stationarity against the alternative of a unit root[END_REF] and[START_REF] Caner | Size distortions of tests of the null hypothesis of stationarity: Evidence and implications for the PPP debate[END_REF].

As suggested by[START_REF] Zivot | Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis[END_REF], the break date is searched between the 15 th and the

th percentile of the sample.

This is so in spite of[START_REF] Galí | On the role of technology shocks as a source of business cycles: Some new evidence[END_REF] using employment rather than hours as the measure of labor input.

Notice that changing the definition of the labor input requires re-specifying productivity and nominal wage variables accordingly.

This latter measure is stationarised using a broken trend with a break date of 1974Q1. As indicated by the Zivot-Andrews test, the resulting variable appears to be stationary.


Maria Rosa Borges
email: mrborges@iseg.utl.pt

Random Walk Tests for the Lisbon Stock Market

Keywords:

come

Introduction

Efficient market theory and the random walk hypothesis have been major issues in financial literature, for the past thirty years. While a random walk does not imply that a market can not be exploited by insider traders, it does imply that excess returns are not obtainable through the use of information contained in the past movement of prices. The validity of the random walk hypothesis has important implications for financial theories and investment strategies, and so this issue is relevant for academicians, investors and regulatory authorities. Academicians seek to understand the behavior of stock prices, and standard risk-return models, such as the capital asset pricing model, depend of the hypotheses of normality or random walk behavior of prices. For investors, trading strategies have to be designed taking into account if the prices are characterized by random walks or by persistence in the short run, and mean reversion in the long run. Finally, if a stock market is not efficient, the pricing mechanism does not ensure the efficient allocation of capital within an economy, with negative effects for the overall economy. Evidence of inefficiency may lead regulatory authorities to take the necessary steps and reforms to correct it.

Since the seminal work of [START_REF] Fama | Efficient Capital Markets: A Review of Theory and Empirical Work[END_REF], several studies show that stock price returns do not follow a random walk and are not normally distributed, including [START_REF] Fama | Permanent and Temporary Components of Stock Prices[END_REF] and [START_REF] Lo | Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test[END_REF], among many others. The globalization markets spawned interest on the study of this issue, with many studies both on individual markets and regional markets, such as Latin America [START_REF] Urrutia | Tests of Random Walk and Market Efficiency for Latin American Emerging Markets[END_REF][START_REF] Grieb | Random Walk Tests for Latin American Equity Indexes and Individual Firms[END_REF], Africa [START_REF] Smith | African Stock Markets: Multiple Variance Ratio Tests of Random Walks[END_REF][START_REF] Magnusson | How Efficient are Africa's Emerging Stock Markets[END_REF], Asia [START_REF] Huang | Do Asian Stock Markets Follow Random Walks? Evidence from the Variance Ratio Test[END_REF]Groenewold and Ariff, 1998), Middle East [START_REF] Abraham | Testing the Random Behavior and Efficiency of the Gulf Stock Markets[END_REF] and Europe [START_REF] Worthington | Random Walks and Market Efficiency in European Equity Markets[END_REF], reporting unconformity with random walk behavior. The list is too extensive for a comprehensive survey, which is beyond the purpose of this study. 3 Previous studies of weak-form efficiency of the Portuguese market include Gama (1998), [START_REF] Dias | Efficiency Tests in the Iberian Stock Markets[END_REF], [START_REF] Smith | Variance Ratio Tests of the Random Walk Hypothesis for European Emerging Stock Markets[END_REF] and [START_REF] Worthington | Random Walks and Market Efficiency in European Equity Markets[END_REF]. Both Gama (1998) and [START_REF] Smith | Variance Ratio Tests of the Random Walk Hypothesis for European Emerging Stock Markets[END_REF] use a variance ratio test and conclude that the Portuguese market was not weak-form efficient until 1998. To our knowledge, the most complete study on Portugal until now is [START_REF] Dias | Efficiency Tests in the Iberian Stock Markets[END_REF] who study daily data of the PSI-20 index from January 1993 to September 2001 and find favorable evidence for a random walk by an augmented Dickey-Fuller test, but find stronger evidence against this hypothesis, using serial correlation and variance ratio tests. [START_REF] Worthington | Random Walks and Market Efficiency in European Equity Markets[END_REF] use more recent data, from August 1995 to May 2003, and mostly find evidence that does not allow the rejection of a random walk, using serial correlation, augmented Dickey-Fuller and variance-ratio tests.

The main contribution of this paper is to add to international evidence on the random walk theory of stock market prices, by testing the Portuguese benchmark index (PSI-20), for the null hypothesis of a random walk. It adds on previous studies for the Portuguese stock market, by demonstrating that the evolution in recent years, until 2006, has been in the direction of increased weak-form efficiency.

Methodology

Serial correlation of returns

An intuitive test of the random walk for an individual time series is to check for serial correlation. If the PSI-20 index returns exhibit a random walk, the returns are uncorrelated at all leads and lags. We perform least square regressions of daily, weekly and monthly returns on lags one to ten of the return series. To test the joint hypothesis that all serial coefficients ( ) t are simultaneously equal to zero, we apply the Box-Pierce Q statistic:

F o r P e e r R e v i e w ( ) = = m t BP t n Q 1 ˆ (1)
where Q BP is asymptotically distributed as a chi-square with m degrees of freedom, n is the number of observations, and m is the maximum lag considered (in this study, m equals ten). We also use a Ljung-Box test, which provides a better fit to the chisquare distribution, for small samples:

( ) ( ) = + = m t LB t n t n n Q 1 2 2
(2)

Runs test

To test for serial independence in the returns we also employ a runs test, which determines whether successive price changes are independent of each other, as should happen under the null hypothesis of a random walk. By observing the number of runs, that is, the successive price changes (or returns) with the same sign, in a sequence of successive price changes (or returns), we can test that null hypothesis.

We consider two approaches: in the first, we define as a positive return (+) any return greater than zero, and a negative return (-) if it is below zero; in the second approach, we classify each return according to its position with respect to the mean return of the period under analysis. In this last approach, we have a positive (+) each time the return is above the mean return and a negative (-) if it is below the mean return. This second approach has the advantage of allowing for and correcting the effect of an eventual time drift in the series of returns. Note that this is a non-parametric test, which does not require the returns to be normally distributed. ( )

1 , 0 N R Z R R = µ (3) where 1 2 + = + n n n R µ and ( ) ( ) 1 2 2 2 = + + n n n n n n n R .

Unit Root Tests

Our third test is the augmented Dickey-Fuller (ADF) test which is used to test the existence of a unit root in the series of price changes in the stock index series, by estimating the following equation through OLS: 6 difference scaled by q to the variance of the first difference tends to equal one, that is, the variance of the q-differences increases linearly in the observation interval, ( ) ( ) ( )

= + + + + = q i it i it i
1 2 2 q q VR = (5)
where ( )

q 2
is q / 1 the variance of the q-differences and ( )

1 2
is the variance of the first differences. Under the null hypothesis ) (q VR must approach unity. The following formulas are taken from [START_REF] Lo | Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test[END_REF], who propose this specification test, for a sample size of 1 + nq observations (

nq P P P ,..., , 1 0 ):
( ) ( )

2 2 1 = = nq q t q t t q P P m q µ (6)
where ( )

+ = nq q q nq q m 1 1 and µ ˆ
is the sample mean of ( ) [START_REF] Lo | Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test[END_REF] generate the asymptotic distribution of the estimated variance ratios and propose two test statistics, ( ) q Z and ( ) q Z * , under the null hypothesis of homoskedastic increments random walk and heteroskedastic increments random walk respectively. If the null hypothesis is true, the associated test statistic has an asymptotic standard normal distribution. Assuming homoskedastic increments, we have

1
( ) ( ) ( ) ( ) 1 , 0 1 N q q VR q Z o = (8)
where

( ) ( )( ) ( ) 2 1 3 1 1 2 2 = nq q q q q o . Assuming heteroskedastic increments, the test statistic is F o r P e e r R e v i e w ( ) ( ) ( ) ( ) 1 , 0 1 * N q q VR q Z e = (9)
where ( )

2 1 1 1 1 4 = = q t t e q t q and ( )( ) ( ) 2 1 2 1 1 2 1 2 1 ˆˆ = = + = nq j j j nq t j t j t j j j t P P P P P P µ µ µ .
which is robust under heteroskedasticity, hence can be used for a longer time series analysis. The procedure proposed by [START_REF] Lo | Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test[END_REF] is devised to test individual variance ratio tests for a specific q-difference, but under the random walk hypothesis, we must have

( ) 1 = q VR
for all q. A multiple variance ratio test is proposed by Chow and Denning (1993). Consider a set of m variance ratio tests

( ) { } m i q M i r ,..., 2 , 1 = where ( ) ( ) 1 = q VR q M r
, associated with the set of aggregation intervals { }

m i q i ,..., 2 , 1 =
. Under the random walk hypothesis, there are multiple subhypotheses:

( ) 0 : 0 = i r i q M H for m i ,..., 2 , 1 = ( ) 0 : 1 i r i q M H for any m i ,..., 2 , 1 =
The rejection of any or more i H 0 rejects the random walk null hypothesis. In order to facilitate comparison of this study with previous research [START_REF] Lo | Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test[END_REF][START_REF] Lo | Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test[END_REF]Campbell et al. 1997) on other markets, the q is selected as 2, 4, 8, and 16.

For a set of test statistics ( )

{ } m i q Z i ,..., 2 , 1 =
, the random walk hypothesis is rejected if any one of the

( ) i q VR
is significantly different than one, so only the maximum absolute value in the set of test statistics is considered. The Chow and Denning (1993) multiple variance ratio test is based on the result: . Chow and Denning (1993) control the size of the multiple variance ratio test by comparing the calculated values of the standardized test statistics, either ( )

( ) ( ) ( ) ( ) { } 1 ; ; ,..., max 1 T m SMM q Z q Z PR m (10)
q Z or ( ) q Z *
with the SMM critical values.

If the maximum absolute value of , say, ( )

q Z
is greater than the critical value at a predetermined significance level then the random walk hypothesis is rejected.

The Data

Our testing of periods has also the advantage of allowing for structural changes, so that the market may follow a random walk in some of the periods while in other periods that hypothesis may be rejected. A similar approach of arbitrarily-chosen periods is taken by [START_REF] Wheeler | The Efficiency of the Warsaw Stock Exchange Stock Exchange: The First Few Years 1991-1996[END_REF] in their analysis of the Warsaw Stock Exchange.

Roughly, each of the periods has duration from one and a half years to four years.

We are particularly interested in period 5, from March 2003 to December 2006, both because it has not been covered by previous studies, and because it is "now".

INSERT FIGURE 1

Non-trading is not a problem for the statistical tests since all the companies included in the index are only very rarely not traded on any given day, and the index is bound to fluctuate on every trading day. We use the daily closing prices to compute also weekly and monthly data. The weekly price series is constructed with the closing price on Wednesdays, to minimize day-of-the-week effects. 10 computed as the logarithmic difference between two consecutive prices in a series.

Table 1 shows the descriptive statistics for the returns of the PSI-20 stock index.

INSERT TABLE 1 The mean returns in the five periods are very different, reflecting the visual criteria used to define those periods. The returns are negatively skewed in almost all periods, and for daily, weekly and monthly data, which means that large negative returns tend to be larger than the higher positive returns. The level of kurtosis is high in the whole sample, but with a tendency to decrease in the later periods. The Jarque-Bera statistic rejects the hypothesis of a normal distribution of returns in all periods and types of data, at a significance level of 1%. The distribution of returns is in fact leptokurtic, as can be confirmed visualy in Figure 2, although it has been approaching the normal distribution in the most recent periods.

INSERT FIGURE 2

Results

Serial Correlation

The results for the tests on serial correlation, Box-Pierce and Ljung-Box statistics are presented in Table 2, for daily, weekly and monthly returns.

INSERT TABLE 2

The daily returns exhibit serial correlation at a significance level of 1% for the total sample and for all the periods, except in period 2, where the B-P and L-B of the lagged variables are significant in one or another period, but the evidence is stronger for lag 4 and, specifically, for lag 1. The regressions strongly prove that the daily return of day t is positively correlated with the return of day t-1, with a coefficient of around 0.17 for the whole sample 1993 to 2006. One important note is that all the significant coefficients, in all regressions, have a positive sign, thus adding to the global evidence of positive correlation of returns. However, the positive correlation of lag 1 in daily returns has decreased in period 4, and then again in period 5, which may be interpreted has a smaller deviation from the independence of returns inherent in the random walk hypothesis.

The evidence of serial correlation decays as the lag length increases, as it is milder for the weekly data, and for monthly data the overall serial correlation of returns is not significant. This means that the larger the interval of the observations of prices, the less important is the lagged price for explaining future prices. This is consistent with the findings of several other studies including Fama (1965), [START_REF] Panas | The Behaviour of Athens Stock Prices[END_REF] and [START_REF] Ma | Are China's Stock Markets Really Weak-form Efficient?[END_REF]. Lastly, we should be cautious in the interpretation of these results, as they assume normality, which we have shown that is not a valid assumption for the distribution of daily returns of the PSI-20 index, in the period 1993 to 2006.

Runs Test

The results of the runs test, which do not depend on normality of returns, are presented in Table 3, for daily, weekly and monthly returns.

INSERT The number of runs is always less than the expected number of runs, for daily, weekly and monthly data, and for all periods, in line with findings of several international studies [START_REF] Worthington | Random Walks and Market Efficiency in European Equity Markets[END_REF]Higgs 2004, Abraham et al. 2002). This difference is significant at the 1% level for the daily data, for periods 1 to 3 (January 1993 to March 2000). In periods 4 and 5 (March 2000 to December 2006), the number of runs is not statistically different from the expected number of runs, which is consistent with a random walk. The low number of runs in the weekly and monthly returns also refutes the random walk hypothesis, except in the periods 4 and 5, for the weekly data.

Unit Root Tests

In our third test we compute the Augmented Dickey-Fuller statistic to test the null hypothesis of a unit-root in the PSI-20 index prices. We show results is Table 4.

INSERT TABLE 4

The number of lagged variables was determined by the Akaike Info Criterion, from a maximum of 10 lags allowed. The results are very clearly in favor of the random walk hypothesis, as the null hypothesis of a unit-root is not rejected for any type of returns (daily, weekly, monthly) or any period. Again, this evidence is consistent with similar findings for the Portuguese stock market, by [START_REF] Dias | Efficiency Tests in the Iberian Stock Markets[END_REF] and [START_REF] Worthington | Random Walks and Market Efficiency in European Equity Markets[END_REF]. In any case we have to be cautious about these results, as [START_REF] Liu | A Variance-Ratio Test of Random Walks in Foreign Exchange Rates[END_REF] show that unit root tests may not detect departures from a random walk. Variance Ratio Tests [START_REF] Lo | Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test[END_REF] show that the variance ratio test is more powerful than the Dickey-Fuller unit root test, and Ayadi and Pyun (1994) also argue that the variance ratio has more appealing features than other procedures. Table 5 presents the results of the variance ratios tests for PSI-20 stock index prices. In order to facilitate comparisons with other studies, we adopt the common procedure of selecting lags 2, 4, 8 and 16.

INSERT TABLE 5

Except for the daily data in period 4, all variance ratios are larger than unity, which indicates that the variances grow more than proportionally with time. This could be due to heteroskedasticity of stock index prices in some cases, but the ( ) q Z * statistic also shows robust results in some of the cases, which is additional proof of autocorrelation in the data. This is consistent with the results of Table 1. The hypothesis of a random walk is rejected by the sample of daily prices for the whole period 1993 to 2006, and the same evidence also applies for the period 1, period 3 and period 5 sub-samples. Period 2 provides mixed evidence, under the assumption of heteroskedasticity which we deem more appropriate, as the variance ratio tests are significant at the 5% level, but the Chow-Denning does not allow the rejection of the null hypothesis of a random walk. In period 4, the random walk is not rejected by any of the tests, and the same is true for periods 4 and 5 together, that is, from March 6 summarizes the results of all the tests performed.

INSERT TABLE 6

Apart from the ADF test, which is very clearly favorable to the random walk hypothesis, all other tests provide mixed evidence. Serial correlation is strong in -1993- 01-01-1993- 02-01-1997- 23-04-1998- 13-03-2000- 01-10-2002- 01-01-1993- 01-01-1993- End 31-12-2006- 31-12-1996- 22-04-1998- 10-03-2000- 30-09-2002- 31-12-2006- 31-12-2006 Notes: Variance ratio tests for daily, weekly and monthly PSI-20 index prices. The variance ratios, VR(q), are reported in the first rows, and the variance-ratio test statistics, Z(q) for homoskedastic increments and Z * (q) for heteroskedastic increments, are reported in parentheses. The null hypothesis is that the variance ratios equal one, which means that the stock index prices follow a random walk. We also show the Chow and Denning (1993) statistic, which tests all the Z * (q) together, * Null hypothesis rejection significant at the 5% level. ** Null hypothesis rejection significant at the 1% level.

F

  distribution with parameters m and T (sample size) degrees of freedom.

  data are daily closing values of the PSI-20 index, which is the Portuguese benchmark index, a cap weighted index reflecting the evolution of the prices of the 20 largest and most liquid shares selected from the universe of companies listed on the Portuguese Main Market. The PSI-20 also serves the purpose of acting as the underlying for futures and options contracts and other index linked products. The source of all data is Reuters, and it includes observations from 1 January 1993 to 31 December 2006, during which the index has shown significant fluctuations, as shown in Figure 1.We apply the tests to the whole sample, but also separately to five periods which are defined by different trends in the market index. In period 1 (from 1-Jan-1993 to 31-Dec-1996) the index showed a trend of slow growth, which accelerated in the period 2 (from 2-Jan-1997 to 22-Apr-1998) reaching a peak in this last day. In period 3(from 23-Apr-1998 to 10-Mar-2000) the index first declined sharply, and then grew very strongly reaching an all-time peak on 10-Mar-2000. Period 4 (from -2000 to 30-Sep-2002) was a depressive period for the market, with a steady trend of descent. In the final period(from 01-Oct-2002 to 29-Dec-2006) the market again recovered a steady growth trend. It is important to clarify that this sub-periods are not defined in terms of any institutional change, and do not reflect any statistical criteria; it is a naïve criterion reflecting only visual trend changes of the market. The

  significant to reject the null hypothesis of zero serial correlation. Some

  2000 to December 2006. With weekly data, most of the individual tests reject the null, but the global test provided by the Chow-Denning statistic does not reject a random walk. Monthly data does not reject a random walk behavior.

  FIGURE 1 Lisbon Stock Market PSI 20 Index -Closing Prices

  If theWednesday observation is not available, due to market closing, we use the Tuesday observation, and if that is also not available, we use Thursday. For the monthly price series, we use the observations of day 15 of each month. In case of a missing observation on day 15, we use day 14. If day 14 is missing, we use day 16. If day 16 is missing we use day 13, and so on. From the sample of 3490 daily observations, we generate 730 weekly observations and 168 monthly observations. The returns are

	F o r
	P
	e
	e r
	R e v
	i e
	w

TABLE 3

 3

	F o r
	P
	e
	e r
	R e v
	i e
	w
	12

  Bera test is a goodness-of-fit measure of departure from normality, based on the sample kurtosis and skewness, and is distributed as a chi-squared with two degrees of freedom. The null hypothesis is a joint hypothesis of both the skewness and excess kurtosis being 0, since samples from a normal distribution have an expected skewness of 0 and an expected excess kurtosis of 0. As the definition of JB shows, any deviation from this increases the JB statistic. * Null hypothesis rejection significant at the 5% level. ** Null hypothesis rejection significant at the 1% level.

									31-12-2006
	Observations	3489	989	322	467	626	1085	729	167
	Mean return	0.0004	0.0005	0.0032	0.0001	-0.0017	0.0007	0.0018	0.0080
	Annualised return	0.0990	0.1461	1.2074	0.0152	-0.3447	0.1982	0.5738	6.3018
	Maximum	0.0694	0.0327	0.0694	0.0540	0.0430	0.0384	0.1212	0.1678
	Minimum	-0.0959	-0.0706	-0.0640	-0.0959	-0.0457	-0.0355	-0.1132	-0.2040
	St. Deviation	0.0099	0.0068	0.0115	0.0153	0.0121	0.0067	0.0251	0.0569
	Skewness	-0.6266	-1.0525	0.0319	-0.8105	-0.2077	-0.0146	-0.2868	-0.3348
	Kurtosis	11.0192	17.3398	10.2941	7.9883	4.1061	6.0520	5.8571	4.1283
	Jarque-Bera	9576.9**	8656.3**	713.9**	535.3**	36.4**	421.2**	257.9**	12.0**
	JB p-value	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0025
	Notes: The Jarque-F o r
			P
			e
			e r
				R e v
					i e
					w

TABLE 2 Serial Correlation Coefficients and Q-statistics for Returns of the PSI-20 stock index: January 1993 to December 2006

 2 Both the Box-Pierce statistic and the Ljung-Box statistic test the null hypothesis of overall zero serial correlation coefficients for lags 1 through 10, and are distributed as a chi-square distribution with ten degrees of freedom. For small samples, the Ljung-Box statistic provides a finite-sample correction that yields a better fit to the chi-square distribution. * Null hypothesis rejection significant at the 5% level. ** Null hypothesis rejection significant at the 1% level.

		Daily	Daily (Period 1)	Daily (Period 2)	Daily (Period 3)	Daily (Period 4)	Daily (Period 5)	Daily and 5) (Periods 4	Weekly	Monthly
	Observations	3479	979	312	457	616	1075	1701	719	157
	Lag 1	0.1694**	0.2807**	0.1591**	0.2789**	0.0965*	0.0552	0,0766**	0.0695	0.2143*
	Lag 2	-0.0231	0.0380	-0.0413	-0.0578	-0.0915*	0.0283	-0,0361	0.0784*	0.0050
	Lag 3	0.0223	-0.0823*	0.0738	-0.0653	0.1338**	-0.0261	0,0821**	0.0322	0.0693
	Lag 4	0.0496**	0.0526	-0.0491	0.1558**	-0.0041	0.0982**	0,0465	0.0634	-0.0329
	Lag 5	0.0009	0.0027	0.0314	-0.0599	-0.0109	-0.0131	-0,0064	0.0199	-0.0404
	Lag 6	-0.0249	-0.0347	-0.0415	-0.0108	-0.0814*	-0.0377	-0,0505*	0.0029	0.0067
	Lag 7	0.0294	0.0895**	-0.0596	0.0320	0.0094	0.0607*	0,0287	0.0173	0.0889
	Lag 8	0.0400*	-0.0349	0.0549	-0.0039	0.0649	0.0213	0,0635**	0.0051	-0.0443
	Lag 9	-0.0288	0.0428	0.0196	-0.0938*	0.0127	-0.0394	-0,0029	0.0592	0.0478
	Lag 10	0.0249	0.0704*	-0.0133	0.0350	-0.0155	0.0551	0,0149	-0.0930*	0.0476
	Box-Pierce Stat.	127.769** 104.770**	13.952	56.875**	28.977**	26.320**	40,427**	20.780*	10.661
	p-value	0.0000	0.0000	0.1752	0.0000	0.0013	0.0033	0,0000	0.0227	0.3845
	Ljung-Box Stat.	127.920** 105.238**	14.168	57.441**	29.231**	26.498**	40,568**	20.999*	10.994
	p-value	0.0000	0.0000	0.1655	0.0000	0.0011	0.0031	0,0000	0.0211	0.3580
	Notes:

TABLE 3 Runs Tests for Daily, Weekly and Monthly Returns of the PSI-20 stock index: January 1993 to December 2006

 3

		Daily	Daily (period 1)	Daily (period 2)	Daily (period 3)	Daily (period 4)	Daily (period 5)	Daily (periods 4 and 5)	Weekly	Weekly (periods 4 and 5)	Monthly	Monthly (periods 4 and 5)
	Panel A: positive/negative returns defined relative to zero
	n+	1836	526	208	226	272	602	875	415	195	96	43
	n-	1653	463	113	240	353	482	835	314	159	71	38
	R R µ	1546 1740.7	413 493.5	117 147.4	199 233.8	292 308.3	525 536.4	817 855.5	324 358.5	162 176.2	59 82.6	28 41.3
	R	29.448	15.652	8.158	10.772	12.280	1 6.253	20.659	13.231	9.297	6.297	4.455
	Z	-6.6116** -5.1426** -3.7314** -3.2296**	-1.3234	-0.6988	-1.8652	-2.6077**	-1.5241	-3.7526** -2.9960**
	p-value	0.0000	0.0000	0.0002	0.0012	0.1857	0.4847	0.0622	0.0091	0.1275	0.0002	0.0027
	Panel B:

positive/negative returns defined relative to the mean return n+

  The runs test tests for a statistically significant difference between the expected number of runs vs. the actual number of runs. A run is defined as sequence of sucessive price changes with the same sign. The null hypothesis is that the successive price changes are independent and random. In Panel A, we define as a positive/negative return any return above/below zero. In Panel B, we define as a positive/negative return any return above/below the mean return. * Null hypothesis rejection significant at the 5% level. ** Null hypothesis rejection significant at the 1% level.

		1753	482	162	226	312	546	887	386	202	84	45
	n-	1736	507	159	240	313	538	823	343	152	83	36
	R R µ R Z p-value Notes: F 1546 413 1745.5 495.2 29.529 15.706 -6.7547** -5.2326** -4.9741** -3.2296** 117 199 161.5 233.8 8.943 10.772 0.0000 0.0000 0.0000 0.0012 o r	292 313.5 12.490 -1.7213 0.0852	525 543.0 16.454 -1.0922 0.2747	817 854.8 20.641 -1.8314 0.0670	324 364.2 13.444 -2.9926** 0.0028	162 174.5 9.206 -1.3544 0.1756	59 84.5 6.442 -3.9581** -2.9439** 28 41.0 4.416 0.0001 0.0032
				P
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					e r
						R e v
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												22

TABLE 4 Augmented Dickey-Fuller tests for the PSI-20 Stock index: January 1993 to December 2006

 4 Fuller statistics test the null hypothesis of a unit root in the stock price series. Failure to reject the null hypothesis means that the random walk hypothesis is not rejected. The number of lags included in the regression is determined by the Akaike Info Criterion. * Null hypothesis rejection significant at the 5% level. ** Null hypothesis rejection significant at the 1% level.

		Daily	Daily (Period 1)	Daily (Period 2)	Daily (Period 3)	Daily (Period 4)	Daily (Period 5)	Weekly	Monthly
	ADF test statistic	-1,3995	-2,0824	1,3256	-0,0597	-2,8460	-1,6406	-1,5521	-1,7205
	p-value	0,9990	0,5545	1,0000	0,9954	0,1814	0,7764	0,8107	0,7380
	Included observations	3485	979	319	457	616	1074	727	166
	Number of lags	4	10	2	9	9	10	2	1
	Notes: Augmented Dickey-

TABLE 5 Variance Ratio Tests for Lags 2, 4, 8 and 16 for Price Increments of the PSI-20 stock index: January 1993 to December 2006

 5

									Lag 2	Lag 4	Lag 8	Lag 16	Chow-Denning
	Daily	VR	(q	)	1,148	1,224	1,346	1,538
		Z	(q	)	(8,733)**	(7,066)**	(6,905)**	(7,219)**
		Z	( * q		)	(3,991)**	(3,329)**	(3,546)**	(3,628)**	(3,991)**
	Daily (Period 1)	VR	(q	)	1,266	1,495	1,682	2,086
		Z	(q	)	(8,375)**	(8,320)**	(7,251)**	(7,761)**
		Z	( * q		)	(5,081)**	(5,035)**	(4,643)**	(5,173)**	(5,173)**
	Daily (Period 2)	VR	(q	)	1,200	1,291	1,436	1,747
		Z	(q	)	(3,587)**	(2,786)**	(2,642)**	(3,042)**
		Z	( * q		)	(1,724)*	(1,492)	(1,702)*	(2,219)*	(2,219)
	Daily (Period 3)	VR	(q	)	1,194	1,258	1,381	1,478
		Z	(q	)	(4,197)**	(2,977)**	(2,781)**	(2,343)**
		Z	( * q		)	(2,957)**	(2,212)*	(2,284)*	(1,899)*	(2,957)*
	Daily (Period 4)	VR	(q	)	1,034	0,959	0,934	0,912
		Z	(q	)	(0,843)	(0,553)	(0,556)	(0,501)
		Z	( * q	)	(0,631)	(0,390)	(0,404)	(0,349)	(0,631)
	Daily (Period 5)	VR	(q	)	1,050	1,130	1,273	1,474
		Z	(q	)	(1,656)*	(2,282)*	(3,034)**	(3,546)**
		Z	( * q		)	(1,410)	(1,896)*	(2,658)**	(3,068)**	(3,068)**
	Daily (Periods 4 and 5)	VR	(q	)	1,051	1,036	1,095	1,212
		Z	(q	)	(2,110) *	(0,790)	(1,333)	(1,988) *
		Z	( * q		)	(1,218)	(0,431)	(0,745)	(1,069)	(1,218)
	Weekly	VR	(q	)	1,066	1,225	1,455	1,654
		Z	(q	)	(1,792)*	(3,250)**	(4,152)**	(4,009)**
		Z	( * q		)	(1,084)	(1,894)*	(2,452)**	(2,275)*	(2,452)
	Monthly	VR	(q	)	1,226	1,398	1,458	1,807
		Z	(q	)	(2,921)**	(2,748)**	(2,001)*	(2,368)**
		Z	( * q		)	(1,551)	(1,515)	(1,206)	(1,506)	(1,551)

TABLE 6 Summary of Test Results: Random Walk Hypothesis Rejected?

 6

										Page
	Test	Daily	Daily (Period 1)	Daily (Period 2)	Daily (Period 3)	Daily (Period 4)	Daily (Period 5)	Daily and 5) (Periods 4	Weekly	Monthly
	Serial Correlation Tests
	Lag 1 return significant	YES	YES	YES	YES	YES	NO	YES	NO	YES
	B-P and L-B Statistic	YES	YES	NO	YES	YES	YES	YES	YES	NO
	Runs Test	YES	YES	YES	YES	NO	NO	NO	YES	YES
	Augmented Dickey-Fuller	NO	NO	NO	NO	NO	NO	NO	NO	NO
	Variance Ratio Test	YES	YES	NO	YES	NO	YES	NO	NO	NO
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24 of 24 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript