Peter Egger

Stephan Gruber

Michael Pfaffermayr
email: michael.pfaffermayr@uibk.ac.at.

Conditional Convergence Evidence from Disaggregated Data

Keywords: Purchasing Power Parity, Sigma-Convergence, Beta-Convergence JEL classification: F31, E31, C21 Purchasing power parity, σ-convergence, β-convergence

This paper contributes to the literature on the Law of One Price (LOP) and absolute Purchasing Power Parity (PPP) in two ways. First, it uses a novel set of PPP data from the International Comparison Programme for OECD countries and 195 internationally comparable products from 1980 to 1996. Second, it derives and applies a test of conditional σ-convergence, which does not require long time spans or high frequency data. Between 1990 and 1996 for 10 out of 23 countries a significant reduction in the variance of the deviations from LOP is found for tradeables, but none in case of non-tradeables. For the former, the deviations from LOP close out at half-lives between 2.2 and 6.3 years. However, there are also persistent country-specific deviations from LOP parities.

Introduction

The overwhelming majority of the empirical literature on the Law of One Price (LOP) and Purchasing Power Parity (PPP) is based on time series data of national consumer prices. 1 "That is, because most data take the form of index numbers, most of what we know about LOP and PPP deviations involves the volatility and persistence of changes in relative prices.

We know relatively little about the absolute relative prices themselves. This is particularly troublesome given that economic theory places much starker restrictions on absolute LOP deviations than on their changes." (Crucini, Telmer, Zachariadis, 2005, p.

724). 2

A considerable part of the empirical LOP research concentrates on individual products. Cumby (1996) studies prices of Big Mac hamburgers, [START_REF] Gosh | Pricing in International Markets: Lessons from the Economist[END_REF] investigate the price of The Economist magazine, [START_REF] Haskel | The Law of One Price -A Case Study[END_REF] look at the prices of IKEA furniture, [START_REF] Froot | The Law of One Price over 700 Years[END_REF] concentrate on basic commodities like wheat, butter and charcoal, and [START_REF] Lutz | Pricing in Segmented Markets, Abritrage Barriers and the Law of One Price: Evidence from the European Car Market[END_REF] assesses automobile prices. 3 Most of the available LOP-studies use price data of cities in North America or the European Union. Only recently, serious efforts have been undertaken to reinvestigate LOP consider-1 The Law of One Price (LOP) states that identical goods sold in different countries and/or locations should exhibit the same price expressed in common currency units, while Purchasing Power Parity (PPP) implies that similar baskets of goods should be identically priced once expressed in common units.

2 In particular, the analysis of changes in relative prices (relative PPP) is not sufficient to show that international price differences cancel out over time in levels (absolute PPP). Here, we concentrate on LOP, i.e. absolute PPP. Recent contributions to relative PPP literature are [START_REF] Obstfeld | Foundations of International Macroeconomics[END_REF], O'Connel (1998aO'Connel ( , 1998b)), [START_REF] Cheung | On the Purchasing Power Parity Puzzle[END_REF]Lai (2000a, 2000b), [START_REF] Higgins | Purchasing Power Parity: Three Stakes Through the Heart of the Unit Root Null[END_REF], [START_REF] Baum | Nonlinear Adjustment to Purchasing Power Parity in the Post-Bretton Woods Era[END_REF], Taylor and Peel (2000), Taylor (2001Taylor ( , 2002)), Murray and Pappell (2002), Taylor and Taylor (2004).

3 See also [START_REF] Isard | How far can we push the Law of One Price?[END_REF], [START_REF] Giovannini | Exchange Rates and Traded Goods Prices[END_REF], and [START_REF] Knetter | International Comparison of Pricing to Market Behavior[END_REF][START_REF] Knetter | International Comparison of Pricing to Market Behavior[END_REF] for early but influential contributions. support of national statistical agencies, EUROSTAT, and the OECD at a three to five years frequency (also [START_REF] Bergstrand | Structural Determinants of Real Exchange Rates and National Price Levels[END_REF]Bergstrand, , 1992, recommends the use of ICP data). This paper represents a first attempt to exploit the ICP data for OECD countries at the disaggregated level of 195 internationally comparable products (basic headings) to study deviations from LOP.

Second, by relying on dynamic panel data methods and panel unit root tests, recent research on PPP implicitly assumes that the data generating process of prices in common currency already runs for an infinite number of periods. Although this is a possible working hypothesis, political shocks such as World War II or the oil crises in the late 70s and early 80s may represent distortions, which are at odds with this assumption. Following [START_REF] Hart | Galtonian Regression across Countries and the Convergence of Productivity[END_REF], we argue that looking at β-convergence, i.e., the absence of a unit root, is not sufficient to detect σ-convergence so that the distribution of real exchange rates indeed collapses over time. We may observe a rising variance (i.e., no σ-convergence), in spite of β-convergence. This is the case if the variance in the initial period is above its long run steady state level as implied by the corresponding stationary AR(1)-process and if the variance of the innovations is large, too. Hence, in this likely relevant case we need a test of σ-convergence in addition to estimates of the β-convergence coefficient.

Only if the null of no σ-convergence is rejected, the half-lives based on the Since our ICP data form a large cross-section (195 products in 23 countries), but exhibit a small time dimension (only 5 observations between 1980 and 1996 are available), we look at both β-and σ-convergence over a given period of time without relying on large T asymptotics as unit root tests do.

Accordingly, we test whether the variance of the real exchange rate has declined over time (see [START_REF] Hart | Galtonian Regression across Countries and the Convergence of Productivity[END_REF]. To accomplish this task, we design and apply a likelihood ratio test of conditional σ-convergence. This test is a generalization of the test of unconditional σ-convergence by [START_REF] Carree | Testing the Convergence Hypothesis: A Comment[END_REF]. 4 In addition, we estimate the associated speed of adjustment from the β-convergence regressions.

Summing up the main results, the ICP data provide clear evidence that international goods prices expressed in common currency units deviate remarkably from LOP parities at any considered period. Concerning σ-convergence we find that over the period 1990-1996 10 out of 23 countries experienced a significant reduction in the variance of the deviations from LOP for tradeable goods, but no reduction in case of non-tradeables. For the longer period between 1980 and 1996 10 out of 16 countries show a significant decrease of the variance of LOP deviations for tradeable goods. The corresponding β-convergence estimates exhibit half-lives for LOP deviations of between 3.4 and 8.1 years over the period 1990-1996, while the adjustment towards parity occurs faster (2.2 to 6.3 years) for tradeable goods at competitive markets.

Over the longer period, our half-live estimates are somewhat bigger, hence, conditional convergence is slower. The OECD provided the authors with the PPP-data at the basic heading level covering the comparison rounds 1980, 1985, 1990, 1993 and 1996 for 24 OECD countries from the joint OECD/EUROSTAT PPP Programme. 6

Official annual average US-Dollar nominal exchange rates (s) published by the International Monetary Fund have been used to calculate real exchange rates (r) for each price observation and each comparison year (t). Real exchange rates at the basic heading level are defined as: r cit =s ct +p cit -p U Sit , where all variables are in logs, c denotes the country index and i is the basic heading index.

Furthermore we have constructed a dummy variable to classify each basic heading as a "tradeable"(tr) or "non-tradeable"(ntr) good. Those basic headings that are tradeable in principle, but not traded at competitive market prices (e.g., town gas, books, etc.) have been classified as non-tradeable. Details about this classification and the data set are available from the authors upon request. 6 The country sample consists of Germany, France, Italy, Netherlands, Belgium, Luxembourg, United Kingdom, Ireland, Denmark, Greece, Spain, Portugal, Austria, Switzerland, Finland, Iceland, Norway, Sweden, Turkey, Australia, New Zealand, Japan, Canada, USA. Note, due to small changes in the classification, a few basic headings are not available in all of the rounds. The time series dimension of the panel data set consists of three to five observations over time for each country and basic heading. So we are faced with a fat panel with N large and T small, and panel unit root tests are not applicable in the present context. Furthermore, if the data generating process exhibits fixed starting values, the absence of a unit root is not sufficient to detect convergence in the sense that the distribution of relative prices collapses in the course of time. As [START_REF] Hart | Galtonian Regression across Countries and the Convergence of Productivity[END_REF] illustrates, it may well be that relative prices are stationary and follow an AR(1) process. However, their variance only shrinks over time (i) if the initial variance is sufficiently large, (ii) there is β-convergence and (iii) the yearly innovations exhibit a low enough variance. Therefore, it is possible to observe a constant variance along with β-convergence. In short, in order to detect convergence in the sense of a collapsing (unimodal) distribution, β-convergence is necessary but not sufficient (see also [START_REF] Barro | Economic Growth[END_REF]. To see this, define π = 1+b and write the β-convergence equation as (ignoring the constant, the dummies and the index for countries and basic headings for the moment):

y T = (1 + b) T -1 y 1 + T t=2 (1 + b) T -t ν t = πy 1 + u T (1) so that σ 2 T = π 2 σ 2 1 + σ 2 uT (2)
We assume that ν is iid N (0, σ 2 ν ) so that u T is also iid normal with variance σ 2 uT and y 1 ∼ iid N (0, σ 2 y 1 ). Hence, for fixed T we have a unique mapping of the time series model into the cross-sectional model. From (2) it can immediately be seen that we observe σ-convergence between periods 1 and 7

F o r P e e r R e v i e w T , if σ T < σ 1 or π 2 < π 2 0 = σ 2 1 -σ 2 u σ 2 1
. The latter term (π 0 ) is always lower than 1, if σ-convergence takes place and the process starts at a variance higher than the long run steady state value. 7 Therefore, finding β-convergence with π < 1 is necessary but not sufficient for σ-convergence, since under π 0 < π < 1 the variance of y actually increases. In the Appendix, we propose a likelihood ratio test to test this hypothesis. This test is based on the work of [START_REF] Carree | Testing the Convergence Hypothesis: A Comment[END_REF] and Egger and Pfaffermayr (2006), and allows for conditional convergence by introducing additional dummies. Specifically, we test the hypothesis

H 0 : σ 2 T = σ 2 1 vs. H 1 : σ 2 T < σ 2 1 .
We estimate the β-convergence regressions first for each country separately and then for groups of countries.

The country-specific specification is given by

r ciT = α c + β c r ci1 + µ 1c d i + µ 2c d i r ci1 + ε ciT (3)
where r cit , t = 1, T denotes the real exchange rate of country c and basic heading i in period t. d i is our dummy variable for the tradeability of goods.

The interaction term d i r ci1 captures a possible difference in the speed of convergence between these two groups of goods. Lastly, a constant which is different from zero indicates permanent LOP deviations. The implied halflives can be easily estimated by t i0.5 = -T ln(2)/ln(1

+ β c + µ 2c ).
The second specification pools over country groups. It introduces countryspecific effects, which capture location-specific determinants of the deviations from LOP.

r ciT = α c + β r ci1 + µ 1 d i + µ 2 d i r ci1 + ε ciT (4)
In both cases, we assume that the starting value is independent of the 7 For T approaching infinity we have

σ T → σν -2b-b 2 for b < 0, so that we observe σ-convergence if σ 2 1 > σν -2b-b 2 . 8 F o r P e e r R e v i e w
error in period t > 1 so that r ci1 is exogenous. In case of an AR(1) process in annual time intervals σ u comprises the error terms from period T to 2, while the initial value has date 1. 8 Hence, if the underlying AR(1)-process exhibits white noise errors, exogeneity of r ci1 is guaranteed.

At the country level we look at σ-convergence for tradeables and nontradeables separately, so the test is on convergence to the corresponding steady states. The test for country aggregates is on conditional σ-convergence,

i.e it allows for conditioning on exogenous dummy-variables. In both cases, the test is a likelihood ratio test. We show in the Appendix, that the corresponding test statistic is easy to calculate and it is distributed as χ 2 (1).

LR = -2 (L 1 -L 0 ) = N K ln σ 2 1,0 σ 2 u,0 σ 2 1 σ 2 u . ( 5
)
The estimators of the variances under H 0 , σ 2 1,0 and σ 2 u,0 , as well as those under H 1 , σ 2 1 and σ 2 u , are defined in the Appendix.

Estimation results

A first inspection of the ICP data reveals undoubted evidence for the fact that real exchange rates implied by disaggregated international goods prices deviate widely from LOP parities. Table 1 shows The evidence on σ-convergence is summarized in Tables 234. Between 1990 and 1996, there is σ-convergence in 10 countries in case of tradable goods, but none in case of non-tradeables. Specifically, the integration process in the 90s seems to have induced convergence in prices in Europe. In the long period 1980 to 1996, a significant reduction of the variance of LOP deviations occurs in 9 out of 16 countries for tradeable goods. For 7 countries the H 0 of no σ-convergence is rejected also for the non-tradeables. For all countries together as well as for the EU12-countries and the EFTA-economies the test likewise rejects for both groups of goods for the longer period 1980-1996 as well as for the period 1990-1996. Hence, one can conclude that there is some tendency of convergence in absolute prices, particularly in the case of tradeables.

The findings concerning σ-convergence merit a closer look at the βconvergence regressions. Tables 5-7 report the country-specific estimates.

Generally, we observe much bigger half-lives for non-tradeables lasting between 3.4 and 8.1 years for the period 1990-1996, while for tradeables these figures are between 2.2 and 6.3 years. For the longer period 1980-1996 the estimates for the half-lives are relatively bigger. To some extent, this indicates that new shocks have occurred within this longer period. The slower convergence in the 80s may be an additional reason for this finding (see Ta- estimates for the country groups in Table 8-10 confirm these findings. Over the period 1990-1996 the half-live of the pooled estimation over all countries is 4.0 years for tradeable goods and 5.6 years for non-tradeables, respectively.

Again the corresponding figures for the period 1980-1996 are higher.

For the interpretation of these findings, one has to bear in mind that we allowed for country-specific intercepts (i.e., country-specific deviations from LOP). These persistent deviations are in sharp contrast with the theoretical propositions behind LOP and absolute PPP. Yet they are significant in many empirical LOP-studies. 9 Obviously, country-specific influences such as national tax law and transportation costs may cause permanent differences in international relative price levels, which can not be arbitraged away easily by international goods trade. 10 Our econometric estimates illustrate that it is important to control for these country-specific influences.

Conclusions

This paper uses disaggregated price data collected within the framework of the International Comparison Programme (ICP), which provides a very comprehensive and advanced statistical database on a large set of internationally 9 See Crucini and Shintani (2004, p.16): "In any case it seems fair to say that the absolute convergence hypothesis is flagrantly violated across international cities and at best weakly supported by the U.S. data. ... Our results using absolute price data indicate that the resting point of most Law-of-One-Price deviations is not zero as implied by the theoretical proposition, even across cities within countries." 10 Engel and Rogers (1996, p.1114) mention several reasons as to why deviations from absolute PPP might occur: (i) geographically segmented markets combined with market power (ii) productivity differences, (iii) differences in prices of non-tradeable inputs due to wage differences and (iv) price differences of tradeable intermediates due to trade barriers. If a good is principally tradeable, the degree of trade barriers limits possible arbitrage and defines the width of the possible deviations from PPP. [START_REF] Parsley | Convergence to the Law of One Price Without Trade Barriers or Currency Fluctuations[END_REF] provide evidence for US cities that the lack of tradeability of goods and services is sufficient to drive a permanent wedge between prices at different locations even within the same country (see also O'Connel and Wei, 2002). comparable goods prices. For the OECD countries, price comparisons have been conducted every three to five years. As a result, a panel with a large cross-section, but a very small time dimension is available to compare price levels across countries even at the disaggregated level. In particular, we are able to distinguish between tradeable and non-tradeable goods.

We propose a new test of conditional σ-convergence for fixed and even small T based on [START_REF] Carree | Testing the Convergence Hypothesis: A Comment[END_REF]. Rejecting a unit root in a time series or, for fixed T , finding β-convergence in a cross-section is not sufficient to detect σ-convergence. For a sufficiently high -but lower than unitycorrelation between initial and end year values the variances of price levels might still be growing over time.

Our estimation results indicate that significant σ-convergence took place in the majority of OECD countries, specifically for tradeable goods, once it is controlled for permanent and country-specific deviations from the law of one price. In line with other studies these turn out to be significant and reflect variables such as trade and transportation costs, differences in taxes and other influences, which all warrant further investigation. Overall, one can conclude that the distribution of relative prices at an international level tends to shrink over time and there is some tendency of convergence. In line with previous findings the corresponding half-lives of LOP deviations to close out over time turn out to be much higher for non-tradeables than for tradeables.

12

y ikT = πy ik1 + µ k + u ikT (6)
where µ k are fixed effects (in our case referring to groups of basic headings) and u ikT is iid N (0, σ 2 u ). In contrast to Carree and Klomp, in this specification we control for heterogeneity and allow for conditional convergence with K-specific steady states. The starting values y ik1 are supposed to be iid N (µ k1 , σ 2 1 ) with µ k1 = µ k and E [y ik1 u ikT ] = 0. 11 Based on (6), under the null hypothesis of no σ-convergence we set

σ 2 T = E (y ikT -πµ k1 -µ k ) 2 = E (π (y ik1 -µ k1 ) + u ikT ) 2 = π 2 σ 2 1 + σ 2 u equal to σ 2 1 . Hence, we test H 0 : π 2 = π 2 0 < 1 -σ 2 u σ 2 1
vs. H 1 : π < π 0 . This test is based on the bivariate normal distribution (see [START_REF] Cannon | Galton's Fallacy and Economic Convergence[END_REF][START_REF] Carree | Testing the Convergence Hypothesis: A Comment[END_REF]or Egger and Pfaffermayr, 2006): with 11 This assumption guarantees that, in contrast to the dynamic fixed effects model (Anderson and [START_REF] Anderson | Formulation and Estimation of Dynamic Models Using Panel Data[END_REF][START_REF] Hsiao | Analysis of Panel Data[END_REF], our estimate is consistent and unbiased. Note that the assumption of differing group effects can principally be tested for.

y ik1 y ikT ∼ N µ k1 πµ k1 + µ k , σ 2 1 πσ 2 1 πσ 2 1 π 2 σ 2 1 + σ 2 u (7) using E [(y ikT -πy ik1 -µ k ) (y ik1 -µ k1 )] = E [(π (y ik1 -µ k1 ) + u ikT ) (y ik1 -µ k1 )] = πσ 2 1 . (ii) The likelihood under H 1 is given by L 1 = - N K 2 ln 4Π 2 + ln σ 2 1 σ 2 u + S N K
13

F o r P e e r R e v i e w S = 1 σ 2 1 σ 2 u K k=1 N i=1 y ik1 -µ k1 y ikT -πy ik1 -µ k π 2 σ 2 1 + σ 2 u -πσ 2 1 -πσ 2 1 σ 2 1 y ik1 -µ k1 y ikT -πy ik1 -µ k1 = K k=1 N i=1 (y ikT -πy ik1 -µ k ) 2 σ 2 u + (y ik1 -µ k1 ) 2 σ 2 1 .
The first order conditions are as follows:

∂L 1 ∂µ k1 = 1 2 N i=1 2(y i1 -µ k1) σ 2 1 = 0 ⇒ µ k1 = m 1 (y 1k ), ∂L 1 ∂µ k = 1 2 N i=1 2(y ikT -πy ik1 -µ k ) σ 2 u = 0, ∂L 1 ∂π = 1 2 K i=1 N i=1 2(y ikT -πy ik1 -µ k ) σ 2 u y ik1 = 0,
where m 1 (y k1 ) denotes the group-specific first moments. Defining the (N K × K) dummy variables matrix D and

Q = I N K -D (D D) -1 D , Qy T and
Qy 1 have typical elements y ikT = y ikT -m 1 (y kT ) and y ik1 = y ik1 -m 1 (y k1 ) [START_REF] Fuller | Transformations for Estimation of Linear Models with Nested-Error Structure[END_REF]. Therefore,

π = K i=1 N i=1 y ikT y ik1 K i=1 N i=1 ( y ik1) 2 = c( y 1 y T ) c( y 1)
, µ k = m 1 (y kT ) -πm 1 (y k1 ), where c (.) denotes the second central moment.

Furthermore,

∂L 1 ∂σ 2 1 = -N K 2 1 σ 2 1 σ 2 u + 1 2 K k=1 N i=1 (y ik1 -µ k1 ) 2 σ 4 1 ⇒ -N K 2 σ 2 1 σ 4 u -N 2 σ 4 u K k=1 c(y k1 ) = 0 ⇒ σ 2 1 = c (y 1 ), ∂L 1 ∂σ 2 u = -N K 2 σ 2 1 σ 2 1 σ 2 u + 1 2 K k=1 N i=1 (y ikT -πy ik1 -µ k ) 2 σ 4 u ⇒ -N K 2 σ 4 1 σ 1 u -N K 2 c(u) σ 4 u = 0 ⇒ σ 2 u = c (u), where c (y 1 ) = 1 K K k=1 c (y k1 ). c (u) is analogously defined. Inserting into the likelihood gives L 1 = - N K 2 ln 4Π 2 + ln σ 2 1 σ 2 u + 2 . ( 8
)
(iii) Under H 0 , one has to maximize the restricted likelihood function

L 0 = - N K 2 ln 4Π 2 + ln σ 2 1 σ 2 u + S R N K with 14 F o r P e e r R e v i e w S R = K k=1 N i=1    y ikT -y ik1 (σ 2 1 -σ 2 u ) /σ 2 1 -µ k 2 σ 2 u + (y ik1 -µ k1 ) 2 σ 2 1    .
The corresponding first order conditions read as follows:

∂L 0 ∂µ k1 = 1 2 N i=1 2(y ik1 -µ k1) σ 2 1 = 0 ⇒ µ k1,0 = m 1 (y k1 ), ∂L 0 ∂µ k = 1 2 N i=1 2 y ikT -y ik1 ( σ 2 1,0 -σ 2 u,0 )/ σ 2 1,0 -µ k,0 σ 2 u = 0 ⇒ µ k,0 = m 1 (y kT )-π 0 m 1 (y 1k ).
Now define y ikT = y ikT -µ k,0 and insert for µ k,0 :

∂L 0 ∂σ 2 1 = -N K 2 σ 2 u σ 2 1 σ 2 u -1 2 K k=1 N i=1 1 σ 2 u (- σ 2 1 -σ 2 u σ 2 1 -1 2 σ 2 u σ 4 1 y ik1 y ikT + 2µ k1 y ik1 σ 4 1 - µ 2 k1 σ 4 1 ). σ 2 1,0 = 1 K K k=1 ( 1 π 0 (m (y k1 , y kT ) -m 1 (y k1 )m 1 (y kT )+ π 0 m 1 (y k1 ) 2 )-m 1 (y k1 ) 2 ) = K k=1 1 π 0 c (y 1k , y T k ) = 1 π 0 c (y 1 , y T ). ∂L 0 ∂σ 2 u = -N K 2 σ 2 1 σ 2 1 σ 2 u -1 2 K k=1 N i=1 (- y 2 ikT σ 4 u + σ 2 1 -σ 2 u σ 2 1 σ 4 u -1 2 1 σ 2 1 -σ 4 u -(σ 2 1 -σ 2 u) 2σ 2 u σ 6 u y ik1 y ikT - y 2 ik1 σ 4 u ) = -N Kσ 2 u + K k=1 N i=1 ((y ikT -µ k ) 2 - σ 2 1 -σ 2 u σ 2 1 σ 4 u -1 2 2σ 2 1 -σ 2 u σ 2 1 σ 2 u y ik1 (y ikT -µ k ) + y 2 ik1
) After some manipulations, we get

σ 2 u,0 = c(y T ) -π 0 c (y 1 , y T ) -1 π 0 c (y 1 , y T ) + c(y 1 ). Using σ 2 1,0 -σ 2 u,0 = π 2 0 σ 2 1,0 = π 0 c (y 1 , y T ) = 1 π 0 c (y 1 , y T )-c (y T )+ π 0 c (y 1 , y T )+ 1 π 0 c (y 1 , y T )-c(y 1 ) or 2 π 0 c (y 1 , y T )-c (y T )-c(y 1 ) = 0, we have π 0 = 2c(y 1 ,y T ) c(y T )+c(y 1 )
and σ 2 u,0 = c(y T )+c 2 (y 1 ) 2 -2c(y 1 ,y T ) 2 c(y T )+c(y 1 ) follows. Now, 1. Calculate σ 2 1 = c (y 1 ) and σ 2 u = c (u) for the unrestricted model using the residuals of the β-convergence regression.

1 N K K k=1 N i=1 (y ikT -π 0 y ik1 -µ k,0 ) 2 = 1 N K K k=1 N i=1 (y ikT -m 1 (y kT )- π 0 (y ik1 -m 1 (y 1k ))) 2 = 1 N K K k=1 N i=1 [(y ikT -m 1 (y kT )) 2 -2 π 0 (y ikT -m 1 (y kT ))(y ik1 - m 1 (y 1k ))+ π 2 0 (y ik1 -m 1 (y 1k )) 2 ] = c (y T ) + c(y 1 )-2 π 0 c (y 1 , y T ) + ( π 2 0 - 1)c(y 1 ) = 2 σ 2 u,0 - σ 2 u,0 σ 2 1,0 σ 2 1 . Inserting gives S R = 1 σ 2 u,0 (2 σ 2 u,0 - σ 2 u,0 σ 2 1,0 σ 2 1 )+ σ 2 1 σ 2 1,0 ) = 2 + σ 2 1 σ 2 1,0 - σ 2 1 σ 2 1,0 = 2 and L 0 = - N K 2 ln 4Π 2 + ln σ 2 1,0 σ 2 u,0 + 2 (9) (
2. For the restricted model, we have σ 2 1,0 = 1 π 0 c (y 1 , y T ) and σ 2 u,0 = c(y T )+c 2 (y 1 ) 2 -2c(y 1 ,y T ) 2 c(y T )+c(y 1 ) .

3. Therefore, the likelihood ratio test

LR = -2 (L 1 -L 0 ) = N K ln σ 2 1,0 σ 2 u,0 σ 2 1 σ 2 u . ( 10
)
is asymptotically distributed as χ 2 (1). In small samples, we use N K -2.5 instead of N K (see [START_REF] Morrison | Multivariate Statistical Methods[END_REF]. Note, we have a one-sided test, so the significance levels are doubled to get the critical value.

(v) Table 11 below provides a Monte Carlo simulation to assess the power of the LR-test. We use the following parametrization, which fits our data:

σ 2 u = 0.025, σ 2 1 = 0.085, so that π 0 = 0.84. The dummies µ k1 and µ k are independent draws from a normal distribution with mean 0 and variance 0.25. The error term ε is drawn from a normal distribution with mean 0 and variance σ 2 u . We replicate the experiment 20000 times and calculate the share of rejections at the 5 percent level of significance. For π=π 0 , this gives the size of the LR-test, for π = π 0 , we get the power of the LR-test.

  ." Building on the pioneering work of Gilbert, Kravis, Summers and Heston, the International Comparison Programme (ICP) collects a wide range of goods prices in each participating country with great effort. ICP is a very comprehensive and advanced statistical exercise, coordinated by the United Nations Statistical Department and conducted with

  coefficient can be interpreted sensibly.

  In contrast to national Consumer Price Index (CPI) data, the ICP price information exhibits a few very desirable properties (see UnitedNations, 1992):First, the ICP methodology provides a very detailed description of goods and services for which price data have to be collected. The list of items to be included in the international basket of goods consists of identical specifications throughout all countries participating in the comparison. Furthermore, the ICP framework has elaborated careful instructions for statistical agencies concerning where, what, and when prices have to be monitored in each country. National statistical agencies are prompted to report only prices of goods, which are representative for the local market.Second, ICP provides price information on a wide range of products at a very detailed level of disaggregation. PPPs are calculated for each homogeneous category of goods and services, the so-called "basic headings". Close inspection of the list of basic headings allows to separate tradeable goods from non-tradeable local goods and services. There are 195 basic headings, each consisting of 5 to more than 50 goods and services uniformly specified for all participating countries. 5 Third, PPPs are computed for each basic heading, reference country are the US. No explicit weights are applied to each of the items within a basic heading for calculation of PPPs at the basic heading level (p). Special computational methods (Eltetö-Köves-Szulc (EKS) method; seeDrechsler, 1973) obtain transitive parities throughout the whole set of participating countries. At the level of basic headings, parities are therefore invariant to the choice of the base country. Since ICP data provide homogeneous baskets of goods throughout all comparison countries, tests for absolute PPP in levels rather than relative PPP in growth rates are possible.

  the (unweighted) mean values of PPPs over all goods (basic headings) for each country and ICP comparison round. The mean parities indicate sound LOP deviations. Given that the US-dollar is used as the reference currency, the descriptive statistics point to a substantially overvalued Dollar for the most observations in our sample.

  ble 5). The importance of country-specific intercepts is underlined by the corresponding t-tests of the constant and the tradeability dummy. In case of non-tradeable goods, these are significant for 16 out of 23 countries over the period 1990-1996. More surprisingly, there are also 10 countries where significant intercepts are found for the group of tradeable goods. The corresponding 10

  Similar to[START_REF] Carree | Testing the Convergence Hypothesis: A Comment[END_REF], we formulate the convergence equation by the autoregressive process for fixed T , by substituting T periods backwards to get

  iv) Summing up, the LR test is calculated as follows 15

Table 11

 11 exhibits the power function for different numbers of groups (K) and observations per group (N ). The Monte Carlo simulation shows that at K = 2 the power of the LR-test is reasonable, if N is sufficiently large (which is the case in our application). In addition the test is properly sized.

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Table 1 :

 1 Country Specific Mean Parities

	Country	1980	1990	1996
	Belgium	-0.33	-0.26	-0.24
	Denmark	-0.43	-0.49	-0.47
	France	-0.35	-0.28	-0.31
	Germany	-0.37	-0.27	-0.30
	Greece	-0.05	-0.02	-0.07
	Ireland	-0.23	-0.19	-0.18
	Italy	-0.06	-0.30	-0.16
	Luxembourg	-0.28	-0.24	-0.27
	Netherlands	-0.33	-0.20	-0.20
	Portugal	0.06	0.11	-0.03
	Spain	-0.09	-0.23	-0.12
	United Kingdom	-0.31	-0.14	-0.10
	EU12	-0.23	-0.21	-0.20
	Austria	-0.32	-0.31	-0.33
	Finland	-	-0.62	-0.36
	Iceland	-	-0.48	-0.31
	Norway	-0.47	-0.54	-0.44
	Sweden	-	-0.53	-0.46
	Switzerland	-	-0.49	-0.51
	EFTA	-0.40	-0.49	-0.40
	Australia	-	-0.09	-0.09
	Canada	0.05	-0.13	0.07
	Japan	-0.21	-0.37	-0.55
	New Zealand	-	-0.01	-0.13
	Turkey	-	0.33	0.43
	Rest of OECD	-0.08	-0.06	-0.05
	All Countries	-0.23	-0.25	-0.22
	Note: Table 1 shows mean purchasing power parities over all
	goods (basic headings) for each country and country groups.
		17

Table 2 :

 2 Sigma Convergence -1980 to 1990 ∆σ ntr LR ntr ∆σ tr LR tr

	Belgium	-0.11	0.87	0.04 0.20
	Denmark	-0.14	1.05	0.05 0.24
	France	-0.04	0.09	0.03 0.13
	Germany	-0.33	8.34 * * -0.06 0.51
	Greece	-0.06	0.20	0.02 0.03
	Ireland	-0.35	9.96 * * -0.02 0.03
	Italy	-0.23	3.27 *	0.09 1.04
	Luxembourg	-0.17	1.96	-0.08 0.94
	Netherlands	-0.22	3.15 * -0.05 0.28
	Portugal	-0.16	1.54	-0.06 0.64
	Spain	-0.06	0.20	-0.01 0.03
	United Kingdom -0.00	0.00	0.08 0.71
	EU12	-0.18 22.18 * * 0.00 0.01
	Austria	-0.20	2.58	-0.05 0.28
	Norway	-0.34	8.12 * * 0.02 0.07
	EFTA	-0.27 10.57 * * 0.03 0.02
	Japan	-0.05	0.13	0.10 1.20
	Canada	1.21 24.41 * * 0.59 25.25 * *
	Rest of OECD	0.08	0.48	0.13 7.04 * *
	All Countries	-0.17 27.68 * * 0.01 0.51

Note: * * significant at 5%; * significant at 10%.

Table 3 :

 3 Sigma Convergence -1990 to 1996 ∆σ ntr LR ntr ∆σ tr LR tr

	Belgium	-0.06	0.29	-0.16 4.69 * *
	Denmark	-0.01	0.01	-0.06 0.63
	France	-0.02	0.05	-0.20 7.58 * *
	Germany	0.09	0.54	-0.12 2.46
	Greece	0.22	2.93 * -0.10 2.06
	Ireland	-0.17	2.24	-0.10 1.80
	Italy	0.02	0.03	-0.22 9.81 * *
	Luxembourg	-0.12	1.17	-0.09 1.47
	Netherlands	-0.03	0.08	-0.11 1.85
	Portugal	-0.03	0.11	-0.20 8.93 * *
	Spain	-0.06	0.27	-0.20 8.30 * *
	United Kingdom -0.12	1.44	-0.20 6.62 * *
	EU12	-0.05	0.82	-0.15 51.29 * *
	Austria	-0.13	1.61	-0.11 2.14
	Finland	0.00	0.00	-0.14 3.33 *
	Iceland	0.12	0.82	0.00 0.00
	Norway	-0.09	0.69	-0.12 2.70
	Sweden	-0.04	0.10	-0.20 7.48 * *
	Switzerland	0.11	0.78	-0.13 2.95 *
	EFTA	0.02	0.03	-0.13 14.34 * *
	Australia	0.32	4.65 * * 0.11 1.69
	Canada	0.24	3.17 *	0.17 3.38 *
	Japan	-0.02	0.04	-0.02 0.10
	Turkey	-0.01	0.01	-0.28 18.40 * *
	New Zealand	-0.02	0.04	-0.01 0.00
	Rest of OECD	0.04	0.57	0.04 4.20 * *
	All Countries	-0.02	0.08	-0.11 61.29 * *
	Note: * * significant at 5%; * significant at 10%.
		19

Table 4 :

 4 Sigma Convergence -1980 to 1996 ∆σ ntr LR ntr ∆σ tr LR tr

		Submitted Manuscript
	Belgium	-0.16	1.74	-0.12 1.92
	Denmark	-0.14	1.32	-0.03 0.11
	France	-0.11	0.60	-0.16 3.81 *
	Germany	-0.26	4.59 * * -0.18 4.90 * *
	Greece	0.14	1.12	-0.07 0.63
	Ireland	-0.46 18.16 * * -0.13 2.25
	Italy	-0.20	2.66	-0.15 3.30 *
	Luxembourg	-0.27	4.66 * * -0.16 3.94 * *
	Netherlands	-0.24	3.86 * * -0.16 3.58 *
	Portugal	-0.21	2.84 * -0.24 9.96 * *
	Spain	-0.11	0.80	-0.20 6.54 * *
	United Kingdom -0.09	0.46	-0.14 2.88 *
	EU12	-0.22 29.57 * * -0.15 39.99 * *
	Austria	-0.33	7.93 * * -0.16 3.61 *
	Norway	-0.40 12.76 * * -0.13 2.55
	EFTA	-0.38 21.46 * * -0.13 6.12 * *
	Japan	-0.07	0.27	0.05 0.29
	Canada	1.89 42.52 * * 0.90 47.33 * *
	Rest of OECD	0.26	1.75	0.36 8.58 * *
	All Countries	-0.19 36.37 * * -0.08 27.13 * *

Note: * * significant at 5%; * significant at 10%. 20 Page 21 of 32 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 5 :

 5 Beta Convergence -1980 to 1990

	HL ntr HL tr	6.16 3.76	10.76 4.78	--	6.00 2.77	8.34 6.26	5.91 3.79	5.17 5.30	5.92 3.45	5.92 6.40	8.67 7.42	7.23 4.15	6.69 2.84	4.93 3.94	6.09 6.53	6.93 5.36	7.26 3.64
	µ 1 t µ 1	-0.06 0.70	-0.26 3.12 * *	-0.18 2.95 * *	-0.12 1.22	-0.16 1.87 * *	-0.07 0.92	-0.42 4.43 * *	-0.24 3.05 * *	-0.11 1.23	-0.18 1.84 * *	-0.16 2.03 * *	-0.20 2.46 * *	-0.19 1.97 * *	-0.30 2.62 * *	-0.23 2.58 * *	-0.18 1.99 * *
	cons t cons	-0.22 3.33 * *	0.01 0.21	0.00 0.04	-0.38 5.31 * *	-0.06 0.80	-0.15 2.34 * *	0.28 3.43 * *	0.01 0.08	-0.20 2.71 * *	-0.17 2.12 * *	-0.07 1.14	-0.02 0.32	-0.35 4.67 * *	0.27 2.64 * *	-0.06 0.83	0.02 0.34
	β t β µ 2 t µ 2	Austria 0.32 3.43 * * -0.17 1.26	Belgium 0.52 5.04 * * -0.29 2.08 * *	Canada -0.02 0.09 -0.01 0.02	Denmark 0.31 3.02 * * -0.23 1.60 *	France 0.44 3.42 * * -0.11 0.68	Germany 0.31 3.47 * * -0.15 1.22	Greece 0.26 2.06 * * 0.01 0.06	Ireland 0.31 4.14 * * -0.18 1.46 *	Italy 0.31 3.03 * * 0.03 0.20	Japan 0.45 3.92 * * -0.06 0.38	Luxembourg 0.38 4.03 * * -0.19 1.51 *	Netherlands 0.35 3.80 * * -0.27 2.05 * *	Norway 0.25 2.81 * * -0.07 0.57	Portugal 0.32 3.19 * * 0.03 0.20	Spain 0.37 3.35 * * -0.09 0.65	United Kingdom 0.38 3.48 * * -0.24 1.52 *	Note: * * significant at 5%; * significant at 10%.
							21

Table 6 :

 6 BetaConvergence -1990Convergence - to 1996

	HL ntr HL tr	4.83 4.63	5.70 3.59	7.23 2.72	7.69 3.39	5.96 3.89	3.62 3.79	6.68 3.35	8.11 3.43	8.16 5.18	5.12 6.27	3.40 3.37	7.11 3.75	7.68 6.29	5.32 3.92	6.20 2.75	4.01 4.26	4.91 4.42	7.79 4.95	5.36 3.99	3.52 3.56	7.21 3.42	4.27 3.99	6.59 2.23
	µ 1 t µ 1	-0.13 2.03 * *	0.00 0.04	-0.17 2.84 * *	-0.13 2.16 * *	-0.09 1.11	-0.08 0.88	-0.08 1.28	-0.09 1.51 *	-0.17 2.13 * *	-0.16 1.85 * *	-0.14 2.28 * *	-0.23 3.56 * *	-0.16 2.01 * *	-0.04 0.77	-0.10 1.72 * *	-0.14 2.23 * *	-0.04 0.53	-0.08 1.05	-0.08 1.23	0.01 0.10	-0.03 0.38	-0.40 3.41 * *	-0.20 3.62 * *
	cons t cons	0.03 0.64	-0.22 4.55 * *	0.22 4.59 * *	-0.06 1.41 *	-0.21 3.27 * *	-0.10 1.33 *	-0.16 3.55 * *	-0.14 2.75 * *	0.04 0.56	0.05 0.74	-0.02 0.52	0.11 2.17 * *	-0.24 3.92 * *	-0.16 3.47 * *	-0.07 1.54 *	-0.02 0.41	-0.20 3.13 * *	-0.04 0.61	0.02 0.31	-0.30 4.43 * *	-0.30 4.31 * *	0.59 5.37 * *	0.07 1.54 *
	β t β µ 2 t µ 2	Australia 0.42 2.78 * * -0.02 0.08	Austria 0.48 5.30 * * -0.17 1.44 *	Canada 0.56 4.26 * * -0.35 2.05 * *	Belgium 0.58 6.45 * * -0.29 2.52 * *	Denmark 0.50 5.04 * * -0.15 1.21	Finland 0.32 2.87 * * 0.02 0.13	France 0.54 5.37 * * -0.25 2.06 * *	Germany 0.60 5.46 * * -0.30 2.27 * *	Greece 0.60 5.25 * * -0.15 1.11	Iceland 0.44 3.99 * * 0.07 0.51	Ireland 0.29 3.27 * * -0.00 0.03	Italy 0.56 5.68 * * -0.23 1.92 * *	Japan 0.58 5.95 * * -0.07 0.54	Luxembourg 0.46 5.06 * * -0.11 0.94	Netherlands 0.51 5.29 * * -0.29 2.32 * *	New Zealand 0.35 3.06 * * 0.02 0.15	Norway 0.43 4.28 * * -0.04 0.31	Portugal 0.59 7.02 * * -0.16 1.46 *	Spain 0.46 5.31 * * -0.11 0.95	Sweden 0.31 3.09 * * 0.00 0.04	Switzerland 0.56 4.72 * * -0.27 1.90 * *	Turkey 0.38 4.29 * * -0.02 0.22	United Kingdom 0.53 6.99 * * -0.38 3.48 * *	Note: * * significant at 5%; * significant at 10%.
										22

Table 7 :

 7 Beta Convergence -1980 to 1996

	HL ntr HL tr	8.42 4.88	14.53 6.06	-6.77	13.15 4.99	9.37 5.69	9.21 5.94	29.99 5.83	7.74 4.82	10.17 5.36	12.40 9.16	7.81 6.50	10.36 4.94	7.77 5.01	11.16 7.53	16.20 6.11	8.13 -
	µ 1 t µ 1	0.04 0.54	-0.23 2.99 * *	-0.26 3.49 * *	-0.11 1.32 *	-0.10 1.25	-0.07 0.90	-0.23 2.58 * *	-0.18 2.63 * *	-0.17 2.17 * *	-0.24 2.48 * *	-0.08 1.04	-0.16 2.12 * *	-0.12 1.43 *	-0.09 0.93	-0.05 0.64	-0.29 3.75 * *
	cons t cons	-0.32 5.60 * *	-0.01 0.12	0.26 3.94 * *	-0.33 4.99 * *	-0.19 3.02 * *	-0.19 3.10 * *	0.04 0.51	-0.04 0.76	-0.06 0.86	-0.33 4.28 * *	-0.17 2.99 * *	-0.03 0.50	-0.32 4.94 * *	-0.03 0.37	-0.11 1.75 * *	0.11 1.81 * *
	β t β µ 2 t µ 2	Austria 0.27 3.25 * * -0.16 1.44 *	Belgium 0.47 5.01 * * -0.31 2.44 * *	Canada -0.69 1.96 * * 0.88 2.22 * *	Denmark 0.43 4.47 * * -0.32 2.39 * *	France 0.31 2.72 * * -0.16 1.19	Germany 0.30 3.54 * * -0.15 1.25	Greece 0.69 5.67 * * -0.54 3.61 * *	Ireland 0.24 3.63 * * -0.14 1.31 *	Italy 0.34 3.76 * * -0.21 1.71 * *	Japan 0.41 3.61 * * -0.11 0.75	Luxembourg 0.24 2.76 * * -0.06 0.51	Netherlands 0.34 4.02 * * -0.24 1.99 * *	Norway 0.24 3.19 * * -0.13 1.18	Portugal 0.37 4.27 * * -0.14 1.24	Spain 0.50 5.50 * * -0.34 2.84 * *	United Kingdom 0.26 2.68 * * -0.30 2.20 * *	Note: * * significant at 5%; * significant at 10%.
							23

Table 8 :

 8 Beta Convergence -1980 to 1990

	HL ntr HL tr	6.59 4.73	5.44 3.85	8.02 6.34	6.49 4.76		HL ntr HL tr	6.29 3.74	4.76 4.09	5.19 4.46	5.63 4.00		HL ntr HL tr	10.91 5.68	8.04 4.95	10.39 8.81	10.36 5.87
	F cµ 1	1.51 * *	2.11 *	0.07	1.41 *		F cµ 1	1.00	3.66 * *	1.24	1.86 * *		F cµ 1	1.31	3.20 * *	0.15	1.70 * *
	β t β µ 2 t µ 2 cons t cons F µ 1 F c	EU12 0.35 11.98 * * -0.12 2.99 * * -0.14 2.09 * * 0.58 6.48 * *	EFTA 0.28 4.38 * * -0.11 1.24 -0.23 3.41 * * 0.35 1.34	Rest of OECD 0.42 4.51 * * -0.09 0.72 -0.17 2.53 * * 5.82 * * 2.16 *	All Countries 0.34 13.46 * * -0.11 3.19 * * -0.14 2.12 * * 0.53 6.30 * *	Table 9: Beta Convergence -1990 to 1996	β t β µ 2 t µ 2 cons t cons F µ 1 F c	EU12 0.52 19.16 * * -0.19 5.46 * * -0.16 3.26 * * 1.31 4.46 * *	EFTA 0.42 9.77 * * -0.06 1.06 -0.24 4.83 * * 0.33 9.21 * *	Rest of OECD 0.45 9.43 * * -0.06 0.94 0.52 7.06 * * 16.47 * * 19.46 * *	All Countries 0.48 23.09 * * -0.12 4.79 * * -0.16 3.28 * * 0.70 10.84 * *	Table 10: Beta Convergence -1980 to 1996	cons t cons F µ 1 F c β t β µ 2 t µ 2	-0.17 2.86 * * 1.78 * * 5.12 * * EU12 0.36 13.83 * * -0.22 6.21 * *	-0.32 5.61 * * 0.38 0.02 EFTA 0.25 4.55 * * -0.15 1.84 * *	-0.33 4.73 * * 7.92 * * 27.03 * * Rest of OECD 0.34 3.47 * * -0.06 0.47	-0.18 2.90 * * 1.38 * 7.70 * * All Countries 0.34 14.66 * * -0.19 6.03 * *	Note: * * significant at 5%; * significant at 10%.
								24

Table 11 :

 11 Monte Carlo Simulation Results for the Power of the LR-Test (20.000 Replications) On Cross-Countries Differences in the Persistence of Real Exchange Rates," Journal of International Economics, 2000 b, 50, 375-397. Crownover, C., J. Pippenger, and D.G. Steigerwald, "Testing for Absolute Purchasing Power Parity," Journal of International Money and Finance, 1996, 15, 783-796. Crucini, M.J. and M. Shintani, "Persistence in Law-of-One-Price Deviations: Evidence from Micro-data," 2004. Working Paper No. 02-W22R, Vanderbilt University. , C.I. Telmer, and M. Zachariadis, "Dispersion in Real Exchange Rates," 2000. Working Paper No. 00-W13, Vanderbilt University. American Economic Review, 2005, 95 (3), 724-738. Cumby, R., "Forecasting Exchange Rates and Relative Prices with the Hamburger Standard: Is what you want what you get with McParity?," NBER Working Paper, 1996, 5675. Drechsler, L., "Weighting of Index Numbers in International Comparisons," Review of Income and Wealth, 1973. Explaining the Border Effect: The Role of Exchange Rate Variability, Shipping Costs and Geography," Journal of Intenational Economics, 2001, 55 (1), 87-105. Rogers, J., "Price Level Convergence, Relative Prices, and Inflation in Europe," Federal Reserve Board International Finance Discussion Paper, 2001, 699. Rogoff, K., "The Purchasing Power Parity Puzzle," Journal of Economic Literature, 1996, 34, 647-668. Taylor, A.M., "Potential Pitfalls for the Purchasing Power Parity Puzzle? Sampling and Specification Biases in Mean-Reversion Tests of the Law of One Price," Econometrica, 2001, 69, 473-498. , "A Century of Purchasing Power Parity," Review of Economics and Statistics, 2002, 84, 139-150. and M. P. Taylor, "The Purchasing Power Parity Debate," Journal of Economic Perspectives, 2004, 18 (4), 135-158. Taylor, M.P. and D.A. Peel, "Nonlinear Adjustment, Long-Run Equilibrium and Exchange Rate Fundamentals," Journal of International Money and Finance, 2000, 19, 33-53. United-Nations, Handbook of the International Comparison Programme number F 62. In 'Studies in Methods.', United-Nations, 1992.

	Number of observations per group (N)	5 10 20 50 100 200 5 10 20 50 100 200	σ 2 T p K = 2 K =5	0.064 0.68 0.113 0.171 0.288 0.615 0.893 0.995 0.220 0.364 0.614 0.946 0.999 1.000	0.068 0.71 0.096 0.125 0.204 0.440 0.730 0.955 0.166 0.248 0.442 0.823 0.985 1.000	0.072 0.75 0.087 0.097 0.138 0.275 0.488 0.780 0.124 0.167 0.287 0.585 0.868 0.991	0.076 0.78 0.073 0.074 0.087 0.149 0.254 0.449 0.100 0.107 0.158 0.304 0.542 0.831	0.081 0.81 0.068 0.064 0.062 0.078 0.100 0.152 0.080 0.074 0.077 0.114 0.175 0.312	0.085 0.84 0.067 0.056 0.054 0.049 0.051 0.049 0.075 0.058 0.054 0.052 0.053 0.051	0.089 0.87 0.070 0.060 0.064 0.072 0.100 0.147 0.082 0.073 0.077 0.112 0.176 0.311	0.093 0.90 0.074 0.071 0.091 0.149 0.251 0.451 0.102 0.110 0.151 0.312 0.539 0.829	0.098 0.93 0.084 0.096 0.138 0.273 0.492 0.782 0.128 0.169 0.284 0.579 0.862 0.991	0.102 0.95 0.096 0.122 0.210 0.445 0.731 0.952 0.168 0.257 0.444 0.824 0.984 1.000	0.106 0.98 0.109 0.161 0.290 0.618 0.892 0.996 0.218 0.360 0.617 0.945 0.999 1.000	K = 10 K = 20	0.064 0.68 0.368 0.617 0.892 0.999 1.000 1.000 0.608 0.881 0.994 1.000 1.000 1.000	0.068 0.71 0.267 0.442 0.732 0.983 1.000 1.000 0.455 0.722 0.951 1.000 1.000 1.000	0.072 0.75 0.181 0.289 0.490 0.865 0.992 1.000 0.305 0.492 0.783 0.990 1.000 1.000	0.076 0.78 0.123 0.158 0.257 0.541 0.828 0.985 0.177 0.267 0.454 0.826 0.984 1.000	0.081 0.81 0.088 0.090 0.105 0.180 0.309 0.541 0.102 0.111 0.156 0.306 0.538 0.826	0.085 0.84 0.079 0.065 0.057 0.052 0.052 0.051 0.080 0.063 0.057 0.050 0.053 0.052	0.089 0.87 0.088 0.085 0.101 0.185 0.311 0.538 0.102 0.113 0.159 0.307 0.543 0.827	0.093 0.90 0.127 0.160 0.256 0.543 0.830 0.984 0.177 0.257 0.449 0.829 0.984 1.000	0.098 0.93 0.184 0.286 0.489 0.868 0.990 1.000 0.297 0.496 0.785 0.992 1.000 1.000	0.102 0.95 0.265 0.444 0.731 0.983 1.000 1.000 0.451 0.721 0.953 1.000 1.000 1.000	0.106 0.98 0.368 0.619 0.891 0.999 1.000 1.000 0.605 0.881 0.995 1.000 1.000 1.000	Note: σ 2 1 =0.085 and σ 2 u =0.025, so under H 0 p=0.84. The figures refer to the share of rejections with a critical χ 2 -value of 3.84.
												25

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

31 Page 32 of 32 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

InEgger and Pfaffermayr (2006) an even more general Wald-test on conditional σconvergence is provided. As long as all explanatory variables are dummy variables as in our application, the here derived LR-test on conditional σ-convergence is sufficient.

The final list of products narrowly specified for the 1996 OECD/EUROSTAT comparison covers over 2900 consumer goods and services, 34 occupations in government, education and health services, 186 equipment goods and 20 construction projects.

If there is an infinite history of the process with a switch in the variance at period 1, the MA-representation implies that the starting value includes all previous errors up to period 1.


Mario Quagliariello
email: mario.quagliariello@bancaditalia.it

E44 -Financial Markets and the Macroeconomy < E4 -Money and Interest Rates < E -Macroeconomics and Monetary Economics, G28

Keywords: JEL classification numbers: E44, G21, G28 bank, business cycle, uncertainty, lending decisions, GARCH

whether they are published or not. The documents may come

INTRODUCTION

In recent years, the increasing interest in financial stability as an autonomous policy target, along with monetary and microeconomic stability, has encouraged the analyses of the linkages between the macroeconomic environment and the soundness of the banking system. This stream of work is usually referred to as research on the "procyclicality" of banks' operations. Most of these studies use the current status of macroeconomic conditions as the main exogenous cause for the state of health of banks. The goal is to assess to what extent macroeconomy affects banks' performance (cyclicality) and whether, in turn, banks' reaction to changing macroeconomic conditions further affects the macroeconomy, reinforcing cyclical fluctuations (procyclicality). These studies generally confirm that banks' balance sheets are affected, simultaneously or with some delay, by the business cycle and claim that banks' behaviour is procyclical.

However, the change in banks' behaviour through the business cycle is not explicitly modelled, but it is simply inferred looking at the reduced-form relationships between micro and macro variables. Bank-specific indicators are frequently included as regressors, but they are only used as control variables, while the focus rests on the proxies for macroeconomic conditions. Furthermore, apart from some recent papers on North-American banks, no study assesses the role that macroeconomic uncertainty plays in determining banks' behaviour: in other words, no attention is paid on the second moments of macroeconomic variables.

Summing up, the current state of the art is unsatisfactory for two main reasons: i) no attempt is made in order to model how banks' management varies in changing This paper tries to fill these gaps. In particular, it aims at disentangling which are the determinants of banks' willingness to invest in risky loans as opposed to riskfree assets and understanding whether macroeconomic uncertainty plays a role in this choice. Following the portfolio model proposed by [START_REF] Baum | The second moments matter: The response of bank lending behavior to macroeconomic uncertainty[END_REF], which provides an explicit link between the cross-bank dispersion of the share of loans held in portfolio and uncertainty, the paper discusses how Italian banks choose between loans and riskfree assets when the uncertainty of macroeconomic conditions increases. Given the substantial differences between North-American and European banking sectors, this paper can provide useful insights on the reliability of the model when a different sample of intermediaries is used.

An important innovation of this work is that, with respect to Baum et al., who neglect the role of idiosyncratic factors in the econometric specification, the impact of macroeconomic uncertainty is assessed after controlling for bank-specific sources of uncertainty. In particular, introducing an appropriate proxy, I explicitly consider the importance of idiosyncratic uncertainty in determining banks' portfolio decisions. This paper also enriches the existing evidence employing different proxies for macroeconomic uncertainty; this also allows me to extensively assess the robustness of the results. Finally, the paper provides evidence on banks' sub-samples, testing whether the theory is valid at different levels of aggregation.

The econometric results confirm that macroeconomic uncertainty plays a significant role in Italian banks' investment decisions. In periods of increasing turmoil, behave more homogenously, as shown by the reduction of the loan-to-asset ratio crosssectional dispersion. Idiosyncratic uncertainty is also relevant and its coefficient takes on a positive sign. When returns on specific investments are less easily predictable, better informed banks can exploit their competitive advantage and behave in a different way with respect to poorly informed intermediaries.

The remainder of the paper is organized as follows. Section 2 and 3 provide a short survey of the literature and describe the portfolio model used in the analysis.

Section 4 reports the empirical results. Conclusions are provided in Section 5.

REVIEW OF THE LITERATURE

There is a huge literature supporting the hypothesis that macroeconomic conditions affect the performance of the banking sector 1 . Most of the work focuses on measures of central tendency rather than on those of uncertainty. [START_REF] Salas | Credit risk in two institutional settings: Spanish commercial and saving banks[END_REF], for instance, observe that macroeconomic shocks are quickly transmitted to Spanish banks' balance sheets. During economic booms, intermediaries tend to expand their lending activity, often relaxing their selection criteria; in the following downturns, bad loans remarkably increase, producing losses.

Using a panel of Italian banks, Quagliariello (in press, b) finds that loan loss provisions and bad debts increase in bad macroeconomic times. [START_REF] Pesola | The role of macroeconomic shocks in banking crises[END_REF] shows that the high level of both corporate and households' indebtedness, along with shortfalls of GDP growth below forecast levels, contributed to the banking crises in the Nordic countries.

Similar evidence is provided in cross-country comparisons by [START_REF] Bikker | Cyclical Patterns in Profits, Provisioning and Lending of Banks[END_REF], 4 Laeven and Majnoni (2003) and [START_REF] Valckx | What determines loan loss provisioning in the EU?[END_REF]. [START_REF] Gambera | Simple Forecasts of Bank Loan Quality in the Business Cycle[END_REF] and [START_REF] Meyer | Are small Rural Banks Vulnerable to Local Economic Downturns?[END_REF] document that a small number of macroeconomic variables are good predictors for non-performing loan ratio in the US. Similarly, [START_REF] Hoggarth | Stress tests of UK banks using a VAR approach[END_REF] provide evidence of a clear link between the state of the UK business cycle and banks' writeoffs.

While there is an extensive literature on the investment decisions of nonfinancial firms under macroeconomic instability (for a survey, see [START_REF] Carruth | What Do We Know about Investment under Uncertainty?[END_REF]), the role of uncertainty regarding future macroeconomic conditions has been largely neglected for banks instead. This is an important issue, since banks' investments fuel those of the rest of the economy. To the best of my knowledge, there are only two recent papers that investigated this issue. [START_REF] Baum | The second moments matter: The response of bank lending behavior to macroeconomic uncertainty[END_REF] use a portfolio model and a sample of US banks. Their results suggest that when macroeconomic uncertainty -proxied by the conditional variance of a relevant macroeconomic variable -increases, the cross-bank dispersion of the share of risky loans to total assets diminishes, as uncertainty hinders bank ability to foresee investment opportunities. In other words, they claim that higher uncertainty makes the signals on expected returns noisier. Uncertainty would therefore push banks to rebalance the composition of their assets according to new (worse) signals provided by credit markets, adversely affecting the allocation of financial resources. This fosters herding behaviour and leads banks to behave more homogeneously than in quiet periods. This evidence is robust to the inclusion of several control variables and holds for total bank loans as well as for their main components. 1 For a survey, see Quagliariello (in press, a). Adopting the same approach, [START_REF] Garcia | Banks Lending and Macroeconomic Uncertainty: The Case of Canada[END_REF] reach similar conclusions for the Canadian banking system. Their results, though based on univariate regression, confirm that there is a negative relationship between the proxy for macroeconomic uncertainty and the cross-bank variance of the loan-to-asset ratio. In other words, also Canadian intermediaries show herding behaviour when they deal with more pronounced aggregate uncertainty.

BANKS' LENDING DECISIONS UNDER UNCERTAINTY: BAUM, CAGLAYAN AND OZKAN'S MODEL

Adapting the framework for non-financial firms' investments presented by [START_REF] Beaudry | Monetary Instability, the Predictability of Prices, and the Allocation of Investment: An Empirical Investigation Using UK Panel Data[END_REF], [START_REF] Baum | The second moments matter: The response of bank lending behavior to macroeconomic uncertainty[END_REF] propose a model that describes how banks set the optimal composition of their portfolios.

In their scheme, banks' managers operate in a risky environment and, in each period, can invest deposits into two different assets: loans and bonds. The investment in bonds is assumed free of default risk, but it bears market risk since the value of the securities may change as a result of varying market conditions. This risk is predictable and, more importantly, it can be managed and hedged. The return of such an investment is the risk free rate (r f ).

Loans to customers entail the exposure to both market and default risk. The latter is the result of an idiosyncratic component -due the probability that the specific customer will default in the future without repaying the debt -and systemic factors, correlated to the status of the economy. In fact, it is well documented that in bad macroeconomic times the riskiness of bank portfolios tends to increase. For bank i at time t, the investment in risky loans provides a stochastic return (r i ) equal to the risk free rate plus a risk premium (rp i ), which is equal for all loansassumed to be homogeneous -and does not depend on the riskiness of the single borrower 2 :

i f i rp r r + = (1)
The risk premium has an expected value E(rp i )=ρ and a variance Var(rp i )=σ 2 ε .

The return on loans can thus be expressed as:

i f i r r ε ρ + + = , (2)
where ε i is a random component distributed as N(0, σ 2 ε ). Baum et al. also assume that each bank has a specific portfolio with different riskiness structure and, hence, the random components of return across different intermediaries are not correlated (E[ε i ε j ]=0).

Banks' managers deal with a portfolio optimization problem. They rebalance the composition of their assets in order to obtain the preferred combination of risk and expected return. According to their utility functions, they choose the shares α i and (1α i ) of their assets to invest respectively in loans and bonds. However, before taking the decision, banks observe neither the actual risk premium nor the random component ε i , but only a noisy signal of them:

ν ε + = i i S , (3)
2 The subscript t is omitted since the model describes a one period problem. where ν is a random variable independent of ε i with a normal distribution N(0, σ 2 ν ). By assumption, the noise component (ν) of the signal banks receive is identical for all of them, while the overall signals remain different across intermediaries because of ε i . In fact, even though all banks are believed to have the ability to overcome asymmetric information problems, cross-sectional differences in their private information set remain. In principle, ν may be observed and uncertainty eliminated if all banks would share their private information. However, information sharing is unlikely to hold in the credit market.

The noise ν can be interpreted as the degree of uncertainty on future macroeconomic conditions. Its impact on banks is homogeneous, regardless of managers' ability to predict the random component of loan return ε i . When macroeconomic uncertainty increases, the higher variance of ν makes the estimates of the true return of loans less precise. The assumption of independence between ν and ε i is now clearer; indeed, it is plausible that the aggregate macroeconomic shock is not correlated with the idiosyncratic component of loan returns.

In order to determine the expected return on loans (r i ), bank managers have to predict the value of ε i . Without observing the noisy signal, banks' unconditional forecast on ε i would be the mean of its distribution, i.e. zero. However, banks do observe the signal and can extract additional information from it. The expected value of the return from loans conditional upon S i , E[ε i |S i ], is assumed to be a constant proportion (λ) of the signal, where λ represents a linear regression coefficient of ε i on S i :

F o r P e e r R e v i e w 8 ) ( ] | [ ν ε λ λ ε + = = i i i i S S E
, where

2 2 2 ν ε ε σ σ σ λ + = (4)
The conditional expected return of the i-th bank's portfolio E[R i |S i ] is therefore:

f i i f i f i i i f i i i r r r S E r S R E ) 1 ( )) ( ( ) 1 ( ]) | [ ( ] | [ α ν ε λ ρ α α ε ρ α - + + + + = - + + + = (5) and the conditional variance Var[R i |S i ] is: 2 2 ] | [ ν λσ α i i i S R Var = (6)
Risk-averse banks are assumed to have the following utility function:

] | [ 2 ] | [ ] | [ i i i i i i S R Var S R E S U E ω - = , (7)
which is increasing in expected return and decreasing in return volatility (and ω is the coefficient of risk aversion).

Employing the portfolio's mean/variance equations, it is straightforward to derive the optimal loan-to-asset ratio (α i ) for bank i and the associated cross-sectional dispersion:

2 ν ωλσ λ ρ α i i S + = (8) 4 2 2 ) ( ν ε σ ω σ α = i Var (9)
The variance of the cross-sectional distribution of the loan-to-asset ratio is negatively correlated to the level of macroeconomic uncertainty σ 2 ν . Indeed, taking the first derivative of the variance of α i with respect to σ 2 ν , it derives:

F o r P e e r R e v i e w 9 0 2 ) ( 6 2 2 2 < - = ∂ ∂ ν ε ν σ ω σ σ α i Var , (10)
which provides a testable implication of the hypothesis that the cross-sectional variance of the loan-to-asset ratio narrows as macroeconomic uncertainty increases.

The variance of α i instead widens when the variance of the idiosyncratic component increases:

0 1 ) ( 4 2 2 > = ∂ ∂ ν ε σ ω σ α i Var (11)
Therefore, it is essential to control for this component when testing for the impact of macroeconomic uncertainty. The choice of a proper proxy for idiosyncratic risk may however represent a major problem.

In sum, I have two hypotheses to test:

EMPIRICAL EVIDENCE

Data

The impact of macroeconomic uncertainty on banks' lending decisions can be empirically tested using the following specification:

F o r P e e r R e v i e w 10 t t t t i t u c b a LTA Var + + + = 2 , 2 , , ) ( ε ν σ σ , (a)
where LTA i,t is the loan-to-asset ratio and Var t (LTA i,t ) is its cross-sectional variance at time t; σ 2 ν represents macroeconomic uncertainty evaluated at time t; σ 2 ε is the idiosyncratic uncertainty and u t is the error term.

The cross-bank variance of the LTA is built up using quarterly data (1990q1-2005q1) for a sample of more than 900 intermediaries, representing virtually the entire Italian banking system in each quarter. Before 1990 the Italian credit market was intensely regulated by public authorities for both monetary policy and supervisory purposes 3 . The results of the analysis would be therefore not reliable. Furthermore, formal Chow tests reject at the 1 per cent level the null hypothesis of no structural break in 1990 and advice to split the sample.

Overall, the dataset includes 58.146 bank/quarter observations. A summary of the characteristics of the sample is provided in table 1.

3 For example, as far as monetary policy is concerned, in 1986q1-1986q2 and 1987q4-1988q1 ceilings on the supply of credit that each bank may provide to the aggregate of its customers (so-called massimale sugli impieghi) were still in use. These constraints, intended for controlling liquidity and aggregate demand, limited banks' ability to expand their shares in the loan market. As a result, even in the absence of specific obligations for banks to subscribe Treasury bonds, the banking system's demand for bonds increased (see [START_REF] Cotula | La politica monetaria in Italia[END_REF]. Also supervisory rules were particularly strict.

According to the banking law, intermediaries were classified as retail banks, providing only short-term credit, and medium and long-term credit institutions. Until 1990, when the Bank of Italy decided to liberalize banks' branch networks, the opening of new branches was limited by the system of "branch plans". These legal barriers between different categories of banks and the administrative constraints on the opening of branches were an obstacle to the enlargement of banks' activity (see [START_REF] Ciocca | The Italian Financial System Remodelled[END_REF]. Accounting ratios for the individual institutions are built up using the statistics that intermediaries are required to report to the Bank of Italy; the macroeconomic variables are drawn from the OECD main economic indicators (MEI) dataset.

F
The evolution of the loan-to-asset ratio over time is drawn in figure 1. The ratio shows a clear ascending trend, witnessing the increasing liberalization of the Italian credit market and the competitive incentives to improve market positions that banks received during the '90s. Also, the gradual reduction of interest rates made the subscription of securities, especially government bonds, less profitable for banks.

[Insert figure 1 here]

Given the relevance of this trend, I estimate model (a) using the coefficient of variation rather than the variance of the LTA in order to have a unit-free measure of dispersion:

t t t t i t u c b a LTA CV + + + = 2 , 2 , , ) ( ε ν σ σ (b)
Table 2 shows the main descriptive statistics for the loan-to-asset ratio (LTA), its variance (STDLTA) and coefficient of variation (CVLTA). Data refer to the whole banking system as well as to 5 dimensional breakdowns 4 .

[Insert table 2 here]

4 According to the classifications provided by the Bank of Italy, banks are grouped into five categories depending on the size of their total assets: major banks (total assets greater than 45 billion euros), large (total assets between 20 and 45 billion euros), medium-sized (7-20), small (1-7), minor (total assets up to 1 billion euros). [Insert figure 2 here] Large banks are those that, in the period under examination, have the smaller cross-sectional dispersion of the indicator. By contrast, small banks are those that behave more heterogeneously. However, in the last two years, major and large banks show the highest levels of the variation coefficients. This may indicate that recently large banks have adopted more diversified growth strategies.

Certainly, the evolution of LTA is also affected by the consolidation process of the Italian banking system. Indeed, after M&As, the newly established banking groups tended to reorganize their portfolios and, in some cases, to allocate specific assets to specialized entities of the group. This is particularly evident for major banks in 1995 and 1998.

Measuring macroeconomic uncertainty

There are several ways to measure macroeconomic uncertainty. According to [START_REF] Driver | Cross-Section vs. Time Series Measures of Uncertainty using UK Survey Data[END_REF] and Sepulveda-Umanzor ( 2004) two approaches are dominant: the "survey based approach" (or cross-section dispersion forecasts) and the "model based approach" (or time series conditional volatility).

The first one exploits the surveys on the expectations on relevant macroeconomic variables and obtains a measure of uncertainty as the intra-personal dispersion of the expectations. A main shortcoming of this methodology is that the intra-personal dispersion (i.e., the true uncertainty) is not observable and may only be 13 approximated using the inter-personal dispersion of the expectations (the disagreement across forecasters). This procedure is reliable only if the latter measure is a valid proxy for the former [START_REF] Zarnowitz | Consensus and Uncertainty in Economic Prediction[END_REF][START_REF] Bomberger | Disagreement as a Measure of Uncertainty[END_REF]. In fact, it is possible that, even if each forecaster is extremely uncertain about future events, all of them submit similar estimates. Then, this measure would fail to capture the amount of existing uncertainty [START_REF] Grier | The Effects of Real and Nominal Uncertainty on Inflation and Output Growth: Some GARCH-M Evidence[END_REF].

The second metric is obtained employing the realised values of the macroeconomic variables in order to get statistical or econometric estimations of their variability. Even though many measures of uncertainty can be implemented from timeseries (unconditional variance, one-step ahead forecast errors, etc.), the conditional heteroskedasticity estimated with G(ARCH) models is one of the most widely used [START_REF] Engle | Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation[END_REF][START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF]. In the case of GARCH (1,1), the conditional variance of a variable (h t ) can be estimated, along with its mean, using the following specification:

t t t x y η β + = 1 2 1 - -+ + = t t t gh a c h η
Provided that the coefficients on the ARCH and GARCH effects are statistically significant, the fitted values of h t can be employed as proxies for uncertainty. This approach has two potential drawbacks. First, as highlighted by [START_REF] Sepulveda-Umanzor | The Relation between Macroeconomic Uncertainty and the Expected Performance of the Economy[END_REF], "rather than measuring uncertainty, the model based approach really measures volatility. The former is a feature that forward looking agents face when confronting any decision, the latter is a characteristic of the data once uncertainty has been solved".

Second, the time-series used in the analyses do not necessarily exhibit (G)ARCH processes. Given the complexity of defining reliable proxies for uncertainty, ideally one would use both the approaches recalled above. However, due to lack of data on survey outputs, in this paper I use the latter and calculate the conditional heteroskedasticity of relevant macroeconomic variables as proxies for macroeconomic uncertainty5 . In spite of some of its shortcomings, this approach is commonly used in the literature. As a robustness check, I also use the one-step ahead forecast error as an alternative metric.

I use the OECD monthly series of the composite leading indicator, industrial production and consumer price index (CPI). The time span of the series is 1975m1-2005m3. The composite leading indicator is an aggregate series that shows a leading relationship with the reference series for the business cycle in a given country; the indicator can therefore be used as an early signal of the main turning points of the aggregate economic activity. The industrial production index measures the increases and decreases in production output. Given the impact that fluctuations in the level of industrial activity have on the remainder of the economy, the variable is frequently used as a short-term indicator of the business cycle. The indicator is seasonally adjusted.

As pointed out by [START_REF] Carruth | What Do We Know about Investment under Uncertainty?[END_REF], often the literature that uses conditional variance as a measure of uncertainty assumes, rather than testing, both the stationarity of the time-series and the presence of (G)ARCH effects. In this paper, the original monthly series are transformed in order to obtain the monthly percentage rates of 15 consistently estimate their conditional variance. For each variable, the specification of the auxiliary regression employed for the ADF tests has been selected via a general tosimple-approach, as suggested by [START_REF] Enders | Applied Econometrics -Time Series[END_REF], starting with 6 lags, a trend and a constant.

[Insert table 3 here]

The results of the Lagrange Multiplier tests for null hypothesis of no ARCH effects suggest that both industrial production growth and CPI inflation exhibit significant conditional heteroskedasticity; for the series of the composite leading indicator, the test is not significant instead (table 4). Accordingly, I exclude the latter variable from the analysis and estimate the GARCH models only for the former two variables.

[Insert table 4 here]

Table 5a and 5b provide the results of the unrestricted GARCH(1,1) models performed on industrial production growth (INDPRODC) and inflation (INFL). In the variance equation, the coefficients of the ARCH and GARCH effects are significant respectively at the 5 and 1 per cent levels for both the variables.

[Insert tables 5a, 5b here]

The conditional variance (h t ) derived from each GARCH model is finally averaged to quarterly frequency in order to obtain the measure of macroeconomic uncertainty The LM tests cannot reject the null hypothesis of no additional ARCH effects at any conventional level. These results confirm that the specification is able to capture all the conditional heteroskedasticity present in industrial production and inflation series and corroborate the choice of the parsimonious GARCH (1,1). The correct specification of the generating regressions should ensure parameter consistency also in the derived model [START_REF] Pagan | The Econometric Analysis of Models With Risk Terms[END_REF].

Figures 3 and4 plot the coefficient of variation of the loan-to-asset ratio and the measures of industrial production and inflation uncertainty respectively.

[Insert figures 3, 4 here] In the aftermath of the EMS crisis in the first half of the '90s, the coefficient of variation of the LTA ratio reaches the lowest figure. In the same period, macroeconomic -especially inflation -uncertainty rises markedly. 6 Strictly speaking this measure is not the quarterly volatility, but the average of the monthly volatilities in a given quarter. As an alternative measure I also use the quarter-end conditional variance. Results are unchanged. The dispersion of the LTA shows a significant increase during the second half of the decade, a period characterized by the vigorous process of consolidation, which encouraged banks to look for new markets and more profitable activities. This may help explain more heterogeneous behaviour across intermediaries. Both measures of economic uncertainty show a descending trend during the '90s; for inflation, this tendency is particularly noticeable at the end of those years. This is likely one of the benefits of the EMU convergence process.

Econometric specification and results

The final specification of model (b) is the following:

t t t t t t t i t u SHAREC f INDPRODC e INFL d STDNPL c h b a LTA CV + + + + + + + = +1 , ) ( (c)
where h t is the estimated conditional variance of either industrial production growth (INDPRVA) or inflation (INFLVA). According to hypothesis 1), the expected sign of the coefficient of this regressor is negative. Along with the proxy for macroeconomic uncertainty (h t ), I also introduce some control variables.

First, I employ the 1-quarter lead of the cross-sectional variance of the nonperforming loan ratio (STDNPL). This indicator is a proxy for the idiosyncratic uncertainty. If at time t the idiosyncratic uncertainty (ε i ) increases allowing "good" banks to correctly predict the expected return/risk of different investments, in the following periods the riskiness of those banks should be significantly lower than that of poorly informed intermediaries. Therefore, the variance of the non-performing loans ratio across intermediaries should widen. Of course, the NPL ratio is also affected by systemic factors; however, its cross-sectional dispersion may still provide some useful to test the robustness of the results in the presence of the levels of the variables [START_REF] Huizinga | Inflation Uncertainty, Relative Price Uncertainty, and Investment in U.S. Manufacturing[END_REF]. Inflation is frequently found as an indicator of macroeconomic mismanagement; high inflation may therefore, per se, imply uncertainty on the future. A negative coefficient is therefore my a priori belief. By contrast, it is not immediately clear the impact of industrial production growth on the dependent variable. Indeed, it is difficult to say whether in good times banks tend to behave more or less homogeneously.

Lastly, I introduce a variable (SHAREC) aiming at assessing the impact of the conditions on financial markets, since this is another factor that may influence banks' decisions. This variable is the quarterly rate of change of the "all shares" index calculated by the OECD from daily closing quotations.

Tables 6, 7 and 8 summarize the variables used in the estimations, show the main descriptive statistics of the regressors and illustrate the pair-wise correlations between the variables. It is interesting to note that the correlations between the dispersion of the loan-to-asset ratio -both in terms of variance and coefficient of variation -and uncertainty measures are negative and statistically significant at the 5 per cent level.

[Insert tables 6, 7, 8 here] Table 9 presents the instrumental-variable regression results. Newey-West standard errors are calculated assuming an autocorrelation up to the order 4.

[Insert table 9 here]

In both the specification, the proxies for macroeconomic uncertainty show the expected negative sign and are statistically significant, although their economic relevance is different. In particular, the magnitude of the coefficient on INFLVA (-1.23) suggests that this variable is an important determinant of LTA dispersion.

Overall, this confirms that Italian banks behave more homogenously when the perspectives of the economy are unclear. This is consistent with the evidence regarding US and Canadian banks reported respectively by Baum et al. (2004) and [START_REF] Garcia | Banks Lending and Macroeconomic Uncertainty: The Case of Canada[END_REF] and with the results of [START_REF] Beaudry | Monetary Instability, the Predictability of Prices, and the Allocation of Investment: An Empirical Investigation Using UK Panel Data[END_REF] for UK non-financial firms.

However, it is worth pointing out that, with aggregate data, no inference can be drawn on the allocation of funds to particular loans and hence the credit risk of the loan book. In other words, the model implicitly assumes that bank's loan portfolio is homogenous. These assumptions should be ideally tested providing results for several loan categories, along with the aggregate ones 7 . Unfortunately, due to data limitations, this is not feasible at this stage.

The proxy for idiosyncratic uncertainty is also statistically significant and takes on a positive sign. When the returns on specific investments are less easily predictable, better informed banks can exploit their competitive advantage and behave in a different way with respect to poorly informed intermediaries.

7 I am grateful to the referee for raising this point. As far as the other control variables are concerned, industrial production growth is never significant. The inclusion of different variables aiming at controlling for the evolution of the business cycle, such as the leading indicator, demand and consumption growth or changes in the interest rates, does not change this evidence. This result indicates that the level of aggregate economic activity does not have any impact on the cross-sectional variability of the share that banks decide to invest in risky loans.

However, the significance of the control variables must be interpreted with caution since the proxy for uncertainty is a generated regressor and the coefficient on it is significantly different from zero [START_REF] Pagan | Econometric Issues in the Analysis of Regressions with Generated Regressors[END_REF][START_REF] Oxley | Econometric issues in macroeconomic models with generated regressors[END_REF].

The coefficient on inflation is not significant as well. The changes of the stock exchange index are slightly significant in the model with industrial production uncertainty.

Robustness checks

Size breakdowns

The dataset with banks' size breakdowns is used for robustness checks. The model is estimated using panel data techniques that exploit the cross-sectional dimension and allow obtaining more robust results8 . For the estimation, I can exploit 10.

[Insert table 10 here]

They appear consistent with those obtained in the previous paragraph.

Macroeconomic uncertainty has a significant negative impact on the cross-bank dispersion of the loan-to-asset ratio, while the proxy for idiosyncratic uncertainty shows a positive sign. In these specifications, industrial production growth is significant, though at the 10 per cent level, and takes on a negative sign. According to this evidence, banks would take more uniform portfolio decisions during expansions. Inflation remains not significant, contrasting the idea that higher inflation levels are perceived as signals of economic turmoil. The Hausman tests for both the regressions imply that the random effects are appropriate for this sample.

Uncertainty based on the one-step ahead forecast error

In this paragraph, I use a different proxy for macroeconomic uncertainty in order to check whether my previous results are robust to different measures. The metric proposed here is based on the 1-step ahead forecast errors of the autoregressive models of industrial production growth and inflation. As suggested by [START_REF] Serven | Macroeconomic Uncertainty and Private Investment in LDCs: An Empirical Investigation[END_REF], to ensure that the predictions use no more information than that available at the time they are formulated, they are computed from recursive estimation of the autoregressions.

Therefore, for each variable, I estimate recursively the following auto-regressive model of order 2:

Unfortunately, this is not necessarily the case, since Italian banks are not specialized according to their size.

F

β β β + + + = - - 2 2 1 1 0
The initial length of the series is set at 50. Since the dataset contains 362 observations (quarters) and I introduce 2 lags, I estimate 310 (362-2-50) regressions, in order to compute the 1-step ahead forecast errors. Using the recursive parameter estimates, I finally obtain the alternative uncertainty measure as the quarterly mean absolute values of the monthly one-step ahead forecast errors for industrial production and inflation (respectively INDPRODFE and INFLFE).

Table 11 provides the results of the regression of LTA variation coefficient on

INDPRODFE and INFLFE.

[Insert table 11 here]

The role of uncertainty is confirmed by this exercise. Both additional proxies for macroeconomic uncertainty are statistically significant and negative. In particular, inflation unpredictability seems to be the most relevant determinant of banks' herding behaviours. Idiosyncratic uncertainty remains significantly positive.

CONCLUSIONS

This paper tries to disentangle which are the determinants of banks' decisions regarding the allocation of their portfolios; in particular it aims at assessing whether macroeconomic uncertainty plays a role in this choice. Following the portfolio model proposed by Baum et al., I discuss 7 1990q1 1991q1 1992q1 1993q1 1994q1 1995q1 1996q1 1997q1 1998q1 1999q1 2000q1 2001q1 2002q1 2003q1 1990q1 1991q1 1992q1 1993q1 1994q1 1995q1 1996q1 1997q1 1998q1 1999q1 2000q1 2001q1 2002q1 2003q1 43 1990q1 1991q1 1992q1 1993q1 1994q1 1995q1 1996q1 1997q1 1998q1 1999q1 2000q1 2001q1 2002q1 2003q1 43 1990q1 1991q1 1992q1 1993q1 1994q1 1995q1 1996q1 1997q1 1998q1 1999q1 2000q1 2001q1 2002q1 2003q1

  ii) the effect of the uncertainty regarding future macroeconomic conditions is typically neglected.

  signals on the expected returns of loans and, therefore, tend to

Hypothesis 1 ):

 1 When macroeconomic uncertainty increases, banks tend to allocate assets in their portfolios more homogeneously (the variance of α across banks reduces); Hypothesis 2): When idiosyncratic uncertainty increases, banks tend to behave more heterogeneously (the variance of α across banks rises).

  table, there is no evidence of a clear relationship between LTA dispersion and bank size during the fifteen years under observation. Fig.2provides a clearer picture of this relationship.

  change of industrial production (INDPRODC), CPI (INFL) and the leading indicator (LEADINDC). The transformation allows to work with stationary series (table 3

  in model (b) as generated regressors. They measure with noise the true, though unobservable, regressor ( 2 ν σ ). The estimates for model (b) can therefore be biased and inconsistent if the ARCH-type model employed is misspecified.Therefore, I use the Lagrange Multiplier in order to test for neglected serial correlation of up to order 12 in the standardized squared residuals of the ARCH models.

  2). The expected sign of this variable is positive. Since the variable may be endogenous, in the estimation it is instrumented with the lagged value of STDNPL.Second, I use the first moment of inflation (INFL) and industrial production growth (INDPRODC) in order to control for the overall macroeconomic conditions and

  results confirm that systemic uncertainty, both on the real and the financial side of the economy, induces intermediaries to herding behaviour. In contrast, individual uncertainty leads banks to behave more heterogeneously when deciding the allocation of their assets.

  60 quarters / 5 size breakdowns). The results of the random effect model are presented in table

  how Italian banks choose between loans and riskfree assets when the uncertainty on macroeconomic conditions increases. With respect to previous work, which neglects the role of idiosyncratic factors, the impact of assessed, taking also into account bank-specific sources of uncertainty.The econometric results confirm that macroeconomic uncertainty does have a role in Italian banks' investment decisions. In periods of increasing turmoil, banks receive noisier signals on the expected returns of loans and, therefore, tend to behave more homogeneously, as shown by the reduction of the cross-sectional dispersion of the loan-to-asset ratio. A plausible corollary is that the allocation of bank credit becomes less efficient. Idiosyncratic uncertainty is also relevant and its coefficient takes on a positive sign. When the returns on specific investments are less easily predictable, better informed banks can exploit their competitive advantage and behave in a different way with respect to poorly informed intermediaries. These results are generally confirmed when bank size breakdowns are employed in the estimation. Furthermore, they are robust to the inclusion of several control variables and the use of different measures of macroeconomic uncertainty.Macroeconomic uncertainty is therefore an important determinant of banks' lending decisions and a cause of potential disturbances in financial resource allocation.Since bank loans are a relevant source of financing for the non-financial sector, central banks and supervisory authorities should monitor the degree of uncertainty on the evolution of the main economic aggregates in order to strengthen macroeconomic and financial stability.

Fig. 2

 2 Fig. 2 CROSS-SECTIONAL DISPERSION OF LTA: SIZE BREAKDOWN (coefficient of variation, percentage values)

  Fig. 1 LOAN-TO-ASSET RATIO (percentage values)

  Fig. 4 CROSS-SECTIONAL DISPERSION OF LTA vs. INFLATION UNCERTAINTY (percentage values)

Table 1 no

 1

		THE SAMPLES: SUMMARY
		. banks per quarter (average)	no. obs
	FULL SAMPLE	953	58146
	MAJOR BANKS	14	870
	LARGE BANKS	16	972
	MEDIUM BANKS	36	2203
	SMALL BANKS	143	8740
	MINOR BANKS	744	45361
	TIME-SPAN	1st quarter 1990 -1st quarter 2005

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Table 2

 2

	LOAN-TO-ASSET RATIO: SIZE BREAKDOWN
	Variable	N. obs	Mean	STD	Min	Max
			TOTAL
	LTA	61	44.81	6.68	36.23	58.10
	STDLTA	61	15.81	3.91	10.51	21.52
	CVLTA	61	0.35	0.04	0.27	0.40
			Major Banks
	LTA	61	44.16	4.33	32.95	51.58
	STDLTA	61	13.12	8.56	4.00	28.47
	CVLTA	61	0.29	0.18	0.09	0.62
			Large Banks
	LTA	61	46.57	3.27	39.38	53.89
	STDLTA	61	14.39	6.98	4.67	28.26
	CVLTA	61	0.31	0.14	0.10	0.63
			Medium Banks
	LTA	61	50.00	7.27	37.79	62.92
	STDLTA	61	16.53	4.18	7.56	22.01
	CVLTA	61	0.33	0.06	0.17	0.45
			Small Banks
	LTA	61	48.28	7.82	36.87	62.90
	STDLTA	61	19.00	5.13	10.55	24.43
	CVLTA	61	0.39	0.07	0.26	0.51
			Minor Banks
	LTA	61	43.86	6.51	35.86	57.34
	STDLTA	61	14.85	3.62	10.33	20.68
	CVLTA	61	0.33	0.04	0.27	0.40
	Notes: LTA is the average of the loan-to-asset ratio; STDLTA is the cross-sectional variance
	of the loan-to-asset-ratio; CVLTA is the cross-sectional coefficient of variation of the loan-to-
	asset-ratio.

Table 4 Variable

 4

		LM TEST FOR ARCH EFFECTS (1)
		ARCH(1)		ARCH(4)
		Chi-squared	df	Chi-squared	df
	INDPRODC	14.206***	1	24.606***	4
	INFL	3.833**	1	52.922***	4
	LEADINDC	0.005	1	6.928	4

*, **, *** significant at the 10, 5 and 1 per cent level respectively. Notes: (1) Lagrange Multiplier test for ARCH effects. 4 lags included in the auxiliary autoregression.

Table 6

 6

		SELECTED VARIABLES
	Name	Description	Source
	LTA	Loan-to-asset ratio (%)	BoI Sup.statistics
	STDLTA	Cross-sectional STD of LTA (%)	BoI Sup.statistics
	CVLTA	Coefficient of variation of LTA (STDLTA/MLTA)	BoI Sup.statistics
	STDNPL	Cross-sectional STD of non-performing loan ratio (%)	BoI Sup.statistics
	INDPRODC	Industrial production change (%)	OECD
	INFL	CPI inflation (%)	OECD
	SHAREC	Share price change (%)	OECD
	INDPRVA	Conditional variance of INDPRODC	OECD
	INFLVA	Conditional variance of INFL	OECD

Table 7

 7

	REGRESSORS: DESCRIPTIVE STATISTICS
	Variable	N. obs	Mean	STD	Min	Max
	STDNPL	61	9.24	0.93	7.60	11.26
	INDPRODC	61	0.04	0.53	-1.18	1.19
	INFL	61	0.28	0.16	0.03	0.69
	SHAREC	61	0.60	3.80	-8.15	11.67
	INDPRVA	61	0.86	0.37	0.41	2.22
	INFLVA	61	0.03	0.01	0.01	0.06

Table 8

 8

		CORRELATION COEFFICIENTS
		LTA STDLTA CVLTA	STDNPL INDPRVA INFLVA
	LTA	1.000
	STDLTA	0.948	1.000
	CVLTA	0.696	0.888	1.000
	STDNPL	0.150	0.398	0.700	1.000
	INDPRVA	-0.620	-0.676 -0.637	-0.415	1.000
	INFLVA	-0.777	-0.728 -0.526	-0.208	0.757	1.000
	Coefficients in bold are significant at the 5 per cent level.

Table 9 ECONOMETRIC RESULTS INSTRUMENTAL VARIABLE REGRESSION OF CV(LTA)

 9 INDPRVA+b 2 LEAD(STDNPL)+b 3 INFL+b 4 INDPRODC+b 5 SHAREC+u CV(LTA)=intercept+b 1 INFLVA+b 2 LEAD(STDNPL)+b 3 INFL+b 4 INDPRODC+b 5 SHAREC+uNotes: Newey-West robust standard errors are reported; the errors are assumed to be heteroskedastic and autocorrelated up to 4 lags. LEAD(STDNPL) instrumented with LAG(STDNPL).

	Uncertainty on industrial production	Uncertainty on inflation
	60	60
	F(5,54)=12.39	F(5,54)=16.40

*, **, *** significant at the 10, 5 and 1 per cent level respectively. The estimated regressions are: CV(LTA)=intercept+b 1

Table 10 ROBUSTNESS CHECKS INSTRUMENTAL VARIABLE PANEL REGRESSION OF CV(LTA)(1)

 10 , **, *** significant at the 10, 5 and 1 per cent level respectively. The estimated regressions are: CV(LTA)=intercept+b 1 INDPRVA+b 2 LEAD(STDNPL)+b 3 INFL+b 4 INDPRODC+b 5 SHAREC+a+u CV(LTA)=intercept+b 1 INFLVA+b 2 LEAD(STDNPL)+b 3 INFL+b 4 INDPRODC+b 5 SHAREC+a+u Notes: (1) Random effect model. LEAD(STDNPL) instrumented with LAG(STDNPL). Hausman test for random effects is reported (Ho: RE).

	Submitted Manuscript	Page 34 of 37
	Uncertainty on industrial production	Uncertainty on inflation
	300	300
	Chi2(5)=255.06	Chi2(5)=294.35
	0.437	0.477
	Chi2(5)=0.01	Chi2(5)=0.01

*

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

Table 11 ROBUSTNESS CHECKS IV REG. OF CV(LTA) WITH DIFFERENT PROXIES FOR UNCERTAINTY

 11 , **, *** significant at the 10, 5 and 1 per cent level respectively. The estimated regressions are: CV(LTA)=intercept+b 1 INDPRODFE+b 2 LEAD(STDNPL)+b 3 INFL+b 4 INDPRODC+b 5 SHAREC+u CV(LTA)=intercept+b 1 INFLFE+b 2 LEAD(STDNPL)+b 3 INFL+b 4 INDPRODC+b 5 SHAREC+uNotes: Newey-West robust standard errors are reported; the errors are assumed to be heteroskedastic and autocorrelated up to 4 lags. LEAD(STDNPL) instrumented with LAG(STDNPL).

	Page 35 of 37	Submitted Manuscript
		Uncertainty on industrial
		production

*

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

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

For Italy, a quarterly survey on firms' expectations regarding future inflation is available only since December 1999.

In principle, if bank's size is a proxy for the type of loans a bank grants, the evidence provided in this section might partly overcome the shortcomings related to loan heterogeneity mentioned above.

ACKOWLEDGEMENTS

I am grateful to F. Cannata, A. De Vincenzo, C. Guerzoni, S. Iannotti, J. Marcucci for their comments. An anonymous referee provided very helpful suggestions for improving the paper. My special thanks go to P. N. Smith for his continuous guidance. All remaining errors are my own. The opinions expressed herein are those of the author and do not necessarily reflect those of Banca d'Italia.

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