Procyclicality and Strategic Complementarities in Bank Regulation

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Introduction

This thesis is a collection of three distinct works around two common topics: how banking regulation affects banks’ behavior, and the role of the banking system in funding the real economy. These two topics are often studied separately, overlooking that banking regulation may affect banks’ behavior and their choices of lending to corporations, households, and the entire eco-nomic and social fabric. A narrow point of view on either regulation or bank lending practices limits the ability to see the big picture and can lead to designing a solution optimal for the problem analyzed, but suboptimal for the whole economic system. As an example, from the regulator’s point of view a new rule that promotes higher credit standards can be optimal, but if this causes a credit crunch, then the general equilibrium may not be socially optimal. This simple example highlights the relevance of a comprehensive understanding of regulation’s effects on banks’ behavior, especially regard-ing the flow of credit to the economy. Several aspects of the credit flow are crucial for economic development: the amount of credit, the relationship with other sources of funds, the banks’ process of creditworthiness evalua-tion and many others, all depend on the behavior of banks and, therefore, on banking regulation. Furthermore, it is worth noting that, the financial and economic environment is not steady but evolves over time. Banks, cor-porations, and all economic agents adapt their behavior to changes caused by financial innovations, political decisions, macroeconomic situation, and many others factors. Lastly, crises have shown that there are often strategic complementarities in the banking system as the decision of a bank is linked to the decisions of all other banks.

In light of the above considerations, establishing the optimal regulation is a puzzle that demands careful evaluation of any action by authorities, as has been established in the literature. For example, a milestone paper on the impact of the regulation on banks’ behavior is by Kim and Santomero (1988), where the authors cast doubts on the relationship between capital requirements and the riskiness of banks and challenge the older paradigm that higher capital requirements result in more secure banks. Instead, they point out that higher capital requirements may cause banks to look for higher interest rate and riskier activities. The complexity of the financial system makes it difficult to forecast the final result of an authority’s decision,

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and measures that should have clear and sound effects can end up with an undesirable outcome once implemented.

To aide our systemic understanding of the implications of banking reg-ulations onto the real economy, we dedicate the first two chapters of this thesis to shed some light on how the banking system evolved over the years, and how this evolution changed the sources of funding for corporations. In particular, we analyze the US banking system from the 1950s to the present day, using a multiresolution analysis framework (Wavelet analysis). Wavelet analysis is a relatively new mathematical technique that allows the exami-nation of the relationship among variables in time and frequency domains. Economically speaking, this allows us first to understand if a variable, or a relationship among variables, characterizes a specific frequency (e.g. the business cycle frequency), and second to contextualize a variable or a rela-tionship in its historical background.

The first chapter studies the relationship between external sources of corporate funding, where we identify a structural break-point in the 1980s. Earlier, loans and debt securities were substitutes at business cycles frequen-cies, and complementary at lower frequencies. After the 1980s, the relation-ship at lower frequencies disappears while at business cycles frequencies we detects alternating periods of substitutability and periods of complementar-ity. Our finding is consistent with the pecking order theory (Myers, 1984) for the periods until the break-point, but only new innovative theories (Holm-strom and Tirole, 1997, Allen and Gale, 1997, and Song and Thakor, 2010) can explain the more recent ones. In general, the idea behind is that the evolution of the financial system changed banks’ role and activities, leading them to specialize in the evaluation and monitoring of corporations’ credit-worthiness. Meanwhile, the introduction of new financial tools such as the securities increased the interconnections among banks, capital market and other financial institutions like insurances and pension funds. The change of the relationship between sources of funds is a natural consequence of the foregoing.

The second chapter focuses on financial accelerator theory. We analyze the relationships among bank loans, the real net worth of corporations, and the corporates’ credit spread to investigate whether data from the US bank-ing system underpin the financial accelerator mechanism. The study finds relationships consistent with the financial accelerator theory from the end of the 1980s to present. The timing of the relationship’s start drives us to hy-pothesize that the financial accelerator in the US may have been triggered by the deep regulation reform, following the serious crisis of the banking system of the 1980s and the financial development of the early 1990s. With regards to the regulation reform, the FED introduced the Basil I directives in the end of the 1980s with several acts, introducing risk-weighted assets and a more cautious assessment of corporate financial condition. There-fore, it changed the process of evaluation and monitoring the risk, reducing

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banks’ risk taking behavior and impairing of loans, but on the other hand, it tightened the link between loans and the corporates’ indicators such as net worth, leverage or other hard information. As to financial development, it drove banks to use standardized models for evaluation and monitoring of risk as Value at Risk, which aligned banks decisions, enhancing the financial accelerators effects (Danielsson and Shin, 2003).

Our analysis suggests that bank regulation, financial innovation, and technological development profusely affected bank’ behavior. The evolu-tion in the financial and economic environments also explains why business cycles have become more severe in the past thirty years. Indeed, the com-plementarity between loans and debt securities has enhanced the stress of the system, thus making even harder to find alternative sources of funds in distress periods. Moreover, in the downturn when corporations’ needs of external funds grows as the net worth drops, businesses ability to access finance is reduced.

The 2007 crisis revealed all these fragilities within the financial and eco-nomic system and nowadays, there is broad consensus regarding the need for a new macro-prudential framework to resist procyclicality (Schoenmaker, 2014). To achieve this objective, new rules should aim to contain the shift of risk appetite, as explained by Borio et al. (2001), and Borio and Zhu (2012), while at the same time limiting the strategic complementarity of forces within the banking and financial systems, as indicated by Aikman et al. (2015). Although these two problems go hand in hand and are of-ten overlapping, they have to be jointly considered if the authority’s goal is to reduce procyclicality and Central Banks worldwide have already taken some decisions to address the issue. Firstly, they have introduced a new bank regulation (Basil III) which brings macro-prudential measures, like the countercyclical buffer (Basil, 2011). Secondly, they have used monetary policy extensively to affect the economy (Stein, 2012, Fawley and Neely, 2013).

To increase the general understanding of how these new tools work, in the third chapter we present a model in which banks suffer from strategic complementary. We derive the conditions for strategic complementarities in the behavior of banks in a banking system in which the supervisory authority has a budget constraint on the resources to allocate for monitoring, and supervision is costly for banks. The strategic complementarity, in turn, can lead to homogeneous private decisions on risk-taking, setting the scene for corner solutions as excessive credit or a credit crunch. The model explains how the countercyclical buffer and the loans support program work. In such a framework, the goal of macro-prudential policies consists of simultaneously restraining the incentive of banks in extending an excessive or an insufficient amount of loans. We show that the countercyclical buffer is a proper tool to reduce the probability of a credit boom, while a loans support program can decrease the probability of a credit crunch. Our analysis finds that

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these macro-prudential measures reduce the chances of the system shifting towards a not optimal corner solution. However, limiting the probability of extreme events does not only concern the macro-prudential policy but it has to be coordinated with monetary and fiscal policy decisions. Paraphrasing Borio (2014a), the prudential policy is not a panacea, and a macro-prudential approach has to be used in each policy decisions to foster financial and economic stability.

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Chapter 1

Are bank loans and debt

securities complements,

substitutes or both?

A wavelet analysis

1.1 Introduction and Related Literature

Are bank loans and debt securities complement, substitutes or both? Are the financial innovation process and deregulation influencing this relation-ship? Recent developments in financial and non-financial corporations fi-nancing have brought the question on the substitutability or complementar-ity among external sources of finance to the focus of policy debate. One of the main questions concerns the impact of financial innovation and deregu-lation. There have been several changes in the last decades, for instance: the development of pension funds and insurance companies, many changes on financial and bank regulation, the computational ability improvement, the introduction of securitization, and many others. All these changes may have remarkably modified the financial system, and the non-financial corporation financing should reflect them.

The financial crisis of 2007-09 has provided renewed interest on several questions such as the relevance of financing and credit conditions in turbu-lent times, and therefore the relative roles of banks and financial markets to invest in the economic activity, and the sub. Developing a financial system that offers a broader range of financing alternatives and instruments is es-sential to structural policies aiming at enhancing the resiliency properties of the financial system in financial distress periods. Indeed, a financing alter-native could alleviate the real economic effects of a credit crunch (Schularick and Taylor, 2009).

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Financial systems are generally classified as bank- or market-based de-pending on the share of banks and other intermediaries in total financing. Developments in the financial system are generally considered to have a pos-itive impact on real-sector growth. More controversial is whether the bank-or market-based nature of the financial systems matter fbank-or the real sectbank-or (Beck and Levine (2002), Levine (2002), Gambacorta et al. (2014) and Allen et al. (2018)).

Traditional theories share with “the multiple avenues of financial inter-mediation” approach (Greenspan, 1999) the view that banks and debt se-curities financing are substituted for corporations (e.g. Bolton and Freixas (2000) and Boot and Thakor (2000)). This implies that they represent alter-native and competing sources of financing, and banks and markets compete for firms’ demand for external financing. This normally leads borrowers to shift from one source of financing to the other one.

On the other hand, recent studies suggest that banks and markets ex-hibit two types of behavior: competition and complementarity. Holmstrom and Tirole (1997) cast doubt on whether bond issuance can be fully sub-stituted by bank lending when banking systems enter in crisis. They stress the complementarity between these two forms of external financing given the monitoring role of bank lending for securities issuance. Allen and Gale (1997) note that intermediaries may complement markets by providing indi-viduals insurance against unforeseen contingencies in obscure states, thereby eliminating the need for individuals to acquire costly state information and reducing their market participation costs.

Finally, Song and Thakor (2010) support the idea that bank and capital market exhibit three forms of interaction: competition, complementarity, and co-evolution. The key conditions for this three-dimensional interaction are securitization and bank capital. Securitization creates a tool by which bank evolution is good for markets since the improved bank screening, there-fore bank evolution enhances the general credit quality of borrowers going to the capital market via securitization. Whereas, bank capital produces a mechanism though which the evolution of markets is good for banks since it reduces the cost of bank equity capital, leading banks to hold more cap-ital, thus decreasing the rationing of potentially creditworthy relationship borrowers and increasing bank lending scope.

In contrast to the extensive theoretical work set out above, there have been rather few empirical studies explicitly addressing the choice between bank lending and securities issuance, especially at the aggregate level. Previ-ous econometric studies using aggregate data include the papers by Kashyap et al. (1993), Davis (2001) and Davis and Ioannidis (2004) and, more re-cently, Becker and Ivashina (2014) and Grjebine et al. (2018).

Kashyap et al. (1993) show that after a tightening of monetary policy, commercial paper issue rises while bank lending is flat, either because firms are rationed from bank lending and increase their demand for commercial

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paper, or owing to the rationing of those categories of borrowers limited to bank lending markets. The result also suggests that investment is affected by a change in the financing mix, even when controlling for interest rates and output, which implies that the substitution that takes place is costly.

Davis (2001) considers straight the substitution of bank loans for mar-ket finance in periods of financial marmar-ket stress. Using data of four main economies, United States, United Kingdom, Japan, and Canada, he tests the benefits of the stability of corporate financing stemming from the existence of active securities markets alongside banks, both during cyclical downturns and during market crisis periods. However, in a more recent paper, Davis and Ioannidis (2004) find that bond finance complemented declines in the supply of bank loans to the corporate sector using US flow of funds data. They interpret this as evidence that the impact on the economy of any de-cline in bank loan supply will be exacerbated by a contraction in market finance.

Finally, Becker and Ivashina (2014) compare the growth rates of the market and bank debts at the aggregate level since the early 1950s for the US economy, whereas Grjebine et al. (2018) provide a cross-country study of the business cycle behavior of the corporate debt structure and the substitution process between debt instruments.

The typical procyclical pattern of bank loans has favored the empirical analysis of the composition of corporate borrowing between bank loans and market debt over the business cycle. Recent findings by Drehmann et al. (2012) and Borio (2014b) stress that credit aggregates tend to display a greater amplitude and a lower frequency in comparison to business cycle fluctuations, with peaks associated with systemic banking crises or serious financial stress periods. Moreover, bond financing behaves very differently from loans during business cycle upturns and downturns (Adrian et al., 2012).

The goal of this paper is to provide new evidence on the substitutabil-ity/complementarity issue between bank loans and debt securities by using the lens of wavelet analysis. In this context, we apply wavelet analysis to separate different time-scales in the data and analyze the relationship be-tween external financing sources at different scale levels using flow of funds aggregate quarterly data from 1953:2 to 2017:4 for external credit-market borrowing in the US corporate sector. Moreover, we exploit the partial wavelet coherence to disentangle the securitization effect since its introduc-tion in 1983:2. We choose the US since it is the main example of a country where debt securities issuance and bank borrowing are widely available as a source of external debt financing. To our knowledge, this is the first study investigating the complementarity and substitutability issue between direct and indirect sources of external financing focusing on the relationship be-tween bank loans and debt securities, over different time horizons.

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among external forms of financing is scale-dependent. At scales correspond-ing to business cycle frequencies, there is evidence of substitutability for corporations between bond finance and bank loans, whereas at longer time scales a complementarity relationship emerges. These statistically signifi-cant negative and positive relationships at different time scale between bank lending and bond issuance came to an end respectively in the mid-1970s, and in the early 1990s due to a structural change of the relationship. At the end of the 1980s at the business cycle frequencies and at higher ones a new relationship between loans and securities started. While for frequencies higher than the business cycle the relationship still always of substitutabil-ity, for the business cycle frequencies there is evidence of complementarity since the mid-1990s. This complementarity relationship turns out to be of substitutability in the partial-coherence if we control respect of ABS (As-set Backed Securities), suggesting that securitization has likely changed the financing decision of corporations and the banks’ activities.

What emerge are two eras in the relationship between bank loans and debt securities: a first period where external financing sources display both substitutability and complementarity patterns depending on scale frequency, and a second one, after the securitization and deregulation processes, in which we observe substitutability, complementarity and co-evolution on the same frequencies range but in different time.

We conclude that the relationship among external sources of funding depends on financial innovation level, deregulation, and scales. These re-sults are useful as they allow some economic remarks about how financial and economic contexts change impact on the behavior of all economic play-ers such as banks and corporations. In particular, they underpin the idea that financial innovation and change of regulation (deregulation) modify the behavior, the decisions, the decision process and even the tasks of banks, financial institutions, and corporations.

1.2 Theory of external financing and the loan-bond

choice

Traditional theories of corporate finance tend to focus on isolating the factors that influence firms demand for alternative sources of funds. With regards to the sources of funds to finance such expenditures, the traditional “pecking order” theory of corporate finance (Myers, 1984) suggests that external debt finance, either in the form of securities or lending, ranks highly as a source of funds for borrowing firms, and hence demand for it is also closely linked to the cycle and interest rates. Internal funds are cheaper but generally limited by the scale of expenditures (including dividends) that tend to increasingly outstrip such internal funds during a cyclical upturn, while cash flows shrink in a downturn. Thus, internally generated funds are typically preferred to

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other funds, followed by debt finance and finally equity finance.

However, these theoretical models are generally applicable only when the circumstances for raising funds are normal. Moreover, these partial equilib-rium models do not capture changes in the suppliers’ behavior of external finance. In particular, the asymmetric information that exists between bor-rowers and lenders, and the lenders’ inability to write complete contracts give rise to well-known agency problems associated with debt contracts (i.e. adverse selection ahead of providing funds and moral hazard once funds have been provided). Anticipating such potential actions by borrowers, the supply of external funds will duly be affected. Variations in agency costs affecting credit supply may occur via a number of channels. For instance, a decrease in the valuation of assets by lowering collateral values increases sharply ad-verse selection and moral hazard for lenders. Reductions in credit supply will impinge more on low-quality borrowers for whom there is asymmetric information. Such patterns give rise to a “financial accelerator” (Bernanke et al., 1999) as changes in cash flow or asset prices over the cycle give rise to procyclical feedback effects of agency costs on the cost of external finance and hence on real corporate expenditures. This will operate especially via borrowers whose net worth is most heavily affected during a recession, and for borrowers whose activities are riskier or harder to be monitored.

The theories of corporate finance, agency costs and the financial accel-erator outlined above tend to apply to debt finance in general rather than distinguishing intermediated and non-intermediated finance. Hence they need to be supplemented to understand the forces underlying the choice of borrowers between banks and securities as a source of such external finance, as well as possible asymmetries in credit rationing. There are several “the-ories of intermediation” (Davis and Mayer, 1991) that cast light on this issue, highlighting, in general, the monitoring advantages of banks to offset a higher price of loans and the consequent substitutability between bank and bond finance for higher quality firms. As a corollary, they suggest that the determinants of intermediated and market financing may differ signifi-cantly, benefiting those firms able to access both types of finance (e.g. Bolton and Freixas, 2000). What do theories of intermediation suggest specifically about debt securities finance? Diamond (1984) points out that financial intermediaries act as delegated monitors to overcome asymmetric informa-tion, whereby diversification reduces monitoring costs. Therefore, market finance is only available to borrowers with a reputation, as it is a capital as-set which would depreciate if firms act opportunistically. Hence, banks will serve small firms with low levels of public information, while larger firms with a higher degree of public information will have the option to be served by securities markets. Substitution can occur between loans and bonds for the latter only, but there is no element of complementarity for either type of firm.

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the conditions under which bank loans and bond issues are complements for the financing of corporate investment projects. They develop a theoretical model in which bank monitoring plays an important role in facilitating bond finance. As banks are assumed to be informed investors, when the supply of bank loans falls, bond finance also declines since there is less monitoring being undertaken. Thus bonds follow changes in bank loans. In contrast, after a decline in the availability of bond finance, banks, as informed in-vestors, can take up the slack through new loans. If so, this would imply loans substitute for bond finance.

Lastly, Song and Thakor (2010) point out that the securitization may be the missing link to explain the modern relationship between loans and secu-rities. They argue that banks and capital markets complement each other and coevolve through securitization and bank equity capital. Their model indicates that banks and markets have different comparative advantages, which means that they compete just when they are viewed in isolation, yet not at the expense of the other.

To sum up, the existing theoretical models show that banks and markets may play different roles in those regards. Theory suggests that banks and markets exist to mitigate agency and asymmetric information problems in different ways. It turns out that banks play an important role in delegate monitoring, allocating capital and sharing risks in the economy. While mar-kets potentially perform better in price debt if there are enough information and the market is depth enough.

Empirical works on the loan-bond choice have focused on the identifica-tion issue related to the fact that observed changes in corporate bank and non-bank finance are likely to reflect movements in both the supply of and demand for external funds.

In what follows we separate short-run and long-run movements in aggre-gate bank loans and securities finance to test whether banks and markets are substitutes, complements or both. In other terms, to provide new evi-dence on the substitutability/complementarity issue between bank lending and security issuance, we look at the relationship between banks and mar-ket finance over different frequency ranges or time scales. Indeed, the choice between intermediated and non-intermediated finance is likely to depend on short-term cyclical factors as well as on long-term contractual commitments between borrowers and lenders. We check if data support any economic the-ory and finally, we contextualize the relational changes between securities and loans in the historical moment to guess which factors could explain the variation that we observe.

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1.3 Wavelet-based exploratory research analysis

The raw material for this work is flow data on funds raised in the credit market by non-financial corporations, drawn from the flow-of-funds data for the US over 1953:2-2017:3, deflated by the Consumer Price Index (CPI)1. Loans comprise bank loans, mortgages and other loans to companies while bonds include also commercial paper (CP), all at market value. Figure 1.1 shows the annual real growth rate over the whole period of analysis, that is 1953:2-2017:3. 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 -0.3 -0.15 0 0.15 0.3 rSecurities rLoan

Figure 1.1: Annual real growth rate of debt securities (red line) and bank loans (blue line), quarterly frequencies.

The objective of this section is to exploit the benefits of the wavelet approach to analyze the relationships between external sources of finance, namely debt securities and loans. The multi-resolution decomposition anal-ysis, provided by the continuous and discrete wavelet transforms, allows the researcher to answer the following question: to what extent do we need to consider different time horizons when examining the relationships between two variables? Eventually, the task of economists is to explain why we ob-serve those relationships. To answer these questions we perform a wavelet-based exploratory data study using the wavelet tools suitable for the analysis of time-frequency dependencies based on the continuously labeled decom-positions, i.e. the wavelet power spectrum and the wavelet coherence, and then their discretized counterpart, i.e. multi-resolution decomposition anal-ysis. Wavelets contextualize the relationship along the time series helping economists to guess factors that can account for it. Moreover, they identify frequencies at which there are relationships, allowing economists to distin-guish relationships among different economic cycles.

Consistently with theories analyzed, we decide to use aggregate data to underline the macroeconomic perspective. We intend to study how, in

1_{To whom was interest the time series are: Loans (NCBLL), Securities (NCBDSL) and}

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the whole economy, the volume of loans and securities change. Moreover, aggregate data fits wavelets’ needs indeed they are data demanding both in term of frequencies and time series length.

1.3.1 Wavelet Power Spectrum, and Wavelet Coherence

The squared (absolute) values of CWT wavelet coefficients, denoted as |Wx(s, u)|2, show the power of the projection of the signal onto the wavelet

transform at the indicated level of scaling.2 The wavelet power spectrum can be interpreted as the energy density in the time-frequency plane and can be quite revealing about the structure of a particular process, such as the presence of multiscale features. Wavelet power is plotted against both time and frequency and is visualized by using contour plots where the color of each point measures the amount of signal energy contained at a specific scale and location. Hence, reading across the graph at a given value for the wavelet scaling, one sees how the power of the projection varies over time at a given scale, while reading down the graph at a given point in time one sees how the power varies with the wavelet scale (Ramsey et al., 1995).

The color code for power ranges from dark blue (low power) to dark red (high power), with regions with a warmer color corresponding to areas of higher power, that is regions with wavelet transform coefficients of the large modulus. As a result, by showing how the power of the projection of the signal varies with the scale of observation, the wavelet transform can get an indication about the underlying structure of the processes and, in particular, the dominant scales of variation in the data or “characteristic scales” according to the definition of Keim and Percival (2015).

In Figures 1.2 and 1.3,3 _{we show the wavelet power spectra for the}

growth rates of real debt securities and real loans, respectively, where time is recorded on the horizontal axis and the vertical axis gives us the peri-ods and the corresponding scales of the wavelet transform. Hence, reading across the graph at a given value for the wavelet scaling, one sees how the power of the projection varies across the time domain at a given scale. At the same time reading down the graph at a given point in time, one sees how the power varies with the scaling of the wavelet (Ramsey et al., 1995). The power of the projection of the signal onto the wavelet transform at the indi-cated level of scaling is indiindi-cated by color coding,4 _{so that we may evaluate}

the scaling characteristics of the data by examining the color plots of the continuous wavelet transform. This color coding can provide an objective

2

See Appendix 1.A.1 for a formal theoretical presentation of the CWT.

3_{To compute wavelet we use the code of Lu´ıs Aguiar-Conraria and Maria Joana Soares}

(Apr. 1, 2018). ASToolbox. Version 2018.

4_{The color code for power ranges from blue (low power) to red (high power). Regions}

with warmer colors (red, orange and bright green) correspond to areas of high power, that is regions with wavelet transform coefficients of large modulus.

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method for determining the principal time scales present in a signal and also for providing information about the scales at which important features pro-vide a significant contribution. We test the wavelet power 5% significance level5 against the null hypothesis that the data generating process is a red noise AR(1), a black contour line identify the significant area. Moreover, a black contour demarcates the cone of influence corresponding to the region affected by edge effects.

Several things are worth noting from estimated wavelet power spectra. First, the growth rates of real debt securities and real loans have similar modes of variability as it is evident from the distribution of high and low power regions at different time-scales. Low power regions are concentrated at shorter time-scales, namely scales corresponding to periods less than one year, and regions with high power are concentrated at time-scales corre-sponding to the business cycle and longer frequencies. However, if high power regions for real debt securities are mostly concentrated between the mid-1960s and early 2000s, regions with high power for real bank loans are evident in two separate areas: at scales corresponding to higher business cycle frequencies, i.e. 2-4 years, between early 1950s and early 1980s, and in the lower right corner of the time-scale space. In this area, the high power region starting in the early 1990s at intermediate scales, i.e. scales greater than 8 years, gradually extends to include scales corresponding to business cycle frequencies since the early 2000s.

Secondly, the white line shows the maxima of the undulations of the wavelet power spectrum, suggesting the most probable cycle in that window of time (Aguiar-Conraria and Soares, 2014). Concerning the real growth rate of debt, we observe three distinct steady cycles: the first one around 4 years, the second close to 8 and the last just below 16. With regards to the real growth rate of loans, there are only two: one around 4 years and the other close to 8 years which start in the mid-1980s. It deserves to be highlighted that, in some periods, these cycles are nearly overlapping.

The wavelet power spectrum can be very useful to determine the domi-nant frequency modes of variability and to reveal structures at various scales, such as multi-scale features, by identifying the temporal locations of signif-icant events altering the structure or volatility of the sequences. However, the presence of regions with high common power, as is for debt securities and bank loans at business cycle frequencies and longer, does not provide evidence of co-movements at the indicated time-scales. Therefore, although useful for revealing potentially interesting features in the data like charac-teristic scales, the wavelet power spectrum is not the best tool to deal with the time-frequency dependencies between two variables. Indeed, even if two

5_{The statistical significance of the results obtained through wavelet power analysis}

was first assessed by Torrence and Compo (1998) by deriving the empirical (chi-squared) distribution for the local wavelet power spectrum of a white noise signal using Monte Carlo simulation analysis.

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1960 1970 1980 1990 2000 2010 -0.2 -0.1 0 0.1 0.2

(a) Real growth rate of debt securities

(b) Wavelet Power Spectrum

1960 1970 1980 1990 2000 2010 Time 1 2 4 8 16 Period

Figure 1.2: Rectified wavelet power spectrum for the growth rate of real debt securities

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1960 1970 1980 1990 2000 2010 -0.2 -0.1 0 0.1 0.2

(a) Real growth rate of loan

1960 1970 1980 1990 2000 2010 Time 1 2 4 8 16 Period

Figure 1.3: Rectified wavelet power spectrum for the growth rate of real bank loans

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series share similarities as to their high power regions, one cannot infer that their relationship is strong in that regions. To detect and quantify rela-tionships between external financing sources cross-wavelet tools like wavelet coherence and wavelet phase-difference have to be used.6

Figure 1.4 displays the wavelet coherence between the real growth rates of debt securities and bank loans. Between 1-4 years, there is not a steady relationship but it is clearly of substitution. Loans and debt securities alter-nate as a leader variable. The frequencies between 4-8 years are character-ized by significant areas before the early-1970s and after the 1990s. There is a substitution relationship until the mid-1990s and then a complementary relationship to the end of the series. As regards the multi-decadal frequen-cies, there is a strong significant area which ends in the early 1990s and the respective phase-differences is in phase throughout the affected area. Thus at the lowest frequency analyzed we detect a complementary relationship.

The coherence wavelet analysis shows two main findings. First, the rela-tionship is scale-dependent. Statistically significant high power regions are evident at scales corresponding to short economic cycle, business cycle fre-quencies and at the highest time scales, namely scales corresponding to peri-ods between 12 and 16 years. The sign of the relationship is scale-dependent too. Frequencies between 1-4 years are characterized by substitution, multi-decadal by complementarity, while at the business cycle frequencies wavelet detects both complementarity and substitution. Second, the relationship is time dependent, until the end of the 1980s significant areas are mainly at scales corresponding the short economic cycle and at multidecadal cycle; whereas, from the early 1990s significant areas are at the frequencies of the short economic cycle and business cycle.

1.3.2 An economic perspective of the coherence

The analysis of Figure 1.4 suggests that at short economic cycle frequen-cies, loans and securities are alternative sources of funds, result consistent with the “pecking order” theory of corporate finance (Myers, 1984). Cor-porations resort to the most economical source of funds thus loans and se-curities compete for corporations’ demand for funds. The complementarity relationship detected between 12-16 years may be due to a macroeconomic dynamic. Instead, less clear is the relationship at the business cycle fre-quencies. From the mid-1990s phase-differences between 4-8 years turn to be in phase and therefore there is a complementarity relationship between securities and loans, contradicting the traditional theory.

Recent studies suggest why in the last decades banks and markets ex-hibit two types of behavior: competition and complementarity. Holmstrom

6_{See Appendix 1.A.2 for a formal theoretical presentation of the coherence and the}

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Wavelet Coherency (Securities and Loans) 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2 4 8 16 Period 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 -- /2 0 /2

Phase-Differences 1~4 frequency band

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

-- /2

0 /2

1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 Time -- /2 0 /2

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and Tirole (1997) point out that the technological progress could have al-lowed banks to process data and to share information efficiently, making the bank decision a signal for other financial institution. In particular, they cast doubt on whether bond issuance can be fully substituted by bank lend-ing when banklend-ing systems enter in crisis. They stress the complementarity between these two forms of external financing given the monitoring role of bank lending for securities issuance. The idea is supported by the lack of relationship in the 1980s, decade characterized by a serious banking crisis7 in US. On the other hand, from the 1990s, the financial and bank system was characterized by the financial innovation and the so call “deregulation”. These could explain features of the new relationship. Allen and Gale (1997) note that a more complex financial system may complement markets by providing individuals insurance against unforeseen contingencies in obscure states, thereby eliminating the need for individuals to acquire costly state information and reducing their market participation costs. Finally, one of the main novelty concerning both financial innovation and deregulation was the securitization and, since the earliest dates back to the early 1980s, the complementarity observed in the 1990s could be consistent even with the theory of Song and Thakor (2010). Indeed, they argue that thanks to secu-ritization, banks could increase the depth of capital market decreasing the cost of bank equity capital and encouraging banks to hold more capital, so diminishing the rationing of potentially creditworthy relationship borrowers. Even though the analysis of the coherence does not allow the proof or the contradiction of any theories, data confirm that there is a complementarity relationship between loans and securities in the last decades. Having a com-prehensive understanding of the relationship between securities and loans is challenging since it is affected by a multitude of factors. Nevertheless, the finding is consistent with new theories, and the timing suggests clearly that the regulation (deregulation) and financial and technological innova-tions are part of this story. In a nutshell, they have altered the behavior, the decisions, the decision process and even the tasks of banks, financial institutions (e.g. pension funds and insurance companies), and corporates; and the mirror-image of the evolution of the financial system is that change of the sources of funding of corporations.

7_{The Banking Crisis was tremendous leading to 1617 bank failures between 1980 and}

1994, the failure of the 9.14% of banks. The mean of the ROE of the whole sector was negative both in 1987 and 1990. In 1990 the Price-to-Book Value per Share reached its negative record of 50% (FDIC, 1997).

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1.4 The wavelet-based correlation analysis and the

structural break test

Although the evidence provided by wavelet-based exploratory data analysis is not based on subjective interpretation, it may be useful to perform a robustness analysis using more formal alternative testing methods. Hence, we first test for the presence of scale-dependency in the relationship between bank loans and debt securities using wavelet-based correlation analysis.8

Then, we test for the presence of a structural break in their relationship by applying the Quandt likelihood ratio (QLR) test (Quandt, 1960 and Andrews et al., 1996).9

1.4.1 Wavelet-based cross-correlation analysis

After decomposing the real growth rates of debt securities and real loans into their different time scale components using the MODWT we explore the scale-by-scale patterns in the data using cross-correlation analysis. In Figure 1.5 we report the MODWT-based wavelet correlations and cross-correlation coefficients, with the corresponding approximate confidence in-tervals, against time leads and lags for several scales, calculated over the period 1953:4-2017:3. As shown in Table 1 each scale is associated with a particular time period: scale 2 to 4-8 quarter periods, scale 3 to 8-16 quarter periods, and so on. In particular, the correlation coefficient of the growth rate of real debt securities at time t is plotted against the value of the growth rate of real bank loans from t-12 to t+12 quarters. The results from the wavelet cross-correlation analysis are fully consistent with those detected using wavelet coherence analysis. The values of the wavelet correlation coef-ficients at lag 0 are indicative of a negative or positive relationship between the two series. At scales 2 (1-2 years), 3 (2-4 years) and barely 4 (4-8 years) the cross-correlation confirms the substitution and the largest correlation coefficient occurring at scale 3. At scales beyond business cycle frequencies namely scale 5, the sign of the contemporaneous correlation coefficient shifts from negative to positive and denotes a strong positive association between the growth rates of bank loans and debt securities.

With the CWT analysis, we conclude that the relationship in the fre-quency of business cycle changes as a result of the deregulation, the financial innovation and the introduction of securitization. Thus, we can split up the sample into two periods to study if, in that frequencies, we observe a change of relationship between market securities and loans. Figure 1.6 displays the cross-correlation but dividing the sample into two periods before and after 1980:1. Although at Level 3 the relationship between securities and loans still to be of substitution but less correlate, at Level 4 it is evident how the

8_{See Appendix 1.A.4 for a formal theoretical presentation of MODWT.}

9

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-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -1 -0.5 0 0.5 1 Level 2 (a) -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -1 -0.5 0 0.5 1 Level 3 (b) -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -1 -0.5 0 0.5 1 Level 4 (c) -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -1 -0.5 0 0.5 1 Level 5 (d)

Figure 1.5: Wavelet cross-correlation coefficients between debt securities and loans at different time scales

relationship turns out to be of complementarity accordingly with previous findings.

In sum, as to scale-dependency, the results from wavelet cross-correlation analysis confirm the evidence provided by wavelet-based exploratory analysis through the wavelet coherence and phase analysis.

1.4.2 Estimation of break dates with the QLR test

We perform the QLR test to verify if there is a structural break in the rela-tionship between the growth rates of bank loans on debt securities. Figure 1.7 shows the result of the test applied to the relationship using aggregate data (top panel), and the sum of the D3+D4 detail components (bottom

panel). If there is a distinct break in the regression function, the date at which the largest value of the statistic occurs is an estimator of the break date.

The maximum QLR statistics, χ2 _{= 50.75 (p-value: 3.41477e-010) and}

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-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -1 -0.5 0 0.5 1 Level 3 (a) -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -1 -0.5 0 0.5 1 Level 3 (b) -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -1 -0.5 0 0.5 1 Level 4 (c) -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -1 -0.5 0 0.5 1 Level 4 (d)

Figure 1.6: Wavelet cross-correlation coefficients between debt securities and loans at different time scales. On the left between 1953:4 and 1980:1 while on the right since 1980:1 to 2017:3.

data and detail components, respectively, indicate that the null hypothesis of no structural change can be rejected as their values exceed the critical value at the 1% significance level (critical value = 7.78, see Stock and Watson (2008), p. 471). Given the perfect coincidence of the estimated break dates, along with the absence of any structural break detected at shorter, D1 and

D2, and longer, D5, time scales, the structural break test indicates that

the structural change occurred in the aggregate relationship between bank loans and debt securities can be reasonably attributed to the change in the relationship at detail scale levels corresponding to business cycle frequencies, D3 and D4. The timing is compatible with the above.

1.5 The role of ABS

The robust statistical analysis fully corroborates the previous conclusion. We highlight that since the 1990s, at business cycle frequencies, the

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0 5 10 15 20 25 30 35 40 45 50 55 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Test di Wald robusto per un break strutturale

5% QLR critical value 0 5 10 15 20 25 30 35 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Test di Wald robusto per un break strutturale

5% QLR critical value

Figure 1.7: QLR test of the relationship between debt securities and loans: aggregate (top) and D3+D4 detail components (bottom)

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ship between securities and loans turns to be of complementary according to new theories of the external source of funding (Holmstrom and Tirole, 1997, Allen and Gale, 2000 and Song and Thakor, 2010). The background idea is that financial innovation has increased the ability to process big amounts of data and exchange information. Meanwhile, the deregulation has increased the complexity of financial conglomerate, and it has also made banks closely linked to others financial institution (e.g. pension funds and insurance com-panies). The combination of the two led banks to specialize in the evaluation and monitoring the risk and specifically credit risk. As a consequence, banks’ decisions became a signal for the market, and it coordinated securities and loans making them complementary. A perfect example of this mechanism is securitization. Indeed securitization allowed financial institutions to fund companies whose risk was evaluated beforehand by a bank.

To study the securitization effects on the relationship between securities and loans we use partial wavelet. 10 As explained by Mihanovi et al. (2009), Partial wavelet allows the study of the cycle of the relationship between two series after controlling the cycle of another series. Therefore, if a third factor generates a common cycle, through this powerful tool, we can study the original relationship “clean” by the effect of the third factor.

We analyze the relationship between securities and loans since 1986:1 to 2017:3 controlling for the ABS (Asset Backed Securities) which are the result of securitizations. As wavelet analysis is data demanding both in frequencies and time, we cannot use the ABS outstanding series, since the monthly data is available just from 2001. Instead, we use the aggregate liabilities of the issuers of Asset-backed Securities which is available since 1985. The pools of assets of the issuers of ABS include home, multifamily, and commercial mortgages, consumer credit (such as automobile and student loans and credit card receivables), trade credit, Treasury securities, agency- and GSE-backed securities, other loans and advances, and miscellaneous assets. Instead, the Liabilities of this sector are the securities issued by the SPVs and are typical ABS11_{. Figure 1.8 displays the time series of the real growth rate of ABS}

and the corresponding continuous wavelet.12 Even if it has no significant area, the most probable cycles are two steady cycles one at 8 years and the other at 16 years which are nearly overlapping with those in the CWT of securities and loans.

Figure 1.9 displays, on the left-hand side, the wavelet coherence between 10

See Appendix 1.A.3 for a formal theoretical presentation of the partial wavelet trans-form.

11

To whom was interest the code of the time series is IABSDSL and we use the dataset Fred (Federal Reserve Bank of St. Louis dataset). For a more punctilious definition of Issuers of ABS see BGFRS (2018).

12_{The series presents an issue, it starts from 0 in 1983:2, and thus the first years looks}

like it has a big grow when actually the volume of ABS is negligible. For this reason we exclude the first three and half of observations.

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1990 1995 2000 2005 2010 2015 -0.5 -0.25 0 0.25 0.5

(a) Real growth rate of ABS

1990 1995 2000 2005 2010 2015 Time 1 2 4 8 16 Period

Figure 1.8: Rectified wavelet power spectrum for the growth rate of real ABS

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ABS and securities, while on the right-hand side, the wavelet coherence between ABS and loans. At frequencies within 1-4 years, there is a significant (in phase) relationship between ABS and securities from the mid-2000s to the early 2010s. At longer frequencies (4-8 years), the relationship starts in the mid-2000s, and it is steady until the end of the series. Concerning the wavelet coherence between ABS and loans, at frequencies between 1-4 years, there is just a small significant stain from the end-1990s to the mid-2000s, which is before out-of-phase and after in phase, and ABS leading. Instead, at 4-8 years frequencies, there is a steady and significant relationship from the mid-1990s to the mid-2010s. The phase difference is in phase and ABS leading throughout the significant area.

W. Coherency ABS and Securities

1990 1995 2000 2005 2010 2015 2 4 8 16 Period 1990 1995 2000 2005 2010 2015 -- /2 0 /2

W. Coherency ABS and Loan

1990 1995 2000 2005 2010 2015 2 4 8 16 Period 1990 1995 2000 2005 2010 2015 -- /2 0 /2

1990 1995 2000 2005 2010 2015

-- /2

0 /2

1990 1995 2000 2005 2010 2015

-- /2

0 /2

Figure 1.9: On the left, the wavelet coherence between ABS and securities, while on the right the wavelet coherence between ABS and loans.

The idea is to verify if ABS have any relevance in the relationship be-tween securities and loans. If the relationship changes, apart from some years, we have empirical evidence that securitization has influenced the re-lationship.

In Figure 1.10 we plot the wavelet coherence between the real growth rates of debts securities and banks’ loans on the left-hand side, and the par-tial coherence after controlling for the real growth rate of ABS from 1986:4

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W. Coherency Securities and Loan 1990 1995 2000 2005 2010 2015 2 4 8 16 Period

W. Partial Coherency S., L.; ABS

1990 1995 2000 2005 2010 2015 2 4 8 16 Period 1990 1995 2000 2005 2010 2015 -- /2 0 /2

1990 1995 2000 2005 2010 2015

-- /2

0 /2

1990 1995 2000 2005 2010 2015

-- /2

0 /2

1990 1995 2000 2005 2010 2015

-- /2

0 /2

Figure 1.10: On the left: wavelet coherence between real debt securities and real loans. On the right: partial wavelet coherence controlled per ABS

to 2017:3 on the right-hand side. Two main findings concerning the re-lationship between securities and loans and the effect of securitization are evident. First, there is a consistent change in the relationship if we control for ABS. Second, the significant area in the partial coherence is different from the simple coherence. At frequencies between 1-4 years, there are just minus variations. There are yet again two significant areas out-of-phase, but they are wider (from the early 1990s to the mid-1990s and from the early 2000s to the early 2010s). Instead, at frequencies within 4-8 years, ABS effect both the relationship and its sign. First, in the coherence, there is a complementary relationship since the beginning and throughout the significant area, while in partial coherence the relationship is of substitu-tion. Second, the significant area that starts in the mid-2000s in the partial wavelet disappears, this means that securities and loans have a common cycle with ABS during that period. This conclusion is even more evident in Figure 1.11 which reports the MODWT between ABS and securities and loans respectively.

Thus, findings suggest that ABS played an influential role in the rela-tionship between securities and loans. At the business cycle frequencies,

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-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -1 -0.5 0 0.5 1 Level 4 (a) -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -1 -0.5 0 0.5 1 Level 4 (b)

Figure 1.11: Wavelet cross-correlation coefficients between ABS and debt securities on the left-hand side and between ABS and loans on the right-hand side, both at frequencies of 4-8 years.

after controlling for the ABS, there is a significant area in which the rela-tionship between securities and loans changes, and an area that is no longer significant. These empirical results are consistent with new theories that forecast a complementary and a co-evolution relationship. In particular, these results confirm the idea that a more interlinked financial system and the last decades of deregulation have remarkably changed the financial sys-tem, the bank system and the way of funding of corporation. Understanding how the financial innovation and the deregulation impacted on the behavior, the activities and the interaction of banks, corporations, financial institu-tions, and even authorities are challenging but mandatory to improve our comprehension of the whole economic and financial system.

1.6 Conclusion

In this paper, we investigate the substitutability/complementarity issue be-tween bank lending and bond issuance using the time-frequency lens of wavelet analysis. The analysis pursues a twofold goal, to improve the knowl-edge about the relationship between the external source of corporate fund-ing, and to contextualize the relationship understanding what factors may explain changes.

Regarding the wavelet analysis, two main findings emerge. First, the relationship between bank loans and debt securities is scale-dependent: at scales corresponding to business cycle frequencies, there is evidence of sub-stitutability for corporations between bond finance and bank loans, whereas at longer time scales a complementarity relationship appears. However, these statistically significant negative and positive relationships at different

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time scales between bank lending and bond issuance end respectively in the mid-1970s, and in the early-1990s as a consequence of a structural change in the relationship. Secondly, in the early-1990s a more complex relationship starts between 4-8 years frequencies in which securities and loans are both substitutes and complementary at different times.

Therefore, what stands out are two distinctly different eras, the first is scale dependent while the second is time-dependent. We suppose that the complexity of the new relationship is also due to financial innovation and the deregulation process. We test this hypothesis by exploiting the partial wavelet properties to disentangle the securitization effect. In particular, we study the relationship between securities and loans after controlling for ABS. The result corroborates the idea that the financial innovation and the deregulation have remarkably changed the relationship and they have played an influential role in made it of complementarity and co-evolving.

With regards the main implication for theoretical models aiming to ex-plain the complex relationship between external financing sources, our re-sults are consistent with the new theories developed by Holmstrom and Tirole (1997), Allen and Gale (1997) and Song and Thakor (2010). The analysis confirms that the modern relationship between securities and loans is not only of substitution, but it can be of complementarity or co-evolution. The study of the ABS’s effects underpins the idea that the introduction of the securitization has undoubtedly tighten the relationship among banks, other financial institution (e.g. pension fund and insurance company) and the financial market. But more in general, the analysis suggests how the eco-nomic and financial context influences the bank and financial system. Any significant change in the boundary condition (e.g. technological innovation and regulation reform) can alter the decisions of banks and corporations, and thereby the relationship between securities and loans.

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1.A

Theorical presentation of Wavelet

The wavelet transform uses a basis function that is dilated or compressed and shifted along the signal and is, therefore, able to attain an optimal trade-off between time and frequency resolution levels (Lau and Weng, 1995, Mallat, 1999). The wavelet transform provides a flexible time-scale window that narrows when focusing on small-scale features and widens on large-scale features. Indeed, when the scale increases (decreases) the wavelet function becomes wider (smaller) and describes only the broad (fine) features of the signal, i.e. coarse (higher) resolution, thus displaying good time resolution and poor frequency resolution at high frequencies, and good frequency reso-lution but poor time resoreso-lution for low frequencies. The good frequency and time localization properties of the wavelet transform, by allowing to study features of the signal locally with a detail matched to its scale, make wavelets a powerful tool for analyzing complex signals. Namely non-stationary sig-nals which have short-lived transient components and features at different scales or singularities.

Wavelets provide a unique method for the analysis of economic rela-tionships on a scale-by-scale basis by using the wavelet tools based on the continuously labeled decompositions, i.e. the wavelet power spectrum, co-herence, partial-coherence and phase, and then their discretized counterpart, i.e. multiresolution decomposition analysis. Specifically, the continuous wavelet transform allows a process of exploratory data analysis that preserve its main advantages, i.e. to look for flexible ways to examine data without preconceptions, to let data suggest questions and generate hypothesis, and to promote a deeper understanding of the estimation results, while simulta-neously avoiding the main disadvantages of exploratory analysis (Gallegati and Ramsey, 2012). Moreover, through the time scale decomposition prop-erty of wavelet analysis, it is possible to test directly for parameters variation across frequencies in a regression model by using time scale regression anal-ysis (e.g. Gallegati et al., 2011).

1.A.1 The continuous wavelet transform (CWT)

The wavelet transform uses a set of orthogonal basis functions which are local, not global. Thus, wavelet analysis, by dealing with local aspects of a signal, provides us with a method having the ability to handle a variety of non-stationary and complex signals.

The essential characteristics of wavelets are best illustrated through the development of the continuous wavelet transform (CWT). We seek “small wave”functions13 ψ(u) such that:

13_{There are “large wave”(e.g the cosine function) namely wave that its integral of the}

square does not converge to 1, even though its integral is zero, while a “small wave”obeys both constraints.

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Z

ψ(u)du = 0, Z

ψ(u)2du = 1.

The continuous wavelet transform (CWT) of a signal x(t) with respect to the wavelet function ψ is a function Wx(s, u)

Wx(s, u) =

Z _∞

−∞

x(t)ψ(s,u)(t)dt,

where the wavelet basis, called “mother wavelet”, defined as

ψ_(s,u)(t) = _√1 sψ t − u s ,

is a functions of two parameters s and u. The first is a scaling or di-lation factor that controls the length of the wavelet, the latter a location parameter that indicates where the wavelet is centered along the signal. The set of CWT wavelet coefficients, each representing the amplitude of the wavelet function at a particular position and for a particular wavelet scale, is obtained by projecting x(t) onto the family of “wavelet daughters” ψ(s,u)

obtained by scaling and translating the “mother wavelet” ψ by s and u, respectively.

The application of the continuous wavelet transform requires the spec-ification of the wavelet function and the treatment of boundary conditions as the continuous wavelet transform, with other types of transforms, suffers from a distortion problem. This is due to the finite time series length which affects wavelet transform coefficients at the beginning and end of the data series. Wavelet transform coefficients are then calculated using the Morlet wavelet, a widely used wavelet among the numerous types of wavelet fami-lies available, e.g. Mexican hat, Haar, Daubechies, etc.. The Morlet wavelet has optimal joint time-frequency concentration as it attains the minimum uncertainty value of the corresponding Heisenberg box. Moreover, being a complex wavelet, it produces complex transforms and thus can provide us with information on both amplitude and phase.

ψ(t) = π−14eiω0t− e− t2

2 .

with ω0= 6 (where ω0 is a dimensionless frequency) since this particular

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Grinsted et al., 2004) and also simplifies the interpretation of the wavelet analysis because the wavelet scale, s, is inversely related to the frequency, f ≈ 1/s.

As with other types of transforms, the CWT applied to a finite length time series inevitably suffers from border distortions. This is due to the fact that the values of the transform at the beginning and the end of the time series are always incorrectly computed, in the sense that they involve missing values of the series which are then artificially prescribed, the most common choices are zero padding extension of the time series by zeros or periodization. Since the effective support of the wavelet at scale s is pro-portional to s, these edge effects also increase with s. The region in which the transform suffers from these edge effects is called the cone of influence. In this area of the time-frequency plane, the results are unreliable and have to be interpreted carefully (see Percival and Walden, 2006).

1.A.2 Wavelet coherence, and Phase-Differences

Let Wx and Wy be the continuous wavelet transform of the signals x(·)

and y(·), their cross-wavelet power is given by |Wxy|=|WxWy| and depicts

the local covariance of the two time series at each scale and frequency (see Hudgins et al., 1993). Being the product of two non-normalized wavelet spectra, the cross-wavelet can identify the significant cross-wavelet spectrum between two time series, although there is no significant correlation between them. Defined the wavelet cross spectrum (complex wavelet coherency), ςxy,

by ςxy = S(s−1_W xy(s, u)) [S(s−1_|W_x_{(s, u))|}2_)S(s−1_|W_y_{(s, u))|}2_)]12 , (1.1)

the wavelet coherence is defined as the absolute value of the wavelet cross spectrum normalized by the wavelet spectra of each signal,

Rxy = |S(s

−1_W

xy(s, u))|

[S(s−1_|W_x_{(s, u))|}2_)S(s−1_|W_y_{(s, u))|}2_)]12

, (1.2)

where S is a smoothing operator (see Torrence and Webster, 1999). The squared wavelet coherence coefficient R2

xy, ranging between 0 and 1, is

anal-ogous to the squared correlation coefficient in linear regression. It can be considered a direct measure of the local correlation between two time series at each scale (Chatfield, 1989) and used to detect the time and frequency intervals where two phenomena have strong interactions. Moreover, from the imaginary and real parts of the cross wavelet transform we can get in-formation about the relative position of the two series through the phase difference, defined as:

φxi,xj = tan−1

ℑ[W x_i, x_j(s, u)] ℜ[W xi, xj(s, u)]

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coherency. However, we are not interested in deepening the analysis of the partial wavelet from a theory point of view. We confine ourselves to report the partial complex coherency in the case of three variables that is our case:

ςxy,z = ςxy− ςxzςyz∗ q (1 − R2 xz)(1 − Ryz2 ) (1.4)

where ςxixj is defined in Equation 1.1 and Rxz by Equation 1.2

14_.

Mean-while, ς∗

yz is the complex conjugate and therefore it is equal to ςzy. Finally,

we denote the partial coherency with rxy,z = |ςxy,z| and the partial phase

difference with φxy,z = Arctan S(ς_xy,z) R(ςxy,z) . (1.5)

The partial wavelet tool allows going beyond bivariate wavelet analysis to higher order variate wavelet analysis.

1.A.4 Discrete Wavelet Transform (DWT)

The CWT contains a high amount of redundant information so that it is computationally impossible to analyze a signal using all wavelet coefficients. A more parsimonious representation of the evolution over time of the peri-odic components of a signal is provided by the discrete wavelet transform (DWT) which discretizes the transform over scale and over time through the dilation and location parameters. In the DWT only a limited number of translated and dilated versions of the mother wavelet are used to decom-pose the original signal by selecting t and λ in a way that the information contained in the signal can be summarized in a minimum number of wavelet coefficients.

The general formulation for the continuous wavelet transform can be restricted to the definition of the discrete wavelet transform (DWT) by dis-cretizing the parameters s and u. In order to obtain an orthonormal basis a transform of the scaling parameter, s = sj₀, and the Nyquist sampling rule, u = ksj₀T , are used. The key difference between the CWT and the DWT lies in the fact that the DWT uses only a limited number of translated and dilated versions of the mother wavelet to decompose the original signal. In others words, a continuous wavelet transform can be restricted to the defini-tion of the discrete wavelet transform when the computadefini-tion is done octave by octave, i.e. λ0= 2, to get the equation for the mother wavelet:

ψj,k(t) = 2−j/2ψ t − 2 j_k

2j

. (1.6)

14_{The interested reader can find all the details and the proof in Aguiar-Conraria and}

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This function represents a sequence of rescaleable functions at a scale of λ = 2j, j = 1, 2, ...J, and with time index k, k = 1, 2, 3, ...N/2j. The wavelet transform coefficient of the projection of the observed function f (t) for i = 1, 2, 3, ...N, N = 2J on the wavelet ψj,k(t) is given by:

dj,k ≈

Z

ψj,k(t)f (t)dt,

j = 1, 2, ..J. (1.7)

For a complete reconstruction of a signal f (t), one requires a scaling func-tion, φ(·), that represents the smoothest components of the signal. While the wavelet coefficients represent weighted “differences” at each scale, the scaling coefficients represent averaging at each scale. One defines the scaling function, also know as the “father wavelet”, by

φJ,k(t) = 2−J/2φ t − 2 J_k

2J

, (1.8)

and the scaling function coefficients vector is given by: sJ,k ≈

Z

φJ,k(t)f (t)dt. (1.9)

By construction, we have an orthonormal set of basis functions, whose detailed properties depend on the choices made for the functions φ(·) and ψ(·) (see for example the references cited above as well as Daubechies (1992) and Silverman (1999)). At each scale, the entire real line is approximated by a sequence of “non-overlapping” wavelets. The deconstruction of the function f (t) is therefore: f (t) ≈ X k sJ,kφJ,k(t) + X k dJ,kψJ,k(t) + X k d_J−1,kψ_J−1,k(t) + ... +X k d1,kψ1,k(t) (1.10)

The above equation is an example of the Discrete Wavelet Transform, based on an arbitrary wavelet function, φ(.). For the DWT, where the number of observations is N, N = 2J_{, the number of coefficients at each}

scale is:

N = N/2J+ N/2J + N/2J−1+ ...N/4 + N/2. (1.11) That is, there are N/2J _{coefficients s}

J,k, N/2J coefficients dJ,k, N/2J−1

coefficients d_J−1,k... and N/2 coefficients d1,k.

Further, the approximation can be re-written in terms of collections of coefficients at given scales. Define