Pierre Pessarossi
email: pierre.pessarossi@acpr.banque-france.fr

Frédéric Vinas
email: frederic.vinas@psemail.eu

The supply of long-term credit after a funding shock: evidence from 2007-2009

Keywords: financial institutions, liquidity risk, loan maturity JEL: G01, G21, G28

We study banks' supply of long-term credit after a negative funding shock. Thanks to a unique database at bank-firm level, we take advantage of the exogenous interbank market freeze in 2007-2008 to assess the causal relation between French banks' liquidity risk and their lending. We find that banks with higher funding risk and more maturity transformation provided a lower supply of long-term loans after the shock, even controlling for credit demand. Short-term lending supply is however unaffected. These findings help explain the severity of the recession that followed the liquidity crisis. And they support Basel III liquidity regulation. This regulation should have a stabilising effect on long-term lending in times of funding stress.

Introduction

Long-term loans allow firms to switch toward more productive investment. By promoting such assets, banks contribute to economic growth [START_REF] Bencivenga | Financial intermediation and endogenous growth[END_REF]. An important question is how banks grant long-term loans in times of funding crisis. Empirical literature on the transmission of liquidity shocks to the real economy [START_REF] Iyer | Interbank liquidity crunch and the firm credit crunch: Evidence from the 2007-2009 crisis[END_REF][START_REF] Schnabl | Financial globalization and the transmission of credit supply shocks: Evidence from an emerging market[END_REF][START_REF] Cornett | Liquidity risk management and credit supply in the financial crisis[END_REF]) has already shown that banks reduce their supply of credit in times of liquidity crisis. Surprisingly, the empirical literature is silent on the reallocation of maturities. They show that banks that were more exposed to the liquidity shock decreased their credit supply, but they do not analyse composition effects on the maturity. If, in addition to the volume effect, banks reallocate their portfolio from long-term loans to short-term loans in times of crisis, it would result in a larger drop in output. Such an amplifying mechanism of the initial shock would further impact long-run productivity. Banks might have incentives to reduce the supply of long-term loans (i) because such loans constitute one of the most illiquid claims in their balance sheet; (ii) because the value of firms' collateral is decreasing at that time [START_REF] Bernanke | Agency Costs, Net Worth, and Business Fluctuations[END_REF][START_REF] Bernanke | The financial accelerator in a quantitative business cycle framework[END_REF], which could be particularly relevant for long-term loans; or (iii) because higher shortterm profit opportunities may have motivated strategic liquidity hoarding during the crisis [START_REF] Radde | Flight to liquidity and the great recession[END_REF]. So it is of primary interest to analyse the effects of a funding shock on bank loan maturities.

The paper is also motivated by the post-crisis regulation Basel III. This regulation introduces liquidity constraints on banks, the so-called liquidity coverage ratio (LCR) and net stable funding ratio (NSFR). The LCR obliges banks to store liquid assets to mitigate short-term funding stress, while the NSFR obliges banks to fund long-term loans from long-term Documents de travail du Centre d'Economie de la Sorbonne -2015.73 funding. The paper tests empirically the effectiveness of a stable funding ratio in a time period that regulation did not exist.

The paper proceeds in three steps. First, we analyse whether banks with less stable funding reduce their supply of long-term loans after the shock. Second, we test whether higher maturity transformation impacts the supply of long-term loans. Third, we analyse whether funding constraints on long-term loans differ depending on firm size.

We use the Great Recession as a natural experiment of a funding shock impacting the French banking system from 2007Q3. From August 2007, interbank refinancing spreads increased dramatically [START_REF] De Socio | The interbank market after the financial turmoil: squeezing liquidity in a "lemons market" or asking liquidity "on tap[END_REF]; the repo markets and unsecured money market funds stopped rolling over banks short-term debts [START_REF] Gorton | Securitized banking and the run on repo[END_REF][START_REF] Duffie | The failure mechanics of dealer banks[END_REF]. This situation caught banks by surprise as they were used to an environment of abundant liquidity [START_REF] Brunnermeier | Deciphering the liquidity and credit crunch 2007-2008[END_REF]. The funding shock culminated in a paralysis of the interbank market after the failure of Lehman Brothers in September 2008. Our paper considers the funding shock as the main transmission channel constraining the French banking sector because there was no bursting of a real estate bubble in France that could have impacted banks on their asset side, unlike other banking systems such as in the USA, Spain and Ireland. Despite this, the real sector was affected by the financial crisis with a severe output loss in the following years: the cost of the crisis in France is estimated at -1.2% of GDP in 2008and -4.2% of GDP in 2009[START_REF] Bricongne | De la crise financière à la crise économique, l'impact des perturbations financières de 2007-2008 sur la croissance de sept pays industrialisés[END_REF]. This framework provides an appropriate set-up to test the effect of banks' funding risk on their long-term credit supply to non-financial firms.

To properly assess the behaviour of banks, a major challenge is to differentiate supply effects from demand effects. After August 2007, the deterioration of the national and international economic outlook decreased investment opportunities for firms, lowering their demand for credit. To control for change in credit demand and properly identify the change in credit supply, [START_REF] Khwaja | Tracing the impact of bank liquidity shocks[END_REF] proposed a methodology based on bank-firm credit Documents de travail du Centre d'Economie de la Sorbonne -2015.73 relationships. We implement their approach with a highly disaggregated loan database from the Banque de France that reports each bank-firm credit exposure quarterly. We take advantage of the numerous firms that have multi-bank relationships to measure the change in credit for a given firm. When several banks lend to the same counterparty, this allows us to measure how banks with different characteristics change their lending towards the same firm.

In this empirical framework, we are thus able to control for firms' characteristics such as firm's credit risk and credit demand. In a nutshell, we analyse the change in banks' credit supply between 2007 and 2009 for several banks lending to the same firm.

Our framework compares the change in the supply of long-term credit for a given counterparty before and after the shock. To measure how banks with different liquidity risk reacted to the shock, we merge our loan data with balance sheet data collected from the French prudential authority, the ACPR. [START_REF] Drehmann | Funding liquidity risk: definition and measurement[END_REF] define funding liquidity risk as 'the possibility that over a specific horizon the bank will become unable to settle obligations with immediacy'. To capture this idea, we use banks' ratio of retail deposits to assets. Retail deposits appeared to be a resilient source of funding during the 2007-2008 financial turmoil, especially compared to other sources of short-term funding [START_REF] Cornett | Liquidity risk management and credit supply in the financial crisis[END_REF][START_REF] Gorton | Securitized banking and the run on repo[END_REF][START_REF] Bcbs | Basel III: the net stable funding ratio[END_REF]. In France, there was no bank run by retail depositors during the crisis, indicating that banks that relied heavily on that source of funding should be more immune from the funding shock. We thus expect them to maintain their supply of long-term loans throughout the period.

We complement this measure of funding risk with a maturity mismatch measure, namely the ratio of long-term lending to deposits. This alternative measure captures the proportion of illiquid assets in the bank's stable funding resources. We expect that banks with a higher proportion of illiquid assets will be more constrained in their lending after the funding shock. As robustness checks, we compare our results with alternative measures of funding risk and maturity mismatch.

Theoretical papers of financial frictions show that liquidity shocks are transmitted to the real economy through frictions at the bank level and firm level (Bernanke and Blinder, 1988;Bernanke, Gertler and Gilchrist, 1999;[START_REF] Holmström | Financial intermediation, loanable funds, and the real sector[END_REF]. Empirical papers analyse the credit channel using correlation at the aggregate level [START_REF] Bernanke | Nonmonetary effects of the financial crisis in propagation of the great depression[END_REF], crosssection analysis at the bank level [START_REF] Kashyap | What do a million observations on banks say about the transmission of monetary policy?[END_REF] or natural experiments (Khwaja and Mian,2008[START_REF] Schnabl | Financial globalization and the transmission of credit supply shocks: Evidence from an emerging market[END_REF][START_REF] Iyer | Interbank liquidity crunch and the firm credit crunch: Evidence from the 2007-2009 crisis[END_REF][START_REF] Kapan | Liquidity shocks and the supply of credit after the 2007-2008 crisis[END_REF]. [START_REF] Khwaja | Tracing the impact of bank liquidity shocks[END_REF], [START_REF] Schnabl | Financial globalization and the transmission of credit supply shocks: Evidence from an emerging market[END_REF] and [START_REF] Iyer | Interbank liquidity crunch and the firm credit crunch: Evidence from the 2007-2009 crisis[END_REF] use credit register data to determine how liquidity shocks affect lending supply. We follow the same empirical strategy by using the 2007-2009 interbank market freeze as an exogenous liquidity shock and by relying on credit register data to estimate the change in loan supply.

The closest papers to our work are [START_REF] Iyer | Interbank liquidity crunch and the firm credit crunch: Evidence from the 2007-2009 crisis[END_REF], [START_REF] Kapan | Liquidity shocks and the supply of credit after the 2007-2008 crisis[END_REF], [START_REF] Cornett | Liquidity risk management and credit supply in the financial crisis[END_REF] and [START_REF] Ivashina | Bank lending during the financial crisis of 2008[END_REF]. These papers analyse bank lending supply during the 2007-2008 financial crisis, but they do not highlight the causality between banks' funding risk and the maturity of the loans they grant after the shock. We contribute to the literature in several ways. First, our approach uses several measures of funding risk and maturity mismatch. Notably, this paper is the first (to the best of our knowledge) to use the residual maturity of banks' balance sheets to test how funding risk and maturity mismatch at bank level affects lending. It thus contributes to the debate on liquidity regulation like the NSFR, which is aimed at building a more resilient banking sector by controlling banks' maturity mismatch and funding risk. 1Secondly, this paper analyses the effect of a funding shock on the maturity of credit. Nonfinancial firms, especially smaller ones with limited access to capital markets, rely heavily on long-term loans for their investment in fixed assets. Distinguishing the effect of a funding shock on long-term loans and short-term loans can highlight the composition effect that a broad view of loan supply would not. It helps explain the observed consequences of the global financial crisis of 2007-2008 for the real sector.

This paper is also related to an emerging literature that tries to capture the strengths and weaknesses of bank's balance sheets with regard to liquidity (e.g. [START_REF] Berger | Bank liquidity creation[END_REF]Brunnermeier, Gorton andKrishnamurthy, 2011, 2012). In this paper, we compare how different characteristics of banks' balance sheets make them more resilient to a liquidity shock.

Our main results show that (i) banks with fewer deposits and more maturity mismatch provided a smaller supply of long-term loans compared to other banks, (ii) on the other hand, no significant difference exists between banks regarding the supply of short-term loans, (iii) smaller firms bear the brunt of the decline in the supply of long-term loans from non-resilient banks.

Some policy implications arise from this paper. The Basel III NSFR liquidity regulation mainly relies on deposits and long-term liabilities [START_REF] Bcbs | Basel III: the net stable funding ratio[END_REF]. The paper gives support to this regulation and shows how the future liquidity regulation might sustain economic growth during times of funding stress. By insulating banks from funding stresses, these shocks are less likely to be transmitted to banks' supply of long-term loans. This liquidity regulation thus has positive implications for economic growth in periods of downturn.

The rest of the paper is organised as follows. Section 2 presents the data and identification strategy. Section 3 describes the results. Robustness checks are set out in Section 4. Section 5 concludes.

Documents de travail du Centre d'Economie de la Sorbonne -2015.73

Data and methodology

Data

We employ three different databases in the study. The loan-level data comes from the Banque de France's credit register. The bank-level database is supplied by the French prudential authority (the Autorité de Contrôle Prudentiel et de Résolution -ACPR). Finally, we also have access to the Banque de France's database on non-financial firms that provide information on firm size.

Loan-level data

The Banque de France's central credit register database describes loans granted by banks to all firms in France. Each line describes the credit exposure of a bank to a firm where the total credit exposure is greater than EUR 25,000. The database is updated quarterly. Information on maturity is provided and we can distinguish between short-term and long-term loans (i.e. loans with a maturity greater than one year). To measure the change in long-term lending through the funding crisis, we consider two dates: 2007Q2 and 2009Q4. The 2007Q2 exposures describe banks' exposure to non-financial firms just before the shock. The 2009Q4 exposures describe banks' exposure after the shock. 2We remove financial institutions from the sample of borrowers. For the remaining firms, we take all bank-firm relationships that appear at both dates and measure the change in lending at the bank-firm level by the change in their respective credit exposure between 2007Q2 and 2009Q4. Hence we focus here on the intensive margin. Finally, we remove firms that have a single banking relationship as our identification strategy relies on firms borrowing from several banks. The change in lending is then computed as the difference in the logarithm of credit exposures between 2007Q2 (before the funding shock) and 2009Q2 (after the funding shock). The variable is then winsorised at the 1% and 99% levels to avoid the effect of potential outliersextreme credit growth figureson the results. We employ the same methodology to compute the change for long-term and short-term lending.

Panel A of Table 2 presents descriptive statistics at the loan level. The average long-term credit exposure in 2007Q2 is EUR 0.769 million, with a huge variation between counterparties (the standard deviation is EUR 5.786 million). This reflects the fact that the database covers borrowers from very small entities to large multinational firms.

The change in long-term lending between 2007Q2 and 2009Q4 is strongly negative, at -11.9% on average. This confirms that long-term credit to firms fell sharply during the crisis in France. However, the change in short-term credit exposures was 'only' -3% on average. This suggests that the decrease in loan supply after the funding shock was not alike for short-term and long-term loans. Next, we match this loan-level dataset with bank level data.

Bank-level data

The bank-level database contains financial statements of French domestic banks at the unconsolidated level. The sample covers all commercial and cooperative banks operating in France during 2007Q2-2009Q4. The data covers detailed information on balance sheets and financial statements. In addition, the data describes residual maturities for assets and liabilities.

Documents de travail du Centre d'Economie de la Sorbonne -2015.73 Some banks merged between 2007Q2 and 2009Q4. This could lead to an artificial increase in credit exposures to certain borrowers. To correct for this potential bias, we take into account mergers and acquisitions between banks by aggregating the data of entities concerned prior to the merger.

We exclude banks with marginal lending activity, i.e. banks that have less than a thousand bank-firm relationships in the credit register. Our sample ends up with 133 banks.

Panel B in Table 2 gives some descriptive statistics on bank-level data. All balance sheet data is taken in 2007Q2, prior to the funding shock. In our sample, banks have an average asset size of EUR 39 billion and the median is EUR 8.9 billion. A large proportion of their activity is dedicated to lending: the median share of loans to customers in their total assets stands at 67.8%. Deposits constitute a large share of their liabilities, with a median value of 40.6% of their total assets. The median ratio of long-term loans to deposits is 1.282, indicating that banks rely on other sources than deposits to fund their long-term credit. This ratio decreases to 0.822 when we take into account both deposits and liabilities with a residual maturity over one year.

Firm size data

The firm-level database reports annual information on firms operating in France. To distinguish firms by size, we consider firms' turnover in 2006Q4 (see Panel C of Table 2). This reporting date is the latest available before the funding shock. There are 119,041 firms in our sample. These firms borrow before and after the funding shocks and have multi-bank relationships. Firms with a turnover below EUR 10 million or with no information on turnover 3 are deemed to be small. They represent 91.9% of our sample. Firms with a turnover between EUR 10 and EUR 100 million are deemed to be medium-sized. They represent 7.1%.

Firms with a turnover over EUR 100 million are deemed to be large. They represent the remaining 0.9% of the sample.

Methodology

In our empirical strategy we want to determine how banks with high funding risk and maturity mismatch adjust their supply of long-term loans after the funding shock.

Measuring banks' funding risk and maturity mismatch

Our goal is to estimate the behaviour of banks' long-term lending during a funding crisis. We thus need to measure the exposure of banks to funding stress. To this end, we compute measures of funding risk and maturity mismatch.

Our funding risk measure is the deposits to assets ratio. Traditionally, deposits have been regarded as a fragile source of funding for banks [START_REF] Diamond | Bank runs, deposit insurance, and liquidity[END_REF]. Recent evidence however suggests that retail depositors are a stable liability. [START_REF] Cornett | Liquidity risk management and credit supply in the financial crisis[END_REF] show that deposits were a stable source of funding during the 2007-2009 financial crisis.

Explicit deposit guarantee schemes largely explain the stability of non-maturing deposits.

Government subsidies on certain deposit contracts also play a role in this stability4 [START_REF] Schlueter | Bank funding stability, pricing strategies and the guidance of depositors[END_REF].

However, measuring the funding risk might not be sufficient to capture the banks' resilience to a liquidity shock. Banks might have a low funding risk but a high maturity mismatch. We introduce a maturity mismatch ratio to capture this feature of banks' balance sheets. Our maturity mismatch measure is the long-term loans to deposits ratio. This ratio captures the relation between long-term assets and stable funding. A bank with a higher proportion of (illiquid) long-term loans needs a larger proportion of stable funding in order to be immune to the funding shock. If its maturity mismatch is too large, the bank will face difficulties in rolling over its debt (e.g. due to a lack of assets to be pledged or sold). The bank can thus be forced to reduce lending due to the loss of a funding source. The bank is also incentivised to hoard liquidity in order to build up a larger liquidity buffer after the shock.

In addition to these measures, we also check the robustness of our results against alternative definitions of funding risk and maturity mismatch (see Section 4).

Identification strategy

We study how banks adjust their long-term lending during a funding crisis. The funding shock is expected to be transmitted to the real sector by a decrease in long-term lending from banks which were not insulated from the funding shock. We hypothesise that the funding risk and maturity mismatch of banks before the shock (2007Q2) explain their lending behaviour after it (captured through the exposures of 2009Q4). Hence, banks with a larger deposit and a lower maturity mismatch should have a greater supply of long-term credit throughout the period. Obviously, this graphical evidence is polluted with other factors affecting long-term credit exposures during this period. A major challenge is to disentangle firms' demand effects from the banks' supply effect we want to estimate. It might be that demand was much lower for banks having a lower deposit ratio. In that case, the decline in loan supply would result from firms' choices and not bank supply. The next step in our identification strategy is thus to disentangle supply from demand effects in bank lending.

To cope with this issue, we follow the methodology proposed by [START_REF] Khwaja | Tracing the impact of bank liquidity shocks[END_REF] and [START_REF] Gan | The real effects of asset market bubbles: loan and firm level evidence of a lending channel[END_REF]. The identification strategy relies on the fact that a large proportion of borrowers have several banking relationships over the period considered. Thus, it is possible to study how several banks adjust their lending to the same counterparty. By doing so, we control for possible demand bias (if demand was low for a particular group of banks) and other borrower characteristics such as risk and investment opportunities. In practice, it consists in including firm fixed effects in the specification. Fixed effects capture firms' heterogeneity and remove the suspicion of an omitted variable bias that could be correlated with bank characteristics.

As many banks in our sample belong to larger banking groups, one last concern is that banking groups were not necessarily exposed to the same extent to the funding shock. biasing our results. To this end, we add banking group fixed effects into our specification. 5

Thus, in our most constrained specification, we analyse the difference in long-term lending for banks belonging to the same banking group and lending to the same counterparty but with different liquidity risk. We are thus confident that a significant coefficient in such a specification properly identifies a funding risk (or maturity mismatch) effect and is not due to correlations with banking group effects or firms' heterogeneity.

Given the unexpected nature of the shock, this strategy makes it possible to identify a causal relation between the level of funding risk or maturity mismatch and the change in long-term credit supply.

Finally, we take into account the possibility that funding risk and maturity mismatch may be correlated with other balance sheet characteristics which also affect lending. We consider five bank characteristics that could influence loan supply: size (logarithm of total assets), profitability (ROA), the non-performing loans ratio (non-performing loans to total assets), the business model (credit to total assets) and the capital ratio (equity to total assets). Larger banks might be able more easily to substitute funding sources, which could make them more insulated to the funding shock. On the other hand, these banks tend to rely more on market funding. Profitability might also be related to bank liquidity risk as banks that rely on shortterm markets to fund illiquid assets might have a higher ROA before a funding shock. Credit to total assets captures the business model orientation of banks, where a higher share indicates that banks perform traditional banking activities. Non-performing loans and the capital ratio 5 Including banking group fixed effects forces us to remove from our sample foreign subsidiaries when they do not constitute a banking group in France, and a few stand-alone domestic entities. Thus, for most specifications, we report results with and without banking group fixed effects. Note that foreign subsidiaries and stand-alone entities do not account for a large share of the credit market in France. Moreover, empirical evidence suggests that excluding foreign subsidiaries should result in more conservative estimates as these subsidiaries tend to behave pro-cyclically (see e.g. [START_REF] Albertazzi | Foreign bank lending: Evidence from the global financial crisis[END_REF]. indicate the quality of the loan portfolio and banks' solvency respectively. There are potential interactions between credit risk and liquidity risk [START_REF] Distinguin | Bank regulatory capital and liquidity: evidence from US and European publicly traded banks[END_REF][START_REF] Imbierowicz | The relationship between liquidity and credit risk in banks[END_REF]Zheng, 2006). A higher capital ratio might also insulate banks from a run as investors might continue to roll over the debt of more solvent institutions. We thus control for the bank's solvency and the quality of its loan portfolio to be confident that we indeed capture a liquidity risk effect. Equation ( 1) describes the main model of interest in the paper:

Documents
LT Lending b,f = ∝ f + δ g + β . Bank ′ s liquidity risk b + Control variables b . γ + ε b,f (1)
Where LT Lending b,f is the difference in the log of long-term lending by bank b to firm f between 2007Q2 and 2009Q4. The Bank's liquidity risk b variable is alternatively the deposits to total assets ratio or the long-term loans to deposits ratio. 𝛽 is our main parameter of interest. Control variables b are bank characteristics, namely size (logarithm of total assets), profitability (ROA), the non-performing loans ratio (non-performing loans to total assets), the business model (credit to total assets) and solvency (equity to total assets). 𝛾 is a vector of parameters to be estimated.

Results

Main results: funding risk, maturity mismatch and long-term lending

We analyse in turn the effect of funding risk and maturity mismatch on long-term lending.

Both are measured at bank level in 2007Q2, prior to the funding shock.

Funding risk and long-term loan supply after the funding shock

We start our analysis by considering how the funding risk of banks affects their supply of long-term credit after 2007Q2. The deposits ratio measures banks' funding risk, with a higher ratio indicating that banks should be more immune to the funding shock. Table 2 reports the results of our model. We start our estimations by regressing the change in long-term lending without including borrowers' fixed effects (columns 1 and 2). We report these columns to allow comparison with the regressions where borrowers' heterogeneity is controlled for.

When we only include the deposits to assets ratio, the coefficient is statistically significant (almost at 5%). The magnitude and statistical significance of the coefficient increases when we control for other bank characteristics. So in the pooled regression, a higher deposits ratio has a positive effect on the long-term lending of banks.

Next, we include firms' fixed effects to control for firms' heterogeneity (i.e. change in loan demand, risk or business opportunities). Remember that in the first two columns, demand effects potentially drive the results if firms with lower demand tend to borrow more heavily from one particular group of banks. Columns 3 and 4 confirm that banks with lower funding risk extend more long-term loans after a funding shock, even when controlling for firms' heterogeneity: the coefficients are both positive and highly significant. In other words, Documents de travail du Centre d'Economie de la Sorbonne -2015.73

demand effects do not seem to drive our results. Interestingly, the coefficients for the deposits to assets variable are very similar to those obtained in the regressions without firms' fixed effects. This result brings additional support to the assumption that the shock was unanticipated: if firms had anticipated the shock, we could have observed a shift towards more resilient banks prior to the shock.

Other bank characteristics are also significant, with overall expected signs. Larger banks with more capital, a higher ROA and a business model oriented towards traditional credit activities are more likely to increase long-term lending after the funding shock.

Finally, we also include banking group fixed effects in our specification. Banking group heterogeneity could drive our results to some extent as some banking groups were more affected by the subprime crisis via foreign subsidiaries. Moreover, internal liquidity management could vary between banking groups. Column 5 reports the results with the inclusion of banking group fixed effects. As a result, most bank characteristicsapart from ROAbecome non-significant. This can be explained by the fact that size, capital ratio and loans to assets do not vary much within a banking group. Thus, banking group effects soak up the significance of these variables. A very interesting result however is that our deposits to assets variable remains highly significant with a similar coefficient as those obtained in the previous specifications. Thus, we find that long-term lending differs after a funding shock depending on the level of deposit funding even for banks lending to the same counterparty and belonging to the same banking group. Given the number of constraints we impose in the last specification, this result makes us confident that we are identifying a proper funding risk effect that is not related to other bank or firm characteristics.

The economic importance of the estimated coefficients also appears significant. If we take the coefficient obtained in column 5, a one-standard deviation in the deposits to assets ratio causes long-term lending to decrease on average by 0.128 × 0.213 = 2.7%.

Documents de travail du Centre d'Economie de la Sorbonne -2015.73

Maturity mismatch and the supply of long-term loans after the funding shock

We next consider how maturity mismatch predicts banks' behaviour after the shock. So far, we have only considered how the liability side of the bank's liquidity risk affects long-term lending. However, considering funding risk alone does not take into account the liquidity risk from the asset side. A bank with a higher proportion of illiquid assets will need more stable funding in order to be immune from the funding shock.

We test whether considering maturity mismatch brings additional information about banks' resilience to a funding shock. Maturity mismatch is defined as the ratio of long-term loans to deposits. A higher ratio indicates that the bank has a relatively more illiquid balance sheet on the asset side. Table 3 reports regression results for our maturity mismatch ratio. As for our funding risk measure, columns 1 and 2 report results without controlling for firms' fixed effects. We find, as expected, a negative correlation between banks' maturity mismatch and their long-term lending, i.e. banks that rely on alternative sources than deposits to fund their portfolio of long-term loans reduce their supply of long-term loans after the shock more. Not too surprisingly given our previous results, we find in columns 3, 4 and 5 that these results are robust to the inclusion of firms' fixed effects and banking group fixed effects. The relation between maturity mismatch and long-term lending thus appears to be causal, with a highly significant coefficient in all specifications.

We do not find that other characteristics affect long-term lending in these specifications. Only the NPL ratio is significant in the OLS model (column 2). In this specification, a higher NPL ratio prior to the shock causes long-term lending to decrease over the period.

Documents de travail du Centre d'Economie de la Sorbonne -2015.73

Our results are consistent with the fact that the asset side also matters for banks' resilience to funding shocks. Banks with a higher relative proportion of illiquid assets are less immune to the shock.

In terms of economic significance, our regressions provide estimates of an average decrease in long-term lending of the same magnitude as above for the deposits to assets ratio. A one standard deviation in the long-term lending to deposits ratio leads to a decrease in lending of 0.002 × 16.208 = 3.2%.

Additional analyses

So far, we have shown that funding risk and maturity mismatch cause banks to reduce longterm lending after a funding shock. These results fit well with the macroeconomic effects observed after the financial crisis. A credit rationing of long-term loans might be more severe for the real sector than a temporary reduction in short-term lending. However, we have not shown that banks decrease long-term loans exclusively. Banks might have decided to ration credit for all credit maturities without distinguishing between short-term and long-term loans.

In this sub-section, we test whether the funding shock also affected the supply of short-term loans.

Change in short-term lending after the funding shock

Table 4 reports the same sets of results for the change in short-term lending. The dependent variable is defined as the change in logarithm of short-term credit exposures between 2007Q2 and 2009Q4. Short-term credit exposures are exposures with a maturity of one year or less.

For the sake of brevity, Table 4 only reports regressions with firms' fixed effects included.

Documents de travail du Centre d'Economie de la Sorbonne -2015.73

However, the results are qualitatively unchanged for the other set of OLS regressions without firms' fixed effects.

Columns 1 and 2 report the results for the deposits to assets ratio (funding risk), while columns 3 and 4 report the results for the long-term loans to deposits ratio (maturity mismatch). The coefficients for the deposits to assets ratio are of the same magnitude as those obtained for long-term lending, but not significant. It is thus not possible to conclude that banks with different deposits ratio behave differently after the funding shock. Coefficients for the long-term loans to deposits ratio are very close to zero and also not statistically significant. In a nutshell, the table shows that funding risk and maturity mismatch do not seem to affect banks' supply of short-term loans, as we do not find evidence that banks with higher liquidity risk behave differently to the other group of banks.

This set of results suggests that banks with high liquidity risk shortened the maturity of their loan portfolio after the shock: compared to other banks, they reduced their supply of longterm loans but continued to lend short-term loans at a similar pace. The results might thus be driven by a strategy of banks highly exposed to liquidity risk of reducing their maturity mismatch after the shock. This also holds true after controlling for firms' demand effects.

To sum up, the change in lending seems to be driven by the change in long-term lending, whereas we do not find that banks with high liquidity risk particularly reduced short-term lending compared to other banks. This result has potentially important macroeconomic implications. Whereas firms might be able to cope with short-term credit rationing by turning to trade credit or factoring, they cannot use these substitutes for long-term credit. Only capital markets are a possible alternative for the external financing of firms. This option is however only credible for the largest firms in our sample. To properly understand how the funding shock affects the ability of firms to fund their long-term assets, we next analyse whether the credit rationing depends on the size of the borrowing firm.

Documents de travail du Centre d'Economie de la Sorbonne -2015.73

Do smaller firms bear the brunt of the rationing of long-term loans?

Table 5 shows the proportion of LT loans in total credit exposures in the Banque de France central credit register database. The mean share of long-term loans is the highest for small firms: it represents 79.7% of their bank funding. This share decreases to 40.2% for mediumsized firms. Finally, this share is only 24.3% for large firms. This is unsurprising given that large firms rely more on capital markets for their long-term borrowing and mainly use bank debt to refinance their working capital. Large firms also have access to bank lending abroad via the international syndicated loan market (which is not captured in our data when the lending bank is not a domestic credit institution). An important point to note with these figures is that the smaller a firm is, the more it will be affected by a rationing of long-term loans.

Given this feature, we go deeper in our analysis to identify if borrower size is an important factor in explaining the decrease in the supply of long-term loans after the shock. The premise of this analysis is that banks tend to ration opaque customers first of all to avoid problems of adverse selection and moral hazard [START_REF] Stiglitz | Credit rationing in markets with imperfect information[END_REF].

To capture this theoretical prediction, we consider the size of borrowers. We create three dummy variables capturing size. The 'small firm' dummy is equal to one when a firm's turnover in 2006Q4 is below EUR 10 million, and zero otherwise. The 'medium-sized firm'

dummy is equal to one when a firm's turnover in 2006Q4 is between EUR 10 and 100 million, and zero otherwise. Finally, the 'large firm' dummy is equal to one when a firm's turnover in 2006Q4 is above EUR 100 million, and zero otherwise.

To capture a size effect, we interact each dummy with our main variables of interest: the deposits to assets ratio and the long-term lending to deposits ratio. For the sake of brevity, we Documents de travail du Centre d'Economie de la Sorbonne -2015.73

only report results including firms' fixed effects. However, the results are robust when these firm fixed effects are not included.

Table 6 reports the regressions of the change in long-term lending on the interaction between firm size and banks' liquidity risk. For small firms, we find that banks' funding risk (deposits ratio) and maturity mismatch (long-term loans to deposits ratio) prior to the shock have a significant effect on long-term lending. We find mixed evidence for firms belonging to the medium-sized category: the long-term loans to deposits ratio is always significant but not the deposits to assets ratio.

Finally, for large firms, the coefficients are not statistically significant in all regressions. In the case of these large firms, these results indicate that on average banks with high liquidity risk do not behave differently after a funding shock compared to other banks: they continue to lend at the same pace.

The results show a clear gradation in the rationing of firms. Small firms clearly bear the largest burden in terms of rationing: they receive fewer long-term loans after the shock from non-resilient banks as measured by funding risk or maturity transformation. Medium-sized firms are also rationed but especially by banks which have a higher proportion of illiquid assets relative to their deposits. Finally, large firms do not appear to be rationed on average by non-resilient banks.

Small firms in our sample rely heavily on long-term bank loans as an external funding source.

They are also less likely to substitute capital market funding for these bank loans due to information asymmetry problems and the fixed costs of public issuance. These results confirm that a funding shock can have major macroeconomic consequences.

Documents de travail du Centre d'Economie de la Sorbonne -2015.73

Robustness checks

Placebo test: 2006Q1-2007Q1

Our identification strategy is based on the assumption that the funding shock was exogenous to a group of banks that showed a similar trend in lending before the shock. Figure 1 is supportive of this assumption as banks were all increasing their long-term lending before the funding shock, while the trend abruptly changed for the less resilient banks after the shock.

In this sub-section, we formally test this assumption by performing a placebo test, i.e. running the analysis on a period prior to the shock. In this period, banks' lending should not be affected by their funding risk or maturity mismatch as no shock occurs. We thus expect that the main coefficients of interest to be non-significant in these regressions.

We apply the same methodology as before by computing the change in the logarithm of longterm credit exposures. We choose to compute the change in long-term lending between In conclusion, we do not find that banks with different funding risk or maturity mismatch behave differently before the funding shock. This test gives us confidence in our previous results, indicating that they were indeed driven by an exogenous shock during the period 2007Q2 to 2009Q4.

Alternative definitions of funding risk and maturity transformation

In this sub-section, we compare our main results with alternative definitions of funding risk and maturity mismatch. Namely, we expand our definition of stable funding by also taking into account banks' long-term debt (defined as bank balance sheet liabilities with a residual maturity over one year).

We define the sum of deposits and long-term debt as 'stable funding'. Our funding risk measure then becomes the ratio of stable funding to total assets. Our maturity mismatch metric similarly becomes the long-term loans to stable funding ratio. The rationale for these measures is that long-term debt might also immunise banks from the funding shock and act as a substitute for deposits. Table 1 gives descriptive statistics for these two liquidity risk measures. The average stable funding ratio is much higher for banks than the deposit ratio, at 63.4%. The long-term loans to stable funding ratio averages 79.5%. This indicates that banks' stable funding covers the portfolio of long-term loans.

Table 8 shows the results for our alternative definitions of funding risk and maturity mismatch. The coefficient for the stable funding ratio is always significant and positive, indicating that banks with a lower ratio reduce their supply of long-term loans. Magnitudes of the coefficients are of the same order as those obtained for the deposits to assets ratio.

The long-term loans to stable funding ratio is also highly significant, except in the first specification where we do not include other control variables and banking group fixed effects.

Documents de travail du Centre d'Economie de la Sorbonne -2015.73

Given the overall results of the paper, this indicates that this ratio is correlated with other bank characteristics. When we do not control for bank characteristics, the ratio fails to reveal the effect of the funding shock. Thus, this ratio appears slightly inferior to the deposits to assets ratio in terms of predicting banks' resilience to the funding shock.

Overall, the alternative funding metrics confirm the credit rationing of long-term loans for firms, although stability of funding only defined in terms of deposits appears to be a slightly better predictor of banks' supply of long-term loans after the funding shock.

Conclusion

This study belongs to the literature that analyses the transmission of financial shocks to the real sector. It is the first to assess how banks adjust the maturity of their lending after a funding shock. The question is of great importance as a decline in long-term lending is detrimental for firms' long-term investment.

We take advantage of the international interbank market freeze which started in August 2007

and was exogenous to the domestic French banking sectorto analyse how banks hit by a funding shock change their supply of long-term loans depending on their level of liquidity risk. To measure the change in long-term lending, we have access to a rich bank-firm credit register dataset on a quarterly basis, with information on credit maturity. This data allows us to disentangle loan demand from loan supply effects by estimating the change in lending between two banks with different liquidity profiles with respect to the same counterparty. We match this data with residual bank balance sheet maturity to measure funding risk and maturity mismatch at the bank level. Finally, we also collect information on firm size to assess the interaction between borrower size and the transmission of the funding shock.

Documents de travail du Centre d'Economie de la Sorbonne -2015.73

We find that banks with less deposit funding reduce their supply of long-term loans after the shock. Maturity mismatch also explains banks' supply of long-term credit after the shock.

However, all banks continued to grant short-term loans at the same pace.

The decline in long-term lending is borne by the smallest firms, which rely more on long-term bank loans as a source of external financing and are less likely to find alternative sources to fund their long-term assets.

Moreover, these results are robust to the inclusion of controls capturing other bank characteristics and banking group fixed effects. They are also robust to alternative definitions of funding risk and maturity mismatch.

These findings help us to understand some of the mechanisms behind the global recession that followed the massive liquidity shock in 2007-2008. The rationing of long-term loans to the smallest firms has important macroeconomic implications. In essence, they also give support to the future liquidity regulations. The Basel III NSFR will require banks to finance their illiquid assets with a minimum share of stable funding. Our empirical results support the rationale of this regulation by showing that lower funding risk and a lower maturity mismatch immunise banks from funding shocks. This in turn allows banks to continue extending longterm loans to firms in times of funding stress. Such a regulation might thus support economic growth during periods of downturn by ensuring that banks continue their role of long-term funding providers for firms that cannot access alternative financing sources. This table reports the results of regressions where the dependent variable is the difference in the log of longterm credit exposures between 2009Q4 and 2007Q2 for each bank-firm in the sample. The explanatory variable of interest is the long-term loans to deposits ratio computed as loans with maturity over 1 year divided by deposits, measured before the funding shock in 2007Q2. Other control variables are bank size computed as the logarithm of banks' total assets in 2007Q2, the capital ratio computed as equity divided by total assets in 2007Q2, the NPL ratio computed as non-performing loans divided by total assets in 2007Q2, the ROA computed as net income divided by total assets in 2007Q2, and the loans to assets ratio computed as loans to non-financial customers divided by total assets in 2007Q2. Standard errors are clustered at the bank level and robust to heteroscedasticity. '***' indicates statistically significant coefficients at the 1% level.

Change in long-term lending (1)

( This table reports the results of regressions where the dependent variable is the difference in the log of shortterm credit exposures between 2009Q4 and 2007Q2 for each bank-firm in the sample. Short-term credit exposures are exposures with a maturity of less than one year. The explanatory variables of interest are the deposits to assets and long-term loans to deposits ratios computed as loans to non-financial customers with maturity over 1 year divided by deposits. Both variables are measured before the funding shock in 2007Q2. Other control variables are bank size computed as the logarithm of banks' total assets in 2007Q2, the capital ratio computed as equity divided by total assets in 2007Q2, the NPL ratio computed as non-performing loans divided by total assets in 2007Q2, the ROA computed as net income divided by total assets in 2007Q2, and the loans to assets ratio computed as loans to non-financial customers divided by total assets in 2007Q2. Standard errors are clustered at the bank level and robust to heteroscedasticity. '**' and '*' indicate statistically significant coefficients at the 5% and 10% levels respectively.

Change in short-term lending (1)

( This table reports the results of regressions where the dependent variable is the difference in the log of long-term credit exposure between 2009Q4 and 2007Q2 for each bank-firm in the sample. Long-term credit exposures are exposures with a maturity of 1 year or greater. The explanatory variables of interest are the deposits to assets and long-term loans to deposits ratios computed as loans to non-financial customers with maturity over 1 year divided by deposits. Both variables are measured before the funding shock in 2007Q2. Other control variables are bank size computed as the logarithm of banks' total assets in 2007Q2, the capital ratio computed as equity divided by total assets in 2007Q2, the NPL ratio computed as non-performing loans divided by total assets in 2007Q2, the ROA computed as net income divided by total assets in 2007Q2, and the loans to assets ratio computed as loans to non-financial customers divided by total in 2007Q2. Small firms is a dummy variable equal to one if a firm has a turnover below EUR 10 million, and zero otherwise. Medium-sized firms is a dummy variable equal to one if a firm has a turnover between EUR 10 and 100 million and zero otherwise. Large firms is a dummy equal to one if a firm has a turnover above EUR 100 million and zero otherwise. Firms' turnover is measured in 2006Q4. This table reports the results of regressions where the dependent variable is the difference in the log of long-term credit exposures between 2007Q1 and 2006Q1 for each bank-firm in the sample. The explanatory variables of interest are the deposits to assets and long-term loans to deposits ratios computed as loans to non-financial customers with maturity over 1 year divided by deposits. Both variables are measured at the beginning of the placebo test in 2006Q1. Other control variables are bank size computed as the logarithm of banks' total assets in 2006Q1, the capital ratio computed as equity divided by total assets in 2006Q1, the NPL ratio computed as non-performing loans divided by total assets in 2006Q1, the ROA computed as net income divided by total assets in 2006Q1, and the loans to assets ratio computed as loans to non-financial customers divided by total assets in 2006Q1. Standard errors are clustered at the bank level and robust to heteroscedasticity. '**' and '*' indicate statistically significant coefficients at the 5% and 10% levels respectively.

Change in long-term lending (1)

( This table reports the results of regressions where the dependent variable is the difference in the log of long-term credit exposures between 2009Q4 and 2007Q2 for each bank-firm in the sample. Long-term credit exposures are exposures with a maturity of 1 year or greater. The explanatory variables of interest are the stable funding ratio defined as the sum of deposits and liabilities with maturity greater than 1 year and the long-term loans to stable funding ratio computed as loans to non-financial customers with maturity over 1 year divided by the sum of deposits and liabilities with maturity greater than 1 year. Both variables are measured before the funding shock in 2007Q2. Other control variables are bank size computed as the logarithm of banks' total assets in 2007Q2, the capital ratio computed as equity divided by total assets in 2007Q2, the NPL ratio computed as non-performing loans divided by total assets in 2007Q2, the ROA computed as net income divided by total assets in 2007Q2, and the loans to assets ratio computed as loans to non-financial customers divided by total assets in 2007Q2. Standard errors are clustered at the bank level and robust to heteroscedasticity. '***', '**' and '*' indicate statistically significant coefficients at the 1%, 5% and 10% levels respectively.

Change in long-term lending (1)

(

Figure 1 Fig. 1 :

 11 Figure1summarises our approach for the deposit ratio. The chart shows the average

  de travail du Centre d'Economie de la Sorbonne -2015.73

  2006Q1 and 2007Q1. This choice is driven by two considerations. First, our period needs to end before the funding shock. Second, the Banque de France central credit register changed the minimum threshold for inclusion in the database from EUR 76,000 to 25,000 in the first quarter of 2006. Choosing a date prior to 2006Q1 would result in selection bias towards the largest firms. We thus prefer to take 2006Q1 in order to keep the same threshold of inclusion in the central credit register database as in our main sample.

Table 7

 7 reports the results of our placebo test. Again, we only report results with the firms' fixed effects. OLS regressions give similar results. In all regressions, neither the deposits to assets ratio nor the long-term loans to deposits ratio are statistically significant. These results

notably hold with or without controlling for banks' other characteristics or banking group fixed effects. Documents de travail du Centre d'Economie de la Sorbonne -2015.73

  Documents de travail du Centre d'Economie de la Sorbonne -2015.73

		Table 1
		Sample descriptive statistics
	Variables	Description	N. obs. Median	Mean	Std. Dev.
	Panel A: Loan level
	Change in long-term lending	Change in log of long-term credit exposures (over 1 year) between 2007Q2 and 2009Q4	233,212	-0.201	-0.119	0.900
	Change in short-term lending	Change in log of short-term credit exposures (less than 1 year) between 2007Q2 and 2009Q4	74,291	0	-0.030	1.398
	Long-term credit exposure 2007Q2	Long-term credit exposure in 2007Q2 in thousands of euro	233,212	153	769	5786
	Short-term credit exposure 2007Q2	Short-term credit exposure in 2007Q2 in thousands of euro	74,291	78	553	13437
	Panel B: Bank level
	Deposits to assets	Deposits to total assets in 2007Q2	133	0.406	0.373	0.213
	Long-term loans to deposits	Loans with residual maturity greater than 1 year to deposits in 2007Q2	133	1.282	4.654	16.208
	Stable funding to assets	Deposits and liabilities with residual maturity greater than 1 year to total assets in 2007Q2	133	0.647	0.634	0.172
	Long-term loans to stable funding	Loans with residual maturity greater than 1 year to stable funding in 2007Q2	133	0.822	0.795	0.326
	Bank size	Ln of total assets in 2007Q2	133	15.996	15.993	1.375
	Total assets	Total assets in 2007Q2 in thousands of euro	133	8,852,190 38,785,240 146,396,931
	Capital ratio	Equity to total assets in 2007Q2	133	0.093	0.092	0.035
	NPL ratio	Non-performing loans to total assets in 2007Q2	133	0.007	0.009	0.007
	ROA	Net profit to total assets in 2007Q2	133	0.004	0.005	0.003
	Loans to assets	Loans to total assets in 2007Q2	133	0.678	0.593	0.212
	Panel C: Firm level
	Large firms	Dummy equal to 1 if a firm has a turnover over EUR 100 million in 2006Q4; 0 otherwise	119,041		0.009
	Medium-sized firms	Dummy equal to 1 if a firm has a turnover between EUR 10 and 100 million in 2006Q4; 0 otherwise	119,041		0.071
	Small firms	Dummy equal to 1 if a firm has a turnover under EUR 10 million in 2006Q4; 0 otherwise	119,041		0.919

Documents de travail du Centre d'Economie de la Sorbonne -2015.73

Table 2 Funding risk and long-term credit supply after the funding shock

 2 This table reports the results of regressions where the dependent variable is the difference in the log of long-term credit exposures between 2009Q4 and 2007Q2 for each bank-firm in the sample. The explanatory variable of interest is the deposits to assets ratio, measured before the funding shock in 2007Q2. Other control variables are bank size computed as the logarithm of banks' total assets in 2007Q2, the capital ratio computed as equity divided by total assets in 2007Q2, the NPL ratio computed as non-performing loans divided by total assets in 2007Q2, the ROA computed as net income divided by total assets in 2007Q2, and the loans to assets ratio computed as loans to nonfinancial customers divided by total assets in 2007Q2. Standard errors are clustered at the bank level and robust to heteroscedasticity. '***', '**' and '*' indicate statistically significant coefficients at the 1%, 5% and 10% levels respectively. Documents de travail du Centre d'Economie de laSorbonne -2015.73

				Change in long-term lending
		(1)	(2)	(3)	(4)	(5)
	Deposits to assets	0.115*	0.151***	0.137**	0.163***	0.128***
		(0.059)	(0.045)	(0.053)	(0.053)	(0.045)
	Bank size		0.004		0.015***	0.012
			(0.008)		(0.005)	(0.007)
	Capital ratio		0.677**		0.417*	0.300
			(0.281)		(0.227)	(0.449)
	NPL ratio		-7.901***		-1.767	-1.230
			(1.656)		(1.130)	(1.334)
	ROA		4.238		6.308*	5.623*
			(3.551)		(3.375)	(3.321)
	Loans to assets		0.129**		0.142***	0.035
			(0.051)		(0.047)	(0.074)
	Constant	-0.155***	-0.322*	-0.162***	-0.544***	-0.412**
		(0.024)	(0.180)	(0.021)	(0.132)	(0.176)
	Firm FE	No	No	Yes	Yes	Yes
	Banking group FE	No	No	No	No	Yes
	Observations	233,212	233,212	233,212	233,212	218,232
	R 2	0.001	0.004	0.484	0.485	0.507

Table 3 Maturity mismatch and long-term credit supply after the funding shock

 3

Table 4 Short-term credit supply after the funding shock

 4

Table 5 Bank loan maturity of French non-financial firms in 2007Q2 This

 5 table reports mean exposures in thousands of euro for long-term loans, i.e. loans with a maturity over 1 year, and short-term loans, i.e. loans with a maturity below 1 year for non-financial firms in France. Small firms are non-financial firms with a turnover below EUR 10 million in 2006Q4. Medium-sized firms are non-financial firms with a turnover between EUR 10 and 100 million in 2006Q4. Large firms are non-financial firms with a turnover above EUR 100 million in 2006Q4. Source: Banque de France credit register and authors' own calculations.

	Small firms (N = 1,874,632)	Medium-sized firms (N=20,111)	Large firms (N=2,487)

Documents de travail du Centre d'Economie de la Sorbonne -2015.73

Table 6 Borrower size and change in long-term lending

 6

  Standard errors are clustered at the bank level and robust to heteroscedasticity. '***', '**' and '*' indicate statistically significant coefficients at the 1%, 5% and 10% levels respectively.

				Change in long-term lending
		(1)	(2)	(3)	(4)	(5)	(6)
	Deposits to assets × Small firms	0.134**		0.165***		0.130***
		(0.055)		(0.056)		(0.048)
	Deposits to assets × Medium-sized firms	0.167		0.154*		0.120
		(0.104)		(0.093)		(0.117)
	Deposits to assets × Large firms	0.087		0.083		0.090
		(0.181)		(0.178)		(0.215)
	Long-term loans to deposits × Small firms		-0.002***		-0.001**		-0.002**
			(0.001)		(0.001)		(0.001)
	Long-term loans to deposits × Medium-sized firms		-0.004***		-0.004***		-0.005***
			(0.001)		(0.001)		(0.001)
	Long-term loans to deposits × Large firms		-0.006		-0.006		-0.008
			(0.006)		(0.005)		(0.006)
	Bank size			0.015***	-0.002	0.012	-0.005
				(0.005)	(0.006)	(0.007)	(0.008)
	Capital ratio			0.416*	0.217	0.298	0.265
				(0.228)	(0.205)	(0.450)	(0.430)
	NPL ratio			-1.759	-0.352	-1.223	0.096
				(1.124)	(0.955)	(1.339)	(1.260)
	ROA			6.296*	4.493	5.619*	3.533
				(3.374)	(2.949)	(3.318)	(2.804)
	Loans to assets			0.143***	0.080	0.035	-0.051
				(0.048)	(0.052)	(0.074)	(0.079)
	Constant	-0.161*** -0.110*** -0.546***	-0.159	-0.412**	-0.025
		(0.021)	(0.006)	(0.133)	(0.136)	(0.175)	(0.185)
	Firm FE	Yes	Yes	Yes	Yes	Yes	Yes
	Banking group FE	No	No	No	No	Yes	Yes
	Observations	233,212	233,212	233,212	233,212	218,232	218,232
	R 2	0.484	0.484	0.485	0.485	0.507	0.507

Documents de travail du Centre d'Economie de la Sorbonne -2015.73

Table 7 Placebo test 2006Q1-2007Q1

 7

Table 8 Alternative definition of funding risk and maturity mismatch

 8

	2)	(3)	(4)	(5)	(6)

Documents de travail du Centre d'Economie de la Sorbonne -2015.73

The Basel III Net Stable Funding Ratio (NSFR), which will come into force on 1 January

2018, will require banks to fund long-term and illiquid assets with stable funding such as deposits and liabilities with residual maturity over one year (BCBS 2014). Documents de travail du Centre d'Economie de la Sorbonne -2015.73

In unreported results, we also tested other periods, namely 2007Q2-2009Q1, 2007Q2-2009Q2, 2007Q2-2010Q2 and 2008Q3-2009Q2. The main results, which are available upon request, remain qualitatively unchanged. Documents de travail du Centre d'Economie de la Sorbonne -2015.73

It is not compulsory for firms with a turnover below EUR 0.75 million to report detailed financial accounts. Documents de travail du Centre d'Economie de la Sorbonne -2015.73

This is, for example, the case in France where certain deposit contracts still benefit from a tax shield and interest rates are set by central regulation.Documents de travail du Centre d'Economie de la Sorbonne -2015.[START_REF] Distinguin | Bank regulatory capital and liquidity: evidence from US and European publicly traded banks[END_REF]

Zheng, H., 2006. Interaction of credit and liquidity risks: modelling and valuation, Journal of Banking and Finance, 30, 391-407. Documents de travail du Centre d'Economie de la Sorbonne -2015.73


Jean-Baptiste Fiot

Francesco Dinuzzo

Electricity Demand Forecasting by Multi-Task Learning

We explore the application of kernel-based multi-task learning techniques to forecast the demand of electricity in multiple nodes of a distribution network. We show that recently developed output kernel learning techniques are particularly well suited to solve this problem, as they allow to flexibly model the complex seasonal effects that characterize electricity demand data, while learning and exploiting correlations between multiple demand profiles. We also demonstrate that kernels with a multiplicative structure yield superior predictive performance with respect to the widely adopted (generalized) additive models. Our study is based on residential and industrial smart meter data provided by the Irish Commission for Energy Regulation (CER).

Introduction

Electricity cannot be stored efficiently in large quantities, therefore it is critical to ensure that the amount generated at a given time is sufficient to meet the load plus the losses while not exceeding this amount significantly. Predictive methods for accurately forecasting the demand of electricity have thus become important tools that guide planning and operation of utility companies. While electric load forecasting is a well-established, several decades old research area in engineering, new modeling problems keep appearing as technological and legislative transformations affect the power industry. With the advent of smart grids and meters, larger and richer sources of data are becoming available, making it possible to build more sophisticated models that enable more accurate billing of electricity and dynamic pricing.

A variety of tools from time series analysis, statistics, and more recently machine learning, have been employed for electricity load forecasting. For an overview on the vast body of available literature on the subject, we refer the reader to the recent book by [START_REF] Soliman | Electrical load forecasting: modeling and model construction[END_REF]. Classical techniques include linear and non-linear regression models estimated by means of variants of least squares fitting, and various types of ARMAX models expressing the forecast as a function of previously observed values of the load and possibly other weather or social variables. Techniques inspired by Artificial Intelligence research such as expert systems, fuzzy logic, and neural networks have also been applied to load forecasting. In particular, black-box models based on neural networks have been extensively analyzed, see the influential review by [START_REF] Hippert | Neural networks for short-term load forecasting: A review and evaluation[END_REF].

In recent years, Generalized Additive Models (GAM) [START_REF] Hastie | Generalized additive models[END_REF] have established themselves as state of the art tools for electricity load forecasting [START_REF] Fan | Short-term load forecasting based on a semi-parametric additive model[END_REF][START_REF] Ba | Adaptive learning of smoothing functions: application to electricity load forecasting[END_REF][START_REF] Nedellec | Gefcom2012: Electric load forecasting and backcasting with semiparametric models[END_REF], due to the existence of efficient and scalable training algorithms and the interpretability of the model, which allows to clearly visualize the effect of individual variables on the load by means of simple longitudinal plots. Meanwhile, kernel methods have been employed with great success in the last decade. Already back in 2001, a kernel-based Support Vector Regression (SVR) approach was employed to win a competition on electricity load forecasting [START_REF] Bj | Load forecasting using support vector machines: A study on EUNITE competition 2001[END_REF] organized by EUNITE (European Network on Intelligent Technologies for Smart Adaptive Systems). Later on, various types of kernelbased regularization methods and Support Vector Machines have been applied to predict the demand of electricity, see for instance [START_REF] Espinoza | Electric load forecasting[END_REF][START_REF] Hong | Electric load forecasting by support vector model[END_REF][START_REF] Elattar | Electric load forecasting based on locally weighted support vector regression[END_REF].

Most research articles on electricity load forecasting focus on predicting a single time series representing the electricity load aggregated over a large number of nodes of the electricity network. Due to aggregation, such time series exhibit high regularity and are therefore significantly easier to forecast than load profiles at lower levels of the network. Nevertheless, making forecasts of the loads at lower levels is becoming increasingly feasible due to the availability of rich smart meter datasets, therefore the problem is attracting consider-able interest in the industry.

Forecasting electricity demand at low levels of the network (such as the demand of an individual household) presents several challenges. First of all, it involves analyzing a much larger number of time series, calling for scalable techniques that can handle a very large amount of measurements. In addition, demand profiles at lower levels of the electricity network are much less regular and thus harder to predict. To tackle these challenges, recent works have investigated the use of clustering techniques for automatically aggregating multiple load time series, reporting improved predictive performance at aggregated level [START_REF] Alzate | Identifying customer profiles in power load time series using spectral clustering[END_REF][START_REF] Alzate | Improved electricity load forecasting via kernel spectral clustering of smart meters[END_REF][START_REF] Humeau | Electricity load forecasting for residential customers: Exploiting aggregation and correlation between households[END_REF].

In this paper, we study the problem of electricity load forecasting at low network level, and we suggest to solve it by means of kernel-based multi-task learning techniques that can discover and take advantage of the relationships between multiple profiles. Kernel based multi-task learning has been studied in a variety of papers [START_REF] Micchelli | Kernels for multi-task learning[END_REF][START_REF] Evgeniou | Learning multiple tasks with kernel methods[END_REF][START_REF] Argyriou | Multitask feature learning[END_REF][START_REF] Bonilla | Multi-task Gaussian process prediction[END_REF] while, in recent years, the problem of learning and exploiting relationships between multiple tasks is a topic that is attracting considerable attention in the machine learning literature [START_REF] Jacob | Clustered multi-task learning: A convex formulation[END_REF][START_REF] Zhang | Flexible latent variable models for multi-task learning[END_REF][START_REF] Zhang | A convex formulation for learning task relationships in multi-task learning[END_REF][START_REF] Dinuzzo | Learning output kernels with block coordinate descent[END_REF][START_REF] Archambeau | Sparse bayesian multi-task learning[END_REF][START_REF] Kang | Learning with whom to share in multi-task feature learning[END_REF][START_REF] Saha | Online learning of multiple tasks and their relationships[END_REF][START_REF] Kumar | Learning task grouping and overlap in multi-task learning[END_REF][START_REF] Romera-Paredes | Exploiting unrelated tasks in multi-task learning[END_REF][START_REF] Dinuzzo | Learning output kernels for multi-task problems[END_REF].

Herein, we develop and compare a variety of kernelbased models for long-term electricity demand forecasting in multiple nodes, with the goal of identifying the best way to capture the complex seasonal effects that characterize such demand patterns. We design kernels specifically tailored to capture the seasonal effects present in electricity load data. By doing so, we expose the performance limits of the very popular additive models, showing that they are often outperformed by multiplicative kernel models. We formulate the problem of forecasting the demand in multiple nodes of the network as a multi-task learning problem, illustrating the usefulness of jointly learning and exploiting similarities between multiple load profiles. Finally, we show how recently developed multi-task learning techniques can be used to gain insights and interpretability on real demand data, while achieving state of the art predictive performance. Our experimental analysis is based on data provided by the Irish Commission for Energy Regulation (CER). sions of significantly different nature. It is therefore common to classify forecasting problems in three categories: short-range forecasting (several minutes up to one week ahead), medium-range forecasting (up to 10 years ahead), and long-range forecasting (as far as several decades ahead), see [START_REF] Soliman | Electrical load forecasting: modeling and model construction[END_REF] for a more comprehensive discussion.

Forecasting models are built starting from datasets containing one or multiple time series, each of them representing the load measurement at a specific location and level in the network, ranging from highly aggregated loads in the transmission network down to the distribution network and to the loads of individual users. Missing measurements and different sampling rates contribute to make these data noisy and challenging to analyze. Moreover, defective meters at a low level in the network are hard to detect, and faulty meters can report wrong measurements before being replaced.

Being mostly driven by human activity, a variety of temporal patterns can be observed in the load data. The top panel of Fig. 1 shows a typical profile for aggregated electricity load over several years (data source: French Réseau de Transport d'Electricité1 ), from which a clear yearly seasonal pattern can be observed, with higher demand in winter and lower demand in the summer. A closer look at this data also reveals typical weekly (Fig. 1, middle panel) and daily (Fig. lem, which can be dealt with by properly extracting and utilizing temporal and calendar features. The type of day of the week can be also taken into account: the bottom panel of Fig. 1 shows a specific week where all days have a similar profile but a difference between week days and weekend can be clearly noticed. Forecasting is particularly challenging on public holidays, and different public holidays may exhibit significantly different load profiles. Fig. 2 illustrates the difficulty of fitting and forecasting the demand in correspondence of public holidays and special events: without including specific information in the model, the prediction error can be particularly high in corrispondence with such events.

A variety of additional features can be typically extracted from the data or obtained from other sources and utilized to forecast the electricity demand. For instance, the electricity consumption is affected by weather conditions (particularly due to heating and air-conditioning), therefore variables such as temperature, humidity and irradiance are often taken into account by forecasting models. Economic indicators such as gross domestic product can be used to model trends in long-range scenarios. Finally, short term forecasting models are typically based on time series techniques, where auto-regressive lagged values of the load itself are incorporated in the model and used to track shortrange trends and deviations from stationarity.

In summary, typical forecasts of the electricity demand may depend on a variety of features that include time and calendar variables, weather and economic conditions, previously observed values of the load, and information about the node of the network where the forecast is required. A general model that takes into ac-count the previously discussed features takes the form

Forecast = f      t, d, c Time / Calendar features , y l , u l Dynamic features , k, s k Node features      , (1)
where the dependent variables are the following: [START_REF] Micchelli | Kernels for multi-task learning[END_REF] is the time of day expressed in hours,

• t ∈ [0,
• d ∈ {1, 2, . . . , 365, 366} is the day of the year,

• c is the type of day, e.g. Monday to Sunday, weekday/weekend, holiday,

• y l is a real vector containing lagged values of the measured electric load,

• u l is a real vector containing measurements of lagged values of exogenous variables other than the load (such as temperature),

• k is the node ID in the electricity network, e.g. meter/customer/zone ID,

• s k is a vector of features describing the node type, e.g. customer/zone type.

In Section 4, we analyze one of the many possible multi-task learning problems that naturally appear within this framework, namely the problem of simultaneously predicting the load in each node of the network. This amount to disaggregate the overall dataset over the multiple smart meters and treat each node k of the network as a different learning task. In the next section, we briefly recall the standard setup of multitask regression and review the techniques that will be employed in Section 4 to solve the problem sketched above.

Kernel-based multi-task regression

In the following, we focus on multi-variate (multi-task) regression problems where the goal is to learn multiple functions

f j : X → R from multiple datasets of pairs (x ij , y ij ) ∈ X × R.
Here, X is a set of input features, m denotes the number of tasks, and j the number of examples for the j-th task. Letting f : X × R m denote the vector-valued function with components f j , we are going to search f by minimizing the following regularization functional

R(f, L) = m j=1 j i=1 (y ij -f j (x ij ))) 2 + λ f 2 H L .
where λ > 0 is a regularization parameter, and H L is a Reproducing Kernel Hilbert Space (RKHS) of vectorvalued functions with (matrix-valued) kernel

H(x i , x j ) = K(x i , x j )L.
Here, K : X × X → R is a positive semidefinite kernel called input kernel, and the square matrix L ∈ R m×m is the output kernel matrix whose entries L jk express the similarity between the tasks (output components) j and k. In view of the representer theorem, there exist functions fj minimizing R(f, L) in the form:

fj (x) = m k=1 L jk k i=1 c ik K(x ik , x). (2)
We refer to [START_REF] Micchelli | On learning vectorvalued functions[END_REF] for more details about RKHS of vectorvalued functions and the corresponding representer theorem.

Fixing L = I (independent kernel ridge regression)

Expression [START_REF] Alzate | Improved electricity load forecasting via kernel spectral clustering of smart meters[END_REF] shows that inter-task transfer is possible only when off-diagonal elements of the output kernel matrix are different from zero. Indeed, by choosing L equal to the identity matrix all the tasks are learned independently by solving a standard kernel regularized least squares problem fj = arg min

fj ∈H   j i=1 (y ij -f j (x ij )) 2 + λ f j 2 H   , (3)
where H is the RKHS of scalar functions with kernel K. This single-task baseline is referred to as independent kernel ridge regression.

Learning L (output kernel learning)

In this subsection, we review a kernel-based multitask regression approach called low-rank Output Kernel Learning (OKL), recently developed in [START_REF] Dinuzzo | Learning output kernels for multi-task problems[END_REF]. In such approach, the functions f j and the output kernel L are jointly optimized by solving the following problem min

L∈S m,p + min f ∈H L R(f, L) + λtr(L), (4)
where S m,p + is the cone of positive semidefinite matrices with rank less than or equal to p. Instead of imposing a low-rank constraint or regularizing the trace of the output kernel, other type of regularizers could be tried, see e.g. [START_REF] Dinuzzo | Learning output kernels with block coordinate descent[END_REF][START_REF] Dinuzzo | Output kernel learning methods[END_REF][START_REF] Ciliberto | Convex learning of multiple tasks and their structure[END_REF]. The low-rank approach has the advantage of allowing us to tightly control the memory required to store the models.

The representer theorem (2) still applies to the inner minimization problem of [START_REF] Argyriou | Multitask feature learning[END_REF]. By plugging the expression (2) into (4), one obtains a functional that is convex quadratic with respect to both the coefficients c ik and L. Although the resulting problem is not jointly convex, the alternating minimization procedure described in [START_REF] Dinuzzo | Learning output kernels for multi-task problems[END_REF] can be applied to obtain a minimizer. An important aspect of the method is that, by selecting the rank parameter p, is it possible to control the overall number of parameters of the model, as well as the memory requirements and the computation time to obtain a solution. More specifically, letting A = ∪ j ∪ i {x ij }, one can show that the solution (4) can be rewritten as

fj (x) = p k=1 b jk g k (x), g k (x) = i=1 a ik K(x i , x), (5)
where = #A, x i ∈ A, i = 1, . . . , , and the coefficients b jk form a low-rank factor of L. It is therefore sufficient to store and optimize ( + m)p parameters, which can be much smaller than m j=1 j .

Electricity demand forecasting in multiple nodes by multi-task learning

In this section, we focus on predicting the demand of electricity in multiple nodes of an electricity network, a multi-task learning problem where each task corresponds to one of the measured smart meters in the network.

To analyze our long-term forecasting approach, we adopt data provided by the the Irish Commission for Energy Regulation (CER)2 , containing electric load measurements from 6435 smart meters, half-hourly sampled from July 14, 2009 to December 31, 2010 (536 days). These meters include residential customers and small-to-medium industrial sites. We consider a midterm test scenario where the goal is to forecast the load in multiple nodes over a time horizon of 171 days, using one year of measurements to build the model. Due to the long forecasting horizon, dynamic features are not available and are therefore dropped from the general model in Eq. ( 1). Such model does not rely on recent measurements of the load, therefore it is able to make predictions over an arbitrarily long horizon. The load forecast for the k-th node is thus simply given by ŷ = f k (t, d, c), where f k are the multiple functions to be learned, taking into account time and calendar features.

From the original CER dataset, several pre-processing steps were performed. The day of the year and time of the day were extracted from the five-digit timestamps. In this dataset, the time of day is an non-zero integer indexing the number of half-hours, and therefore it should be normally in the set {1, 2, . . . , 48}. Two meters containing time of days higher than 50 half-hours were discarded, as it was unclear how to interpret these measurements. The dataset also contains days with 46 and 50 measurements and time of days up to 50. These inconsistencies are caused by the start and the end of daylight saving time (DST) and are easily fixable. When DST starts in Ireland

MAPE(i, G) = 100 j∈Gi y ij -j∈Gi f j (t i , d i , c i ) j∈Gi y ij , NMAE(i, G) = j∈Gi |y ij -f j (t i , d i , c i )| j∈Gi y ij .
MAPE(i, G) measures the relative absolute percentage error incurred when forecasting the aggregated demand using the sum of the forecasts. On the other hand, NMAE(i, G) is the sum of the forecasting errors over individual tasks, relative to the naive baseline of predicting f j (t i , d i , c i ) = 0 for all i, j. Since the demand values y ij are always non-negative, the two metrics are undefined only for those groups on meters for which the cumulative demand in the i-th time slot is identically zero, or no measurements are available for any of the meters. We compute the average and standard deviation of these two metrics over all the time slots in the test period.

One advantage of the long-term forecasting approach is that it allows to naturally incorporate and handle time series with missing observations, without resorting to inputing techniques or discarding data. In the following, we analyze a variety of kernel based models to solve this multi-task regression problem. First of all we introduce kernels based on the time/calendar features of model ( 1),

K t (t 1 , t 2 ) = exp (-h T (|t 1 -t 2 |)/σ t ) , K d (d 1 , d 2 ) = exp (-h D (|d 1 -d 2 |)/σ d ) , K c (c 1 , c 2 ) = 1 if c 1 = c 2 0 if c 1 = c 2 . ,
where h P (x) = min{x, P -x} is a change of variable that yields P -periodic kernels over the square [0, P ] 2 . By observing that the Fourier transform of exp(-|x|) is non-negative, it can be easily shown that periodized kernels such as K t and K d are positive semidefinite, see [START_REF] Schölkopf | Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond[END_REF]. In our experiment, σ t and σ d were respectively set to 4 hours and 120 days. In order to define K((t 1 , d 1 , c 1 ), (t 2 , d 2 , c 2 )), we combine these three kernels to define a variety of models:

• Additive Models K t (t 1 , t 2 ) + K d (d 1 , d 2 ) , (6)
K t (t 1 , t 2 ) + K d (d 1 , d 2 ) + K c (c 1 , c 2 ) , (7)
• Semi-Additive Models

K d (d 1 , d 2 ) + K t (t 1 , t 2 ) • K c (c 1 , c 2 ) , (8)
K t (t 1 , t 2 ) + K d (d 1 , d 2 ) • K c (c 1 , c 2 ) , (9)
• Multiplicative Models

K t (t 1 , t 2 ) • K d (d 1 , d 2 ) , (10)
K t (t 1 , t 2 ) • K d (d 1 , d 2 ) • K c (c 1 , c 2 ) , (11)
First of all, we have trained independent kernel ridge regression models (see Sec. 3.1) for each measured smart meter using all the kernels from ( 6) to [START_REF] Dinuzzo | Output kernel learning methods[END_REF]. We compare these models against a multi-task learning approach that simultaneously performs estimates All forecasts are obtained using a multiplicative kernel model [START_REF] Dinuzzo | Output kernel learning methods[END_REF].

for all the meters, and also allows us to exploit the available meter grouping information in the dataset. Specifically, we have trained two separate multi-task output kernel learning (OKL) models (see Sec. 3.2): the first is trained over all the residential meters (the union of meters labeled "residential" and "others"), and the second over industrial (SME) meters. The maximum rank constraint for the first model was set to p = 200 to obtain a compact model that fits into memory, while the OKL model for SME meters was trained with full rank p = 485. We refer the reader to [START_REF] Dinuzzo | Learning output kernels for multi-task problems[END_REF] for a discussion on the effect of this parameter. Both OKL models utilize the multiplicative input kernel [START_REF] Dinuzzo | Output kernel learning methods[END_REF], as it proves to be the most effective at capturing the seasonal effects.

Figure 3 illustrates the challenges of long-term forecasting at low network level versus forecasting aggregated demands. We analyze the measured load and the corresponding forecast over a window of 5 weeks within the test period. In the top panel, one can see the aggregated load and the corresponding forecast obtained as the sum of all disaggregated forecasts obtained using model [START_REF] Dinuzzo | Output kernel learning methods[END_REF]. The kernel-based forecasts are rather accurate overall, only slighly estimating the total load during the Christmas week, a particularly problematic period to predict. In the middle panel, the measured load for a single SME meter is compared with the corresponding forecast. The varying demand profiles of different days of the week are captured rather well by the model. Again, there is a larger error over the Christmas week, caused by a sudden drop of the demand to a low baseline value (probably due to interruption of business activities followed by a slow resumption in the subsequent days). This leads the mod-els to over-estimate the load, though the model learned with a multi-task approach is less affected. Finally, the bottom panel shows the electricity demand of a residential customer, characterized by rapid variations with sharp consumption peaks and irregular patterns, that make the forecast even more difficult.

Table 2 reports the performance of all methods over the full set of 6433 smart meters, while Tables 3,4 and 5 report disaggregated performance measures over each group from Tab. 1. We start by analyzing the performance of the additive models, which are probably the most widely adopted in the literature. By comparing the performance of models ( 6) and ( 7), we can observe that adding a constant bias specific to the type of the day of week (kernel K c ) does not necessarily improve the accuracy of the model. The overall NMAE and MAPE are in fact higher for model [START_REF] Bj | Load forecasting using support vector machines: A study on EUNITE competition 2001[END_REF], see Tab. 2. Semi-additive models where the type of day of the week is utilized to switch between different profiles yields a significant improvement in performance.

The two semi-additive models ( 8) and ( 9) achieve similar performance over the groups residential and others (see respectively Tab. 3 and Tab. 5). However, for the SME customer group, model ( 9) is better in terms of both NMAE and MAPE (see Tab. 4). In previous works such as [START_REF] Ba | Adaptive learning of smoothing functions: application to electricity load forecasting[END_REF], semi-additive models of the form [START_REF] Ciliberto | Convex learning of multiple tasks and their structure[END_REF] have been proposed to switch between different daily patterns, depending on the type of day. Interestingly, our results show that in certain situations, such as when modeling industrial customers, it is even better to switch the overall sum of the daily pattern and the yearly pattern. We took a step even further by utilizing fully multiplicative models [START_REF] Dinuzzo | Learning output kernels with block coordinate descent[END_REF] and [START_REF] Dinuzzo | Output kernel learning methods[END_REF]. The multiplicative model [START_REF] Dinuzzo | Learning output kernels with block coordinate descent[END_REF] pools over different days of the week, while (11) learns independent models for each day. While the former is not always better than the semi-additive models, the latter significantly outperforms them. We can conclude that a multiplicative kernel structure [START_REF] Dinuzzo | Output kernel learning methods[END_REF] is the best at capturing yearly, weekly and daily seasonal effect, both overall and for each customer group. Such conclusion is aligned with recent results presented in [START_REF] Guerini | Long-term electric load forecasting: A torus-based approach[END_REF], where tensor product basis functions were utilized to capture weekly and yearly seasonalities in the simpler context of load forecasting for a single highly aggregated time series. A further performance improvement can be obtained by utilizing a multi-task learning approach, where correlation between electricity demand behavior of multiple customers is learned and exploited. As the Tables show, the multi-task OKL approach provides the lowest mean NMAE over all the meters performance, residential and others, and second lowest mean NMAE for SME (only 2% higher than the lowest for this group). The multi-task learning approach also provides the lowest mean aggregated MAPE, overall and for each [START_REF] Dinuzzo | Output kernel learning methods[END_REF].

In addition to improving forecasting accuracy, the lowrank multi-task learning model is significantly more compact in terms of number of parameters. For all the single-task methods (with additive, semi-additive and multiplicative kernels), the number of parameters is equal to the overall number of training observations m j=1 j . In our experiment, this amounts to about 13 million parameters (precisely 12785524 parameters). The low-rank output kernel learning method models each prediction function fj as a linear combination of p latent functions, shared by all tasks (see Sec. 3.2). These p functions g k can be seen as typical load profiles. As a consequence, only ( + m)p parameters are required to learn the prediction functions for all smart meters. In our experiment, this gave a total of about 3 million parameters (precisely 3016310) thus producing a model that is about 4.24 times more compact, in addition to being more accurate.

Discussion and conclusions

Our analysis shows that kernel-based multi-task learning is effective for the resolution of electric load forecasting problems. Focusing on the challenging problem of forecasting the electric load of individual customers, we designed kernels that take into account relevant multiple seasonality patterns. We demonstrated the clear benefits of multiplicative kernel models over additive or semi-additive models. Our results suggest a new modeling direction, as opposed to the (generalized) additive models, widely employed in the energy community. We illustrated further performance gain made possible by using a multi-task learning approach over a large number of single-tasks baselines. While recent studies reported MAPE around 3% for the short term forecasting of an aggregated signal of a few thousands of smart meters e.g. [START_REF] Alzate | Improved electricity load forecasting via kernel spectral clustering of smart meters[END_REF], our method achieves a MAPE of 4% on a long term forecasting scenario, which is a much harder problem as auto-regressive terms are not available.

The ideas and results presented in this paper open a wide range of considerations. First of all, they suggest that electricity demand data can be used as natural test benchmarks for multi-task learning methods. In addition, these problems motivate developing new techniques that allow to incorporate more complex task relationships structures taking into account, for instance, topological and physical constraints from the electricity network. The development of online methods that can automatically discover relationships between multiple tasks seems to be particularly important for short-term load forecasting scenarios. Finally, combining online multi-task learning methods with topological network constraints would allow to start tackling very complex scenarios such as forecasting on a full electricity network with dynamic reconfigurations.

Figure 3 :

 3 Figure3: CER data: measured load (blue curves) and corresponding forecast with independent (red curve) and multi-task (black dashed curve) for the aggregated demand (top panel), a single SME meter (mid panel), and a single residential meter (bottom panel). Measurements are shown over 5 weeks of the test period. All forecasts are obtained using a multiplicative kernel model[START_REF] Dinuzzo | Output kernel learning methods[END_REF].

Table 1 :

 1 Number of meters and sparsity for each customer group in the Irish CER dataset

	Customer group Meters Sparsity
	Residential	4225	0.028%
	Industrial (SME)	485	0.035%
	Others	1723	17%

Table 2 :

 2 Overall accuracy of the different long-term forecasting methods on the CER dataset

	Method	Kernel	NMAE Mean	Std	MAPE Mean	Std
	Additive Models	K d + K t	0.4829	0.0834	8.0384	7.2644
		K d + K t + K c	0.4897	0.0846	8.0435	6.9948
	Semi-Additive Models K d + K t • K c	0.4546	0.0682	7.1413	6.6078
		K d + K t • K c	0.4507	0.0649	7.2921	6.5868
	Multiplicative Models K d • K t	0.4663	0.0771	5.5284	5.4252
		K d • K t • K c	0.4237	0.0528	4.2917	4.3055
	Multi-Task OKL	K d • K t • K c	0.4226 0.0487	4.0222 4.0541

Table 3 :

 3 Accuracy of the different long-term forecasting methods on the residential customer group

	Method	Kernel	NMAE Mean	Std	MAPE Mean	Std
	Additive Models	K d + K t	0.5114	0.0929	13.1083	9.9519
		K d + K t + K c	0.5157	0.0941	13.0127 10.2067
	Semi-Additive Models K d + K t • K c	0.5005	0.0900	10.7139	9.9966
		K d + K t • K c	0.4977	0.0846	10.8144	9.6131
	Multiplicative Models K d • K t	0.5058	0.0844	8.4289	7.1833
		K d • K t • K c	0.4776	0.0721	5.2692	5.6305
	Multi-Task OKL	K d • K t • K c	0.4711 0.0716	4.9166 5.2588

Table 4 :

 4 Accuracy of the different long-term forecasting methods on the SME customer group

	Method	Kernel	NMAE Mean	Std	MAPE Mean	Std
	Additive Models	K d + K t	0.4517	0.2204	15.0748 23.1005
		K d + K t + K c	0.4590	0.1769	16.6668 17.0665
	Semi-Additive Models K d + K t • K c	0.3704	0.1238	8.1880 10.8081
		K d + K t • K c	0.3646	0.1305	8.1422 11.7269
	Multiplicative Models K d • K t	0.4006	0.1923	12.5359 20.4140
		K d • K t • K c	0.3127	0.1105	5.9289 10.3842
	Multi-Task OKL	K d • K t • K c	0.3194 0.0725	5.2940 5.8227

Table 5 :

 5 Accuracy of the different long-term forecasting methods for the smart meters labeled as "others" Finally, the multi-task approach is more robust, as the temporal standard deviation is the lowest for both NMAE and MAPE. Again, such robustness can be observed overall the customers as well as for each customer group. In particular, it is worth mentioning a 44% improvement of the standard deviation of the MAPE for SME meters, compared to the best single task model that uses the multiplicative model

	Method	Kernel	NMAE Mean	Std	MAPE Mean	Std
	Additive Models	K d + K t	0.4948 0.0779	7.4672	6.9361
		K d + K t + K c	0.5001 0.0804	7.4141	6.4660
	Semi-Additive Models K d + K t • K c	0.4659 0.0638	6.4749	5.9244
		K d + K t • K c	0.4601 0.0605	6.6273	5.9189
	Multiplicative Models K d • K t	0.4801 0.0710	5.4504	5.5997
		K d • K t • K c	0.4370 0.0517	4.0279	4.2522
	Multi-Task OKL	K d • K t • K c	0.4361 0.0490	3.7450 3.9503
	customer groups.

http://clients.rte-france.com/lang/fr/ visiteurs/vie/vie_stats_conso_inst.jsp

http://www.ucd.ie/issda/data/ commissionforenergyregulationcer/

http://www.timeanddate.com/time/change/ ireland/dublin