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Economic Growth and Cycles in Italy, 1881-1913 Luigi Einaudi’s Interpretation Revisited

Sella L., Marchionatti R. e F. Cassata

Dipartimento di Economia “S. Cognetti de Martiis”

Centro di Studi sulla Storia e i Metodi dell’Economia Politica "Claudio Napoleoni"

(CESMEP)

Working paper No. 05/2010

Università di Torino

Economic Growth and Cycles in Italy, 1881-1913

Luigi Einaudi’s Interpretation Revisited1

Lisa Sella, Roberto Marchionatti and Francesco Cassata Department of Economics ‘S. Cognetti de Martiis’, University of Torino, Torino, Italy

ABSTRACT

Since the 1950s, beginning with the Gerschenkron’s classical article, a great number of studies have dealt

with the issue of the Italian economic growth in the post-unification period. In this regard, an important work

of reconstructing of historical data series has been accomplished, particularly by Fenoaltea. However,

Fenoaltea’s analysis underestimates the short-term cyclical variability of the Italian economy in the sub-

period 1893-1913, a phenomenon on the contrary emphasized in the neglected qualitative-quantitative

analyses conducted by Luigi Einaudi and colleagues in the early twentieth century. This paper uses various

advanced spectral methods to reconsider Italian economic growth and cycles in the post-Unification period

in an attempt to evaluate Einaudi’s interpretation of Italian cyclical economic growth. Our analysis

substantially confirms the soundness of Einaudi’s interpretation.

Keywords: Economic Growth, Business cycles, Italian economy, Luigi Einaudi, Spectral methods.

1. Introduction

The work of Luigi Einaudi and his colleagues of the Turin school of economics at the beginning of

the twentieth century represents the first systematic analysis of Italian economic growth in the post-

unification period until the end of the Giolitti’s era and World War I, and it concentrated on in the

period between the 1880s and the war. This was a qualitative-quantitative analysis based on a

limited amount of official data, particularly at the macroeconomic level, augmented by statistical

enquiries furthered by Einaudi himself (see Bachi, 1908-1921) and a theoretically sound analysis of

facts.2 Indeed, according to Einaudi, being an economic historian requires one to “review and

rewrite the history of past facts” through the lens of the contemporary economic theory, which

enabled them to be put “in perspective so as to provide a logical interpretation” (Einaudi 1936:

157). Einaudi’s analysis showed that the Italian economy had undergone a growth trend

��������������������������������������������������������1 We are grateful to Stefano Fenoaltea for his comments. 2 The importance of this approach was first recognized by Schumpeter, who, in his History of Economic Analysis, wrote that Einaudi’s work “really fertilized general economics and did not, as in Germany, conflict with theory” (Schumpeter 1954: 855).

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characterized by a 20-year cycle and a 10-year periodicity, as follows : 1881-87: growth; 1887/8-

1898: crisis and depression reaching its lowest point in 1893, with 1897 identified as the starting

point of the recovery; 1898-1908/9: sustained growth interrupted by a brief crisis; 1909-13: a

reversal of trend considered not a real crisis, but rather “a malaise”. In this account, the upward

phase of 1898-1908 period, so-called Giolittian growth, corresponded to the phase of general

international prosperity, whose main features were the diffusion of technological innovations in key

sectors and the attraction of foreign resources and capital, especially via the German-style mixed

banks.3

Postwar studies of the Italian post-unification economic growth, which started with Gerschenkron’s

classic article (Gerschenkron 1955) and were developed by many authors,4 refer only marginally to

Einaudi’s analysis, even though Einaudi’s methodological contribution is recognized as important.

In particular, Fenoaltea (2004) recognizes the contiguity of his method with the Einaudian

theoretical-quantitative approach. We are indebted to Fenoaltea for his valuable reconstructions of

historical data series after the early Istat-Vitali estimates, considered seriously distorted. Fenoaltea

(2005) proposes a new reconstruction of Italian aggregate GDP by combining Federico’s (2003)

series for agriculture and his own series for industries and services (Fenoaltea 2003 and 2005). As

Fenoaltea points out, the reconstruction exhibits a higher and much steadier long-term growth than

do the previous Istat-Vitali (1957) estimates. Fenoaltea’s estimates identify a growth phase between

the late 1870s and the late 1880s, followed by stagnation until the second half of the 1890s, and

then rapid growth until 1908, followed by slower growth. This analysis of the cyclical growth of the

Italian economy appears to be consistent with the outcome of Einaudi’s analysis. However,

Fenoaltea’s new series underestimate the cyclical variability of the Italian economy in the sub-

period 1893-1913, probably because of a statistical artifact (Fenoaltea, 2003, 2005; Ciccarelli &

Fenoaltea, 2007). By contrast, this phenomenon is emphasized in the qualitative analyses by

Einaudi and his school, who stressed in particular a short-run variability in the period 1898-1908.

This paper investigates Italian economic growth and cycles in the post-Unification period in order

to attempt an evaluation of the Einaudi’s analysis of Italian cyclical growth. Our empirical

investigation is performed by means of several advanced spectral methods (Ghil, et al., 2002),

which provide valuable time-and-frequency-domain tools that complement the traditional time-

domain analysis of time series. In particular, these tools enable the identification and description of

��������������������������������������������������������3 Einaudi’s analysis is reconstructed and discussed in Cassata and Marchionatti (2010). 4 Most notably: Bonelli, (1978), Cafagna (1989), Toniolo, (1990), Zamagni (1993 and 2003), Cohen and Federico (2001), Fenoaltea (2003, 2005, 2006, 2010), Ciccarelli and Fenoaltea (), Delli Gatti and Gallegati (), Ciocca (2007).

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nonlinear trends and dominant cycles that characterize the behaviour of the series. They are

consequently well suited to the analysis of short and noisy data, like ours.

Our time-and-frequency-domain analysis substantially confirms the soundness of the Einaudi’s

interpretation of Italian economic growth and fluctuations in the period 1861-1913. In particular, it

identifies the weakness of Italian economy until the end of the nineteenth century, followed by its

more rapid and robust development during the so-called Giolittian growth period. Moreover, the

analysis shows both the 20-year and 10-year periodicities identified by Einaudi. The corresponding

time-domain reconstructions show excellent agreement with the Einaudian periodization of the

Italian economic development. Finally, our analysis finds an increase in the amplitude modulation

of all the significant short-run fluctuations detected, which may explain the short-run variability

observed by the Einaudians during the Giolittian growth period of 1898-1908.

��

�2. Italian economic growth 1881-1913. A spectral analysis  2.1. Premise

In this section we present a quantitative time-and-frequency-domain analysis of Italian economic

growth in the 1881-1913 period, based on the annual GDP series recently reconstructed by

Fenoaltea (2005) and Rossi et al. (1993). The time span 1881-1913 is in fact too short for a reliable

time-and-frequency-domain analysis on annual data. Consequently, we constructed an extended

1861-1951 GDP series by jointing together Fenoaltea’s series for the sub-period 1861-1913, with

Rossi et al.’s series for the sub-period 1913-1951. This option was preferred to the reconstructions

available for the whole time span (Maddison, 1995; Bardini, Carreras, & Lains, 1995), since these

generally underestimate the substantial growth of industrial output, agricultural production, and

food consumption at the end of the nineteenth century, which was erroneously identified as the

“crisis of the 1880s” (Fenoaltea, 2003; Cohen & Federico, 2001).

Our analysis points out that Fenoaltea’s new series underestimate the cyclical variability of the

Italian economy in the sub-period 1893-1913 (Figure 1). As shown by the author himself

(Fenoaltea, 2003, 2005; Ciccarelli & Fenoaltea, 2007), on the one hand this artifact is probably due

on the one side to Federico’s reconstruction of agricultural series, which yields too short and mild

cyclical variations despite it accounts for both producers’ and consumer’s reactions to price and

income changes (Federico, 2003); on the other hand, Fenoaltea’s (2005) industry and services series

show slighter fluctuations than the Istat-Vitali (1957) series, since the author doesn’t attribute to the

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missing pieces the variation of the available components of the series (Fenoaltea 2010). In addition,

some data points are simple interpolations of a small number of census benchmarks.

Figure 1 –Residuals of smooth Hodrick-Prescott filtering for the natural logarithm of Italian GDP 1861-1951 in mean deviation. The result is robust to different filtering procedures. �

However, the spectral approach adopted here mitigates this shortcoming. In fact, it enables the

identification of hidden regularities even when they have low explanatory power somewhere in the

series. Therefore, it is not surprising that we are able to find, in accordance with Eianudi’s analysis,

some evidence of the short-run variability in the Giolittian growth period 1898-1908.

�

2.2. Methodology Singular Spectrum Analysis (SSA) and other advanced spectral methods (Ghil, et al., 2002; Sella,

2008b) are applied in order to characterize GDP trend and cyclical components. These techniques

allow precise identification and description of the main regularities characterizing short and noisy

time series, i.e. nonlinear trends, low-frequency components, dominant cycles, and seasonalities.

Thus, this time-and-frequency-domain analysis represents a complementary approach to traditional

time-domain investigation (Ciccarelli & Fenoaltea, 2007; Delli Gatti, Gallegati, & Gallegati, 2005;

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Gallegati & Stanca, 1998), an it sheds new light on Italian economic growth in the period 1881-

1913.

Because the GDP series are heavily trended, reduction to stationarity and trend extraction turn out

to be fundamental crucial issues. This work reports the results from three different filtering

methods, namely a Fourier low-pass filter, a flexible version of the classic Hodrick-Prescott (1997)

filter (Pollock, IDEOLOG: A program for filtering econometric data - A synopsis of alternative

methods, 2008), and an SSA-based detrending (Ghil & Vautard, 1991).

The Fourier low-pass filter is a method of signal extraction based on the Fourier representation of a

stationary time series in the frequency domain, i.e. a linear decomposition of the series into a sum

of sines and cosines of increasing harmonics. The filter is obtained by setting to zero the

coefficients of the sinusoidal functions belonging to the stop band, thus causing leakage effects in

its finite implementation (Pollock, IDEOLOG: A program for filtering econometric data - A

synopsis of alternative methods, 2008).

The Hodrick-Prescott (1997) filter is standard in macroeconomics. It implies a second order

differentiation of the original signal, which is handled like a second-order random walk affected by

white noise. The smoothing parameter defines the cut frequency of the filter (Gómez, 2001). The

flexible version adopted here makes it possible to handle structural breaks (Pollock, The frequency

analysis of the business cycle, 2008).

Finally, SSA is a nonparametric data-adaptive spectral method analyzing time series structure from

the point of view of their variability. In particular, it embeds the series in a new coordinate system,

which represents the M directions of most variability in the series. When the object is projected onto

the new coordinate system, its Principal Components (PCs) are obtained. Their behavior can be

reconstructed in the time domain, yielding the so called Reconstructed Components (RCs, see

appendix). The extrapolation of components associated with trend behavior is tantamount to a low-

pass data-adaptive filter on the original series (Vautard, Yiou, & Ghil, Singular-Spectrum Analysis:

a toolkit for short, noisy chaotic signals, 1992; Penland, Ghil, & Weickmann, 1991).

Our pluralistic approach is justified on the one hand by the huge discretion of trend/cycle

decomposition in economic time series (Pollock, Investigating economic trends and cycles, 2008;

Pollock, The frequency analysis of the business cycle, 2008; Pollock, IDEOLOG: A program for

filtering econometric data - A synopsis of alternative methods, 2008). On the other hand, the

assumptions underlying spectral methods are hard to fulfill in practice, and the recommended

modus operandi is to apply several independent techniques before drawing any conclusion about

the spectral properties of the data generating process.

 

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2.3. Analysis Our investigation is based on annual GDP 1861-1951 in natural logarithms (Figure 2, left panel).

This section mostly presents the results for the span 1881-1913. Data are expressed in mean

deviation, in order to avoid spurious effects on the low-frequency bands (Estrella, 2007).

The heavy trend masks any contribution to the variability of the series by periodic and quasi-

periodic components characterized by high fundamental frequencies (Granger, 1966). As a

consequence, the periodogram of the original series, which describes how the power (or variance)

of the series is distributed against frequency (see appendix), exhibits a large bump at the zero-

frequency band (Figure 2, right panel), while contributions from any further component are

indistinguishable.

Figure 2 – Natural logarithm of Italian GDP 1861-1951 in mean deviation (left panel) and the corresponding periodogram (right panel). Source: see text.

By contrast, when the trend is removed, contributions from dominant oscillatory components can be

observed. Figure 3 collects the periodograms of the residuals of three different detrending

procedures applied to Italian GDP 1861-1951. More specifically, the straight line shows the spectral

peaks obtained from the residuals of a Fourier low-pass filter5 characterized by π/4 cut frequency,

which allows the total filtering of components with frequency higher than 0.785 rads. Thus, the

residuals include all cyclical movements characterized by periodicities in the 2-8 years range, i.e.

the classic business cycle range.

The broken line represents the periodogram of the residuals of a smoothed Hodrick-Prescott (HP)

filter designed to accommodate the structural breaks during World War I and II. More in detail, the

smoothing parameter λ=1 has been applied in the regions 1914-1920 and 1938-46, while the rest of

��������������������������������������������������������5 This method assumes stationary time series with coincident starting and ending points (Pollock, Investigating economic trends and cycles, 2008). Therefore, our data have been preprocessed by double differencing the residuals of a linear extrapolation obtained by weighted least squares.

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the series has been filtered by λ=6. The plot shows that this procedure filters out6 all components

with a periodicity higher than 10 years.

Figure 3 – Periodograms of residuals from low-pass filtering (straight line), smoothed Hodrick-Prescott filtering (broken line), and SSA detrending (dotted line).

Finally, the dotted line is the power spectrum estimate of the residuals from an SSA-based

detrending. The SSA residuals have been obtained by extrapolating the first two RCs from the

original GDP series for M=30. RCs 1-2 are classified as trend components by Kendall’s tau

nonparametric test for global trend identification (Kendall & Stuart, 1968) and they jointly explain

92.7% of the series total variance. This extrapolation is tantamount to a low-pass data-adaptive

filtering including in its residuals all components with 20-year periodicity and lower.

The spectral estimates of the three residual series are in quite good agreement, showing some short-

run peaks particularly evident in the low-pass and HP residuals, which are partially masked in the

SSA residuals by more informative long-run behavior. Figure 4 plots the corresponding trends (left

panel). As expected, the low-pass and HP trends closely follow the series along the structural

breaks. By contrast, the SSA trend captures the global behavior of the series.

��������������������������������������������������������6 Gómez (2001) shows the correspondence between the smoothing parameter λ and the cut frequency of the HP filter. Note that when the filter is applied to difference stationary series, it is tantamount to first differencing the data and applying an asymmetric moving average filter. This operation amplifies growth cycles at business cycle frequencies and dampens both short- and long-run fluctuations (Cogley & Nason, 1995).

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Figure 4 – Left panel: raw GDP series and estimated low-pass (straight line), Hodrick-Prescott (bold line) and SSA

trends (dotted line). Right panel: SSA trend for the period 1881-1913.

3. Einaudi’s interpretation reconsidered � The trend reconstructed by SSA (Figure 4, right panel) has the same features as identified by

Einaudi and his school in the period 1881-1913, i.e. a weak development of Italian economy until

the end of the nineteenth century, followed by more rapid and robust development during the so-

called Giolittian growth period. The kink is at around 1897, which Einaudi identified as the starting

point of the recovery from the end-of-the-century crisis.

Einaudi identified two cycles of about 20-year and 10-year periodicity around this growth trend.

SSA detects both of them in the SSA residuals. Vautard & Ghil (1989) demonstrate how this

method can efficiently characterize anharmonic oscillations embedded in the data (see appendix).

The middle panel of Figure 5 shows the reconstruction of these two fluctuations (RCs 1-2 and 3-4;

M=30), which jointly capture the 53% of the residual series variance. Their interaction with the

growth trend (Figure 5, upper panel) shows an excellent agreement with the Einaudian

periodization of Italian economic development: expansion in 1881-86, crisis in 1886-1898, a faster

expansion in 1898-1907, and the subsequent slowdown in 1907-09. Moreover, our analysis

highlights that the features of Italian development outlined by Einaudi until the end of the

nineteenth century (deep crisis, 1886-93, depression, 1893-94, reconstruction, 1893-1901) can be

traced back to the 20-year cycle, which clearly exhibits analogous behavior (Figure 5, middle left

panel). Similarly, the 10-year oscillation (Figure 5, middle right panel) matches Einaudi’s finding

for the span 1898-1908. In fact, the combined reconstruction of the two cycles (Figure 5, lower

panel) shows the predominance of the 20-year movement during the first sub-period, while the 10-

year component predominates thereafter.

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Figure 5 – SSA detrended series (7.3% of raw GDP variance). Upper panel: interaction growth trend / long-run cycles.

Middle panels: time-domain reconstructions of 20-year cycle (RCs 3-4, 19% variance) and 10-year cycle (RCs 1-2,

34% variance). Lower panel: combined reconstruction of long-run cycles (RCs 1—4, 53% variance). Elaborations for

M=30.

Finally, Einaudi and his colleagues identified the effects of several short crises during the Giolittian

growth period of 1898-1908, which cannot be captured by the long-run regularities discussed.

However, our analysis finds a corresponding increase in amplitude modulation for all the significant

short-run fluctuations detected, which may explain the short-run variability observed by the

Einaudians. Figure 6 shows the SSA reconstructions for the 6-year, 4.8-year, and 3.5-year

periodicities detected in low-pass residuals (RCs 1-2, 3-4, and 7-8 respectively; M=30). Monte

Carlo tests confirm their significance against both red and white noise null hypotheses at the 95%

level, while the classical periodograms (Figure 3) confirm their detection.7 These short-term

movements have been detected in all residual series as well. However, in SSA residuals the short-

run behavior is partially masked by more powerful 20-year and 10-year periodicities. Although

further sectorial analysis is needed to investigate their origin, we surmise that such short-run

behavior is connected with increasing industrialization.

��������������������������������������������������������7 Similar results have been obtained by multi-taper analysis (Ghil et al., 2002).

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 Figure 6 – Short-run behavior detected by SSA in low-pass residuals. Elaborations for M=30. 4. Some final considerations

�One hundred years after its formulation, the analysis of the Italian economic growth by Einaudi and

his school appears surprisingly sharp notwithstanding the objective limitations, in particular the

lack of official statistics available, in identifying growth cycles in a complex, but crucial, period of

Italian economic history. Contemporary advances in statistical techniques support the Einaudians’

findings. How do we explain this result? We believe that a possible explanation lies in Einaudi’s

theoretical-quantitative approach. In fact that analysis is the result of the application of a method of

Marshallian origin, seemingly loose if compared with the contemporary cliometrics, but which

proves to be highly sophisticated, and admirably able to link theoretical, statistical and historical

analysis closely together. Einaudi conducted his analysis of economic growth and fluctuations by

using a considerable set of data and information8 - price, production, interest rates series, the trend

of Stock Exchange and company profits, banks’ balance sheets, fluctuations in the exchange rate,

international trade movements, together with information from the business community -, to which

the eye of the historian and the economist give significance. This is probably a case in which, as

Schumpeter wrote, “we stand to profit from visits to the lumber room” (Schumpeter 1954: 4), of

course “provided we do not stay there too long” (ibid.).

Appendix: a methodological note

��������������������������������������������������������8 For examination of the statistical data and statistical tools used by Einaudi and his colleagues see Cassata - Marchionatti (2010) and Marchionatti, Cassata, Becchio & Mornati (2010).

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The spectral analysis of time series is essentially based on the estimation of the power spectrum, which

describes how the power (or variance) of the series is distributed with frequency. The periodogram is a

classical estimator of the power spectrum. It is asymptotically unbiased but inconsistent. Thus, various

smoothing operations have been devised (Pollock, 1999), together with other spectral estimation methods.

Since the assumptions underlying such methods are hard to fulfill in practice, it is recommended to apply

several independent techniques before drawing any conclusion about the spectral properties of the data

generating process (Ghil, et al., 2002).

SSA is a nonparametric data-adaptive spectral method (Broomhead & King, 1986b; Vautard & Ghil, 1989;

Golyandina, Nekrutkin, & Zhigljavsky, 2001). It involves three basic steps: (1) embedding a N-length time

series into a M-dimensional vectorial space, defining a new coordinate system; (2) computing the MxM lag-

covariance matrix CD of the series; (3) diagonalizing CD, which provides its eigenelements. Each of the M

eigenvalues represents the partial variance carried out in the direction defined by the corresponding

eigenvector. Projection of the original series onto each eigenvector yields the Principal Components (PCs),

which are reciprocally independent time series of length N-M+1. They can be interpreted as filtered versions

of the original series, where the choice of M represents the classical trade-off resolution vs. variance (Ghil, et

al., 2002). Vautard, Yiou, & Ghil (1992) demonstrate that the sum of the spectra of all PCs equals the power

spectrum of the original series, thus no information is lost by the filtering process. Given a subset of

eigenvalues, it is possible to extract N-length time series by combing the corresponding PCs. These are

called Reconstructed Components (RCs) and they capture the variability associated with the eigenvalues

selected. Vautard & Ghil (1989) show that anharmonic oscillations embedded in the data can be efficiently

characterized by pairs of nearly-equal eigenvalues, whose eigenvectors are in phase quadrature. This is an

advantage with respect to classical Fourier methods, which would require an high number of harmonics and

sub-harmonics to describe the same oscillation. This is due to the data-adaptive features of the SSA method.

Monte Carlo tests have been designed to assess the reliability of trend and oscillatory modes identified by

means of SSA ( (Allen & Smith, 1996).

References

Bardini, C., Carreras, A., & Lains, P. 1995. The National Accounts for Italy, Spain and Portugal. Scandinavian Economic History Review , 43, 115-47.

Bonelli, F. 1978. Il capitalismo italiano. Linee generali di interpretazione. In R. Romano, C. Vivanti (editors), Storia d’Italia. Annali 1, Einaudi, Torino, 1193-1255;.

Cafagna, L. 1989. Dualismo e sviluppo nella storia d’Italia, Marsilio, Venezia.

Cassata, F. & Marchionatti, R. 2010. Cronache economiche di un trentennio. Lo sviluppo dell’economia italiana 1881-1913 nell’interpretazione di Luigi Einaudi e la sua scuola. Rivista di Storia Economica, 26 (2), 161-208.

11

Ciccarelli, C., Fenoaltea, S. 2007. Business fluctuations in Italy, 1861-1913: The new evidence. Explorations in Economic History , 44 (3), 432-51.

Ciocca, P. 2007. Ricchi per sempre? Una storia economica d’Italia (1796-2005), Bollati Boringhieri, Torino. Cogley, T., Nason, J. M. 1995. Effects of the Hodrick-Prescott filter on trend and difference stationary time series. Implications for business cycle research. Journal of Economic Dynamics and Control, 19, 253-78.

Cohen, J., Federico, G. 2001. Lo Sviluppo Economico Italiano, 1820-1960. Il Mulino, Bologna.

Delli Gatti, D., Gallegati, M., & Gallegati, M. (2005). On the nature and causes of business fluctuations in Italy, 1861-2000. Explorations in Economic History , 42, 81-100.

Einaudi, L. 1936. Lo strumento economico nella interpretazione della storia. Rivista di Storia Economica, I, 149-158.

Estrella, A. 2007. Extracting business cycle fluctuations: What do time series filters really do? Federal Reserve Bank of New York, Staff Report (289).

Federico, G. 2003. Le nuove stime della produzione agricola italiana, 1860-1910: primi risultati e implicazioni. Rivista di Storia Economica , 19, 357-81.

Fenoaltea, S. 2003. Notes on the rate of industrial growth in Italy, 1861-1913. Journal of Economic History, 63, 695-735.

Fenoaltea, S. 2004. Einaudi commentatore e protagonista della politica economica: aspetti dell’età giolittiana. Rivista di Storia Economica, XX, 3, 271-76.

Fenoaltea, S. 2005. The growth of the Italian economy, 1861-1913: Preliminary second-generation estimates. European Review of Economic History , 9, 273-312.

Fenoaltea, S. 2010. The reconstruction of historical national accounts: the case of Italy. PSL Quarterly Review, 63 (252), 77-96.

Gallegati, M., Stanca, L. M. 1998. Le Fluttuazioni Economiche in Italia, 1861-1995 ovvero il camaleonte e il virus dell'influenza. Giappichelli, Torino.

Gerschenkron, A. 1955. Notes on the Rate of Industrial Growth in Italy, 1881-1913. The Journal of Economic History, 15 (4), 360-75. Ghil, M., Vautard, R. 1991. Interdecadal oscillations and the warming trend in global temperature time series. Nature (350), 324-27.

Ghil, M., Allen, M. R., Dettinger, M. R., Ide, K., Kondrashov, D., Mann, M. E., et al. 2002. Advanced Spectral Methods for Climatic Time Series. Review of Geophysics , 40 (1), 1-41.

Gómez, V. 2001. The use of Butterworth filters for trend and cycle estimation in economic time series. Journal of Business & Economic Statistics , 19 (3), 365-73.

Granger, C. W. 1966. The typical spectral shape of an economic variable. Econometrica , 34 (1), 150-61.

Hodrick, R. J., Prescott, E. C. 1997. Postwar U.S. business cycles: An empirical investigation. Journal of Money, Credit and Banking , 29 (1), 1-16.

12

13

ISTAT. 1957. Indagine sullo sviluppo del reddito nazionale dell'Italia dal 1861 al 1956. Annali di Statistica, serie VIII , 9.

Kendall, M. G., Stuart, A. 1968. The Advanced Theory of Statistics. Griffin, London.

Maddison, A. 1995. Monitoring the World Economy 1820-1992. OECD, Paris.

Marchionatti, R., Cassata, F., Becchio G., Mornati, F. 2010. When Italian Economics “was second to none”. Luigi Einaudi and the Turin School of Economics. Working Paper CESMEP, University of Turin. �Penland, C., Ghil, M., Weickmann, K. 1991. Adaptive filtering and maximum entropy spectra, with application to changes in atmospheric angular momentum. Journal of Geophysical Research , 96, 22659-71.

Pollock, D. S. 2008. IDEOLOG: A program for filtering econometric data - A synopsis of alternative methods. University of Leicester Discussion Papers in Economics 08/21.

Pollock, D. S. 2008b. Investigating economic trends and cycles. In T. C. Mills, & K. Patterson, Palgrave Handbook of Econometrics: Vol. 2 Applied Econometrics. Palgrave Macmillan, Houndmills, Basingstoke.

Pollock, D. S. 2008c. The frequency analysis of the business cycle. University of Leicester Discussion Papers in Economics 08/12.

Rossi, N., Sorgato, A., Toniolo, G. 1993. I conti economici italiani: una ricostruzione statistica, 1890-1990. Rivista di Storia Economica , 10 (1), 1-47.

Sella, L. 2008. Old and new spectral techniques for economic time series. Department of Economics "Cognetti de Martiis" Working Papers, Q.R. 08 (9).

Toniolo, G. 1990. An Economic History of Liberal Italy, 1850-1918, Routledge, London.

Vautard, R., Ghil, M. 1989. Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series. Physica D, 35, 395-424.

Vautard, R., Yiou, P., & Ghil, M. 1992. Singular-Spectrum Analysis: a toolkit for short, noisy chaotic signals. Physica D , 58, 95-126.

Vitali, O. 1969. La stima del valore aggiunto a prezzi costanti per rami di attività. In G. Fuà, Lo sviluppo economico in Italia (Vol. 3). Franco Angeli, Milano.

Zamagni, V. 1992. Il valore aggiunto del settore terziario italiano nel 1911. In G. M. Rey, I conti economici dell'Italia, vol. 2. Laterza, Bari.

Zamagni, V. 2003. The Economic History of Italy, 1860-1990: Recovery after Decline, Clarendon Press, Oxford.

Zamagni, V., Battilani, P. 2000. Stima del valore aggiunto dei servizi. In G. M. Rey, I conti economici dell'Italia, vol. 3**. Laterza, Roma-Bari.