ESTUDIOS SOBRE LA ECONOMIA ESPAÑOLA

Albert Marcet Morten O. Ravn

EEE 100

May, 2001

FEDEA Fundación de Estudios de Economía Aplicada

http://www.fedea.es/hojas/publicado.html

Albert Marcet Universitat Pompeu Fabra, CREI and the CEPR

and

Morten O. Ravn London Business School and the CEPR

March 22, 2001

Abstract

Many empirical studies of business cycles have followed the practise of applying the Hodrick-Prescott filter for cross-country comparisons. The Hodrick-Prescott filter involves the weight, , which determines the “smoothness” of the trend and is a standard choice for quarterly data. We show that this choice might distort the results and that care should be taken in checking if the results are “reasonable” in the light of common wisdom. For example, for Spanish data we show that, contrary to conventional wisdom, the results imply that the Spanish cycle is very smooth and that the period of 1975-1985 was one of macroeconomic tranquility. We propose a method for adjusting the smoothing parameter in the HP-filter by re-writing the HP-filter as a constrained minimization problem which selects endogenously a value of that imposes cross-country consistency on the detrending. Our proposed method is easy to apply, retains all the virtues of the standard HP-filter. When applied to Spanish data the results imply a return to conventional wisdom and we find results in line with economic historian’s views. We also look at data for a number of OECD countries and find that, with the exception of Spain, Italy and Japan, the standard choice of is sensible.

*We are grateful to Fabio Canova, Albert Carreras, Wouter den Haan, Jordi Gali, Gary D. Hansen, Edward C. Prescott, and Andrew Scott for helpful discussions of this paper. Marcet acknowledges support from DGES. The addresses of the authors are: Albert Marcet, Department of Economics and Business, Universitat Pompeu Fabra, Ramon Trias Fargas, 25-27, 08005, Barcelona, Spain, email: albert.marcet@econ.upf.es. Morten O. Ravn, Department of Economics, London Business School, Regent's Park, London NW1 4SA, England, email: mravn@london.edu.

1 Introduction

The purpose of this paper is to propose a method that enhances one's ability to make international comparisons of the properties of business cycle fluctuations. Following Lucas' (1977) concept of an international business cycle, many researchers have attempted to document similarities and differences across countries in the fluctuations over the business cycle. This is an important line of research because it influences the way in which theories of fluctuations are developed to address the empirical regularities and anomalies that are uncovered.

At a practical level this line of research is faced with the problem of finding the appropriate way in which to isolate the business cycle movements from the data. There are many different statistical techniques available for this purpose each having its advantages and disadvantages.. However, in the present context, one natural requirement is that similar procedures are applied to data for different countries. Partly for that reason, and partly because of ease of computation and reproduction, many researchers have adopted the Hodrick and Prescott (1980, 1997) detrending method (the HP-filter from now on) in empirical investigations of the properties of business cycle fluctuations.. Hodrick and Prescott (1980) originally applied this procedure to US post-war quarterly data and their findings have since been updated and extended in a number of papers including Kydland and Prescott (1990) and Cooley and Prescott (1995) (as well as in Hodrick and Prescott, 1997).

A large number of studies have examined data for other countries and in many cases compared the statistics with those obtained for the US data (in order to assess commonalities and differences in the business cycle properties). Blackburn and Ravn (1992) investigate UK business cycles, Brandner and Neusser (1992) study German and Austrian business cycles, Danthine and Girardin (1989) look at Swiss data, Dolado, Sebastián and Vallés (1993), Puch and Licandro (1997) and Borondo, González and Rodríguez (1999) study Spanish data, and Kim, Buckle and Hall (1994) look at data from New Zealand. Other studies have directly looked at cross-country comparisons, see e.g. Fiorito and Kollintzas (1994) and Blackburn and Ravn (1991) for studies of post war business cycles in a cross section of OECD countries. In an interesting analysis, Backus and Kehoe (1992) look at the historical business cycle moments for a cross-section of OECD countries and compare the business cycle features both across countries and across different periods of time. Yet other studies have looked at how business cycles are related across countries, see e.g. Backus, Kehoe and Kydland (1992), Ravn (1997) or Ambler, Cardia and Zimmermann (1999).

These studies have all applied the HP-filter in isolating the business cycle movements. We will argue that the way in which the HP-filter has been applied may not allow for a straightforward cross-country comparison of the business cycle moments, but rather than proposing an alternative filtering method we suggest a simple modification in the application of the filter that improves the properties of the results along the lines of allowing for cross-country comparisons of the statistical moments of the business cycle components. Hence, the paper does not follow in the lines of previous papers that have criticized the HP-filter (see e.g. Canova, 1998, Cogley and Nason, 1995, Harvey and Jaeger, 1995, Kaiser and Maravall. 1999, King and Rebelo, 1993, or Soderlind, 1994) but rather as a practical suggestion for improving the filter's properties in cross-country comparisons of business cycle moments.

The HP-filter involves removing the trend component of the data by use of a low frequency filter which is implemented by minimizing an objective function that depends on the weighted average of two components: the squared sum of the business cycle component (the deviation from trend) and the squared sum of the acceleration of the trend component weighted by a parameter (usually denoted by ). The choice of the value of the smoothing parameter in the HP-filter is usually left unaddressed and most researchers tend to follow the suggestion of Hodrick and Prescott (1980, 1997) and set this parameter equal to 1600 for quarterly data. In some studies, sensitivity analyses are carried out to the choice of , but they have limited their discussion to showing that some key calculations did not change when changed slightly. Hodrick and Prescott (1980, 1997) for one, themselves carry out such a sensitivity analysis and show that the results are reasonably robust whereas, for example, Canova (1998) arrives at a more negative result.

Hodrick and Prescott's (1980, 1997) choice of 1600 itself was guided partly by a prior on the variability of the trend and the cycle components of the US data and partly by the fact that with this value of the smoothing parameter produces “reasonable results” in the sense that the cyclical component agrees to a large extent with “conventional wisdom” on the US business cycle (see the quote from Kydland and Prescott, 1990, below). However, most subsequent studies have simply adopted the value of 1600 without considering either the sensitivity of the results to choices of this parameter or the sensibility of the results in the light of “traditional wisdom”. This practice is problematic since - as is well known - the HP-filter (as any other filter applied to finite samples) assigns parts of the low frequency fluctuations to the trend. This can create practical problems if the properties of the “trend” component differ substantially across countries. If, for example, a given country has experienced longer cycles than other countries, the trend component for this country will absorb a larger part of the cyclical component if not taken into account. Another way to express this is that since the HP-filter is an approximation to a low pass filter, the quality of this approximation depends on the mass of the spectrum of the data that is subject to the approximation error. This problem may make international comparisons of business cycle moments difficult if the trend properties of the data differ substantially.

A similar problem relates to how one chooses to adjust the HP-filter when applied to data that are sampled at different frequencies. Ravn and Uhlig (1997) discuss in detail how such adjustments to the frequency of observations should be carried out.

We initially illustrate the pitfalls of following this convention by looking at quarterly data for Spain. We show that, even though this has been the choice in many applied studies for Spanish data, if is used for this data one arrives at conclusions completely at odds with the view that any historian would have on the modern history of economic fluctuations in Spain. We argue that this is precisely because there have been very strong low frequency movements in Spain in the post-war time. Since the fall of the Franco regime the Spanish economy has gone through a process of restructuring and Spain has been subject to a protracted transitional process. Furthermore, the consensus among economic historians is that during and after the oil shocks of the 70's, Spain experienced a very long recession, lasting from 75 to 85, which has not been experienced in other countries in the same way. But to the contrary, the HP-filter with delivers the result that ... the Spanish cycle was unaffected by the oil shocks. An odd conclusion for a country that imports most of the oil it consumes and that did not follow reasonable policies to adapt to the oil shocks.

Our suggestion is to apply a simple method for choosing in a systematic way, that will be valid in international comparisons. The idea is to choose that generates a comparable level of volatility in the trend in each country. We show that, if we reinterpret the HP-filter as the solution to a constrained minimization problem, our procedure is consistent with imposing the same constraint across countries, while the usual practice of keeping constant (and equal to 1600) amounts to changing the constraint across countries. Furthermore, applying our procedure to Spain, the cyclical component becomes “reasonable” and it agrees with the story that an economic historian would tell. We also extend this to look at data for other countries. The results are very encouraging and tend to confirm our claim that the constrained minimization is a better way to interpret the HP-filter in international comparisons.

In section 2 we describe the HP-filter, and the reasoning that lead several authors to use . Section 3 discusses the reinterpretation of the filter and present results for Spain. Section 4 provides more empirical evidence, and section 5 concludes.

2 The Hodrick-Prescott Filter and Its Implications

In this section we will outline the Hodrick and Prescott (1980, 1997) filter, how the choice of the value of the smoothing parameter is usually made, and the implications for the measurement of the cycle in Spanish GDP. This will set the scene for our suggestion of how to modify this filter that we propose in the next

section.

Let denote the natural logarithm of a given time-series which is observed over the sample from t = 1 to T. Consider decomposing this series into a trend component, denoted by , and a cyclical component, denoted by , so that:

\[y _ {t} = y _ {t} ^ {t r} + y _ {t} ^ {c}\tag{1}\]

Much of the business cycle literature has applied the method put forward by Hodrick and Prescott (1980, 1997), and given its dominance in the empirical literature, this method shall be our concern from now on. These authors proposed to carry out the decomposition by making use of a low frequency filter. It involves estimation of the trend component by solving the following minimization problem:

\[\min _ {\left\{y _ {t} ^ {t r} \right\} _ {t = 1} ^ {T}} \sum_ {t = 1} ^ {T} \left(y _ {t} - y _ {t} ^ {t r}\right) ^ {2} + \lambda \sum_ {t = 2} ^ {T - 1} \left(\left(y _ {t + 1} ^ {t r} - y _ {t} ^ {t r}\right) - \left(y _ {t} ^ {t r} - y _ {t - 1} ^ {t r}\right)\right) ^ {2}\tag{2}\]

Here the first term in the objective function is a measure of the “goodness-of-fit” since it measures the average closeness of the trend to the actual data. The second component punishes accelerations in the trend component. is a key parameter in the filter: it determines the trade-off between “goodness-of-fit” and the smoothness of the trend. Variations in this parameter will determine the properties of the trend component and, thus, the properties of the cyclical component. In the limit as the trend becomes linear thereby allowing for large fluctuations in the cyclical component whereas leads to the trend component being equal to the data series , and it sets the cyclical component to zero. Alternatively, one can think of the filter in terms of a low pass filter (see e.g. King and Rebelo, 1993, Ravn and Uhlig, 1997, or Baxter and King, 1999) for discussions along this line.

The HP-filter can be seen as forcing the trend component to lie somewhere in the middle between a linear trend and the actual series, and the parameter governs how close we are to one or the other. In the formulation of Hodrick and Prescott (1980, 1997) is taken as a fixed parameter, which they set equal to 1600 for US quarterly data. Their choice of this value was based upon a prior about the variability of the cyclical part relative to the variability of the change in the trend component. Hodrick and Prescott (1997) state that:

Baxter and King (1999) provide an alternative method in the spirit of the HP-filter based explicitly on a band-spectrum technique, a technique that the HP-filter is often related to. It remains to be seen whether the Baxter and King procedure will gain widespread use. In any case, for cross country comparisons, one would find similar problems to the ones we encounter in choosing a fixed band across countries, and solutions similar to the one we propose here for the HP filter could be applicable to the Baxter and King filter. Christiano and Fitzgerald (1999) propose an alternative band-pass filter.

Other popular methods for making this decomposition include polynomial trends, ARIMA decompositions (such as the Beveridge and Nelson, 1981, method), unobserved components methods (c.f. Harvey, 1985 or Watson, 1986) or multivariate methods, see Canova (1998) for a comprehensive discussion and evaluation.

“If the cyclical components and the second differences of the growth components were identically and independently distributed, normal variables with means zero and variances and (which they are not), the conditional expectation of the , given the observations, would be the solution to program (2) when , …“Our prior view is that a 5 percent cyclical component is moderately large, as is a one-eight of 1 percent change in the growth rate in a quarter. This led us to select or ”.

This quote brings out what will be our main message: The choice of the smoothing parameter depends on the properties of the trend component, i.e. differences in the low frequency characteristics of will have an effect on the choices one makes about the value of the smoothing parameter.

Kydland and Prescott (1990) argue further in favor of the choice of for quarterly post war US data because:

“With this value, the implied trend path for the logarithm of real GNP is close to the one that students of the business cycle and growth would draw through a time plot of the series”

This argument says that the results should to some extent be judged against “conventional wisdom” on trends and business cycles. Indeed, one could hardly disagree. However, in most (all) cases, such conventional wisdom is not precise enough to allow one to make the statistical decomposition, a point also stressed by Hodrick and Prescott (1980, 1997). Nevertheless, one may take this as at litmus test: If the results are very different from those argued for by economic historians and other students of business cycle one should be careful with the interpretation of the results.

For example, consider a time time-series process for some variable given by an AR(1) specification:

AR(1)

1

2,

where is i.i.d. But now consider two different cases (letting the variance of be equal in both cases). In Case 1 is close to one (and the variance of is low) while in Case 2, (and the variance of is high). If we use the same to extract the trend and cyclical components from both series (and the quote of Hodrick and Prescott seems to recommend that, since both series have the same variance), it is clear that Case 1 will display a trend component that oscillates and closely resembles the actual series, while Case 2 will extract a trend that is much closer to zero: the actual trend in both cases. In this example, to discover that the trend component is zero and avoid the oscillations in the trend, one would have to use a much higher in Case 1.

However, the fact that the results for the HP-filter with are “reasonable” for US data does not automatically guarantee that similar good results are obtained for applications to data for other countries. Spanish quarterly data provides a good example of the pitfalls that one might face. To show this, we analyze Spanish real GDP for the period 1970 quarter 1 to 1998 quarter 4. These data were obtained from the Spanish National Institute of Statistics (downloaded from the web page of Instituto Nacional de Estadística at www.ine.es). This is a relatively short period but, unfortunately, quarterly Spanish national accounts data do not exist for a longer sample period. Figure 1 illustrates these data graphically (the figure shows the logarithm of quarterly Spanish GDP in which we have normalized the first observation to 1). Along with the series for real GDP we also plot the estimated Hodrick-Prescott trend component for (Panel A) and the estimated cyclical component (Panel B). This is the same data set that has been used in a number of studies on the Spanish economy.

The plot of the time series for real Spanish GDP illustrates that there have been some large and prolonged fluctuations in the Spanish economy, especially in the period 1975-1985. The consensus view among economic historians (see, for example, Pollard (1990) for an extensive discussion) about the Spanish cycle is as follows: the two oil shocks of the seventies had a large negative impact in Spain, much as they did in other non-oil producing countries. However, the short and sharp recovery that was observed in most industrialized economies in between the two oil shocks, around 1978, did not occur in Spain. But regardless of the disagreement on the precise causes of what happened, all economic historians agree that there was no recovery between the two oil shocks, and that this was caused by the exceptional political circumstances caused by the transition from an authoritarian to a democratic regime.

An alternative way to see this is that while the Spanish GDP per capita relative to the average GDP per person in the four main European economies, Germany, France, Italy, and the U.K., rose from 40.9 percent to 54.3 percent in the period 1965-1975, it fell continuously to 48.9 percent by 1985. In other words, Spain experienced one very long recession from, roughly, 1975 to 1985, while most other countries experienced at least two recoveries during this period. This can be seen, for example, in Figure 2 that displays the US cyclical component.

The results for , however, do not support these views. Panel B of

Although the Spanish National Institute of Statistics does not report how the data have been constructed, one is lead to suspect that the quarterly data have been obtained by interpolating lower frequency data since the GDP series appear to be very smooth. Unfortunately, alternative and better data do not exist and we proceed with the analysis of these data.

Historians might disagree about the precise reasons for the absence of this recovery: some will say that it was due to excessive pressure on wages from the recently legalized trade unions, others will say that the high inflation scared away investment (and that the high inflation was due to a weak government and a loose monetary policy), others will say that the government did not react “appropriately” to the first oil shock, and yet others will say that the fear of a very unstable political situation led to capital flight.

Figure 1 says that, according to the HP-filter, the Spanish business cycle has been rather smooth, and, in particular, the period from 1976 to 1985 was an average period, without much business cycle activity. Compared to the US cycle in Figure 2, and if we confine our attention to the cyclical component, we would say that the oil shocks had little effect on the Spanish cycle. The reason for this is that the HP-filter associates the main part of the first oil shock and its aftermath to the trend component, as can be seen from Panel A, Figure 1.

These observations translate easily into statistical moments. Table 1 lists the standard deviation for the cyclical component of Spanish and US data on real GDP for the same period together with the estimated autocorrelation coefficients of orders 1-5 quarters. As previously reported by Dolado, Sebastian, and Vallés (1993), the results indicate, surprisingly and contrary to economic historians' view, that the Spanish business cycle for 1970-1998 has been very smooth indeed. We find that the HP-filter with implies a standard deviation of the cyclical component of Spanish GDP is 30 percent lower than the corresponding US number. Several authors (see Dolado, Sebastian and Vallés, 1993, Puch and Licandro, 1999, and Borondo et al., 1999) have described the Spanish economy as one with lower output volatility . We believe, however, that results with HP(1600) are a mere artifact of the way that the trend is estimated and should be dismissed, if nothing else, because it leads to a business cycle component very different from the consensus view among students of the business cycle.

3 Choosing the Smoothing Parameter

To insure greater cross-country comparability of the business cycle properties we suggest a method for (i) calculating the trend component without running into the sort of problems highlighted in the previous section, while (ii) keeping the method close to the original HP-filter thereby allowing for straightforward comparability to other data, computational ease and reproduction. Our approach is to select endogenously in order to maintain comparability across countries.

We propose re-cast the Hodrick-Prescott (1980, 1997) as a constrained minimization problem.. We can rewrite the filter as:

\[\min _ {\left\{y _ {t} ^ {t r} \right\} _ {t = 1} ^ {T}} \sum_ {t = 1} ^ {T} \left(y _ {t} - y _ {t} ^ {t r}\right) ^ {2}\tag{3}\]

subject to:

Dolado et al. show calculations for various values of , some of them close to the ones we will find for Spain, but in the conclusion they only discuss the values for .

This general idea can be incorporated in other filters. For example, in the Baxter and King (1999) filter, the band width across countries could be chosen in a similar way.

\[\frac {\sum_ {t = 2} ^ {T - 1} \left(\left(y _ {t + 1} ^ {t r} - y _ {t} ^ {t r}\right) - \left(y _ {t} ^ {t r} - y _ {t - 1} ^ {t r}\right)\right) ^ {2}}{\sum_ {t = 1} ^ {T - 1} \left(y _ {t} - y _ {t} ^ {t r}\right) ^ {2}} \leq V\tag{4}\]

where is a constant determined by the researcher computing the trend.

When formulated this way, V can be thought of as a “target value” for the relative variabilities of the (acceleration in the) trend and cyclical component. The intuitive appeal of this formulation is that the value of V now has a direct interpretation in terms of “fixing” the closeness of the trend to the data. For example, in the case of cross country comparisons, we would set V constant across countries and, in that way, we would ensure that we have the same variability of the trend in all countries, proportionally to the variability of the cyclical component. This ensures comparability across countries, in the sense that the relative variability of the acceleration of the trend and the cyclical component is common to all.

This problem and the HP-filter are equivalent. First of all, notice that if we set V = 0 the HP-filter will result in a linear trend component, while if we let V go to infinity the HP-trend becomes equal to the series . In other words, by changing V, we have the same flexibility as by changing in the standard formulation of the HP-filter.

Second, if we rewrite (4) multiplying both sides of that equation by , it is clear that the Lagrangean of the above minimization problem solves

\[\min _ {\left\{y _ {t} ^ {t r} \right\} _ {t = 1} ^ {T}} (1 - \bar {\lambda} V) \sum_ {t = 1} ^ {T} \left(y _ {t} - y _ {t} ^ {t r}\right) ^ {2} + \bar {\lambda} \sum_ {t = 2} ^ {T - 1} \left(\left(y _ {t + 1} ^ {t r} - y _ {t} ^ {t r}\right) - \left(y _ {t} ^ {t r} - y _ {t - 1} ^ {t r}\right)\right) ^ {2}\tag{5}\]

where is the Lagrange multiplier of the rewritten constraint (4). It is obvious that the solution to this Lagrangean is equivalent with the HP-filter setting

\[\lambda = \frac {\bar {\lambda}}{1 - \bar {\lambda} V}\tag{6}\]

Thus, if we set V equal to the ratio in the left side of (4) using US HP-filtered data with , and then we solve the above minimization problem for US data using that value of V, we would recover the same series as with the HP(1600) filter. The usual value of would then be interpreted as a value of that satisfies (6) when is the Lagrange multiplier of the constraint (4) for the value of V in the US.

Therefore, our approach is to impose a comparable level of variability of the acceleration of the trend and cyclical components across countries. The that will be applied for each country will be endogenously determined when we solve for the Lagrange multiplier of constraint (4) for each country (using equation (6)). It is more desirable to keep constant V rather than across countries, since

V is a parameter that can be easily interpreted. In this sense, the usual practice of keeping constant amounts to changing the constraint (4) across countries.

We refer to this problem, setting V as in the US and using this value for all countries, as “adjustment rule 1”. This adjustment is very easy to compute. Since the mapping between and , is one to one, solving for is equivalent to solving for . Given a value for we compute the trend in the usual way (using the HP-filter with the specified value of ) and, since the trend is a function of , we can define the function

\[F (\lambda) \equiv \frac {\sum_ {t = 2} ^ {T - 1} \left[ \left(y _ {t + 1} ^ {t r} (\lambda) - y _ {t} ^ {t r} (\lambda)\right) - \left(y _ {t} ^ {t r} (\lambda) - y _ {t - 1} ^ {t r} (\lambda)\right) \right] ^ {2}}{\sum_ {t = 2} ^ {T - 1} \left[ \left. y _ {t} - y _ {t} ^ {t r} (\lambda) \right] ^ {2} \right.}\tag{7}\]

where is the trend component that relates to . Now the problem is to find a value that solves the equation . If there is no solution to this equation, the Kuhn & Tucker conditions imply that the constraint is not binding so that the solution to the minimization problem is given by and, therefore, .

A variety of iterative schemes can be used in order to solve the equation . In practice we found no problems with multiplicity of solutions to the first order conditions, and it suffices to adjust upwards when is positive and vice versa. Thus, our proposed adjustment to the standard HP-filter allows for easy computation, and can be replicated in a straightforward manner by other researchers hence maintaining the advantages usually associated with the HP-filter.

The second method that we propose is even closer to the HP-filter as it sets

\[\min _ {\left\{y _ {t} ^ {t r} \right\} _ {t = 1} ^ {T}} \sum_ {t = 1} ^ {T} \left(y _ {t} - y _ {t} ^ {t r}\right) ^ {2}\tag{8}\]

\[\mathrm{s.t.}: \frac {1}{T - 2} \sum_ {t = 2} ^ {T - 1} \left(\left(y _ {t + 1} ^ {t r} - y _ {t} ^ {t r}\right) - \left(y _ {t} ^ {t r} - y _ {t - 1} ^ {t r}\right)\right) ^ {2} \leq W\tag{9}\]

The interpretation of this rule is clear: the constraint now restricts the variability of the acceleration in the trend component directly and now has the interpretation as the Lagrange multiplier on (9). We refer to this problem, setting W as in the US and using this value for all countries, as “adjustment rule 2”. This can be computed in a way analogous to the previous adjustment rule.

The difference between the two rules is clear: rule 2 imposes the same variability of the growth of the trend across countries, while rule 1 allows for a larger variability of the growth rate in countries with a more volatile cyclical component. Rule 2 should be used if the researcher believes, a priori, that there is no reason to expect that the deviation of actual trend from a linear trend should differ across countries. Two countries that share common industrial structures and are subject to similar economic conditions, such as the US and the UK (from the Thatcher period onwards) would be obvious candidates for imposing Rule 2. Rule 1 should be used instead, if the researcher believes deviations from linear trend are larger in some of the countries considered. For example, if some of the countries considered underwent large changes in their economic environment or if they had initially high growth due to convergence from an initially low level of income to a higher steady state income level, a larger deviation from linear trend should be expected in those countries, and Rule 1 would seem more appropriate. Alternatively, two countries might differ in e.g. their exposure to shocks to the economy due to different industrial structures leading one to expect that one country has larger cyclical variations than the other. If no differences in the trend are expected apriori, one would use rule 2. For example countries that are more specialized might be more subject to business cycle variations than other more diversified countries.

In addition to these different interpretations, using the two rules will be useful as a way to test the sensibility of our results to small changes in the interpretation of the constrained minimization problem that we propose.

3.1 Results for Spain

We first concentrate on the discussion of how these adjustment rules affect the computation of the Spanish cycle. The results are reported in Table 1. First, let us evaluate the quantities V and W for . For US data we find that and . Using for the Spanish data implies that and . Thus, the variance of the (acceleration in the) trend component relative to the variance of the cyclical component is around 5 times higher for the Spanish data than for the US data for and the variance of the acceleration in the trend component itself is more than twice as high in the Spanish data. Thus, to match V or W, it is clear that needs to be raised so as to induce less variation in the trend component in Spain. For each adjustment rule we find that is equal to 5385 and increases further to 6369. These are large increases and we now proceed to investigate how they affect the business cycle properties of the data.

We find large changes in the implications for the behavior of the business cycle component: with adjustment rule 1, the percentage standard deviation in Spain increases to 1.81 or 7 percent higher than in the US. This is in stark contrast to the result that the Spanish business cycle is less pronounced than the US business cycle. For adjustment rule 2, we find a standard deviation of the cyclical component of 1.92 percent per quarter which is 13 percent higher than the corresponding US number. Thus, in line with common agreement among economic historians, we now find that the Spanish business cycle has been more variable than the US business cycle.

It is also worth noticing that with HP(1600) we obtain much higher persistence of the cyclical component in Spain than in the US, and that this persistence increases even more with our adjustment rules. These results confirm that one should not use the same in Spain as in the US, given our discussion of the effects of differing low frequency data on the HP filter.

As we said in the introduction and in section 2, a litmus test that any filtering procedure should pass is to yield results that agree with the conventional wisdom about the occurrence and relative importance of recessions and expansions. From Figure 3 we see that the cyclical components for the two adjustment rules are very similar but significantly different from what is implied for . First, we see that both adjustment rules lead to an increase in the “size” of the Spanish business cycle as we have just discussed. Secondly, the general shape for the early 1970’s and post 1987 are very similar for all three cases, the only difference being the magnitude of the cycle. But, for the period from 1974.2 to 1987 the results are rather different. As we said before, had the counterfactual implication that Spain had enjoyed quite a stable period around the oil shocks. When we adjust the smoothing parameter we find, instead, that the period 1974.2-1987 appropriately displays almost one long recession only interrupted by a very mild recovery around 1976-1977. Our adjustment rules now support the conventional wisdom (discussed above) much more than the HP-1600 series, specially if we compare Spain under the adjustment rule 2 with the US.

This also explains why the adjustment rules discussed above yield such different results from common practice: HP(1600) interprets this period as a downturn and then an upturn of the trend (see panel A in Figure 1), leaving the cycle unchanged, and assigning too much volatility to the trend.

This shows that the adjustment rules, in addition to giving more comparable results across countries, yields reasonable results for the implied business cycle components in Spain.

4 International evidence for OECD economies

In this section we extend the analysis above to OECD economies. We look at post war quarterly real GDP for Australia, Canada, France, Italy, Japan, Switzerland, and the United Kingdom. For Australia, Japan and the UK the sample periods are 1960.1 to 1998.4. For Canada the sample period is 1961.1-1998.4, for Italy and Switzerland it is 1980.1-1998.4, while for France it is 1985.1-1998.4.

As above, we look at the results for and for the two adjustment rules when we match V and W to the US data for the same sample period. The results are listed in Table 2. With the exception of Italy and Japan, we find that the choice of appears to be appropriate: we find very small changes in when we apply either of the two adjustment rules and the changes in that do occur, do not significantly affect the variability or the persistence of the cyclical component of GDP in any of these countries. For all these countries, we also find that the two adjustment rules lead to very similar results. In some cases the robustness of the choice of is rather remarkable; for France, for example, we find that and , while for the UK we find and . Thus, many previous studies of the business cycle properties of data from these countries based on the conventional choice of have, perhaps by chance, been consistent in the sense of imposing similar trend properties across countries when comparing the results with those of the US as reported by e.g. Kydland and Prescott (1990).

main OECD economies and because we were able to obtain quarterly data for sufficiently long periods of time. We also have data for Germany but chose to eliminate these because of German Unification. The data were all taken from the OECD national accounts database and relate to GDP in constant prices.

For Italian data we find that the choice of the adjustment rule matters more for the results. Adjustment rule 1 leads to an increase in to 2479. Adjustment rule 2, however, which takes into account only the variability of the trend component, leads to a drop in the value of the smoothing parameter to 1061. The reason for this difference is that the variabilities of both the trend and the cyclical component are quite small for the Italian data but more so as far as the cyclical component is concerned. Thus, adjustment rule 1 leads one to lower the smoothing parameter in order to generate higher volatility of the trend component relatively to the cyclical component in order to match the relative variability observed in the US data. For adjustment rule 2 the cyclical variability does not directly affect the choice of the smoothing parameter and hence we obtain an increase in when using this adjustment rule. Nevertheless, regardless of whether ones uses or either of the two adjustment rules, the variability of the cyclical component is significantly below what is observed for the US and we do not find major changes in the business cycle moments of Italian GDP.

The only other country for which we find large effects of using rules 1 or 2 is Japan. For we find that the Japanese business cycle is slightly smoother than the US business cycle. For the US data we find that and for while the Japanese numbers are and . Thus, as for the Spanish data, our adjustment rules will lead to increases in so as to lower the variability of the trend component. We find and . These increases in the value of the smoothing parameter imply that the Japanese business cycle is more volatile than the US business cycle with the standard deviation of the cyclical component being 13 percent higher than the corresponding US number for adjustment rule 1 and 32 percent higher for adjustment rule number 2. These results are similar to those obtained for Spain.

We find these results compelling in the light of the macroeconomic developments in Japan with the prolonged period of sustained growth in the 1960 and the 1970's and the lengthy period of macroeconomic turbulence of 1990's. With these phenomena become almost exclusively attributed to the trend while the adjustment rules imply that there are also business cycle effects. Also, the case of Japan shows how the two rules differ. We know that Japan grew much faster in the 60's than other countries, and the difference with other countries was less marked later on. The large recession of the Japanese economy in the 1990's have reversed this picture so that Japan now is growing substantially slower than most other OECD countries. To the extent that these phenomena should be attributed to transitional growth, they should be absorbed by the trend component, we should find it natural that a linear trend is not appropriate for Japan, and we should allow for more variability in the Japanese trend, so that rule 1 seems more appropriate.

Note that the numbers for the US are slightly different from those we quoted in the previous section because the sample period is different.

We also want to check if this change in the value gives rise to a different view on the Japanese business cycle as is the case for the Spanish data. Figure 4 illustrates graphically the cyclical components for for the two alternative values of the smoothing parameter. The figure reveals that the change in the value of the smoothing parameter mainly gives rise to an increase in the business cycle volatility. Thus, for Japan, the adjustment rules lead to an increase in business cycle volatility but leaves the business cycle dating unchanged.

5 Summary and conclusions

This paper has proposed a simple method for adjusting the Hodrick and Prescott (1980,1997) detrending procedure, which is by far the most common detrending procedure used nowadays in the business cycle literature. The usual practice of setting has been followed in applications to various countries, without checking the results roughly correspond to “conventional wisdom” about the behavior of the cycle. We argue that this choice hinges on the properties of the trend component. In the spirit of salvaging the HP filter as much as possible, we have proposed a way to choose in a systematic way across countries so as to make results comparable. Our suggestion is simply to re-interpret the HP-filter as a constrained optimization problem that involves minimizing the variance of the cyclical component subject to a constraint on the variability of the acceleration in the trend. In this way the value of the smoothing parameter can be re-interpreted as relating to the multiplier on the imposed constraint. For Italy, Spain and Japan, the results obtained with our approach are quite different from the results of . The standard choice of setting amounts to changing the parameter of the constraints across countries, while our adjustment rule insure that the constraint is the same for all countries.

For Spanish data we showed that the standard choice of leads to counterfactual results: the Spanish cycle is estimated to be very smooth and the period 1975-85 is estimated to be a period of macroeconomic stability, which is in direct contrast to common wisdom among economic historians, who would classify this period as one long recession. However, imposing the adjustment rules that we propose, one would conclude that Spain indeed has large business cycles and that 1975-85 was, to a closer approximation, one long recession . Thus, our adjustment rules make sense not only from a statistical point of view but also, from a practical point of view.

We have described two alternative choices for the exact formulation of the constraint on the variability of the acceleration component, which we dubbed "adjustment rules 1 and 2". These adjustment rules gave rise to very similar results for most countries, showing some robustness to the general idea of recasting the HP-filter as a constrained minimization problem

We also showed that when applied to data for other OECD countries approximates quite well the values that one would have chosen using either of our adjustment rules for most countries. The main exception to this is Japan for which we find results similar to those that we document for the Spanish data. Spain and Japan share the common feature that there have been prolonged business cycle fluctuations which the traditional choice of do not uncover.

In summary, we re-interpret the HP-filter so as to make cross country comparisons in a natural and practical way. The adjustment rule is easily computed. In some countries this rule has little effect on the computations, but in some others it has a large effect and in the right direction.

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6 Tables and Figures

Table 1. Quarterly Data: 1970.1-1998.4

Table 2. International Evidence

Figure 1. Spanish real GDP, lambda=1600

Figure 2. US cycle, lambda=1600

Figure 3. Spanish Cyclical Components

Figure 4. Real GDP for Japan