Did leaving the gold standard tame the business cycle? Evidence from NBER reference dates and real GNP.
1. Introduction
"Our empirical results are striking [...] a clear rejection of the null hypothesis of no postwar duration stabilization." This is how Diebold and Rudebusch (1999, p. 7) summarize the six papers that constitute the second part of Business Cycles: Durations, Dynamics and Forecasting. The view that the U.S. economy has been more stable since World War II is widely held and dates back at least to Arthur Burns' (1960) presidential address to the American Economic Association. Furthermore, Baily (1978) can be credited with singling out 1946 as a date of demarcation. He notes that, from 19001945, the U.S. gross national product (GNP) gap was 381% more volatile than it was from 19461976. Baily chooses this break point to coincide with the U.S. Employment Act of 1946, and it is the basis of the Diebold and Rudebusch (1992; 1999) empirical analyses. (1) However, there are other plausible dates to consider and an important paper by Cover and Pecorino (2005) addresses this issue. They follow Diebold and Rudebusch (1992) in using a ranksum test to evaluate potential break points in the length of business cycles. Expansion and recession lengths are defined primarily in terms of National Bureau of Economic Research (NBER) reference dates. Unlike Diebold and Rudebusch, however, Cover and Pecorino (henceforth "CP") evaluate all potential dates in terms of the associated probability of a structural break. CP find that March 1933 is the most probable break point when considering either NBER dates or the Romer (1994) alternative reference dates. This finding holds whether considering the length of expansions or the ratio of expansion length to the following recession length. CP (2005, p. 467) identify March 1933 with the U.S. departure from the gold standard. Is the CP claim of a March 1933 break point convincing? Of note, it is based on a particular and narrow definition of macroeconomic stability. Specifically, the claim is based on an examination of business cycles and the durations of their stages. However, a large part of macroeconomic analysis instead focuses on (i) growth cycles and/or (ii) the volatility of aggregate time series. The contribution of this article is to evaluate whether or not 1933 remains the most probable break point when macroeconomic stability is assessed from a perspective of (i) and/or (ii). The issues explored in this article are not only important from the perspective of economic history. CP's identification of March 1933 with departure from the gold standard leads them to claim (2005, p. 467) that their findings represent a challenge to real business cycle (RBC) theory, given that stabilization is then viewed as the product of discretionary monetary policy. (2) Furthermore, 1933 can be generally associated with the increased government activism of the Roosevelt administration. So, did discretionary policy of one type or another stabilize the U.S. economy? Or, alternatively, what if the break is actually latersay, 1946, as conventionally assumed? This would correspond to the Full Employment Act. However, it would also correspond to the end of World War II and the start of the diffusion of wartime technologies. Also, returning soldiers and laborers associated with the domestic war mobilization reentered peacetime production with new human capital. These changes in real factors could have resulted in a structural break. The article is organized as follows. Section 2 elaborates on the difference between business and growth cycles both in principle and in practice. Section 3 outlines methodology applied by CP to NBER reference dates. Section 4 reports the results of applying that methodology to both NBER reference dates (business cycles) and reference dates defined using HPfiltered real gross national product (GNP) (growth cycles). Section 5 then contrasts the existing literature on postWorld War II stabilization (focusing on cycle durations) with the literature on the "great moderation" of the 1980s (focusing on aggregate time series volatility). Section 6 outlines two empirical methodologies related to the later literature, and section 7 reports the results of applying them to the question of post World War II stabilization. Section 8 summarizes our conclusions, including (i) the claim of a break point around 1933 is robust to the consideration of growth rather than business cycles but (ii) examining the volatility of real GNP suggests a considerably later break point perhaps as late as the 1950s. [FIGURE 1 OMITTED] 2. Business Cycles versus Growth Cycles The difference between the two cycle concepts is semantically obscured because the term "business cycle" is often used interchangeably for both types. However, the "growth cycles" concept arose subsequently to the business cycle concept. Growth cycles were defined by the NBER in the 1960s as periods of increases and decreases in economic activity around some defined trend (Mintz 1969, 1974; Moore and Zarnowitz 1986). This can be contrasted to the NBER concept of business cycles as absolute increases and decreases in economic activity. (See Figure 1 for a graphical compare and contrast of the two cycle concepts.) Lucas (1983, 1987) popularized the growth cycle concept as a focus of macroeconomics and coopted the "business cycle" terminology for that purpose. (3) Real business cycle theorists, such as Kydland and Prescott (1991, 1996), further entrenched the growth cycle concept as part of accepted macroeconomic methodology. Thus far, the literature on post World War II stabilization summarized by Diebold and Rudebusch (1999) is based on the business cycle concept as stated by the NBER (e.g., "a recession is a significant decline in economic activity spread across the economy, lasting more than a few months, normally visible in real gross domestic product (GDP), real income, employment, industrial production, and wholesaleretail sales.") (4) NBER reference dates are primarily monthly and represent months where recessions turn expansionary or expansions turn recessionary ("troughs" and "peaks," respectively). Growth cycles, on the other hand, are consistent with a macroeconomics based on some variant of the neoclassical growth model (Solow 1956, Cass 1965, and Koopmans 1965) where some (constant or smoothly evolving) growth rate in laboraugmenting technical change defines a balanced growth path (or trend) for the economy. Shocks can cause the output of the economy to be temporarily above or below this trend. One interpretation of such deviations from trend is as periods when the economy is operating above or below its potential level of output. Of note, the economy can be above trend during a time when economic activity is falling (a business cycle recession); likewise, the economy can be below trend when economic activity is rising (a business cycle expansion). The terms expansions and recessions are straightforwardly descriptive of business cycle stages but they can be awkward in application to growth cycles. (Better analogous terms might be periods of general prosperity or depression.) However, expansions and recessions during growth cycles ostensibly consist of periods' of economic activity above and below the trend, respectively, and reference dates would correspond to the turning points. (5) Also of note, business cycle stages can be analyzed purely in terms of their durations. However, when applying a growth cycle concept to actual data, it is almost impossible to abstract entirely from the magnitude of fluctuations. This is because a trend must be defined and the definition of the trend is invariably influenced by the volatility of the relevant data. For example, the popular Hodrick and Prescott (HP) (1997) filter is a decomposition of a time series into a trend (or growth) component and a residual (or cyclical) component. This decomposition arises from the minimization of a function that penalizes according to the level of the cyclical component and the change in the trend growth rate. With quadratic penalties for both, all else equal, a large fluctuation in economic activity will result in a decomposition assigning a larger change in trend at that point in time. (6) The differences between a business cycle and growth cycle view of the macroeconomy can be significant in practice. For example, according to the NBER reference dates from 18751983, the longest recession was the Great Depression beginning in 1929 (III) and lasting 14 quarters. However, examining HPfiltered real GNP data covering the same time period, the U.S. economy was below trend for 19 quarters on two occasions, the first beginning in 1937 (IV) and the second beginning in 1945 (IV). These long periods below trend are the result of the relatively high GNP growth periods following the trough of the Great Depression and during World War II and their (positive) effect on the HPfilter defined trend. In other cases the choice of a business cycle or growth cycle perspective can simply affect how we perceive the timing of events. For the 18751983 time period the NBER reference dates mark 1961 (I) as the beginning of the longest expansion (35 quarters). Detrended GNP data suggests the same basic time period as the longest expansion, but it begins later (1965 [III]) and the duration is shorter (17 quarters). This discrepancy is due to the time needed for the economy, following the decrease in economic activity beginning in 1960 (I), to not only grow but to achieve a level of GNP above trend. (7) For our purposes, we do not wish to argue for or against either cycle concept. We note only that the growth cycle concept is at least as prevalent as the business cycle concept in macroeconomic analyses. Therefore, if CP's claim of a most probable break point around 1933 is robust to considering growth cycle as well as business cycle reference dates, this would strengthen that claim considerably. 3. The Diebold and Rudebuseh/Cover and Pecorino Empirical Methodology Following the example of Diebold and Rudebusch (1992) and Cover and Pecorino (2005), we employ a Wilcoxon ranksum test to evaluate the likelihood of various potential break points associated with both business and growth cycles. We use both the businesscycle chronology of the NBER and expansions and recessions defined from HPfiltered real GNP. For NBER reference dates, we examine the lengths of expansions (periods from troughs to peaks), the lengths of recessions (periods from peaks to troughs), and the ratio of the length of an expansion to the length of the following recession. Using HPfiltered real GNP the same measures are constructed, except that the discrete observations dictate defining expansions in terms of the number of periods above trend and recessions as the number of periods below trend. The null hypothesis is that the distributions of the durations of the U.S. economy before and after a given date are identical. The alternative hypothesis is that, from that given date onward, the average expansions are longer (or recessions shorter; or the ratios of expansions to recessions larger). Following precedent, our prior is that the U.S. economy either became more stable at some point in time or did not change, so a onetail hypothesis test is used. Denote the n observations, in temporal order, in a sample of durations as {[X.sub.1], [X.sub.2], ... , [X.sub.n]} and rank them in descending order. The ranks are then denoted as {[R.sub.1], [R.sub.2]. ... , [R.sub.n]} respectively. (8) For example, the 30th expansion in a given sample ([X.sub.30]) may be the longest in that sample ([R.sub.1]. This would be the case in our sample of expansions based on HPfiltered quarterly real GNP (Table 1; left side). The 30th expansion in the time series began in the third quarter of 1965 and lasted for 17 quarterlonger than any other expansion in the sample. Romer (1994) makes a similar observation concerning the NBER researcher's focus on detrended data pre1929. Recall, on a related note, that Cover and Pecorino (2005) show that the March 1933 break point is robust to Romer's (1994) alternative reference dates, which are based on that observation. The set {[R.sub.1], [R.sub.2], ... , [R.sub.n]} is divided into two samples: {[R.sub.1], [R.sub.2]. ... , [R.sub.m]} and {[R.sub.m+1,] [R.su.b.m+2], ... , [R.sub.n]}. The Wilcoxon test statistic is the sum of the ranks in the second sample: W = [n.summation over (i=m+1)] [R.sub.i]. (1) The Wilcoxon test may be interpreted as a distributionfree t test where we do not need to have normality in the sample distribution. (9) Intuitively, if the distributions are the same pre and post a potential break point, the average rank in the earlier sample should equal that of the later sample. We consider each observation in a given sample as a potential break point and compute the corresponding pvaluethe marginal significance of the test. The lowest pvalue indicates the most probable structural break point in the distribution of durations. In detrending real GNP, the HPfilter is based on defining a given time series, [y.sub.t], as the sum of a growth component [g.sub.t] and a cyclical component [c.sub.t]: [y.sub.t]= [g.sub.1]+[c.sub.t] for t=l, ... , T, (2) by solving the minimization problem, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3) where the parameter [lambda] is a positive number which penalizes variability in the growth component series. The larger the value of [lambda], the smoother is the growth component series defined by the filter. From possible detrending methods we choose the HPfilter both because of its prevalence in macroeconomic applications and because Canova (1999) has demonstrated that, among a range of possible filters, it closely reproduces the NBER business cycle dating. We view this as a virtue because we can ask whether, given that the general view of macroeconomic performance remains similar, the most probable break point hinges on the particular cycle concept considered. We use the quarterly real GNP series provided by Balke and Gordon (1986) for 18751983. This series is based on conventionally used annual series for real GNP. (10) The quarterly values are then interpolated using an algorithm and data described in Gordon and Veitch (1986). (11) Quarterly observations past 1983 could be readily obtained from the U.S. National Income and Product Accounts (NIPA). However, given the existing literature proposing and providing evidence for an important structural break around 1984, excluding the later time period seems reasonable; even more so since, in section 4 below, we demonstrate that doing likewise for the NBER reference dates does not alter the March 1933 result. We HPfilter the real GNP data using [lambda] = 1600. Again, we only justify this parameter value by noting that it is widely used in practice and by Canova (1999). (12) Using it creates the type of trend and deviation time series that are the stuff of most macroeconomic analyses. 4. Results: NBER Reference Dates versus Detrended GNP Tables 24 summarize the results for monthly NBER reference dates, 1879 (I) to 1980 (III). (13) We follow CP and report only pvalues for a selection of dates. (14) In general, reporting begins with the dates closest to 1921 and ends with the latest date in the 1960s; this practice is then applied throughout. (15) The reference date listed in column 1 represents the beginning of the second sample. For expansion duration (Table 2) the most probable break point is 1933 (I). For recession duration (Table 3) the most probable break point is either 1937 (II) or 1945 (1). (16) (1937 [II] represents the first post gold standard recession, so this is not inconsistent with a 1933 break point.) However, the 1945 (I) probability still makes the result ambiguous relative to results obtained when considering expansions. When we turn to the ratio of expansion length to the following recession (Table 4), once again 1933 (I) is the most probable break point. These results are exactly down to every quarterthose reported in Cover and Pecorino (2005) (see their tables 2, 5, and 6). This is true despite the fact that CP report results for the longer time period, 1854 (IV) to 2001 (I). This is important because it demonstrates that their results are robust to considering the shorter time period for which we analyze quarterly real GNP data below. We can state with confidence the following: Considering business cycle durations using NBER reference dates supports the conclusion that the first quarter of 1933 ushered in longer expansions absolutely and relative to adjacent recessions. Table 1 presents the lengths and ranks of expansions and recessions determined from HPfiltered real GNP assuming a value of [lambda] = 1600. Tables 57 then present results of the rank sum tests. Though the results are not as definitive as those using NBER reference dates, it is hard to argue that they are inconsistent with a 1933 break point. In the case of expansions (Table 5) 1935 (III) is the most probable break point. This is only the second expansion subsequent to the departure from the gold standard with the previous expansion (beginning in 1934 [II]) lasting only a single quarter. Furthermore, in the case of recessions (Table 6), 1931 (IV) is the most probable break point. This is the recession previous to, and encompassing, the U.S. departure from the gold standard. Finally, in the case of the expansion to following recession ratio, while 1950 (III) is the most probable break point, 1933 (I) is second with a pvalue only 0.0006 greater. (Given the range of pvalues reported, 0.0006 is an exceedingly small difference.) But, more importantly, for either date the marginal significance level is not less than 20%. The lack of statistical significance in the tests involving the expansion to recession ratio is indicative of symmetric changes in expansion and recession lengths. While the average pre1935 (III) expansion is indeed shorter than the average expansion in the later period (5.2 versus 9 quarters), the average recession length pre1931 (IV) was 5.2 quarters; the average recession length in the later period was longer (9.1 quarters). Even if the two 19quarter recessions beginning in 1937 (IV) and 1945 (IV) are excluded, the average length of recessions from 1931 (IV) onward (7.1 quarters) was still longer than that of the earlier period. (17) We still view the above as not inconsistent with Cover and Pecorino's (2005) hypothesis that a break point near 1933 implies a causal link between the abandonment of the gold standard and relative macroeconomic stability. In terms of growth cycles, if both expansions and recessions became longer, this may imply a smoothing of the evolution of real GNP. Unfortunately, tests involving cycle durations cannot directly speak to this possibility. In summary, though, we note that examining HPfiltered GNP and associated growth cycles seems to be at least weakly supportive of a 1933 break point. The cycle concept considered does not appear to be critical. (18) 5. Postwar Stabilization versus the "Great Moderation"; Durations versus Volatility One downfall of the ranksum test is that it speaks only to durationthe relative length of periods of decline and increase in economic activity, or of periods above and below trend. Even in the case of detrended GNP, where the HPfilter takes into account the volatility of the time series in defining the trend, the ranksum test then considers only relative durations (however they may be defined). Whether using a simple measure of economic activity, or deviations from a defined trend for such a measure, it seems desirable to evaluate a general measure of volatility. Research on postwar stability considering volatility does exist. However, it generally focuses on annual data and the comparison of simple measures of volatility such as sample standard deviations from earlier and later periods. Examples include Baily (1978), Balke and Gordon (1989), Romer (1986, 1989), and Watson (1994). However, more sophisticated tests have been developed and applied in the study of the "great moderation" of the 1980s. Kim and Nelson (1999), McConnell and PerezQuiros (2002), and Stock and Watson (2002), for example, have explored whether or not the U.S. economy became more stable during the 1980s and attempting to establish the most probable break point date associated with that stabilization; sometime during 1984 appears to be the consensus. To our knowledge no existing work applies the tools used to date the "great moderation" of the 1980s to date the earlier break point associated with postWorld War II stability. The available tests allow us to ask when a structural break in the volatility of real GNP occurred and, as well, whether the break is associated with the conditional variance (i.e., the shocks) or conditional mean (i.e., the persistence of the effects of shocks) of the data generating process. 6. The McConnell and PerezQuiros and Stock and Watson Methodologies We begin by following McConnell and PerezQuiros (2000), considering the following system: [DELTA] ln ([y.sub.t]) = [mu] + [phi] [DELTA] ln (y.sub.t1]) + [[epsilon].sub.t]; (4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5) where y is real GDP, [D.sub.1] and [D.sub.2] are dummy variables taking values, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6) and T is a potential break point. If there is no break ([[alpha].sub.1] = [[alpha].sub.2]), then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an unbiased estimator of the standard deviation of [[epsilon].sub.t]. In the case of a structural break at T, [[??].sub.1] and [[??].sub.2] are the estimators of the earlier and later subsample standard deviations, respectively. (19) Since the nuisance parameter, T, is only present under the alternative hypothesis, Lagrangian Multiplier, Likelihood Ratio, and Wald tests of [[alpha].sub.1] = [[alpha].sub.2] do not have standard asymptotic properties. Andrews (1993), therefore, considers the supremum of the F statistics associated with a range of potential break points, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7) He demonstrates the asymptotic properties of this statistic and provides the asymptotic critical values. This statistic allows for the determination of the most probable break point (i.e., the T that maximizes Equation 7). Andrews and Ploberger (1994) also propose the additional test statistics: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9) McConnell and PerezQuiros (2000) use the approximation suggested by Hansen (1997) to compute pvalues to these statistics, which speak to whether or not a break occurred within the range of [T.sub.1] to [T.sub.2]. These statistics are increasing in the average value of F statistics across considered [T.sub.s], but they also impose a penalty for considering a larger range of [T.sub.s]. Following McConnell and PerezQuiros (2000), we estimate the system, Equations 4 and 5, for the quarterly real GNP, 18751983, and compute the test statistics, Equations 7, 8, and 9. The estimation method is generalized method of moments (GMM) with a constant, lagged [DELTA]ln([y.sub.t1]), and [D.sub.1t] and [D.sub.2t] as instruments for period t. This will provide a baseline where the Pfilter plays no role in the results. We then estimate the analogous system using the percent deviation from HPtrend time series using [Dev.sub.t] = [phi] [Dev.sub.t1] + [[epsilon.sub.t], (10) where no constant is included because it is zero by definition of the filter. The test statistics are also computed for these cases. Following McConnell and PerezQuiros, we begin by using [T.sub.1] = 1891 (III) and [T.sub.2] = 1967 (II), where 15% of the quarterly observations are truncated at each end of the time series. We also provide two robustness checks on our resultsone regarding data and one regarding estimation technique. First, we apply the McConnell and PerezQuiros (2000) test to annual real GNP time series (and associated percent deviations from HPfilter trend) that includes, respectively, the corrected prewar observations of Romer (1989) and Balke and Gordon (1989). For each of these annual GNP series we filter separately, using [lambda] = 100 and = 6.25, producing four separate histories of growth cycles. The value of 100 is consistent with the suggestion of Backus and Kehoe (1992) for annual data while the value of 6.25 has been suggested as an alternative by Ravn and Uhlig (2002). (20) Second, we consider the possibility that the reduced postwar GNP volatility is based on a break in the conditional mean of the datagenerating process, rather than (or in addition to) the conditional variance. The conditional variance is determined by the size of the shocks hitting the economy; the conditional mean is based in part on the persistence of shock's effects. Stock and Watson (2002) propose a test based on the specification, [DELTA]ln([y.sub.t]) = [D.sub.1t] [[micro].sub.1] + [D.sub.2t] [[alpha].sub.2] + [D.sub.3t] [[beta].sub.1] [DELTA]1n ([y.sub.t]1) + [D.sub.4t] [[beta].sub.2] [DELTA]1n ([y.sub.t1]) + [[epsilon].sub.t], (11) where, [D.sub.1t];[D.sub.3t] = {1 if t [less than or equal to] T 0 if t > T [D.sub.2t];[D.sub.4t] = {0 if t [less than or equal to] T 1 if t > T, (12) and T is a potential break point in the conditional mean of the series. In practice, we run an ordinary least squares (OLS) regression for each T [member of] [[T.sub.1], [T.sub.2]] where [T.sub.1] = 1891 (III) and [T.sub.2] = 1967 (II) so that we focus on the middle 70% of the sample. For each regression we test [H.sub.0] : [[micro].sub.l] = [micro].sub.2] [[beta].sub.1] = [[beta].sub.2]. The T associated with the largest heteroskedasticityrobust Wald statistic is considered the most probable break point in the conditional mean. Using the T as chosen above, we then save the residuals from that particular regression and consider the additional OLS regression [[??].sub.t] (T) = [[alpha].sub.0] + [[alpha].sub.1] [D.sub.t]+ [v.sub.t], (13) where [D.sub.t] = {1 if t [less than or equal to] [tau] 0 if t > [tau]' (14) for each [tau] [member of] [1891 (III), 1967 (II)]. The x associated with the largest Wald statistic for [H.sub.0]: [[alpha].sub.l] = 0 is considered the most probable break point in the conditional variance. The Stock and Watson (2002) test is performed on both the quarterly real GNP growth rates and the HPfilter percent deviations series. 7. Results: Structural Breaks in the Volatility of Real GNP Table 8 reports the test statistic values and associated pvalues for a structural break between [T.sub.1] = 1891 (III) and [T.sub.2] = 1967 (II). For real GNP growth rates, as well as the deviations from HPfilter trend series, the estimated structural break point indicated by the supremum [F.sub.n] is around the year 1951 (with pvalues of 0.000 in each case). For the simple growth rates, supremum [F.sub.n] is associated with 1950 (III); for the deviation series it is associated with 1951 (III). The exp [F.sub.n] and ave [F.sub.n] statistics confirm that a structural break in the 1891 (III) to 1967 (II) range is very probable. Furthermore, there is no doubt that the estimated breaks are associated with decreases in volatility. The standard deviation of the GNP growth rates pre1950 (III) is 0.0286; almost three times that associated with 1950 (III) onward (0.0108). For the percent deviations from trend series the difference in standard deviations is also dramatic: 0.0512 for pre1951 (III) and then 0.0187 from that point onward. The year 1951 is closer to the 1946 Full Employment Act than to the 1933 departure from the gold standard. As well, it corresponds very closely to the Treasury Federal Reserve Accord of (March) 1951. (21) The Treasury Federal Reserve Accord is associated with a more independent and flexible monetary policy, but in a different sense than the 1933 departure from the gold standard. The 1951 Accord freed the Fed from an obligation to support the market of U.S. Treasuries at fixed prices. This removed a meaningful constraint on the Fed's conduct of monetary policy. What may also be meaningful are the [F.sub.n] values associated with the a priori potential break points. For 1933 (I) the associated [F.sub.n] is 2.15 (pvalue = 0.141) for the simple growth rate series and 4.97 (pvalue = 0.026) for the deviationfromtrend series. For 1946 (I) the corresponding [F.sub.n] values are 65.02 (pvalue = 0.000) and 71.80 (pvalue = 0.000). If we simply asked whether a break point in volatility occurred in 1933, we would reject the null with a good level of confidence; likewise if we asked about 1946. However, when considering all potential dates, 1946 is more likely, and 1951 is most probable. If we consider the two annual data real GNP series incorporating, respectively, the pre1929 corrections of Romer (1989) and Balke and Gordon (1989), the implication is that the break point is even later in the 1950s. Moreover, the similarities between the predictions of the two corrected GNP series are striking (Table 9). In either case, analyzing the growth rates of deviations from a HPfilter trend using [lambda] = 6.25, the most probable break point is in 1959. For deviations using [lambda] = 100, in either case, the most probable break point is slightly earlier: 1955. (All of the associated test statistics are significant at better than the I% level.) Focusing again on the quarterly real GNP series, Table 10 reports results from the Stock and Watson (2002) test. This test allows potentially for breaks in both the conditional variance and conditional mean of the process. For GNP growth rates the most probable break point is in 1940 (II) and is significant at better than the 5% level. For deviations from trend the most probable break point is in 1903 (IV), but is not very probable at all: pvalue = 0.313. Based on these conditional mean break points, the residuals from the resulting regressions for both series suggest a most probable break point in the conditional variance sometime in 1947. These dates are earlier than those implied by the McConnell and PerezQuiros (2000) test, but are still later than the 1933 date suggested by examining durations of reference cycles. A primary conclusion of this section is that if there was a structural break in the volatility of real GNP it was considerably later than the 1933 departure from the gold standard, and that this conclusion is robust to data corrections, frequency changes, and detrending using the HPfilter. 8. Conclusions Cover and Pecorino's (2005) results are remarkable, and their robustness to the substitution of Romer (1994) reference dates, our altering of the time period considered, and the substitution of HPfilter defined growth cycle dates in real GNP is striking. The most probable structural break appears to be around 1933, coinciding with the U.S. departure from the gold standard. The particular cycle concept considered appears not to be critical. On the other hand, when considering the volatility of quarterly real GNP growth, or its deviations from an HPfilter trend, the most probable structural break is in 1947 or 1951. This may or may not weaken the view that improved monetary policy stabilized the U.S. economy. While 1947 is not straightforwardly linked to a structural improvement in monetary policy, the Treasury Federal Reserve Accord of 1951 is associated with monetary policy less constrained by the debtfinance priorities of the Treasury. However, if we utilize annual GNP data in the analysis and incorporate the wellknown pre1929 corrections of Romer (1989) or Balke and Gordon (1989), the most probable break point in volatility is even later in the 1950s. If one were to draw a primary conclusion from this article, then it would seem to be that the choice of growth cycles or business cycles does not yield dramatically different answers as to when the U.S. economy stabilized. Rather, it is the focus on durations or volatility that can make an important difference. We should take some space to comment on what types of conclusions cannot be drawn from this article. First, our results cannot speak, in and of themselves, as to whether focusing on reference cycle durations or the volatility of aggregate time series is a more meaningful way of evaluating macroeconomic performance. Our contribution to this debate is to note an important question for which the two focuses provide significantly different answers. Also, a weakness of the present study involves the ability to deal with the effects of the wartime economy and the Great Depression. Cover and Pecorino (2005) report that, when focusing only on peacetime recessions, the break in durations occurred in either 1937 or 1948 with nearly equal probability. This is an unenlightening result if one's prior is that the economy changed structurally in either 1933 or 1946. (May 1937 corresponds to the first recession to follow the departure from the gold standard; November 1948 corresponds to the first recession to follow the Full Employment Act.) Moving to an examination of real GNP volatility, removing the WWII observations for the United States excludes (at least) 1942 through 1945, taking out a large portion of the time between the two potential break points, and a portion that borders the later potential break point. Similarly, taking out Great Depression observations actually excludes 1933 entirely. How to best deal with these problems remains an important, open question. Received June 2008; accepted January 2009. References Andrews, Donald W. K. 1993. Test for parameter instability and structural change with unknown change point. Econometrica 61:8256. Andrews, Donald W. K., and Werner Ploberger. 1994. Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62:13831414. Backus, David K., and Patrick K. Kehoe. 1992. International evidence on the historical properties of business cycles. American Economic" Review 82:86488. Baily, Martin N. 1978. Stabilization policy and private economic behavior. Brookings Papers on Economic Activity 27:1150. Balke, Nathan S., and Robert J. Gordon. 1986. Appendix B historical data. In The American business cycle: Continuity and change, edited by Robert J. Gordon. Chicago: University of Chicago Press, pp. 78850. Balke, Nathan S., and Robert J. Gordon. 1989. The estimation of prewar gross national product: Methodology and new evidence. Journal of Political Economy 97:3892. Barro, Robert J. 2007. Macroeeonomics: A modern approach. Belmont: SouthWestern College Publishers. Canova, Fabio. 1999. Does detrending matter for the determination of the reference cycle and the selection of turning points? Economic Journal 109:12650. Cass, David. 1965. Optimum growth in an aggregative model of capital accumulation. Review of Economic Studies 32:23340. Chatterjee, Satyajit. 1999. Real business cycles: A legacy of countercyclical policies? Federal Reserve Bank of Philadelphia Business Review January/February: 1727. Cover, James P., and Paul Pecorino. 2005. The length of U.S. business expansions: When did the break in the data occur? Journal of Macroeeonomics 27:45271. Diebold, Francis X., and Glenn D. Rudebusch. 1992. Have postwar economic fluctuations been stabilized? American Economic Review 92:9931005. Diebold, Francis X., and Glenn D. Rudebusch. 1999. Business cycles: Durations, dynamics and forecasting. Princeton: Princeton University Press. Gordon, Robert J., and John M. Veitch. 1986. Fixed investment in the American business cycle. In The American Business Cycle: Continuity and Change, edited by Robert J. Gordon. Chicago: University of Chicago Press, pp. 267358. Hansen, Bruce E. 1997. Approximate asymptotic pvalues for structural change tests. Journal of Business and Economic Statistics 15:607. Hodrick, Robert J., and Edward C. Prescott. 1997. Postwar U.S. business cycles: An empirical investigation. Journal of Money, Credit and Banking 29:116. Kim, ChangJin, and Charles R. Nelson. 1999. Has the U.S. economy become more stable? A Bayesian approach based on a Markovswitching model of the business cycle. Review of Economics and Statistics 81:60816. Koopmans, Tjalling C. 1965. On the concept of optimal economic growth. In The econometric approach to development planning. Amsterdam: North Holland, pp. 22587. Kydland, Finn, and Edward C. Prescott. 1991. The econometrics of the general equilibrium approach. Scandinavian Journal of Economics 1:6381. Kydland, Finn, and Edward C. Prescott. 1996. The computational experiment: An econometric tool. Journal of Economic Perspectives 10:6985. Lucas, Robert E. 1983. Studies in businesscycle theory. Cambridge: MIT Press. Lucas, Robert E. 1987. Models of businesscycles. Oxford: Blackwell. McConnell, Margaret M., and Gabriel PerezQuiros. 2000. Output fluctuations in the United States: What has changed since the early 1980s? American Economic Review 90:146476. Mintz, Ilse. 1969. Dating postwar business cycles: Methods and their application to Western Germany, 19501967. NBER Occasional Paper No. 107. Mintz, Ilse. 1974. Dating United States growth cycles. Explorations in Economic Research 1:113. Moore, Geoffrey H., and Victor Zarnowitz. 1986. The development and role of the national bureau's business cycle chronologies. In The American business cycle: Continuity and change, edited by Robert J. Gordon. Chicago: University of Chicago Press, pp. 73579. Prescott, Edward C. 1986. Theory ahead of business cycle measurement. Federal Reserve of Minneapolis Quarterly Review 10:922. Ravn, Morten O., and Harald Uhlig. 2002. On adjusting the HodrickPrescott filter for the frequency of observations. Review of Economics and Statistics 84:37180. Romer, Christina D. 1986. New estimates of prewar gross national product and unemployment. Journal of Economic History 46:34152. Romer, Christina D. 1989. The prewar business cycle reconsidered: New estimates of gross national product, 18691908. Journal of Political Economy 97:137. Romer, Christina D. 1994. Remeasuring business cycles. Journal of Economic History 54:573609. Solow, Robert M. 1956. A contribution to the theory of economic growth. Quarterly Journal of Economics 70:6594. Stock, James H., and Mark M. Watson. 2002. Has the business cycle changed and why? NBER Working Paper No. 9127. Watson, Mark W. 1994. Business cycle durations and postwar stabilization of the U.S. economy. American Economic Review 84:2446. Young, Andrew T., and Alex K. Blue. 2007. Retail prices during a change in monetary regimes: Evidence from Sears, Roebuck catalogs, 19381951. Managerial and Decision Economics 28:76376. (1) There may also be other reasonable rationales for a similar date. Young and Blue (2007) suggest that the Bretton Woods arrangement (ratified by the United States in 1945 just previously to the Employment Act) may have, by promoting a longrun, upward trend in prices, led to more flexible prices and, therefore, more macroeconomic stability. (2) CPs claim, even conditional on the accuracy of their findings, is not necessarily reasonable. For example, Chatterjee argues that ability of RBC models to match key moments of postwar aggregates may actually be "the legacy of countercyclical policies": "One possibility is that the success of [RBC] theory reflects better postWWII countercyclical policies [that] reduced some of the instabilities that characterized preWWII business cycles" (1999, p. 26). According to this view, countercyclical policies have succeeded in making the U.S. economy an approximate one with competitive markets operating efficiently "and fluctuations in the growth rate of businesssector productivity ... surfaced as the dominant sources of business cycles" (1999, p. 26). (3) Lucas (1983, p. 217) states: "Let me ... review the main qualitative features of economic time series we call 'the business cycle." Technically, movements about trend in gross national product in any country can be well described by a stochastically disturbed difference equation of very low order." (4) http://www.nber.org/cycles.html/ (5) A wellknown example of growthcyclebased macroeconomic analysis is Prescott's (1986) RBC model exercise. Prescott's model abstracts entirely from trend technological change and then compares simulated data from that model to detrended U.S. data. Whether or not economic activity is (in an absolute sense) going up or down is of secondorder importance in practice. (6) Even in the case of fitting a simple linear trend, periods of absolute increase and decrease in economic activity are symmetric in terms of their severity, and volatility will influence the trend. (7) Table 4 reports the HPfilter trend deviationdefined expansions and recessions for the Balke and Gordon (1986) data. (8) When observations tie in terms of rank, we follow Diebold and Rudebusch (1992) and use the average rank. (9) See Diebold and Rudebusch (1992, pp. 4445). (10) These sources are noted in Balke and Gordon (1986. p. 788). (11) The quarterly interpolators are a constant, a linear time trend, and an index of industrial production and trade. (12) Canova (1999) also considers [lambda] = 4. (13) These represent the reference dates that most closely correspond to the beginning and end dates of our real GNP time series. (14) At the very initial or final NBER reference dates we would expect pvalues to be very high given the lack of observations in one sample or the other. (15) In no case does this reporting leave out a pvalue lower than those contained in the tables. (16) The pvalue associated with 1937 (II) is lower but by 0.0002. Relative to the other reported pvalues, this difference is smaller by an order of magnitude, so noting the (approximate) equivalence in probability to the 1945 (I) date seems appropriate. (17) One ad hoc way to deal with the symmetric effects of the HPfilter on expansion and recession lengths is to define a morethanjustbelow trend definition of recessions. For example. Barro (2007, p. 176) defines recessions in terms of HPfiltered GDP as, alternatively, periods either more than 1.5% or more than 3% below trend. One advantage to such a definition is that it may more realistically capture what most people think of as recessions (i.e., significantly bad economic times, rather than mild slowdowns in growth). However, such definitions are arbitrary and take us away from the discipline employed in most macroeconomic analysis. (It should be noted that Barro's use of these definitions is in the context of a textbook.) We experimented with defining expansions and recessions from the filtered GNP using both the 1.5% and 3% rules. Doing so yielded statistically significant results for expansion lengths but not for recession lengths. Using the 1.5% rule, the most probable break point for expansions is in 1942 (Ill); using the 3% rule the most probable date is 1950 (Ill). These are provocative results in that they are considerably later than 1933, we but hesitate to stress them here because of the arbitrariness of the rules. (18) An anonymous referee pointed out that the growth cycle stages may be alternatively conceptualized in terms of periods of growth above and below the trend level of growth (rather than in terms of periods of GNP levels above and below trend level). This alternative conceptualization is, taken literally, difficult to bring to GNP data in a sensible way. This is because quarterly fluctuations are marked, several "expansions" lasting merely a quarter or two occur during sustained periods below trend. Such is similar for "recessions" occurring during periods above trend. However, the referee's conceptualization is consistent with, and made operational according to, Moore and Zarnowitz's (1986, p. 772) statement: "dates mark the approximate time when aggregate economic activity was farthest from its longrun trend level (peak) or farthest below its longrun trend level (trough)." We created alternative growth cycle dates. dating recessions as the largest (absolute value) percentage deviation from trend during a period of GNP below trend: and expansions as the largest percentage deviation from trend during an above trend period. The results are inconsistent across expansion and recession samples, and inconsistent with the previous literature's findings. The most likely expansion break point is 1904 (IV): the most likely recession break point is 1937 (I). (19) McConnell and PerezQuiros (2000) also consider possible structural breaks in the mean of the series, [[mu].sub.1] [not equal to] [[mu].sub.2]. They find no such statistically significant break with quarterly GDP data from 19531999. (20) These values for [lambda] are, again, justified only by convention. However, as we shall see, the estimated most probable break points are identical regardless of which [lambda] value is used (and, for that matter, whether the Balke and Gordon 1989 or Romer 1989 series is used). (21) This is an event that, to our knowledge, has not been previously suggested as important in this literature. We are grateful to an anonymous referee for pointing out the coincidence of the event and estimated break point. Andrew T. Young * and Shaoyin Du [dagger] * Department of Economics, 371 Holman Hall, University of Mississippi, University, MS 38677, USA: Tel.6629155829: Email atyoung@olemiss.edu; corresponding author. [dagger] Department of Economics, University of Mississippi, University, MS 38677, USA: Email sdul@olemiss.edu. We are indebted to two anonymous referees for their constructive comments on a previous draft. We thank James Cover and other participants at the University of Alabama economics seminar series for helpful discussion; we likewise thank Ron Balvers and other participants at the West Virginia University economics seminar series and participants at the San Jose State economics seminar series. We are especially appreciative toward Paul Pecorino for commenting extensively on a previous version of this article. Table 1. Lengths and Ranks of Expansions and Recessions Using HPFiltered Quarterly Real GNP Beginning Expansions Year and (Quarters Quarter of above Expansion Trend) Rank 1875 (I) 3 25 1876 (I) 2 29.5 1879 (IV) 13 3.5 1884 (II) 1 32.5 1886 (II) 7 15 1890 (II) 3 25 1891 (III) 8 13 1895 (I) 4 20 1897 (III) 3 25 1899 (I) 4 20 1901 (I) 5 18 1902 (III) 1 32.5 1903 (I) 3 25 1905 (IV) 8 13 1909 (III) 4 20 1911 (II) 12 5.5 1916 (I) 1 32.5 1917 (IV) 10 8 1922 (IV) 6 16.5 1925 (IIn 8 13 1928 (III) 9 10 1931 (I) 3 25 1934 (II) 1 32.5 1935 (III) 9 10 1942 (III) 13 3.5 1950 (III) 14 2 1955 (I) 11 7 1959 (I) 6 16.5 1962 (I) 3 25 1965 (III) 17 1 1972 (II) 9 10 1977 (II) 12 5.5 1981 (I) 3 25 1983 (III) 2 29.5 Beginning Recessions Year and (Quarters Quarter of below Recession Trend) Rank 1875 (IV) 1 2.5 1876 (III) 13 31 1883 (I) 5 14 1884 (III) 7 21.5 1888 (I) 9 25 1891 (I) 2 5 1893 (III) 6 17.5 1896 (I) 6 17.5 1898 (II) 3 7 1900 (I) 4 10.5 1902 (II) 1 2.5 1902 (IV) 1 2.5 1903 (IV) 8 24 1907 (IV) 7 21.5 1910 (III) 3 7 1914 (II) 7 21.5 1916 (II) 6 17.5 1920 (II) 10 27 1924 (II) 5 14 1927 (III) 4 10.5 1930 (IV) 1 2.5 1931 (IV) 10 27 1934 (III) 4 10.5 1937 (IV) 19 32.5 1945 (IV) 19 32.5 1954 (I) 4 10.5 1957 (IV) 5 14 1960 (III) 6 17.5 1962 (IV) 11 29.5 1969 (IV) 10 27 1974 (III) 11 29.5 1980 (II) 3 7 1981 (IV) 7 21.5 Expansions are ranked longest to shortest, while recessions are ranked shortest to longest; ties are denoted with the average rank. The HPfilter is applied with a smoothing parameter value, [lambda] = 1600. Table 2. Test for Break in the NBER Sample Using Expansions: 1879 (I)1980 (III) Column 1 2 3 Trough Beginning Expansion Wilcoxon Number of in the Second Sample Statistic Expansions in Row (and Length in Quarters) First Sample 1 1921 (III) (7) 239 12 2 1924 (III) (9) 229.5 13 3 1927 (IV) (7) 216 14 4 1933 (I) (17) 208.5 15 5 1938 (II) (27) 186.5 16 6 1945 (IV)(12) 162.5 17 7 1949 (IV) (15) 144.5 18 8 1954 (II) (13) 123.5 19 9 1958 (II) (8) 104.5 20 10 1961 (I) (35) 93 21 4 5 Trough Beginning Expansion Number of Exact Marginal in the Second Sample Expansions in Significance Row (and Length in Quarters) Second Sample Level 1 1921 (III) (7) 14 0.0043 2 1924 (III) (9) 13 0.0022 3 1927 (IV) (7) 12 0.0021 4 1933 (I) (17) 11 0.0005 5 1938 (II) (27) 10 0.0026 6 1945 (IV)(12) 9 0.0129 7 1949 (IV) (15) 8 0.0211 8 1954 (II) (13) 7 0.0486 9 1958 (II) (8) 6 0.0806 10 1961 (I) (35) 5 0.0511 NBER are primarily monthly and so are durations. Lengths in terms of quarters are approximated by rounding. Table 3. Test for Break in the NBER Sample Using Recessions: 1873 (III)1981 (III) Column 1 2 3 Peak Beginning Recession in Wilcoxon Number of the Second Sample Statistic Recessions in Row (and Length in Quarters) First Sample 1 1923 (II) (5) 130 13 2 1926 (III) (5) 116 14 3 1929 (III) (14) 104.5 15 4 1937 (II) (5) 79.5 16 5 1945 (I) (3) 68 17 6 1948 (IV) (4) 64.5 18 7 1953 (II) (3) 56 19 8 1957 (III) (8) 50 20 9 1960 (II) (3) 46.5 21 10 1969 (IV) (4) 40.5 22 1 4 5 Peak Beginning Recession in Number of Exact Marginal the Second Sample Recessions in Significance Row (and Length in Quarters) Second Sample Level 1 1923 (II) (5) 13 0.0090 2 1926 (III) (5) 12 0.0081 3 1929 (III) (14) 11 0.0103 4 1937 (II) (5) 10 0.0011 5 1945 (I) (3) 9 0.0013 6 1948 (IV) (4) 8 0.0068 7 1953 (II) (3) 7 0.0119 8 1957 (III) (8) 6 0.0296 9 1960 (II) (3) 5 0.0914 10 1969 (IV) (4) 4 0.1813 NBER are primarily monthly and so are durations. Lengths in terms of quarters are approximated by rounding. Table 4. Test for Break in the Sample in the NBER Sample Using Expansion to Recession Following Ratio: 1879 (I)1981 (III) Column 1 2 3 Trough Beginning Wilcoxon Number of Expansion Statistic Cycles in Row in the Second Sample First Sample 1 1921 (III) 208 12 2 1924 (III) 197 13 3 1927 (IV) 184 14 4 1933 (I) 183 15 5 1938 (II) 164 16 6 1945 (IV) 139 17 7 1949 (IV) 121 18 8 1954 (II) 101 19 9 1958 (II) 80 20 10 1961 (I) 65 21 1 4 5 Trough Beginning Number of Cycles Exact Marginal Expansion in Second Sample Significance Row in the Second Sample Level 1 1921 (III) 13 0.0173 2 1924 (III) 12 0.0128 3 1927 (IV) 11 0.0123 4 1933 (I) 10 0.0012 5 1938 (II) 9 0.0033 6 1945 (IV) 8 0.0213 7 1949 (IV) 7 0.0369 8 1954 (II) 6 0.0781 9 1958 (II) 5 0.1678 10 1961 (I) 4 0.1843 Table 5. Test for Break in the Real GNP Sample Using Expansions: 1875 (I)1983 (III) Column 1 2 3 Trough Beginning First Wilcoxon Number of Expansion in the Second Statistic Expansions in Row Sample (and Length in Months) First Sample 1 1922 (IV) (18) 328.0 18 2 1925 (III) (24) 309.5 19 3 1928 (III) (27) 287.5 20 4 1931 (I) (9) 262.5 21 5 1934 (II) (3) 252.5 22 6 1935 (III) (27) 250.0 23 7 1942 (III) (39) 225.0 24 8 1950 (III) (42) 193.5 25 9 1955 (I) (33) 160.5 26 10 1959 (I) (18) 132.5 27 11 1962 (I) (9) 114.0 28 12 1965 (III) (51) 104.0 29 1 4 5 Exact Trough Beginning First Number of Marginal Expansion in the Second Expansions in Significance Row Sample (and Length in Months) Second Sample Level 1 1922 (IV) (18) 16 0.0489 2 1925 (III) (24) I S 0.0517 3 1928 (III) (27) 14 0.0692 4 1931 (I) (9) 13 0.1093 5 1934 (II) (3) 12 0.0635 6 1935 (III) (27) 11 0.0163 7 1942 (III) (39) 10 0.0289 8 1950 (III) (42) 9 0.0816 9 1955 (I) (33) 8 0.2077 10 1959 (I) (18) 7 0.3407 11 1962 (I) (9) 6 0.3485 12 1965 (III) (51) 5 0.2182 The HPfilter is applied with a smoothing parameter value, a = 1600. Table 6. Test for Break in the Real GNP Sample Using Recessions: 1875 (IV)1981 (IV) Column 1 2 3 Trough of First Recession Wilcoxon Number of in the Second Sample Statistic Expansions in Row (and Length in Months) First Sample 1 1924 (II) (15) 286.0 18 2 1927 (III) (12) 272.0 19 3 1930 (IV) (3) 261.5 20 4 1931 (IV) (30) 259.0 21 5 1934 (III) (12) 232.0 22 6 1937 (IV) (57) 221.5 23 7 1945 (IV) (57) 189.0 24 8 1954 (I) (12) 156.5 25 9 1957 (IV) (15) 146.0 26 10 1960 (III) (18) 132.0 27 11 1962 (IV) (33) 114.5 28 12 1969 (IV) (30) 85.0 29 1 4 5 Trough of First Recession Number of Exact Marginal in the Second Sample Expansions in Significance Row (and Length in Months) Second sample Level 1 1924 (II) (15) IS 0.1340 2 1927 (III) (12) 14 0.1100 3 1930 (IV) (3) 13 0.0689 4 1931 (IV) (30) 12 0.0191 5 1934 (III) (12) 11 0.0430 6 1937 (IV) (57) 10 0.0212 7 1945 (IV) (57) 9 0.0744 8 1954 (I) (12) 8 0.2000 9 1957 (IV) (15) 7 0.1210 10 1960 (III) (18) 6 0.0835 11 1962 (IV) (33) 5 0.0724 12 1969 (IV) (30) 4 0.1827 The HPfilter is applied with a smoothing parameter value, 7v = 1600. Table 7. Test for Break in the Real GNP Sample Using the Expansion to Following Recession Ratio: 1875 (I)1983 (III) Column 1 2 3 Trough of First Expansion Wilcoxon Number of in the Second Sample Statistic Expansions in Row (and Length in Months) First Sample 1 1922 (IV) (18) 263.0 18 2 1925 (III) (24) 243.0 19 3 1928 (III) (27) 217.0 20 4 1931 (I) (9) 184.0 21 5 1934 (II) (3) 178.0 22 6 1935 (III) (27) 174.0 23 7 1942 (III) (39) 165.0 24 8 1950 (III) (42) 154.0 25 9 1955 (I) (33) 124.0 26 10 1959 (I) (18) 97.0 27 11 1962 (I) (9) 81.0 28 12 1965 (III) (51) 76.0 29 1 4 5 Trough of First Expansion Number of Exact Marginal in the Second Sample Expansions in Significance Row (and Length in Months) Second Sample Level 1 1922 (IV) (18) 15 0.3908 2 1925 (III) (24) 14 0.4321 3 1928 (III) (27) 13 0.4458 4 1931 (I) (9) 12 0.2326 5 1934 (II) (3) 11 0.3712 6 1935 (III) (27) 10 0.4429 7 1942 (III) (39) 9 0.3205 8 1950 (III) (42) 8 0.2320 9 1955 (I) (33) 7 0.4200 10 1959 (I) (18) 6 0.4161 11 1962 (I) (9) 5 0.4297 12 1965 (III) (51) 4 0.3425 The HPfilter is applied using a smoothing parameter value, [lambda] = 1600. Table 8. McConnell & PerezQuiros (2002) Test for a Break in GNP Growth Volatility, 1875(I)1983 (III); [T.sub.1] = 1891 (III) and [T.sub.2] = 1967 (II) Column 1 2 3 [DELTA]ln [Dev.sub.t] Row Statistic ([y.sub.t]) ([lambda]) = 1600) 1 sup [F.sub.n] 79.12 (0.000) 79.81 (0.000) Most probable 1950 (III) 1951 III) break point 2 exp [F.sub.n] 35.45 (0.000) 36.82 (0.000) 3 ave [F.sub.n] 24.60 (0.000) 25.61 (0.005) 4 [F.sub.1933] (1) 2.15 (0.141) 4.97 (0.026) 5 [F.sub.1946 (1) 65.02 (0.000) 71.80 (0.000) The pvalues are reported beside the test statistics in parentheses. The HPfilter is applied using a smoothing parameter value of 7L = 1600. Table 9. McConnell and PerezQuiros (2002) Test for a Break in GNP Growth Volatility Using Romer (1989) and Balke and Gordon (1989) Corrections: 18751983; [T.sub.1] = 1887 and [T.sub.2] = 1965 Column 1 2 3 [DELTA]ln [Dev.sub.t] Row Statistic ([y.sub.t]) ([lambda] = 100 1 Romer (1989) sup [F.sub.n] 19.80 (0.000) 12.84 (0.000) Most probable 1959 1955 break point 2 Balke and Gordon (1989) sup [F.sub.n] 28.42 (0.000) 27.67 (0.000) Most probable 1959 1955 break point 1 4 [Dev.sub.t] Row Statistic ([lambda] = 6.25) 1 Romer (1989) sup [F.sub.n] 14.78 (0.000) Most probable 1959 break point 2 Balke and Gordon (1989) sup [F.sub.n] 30.97 (0.000) Most probable 1959 break point Table 10. Stock and Watson (2002) Test for a Break in GNP Growth Volatility, 1875 (n1983(III); [T.sub.1] = 1891 (III) and [T.sub.2] = 1967 (II) Column 1 2 3 [DELTA]ln [Dev.sub.t] Row Statistic ([y.sub.t]) ([lambda] = 1600) 1 Conditional mean Quandt likelihood 7.63 (0.022) 2.33 (0.313) ratio Most probable break 1940 (II) 1903 (IV) point 2 Conditional variance Wald statistic 48.37 (0.000) 66.94 (0.000) Most probable break 1947 (I) 1947 (II) point The pvalues are reported beside the test statistics in parentheses. The HPfilter is applied using a smoothing parameter value of [lambda] = 1600. 

Reader Opinion