On the Gibson Paradox - SSRN papers

Review of International Economics, 7(1), 117–125, 1999

On the Gibson Paradox Apostolos Serletis and George Zestos* Abstract This paper uses recent developments in the theory of nonstationary regressors to investigate empirical relationships previously taken to support the Gibson paradox, using quarterly data over the 1957:1–1994:4 period on nominal interest rates and prices for eight European Union countries—Belgium, Denmark, England, France, Germany, Ireland, Italy, and The Netherlands. Using the methodology suggested by Kydland and Prescott, it is shown that the (relevant) cyclical nominal interest rate—price level contemporaneous correlations are weak, thereby punching a hole in the Gibson paradox. Evidence is also presented, based on the integration properties of the data, that standard Gibson paradox regressions are spurious.

1. Introduction The strong positive correlation between the price level, as measured by a (log) price index, and the long-term nominal interest rate, as measured by the yield to maturity of long-term bonds, is generally known as the Gibson paradox (after the English banker A. H. Gibson who empirically established it in the 1920s). It is termed a paradox because microeconomic theory predicts that since interest is a cost of capital (which is taken as a fixed cost), output and prices will be unaffected by changes in interest rates. The Gibson paradox is commonly regarded as casting doubt on classical macroeconomic theory, because it seems to contradict classical neutrality of money propositions, such as, for example, the long-run neutrality and superneutrality of money propositions and the Fisherian link between inflation and short-term nominal interest rates. Although many theories have arisen to explain this paradox (e.g., Keynes, 1925; Fisher, 1930; Wicksell, 1962; Sargent, 1973; Shiller and Siegel, 1977; Barsky and Summers, 1988), recently Corbae and Ouliaris (1989), using annual UK and US data over the 1920–87 period, present evidence which suggests that standard Gibson paradox regressions (involving the rate of interest and the price level) are spurious and that the strong positive correlation between nominal interest rates and the price level is nothing but a statistical anomaly. On the other hand, Klein (1995), using US data over the past four decades, presents evidence that is consistent with the Gibson relationship. In this paper, in the spirit of Corbae and Ouliaris (1989) and Klein (1995), we reexamine the relationship between the long-term nominal interest rate and the price level using quarterly data for eight European Union countries. In doing so, we calculate cyclical nominal interest rate–price level correlations using the methodology suggested by Kydland and Prescott (1990) and find that they are strikingly weak. We also use recent advances in applied econometrics and show that the rate of interest and the price level are integrated of different orders, meaning that an ordinary least-squares

* Serletis: University of Calgary, Calgary, Alberta T2N 1N4, Canada. Tel: (403)220-4092; Fax: (403)282-5262; E-mail: [email protected], http://www.ucalgary.ca/,serletis/hp.html. Zestos: Christopher Newport University, Newport News, VA 23606-2998, USA. Tel: (804)594-7067; Fax: (804)594-7808; E-mail: [email protected]. The authors thank an anonymous referee for useful comments, and Bruce Zarobell and Grigorios Zimonopoulos for excellent research assistance. © Blackwell Publishers Ltd 1999, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.

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Apostolos Serletis and George Zestos

regression involving the interest rate and the price level can be interpreted as a spurious regression. The paper is organized as follows. Section 2 briefly discusses the Hodrick and Prescott (HP) filtering procedure for decomposing time series into long-run and business cycle components and presents HP cyclical correlations of the price level with the nominal interest rate. In section 3, we perform unit root tests in the autoregressive representation of each individual time series to determine whether Gibson paradox regressions can be interpreted in a meaningful way. In the last section, we summarize the main results and conclude.

2. Cyclical Correlations Between R and Log(P) We use quarterly data (from the IMF International Financial Statistics) over the 1957:1 to 1994:4 period on the long-term nominal interest rate (as measured by the yield to maturity of long-term bonds) and the price level (as measured by the consumer price index) for eight European Union countries. The countries are Belgium, Denmark, England, France, Germany, Ireland, Italy, and The Netherlands. Figure 1 shows the time paths of the long-term nominal interest rate and the logged price level for each of the eight countries. For a description of the cyclical nominal interest rate–price level correlations, we follow the current practice of using the Hodrick–Prescott (HP) filter to represent growth and business cycle components—see Prescott (1986). For a time series Xt, for t = 1, 2, … , T, this procedure defines the trend or growth component, denoted t t, for t = 1, 2, … , T, as the solution to the following optimization problem: T

2

T -1

min Â ( X t - t t ) + l Â [(t t +1 - t t ) - (t t - t t -1 )] tt

t =1

2

(1)

t=2

so that Xt - t t is the HP filtered series. The larger is l, the smoother the trend path, and as l Æ •, the linear trend results. In our computations, we set l = 1600, as it has been suggested for quarterly data—see, for example, Cooley and Prescott (1995). We measure the degree of comovement of the series by the magnitude of the correlation coefficient r( j), j Œ {0, ±1, ±2, …}. The contemporaneous correlation coefficient, r(0), gives information on the degree of contemporaneous comovement between the series. Also, r(j), j Œ {±1, ±2, …} measures the cross-correlation over time, indicating whether the variables lead or lag one another. In Table 1 we report contemporaneous correlations as well as cross-correlations (at lags and leads of one through six quarters) of the price level with the nominal interest rate. A number near one in the x column indicates strong contemporaneous movements whereas a number near zero indicates weak contemporaneous movements. The numbers in the remaining columns indicate whether there is any evidence of a phase shift relative to the nominal interest rate. The results indicate that except for Italy the cyclical nominal interest rate–price level contemporaneous correlations are strikingly weak, and that the price level always lags the nominal interest rate. This clearly suggests that the well-documented high correlation between the long-term nominal interest rate and the price level does not carry over to their cyclical components and it is nothing but a statistical anomaly. As Corbae and Ouliaris (1989, p. 295) put it, “it should not be used as grounds for acceptance or rejection of the Gibson paradox.” © Blackwell Publishers Ltd 1999

Table 1. Hodrick–Prescott Cyclical Correlations of the Price Level with the Nominal Interest Rate Correlation coefficients x(t - 6)

x(t - 5)

x(t - 4)

x(t - 3)

x(t - 2)

x(t - 1)

x

x(t + 1)

x(t + 2)

x(t + 3)

x(t + 4)

x(t + 5)

x(t + 6)

Belgium

-0.45

-0.43

-0.33

-0.22

-0.11

0.04

0.16

0.25

0.31

0.37

0.39

0.40

0.40

Denmark

-0.20

-0.17

-0.07

0.01

0.11

0.24

0.32

0.37

0.37

0.33

0.24

0.18

0.13

England

-0.35

-0.31

-0.26

-0.17

-0.03

0.11

0.28

0.45

0.59

0.63

0.60

0.55

0.47

France

-0.32

-0.24

-0.15

-0.03

0.10

0.23

0.36

0.46

0.54

0.57

0.53

0.48

0.44

Germany

-0.37

-0.29

-0.21

-0.09

0.05

0.20

0.31

0.42

0.52

0.58

0.60

0.62

0.63

Ireland

-0.34

-0.34

-0.27

-0.12

0.05

0.16

0.31

0.48

0.56

0.52

0.48

0.45

0.41

Italy

-0.11

-0.00

0.11

0.25

0.39

0.50

0.55

0.56

0.51

0.41

0.31

0.24

0.19

The Netherlands

-0.40

-0.34

-0.29

-0.21

-0.09

0.05

0.15

0.23

0.29

0.35

0.38

0.40

0.42

Note: Sample period, quarterly data: 1957:1–1994:4.

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© Blackwell Publishers Ltd 1999

THE GIBSON PARADOX

Country

120 Apostolos Serletis and George Zestos


Figure 1. Long-Term Interest Rates, R, and Logged Price Levels

THE GIBSON PARADOX

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Figure 1. Continued

122


3. Integration and Cointegration Properties of the Data Having investigated cyclical nominal interest rate–price level correlations, we now use recent advances in the theory of nonstationary regressors to provide reliable statistical evidence regarding the nature of the relationship between the long-term nominal interest rate and the price level. In particular, we evaluate empirically the univariate time series properties of the variables involved and test for cointegration to establish whether standard Gibson paradox regressions are spurious or not. Cointegration is a relatively new statistical concept designed to deal explicitly with the analysis of the relationship between nonstationary time series. In particular, it allows individual time series to be integrated, of order one (or I(1) in the terminology of Engle and Granger (1987)), but requires a linear combination of the series to be stationary. Higher orders, of course, are possible but complications arise when the series are integrated of different orders; in such cases two series that are individually integrated of different orders cannot be cointegrated. Therefore, the basic idea behind cointegration is to search for a linear combination of individually nonstationary time series that is itself stationary. Evidence to the contrary provides strong empirical support for the hypothesis that the integrated variables have no inherent tendency to move together over time. In the present context, if Rt and log(Pt) are each integrated of order one and cointegrate, then the standard Gibson paradox regression can be interpreted in a meaningful fashion. If, however, the variables do not cointegrate then ordinary least squares yields misleading results—in fact, Phillips (1987) formally proves that a regression involving integrated variables is spurious in the absence of cointegration. Finally, if the series are integrated of different orders, the Gibson paradox regression cannot be cointegrated and inferences regarding the Gibson paradox will be spurious. We begin by testing for the presence of a unit root in the autoregressive representation of each individual series. Dickey–Fuller (DF) and augmented Dickey–Fuller (ADF) tests of the null hypothesis that a single unit root exists in each series are conducted using the following ADF regression (see Dickey and Fuller (1981) for more details): ,

Dzt = a 0 + a1t + a 2 zt -1 + Â b jDzt - j + e t ,

(2)

j=1

where zt is the series under consideration and , is selected to be large enough so that e t is white noise. The null hypothesis of a single unit root is rejected if a 2 is negative and significantly different from zero. In practice, the appropriate order of the autoregression in the ADF test is rarely known. One approach would be to use a model selection procedure based on some information criterion. However, Said and Dickey (1984) showed that the ADF test is valid asymptotically if the order of the autoregression is increased with sample size T at a controlled rate of T1/3. For the sample used here, this translates into an order of 5. It is to be noted that for an order of zero the ADF reduces to the simple DF test. Also, the distribution of the t-test for a2 is not standard; rather it is that given by Fuller (1976). Tables 2 and 3 contain DF and ADF tests of the null hypotheses that a single unit root exists in each series as well as in the first differences of the series. Based on the “without trend” and “with trend” versions of the ADF test, the null hypothesis of a unit root in levels cannot be rejected except for Germany’s nominal interest rate, while © Blackwell Publishers Ltd 1999

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THE GIBSON PARADOX Table 2. Tests for a Unit Root in Levels Without trend

With trend

Country

Variable

DF

ADF

DF

ADF

Belgium

Rt log(Pt) Rt log(Pt) Rt log(Pt) Rt log(Pt) Rt log(Pt) Rt log(Pt) Rt log(Pt) Rt log(Pt)

-1.378 0.972 -1.439 -0.319 -1.711 0.916 -1.233 -1.403 -2.087 0.385 -1.551 0.236 -1.148 2.484 -1.683 -1.526

-1.996 -0.799 -1.595 -1.013 -1.635 -0.409 -1.695 -0.064 -3.241* -0.411 -1.368 -0.773 -1.678 -0.448 -2.184 -1.003

-0.970 -1.875 -0.971 -0.364 -1.228 -1.804 -0.868 0.498 -2.129 -1.158 -1.197 -1.014 -1.115 -2.970 -1.530 0.531

-2.070 -2.053 -0.984 -0.839 -1.066 -2.141 -1.756 -2.441 -3.395* -3.112 -0.724 -2.156 -2.087 -2.481 -2.339 -1.040

Denmark England France Germany Ireland Italy The Netherlands

Notes: The regression is given by equation (2). Unit-root test results are reported for an ADF statistic of order 5. An asterisk indicates significance at the 5% level. The 95% critical values for the DF and ADF tests are -2.8 for the “without trend” version of the test and -3.4 for the “with trend” version of the test.

Table 3. Tests for a Unit Root in First Differences of Levels Without trend

With trend

Country

Variable

DF

ADF

DF

ADF

Belgium

Rt log(Pt) Rt log(Pt) Rt log(Pt) Rt log(Pt) Rt log(Pt) Rt log(Pt) Rt log(Pt) Rt log(Pt)

-7.258* -4.985* -9.936* -8.894* -10.252* -9.066* -7.491* -4.932* -8.118* -7.943* -11.264* -8.338* -7.636* -4.224* -8.952* -9.661*

-4.784* -1.925 -5.968* -2.611 -4.675* -2.296 -4.342* -2.083 -4.931* -2.761 -5.926* -2.251 -4.147* -1.815 -4.854* -2.650

-7.253* -4.938* -9.988* -8.878* -10.296* -9.103* -7.482* -5.021* -8.089* -7.920* -11.308* -8.319* -7.613* -4.254* -8.945* -9.774*

-4.871* -1.764 -6.258* -2.654 -4.822* -2.191 -4.391* -2.077 -4.920* -2.690 -6.127* -2.190 -4.146* -1.627 -4.885* -2.890

Denmark England France Germany Ireland Italy The Netherlands

Note: See the notes to Table 2. © Blackwell Publishers Ltd 1999

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the null hypothesis of a second unit root is rejected for the interest rates but not for the price series. Hence, we conclude that the interest rate series are characterized as I(1)—i.e., having a stochastic trend—except for Germany’s interest rate series which is characterized as I(0), and the price series are characterized as I(2), meaning that the inflation rate series are all I(1). These results support the hypothesis that the long-term nominal interest rate and the price level do not form a cointegrated system, suggesting that inferences regarding the strength of the relationship between the nominal interest rate and the price level are spurious. This is consistent with the evidence presented by Corbae and Ouliaris (1989) who argue (using annual UK and US data over the 1920–87 period) that although the nominal interest rate and the price level are each I(1), the Gibson paradox regression is not cointegrated.

4. Conclusion We have investigated cyclical nominal interest rate–price level correlations in eight European Union countries using quarterly data from 1957:1 to 1994:4 and the methodology suggested by Kydland and Prescott (1990). We show that the well documented strong positive correlation between the nominal interest rate (as measured by the yield to maturity of long-term bonds) and the price level (as measured by the logged consumer price index) does not carry over to their cyclical components. We argue that this is evidence for rejection of the Gibson paradox. We have also used recent developments in the theory of nonstationary regressors to investigate familiar relationships previously taken to support the Gibson paradox. We show that the interest rate and the price level are integrated of different orders, suggesting that they cannot possibly cointegrate, thus invalidating the use of standard Gibson paradox regressions. The same conclusion has been reached by Corbae and Ouliaris (1989), but for a different reason, however—they show that although prices and interest rates are I(1), they do not cointegrate. Of course, since the Gibson relationship does not hold, as Klein (1995, p. 160) puts it “. . . there are many alternative hypotheses to consider.” One such hypothesis is the Fisherian relationship between inflation and nominal interest rates, according to which nominal interest rates move one-for-one with inflation in the long run (or, equivalently, a permanent change in the rate of inflation has no long-run effect on the level of the real interest rate). So much evidence is available regarding this question that one might reasonably assume that the question has been solved. However, this is not the case. In fact, Fisher and Seater (1993) and King and Watson (1997) contributed to the literature on testing long-run neutrality (and superneutrality) propositions (by developing tests using recent advances in the theory of nonstationary regressors), and showed that meaningful Fisher-effect tests can be constructed only if both the inflation rate and nominal interest rate series satisfy certain nonstationarity conditions and that much of the older literature violates these requirements, and hence has to be disregarded. In particular, they show that Fisher-effect tests are possible if the inflation and interest rate series are integrated of order one and do not cointegrate. This is exactly the case with the dataset used in this paper, suggesting that testing the Fisherian link between inflation and nominal interest rates, using the nonstructural bivariate autoregressive methodology recently proposed by King and Watson (1997), is an area for potentially productive future research. See Koustas and Serletis (1999) for such tests. © Blackwell Publishers Ltd 1999

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