Money demand in Japan and nonlinear cointegration.

THE THEORY OF money demand implies that the money demand function is almost infinitely elastic at extremely low nominal interest rates. This feature of the money demand function has important implications for monetary policy. For example, it suggests that the quantity of money that the central

bank prints will not have any effect on either inflation or output. Both Keynesians and monetarists were interested in this problem which has been called the liquidity trap or the zero interest bound. Because of very low short-term interest rates in Japan today and the lowest short-term interest rates in the United States in the past 45 years, many researchers have become interested in this problem again. (1) Therefore, it is important to understand how to properly incorporate the liquidity trap feature when estimating the money demand function. In the recent literature which uses a cointegration framework to estimate the long-run money demand, the log-level (semi-log) functional form has typically been used (see e.g., Stock and Watson, 1993, and Ball, 2001). However, the log-level form with log money and the level of the interest rate does not incorporate the liquidity trap feature. A notable exception is Hoffman and Rasche (1991), who use a log-log form.

We estimate the long-run money demand for Japan with three functional forms implied by different theories of money demand. Two of these functional forms are nonlinear and allow for the liquidity trap--the log-log form and the form implied by the money in the utility function with the constant elasticity of substitution (the MUFCES form for short). The third functional form, which serves as a benchmark, is the conventional log-level form. One shortcoming of the conventional linear cointegration methods which have been used in the empirical money demand literature is that they require one to make different and mutually incompatible assumptions about the trend properties of the nominal interest for different functional forms. Therefore, in addition to the conventional linear cointegration methods, we use a procedure for nonlinear cointegration developed recently by Bae and de Jong (2004) that does not suffer from this internal inconsistency. Their "Nonlinear Cointegration Least Square" (NCLS) technique allows us to estimate all three functional forms under the same assumption about the trend properties of the nominal interest rate.

In related literature, Anderson and Rasche (2001) and Bae and de Jong (2004) estimate and compare different functional forms of long-run money demand for the United States. Yoshida and Rasche (1990) estimate Japanese money demand using a vector error correction model and the log-level functional form. Miyao (2003) uses structural break tests to study the stability of Japanese money demand with the log-level and log-log functional forms. His empirical results indicate a structural break for the log-level form but not for the log-log form. Fujiki and Watanabe (2004) investigate the stability of Japanese money demand with the log-log form and confirm Miyao's finding that the log-log form is stable. Nakashima and Saito (2005) use cointegration methods that allow for possible structural breaks to approximate the nonlinear shape of the long-run Japanese money demand curve by two linear segments--one for the earlier "high" interest rate period and the other for the more recent "low" interest rate period. They find that the interest rate semi-elasticity is much higher for the recent "low" interest rate period.

Our paper is complementary to this recent literature on Japanese money demand, but differs from it in two important ways. First, the nonlinear cointegration approach used here provides a unified econometric framework to estimate the different functional forms. Second, we compare the different functional forms in terms of out-of-sample prediction performance. (2) Our empirical results indicate that the log-log and MUFCES functional forms that allow for the liquidity trap outperform the log-level functional form in terms of the out-of-sample prediction. The results are qualitatively similar between the log-log and the MUFCES forms and between linear and nonlinear cointegration techniques.

1. ESTIMATING MONEY DEMAND IN JAPAN

1.1 Cointegration Methods

This subsection describes the three functional forms that we estimate, and also provides a brief overview of Bae and de Jong's (2004) nonlinear cointegration procedure. Since the nominal interest rate exhibits a persistent serial correlation, the assumption that the nominal interest rate is I(1) is generally accepted as a good approximation. Therefore, we regard the long-run money demand function as a cointegrating regression. We consider the following three functional forms that are motivated by different theories of money demand (3):

(1) [m.sub.t] = [[beta].sub.0] + [[beta].sub.1] [i.sub.t] + [u.sub.t]

(2) [m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln [absolute value of [i.sub.t]] + [u.sub.t]

(3) [m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln (1 + [absolute value of [i.sub.t]]/ [absolute value of [i.sub.t]]] + [u.sub.t]

Where [m.sub.t](= ln[M.sub.t]/[P.sub.t][Y.sub.t]) is the logarithm of the real money balance and it is the nominal interest rate. We impose the restriction of unit income elasticity of money demand and allow [i.sub.t] and [u.sub.t] to be temporally dependent and [u.sub.t] to be serially correlated.

In Equations (2) and (3) money demand is a nonlinear function of the interest rate. Using conventional linear cointegration methods, such as Fully Modified OLS (FMOLS) and Dynamic OLS (DOLS), requires one to make different assumptions regarding the trend properties of it for different functional forms because [i.sub.t], ln [absolute value of [i.sub.t]] and ln(1 + [absolute value of [i.sub.t]]/[absolute value of [i.sub.t]]) must all be assumed to be I(1) simultaneously. However, if [i.sub.t] is I(1), then ln [absolute value of [i.sub.t]] and ln(1 + [absolute value of [i.sub.t]]/[absolute value of [i.sub.t]]) cannot be I(1) in any meaningful sense and vice versa. Because of this internal inconsistency, estimation results from the conventional linear cointegration methods might not be directly comparable with each other. Therefore, along with the conventional linear cointegration methods, we also consider a nonlinear cointegration method proposed recently by Bae and de Jong (2004). Their NCLS technique enables us to estimate different functional forms under the one assumption that [i.sub.t] is I(1). However, the NCLS estimation method used in this paper has no asymptotic justification for Equation (3). Since a theory has not yet been fully developed, we also report bootstrap confidence intervals along with asymptotic ones.

Since the NCLS technique is relatively new, we illustrate briefly how to implement it for the estimation of [[beta].sub.1] in Equation (2). Let [k.sub.n] be an integer-valued positive sequence that diverges to infinity at a slower rate than n such that [k.sub.n][n.sup.-p-2/3p+[eta]] [right arrow]0 for some [eta] > 0 and p, (4) and [n.sub.j] = [nj/[k.sub.n]] for j = 0,1,2, ..., [k.sub.n]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for [n.sub.j-1] + 1 [less than or equal to] t [less than or equal to] [n.sub.j] for j = 1,2, ..., [k.sub.n]. Then the NCLS estimator [[beta].sub.1] is defined as an IV estimator that uses [z.sub.t] as the instrumental variable for ln [absolute value of [i.sub.t]]. Note that although [[beta].sub.1] is a consistent estimator, it cannot be used for statistical inference unless the limiting Brownian processes associated with [DELTA][i.sub.t] and [u.sub.t] are orthogonal, which is unlikely in the case of the long-run money demand function. Therefore, the following fully modified type NCLS estimation technique is used. The estimation procedure is as follows.

1. Calculate the residual, [u.sub.t], from a regression by the NCLS estimation method.

2. Get a HAC estimate for the long-run covariance matrix of ([u.sub.t], [DELTA][i.sub.t]), [??], by using ([[??].sub.t], [DELTA][i.sub.t]).

3. Calculate [m.sup.[dagger].sub.t] in a way analogous to the FMOLS, [[??].sup.[dagger].sub.t] = [m.sub.t] - [[??].sub.21][[??].sup.-1.sub.22][DELTA][i.sub.t].

4. The fully modified version of the NCLS estimator [[beta].sub.1] is defined as the NCLS estimator that is calculated using the modified dependent variable [[??].sup.[dagger].sub.t] instead of [m.sub.t].

The usual t- and F-tests are now valid because they achieve the correct significance level conditional on [DELTA][i.sub.t].

1.2 Data and Empirical Results

We use a quarterly data set from 1976:1 to 2003:4. (5) Since the data frequency is quarterly, we add quarterly seasonal dummy variables in the regression. In order to check the robustness of the estimation results, we use various measures ot monetary aggregates and nominal interest rates. For monetary aggregates, we use "Cash in Circulation" (CIC), M1, and M2 + CD and for the nominal interest rate, we use the "Call Rate," the "Gensaki Rate," and the "City Bank Lending Rate." The Consumer Price Index (CPI) is used as a proxy for price, (6) and both real GDP and Private Consumption (CON) are used as proxies for income. Since our results are similar with different measures of monetary aggregates and nominal interest rates, we only report results with M1 as the monetary aggregate and the Gensaki Rate as the nominal interest rate. (7)

Tables 1 and 2 report the coefficient estimates for Equations (1)-(3), and the corresponding asymptotic and bootstrap confidence intervals, using GDP and Private Consumption as the respective income variables. To make the comparison across the different functional forms, semi-elasticities of interest rate are computed at the 25th percentile, median, and 75th percentile level of the nominal interest rate. Since no asymptotic and bootstrap confidence intervals contain zero, the coefficient estimates are statistically significant in all combinations of functional forms and estimation methods. Asymptotic and bootstrap confidence intervals are generally similar, though the bootstrap one is larger than the asymptotic one. When we compare the estimation results across the different functional forms, the two nonlinear forms yield similar and much higher interest rate semi-elasticities than the conventional log-level form at lower interest rates. However, the estimation results are robust across the different estimation methods, including the NCLS estimator.

To address the question of which functional form is most appropriate for the Japanese long-run money demand, we investigate out-of-sample prediction performances for the three different functional forms. Table 3 reports the sum of squared error for two different methods of out-of-sample prediction performance. Equations (2) and (3), which correspond to nonlinear functions of the interest rate, clearly outperform the log-level Equation (1) under all estimation methods. These prediction performance results support empirically our conviction that the nonlinear functional forms are more appropriate for modeling the Japanese long-run money demand.

2. CONCLUSIONS

We estimate the long-run money demand for Japan with two nonlinear functional forms that allow for the liquidity trap and compare the empirical results with the standard log-level functional form. The nonlinear functional forms reveal a significantly higher interest rate semi-elasticity of money demand at low interest rates compared to the log-level functional form. They also convincingly outperform the log-level functional form in out-of-sample prediction exercises. These results are robust across both linear and nonlinear cointegration methods and highlight the importance of employing functional forms that allow for the liquidity trap in estimating the long-run Japanese money demand.

We thank Bill Dupor and Hu McCulloch for helpful discussions, and Shinichi Nishiyama at the Bank of Japan for kindly providing the data of monetary aggregates and various interest rate measures. We also thank Ken West (the editor) for his helpful comments and advice. This paper is based on a part of Bae's dissertation.

LITERATURE CITED

Anderson, Richard G., and Robert H. Rasche (2001). "The Remarkable Stability of Monetary Base Velocity in the United States, 1919-1999." Federal Reserve Bank of St. Louis Working Paper No. 2001-008A.

Bae, Youngsoo, and Robert M. de Jong (2004). "Money Demand Function Estimation by Nonlinear Cointegration." Manuscript, Ohio State University.

Bae, Youngsoo, Vikas Kakkar, and Masao Ogaki (2004). "Money Demand in Japan and the Liquidity Trap." Ohio State University Department of Economics Working Paper No. 04-06.

Ball, Laurence (2001). "Another Look at Long-Run Money Demand." Journal of Monetary Economics 47, 31-44.

Benhabib, Jess, Stephanie Schmitt-Grohe, and Martin Uribe (2003). "Backward-looking Interest-rate Rules, Interest-rate Smoothing, and Macroeconomic Instability." Journal of Money, Credit, and Banking 35, 1379-1412.

Eggertsson, Gauti B. (2004). "The Deflation Bias and Committing to Being Irresponsible." Journal of Money, Credit, and Banking, forthcoming.

Eggertsson, Gauti B., and Michael Woodford (2003). "The Zero Bound on Interest Rates and Optimal Monetary Policy." Brookings Papers on Economic Activity 1, 212-219.

Fujiki, Hiroshi, and Kiyoshi Watanabe (2004). "Japanese Demand for M1 and Demand Deposits: Cross-Sectional and Time-Series Evidence from Japan." Monetary and Economic Studies 22, 47-77.

Hoffman, Dennis L., and Robert H. Rasche (1991). "Long-Run Income and Interest Elasticities of Money Demand in the United States." The Review of Economics and Statistics 73, 665-674.

Jung, Taehun, Yuki Terahashi, and Tsutomu Watanabe (2001). "Zero Bound on Nominal Interest Rates and Optimal Monetary Policy." Mimeo, Hitotsubashi University.

Krugman, Paul R. (1998). "It's Baaack: Japan's Slump and the Return of the Liquidity Trap." Brookings Papers on Economic Activity 1998, 137-187.

Miyao, Ryuzo (2003). "Liquidity Traps and the Stability of Money Demand: Is Japan Really Trapped at the Zero Bound?" Manuscript, Kobe University.

Nakashima, Kiyotaka, and Makoto Saito (2005). "Uncovering Interest-elastic Money Demand: Evidence from the Japanese Money Market with a Low Interest Rate Policy." Mimeo.

Orphanides, Athanasios, and Volker Wieland (2000). "Efficient Monetary Policy Design Near Price Stability." Journal of the Japanese and International Economies 14, 327-365.

Stock, James H., and Mark W. Watson (1993). "A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems." Econometrica 61, 783-820.

Taylor, John B. (1999). "A Historical Analysis of Monetary Policy Rules." In, Monetary Policy Rules, edited by John B. Taylor. Chicago: University of Chicago.

Woodford, Michael (2003). Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton University.

Yoshida, Tomoo, and Robert H. Rasche (1990). "The M2 Demand in Japan: Shifted and Unstable? " Bank of Japan Monetary and Economic Studies 8, 9-30.

YOUNGSOO BAE is from the Department of Economics at Ohio State University (E-mail: bae.35@osu.edu). VIKAS KAKKAR is from the Department of Economics and Finance at City University of Hong Kong (E-mail: efvikas@cityu.edu.hk). MASAO OGAKI is from the Department of Economics at Ohio State University (E-mail: mogaki@econ.ohio-state.edu).

(1.) See, for example, Krugman (1998), Orphanides and Wieland (2000), Jung, Terahashi, and Watanabe (2001), Woodford (2003), Eggertsson and Woodford (2003), and Eggertsson (2004). The functional form of Taylor-type interest rate rules used in these papers implicitly depends on the form of the relationship between velocity and the interest rate as in Taylor (1999), among other factors. Therefore, the functional form depends on the shape of the money demand function.

(2.) Another difference with the existing literature is our use of the MUFCES functional form. To our knowledge, Benhabib, Schmitt-Grohe, and Uribe (2003) is the only other paper which estimates a money demand function with this specification. However, they do so for U.S. data rather than Japanese and do not use a cointegration framework.

(3.) Bae, Kakkar, and Ogaki (2004) describe how these functional forms are derived from various theories of money demand.

(4.) The following assumption needs to be made regarding p. Let ut and Air be linear processes given by

[u.sub.t] = [[infinity].summation over i=0][[phi].sub.1,i][[epsilon].sub.1,t-i]

[DELTA][i.sub.t] = [[infinity].summation over i=0][[phi].sub.2,i][[epsilon].sub.2,t-i]

where [[epsilon].sub.t] = ([[epsilon].sub.1,t], [[epsilon].sub.2,t]) is a sequence of independent and identically distributed (i.i.d.) random variables with mean zero. E [absolute value of [[[epsilon].sub.j,t]].sup.P] < [infinity] for some p > 2 for j = 1,2; for details, see Bae and de Jong (2004).

(5.) The data on the various monetary aggregates and nominal interest rates are from the Bank of Japan, whereas the CPI, GDP, and Private Consumption (CON) data were obtained from Datastream.

(6.) In order to incorporate the impact of consumption tax changes in 1989 and 1997 on the CPI, we used moving averages of the CPI series around the time when the consumption tax was changed. This did not affect our results materially.

(7.) A relatively "narrow" monetary aggregate like M1 is more consistent with the microfoundations from which the functional forms estimated here are derived. The "Gensaki Rate" is frequently employed in the empirical studies on Japanese money demand. Estimation results with other measures are available from us upon request.

TABLE 1

ESTIMATION RESULTS FOR [[beta].sup.a.sub.1] (GDP, M1, AND
"GENSAKI RATE")

                               SOLS                DOLS (b)

[m.sub.t] = [[beta].sub.0] + [[beta].sub.1][i.sub.t] + [u.sub.t]

                        -0.0597                -0.0640
Asymptotic (c)         (-0.0819, -0.0376)     (-0.0847, -0.0432)
Bootstrap (e)          (-0.1673, -0.0225)     (-0.1888, -0.0218)

[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln [absolute value of
[i.sub.t]] + [u.sub.t]

                        -0.0987                -0.1054
Asymptotic             (-0.1076, -0.0898)     (-0.1149, -0.0959)
Bootstrap              (-0.1548, -0.0849)     (-0.1782, -0.0874)
Semi-Elasticity (f)
25% (0.27)              -0.3656               -0.3904
50% (3.86)              -0.0256               -0.0273
75% (6.33)              -0.0156               -0.0167

[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln(1 + [absolute value of
[i.sub.t]]/[absolute value of [i.sub.t]]) + [u.sub.t]

                        0.0997                0.1066
Asymptotic             (0.0909, 0.1085)       (-0.0973, 0.1160)
Bootstrap              (0.0859, 0.1559)       (-0.0886, 0.1795)
Semi-Elasticity (f)
25% (0.27)             -0.3683                -0.3938
50% (3.86)             -0.0249                -0.0266
75% (6.33)             -0.0148                -0.0158

                            FMOLS (c)              NCLS (d)

[m.sub.t] = [[beta].sub.0] + [[beta].sub.1][i.sub.t] + [u.sub.t]

                       -0.0640                -0.0621
Asymptotic (c)         (-0.0854, -0.0426)     (-0.0835, -0.0407)
Bootstrap (e)          (-0.1808, -0.0258)     (-0.1767, -0.0217)

[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln [absolute value of
[i.sub.t]] + [u.sub.t]

                        -0.1031                -0.1004
Asymptotic             (-0.1112, -0.0949)     (-0.1081, -0.0926)
Bootstrap              (-0.1594, -0.0886)     (-0.1299, -0.0712)
Semi-Elasticity (f)
25% (0.27)              -0.3819                -0.3719
50% (3.86)              -0.0267                -0.0260
75% (6.33)              -0.0163                -0.0159

[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln(1 + [absolute value of
[i.sub.t]]/[absolute value of [i.sub.t]]) + [u.sub.t]

                        0.1041                  0.1014
Asymptotic             (0.0960, 0.1122)       (-0.0938, 0.1091)
Bootstrap              (0.0897, 0.1603)       (-0.0724, 0.1024)
Semi-Elasticity (f)
25% (0.27)             -0.3845                -0.3745
50% (3.86)             -0.0260                -0.0253
75% (6.33)             -0.0155                -0.0151

(a) Figures in parenthesis are the 95% confidence interval.

(b) The number of leads and lags is 3.

(c) A HAC estimator with Bartlett kernel is used with the bandwidth
parameter of 4.

(d) The bandwidth parameter, [n/[k.sub.n]], is 5. Results are robust
with parameter values between 4 and 8.

(e) The overlapping block bootstrap method is used with the block
size of 5. Bootstrap sample size is 10,000.

(f) For Equation (1), [[beta].sub.1] itself is semi-elasticity.

TABLE 2

ESTIMATION RESULTS FOR [[beta].sup.a.sub.1] (CON, M1, AND
"GENSAKI RATE")

                             SOLS                  DOLS (b)

  [m.sub.t] = [[beta].sub.0] + [[beta].sub.1][i.sub.t] + [u.sub.t]

                       -0.0595               -0.0625
Asymptotic (c)        (-0.0814, -0.0377)    (-0.0831, -0.0419)
Bootstrap (e)         (-0.1666, -0.0205)    (-0.1850, -0.0204)

  [m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln [absolute value of]
                     [i.sub.t]] + [u.sub.t]

                       -0.0986               -0.1054
Asymptotic            (-0.1067, -0.0905)    (-0.1152, -0.0957)
Bootstrap             (-0.1518, -0.0851)    (-0.1832, -0.0872)

Semi-Elasticity (f)

25%(0.27)             -0.3652               -0.3904
50%(3.86)             -0.0255               -0.0273
75%(6.33)             -0.0156               -0.0167

 [m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln(1 + [absolute value of
       [i.sub.t]]/[absolute value of [i.sub.t]]) + [u.sub.t]

                        0.0996              0.1067
Asymptotic            (-0.0916, 0.1076)     (0.0971, 0.1163)
Bootstrap             (-0.0863, 0.1528)     (0.0886, 0.1842)

Semi-Elasticity (f)

25%(0.27)              -0.3679              -0.3941
50%(3.86)              -0.0248              -0.0266
75%(6.33)              -0.0148              -0.0159

                      FMOLS (c)             NCLS (d)

 [m.sub.t] = [[beta].sub.0] + [[beta].sub.1][i.sub.t] + [u.sub.t]

                       -0.0638               -0.0620
Asymptotic (c)        (-0.0852, -0.0424)    (-0.0834, -0.0407)
Bootstrap (e)         (-0.1804, -0.0239)    (-0.1765, -0.0201)

  [m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln [absolute value of
                   [i.sub.t]] + [u.sub.t]

                       -0.1031               -0.1003
Asymptotic            (-0.1107, -0.0955)    (-0.1081, -0.0926)
Bootstrap             (-0.1573, -0.0890)    (-0.1303, -0.0723)

Semi-Elasticity (f)

25%(0.27)             -0.3819               -0.3715
50%(3.86)             -0.0267               -0.0260
75%(6.33)             -0.0163               -0.0158

 [m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln(1 + [absolute value of
        [i.sub.t]]/[absolute value of [i.sub.t]]) + [u.sub.t]

                       0.1041                0.1014
Asymptotic            (0.0966, 0.1116)      (0.0937, 0.1091)
Bootstrap             (0.0902, 0.1582)      (0.0710, 0.1016)

Semi-Elasticity (f)

25%(0.27)             -0.3845               -0.3745
50%(3.86)             -0.0260               -0.0253
75%(6.33)             -0.0155               -0.0151

(a) Figures in parenthesis are the 95% confidence interval.

(b) The number of leads and lags is 3.

(c) A HAC estimator with Bartlett kernel is used with the bandwidth
parameter of 4.

(d) The bandwidth parameter, [n/[k.sub.n]], is 5. Results are robust
with parameter values between 4 and 8.

(e) The overlapping block bootstrap method is used with the block
size of 5. Bootstrap sample size is 10,000.

(f) For Equation (1), [[beta].sub.1] itself is semi-elasticity.

TABLE 3

PREDICTION PERFORMANCE RESULTS (a) (M1 AND "GENSAKI RATE")

                                                     SOLS       DOLS

                        [y.sub.t] = GDP

  Sum of Squared "Stepwise One-Step Ahead" Forecast Errors (b)

[m.sub.t] = [[beta].sub.0] +
  [[beta].sub.1][i.sub.t] + [u.sub.t]                3.2584     3.0930
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln
  [absolute value of [i.sub.t]] + [u.sub.t]          0.2883     0.1770
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln(1 +
  [absolute value of [i.sub.t]]/
  [absolute value of [i.sub.t]]) [u.sub.t]           0.2786     0.1709

Sum of Squared Forecast Errors Based on Fixed Estimation Period (c)

[m.sub.t] = [[beta].sub.0] +
  [[beta].sub.1][i.sub.t] + [u.sub.t]                5.8052     5.8603
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln
  [absolute value of [i.sub.t]] + [u.sub.t]          1.3549     1.0345
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln(1 +
  [absolute value of [i.sub.t]]/
  [absolute value of [i.sub.t]]) [u.sub.t]           1.2925     0.9739

                  [y.sub.t] = Private Consumption

  Sum of Squared "Stepwise One-Step Ahead" Forecast Errors (b)

[m.sub.t] = [[beta].sub.0] +
  [[beta].sub.1][i.sub.t] + [u.sub.t]                3.2837     3.0952
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln
  [absolute value of [i.sub.t]] + [u.sub.t]          0.3099     0.1745
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln(1 +
  [absolute value of [i.sub.t]]/
  [absolute value of [i.sub.t]]) [u.sub.t]           0.3002     0.1695

Sum of Squared Forecast Errors Based on Fixed Estimation Period (c)

[m.sub.t] = [[beta].sub.0] +
  [[beta].sub.1][i.sub.t] + [u.sub.t]                5.8671     5.9844
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln
  [absolute value of [i.sub.t]] + [u.sub.t]          1.4032     1.2064
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln(1 +
  [absolute value of [i.sub.t]]/
  [absolute value of [i.sub.t]]) [u.sub.t]           1.3395     1.1388

                                                     FMOLS      NCLS

                        [y.sub.t] = GDP

  Sum of Squared "Stepwise One-Step Ahead" Forecast Errors (b)

[m.sub.t] = [[beta].sub.0] +
  [[beta].sub.1][i.sub.t] + [u.sub.t]                3.0541     3.1431
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln
  [absolute value of [i.sub.t]] + [u.sub.t]          0.2083     0.2580
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln(1 +
  [absolute value of [i.sub.t]]/
  [absolute value of [i.sub.t]]) [u.sub.t]           0.2029     0.2489

Sum of Squared Forecast Errors Based on Fixed Estimation Period (c)

[m.sub.t] = [[beta].sub.0] +
  [[beta].sub.1][i.sub.t] + [u.sub.t]                5.6086     5.6756
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln
  [absolute value of [i.sub.t]] + [u.sub.t]          0.9780     1.1726
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln(1 +
  [absolute value of [i.sub.t]]/
  [absolute value of [i.sub.t]]) [u.sub.t]           0.9247     1.1107

                  [y.sub.t] = Private Consumption

  Sum of Squared "Stepwise One-Step Ahead" Forecast Errors (b)

[m.sub.t] = [[beta].sub.0] +
  [[beta].sub.1][i.sub.t] + [u.sub.t]                3.0615     3.1416
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln
  [absolute value of [i.sub.t]] + [u.sub.t]          0.2255     0.2746
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln(1 +
  [absolute value of [i.sub.t]]/
  [absolute value of [i.sub.t]]) [u.sub.t]           0.2198     0.2657

Sum of Squared Forecast Errors Based on Fixed Estimation Period (c)

[m.sub.t] = [[beta].sub.0] +
  [[beta].sub.1][i.sub.t] + [u.sub.t]                5.6377     5.6774
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln
  [absolute value of [i.sub.t]] + [u.sub.t]          1.0344     1.1965
[m.sub.t] = [[beta].sub.0] + [[beta].sub.1]ln(1 +
  [absolute value of [i.sub.t]]/
  [absolute value of [i.sub.t]]) [u.sub.t]           0.9772     1.1345

(a) For estimation, the same the specifications as in Tables 1 and 2
are used. The prediction period is from 1998:1 to 2003:4.

(b) In order to get a forecast at time t, [m.sub.t] all observations
up to time t - 1 are used for estimation; i.e. each time, different
coefficient estimates are used.

(c) The coefficients are estimated once by using the sample up to
1997:4. Then these coefficients are used to generate forecasts from
1998:1 to 2003:4.