Journal Issue

Does the Gap Model Work in Asia?

International Monetary Fund. Research Dept.
Published Date:
January 1997
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IT IS GENERALLY accepted that inflation is closely related to the pace of economic activity. The most common formulation is that there is a level of output—referred to as potential or trend output—that is consistent with a stable rate of inflation. In this formulation of the “gap model,” the change in inflation will be related to the level of the output gap, defined as actual minus potential output: inflation will tend to rise if the gap is positive, it will tend to fall if the gap is negative, and it will remain stable if the gap is zero. An alternative version of the gap model focuses on the change in the gap rather than the level of the gap. In this case, the change in inflation will be related to the change in the output gap, implying that the level of inflation will tend to remain stable as long as the level of the gap is unchanged. The gap model is also commonly applied to wage inflation—the Phillips curve—in which case the relevant “gap” is between the actual rate of unemployment and the natural rate of unemployment (or the nonaccelerating rate of unemployment, the NAIRU).

The gap model has proved to be a useful tool for policy analysis since economic policies will generally have direct impacts on the gap. Monetary and fiscal policies, for example, will affect actual output while structural policies will affect potential output. Inflation will, of course, also depend on other factors. Changes in import prices, indirect taxes, inflation expectations, and labor market policies and institutions that affect wage formation can all have direct impacts on inflation. Although variables such as these are usually incorporated into most empirical models, the output or the unemployment gap remains at the center of almost all inflation models (Nickell, 1988; Masson, Symansky, and Meredith, 1990; and Chadha, Masson, and Meredith, 1992).

Considerable empirical support for one or both versions of the gap model is provided by a large number of studies for industrial countries.1 There have, however, been only a few empirical studies of the gap model for developing countries.2 The paucity of empirical studies, together with the very rapid growth sustained by many developing or newly industrializing economies in Asia during the past one or two decades—and overheating problems that some countries have recently had to address— raises the question of whether the gap model is relevant for Asia. The purpose of this paper is to present some evidence on this issue for a group of 13 developing, newly industrializing, and industrial Asian economies: Australia, China, Hong Kong, India, Indonesia, Japan, Korea, Malaysia, New Zealand, the Philippines. Singapore, Taiwan Province of China, and Thailand. This group includes the rapidly growing and, at least in some cases, relatively high-inflation economies of East and Southeast Asia. To be as evenhanded as possible, we adopt a research strategy designed to minimize judgmental input. In the next section, we discuss a new methodology used to estimate trend output and present estimates of the output gap. Empirical tests of the gap model are presented in Section II. Section III concludes.

I. Output Gaps

To test the gap model, it is first necessary to estimate potential or trend output in order to define the gap between actual and potential output. Any test of the gap model is, therefore, a joint test of the estimated gap and of the impact of the gap on inflation. Unfortunately, estimating trend or potential output is more an art than a science. There are many different approaches, and no one method is trouble free (Adams and Coe, 1990: Giorno and others, 1995; and De Masi, 1997). One general approach is to estimate a production function and then calculate potential output as the level of output that would obtain if all factors of production were fully utilized (IMF, 1991; and OECD, 1994). This structural approach has the advantage of explicitly identifying the sources of output growth. In practice, however, these studies commonly focus on capital and labor, with total factor productivity, typically an important source of growth, left largely unexplained.

A less structural, much simpler, approach is to use some type of univariate smoothing technique to define trend output. The most popular is the Hodrick-Prescott (1997) filter, which was first presented in a 1981 working paper. In this paper, we use a nonparametric method to estimate trend output that has the important advantage of allowing the data to determine the degree of smoothing. By contrast, the Hodrick-Prescott filter and other smoothing techniques require the user to specify an arbitrary smoothing parameter, which, in effect, determines the size of the data “window” used to calculate the trend. The larger the window, the smoother will be the trend: the smaller the window, the more the trend will resemble the actual data.3 Based on U.S. data. Kydland and Prescott (1990) suggest smoothing parameters of 1600 for quarterly data and 100 for annual data. These smoothing parameters have, to a large extent, become the industry standard, even for countries where the business cycle may differ substantially from the U.S. business cycle. The absence of statistical criteria to guide the choice of the smoothing parameter is problematic since the estimated trends will depend importantly on the value of this parameter.

The technique used in this paper determines the size of the data window through a statistical procedure that is asymptotically optimal, in the sense that it minimizes a global error criterion and thus ensures that the degree of smoothing is consistent with the cyclical properties of the data. Since the degree of smoothing is determined on statistical grounds and does not require the specification of an arbitrary smoothing parameter or data window size, different researchers analyzing the same data will obtain the same trend estimates. In the Appendix, the method that we use to estimate trend output and output gaps is described and compared with estimates obtained with a Hodrick-Prescott filter.

Real GDP and its estimated trend are displayed in the left panels of Figure 1 (note that the scales differ across countries). The GDP data, as well as the data used in the next section, are from the International Monetary Fund’s International Financial Statistics. For China, the estimated trend output is based on new estimates of real national income from Hu and Khan (1996). which are available for a much longer time period than the published GDP data, which begin only in 1978. Like other univariate estimates, our estimated trends have the property that their average value over the sample period is the same as the average value of actual GDP. The estimated trend at the beginning and the end of the sample may be biased by the absence of data prior to and beyond the sample period. To address this problem, we (1) exclude the first two years of the estimated trends and (2) extend the GDP data two years beyond the last year shown in the figure by using the projections from the World Economic Outlook, May 1996 and then exclude these two years of the estimated trends. The uncertainty of the estimated trends for the most recent period, a problem that is common to all centered-smoothing techniques, limits their usefulness as an operational tool for practical policy implementation purposes.

Figure 1.Output Gaps and Inflation

(Real GDP or real national income indexed by 1980 value of trend and displayed in logarithmic scale)
(Real GDP or real national income indexed by 1980 value of trend and displayed in logarithmic scale)
(Real GDP or real national income indexed by 1980 value of trend and displayed in logarithmic scale)
(Real GDP or real national income indexed by 1980 value of trend and displayed in logarithmic scale)
(Real GDP or real national income indexed by 1980 value of trend and displayed in logarithmic scale)

There are a number of notable features in the development of actual output. First, the majority of countries have experienced remarkably rapid increases in real output, in contrast with Australia, India, New Zealand, and the Philippines, where increases in output have been more modest (Table 1). Second. GDP appears to have accelerated since 1970 in many of the high-growth economies while declining in most of the low-growth countries. Third, growth has been sustained remarkably well since the mid-1960s, with only a few examples of stagnant or negative growth.

Table 1.Output and Prices(Average annual growth rates)
Data window (years)Full sample a1970s1980sEarly 1990sa
Australia113. 72.0
China b87.
Hong Kong98.
New Zealand142.62.48.22
Taiwan Province88.
of China

Data refer to the years shown in Figure 1. typically starting in the early 1960s and ending in 1994.

Based on national income.

Data refer to the years shown in Figure 1. typically starting in the early 1960s and ending in 1994.

Based on national income.

Given these features of actual output developments, the estimated trends are relatively smooth. The data window—the number of years of data used to define the trend—is typically 7 to 9 years, although it is only 3 years for Thailand and as large as 14 years for New Zealand (see the first column of Table 1). The size of the data window tends to be inversely related to the growth of output and positively related to the volatility of output growth (compare Figure 1). The different sizes of the data window indicate that it would be inappropriate to arbitrarily apply the same degree of smoothing to the GDP data for all economies.

The output gaps implied by actual output and the estimated trends are displayed in the right panels of Figure 1; the gap is defined as actual output minus trend output as a percent of trend output. The estimated output gaps are typically in the range of ±3 percent. The largest estimated output gaps are minus 5 percent in Hong Kong in the mid-1970s and the Philippines in the mid-1980s, and ± 10 percent in China in the 1960s. Except for a few years in the mid-1980s, real output growth has been remarkably stable since the mid-1970s in Malaysia, Singapore, and Thailand, so the calculated output gaps are relatively small. As discussed in the Appendix, our estimated trends are less smooth than those obtained from the Hodrick-Prescott filter, and hence our calculated output gaps are generally smaller.4 In the next section, we test the extent to which our estimates of the output gap explain inflation developments in 13 Asian economics.

II. Testing the GAP Model

Annual rates of inflation are displayed in the right panels of Figure 1, together with the estimated output gaps. In most countries, inflation has been volatile, with spikes of 25-50 percentage points not uncommon, particularly during the 1970s and early 1980s. Many, although not all, of these inflation spikes appear to be associated with smaller spikes in the output gaps. Periods of relatively high inflation in the 10-15 percent range are not uncommon, and Indonesia experienced a sustained period of very high inflation in the 1960s. Except for China and Hong Kong, inflation has generally been lower—below 5 percent in Japan, Malaysia, Singapore. Taiwan Province of China, and Thailand—and more stable since the mid-1980s. In China, by contrast, inflation has risen substantially since the mid-1980s: this is also true, albeit to a lesser extent, in Hong Kong. Except for China, Phillips and Perron (1988) unit root tests indicate that inflation, as well as the change in inflation, is stationary.

To assess the importance and significance of the visual evidence of a relationship between inflation and the estimated output gaps, we estimate a simple specification of the gap model

where πt=100(CPI/CPIt11),GAPt=100(GDPtGDPtTR)/GDPtTR, is a stochastic disturbance term, and Δ is the first-difference operator. We include a constant to avoid imposing the constraint that the noninflation-ary level of the output gap is exactly zero. Model 1 can be derived from a simple inflation-expeciations-augmenied Phillips curve with adaptive expectations:

with πte=πt1,andwhereπte is inflation expectations at time t. The assumption of adaptive expectations greatly simplifies the modeling of inflation expectations, allowing us to focus on the role of the output gap as a determinant of the change in inflation.

The coefficients of the gap variable can be interpreted as semielas-ticities that give the percentage point change in inflation implied by a 1 percentage point output gap. We estimate equations with only the contemporaneous value of the gap variable and with up to five lagged values (Y = 5) of the gap variable, using the Schwarz (1978) information criterion to determine the preferred lag length. For given estimates of trend output, this estimation procedure requires no judgment on our part to select the preferred specification. Our test of the gap model is simply an F-test that the β1i’s are jointly significantly different from zero—that is, that the output gap is a significant determinant of the change in inflation— and that Σβ1i0.

It is important to note that model 1 encompasses both the level and the change versions of the gap model discussed at the beginning of this paper. The change version of the gap model is

Model 2 is a special case of model 1, with the constraint that the coefficients of the level of the gap alternate in sign, with each pair equal in absolute value: this implies that they sum to zero, that is, that β11=-β1013=-β12, and soon. Model I also encompasses specifications that allow both the level and the change in the gap to affect inflation, in which case some of the estimated coefficients would be negative but their sum would be positive.

The estimation results for model 1 are summarized in panel A of Table 2. The estimated lags are short: only the contemporaneous value of the gap enters six of the equations while the rest have one or two lags. The F-tests indicate that the level of the gap, whether included contemporaneously or with lags, is a significant determinant of the change in inflation in all economics except China, India, and Thailand.5 For New Zealand, the gap has a significant impact on the change in inflation, but the sum of the estimated coefficients of the gap variables is negative (although almost zero). The estimated elasticity of the change in inflation with respect to the gap is largest for Indonesia. Taiwan Province of China, and Korea, owing to the high rates of inflation compared to the small estimated output gaps in these economies. For the economies where the estimated coefficients are significant, the level of the output gap explains one-fourth to one-half of the total variance in the change in inflation over the sample period.

Table 2.Testing the GAP Model
gap coefficients
Lag lengthSumSignsF-testaR2
A. Model 1
Hong Kong00.97918.554**0.382
New Zealand20.142-+-4.434**0.322
Philippi ncs20.894-+-8.950**0.499
Taiwan Province of China16.069++8.829**0.404
B. Model 2
Hong Kong32.912++++9.174**0.605
New Zealand10.323-+7.077**0.521
Taiwan Province of China13.443++7.726**0.373

** (*) indicates rejection of ihe null hypothesis that the gap variables jointly have no effect on the change in inflation at the 5 (10) percent significance level.

Based on national income.

** (*) indicates rejection of ihe null hypothesis that the gap variables jointly have no effect on the change in inflation at the 5 (10) percent significance level.

Based on national income.

Except for Japan, New Zealand, and the Philippines, there are either no lags on the gap variable or the signs of the estimated coefficients are all positive. This suggests that, based on our criterion for determining the number of lags, the change in inflation is related to the level rather than the change in the output gap. For Japan and the Philippines, there are positive and negative estimated gap coefficients, but their sum is positive, suggesting that both the level and the change in the output gap may be relevant.

Estimation results with the constraints implied by model 2 are presented in panel B of Table 2.6 The estimated equations for model 2 are nested in model 1 for Korea, the Philippines, New Zealand, and Japan; an F-test rejects the constraints implied by model 2 only for Korea.7 Thus, for Japan, New Zealand, and the Philippines, the data suggest that the change specification works as well or better than model 1. Indeed, for New Zealand, the unconstrained estimation results for model 1 are virtually identical to the constrained results for model 2. For most of the other countries, either model 1 or model 2 appears to be broadly consistent with the data. There is no evidence that the change in the output gap is an important determinant of the change in inflation for China, India, or Thailand, which is consistent with the results obtained from model 1.

We now extend model 1 to allow the money supply and the terms of trade to have an impact on the change in inflation in addition to the level or the change in the output gap. In most inflation models, the money supply does not directly affect inflation but instead has an indirect effect through the output gap. The money supply could, of course, affect inflation expectations and thereby have a direct impact on inflation over and above the effect of the output gap. Moreover, a measure of the money supply adjusted for secular movements in the velocity of money may be better able to explain inflation developments than the estimates of the output gap presented above. To test this possibility, we construct a measure of the real money gap analogous to our measure of the output gap: MGAPt=100(M2tM2tTR)/(M2tTR), where M2 is broad money from International Financial Statistics and M2tTR is estimated by the same method used to estimate trend output. Adding this measure of the money gap to model 1 gives8

We use the same procedure to select lag lengths, but constrain both the output and the money gap measures to have the same number of lags.

The estimation results for model 3 are presented in Table 3.9 The most striking result is for India, where the levels of both the output gap and the money gap are very significant, explaining fully two-thirds of the variance of the change in inflation. This is in sharp contrast to model 1, in which the output gap alone is not significant and explains virtually nothing. The level of the money gap is also important in Hong Kong, but with a negative sign, and in New Zealand. For the other economies, adding the money gap makes little difference to the significance or size of the estimated coefficients of the output gap obtained in model 1; the exception is Indonesia, where the estimated coefficient is substantially smaller. There is no evidence that either measure of the output or the money gap is a significant determinant of the change in inflation in Thailand.

We also estimated the gap model with the percent change in the terms of trade included in addition to the output and money gap variables. This specification allows, for example, commodity price shocks or exchange rate changes to have direct effects on inflation. In general, the terms of trade variable was not significant and had little impact on the size or significance of the gap variable. The exceptions were Korea and Japan, where terms of trade growth was significant (for Korea, the output gap became insignificant) when added to models 1 and 2, and Japan and New Zealand, where terms of trade growth was significant when added to model 3 (for New Zealand, the output gap became insignificant).

Table 3.Testing the GAP Model with Money
Lag lengthGAP coefficientsMGAP coefficientsModel
Model 3
Hong Kong10.9297.455**-0.7105.295**8.583**0.569
Now Zealand2-0.0467.113**0.3844.852**5.558**0.572
Taiwan Province17.55210.529**-1.1751.6375.450**0.476
of China

** (*) indicates rejection of the null hypothesis that the gap variables jointly have no effect on the change in inflation at the 5 (10) percent significance level.

** (*) indicates rejection of the null hypothesis that the gap variables jointly have no effect on the change in inflation at the 5 (10) percent significance level.

Diagnostic tests indicate that the residuals from the regressions in models 1-3 are essentially white noise, suggesting the absence of significant specification errors. Both the Breusch-Godfrey lest for autoregressive errors and the Engle test for autoregressive conditional heteroscedaslicity are not significant at the 5 percent level for any country in any of the three models. The Breusch-Pagan test for heteruscedasticity is significant at the 5 percent level in all models for Taiwan Province of China and in model 2 for Indonesia.10

In summary, the estimation results indicate that the output gap is an important determinant of the change in inflation for all of the economies in our sample except China and Thailand. For China, the failure of the gap model may reflect the relatively few observations available since the implementation of economic reforms: for Thailand, the gap model fails partly because output has grown so steadily since the mid-1970s that the estimated gap is very close to zero for much of the sample period (compare Figure 1). For the other economies, model 1. the most basic version of the gap model, explains an average of 34 percent of the variance of the change in inflation over the full sample period; for India, model 3, which includes a measure of the money gap, explains 67 percent. Given that the dependent variable is the second difference of the price level, the goodness of fit of the estimated equations is relatively high. Figure 2 presents the actual and predicted change in inflation for the most recent five years based on our estimates of model 1 and, in the case of India, model 3. Although the estimated gap models do a good job of predicting the change in inflation for some economies, they perform poorly in a number of countries. Other factors, including changes in indirect taxes, import prices, and random shocks, are also important; in many cases, they are more important than the level of the output gap.

III. Conclusions

Double-digit rates of growth are not uncommon in Asia. Some rapidly growing economies, such as Singapore, Taiwan Province of China, and Thailand, have been able to sustain high rates of growth while keeping inflation relatively low, while others have experienced higher rates of inflation, thus raising the question of the relevance of the gap model for Asia. To answer this question as neutrally as possible, we have presented a new procedure to estimate trend output that allows the data to determine the degree of smoothing rather than requiring the researcher to specify this arbitrarily. We then used the output gaps implied by the estimated trends to test a number of versions of the gap model. As discussed in the Appendix, the output gaps implied by our estimates of trend output generally perform better than estimates based on a Hodrick-Prescott filter.

The answer to the question posed in the title to the paper is clear: the gap model works very well in almost all Asian economies. The output gap is a significant determinant of the change in inflation in 11 of the 13 economies studied. The only exceptions are Thailand and China, where there is no evidence that our estimate of the output gap has any impact on the change in inflation. In India, the output gap is an important determinant of the change in inflation when a measure of the broad money gap is included in the estimation. In New Zealand, the change in inflation is more closely related to the change in the output gap than to the level of the output gap: both the level of. and the change in. the output gap appear to be important in a number of countries, including Japan and the Philippines. For almost all economies, the output gap remains a significant determinant of inflation when other possible determinants of inflation, such as changes in the money supply or in the terms of trade, are also included in the estimated equations.

Figure 2.Actual and Predicted Changes in Inflation

(Percentage points)

a Predieted change in inflation is bused on model 1 except for India, which is based on model 3.

Our findings have a number of obvious and closely related policy implications. The first is that accelerating prices are, in general, an indication of an overheating economy. The second implication is that reductions in inflation will generally require the implementation of restrictive macro-economic policies that temporarily reduce the growth of real output. Although the gap model generally does a good job of explaining changes in inflation, it is clear that other factors, such as changes in indirect taxes and random shocks, are also important. Moreover, estimates of the output gap are inevitably most uncertain for the most recent period. Thus, a third implication is that, while estimates of the output gap can be a useful tool for policy analysis, they need to be used in conjunction with other indicators and analyses for practical policy implementation purposes. In this regard, the developing, newly industrializing, and industrial economies of Asia arc no different than the industrial countries of North America and western Europe.

APPENDIX: A Nonparametric Regression Method for Estimating Trends and Gaps

This Appendix outlines a nonparametric regression estimation method to estimate output trends and output gaps without having to specify the functional form of the trend in the underlying output series or the degree of smoothing applied to the actual data.11

The estimation is performed in the spirit of the classical decomposition that splits output into two components, trend output and an output gap, denoted by

where yt is output, m(t) is trend output, t is a time trend, and yt* is the output gap. A typical method of estimating the function m(t) is to regress output on a known parametric form such as β01t. However, modeling trends in economic data often require more complex functions, usually involving higher-order polynomials. The aim of a nonparametric regression estimation of m(t) is to approximate an unknown trend function arbitrarily closely, given a large enough sample. It is not necessary to specify die functional form of m(t), although it is necessary to assume that the trend has an adequate number of derivatives so that it is smooth relative to yt* Consequently, using nonparametric regression Hi estimate m(t) allows truly flexible functional forms to be considered. Of the large number of nonparametric regression estimates available, we use the Nadaraya-Watson estimator m^h(t) ofm(t), which has the form

where Kh(u)=h1K(u/h),h is the bandwidth parameter. T is the sample size, and K(u) is a kernel with support [-1,1] that satisfies K(u)du=1. The bandwidth parameter determines the size of the data “window” used to calculate the smoothed trend series while the kernel determines the distribution of the weights, which can be thought of as the shape of the data window. For a simple moving average, the shape of the kernel is rectangular. We use the Epanechnikov kernel, defined as

where I(|u|≤1) is the indicator function. This is a commonly used parabolic-shaped kernel function that minimizes the mean squared error of all twice-differentiable functions in. In practice, the choice of kernel is not critical since the mean squared error properties of alternative kernels arc almost identical. For example, the Epanechnikov kernel is only 6 percent more efficient than the rectangular-shaped kernel.

The accuracy of kernel smoothers as estimators of m is a function of the kernel K and the bandwidth ft, with the latter being the most important. We use a data-dependent bandwidth selection procedure that minimizes quadratic error measures for the regression. There are three main types of such procedures: (1) cross-validation or leave-out. (2) penalizing functions, and (3) plug-in methods. The first two methods are asymptotically optimal in the sense that they minimize mean squared errors without reference to the smoothness of m. but in practice all three methods yield similar estimates of A for a given data set. We use the cross-validation or leave-out method, which is based on regression smoothers in which one—say, the jth—observation is left out:

where Wht(x)=Kh(xt/T)/[T1t=1TKh(xt/T)]. With these modified smoothers, we define the cross-validation function

The cross-validation function is constructed over a dense grid of h values, with h chosen such that this function is at the minimum. This is called the least squares cross-validation estimate of h. This method validates the ability to predict {yt}j=1t across the subsamples {(t,yt)}tj.

The procedure outlined above is similar to the Hodrick-Prescott filter except that, in the latter procedure, the effective bandwidth or smoothing parameter must be supplied arbitrarily by the user. The estimated trends from the nonparametric regression method are compared to the estimated trends from the Hodrick-Prescott filter in Figure 3 for New Zealand, the Philippines, and Singapore. The Hodrick-Prescott filter results in smoother trends and correspondingly more volatile output gaps. When the output gap based on the Hodrick-Prescott filter is used in model I in place of the kernel smoother, the size of the estimated coefficients of the output gap variables is smaller, owing to the larger swings in the output gaps. In addition, more lags are required on the gap variables because of higher levels of serial correlation in the residuals. Even with the additional lags, however, the model’s explanatory power does not increase, and diagnostic tests show significant serial correlation problems for a number of countries. These results suggest that estimated trends and output gaps based on the nonparametric kernel smoother do a better job of explaining inflation developments than do estimates based on the Hodrick-Prescott filter.

Figure 3.Output Gaps

(Real GDP indexed by 1980 value of trend and displayed in logarithmic scale)

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See, for example. IMF (1991), OECD (1994), and Bryant and others (1988).

Exceptions are Andersen (1991) and IMF (1995). Singh (1996) discusses output gaps and inflation and other indicators of overheating in Asian economies. Agenor (1996) presents a survey of theoretical and empirical studies on labor markets in developing countries.

In a simple moving average, the size of the window is the number of lags over which the data are averaged.

King and Rebelo (1993) and Cogley and Nason (1995) argue that the Hodriek-Prescolt filter oversmooths and tends to add spurious cycles in the resulting output gaps.

For China, the insignificance of the gap variable may simply reflect inclusion in the sample of the prereform period, when many prices were not market determined.

For Australia and New Zealand, the negative sign on the contemporaneous change in the output gap may indicate that the consumer price index contains a component that is directly related to mortgage interest rates. In this case, increased inflation could be the initial effect of a tightening of monetary policy that raised interest rales and reduced actual output and the output gap.

The models are nested only if the lags implied by the Schwarz information criterion in model 2 are one less than the lags in model 1. since the first-difference specification of model 2 incorporates a lag in each gap variable. Nonnested J-tests, which arc discussed in Kennedy (1992, pp. 88-90), indicate for model 1 that the change specification of model 2 is not rejected for any country, while the same tests for model 2 indicate that the level specification of model 1 is rejected, albeit only marginally, for Australia and Hong Kong.

The estimation results discussed below are largely unchanged if MGAP is de-lined instead as the difference between the growth of the money supply and the growth of its trend.

We do not report results for China for the reasons noted in footnote 5.

Engle (1984) discusses these and other diagnostic tests. When the Indonesian sample is restricted to exclude the inflationary period of the 1960s, the tests for het-eroscedasticity are no longer significant while the tests for the significance of the estimated coefficients of the output and money gap variables are unchanged.

Although the nonparametric regression literature is dominated by asymptotic theory results, nonparametric regressions have been used to estimate consumption and production functions. Phillips curves, and forecasting equations for exchange rates and stock market returns: see Hardle (1990) and Ullah and Vinod (1993).

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