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The kernel-location approach: A new non-parametric approach to the analysis of downward nominal wage rigidity in micro data
Economics Letters 97 (2007) 253 – 259 www.elsevier.com/locate/econbase
The kernel-location approach: A new non-parametric approach to the analysis of downward nominal wage rigidity in micro data ☆ Christoph Knoppik 1,2 Sozial- und Wirtschaftswissenschaftliche Fakultät, University of Bamberg, Feldkirchenstrasse 21, D-96052 Bamberg, Germany Received 22 March 2006; received in revised form 1 March 2007; accepted 15 March 2007 Available online 5 July 2007
Abstract The kernel-location approach combines kernel-density estimation and the principle of joint variation of location and shape of the distribution of wage changes, to estimate counterfactual and factual distributions of wage changes and degree and form of downward nominal wage rigidity. © 2007 Elsevier B.V. All rights reserved. Keywords: Downward nominal wage rigidity; Kernel-location approach; Kernel density estimation JEL classification: C14; E24; J30
1. Introduction Evidence on downward nominal wage rigidity (DNWR) from micro data constitutes an important input to debates on whether low inflation increases equilibrium unemployment, but as noted e.g. by RodríguezPalenzuela, Camba-Mendez, and Garcia (2003) this evidence is far from clear cut. Therefore, alternative
☆ I would like to thank Thomas Beissinger, Jürgen Heubes, Hans Ludsteck, Joachim Möller, Walter Oberhofer, and the participants of a session at the AEA 2005 annual meeting, where earlier stages of this work have been presented. 1 Affiliation at the time of submission: Prof. Dr. Christoph Knoppik, University of Bamberg, Germany. Tel.: +49 951 863 2634, e-mail: [email protected]. 2 Affiliation at the time of revision: University of Regensburg, Germany. Tel.: +49 941 943 2700, e-mail: christoph. [email protected]; url: http://www.wiwi.uni-regensburg.de/knoppik/.
0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.03.010
254
C. Knoppik / Economics Letters 97 (2007) 253–259
approaches to the empirical analysis of DNWR seem desirable. The general advantage of non- or semiparametric approaches in this research is to avoid functional assumptions that parametric approaches have to make. However, there are also some drawbacks. The skewness-location approach of McLaughlin (1994) yields no quantitative information on the extent of DNWR in the data. For the symmetry approach of Card and Hyslop (1997), the applicability of the key assumption of symmetry has frequently been denied, e.g. by McLaughlin (1999) for US PSID data. Finally, while the histogram-location approach of Kahn (1997) makes no assumptions on the functional form of the wage change distribution, it does stipulate a functional form for the rigidity, which is itself under debate in the literature, see e.g. Knoppik and Beissinger (2003). This paper proposes a new non-parametric approach to the analysis of downward nominal wage rigidity in micro data that uses kernel density estimates to model the factual and counterfactual distributions of wage changes, and uses the principle of joint variation of location and shape of annual distributions of per cent nominal wage changes for the identification of DNWR. While these two building blocks of analysis are separately used in other approaches, it is their combination that determines the favourable properties of the new approach. Sharing the general advantages of the non-parametric approaches, it avoids the problematic symmetry assumption and it provides a quantitative estimate of the degree of DNWR in the data without imposing a functional form of rigidity. Sections 2 and 3 explain the basic idea behind the kernel-location approach and the organization of data. Sections 4 and 5 present and interpret the estimators for the counterfactual and factual distribution, the rigidity function and the average degree of rigidity. Section 6 illustrates the approach with an application to artificial data. 2. Basic idea The kernel-location approach analyses annual cross-sectional distributions of per cent wage changes. A counterfactual distribution that would prevail in the absence of DNWR and a factual distribution that may be influenced by DNWR are distinguished. DNWR is thought to lead to a thinning over the range of negative nominal wage changes, and a corresponding pile-up at zero, so that both distributions differ for non-positive wage changes if DNWR does exist. Since the counterfactual distribution cannot directly be observed for negative nominal wage changes it cannot be compared to the observed factual distribution. However, under the assumption that the counterfactual distribution has a shape that is invariant over time and only changes its location (reflecting changes in price inflation and other influences) the mediancentered counterfactual distribution is identical over time. 3 Periods with different location provide information on different parts of the centered counterfactual and factual distributions, which can then be compared over the overlapping range. This basic idea is illustrated in column a) of Fig. 1. By assumption, there is a counterfactual distribution with time-invariant shape, there is DNWR, and there are T = 10 periods with different locations of the distribution, with medians mt in ascending order (mmin in period 1 and mmax in period 10). Panels ai) to aiii) of Fig. 1 show three median-centered theoretical distributions
This assumption of invariant shape is crucial to all approaches employing the identification principle of ‘joint variation of location and shape’. One concern is that the dispersion of the distribution is higher in times of higher inflation. Suitable standardization of the data can help in this respect. Another potential problem is that the observable median might be affected by the effects of rigidity. The use of higher quantiles as measures of location may remedy such situations. 3
C. Knoppik / Economics Letters 97 (2007) 253–259
255
Fig. 1. Theoretical and estimated distributions. Theoretical and estimated median-centered distributions by period and aggregated as discussed in text.
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C. Knoppik / Economics Letters 97 (2007) 253–259
under this set up 0(t = 1, 5, 10). In terms of median-centered changes, thinning occurs below −mt. Information on the factual distribution is available only below −mmin, whereas information on the counterfactual distribution is only available above −mT. Therefore, for the range −mmax b x b −mmin there is information on both distributions, which can be used to analyse the effects of DNWR. Only the variation in location, expressing itself in the width of this range, contributes to the identification of DNWR; this is also true of the other approaches employing the identification principle of ‘joint variation of location and shape’. Panels aiv) and av) of Fig. 1 show those parts of the aggregate counterfactual and factual distributions that can be deducted from the period information. Different amounts of information are available in the subdivisions of −mmax b x b −mmin defined by period medians. For example, between −m2 and −m1 there is information on the factual only from period 1, but there is information on the counterfactual from periods 2 to 10. The overlap of the aggregate estimates can be used to reach conclusions with respect to existence, extent and type of DNWR. 3. Organization of data The estimators can be formulated in a transparent way by organizing the data as in the example. x˜it denote observed per cent wage changes, where i runs over the cross-section dimension and t over the time dimension. N and Nt are the number of observations and the number of observations from period t. The median mt of observations x˜it from period t is used in defining median centered values Xit u x˜ it mt :
ð1Þ
It is useful to re-index observations in the time dimension, such that periods with higher time-index τ have higher medians, ms N ms V⇔ s N s V:
ð2Þ
Re-indexed annual medians of uncentered observations are used to define the intervals I2 to IT. Isþ1 ¼ msþ1 ; ms ½; s ¼ 1 N T 1:
ð3Þ
I2 to IT are parts of the ‘core interval’ IC = ]− mT, − m1[ between the lowest and the highest periodmedian in the sample. I1 = ]− m1, ∞[ and IT+1 = ]− ∞, − mT [ are intervals to the right and to the left of the core interval, with only negative nominal changes in IT+1 and only positive nominal changes in I1 in all periods. Signs of nominal wage changes do change over time in the intervals within the core interval Ic. In any period τ there are positive nominal changes in all Is, s ≤ τ and negative nominal changes in all Is, s N τ. 4. Kernel density estimators of the factual and counterfactual distribution The estimators for the distributions suitably weigh the partial period-wise kernel density estimates of median-centered factual and counterfactual distributions in order to obtain overlapping partial estimates of the aggregate factual and counterfactual distributions. Before more refined estimators are proposed that avoid the danger of discontinuity bias, in a first step preliminary estimators are formulated. Information from different periods requires separate treatment since different parts of factual and counterfactual distribution can be estimated in each period and contribute differently to the aggregate, as
C. Knoppik / Economics Letters 97 (2007) 253–259
257
discussed in Section 2. For each period τ, the factual and counterfactual distribution can be estimated for the two disjunctive parts of the domain divided by the negative of the median, −mτ: 1 X Kh ðx Xis Þ x b ms s ¼ 1 N T ; ð4Þ fˆs ðxÞ ¼ Ns Xis b m s
1 X Kh ðx Xis Þ gˆs ðxÞ ¼ Ns Xis N ms
x N ms
s ¼ 1 N T:
ð5Þ
Kh(.) denotes a kernel with bandwidth h. These partial estimates can be aggregated over periods. For each interval within the core interval, Ij, j = 2…T, aggregation is only over those periods that do provide information on the distribution in question. E.g., information on the factual distribution in interval I2 comes only from period τ = 1, whereas information on the counterfactual distribution in interval I2 comes from periods τ = 2…T. The aggregate estimators are: fˆðxÞ ¼
j1 X Ns ˆ f s ðxÞ Q s¼1 j1
xa Ij ;
j ¼ 2 N T;
ð6Þ
gðxÞ ˆ ¼
T X Ns gˆ ðxÞ Rj s s¼j
xa Ij ;
j ¼ 2 N T:
ð7Þ
In Eqs. (6) and (7) weighing is based on numbers of observations, using Qs ¼
s X s¼1
Ns and Rs ¼
T X
Ns for s ¼ 1 N T ;
ð8Þ
s¼s
where Qτ is the number of all observations of periods that only have negative nominal observations in Iτ +1 and R τ is the number of all observations of periods that only have positive nominal observations in Iτ. Because of the discontinuity of the distributions at nominal zero (i.e. at −mτ), some of the probability mass is spilled over the borders of the intervals and lost for estimation in the application of estimators (4)–(7). This causes a downward ‘discontinuity bias’ in the neighborhood of the interval borders. The simple but effective solution proposed here is to not use the period estimates where they are known to be biased. The bias is only present within a distance of b from the discontinuity, where b is equal to half of the total width of the kernel used. The practical implication of this solution is that ‘effective intervals’ have to be used for the aggregate estimation that differ from the intervals Iτ; neither are the effective intervals for the estimation of the factual and the counterfactual the same: f ¼ msþ1 b; ms b½; s ¼ 1 N T 1; Isþ1 g ¼ msþ1 þ b; ms þ b½; s ¼ 1 N T 1: Isþ1
Accordingly, the modified core interval Icb = [−mT + b, −m1 − b] is now 2b smaller than the original core interval Ic = ]−mT, −m1[. The overlap between the estimated counterfactual and factual is reduced to
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C. Knoppik / Economics Letters 97 (2007) 253–259
mT − m1 − 2b. The solution works well, if this loss of overlap is not large. The dependency of the loss of overlap on bandwidth constitutes an argument for under-smoothing. The discontinuity unbiased aggregate estimators for the counterfactual distribution g(x) and the factual distribution f(x) are: fˆðxÞ ¼
j1 X Ns ˆ f s ðxÞ xaIjf ; Q s¼1 j1
gðxÞ ˆ ¼
T X Ns gˆ ðxÞ Rj s s¼j
xaIjg ;
j ¼ 2 N T;
j ¼ 2 N T:
ð9Þ
ð10Þ
The computation of the variance of these estimates has to be adjusted accordingly.4 If the difference between the two estimates is significant, this indicates downward nominal wage rigidity. 5. Form and extent of nominal wage rigidity The rigidity function is a concept that captures the thinning effect at wage reductions of different sizes as the per cent difference between the counterfactual and factual distributions, see Beissinger and Knoppik (2001). For example, ‘proportional rigidity’ assumes a constant share of prevented wage cuts that is independent of their size; ‘threshold rigidity’ assumes a 100per cent of prevented wage cuts for wage cuts smaller than a certain threshold, and of zero per cent for bigger cuts. In other approaches, these functional forms of rigidity have to be assumed, e.g. ‘proportional rigidity’ in Kahn (1997) and ‘threshold rigidity’ in Altonji and Devereux (2000). Instead of assuming a functional form, the kernel-location approach estimates the rigidity function ρˆ(x) as qðxÞ ˆ ¼
gðxÞ ˆ fˆðxÞ for mT þ b b x b m1 b; gðxÞ ˆ
ð11Þ
and thereby helps to shed some light on the question, which functional form of rigidity is most appropriate. The average degree of downward nominal wage rigidity, measuring the proportion of notional wage cuts prevented by DNWR, can be defined as Z m1 b Z m1 b DG DF fˆðzÞdz: ð12Þ gðzÞdz; ˆ DFu qˆ ¼ ; DGu DG mT þb mT þb 6. Illustration with artificial random data Artificial random data was generated using the same specifications underlying the theoretical example discussed and illustrated in Section 2. With 500 observations in each of 10 periods the example roughly mimics the respective orders of magnitude of household panel surveys (PSID, GSOEP, BHPS) which have been used in previous analyses of DNWR. The estimation used the rectangular kernel. Its results are depicted in column b) of Fig. 1. Panels bi) to biii) show estimates of parts of the factual and counterfactual from periods t = 1, 5, 10. In each case −mτ and −mτ ± b are marked by vertical lines. 4
The variance of the estimators is discussed in the longer working paper version of this paper.
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Panels biv) and bv) show aggregate estimates of the counterfactual and factual distributions. Solid vertical lines mark the original core interval (defined by the two extreme medians), dotted vertical lines mark the effective intervals used to avoid discontinuity bias. The aggregate estimates are plotted with varying thickness reflecting the different number of underlying periods in the different subintervals. The plots illustrate the general tendency for aggregate estimates that are smoother and closer to the true model whenever more periods and therefore a larger number of observations underlie their estimation. The aggregate estimates recover the underlying true distributions quite well over most of the effective core interval and correctly point to the high degree of DNWR built into the example. Overall, the example underlines the applicability of the kernel-location approach to the relevant data sets. 7. Conclusions The new kernel-location approach to the analysis of downward nominal wage rigidity combines kernel density estimation and the identifying principle of joint variation of location and shape. The approach provides partial estimates of counterfactual and factual distributions of per cent annual nominal wage changes, of the rigidity function and of the degree of downward nominal wage rigidity. One of the advantages over earlier approaches is that the functional form of rigidity is estimated, rather than assumed. Preliminary analyses in Knoppik (2006) and Knoppik, Beissinger, and Blaes (2007) of US and German micro data find significant nominal rigidity that exhibits the proportional form proposed by Kahn (1997). References Altonji, J.G., Devereux, P.J., 2000. Is there nominal wage rigidity? Evidence from panel data. Research in Labor Economics 19, 383–431. Beissinger, T., Knoppik, C., 2001. Downward nominal rigidity in West-German earnings 1975–1995. German Economic Review 2 (4), 385–418. Card, D., Hyslop, D., 1997. Does Inflation “Grease the Wheels of the Labor Market”? In: Romer, C.D., Romer, D.H. (Eds.), Reducing Inflation — Motivation and Strategy. University of Chicago, Chicago, pp. 71–114. Kahn, S., 1997. Evidence of nominal wage stickiness from microdata. American Economic Review 87 (5), 993–1008. Knoppik, C., 2006. Downward Nominal Rigidity in US Wage Data from the PSID? An Application of the Kernel-Location Approach, University of Regensburg, Discussion Paper. Knoppik, C., Beissinger, T., 2003. How rigid are nominal wages? Evidence and implications for Germany. Scandinavian Journal of Economics 105 (4), 619–641. Knoppik, C., Beissinger, T., Blaes, B., 2007. Nonparametric Evidence on Extent and Functional Form of Downward Nominal Rigidity in Germany: An Application of the Kernel-Location Approach, University of Regensburg, Prepared for the VfS 2007 Annual Meeting. February. McLaughlin, K.J., 1994. Rigid wages? Journal of Monetary Economics 34, 383–414 (December). McLaughlin, K.J., 1999. Are nominal wage changes skewed away from wage cuts? Federal Reserve Bank of St. Louis Review 81 (3), 117–132 (May). Rodríguez-Palenzuela, D., Camba-Mendez, G., Garcia, J.A., 2003. Relevant Economic Issues Concerning the Optimal Rate of Inflation. In: European Central Bank (Ed.), Background Studies for the ECB's Evaluation of its Monetary Policy Strategy. European Central Bank, Frankfurt, pp. 91–126.
The kernel-location approach: A new non-parametric approach to the analysis of downward nominal wage rigidity in micro data ☆ Christoph Knoppik 1,2 Sozial- und Wirtschaftswissenschaftliche Fakultät, University of Bamberg, Feldkirchenstrasse 21, D-96052 Bamberg, Germany Received 22 March 2006; received in revised form 1 March 2007; accepted 15 March 2007 Available online 5 July 2007
Abstract The kernel-location approach combines kernel-density estimation and the principle of joint variation of location and shape of the distribution of wage changes, to estimate counterfactual and factual distributions of wage changes and degree and form of downward nominal wage rigidity. © 2007 Elsevier B.V. All rights reserved. Keywords: Downward nominal wage rigidity; Kernel-location approach; Kernel density estimation JEL classification: C14; E24; J30
1. Introduction Evidence on downward nominal wage rigidity (DNWR) from micro data constitutes an important input to debates on whether low inflation increases equilibrium unemployment, but as noted e.g. by RodríguezPalenzuela, Camba-Mendez, and Garcia (2003) this evidence is far from clear cut. Therefore, alternative
☆ I would like to thank Thomas Beissinger, Jürgen Heubes, Hans Ludsteck, Joachim Möller, Walter Oberhofer, and the participants of a session at the AEA 2005 annual meeting, where earlier stages of this work have been presented. 1 Affiliation at the time of submission: Prof. Dr. Christoph Knoppik, University of Bamberg, Germany. Tel.: +49 951 863 2634, e-mail: [email protected]. 2 Affiliation at the time of revision: University of Regensburg, Germany. Tel.: +49 941 943 2700, e-mail: christoph. [email protected]; url: http://www.wiwi.uni-regensburg.de/knoppik/.
0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2007.03.010
254
C. Knoppik / Economics Letters 97 (2007) 253–259
approaches to the empirical analysis of DNWR seem desirable. The general advantage of non- or semiparametric approaches in this research is to avoid functional assumptions that parametric approaches have to make. However, there are also some drawbacks. The skewness-location approach of McLaughlin (1994) yields no quantitative information on the extent of DNWR in the data. For the symmetry approach of Card and Hyslop (1997), the applicability of the key assumption of symmetry has frequently been denied, e.g. by McLaughlin (1999) for US PSID data. Finally, while the histogram-location approach of Kahn (1997) makes no assumptions on the functional form of the wage change distribution, it does stipulate a functional form for the rigidity, which is itself under debate in the literature, see e.g. Knoppik and Beissinger (2003). This paper proposes a new non-parametric approach to the analysis of downward nominal wage rigidity in micro data that uses kernel density estimates to model the factual and counterfactual distributions of wage changes, and uses the principle of joint variation of location and shape of annual distributions of per cent nominal wage changes for the identification of DNWR. While these two building blocks of analysis are separately used in other approaches, it is their combination that determines the favourable properties of the new approach. Sharing the general advantages of the non-parametric approaches, it avoids the problematic symmetry assumption and it provides a quantitative estimate of the degree of DNWR in the data without imposing a functional form of rigidity. Sections 2 and 3 explain the basic idea behind the kernel-location approach and the organization of data. Sections 4 and 5 present and interpret the estimators for the counterfactual and factual distribution, the rigidity function and the average degree of rigidity. Section 6 illustrates the approach with an application to artificial data. 2. Basic idea The kernel-location approach analyses annual cross-sectional distributions of per cent wage changes. A counterfactual distribution that would prevail in the absence of DNWR and a factual distribution that may be influenced by DNWR are distinguished. DNWR is thought to lead to a thinning over the range of negative nominal wage changes, and a corresponding pile-up at zero, so that both distributions differ for non-positive wage changes if DNWR does exist. Since the counterfactual distribution cannot directly be observed for negative nominal wage changes it cannot be compared to the observed factual distribution. However, under the assumption that the counterfactual distribution has a shape that is invariant over time and only changes its location (reflecting changes in price inflation and other influences) the mediancentered counterfactual distribution is identical over time. 3 Periods with different location provide information on different parts of the centered counterfactual and factual distributions, which can then be compared over the overlapping range. This basic idea is illustrated in column a) of Fig. 1. By assumption, there is a counterfactual distribution with time-invariant shape, there is DNWR, and there are T = 10 periods with different locations of the distribution, with medians mt in ascending order (mmin in period 1 and mmax in period 10). Panels ai) to aiii) of Fig. 1 show three median-centered theoretical distributions
This assumption of invariant shape is crucial to all approaches employing the identification principle of ‘joint variation of location and shape’. One concern is that the dispersion of the distribution is higher in times of higher inflation. Suitable standardization of the data can help in this respect. Another potential problem is that the observable median might be affected by the effects of rigidity. The use of higher quantiles as measures of location may remedy such situations. 3
C. Knoppik / Economics Letters 97 (2007) 253–259
255
Fig. 1. Theoretical and estimated distributions. Theoretical and estimated median-centered distributions by period and aggregated as discussed in text.
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C. Knoppik / Economics Letters 97 (2007) 253–259
under this set up 0(t = 1, 5, 10). In terms of median-centered changes, thinning occurs below −mt. Information on the factual distribution is available only below −mmin, whereas information on the counterfactual distribution is only available above −mT. Therefore, for the range −mmax b x b −mmin there is information on both distributions, which can be used to analyse the effects of DNWR. Only the variation in location, expressing itself in the width of this range, contributes to the identification of DNWR; this is also true of the other approaches employing the identification principle of ‘joint variation of location and shape’. Panels aiv) and av) of Fig. 1 show those parts of the aggregate counterfactual and factual distributions that can be deducted from the period information. Different amounts of information are available in the subdivisions of −mmax b x b −mmin defined by period medians. For example, between −m2 and −m1 there is information on the factual only from period 1, but there is information on the counterfactual from periods 2 to 10. The overlap of the aggregate estimates can be used to reach conclusions with respect to existence, extent and type of DNWR. 3. Organization of data The estimators can be formulated in a transparent way by organizing the data as in the example. x˜it denote observed per cent wage changes, where i runs over the cross-section dimension and t over the time dimension. N and Nt are the number of observations and the number of observations from period t. The median mt of observations x˜it from period t is used in defining median centered values Xit u x˜ it mt :
ð1Þ
It is useful to re-index observations in the time dimension, such that periods with higher time-index τ have higher medians, ms N ms V⇔ s N s V:
ð2Þ
Re-indexed annual medians of uncentered observations are used to define the intervals I2 to IT. Isþ1 ¼ msþ1 ; ms ½; s ¼ 1 N T 1:
ð3Þ
I2 to IT are parts of the ‘core interval’ IC = ]− mT, − m1[ between the lowest and the highest periodmedian in the sample. I1 = ]− m1, ∞[ and IT+1 = ]− ∞, − mT [ are intervals to the right and to the left of the core interval, with only negative nominal changes in IT+1 and only positive nominal changes in I1 in all periods. Signs of nominal wage changes do change over time in the intervals within the core interval Ic. In any period τ there are positive nominal changes in all Is, s ≤ τ and negative nominal changes in all Is, s N τ. 4. Kernel density estimators of the factual and counterfactual distribution The estimators for the distributions suitably weigh the partial period-wise kernel density estimates of median-centered factual and counterfactual distributions in order to obtain overlapping partial estimates of the aggregate factual and counterfactual distributions. Before more refined estimators are proposed that avoid the danger of discontinuity bias, in a first step preliminary estimators are formulated. Information from different periods requires separate treatment since different parts of factual and counterfactual distribution can be estimated in each period and contribute differently to the aggregate, as
C. Knoppik / Economics Letters 97 (2007) 253–259
257
discussed in Section 2. For each period τ, the factual and counterfactual distribution can be estimated for the two disjunctive parts of the domain divided by the negative of the median, −mτ: 1 X Kh ðx Xis Þ x b ms s ¼ 1 N T ; ð4Þ fˆs ðxÞ ¼ Ns Xis b m s
1 X Kh ðx Xis Þ gˆs ðxÞ ¼ Ns Xis N ms
x N ms
s ¼ 1 N T:
ð5Þ
Kh(.) denotes a kernel with bandwidth h. These partial estimates can be aggregated over periods. For each interval within the core interval, Ij, j = 2…T, aggregation is only over those periods that do provide information on the distribution in question. E.g., information on the factual distribution in interval I2 comes only from period τ = 1, whereas information on the counterfactual distribution in interval I2 comes from periods τ = 2…T. The aggregate estimators are: fˆðxÞ ¼
j1 X Ns ˆ f s ðxÞ Q s¼1 j1
xa Ij ;
j ¼ 2 N T;
ð6Þ
gðxÞ ˆ ¼
T X Ns gˆ ðxÞ Rj s s¼j
xa Ij ;
j ¼ 2 N T:
ð7Þ
In Eqs. (6) and (7) weighing is based on numbers of observations, using Qs ¼
s X s¼1
Ns and Rs ¼
T X
Ns for s ¼ 1 N T ;
ð8Þ
s¼s
where Qτ is the number of all observations of periods that only have negative nominal observations in Iτ +1 and R τ is the number of all observations of periods that only have positive nominal observations in Iτ. Because of the discontinuity of the distributions at nominal zero (i.e. at −mτ), some of the probability mass is spilled over the borders of the intervals and lost for estimation in the application of estimators (4)–(7). This causes a downward ‘discontinuity bias’ in the neighborhood of the interval borders. The simple but effective solution proposed here is to not use the period estimates where they are known to be biased. The bias is only present within a distance of b from the discontinuity, where b is equal to half of the total width of the kernel used. The practical implication of this solution is that ‘effective intervals’ have to be used for the aggregate estimation that differ from the intervals Iτ; neither are the effective intervals for the estimation of the factual and the counterfactual the same: f ¼ msþ1 b; ms b½; s ¼ 1 N T 1; Isþ1 g ¼ msþ1 þ b; ms þ b½; s ¼ 1 N T 1: Isþ1
Accordingly, the modified core interval Icb = [−mT + b, −m1 − b] is now 2b smaller than the original core interval Ic = ]−mT, −m1[. The overlap between the estimated counterfactual and factual is reduced to
258
C. Knoppik / Economics Letters 97 (2007) 253–259
mT − m1 − 2b. The solution works well, if this loss of overlap is not large. The dependency of the loss of overlap on bandwidth constitutes an argument for under-smoothing. The discontinuity unbiased aggregate estimators for the counterfactual distribution g(x) and the factual distribution f(x) are: fˆðxÞ ¼
j1 X Ns ˆ f s ðxÞ xaIjf ; Q s¼1 j1
gðxÞ ˆ ¼
T X Ns gˆ ðxÞ Rj s s¼j
xaIjg ;
j ¼ 2 N T;
j ¼ 2 N T:
ð9Þ
ð10Þ
The computation of the variance of these estimates has to be adjusted accordingly.4 If the difference between the two estimates is significant, this indicates downward nominal wage rigidity. 5. Form and extent of nominal wage rigidity The rigidity function is a concept that captures the thinning effect at wage reductions of different sizes as the per cent difference between the counterfactual and factual distributions, see Beissinger and Knoppik (2001). For example, ‘proportional rigidity’ assumes a constant share of prevented wage cuts that is independent of their size; ‘threshold rigidity’ assumes a 100per cent of prevented wage cuts for wage cuts smaller than a certain threshold, and of zero per cent for bigger cuts. In other approaches, these functional forms of rigidity have to be assumed, e.g. ‘proportional rigidity’ in Kahn (1997) and ‘threshold rigidity’ in Altonji and Devereux (2000). Instead of assuming a functional form, the kernel-location approach estimates the rigidity function ρˆ(x) as qðxÞ ˆ ¼
gðxÞ ˆ fˆðxÞ for mT þ b b x b m1 b; gðxÞ ˆ
ð11Þ
and thereby helps to shed some light on the question, which functional form of rigidity is most appropriate. The average degree of downward nominal wage rigidity, measuring the proportion of notional wage cuts prevented by DNWR, can be defined as Z m1 b Z m1 b DG DF fˆðzÞdz: ð12Þ gðzÞdz; ˆ DFu qˆ ¼ ; DGu DG mT þb mT þb 6. Illustration with artificial random data Artificial random data was generated using the same specifications underlying the theoretical example discussed and illustrated in Section 2. With 500 observations in each of 10 periods the example roughly mimics the respective orders of magnitude of household panel surveys (PSID, GSOEP, BHPS) which have been used in previous analyses of DNWR. The estimation used the rectangular kernel. Its results are depicted in column b) of Fig. 1. Panels bi) to biii) show estimates of parts of the factual and counterfactual from periods t = 1, 5, 10. In each case −mτ and −mτ ± b are marked by vertical lines. 4
The variance of the estimators is discussed in the longer working paper version of this paper.
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Panels biv) and bv) show aggregate estimates of the counterfactual and factual distributions. Solid vertical lines mark the original core interval (defined by the two extreme medians), dotted vertical lines mark the effective intervals used to avoid discontinuity bias. The aggregate estimates are plotted with varying thickness reflecting the different number of underlying periods in the different subintervals. The plots illustrate the general tendency for aggregate estimates that are smoother and closer to the true model whenever more periods and therefore a larger number of observations underlie their estimation. The aggregate estimates recover the underlying true distributions quite well over most of the effective core interval and correctly point to the high degree of DNWR built into the example. Overall, the example underlines the applicability of the kernel-location approach to the relevant data sets. 7. Conclusions The new kernel-location approach to the analysis of downward nominal wage rigidity combines kernel density estimation and the identifying principle of joint variation of location and shape. The approach provides partial estimates of counterfactual and factual distributions of per cent annual nominal wage changes, of the rigidity function and of the degree of downward nominal wage rigidity. One of the advantages over earlier approaches is that the functional form of rigidity is estimated, rather than assumed. Preliminary analyses in Knoppik (2006) and Knoppik, Beissinger, and Blaes (2007) of US and German micro data find significant nominal rigidity that exhibits the proportional form proposed by Kahn (1997). References Altonji, J.G., Devereux, P.J., 2000. Is there nominal wage rigidity? Evidence from panel data. Research in Labor Economics 19, 383–431. Beissinger, T., Knoppik, C., 2001. Downward nominal rigidity in West-German earnings 1975–1995. German Economic Review 2 (4), 385–418. Card, D., Hyslop, D., 1997. Does Inflation “Grease the Wheels of the Labor Market”? In: Romer, C.D., Romer, D.H. (Eds.), Reducing Inflation — Motivation and Strategy. University of Chicago, Chicago, pp. 71–114. Kahn, S., 1997. Evidence of nominal wage stickiness from microdata. American Economic Review 87 (5), 993–1008. Knoppik, C., 2006. Downward Nominal Rigidity in US Wage Data from the PSID? An Application of the Kernel-Location Approach, University of Regensburg, Discussion Paper. Knoppik, C., Beissinger, T., 2003. How rigid are nominal wages? Evidence and implications for Germany. Scandinavian Journal of Economics 105 (4), 619–641. Knoppik, C., Beissinger, T., Blaes, B., 2007. Nonparametric Evidence on Extent and Functional Form of Downward Nominal Rigidity in Germany: An Application of the Kernel-Location Approach, University of Regensburg, Prepared for the VfS 2007 Annual Meeting. February. McLaughlin, K.J., 1994. Rigid wages? Journal of Monetary Economics 34, 383–414 (December). McLaughlin, K.J., 1999. Are nominal wage changes skewed away from wage cuts? Federal Reserve Bank of St. Louis Review 81 (3), 117–132 (May). Rodríguez-Palenzuela, D., Camba-Mendez, G., Garcia, J.A., 2003. Relevant Economic Issues Concerning the Optimal Rate of Inflation. In: European Central Bank (Ed.), Background Studies for the ECB's Evaluation of its Monetary Policy Strategy. European Central Bank, Frankfurt, pp. 91–126.