New Keynesian Phillips Curve Estimation: The Case of Hungary (1981-2006)
Aleksandar Vasilev American University in Bulgaria avasilev@aubg. edu
This paper investigates for the presence of a New Keynesian Phillips (nkpc) curve in Hungary in the period 1981:3-2006:2. The empirical model we test features forward-looking firms who pre-set prices for a couple of periods ahead, using Calvo (1983) pricing rule. We also estimate a hybrid version of nkpc, where some of the firms are backward looking, and others are forward-looking in their price-setting behaviour. Real marginal costs and forward-looking behaviour are statistically significant and quantitatively important in the nkpc. However, there are some econometric issues to be considered, such as the weak identification of the parameters of the structural nkpc as well as those of the hybrid nkpc.
Key Words: New Keynesian Phillips curve, Hungary, instrumental
non-linear gmm Estimation, weak identification jel Classification: C22, C2, E24
Introduction
This paper investigates for the presence of a New Keynesian Phillips (nkpc) curve in Hungary in the period 1981:3-2006:2. Hungary is a unique case among the transition economies as a country that traded freely with Western European countries even before the fall of the socialist regime, and thus is an interesting case of study. Under that regime, export firms had to use market prices in order to be competitive and gain market share in Western Europe. In that sense, we can regard the behaviour of exporting firms as closely resembling the behaviour of a profit-maximizing Western firm operating in a competitive environment. Therefore, we will adopt models developed for the us to study the dynamics of inflation in this transition country.
Given the enormous literature on the subject, the paper will not provide a detailed overview of the topic; instead, the interested reader is referred to the recent study in Vasicek (2011) and the references therein. The study follows the methodology proposed by Gali and Gertler (1999), who claim that a potential source of inflation may be the sluggish adjustment of real marginal costs to movements in output. The empirical model
Managing Global Transitions 13 (4): 355-367
356 Aleksandar Vasilev
tested features forward-looking firms who pre-set prices for a couple of periods ahead, using Calvo (1983) pricing rule. In addition, measures of real marginal cost are used instead of the old-fashioned output gap. The reason is that marginal costs are a better proxy for the impact of the productivity gains on inflation, which the ad hoc measure output gap misses. A hybrid version of nkpc, where some of the firms are backward looking, and others are forward-looking in their price-setting behaviour, is also estimated.
Despite the presence of a substantive literature on the subject of nkpc in Hungary (Menyhert 2008; Vasicek 2011; Franta, Saxa, and Smidkova 2010), earlier studies either take a much shorter time span (Menyhert 2008; Vasicek 2011), or focus on inflation persistence (Franta, Saxa, and Smidkova 2010). In this paper the emphasis is on the transition experience of Hungarian economy (hence the time period that is chosen), and not on inflation forecasting. In addition, the paper touches upon the problem of weak identification, which previous studies do not discuss. Therefore, given the different focus of the paper, the results from earlier studies are not directly comparable.
The paper is organized as follows: the second section describes Gali and Gertler's (1999) approach and, thus provides a brief review of the theory that gave rise to the new Phillips curve, and discusses some existing empirical results. The third Section contains the estimates of the new Phillips curve. In the fourth section, the model is extended to allow for backward-looking firms and results of a so-called 'hybrid Phillips curve' are presented. The fifth section concludes.
The New Phillips Curve: Background Theory and Evidence
The setup of the model features monopolistically competitive firms who face some constraints on price adjustments. The price adjustment rule is time-dependent - every period a fraction 1/X of firms set their prices for X periods ahead in the spirit of Taylor (1980). In order to keep track of the histories of all firms we use Calvo pricing (1983) rule, which simplifies the aggregation problem: in any given period, each firm has a fixed probability 1 - d that it may adjust its price during that period. Therefore, the average time over which a price is fixed is gjvenby(1-0)Zto kOk-1 = 1/(1-0).
Another common assumption is that the monopolistically competitive firm faces a constant price elasticity of demand curve. Then, Gali and Gertler (1999) show that the aggregate price level pt evolves as a convex combination of the lagged price level pt-1 and the optimal reset price pt *
Managing Global Transitions
New Keynesian Phillips Curve Estimation 357
(the price selected by firms that are able to change the price at period t). Therefore, the pricing rule takes the following form:
Pt = dp- + (1 - d)pt *.	(1)
Let mc1l be the firm's marginal costs (as a percentage deviation from the steady state) and ¡3 denotes the discount factor. Each firm chooses a price at t to maximize expected discounted profits subject to the Calvo pricing rule, so the optimal reset price is:
TO
pt * = (1 - ¡36)YJ(fiO)kEt {mcnt+k}.	(2)
k=0
Now let nt = ptpt-1 denote the inflation rate. Combining (1) and (2), Gali and Gertler (1999) obtain the following equation for the inflation dynamics, or the 'traditional forward-looking New Keynesian Phillips curve.'
nt = Amct + ¡3Et {nt+i},	(3)
where A = (1-0)(1-fi0)/0 depends on the frequency of price d adjustment and the discount factor ¡3. Iterating forward for inflation they obtain
TO
nt = A^^Et{mct+k} < to.	(4)
k=0
Therefore, the theory says inflation is a discounted stream of expected future marginal costs. Note that the sum above is finite due to the discounting effect and the assumption that marginal costs are bounded in each time period.
Traditional Phillips curve emphasizes the use of a proxy for real activity, namely the 'output gap,' or the observed gdp series less of a trend. In other words, this is a measure, which shows how current gdp differs from the potential one. It is obtained by taking logs from the series, seasonally adjusting the quarterly series, differencing to eliminate the unit root and applying the Hodrick-Prescott filter so that we express it as a percentage change from the steady state. Thus, mct = kxt, where k is the elasticity of the marginal cost. Plugging the expression above into the inflation equation, we obtain
nt = Akxt + fiEt {nt+i}.	(5)
Volume 13 • Number 4 • Winter 2015
358 Aleksandar Vasilev
Substituting forward, the resulting expression becomes
TO
nt = ÄkJ] ßkEt {mc^j.	(6)
k=0
It is widely known fact that conventional measures of the output gap contain a substantial amount of measurement errors. That is primarily because the theoretical measure of 'natural level' of output is not an observable. The gap is estimated by fitting a smooth deterministic trend and subtracting it from the series. This trend fitting itself involves measurement error. Depending on whether supply or demand shocks are predominant in the economy, estimation could lead to counter-intuitive signs of the coefficients.
Gali and Gertler (1999) concentrate on obtaining a measure for real marginal costs, estimated in a way that it is consistent with theory. Their theory is used as a guide for the estimation in this paper: Output is assumed to be produced by A Cobb-Douglas production function, Yt = AtK?kNfn, where At denotes total factor productivity, Kt capital, and Nt labour. Real marginal cost (mo) is the ratio of the real wage to the marginal product of labour (mpl). Thus, Mct = (Wt/Pt)(dYt/dNt) = St/an, where St = (WtNt/PtYt is the labour income share. Using lowercase letters to denote percent deviation from the steady state, the formula becomes mct = st. That measure is obtained by first taking natural logs from the series and then applying the Hodrick-Prescott filter to it. This series, as well as the series for inflation, is stationary: Dickey-Fuller test rejects the presence of a unit root at 1% level of significance.
After plugging the expression for real mo into the inflation equation, we obtain
nt = Äst + ßEt {nt +1/.	(7)
Since this is a rational expectations (re) model, the forecast of nt+1 is uncorrelated with any of the variables in the information set, i.e. variables in time t or earlier. This leads to the following moment condition
Et {(nt - Äst - ßnt+i)zt} = 0,	(8)
where zt is a vector composed of the variables taken from the information set, which are orthogonal to the inflation surprise. The moment condition above is used to estimate the model using the Generalized Method of Moments (gmm).
Managing Global Transitions
New Keynesian Phillips Curve Estimation 359
table 1 Reduced-Form Estimates
Proxy for real mc	ß	A	/-statistic	^-value
LN_ULC_HP	0.1247 (0.1494)	-0.0002 (0.0007)	9.793	0.2006
ln_shlaböe_hp	0.1135 (0.1320)	0.0250 (0.0654)	8.9113	0.2591
DLGDP_SA_HP	0.1321 (0.1453)	-0.0642 (0.1227)	9.1982	0.2591
notes N = 100, df = 7.
An important reason why gmm estimation is used is that non-linear least squares (nlls) will give biased and inconsistent estimates since corr(nt, nt+1) + o, and thus corr(et, nt+1), which violates one of the underlying assumptions for using nlls. Note that using nlls-iv estimation with homoscedasticity assumption and no autocorrelation yields exactly the gmm orthogonality condition.
The data used is quarterly for Hungary over the period 1981:3-2006:2. Estimation results are presented in the next section. For st, natural logarithm of the labour income share is used. Inflation is measured as a percentage change in the Consumer Price Index (cpi), seasonally adjusted and differenced in order to eliminate the unit root in the series. The instrument set includes four lags of inflation, the labour income share, the output gap, the long-short interest rate spread, wage inflation and the growth in money supply (mi aggregate).
The New Phillips Curve: Estimation
We first estimate the reduced from equation, which involves only A and S, but not the structural parameter d, which was the measure of price rigidity. In addition, Appendix 1 checks the identification of the model parameters. Three cases are considered, with log of cyclical component of unit labour costs (ln_ulc_hp), log of cyclical component of the labour share (ln_shlabor_hp), and the differenced log of seasonally adjusted output gap (dlgdp_sa_hp), respectively, as a proxy for real marginal costs. Results are provided in table 1, where the Newey-West estimate of the covariance matrix was used to provide robust standard errors.
Neither the coefficient on the real marginal costs, nor the estimate of the discount factor S is statistically significant. The last result, however, is in line with Gali and Gertler's (1999) findings for us: using output gap
Volume 13 • Number 4 • Winter 2015
360 Aleksandar Vasilev
table 2 Estimates of the New Keynesian Phillips Curve
Mc proxy used		e	ß	A	J-stat	^-value
LN_ULC_HP	(l)	1.0002	0.l24l	-0.0002	9.7527	0.203
		(0.0008)	(0.1493)	(0.0007)		
	(2)	1.0000	0.l247	0	9.7931	0.2006
		(0.0001)	(0.1494)	(0.0000)		
LN_SHLABOR_HP	(l)	0.2963	2.6178	0.1579	8.3573	0.3021
		(0.0908)	(0.8581)	(0.0527)		
	(2)	0.9969	0.1137	0.027	8.9167	0.2587
		(0.0093)	(0.1321)	(0.0001)		
DLGDP_SA_HP	(l)	0.4023	1.3682	0.2687	9.7798	0.2014
		(0.0604)	(0.2757)	(0.0455)		
	(2)	l.0096	0.1368	-0.0083	9.1927	0.2391
		(0.0273)	(0.1453)	(0.0005)		
notes (1) Case 1. (2) Case 2. N = 100, df = 7.
should not generate a nkpo when quarterly data was used. In order to recover the structural estimate of d non-linear instrumental gmm was also estimated. Fuhrer and Moore (1995) show that in small samples gmm is sensitive to the nature of normalization of the orthogonality conditions. In this paper the ones used by Gali and Gertler (1999) are used:
Et{(tort - (1 - 0)(i - Pe)st - efiirt+1)zt}	(9)
and
Et{(nt - 0-1(i - 0)(i - ffist - pnt+1)zt}	(10)
Their claim is that (9) minimizes non-linearities, while in (10) the coefficient of inflation in the current period is restricted to be one. We do each specification for (log) labour share, (log) unit labour costs and output gap.
The results are reported in table 2, where cases [1] and [2] denote specifications (9) and (10), respectively. The first two columns give the estimates of the structural parameters d and 3, and the third provides the estimate for A. Standard errors for A were obtained using the delta method. /-statistic for over-identifying restrictions is also provided. At 5% level of significance, the model is always correctly specified.
The two specifications yield some heterogeneity in the results: the esti-
Managing Global Transitions
New Keynesian Phillips Curve Estimation 361
mate of 6 is unity (all the firms adjust), 0.3 in the case oflog-labour income share, and 0.4 in the regression with the output gap. The estimates for A and S are in the majority of the cases not statistically different from zero.
Generally, estimates are very sensitive to the gmm normalization procedure, and sometimes to the initial values chosen. The problem was that the program gives highly negative and statistically significant S, which is in conflict with the economic logic. The reason is that the reduced form model is identified, while the structural one is not. The latter has multiple solutions, and that is formally shown in the appendix. Therefore doing Continuous Updating (cu) will not solve the problem. Using Maximum Likelihood Estimation (mle) is also of no help since the identification issue is not solved. Mavroeidis (2007) points out that Wald and lr test are not robust to failure of the identification assumption. That is a serious issue to be considered for all Neo-Keynesian economists who have nkpc equation in their models. In a very recent working paper, Boug, Cappelen, and Swensen (2007) show that the estimate surface is flat; this finding is a sign of a weak identification. Hendry (2004) also advises that nkpc specification be used with caution.
In the other camp, Martins and Gabriel (2005) try to save the model by using Generalized Empirical likelihood. Stock and Wright (2000) develop confidence set estimation to fix weak identification. They admit, however, encountering problems with fixing Wald statistics. It is worth noting that Gali and Gertler (1999) do not discuss this econometric problem. They only mention several other reasons that may cause the estimate of 6, to have an upward bias. The first one is statistical: our measures of the real marginal cost are just proxies, and thus contain measurement error. Thus, the parameter A is biased towards zero, and appears insignificant, while in reality mc is an important factor for determining inflation. The second reason lies in the theory, which serves as a basis for the model. It assumes a constant mark-up of prices over mc. If mark-up is allowed to vary over the business cycle, however, then price setting becomes less sensitive to mc, and this explains why A is not statistically significant as well. In a recent paper, Gali, Gertler, and Lopez-Salido (2005) still claim their results are robust, again failing to mention the identification issue.
In the next section, an alternative, called 'hybrid' nkpc, is considered. It is a more sophisticated model of inflation dynamics. Unfortunately, much of the criticism in the paragraphs above is relevant for the hybrid version, as the problem of weak identification is even bigger in that specification.
Volume 13 • Number 4 • Winter 2015
362 Aleksandar Vasilev
Hybrid Phillips Curve
Inflation in data features a significant amount of inertia. Thus, in this section we extend the basic Calvo model, and allow for inertia in inflation. Now the environment includes two groups of firms - not only forward-looking, but also backward-looking ones. The latter use a rule of thumb (behave in an adaptive way) when setting prices. In this case, we can see what share of firms is not optimizing, and therefore not acting rationally.
We the share of the backward-looking firms is denoted by <. The aggregate price level now evolves according to the following formula
pt = 6pt+ (1 - 9)pt*,	(11)
where pt* is an index of the prices that were reset in period t. Let p[ denote the price set by a forward-looking firm at t and ptt the price set by a backward-looking firm. Then the index of the newly set prices may be
expressed as
*
pt * = (1 - o)pft + upht.	(12)
Accordingly, pft may be expressed as
TO
pt = (1 - ßd)J](ßd)kEt {mc^j.	(13)
k=0
Gali and Gerler derive a rule based on the recent pricing behaviour of the competitors as
Pt=Pt-i* + nt-1.	(14)
Then they obtain the hybrid Phillips curve by combining (13) and (14),
nt = Amct + Yf E {nt+i} + Jb^t-i,	(15)
where A = (1 - <)(i - 0)(i - ¡30)(p-\ jf = ¡3Q(p-\ jb = <(p-\ and (p = Q + <[1 - 6(1 -£)].
Note that when < = 0, this means that all the firms are forward-looking, and we are back to the nkpc. While the reduced form in this case is identified, the hybrid nkpc is adds another dimension of non-linearity and makes the identification problem even more severe.
Next, we provide the estimates of the empirical hybrid nkpc and evaluate its overall performance. Log labour share is again used as a measure of mc. To check for robustness, the regression is run with unit labour
Managing Global Transitions
New Keynesian Phillips Curve Estimation 363
table 3 Hybrid nkpc Reduced-Form Estimates
Mc proxy used	yf	yt	A	J-stat	^-value
LN_ULC_HP	0.0575 (0.1733)	0.0946 (0.1073)	-0.0023 (0.0007)	8.6720	0.1929
ln_shlabör_hp	0.1182 (0.1693)	0.1278 (0.1192)	-0.1543 (0.0919)	7.7079	0.2603
DLGDP_SA_HP	0.1828 (0.1522)	0.064 (0.1230)	0.2551 (0.2433)	8.5487	0.2388
notes N = 100, df = 6.
costs and output gap as well. Appendix 2 checks the identification of the model parameters. In this case, the model takes the following form
nt = Ast + 7/ Et (nt+1) + n^t-i + £t.	(16)
As seen from table 3, the gamma coefficients are not significant, while lambda estimates are. However, their sign is negative, which makes no economic sense. Still, the /-test does no reject the null of correct specification.
The paper then proceeds with the structural estimation procedure using again non-linear instrumental gmm estimator. As in the previous sections, two alternatives are presented, where the first specification minimizes non-linearities, while the second restricts the coefficient of inflation in the current period to one.
Et{(0fft - (1 - W)(1 - 0)(i - P6)st - OpntM = 0.	(17)
Et ((nt - (1 - w)(i - 0)(i - mr'st - ep4>~int+i)zt} = 0.	(18)
Results are provided in table 4, where [1] and [2] denote specifications (17) and (18), respectively. The automatic choice of the Newey-West co-variance matrix provided robust standard errors.
The estimate of d is almost everywhere 1, except for the case where the output gap is used, where it is 0.64. All other coefficients are not significant, with the exception for the regression with unit labour cost. That equation, however, gives puzzling results because the share of forward-looking firms is negative, which makes no economic sense. Still, the /-test confirms that the model is correctly specified.
The effect of the output gap was also found to be zero by Roberts (1997; 1999) when quarterly data are used, while Fuhrer (1997) obtains a signif-
Volume 13 • Number 4 • Winter 2015
364 Aleksandar Vasilev
table 4 Estimates of the New Hybrid Phillips Curve
MC	proxy	«	e	ß	yf	yt	A	/-stat	^-value
(a)	(1)	0.3580	1.0010	-0.3517	0.4093	-0.4025	-0.0008		
		(0.1319)	(0.0008)	(0.1279)	(0.1277)	(0.1319)	(0.0000)	8.25	0.22
	(2)	0.7018	1.0054	0.9465	0.4204	0.5700	-0.0005		
		(0.6067)	(0.0310)	(0.8394)	(0.1202)	(0.0399)	(0.0017)	7-59	0.27
(b)	(1)	0.3792	0.9763	-0.3439	0.4420	-0.3913	0.0199		
		(0.1273)	(0.0805)	(0.1031)	(0.1034)	(0.1292)	(0.0794)	8.15	0.23
	(2)	0.0611	1.0140	0.0575	0.0601	0.0573	-0.0129		
		(0.1339)	(0.0737)	(0.1071)	(0.1142)	(0.1068)	(0.0735)	8.63	0.20
(c)	(1)	0.1593	0.6384	-0.1120	0.2327	-0.1044	0.5920		
		(0.0970)	(0.1034)	(0.1544)	(0.153)	(0.0975)	(0.1044)	8.59	0.20
	(2)	0.0428	0.9846	0.0822	0.0433	0.0819	0.0149		
		(0.1264)	(0.1485)	(0.1302)	(0.1292)	(0.0808)	(0.0961)	8.46	0.21
NOTES (a) LN_ULC_HP. (b) LN_SHLABOR_HP. (c) DLGDP_SA_HP. (l) Case 1. (2) Case 2. N = 100, df = 6.
icant effect of the output gap in a model with many restrictions. One explanation, aside from the identification issue, is that compared to the us, Hungary is a small open economy, so firms take international prices as given. In a regime of free trade, those firms have to adjust quickly and act in a very competitive environment, as compared to the us firms, which may be acting indeed as monopolistically competitive producers and can afford to run a band of inaction. Indeed, the degree of backwardness is not statistically different from 0, and mark-up is seriously squeezed (theoretically equals the transportation costs of the foreign import companies).
Conclusion
This paper investigated for the presence of a New Keynesian Phillips (nkpc) curve in Hungary in the period 1981:3-2006:2. The study followed the methodology proposed by Gali and Gertler (1999), who claim that a potential source of inflation may be the sluggish adjustment of real marginal costs to movements in output. The empirical model tested featured forward-looking firms who pre-set prices for a couple of periods ahead, using Calvo (1983) pricing rule. In addition, measures of real marginal cost were used instead of the old-fashioned output gap. The reason was that marginal costs are a better proxy for the impact of the productivity gains on inflation, which the ad hoc measure output gap misses.
Managing Global Transitions
New Keynesian Phillips Curve Estimation 365
A hybrid version of nkpc, where some of the firms are backward looking, and others are forward-looking in their price-setting behaviour, was also estimated. However, there are some econometric issues to be considered, such as the weak identification of the parameters of the structural nkpc as well as those of the hybrid nkpc.
References
Boug, P., A. Cappelen, and A. Swensen 2007. 'The New Keynesian Phillips Curve Revisited.' Discussion Paper 500, Statistics Norway, Research Department, Kongsvinger.
Calvo, G. A. 1983. 'Staggered Prices in a Utility Maximizing Framework.' Journal of Monetary Economics 12 (3): 383-98.
Franta, M., B. Saxa, and K. Smidkova 2010. 'The Role of Inflation Persistence in the Inflation Process in the New eu Member States.' Czech Journal of Economics and Finance 60 (6): 480-500.
Fuhrer, J. C. 1997. 'The (Un)Importance of Forward-Looking Behavior in Price Specifications.' Journal of Money, Credit and Banking 29 (3): 33850.
Fuhrer, J. C., and G. Moore. 1995. 'Inflation Persistence.' Quarterly Journal ofEconomics 440:127-59.
Gali, J., and M. Gertler. 1999. 'Inflation Dynamics: A Structural Estimation Analysis.' Journal of Monetary Economics 44 (2): 195-222.
Gali, J., M. Gertler, and D. Lopez-Salido. 2005. 'Robustness of the Estimates of the Hybrid New Keynesian Phillips Curve.' Journal of Monetary Economics 52 (6): 1107-1118.
Hendry, D. F. 2004. 'Unpredictability and the Foundations of Economic Forecasting.' Economics Papers 2004-W15, Economics Group, Nuffield College, University of Oxford, Oxford.
Martins, L. F., and V. Gabriel 2006. 'Robust Estimates of the New Keynesian Phillips Curve.' Discussion Papers in Economics 02/06, University of Surrey, Guildford.
Mavroeidis, S. 2007. 'Testing the New Phillips Curve Without Assuming Identification.' Working paper, Brown University, Providence, ri.
Menyhert, B. 2008. 'Estimating the Hungarian New Keynesian Phillips Curve.' Acta Oeconomica 58 (3): 295-318.
Roberts, J. M. 1997. 'Is Inflation Sticky?' Journal of Monetary Economics 39
(2): 173-96.
-. 1998. 'Inflation Expectations and the Transmission of Monetary
Policy.' Mimeo, Federal Reserve Board, Washington, dc.
Stock, J., and J. H. Wright 2000. 'gmm with Weal Identification.' Econo-metrica 68 (5): 1055-96.
Volume 13 • Number 4 • Winter 2015
366 Aleksandar Vasilev
Taylor, J. B. 1980. 'Aggregate Dynamics and Staggered Contracts.' Journal
of Political Economy 88 (1): 1-23. Vasicek, B. 2011. 'Inflation Dynamics and the New Keynesian Phillips Curve in Four Central European Countries.' Emerging Markets Finance and Trade 47 (5): 71-100.
Appendix 1 New Keynesian Phillips Curve Identification
We want to show whether E(gt(6)) = 0 only at 6 = 60, where 6 = (A =
(1 - 8)(1 -p8)/p8).
We need to consider two sub-cases:
1.	The reduced-form case
gt(6) = zu(nt - Ast -pEtnt+i + A0st + PoEtnt+i - Aost -PoEtnt+i) = z^st - z^(A - Ao)st - z^(p -Po)Etnt+i. Therefore, E(g(6)) = 0iff A = A0 andp = po. The reduced from model is identified.
2.	The structural parameter case
Here, E(g(6)) = 0iffp = po and (1 - 8)(1 -p9)/9 = (1 - ^(1 -Po&o)/»o. By assumption 0o > 0 (some of the firms always adjust). Therefore, 60(1 - po - e + pe2) = 8(1 - podo - do + poOl,), or do -peOo - eOo -pe2Oo = e -poOoO - OoO -p0oo0e. Cancelling equal terms on both sides, we obtain: Oope2Oo = 8 + poO1O. Imposingp = po, we obtain: Oo -po828o = 8 -poe\8. Thus, (8 - eo)(1 + poeeo), which holds when 8 = Oo U 8 = -(1/poOo).
The second possibility creates a problem in the sense that the structural model is not identified - the t-statistics are not normally distributed.
Appendix 2 Hybrid New Keynesian Phillips Curve Identification
1. The reduced-form case
gt(6) = zu(nt - Ast - YfEtnt+i - Jb^t-i + Aost + yfEtnt+i + Jbo^t-i - Aost - JfEtn(t + 1) - Jbo^t-i) = zust - zu(A - Ao)st - z^(jf - jfo)Etnt+i - z^(jb - Jbo)nt-i. Again, E(g(6)) = 0iff A = Ao, jf = jf o and jb = Jbo. The reduced form is identified.
2. The structural parameter case: Here, E(gt(<5)) = oiff A Yf KYb)

e0+M0[i-e0(i-ß0)}
Ao
Yf 0 KYbo
Managing Global Transitions
New Keynesian Phillips Curve Estimation 367
Note that the derivations for nkpc correspond to a specification with m = 0, and it was not identified. Now we allow for additional layer of non-linearity, therefore this model is not identified either and we can prove this using Monte Carlo simulations.
This paper is published under the terms of the Attribution-NonCommercial-NoDerivatives 4.0 International (cc by-nc-nd 4.0) License (http://creativecommons.org/licenses/by-nc-nd/4.o/).
Volume 13 • Number 4 • Winter 2015