A robust examination of the discount factor on Expected Inflation, the Output Gap and Endogeneity in the ‘new’ Phillips-Curve: Evidence from four economic regions

A robust examination of the discount factor on Expected Inflation, the

Output Gap and Endogeneity in the ‘new’ Phillips-Curve: Evidence

from four economic regions

University of Groningen

Frank Heusinkveld S1831577

Abstract

This paper investigates the ´new´ Phillips-Curve (NPC) for the United States, Europe, Canada and Japan. In particular, three aspects of the NPC are investigated. First, the assumption of a unit discount factor on expected inflation is challenged. Second, the recent empirical failure of a short run trade-off between inflation and the output gap in the NPC is re-examined. Third, the assumption of endogenous regressors is put to the test. Various VAR forecasts are employed to measure expected inflation and serve as a robustness check for the GMM and OLS results. I find that independent of the proxy for expected inflation, the discount factor in the United States, Europe and Canada is on average equal to one, in contrast to Japan where it was found to be significantly lower. The hypothesized short run trade-off between inflation and the output gap in the NPC seems to work reasonably well in Canada, as opposed to the United States, Europe and Japan. None of the economic regions can support the assumption of endogenous regressors on the basis of a formal test. OLS estimation suggests that the discount factor differs significantly from one in all regions. The main implication of this paper is that a correctly specified NPC may require a country-specific framework. The NPC for Japan supports this notion and may be best explained in terms of a ‘liquidity trap’, in which the Japanese economy may currently reside.

JEL classification: C26, C32, E31, E37

1. Introduction

In the early 20th_{century, Fisher (1926) conducted the first statistical investigation of the}

relation between unemployment and price changes. He inferred that increasing prices are likely to result in a decrease of unemployment. Remarkably though, it took 32 years before the problem was re-examined by Phillips (1958) and became known as the Phillips-Curve (PC). The PC refers to the supposed inverse relation between inflation and unemployment. The existence of a PC implies that monetary policy can determine what rate of inflation would be necessary to arrive at the desired level of unemployment. Alternatively it provides insights on the rate of inflation at a given level of unemployment. However, the oil crisis of the 1970’s led to a situation of simultaneous high inflation and high unemployment, which could not occur according to the PC. The occurrence of stagflation made it clear that the supposed trade-off did not hold in the long run. The notion was introduced that in the long run only one unemployment rate could ultimately prevail, commonly referred to as the natural rate of unemployment.1The implication of a natural unemployment rate is that monetary policy can only influence unemployment temporarily by permanently increasing prices, since in the long run the unemployment rate will converge back to its natural rate.

In the traditional short run PC, current inflation is related to a cyclical indicator such as unemployment or an output gap plus lags of inflation.2Two basic concerns have led empirical research to re-examine this specification. First, the stability of the traditional PC across policy regimes was considered questionable. Lucas (1976) argued that the lags of inflation may very well embed expectations of future inflation and that a stable PC should incorporate expectations of future inflation derived from factors such as firm preferences and technological constraints. Second, the observed low levels of inflation and high positive output gaps in the late 1990’s have caused the traditional PC to over-predict inflation. The second concern is inherently related to the first, since it involves the stability of the PC over time. These two concerns have led to an empirical investigation of an expectations-augmented Phillips-Curve. In effect, the expectations-augmented Phillips-Curve is generally referred to as the ‘new’ Phillips-Curve (NPC).

1_{It is expected that the unemployment rate remains fixed at its natural rate in the long run, independent of the}

rate of inflation. Effectively this produces a vertical Phillips-Curve in the long run.

2_{The output gap is defined as the deviation of actual output from its long term trend. For example, when actual}

The NPC specification originates from a ‘staggered price’ model in which firms are only allowed to change prices when they receive a random signal.3_{Since firms may not}

receive the signal when they want to change prices, it becomes important to anticipate on expected future prices (i.e. expected future inflation). The relative importance of expected inflation is measured by a discount factor, which may differ across firms. According to Roberts (1995) the discount factor is on average equal to one for high frequency data. Although various methods exist to measure expected inflation, Roberts assumes a unit discount factor independently of the method used. For instance, Gali, Gertler and Lopez (2001) (henceforth, GGL) provide estimates of the discount factor for the United States and Europe that appear in line with this assumption (eg. 1.012 and 0.990 respectively). On the other hand, Seyfried and Ewing (2004) present estimates for the United Kingdom, Japan and Canada that appear to be rather low in this context (eg. 0.759, 0.790 and 0.929 respectively). Consequently, empirical evidence shows that unrestricted discount factor estimates may differ from unity and across distinct economic regions.

Another important issue concerning the specification of the NPC is to identify the relevant variable that drives inflation. In this regard, the initial empirical investigation of the NPC has moved from unemployment to the output gap. Roberts (1995, 1997) played a prominent role in this process. Recently, however, the output gap has met with considerable criticism. The reason is that when the NPC is estimated with actual data, the output gap appears to have a negative relation to inflation, which is inconsistent with theory.

The failure of the output gap in the NPC is mainly attributed to two factors. First, the existence of an output gap is theoretical. Okun’s Law assumes that in the long run output will be equal to trend output, although it is not certain that it actually will.4Evidently, if there is no long term trend or potential output, the output gap does not exist. Second, many different methods exist to obtain trend output. For instance GGL argue that detrending actual output data results in a significant loss of information, independently of the method used.

Furthermore, a crucial assumption from the ‘staggered price’ theory is that firms expectations of future inflation and their effect on output are endogenous to current inflation. Future inflation expectations of firms are a substantial contributor to actual inflation in the NPC and the amount of goods supplied by a firm depend on these expectations. According to

3_{The term ‘staggered prices’ refers to the persistence of past prices in the current price level. Since firms can}

only change prices when they receive the signal, past prices may persist in the current price level. Price setting by firms is therefore ‘staggered’. The model by Calvo (1983) is addressed in more detail in Section 2.1.

4_{Trend or potential output is related to the natural rate of unemployment in the sense that when output is at its}

Roberts (1995), an unexpected demand shock to the economy (i.e. the residual) may be correlated with output. Thus, the possibility exists that the coefficient estimate of the output gap in an Ordinary Least Squares (OLS) regression suffers from an endogeneity problem. Despite the intuitive theoretical underpinnings of endogeneity in the NPC, a formal test for this crucial assumption may point to a weakness of the empirical NPC.

In this paper, I hypothesize that the role for expected inflation may be less prominent than implied by a unit discount factor. I will construct several terms for expected inflation on the basis of different information sets and by means of Vector Autoregressive (VAR) forecast models. This approach allows for the representation of different perspectives of forecasting agents in the market. For each term of expected inflation, I provide unrestricted estimates of the discount factor. Although Roberts (1995, 1997) assumes that the discount factor is close to one for high frequency data, many macroeconomic data have quarterly frequencies and are therefore limited. The fit of the constructed terms is examined in relation to the output gap and applied to quarterly data of the United States, Europe, Canada and Japan. Data is collected over the period 1976Q1 – 2010Q2 for the United States, Canada and Japan and 1995Q1 – 2010Q2 for Europe. An investigation of the NPC on data from Canada and Japan may prove that distinct economies exhibit empirical differences in terms of the hypothesized short run trade-off between inflation and the output gap. Specifically for Japan, the past decade of near zero inflation rates and negative output gaps are of interest in this context. I detrend actual output data with a method developed by Hodrick and Prescott (1997), commonly referred to as the HP-filter. Provided that many authors employ this method of detrending, this allows for a direct comparison of the results. In addition, I employ a test to determine whether the assumption of endogeneity in the NPC is justified. To my knowledge, few authors have employed statistical tests to infer whether endogeneity is present, although this assumption is crucial for the estimation methodology. An implication of a violation of the endogeneity assumption is that Ordinary Least Squares (OLS) may produce more efficient estimates as opposed to Generalized Method of Moments (GMM).

2. Literature Review

In this section a more dedicated understanding of the microeconomic models from which the NPC originates is developed. The discussion is limited to the connection between price setting by firms and the effect on the general price level in the model by Calvo (1983). Furthermore, a detailed discussion of the variables that constitute the NPC is presented in junction with past results. Finally, important ongoing debates concerning estimation methods, alternative

theories and specifications of the NPC are addressed.

2.1 Microeconomic foundation of the NPC: A ‘Staggered Price’ Economy

Recall from the introduction that Lucas criticized the way in which macroeconomic models were evaluated. The main critique being that until then most macroeconomic models did not properly account for responses of firms and households on changing monetary policy in which the Phillips-Curve is no exception. Evidently, individual responses of firms and households to changing monetary policy constitute the aggregated effect of such changes. Therefore, Lucas advocated that in order to obtain a reliable short run Phillips-Curve relation, a model should be developed that incorporates preferences and limitations of individual firms and households in terms of wages, technology and resources. Then, when monetary policy changes, the effects could be assessed on an individual basis and aggregated to arrive at conclusions for the economy as a whole. Many models have been suggested ever since, of which most recent papers extent on the work of Calvo.

In Calvo’s model, the economy is assumed to consist of monopolistic competitive firms, implying one competitive market of identical firms. Each period, a firm determines its optimal price according to an objective function. Effectively, the objective function

A direct result is that the output gap appears in the NPC as opposed to unemployment. The NPC thus derives the effect of pricing decisions made by individual firms on the general price level in terms of inflation.5_{Provided that all firms are identical and produce one product, the}

expected price change of that product is equivalent to expected inflation.

2.2 Variables of the NPC: Expected Inflation

This section reviews the most important definitions surrounding expectations in the NPC. Unless stated otherwise, expected inflation is defined as inflation expectations formed in period t concerning inflation in period t + 1. Moreover, expected inflation of period t + 1 is assumed to be relevant in explaining inflation in period t. The importance of expected inflation in the NPC is measured by a discount factor which is expected to be close to one. After reviewing the most common methods of measuring expected inflation, results of the unrestricted discount factor estimates are presented.

The existing literature reveals a variety of approaches exists to measure expected inflation. This is not surprising since not one method of measuring expected future inflation can be deemed particularly superior to another method. It follows that an explicit measure of expected inflation is not required. Roberts (1995, 1997) employs results from a survey, which seems appropriate and appealing. The surveys are send to professional forecasters of inflation. Given that professional forecasters are in a good position to acquire all relevant information and have the means to adequately analyze the data, they are likely to provide a good measure of expected inflation. Despite the access to potentially superior information and analyzing methods, two main drawbacks of survey results can be identified. First, professional forecasters have very little incentive to provide thoughtful answers, which is a well known problem for voluntary surveys in general. Second, most surveys of expected inflation are at annual or semi-annual frequencies. In order to properly match the survey data with real inflation data, the investigation has to resort to (semi) annual data. If (semi) annual data is used this limits the sample size and may induce biases which arise from small samples. By employing the actual inflation rate in period t + 1 as a proxy for expectations formed in period t, the measure of expected inflation is at the same frequency as the actual inflation data. A drawback of employing the actual inflation rate in period t + 1 is that it induces additional noise in the estimation of the NPC. The additional noise stems from the fact that it is unlikely that forecasters of inflation in period t will be able to perfectly predict inflation in period t + 1.

5_{Section 3.1 provides the mathematical equation of the relation just described. For a complete mathematical}

This implies that the true expectations of inflation formed in period t are likely to differ from the actual rate of inflation in period t + 1. Note that if forecasters in period t would be able to perfectly predict inflation in period t + 1, it would be impossible for monetary policy to surprise the market and thus temporarily induce lower unemployment.

Seyfried and Ewing (2004) and Senda (2005) assume that expectations formed in period t – 1 concerning inflation in period t is relevant in explaining inflation in period t. According to Gali and Gertler (1999) (henceforth, GG) this is very unlikely to be true. The reason is that all past information is assumed to be available to the public and is thus incorporated in the current price level. Any past information, including expectations formed in the past, are already accounted for and as a consequence irrelevant in explaining current inflation. Also, the emphasis in the model by Calvo is on expected price changes in period t + 1 formed in period t as opposed to expectations formed in period t – 1.

Furthermore, it is common to construct a term for expected inflation by means of an Autoregressive (AR) or Vector Autoregressive (VAR) forecast model. AR and VAR models are constructed on past information only and therefore fit nicely with the assumption of all past information being available to the public. As mentioned by Musy and Pommier (2007), a VAR model requires the researcher to select a set of variables that are believed to be used by inflation forecasters in the market. In relation to the potentially biased survey results of professional forecasters, a VAR model may omit relevant variables that forecasters use. In effect, forecasters of inflation have an exact objective function by which they form

expectations, whereas a VAR model represents an inexact substitute of this objective

function. If, however, it is assumed that the employed variables in the VAR model are a good representation of the true variables used by forecasters, the VAR forecasts work equally well as the surveys do. In addition, it is rather easy to construct several VAR models on basis of different information sets in order to check for robustness of the results.

Seyfried and Ewing provide a different view however, since the results for the United Kingdom and Japan are too low (eg. 0.759 and 0.790 respectively).

This section reviewed the possibilities for proxies of expected inflation in the NPC. Given that explicit measures of expected inflation are not required, the choice for obtaining a proxy may be trivial. The discount factor measuring the importance of expected inflation in the NPC is commonly restricted to 0.99 or one. Past results of the unrestricted discount factor estimates appear largely in line with the expected value near one. In this regard, the AR and VAR forecasts appear superior to the actual inflation rate in period t – 1.

2.3 Variables of the NPC: Output Gap

In addition to expected inflation, a key aspect of the specification of the NPC is to determine what variable drives inflation. This section addresses the output gap as the relevant variable driving inflation. Specifically, attention is devoted to the role of the output gap for monetary policy, the existence of various methods to measure the output gap and the associated uncertainty in its measurement. A few examples of authors that estimate the NPC with the output gap are discussed and show that the results are mixed.

Recall from the introduction that the Phillips-Curve is defined as the inverse relation between inflation and unemployment. The relation between unemployment and the output gap is described by Okun’s Law. According to Okun’s Law, an increase in unemployment will lead to a decrease of real output relative to a long term trend. The difference between real output and its long term trend is defined as the output gap. For example, when real output is above trend output, this results in a positive output gap and is expected to put upward pressure on prices (i.e. inflation). A key assumption in this respect is that in the long term, real output will converge towards trend output. It is important to consider that the possibility exists that real output does not converge to trend output in the long term. If this is the case, there is no such thing as an output gap.

Assuming that the output gap exists, stabilizing real output towards potential or trend output (i.e. the output gap) can be relevant. In this regard, Smets and Gaspar (2002) state that, when the economy is hit by a supply shock6, this has no effect on inflation if the output gap remains unchanged. Thus, when monetary policy aims at stabilizing the output gap, a supply shock can be mitigated from affecting price stability. Stated otherwise, in case of a supply shock, stabilizing the output gap results in stabilizing inflation.

However, Smets and Gaspar also state that, when the economy is hit by a so called cost-push shock, this has a direct impact on inflation, but not on the output gap.7_{Consequently, in}

response to a cost-push shock, monetary policy may have to induce (costly) fluctuations in the output gap in order to mitigate a cost-push shock from significantly affecting price stability. Provided that it is difficult for monetary policy to identify the source of the shock, Smets and Gaspar argue that monetary policy should aim at stabilizing inflation instead of the output gap in order to achieve price stability. An interesting and somewhat contrasting finding of Smets and Ehrmann (2002), however, is that when monetary policy is assumed to be able to identify the source of the shock, the response of monetary policy is the same when it is assumed that the source of the shock cannot be identified.

Another important issue concerning the output gap is the degree of uncertainty with which the output gap can be estimated.8In relation to monetary policy behavior, Smets (2002) concludes that uncertainty in the output gap measurement does not alter monetary policy behavior. Moreover, when uncertainty on the accuracy of the output gap increases, this is off-set by decreasing the weight on the output gap in the decision-making process.9So

effectively, monetary policy behavior is not affected by output gap uncertainty. Concerning various approaches to obtain the output gap, Cayen and Van Norden (2005) track revisions of output gap estimates over time for 12 different methods to measure the output gap. The employed methods consist of univariate as well as multivariate ‘detrending’ techniques and include the commonly employed Hodrick-Prescott (HP) filter.10 11It is observed that the obtained output gap estimates vary widely across methods and over time. More striking is the finding that not only the absolute magnitude of the output gap varies widely, but also the sign of the output gap appears ‘wrong’ in at least 25% of the observations as compared to the final revision (eg. applicable to all methods). Unfortunately, the analysis does not allow concluding what method is ‘best’. It does raise the question of whether the recent failure of the output gap in the NPC is more likely attributable to the difficulty in measuring the output gap or its existence in the first place.

7_{A cost-push shock is defined as a substantial increase in the price of important goods or services with no}

suitable substitute, for instance oil.

8_{Notably the uncertainty mainly arises from the fact that potential or trend output is unobservable.}

9_{The model by Smets (2002) employs the Taylor interest rule, by which monetary policy can derive the optimal}

interest rate. It appears that as uncertainty on the output gap measurement increases, the role for the output gap in determining the optimal interest rate decreases.

10_{By ‘detrending’ what is meant is that real output data is separated into a cyclical and trend component. The}

trend component is generally referred to as potential or trend output. Monetary policy may be inclined to stabilize real output relative to potential or trend output.

11_{For instance, GGL, Neiss and Nelson (2005) and Musy and Pommier (2007) employ the HP-filter to obtain the}

Provided that the output gap has a negative relation to unemployment and thus, a positive relation to inflation in terms of the Phillips-Curve, inflation and the output gap should be observed to vary positively. Roberts (1995, 1997) and Seyfried and Ewing present

significant supporting evidence in this regard, although other authors have found the opposite. For instance, GGL, Neiss and Nelson (2005) (henceforth, NN) and Musy and Pommier find a significant negative relation between inflation and the output gap. The main argument set forth by GGL and NN is that the recently observed low levels of inflation and high positive output gaps have led to the demise of the output gap in the NPC.12_{In addition and}

independent of the method used, ‘detrending’ real output data may result in a significant loss of information inherent in the data. An implication is that the loss of information may render the output gap insensitive to price changes (and vice versa) and thereby reduce the exploitable trade-off between inflation and the output gap for monetary policy.

This section addressed the output gap as the relevant variable driving inflation in the NPC. Although the output gap may not exist, a variety of approaches are available to estimate it. Each method brings with it a similar amount of uncertainty concerning the accuracy of the estimates, hence no single method can be deemed particularly superior to another. Empirical results of the output gap coefficient in the NPC are rather mixed in terms of the theory implied positive relation between inflation and the output gap. It appears that the degree of uncertainty in the output gap measurements have no effect on monetary policy behavior. This can be explained by less weight placed on the output gap when uncertainty increases in specific policy rules by which nominal interest rates are set.

2.4 Estimation methods of the NPC

The dominant practice in estimating the NPC involves estimation by Generalized Method of Moments (GMM).13Other methods include ARCH-type models and Ordinary Least Squares (OLS).14In order to determine what estimation method is likely to be appropriate, an

examination of the estimation properties is required. Recall that OLS requires the explanatory variables to be uncorrelated with the error terms (i.e. exogenous regressors), whereas GMM

12_{Recall that high positive output gaps are expected to put upward pressure on prices.}

13_{In this case, Generalized Method of Moments (GMM) can be viewed as a special case of Instrumental}

Variables (IV). The advantage of GMM over conventional IV is that GMM produces more efficient coefficient estimates in the presence of heteroscedasticity of unknown form. Authors that estimate the NPC with GMM include GG, GGL, NN and Muto (2009).

14_{For instance, Seyfried and Ewing estimate with ARCH-type models and OLS for comparison purposes,}

allows for this correlation by using instruments. This raises the question of whether it is likely that the explanatory variables are correlated with the error term (i.e. endogenous).

To address this issue recall that in a ‘staggered price’ economy as described in Section 2.1, firms anticipate on expected future prices charged by other firms and setting their level of production and prices accordingly. Thus, current expectations of firms depend on the current level of production and prices, whereas the current level of production and prices depend on current expectations of future production and prices. So it is not only current inflation that depends on expectations and production, also expectations are a function of current inflation and production, and current production is adjusted towards expectations for the next period. This constitutes the coherent dynamics of the NPC variables and hence, are likely to be endogenous. Formally, the endogeneity of the regressors can be tested for by means of a Hausman test, although this test is generally not employed in practice according to Brooks (2009).15_{The potential problem of the Hausman test in this regard is that the properties of the}

variables might not be recognized as endogenous, while theory makes it very likely that they are. A consequence is that the Hausman test may justify the use of an OLS regression when this is not what theory suggests. It is possible that due to a lack of power of the Hausman test the regressors appear as exogenous. Nevertheless, it is surprising that besides the potential lack of power of the test, no suitable explanation is available or a more robust test for endogeneity has been developed.

2.5 Alternative specification of the NPC: A ‘hybrid’ model

This section reviews an alternative specification of the NPC as discussed so far. Section 2.1 discussed the origin of expected inflation in the NPC and Section 2.2 reviewed different approaches in order to obtain a measure of expected inflation. In addition, Section 2.3 discussed the output gap as the relevant variable driving inflation. The resulting specification is defined as the NPC, which relates current inflation to expected future inflation and the output gap. Alternatively, this specification of the NPC can be termed a forward looking NPC.

Although the forward looking NPC appeared to be an empirical success mainly due to results published by Roberts (1995, 1997), GG have proposed a different specification. Their specification assumed that a fraction of firms set prices according to a backward looking rule,

15_{Noting that for instance Roberts (1995, 1997), GG, GGL, NN and Muto do not present any results on a formal}

while the remaining fraction of firms are forward looking. Effectively, backward information is added to the purely forward looking NPC and is termed the ‘hybrid’ NPC.

The main reason for the addition of backward information is that inflation is observed to be persistent. In other words, inflation in the current period is observed to be to some degree dependent on inflation in the previous period.16A way to account for this observation in the NPC derived from the ‘staggered price’ economy is to assume that a fraction of firms are backward looking when determining their optimal price. For example, GG, GGL and Dufour, Khalaf and Kichian (2005) present favourable results for the ‘hybrid’ NPC. They come to the conclusion that backward looking firms may play a significant role in the NPC, although forward looking firms appear slightly dominant. Alternatively, Sbordone (2002) examines the properties of a forward looking NPC with Real Marginal Cost (RMC) as the relevant variable driving inflation.17One conclusion is that a purely forward looking NPC explains price behavior quite well. Furthermore, instead of aiming at another specification of the NPC, theoretical and empirical research should attempt to better understand and define the behavior of RMC in relation to inflation. Sbordone suggests that a more reliable definition of RMC will lead to a better fit of the forward looking NPC, implying that augmenting the NPC with backward information may not be appropriate.

Where Sbordone solely considers forward looking behavior of firms, Muto (2009) estimates and compares a forward looking and hybrid NPC. The study by Muto centres on the relevance of a backward looking component in the NPC and concludes that the model rejects the necessity of a backward looking component of inflation for Japan. Moreover, Muto in accordance with Sbordone argues that a poor measure of RMC will lead to an undervaluation of the purely forward looking NPC and consequently overstates the importance of a backward looking component. The results of Sbordone and Muto have two notable implications. First, a purely forward looking NPC appears to be the appropriate specification instead of a ‘hybrid’ NPC, however, may require a better understanding of the variable driving inflation (i.e. RMC or the output gap) Second, extending the likeliness of measurement error in RMC to that of the output gap, this may point to the use of lagged inflation which may not be appropriate.

16_{This implies that the role for expected future inflation in the NPC may be less prominent than generally}

assumed. Stated otherwise, if backward information is included, the assumption of a unit discount factor on expected future inflation becomes invalid.

2.6 Alternative specification of the NPC: Inclusion of a constant

A well-known property of regression analysis is that the inclusion of a constant ensures the expected value of the errors is zero on average. The specifications of the NPC discussed thus far generally assumed the constant equals zero. But what justifies this approach? Concerning the inclusion of a constant, Vinod (2010) presents OLS estimates of the hybrid NPC, with and without a constant and finds that the constant is statistically insignificant. Besides the

insignificant constant, the adjusted R-squared of the OLS regression without a constant equals 0.8819 and 0.6728 when the constant is included. Derived from the adjusted R-squared statistic, the NPC appears to have more explanatory power when the constant is omitted. In addition, the estimates of the output gap coefficient are positive, but statistically insignificant. Vinod (2010) therefore argues that it may be appropriate to force the regression through the origin, however notes that since the output gap coefficient is retained in the model (given statistically insignificant estimates), the same rein of thought may be applied to the insignificant constant. It can therefore be said that both approaches can be justified,

3. Methodology

This section presents the equation of the NPC, derived from the previously discussed literature in Sections 2.1 – 2.6. In addition, the hypothesis that is tested is defined. Furthermore, this section explains how the variables described in Section 2.2 and 2.3 that comprise the NPC are constructed. What follows is a description of the preliminary step (i.e. formation of an instrument set) required to estimate the NPC and the statistical tests to infer its validity. Section 3 concludes with a description of data sources and includes a graphical display of the co movements between inflation and the output gap. It is useful to graphically consider these co movements for the reason that the hypothesized relation in Section 3.1 dictates a positive relation. By graphically observing these movements over time, valuable insights can be gained on what is to be expected when the NPC is estimated.

3.1 Equation and hypothesis

Section 2.1 discussed the origin of the NPC, which relates inflation to expected future inflation and the output gap. The derived relation stems from individual pricing decisions made by firms, who are induced to anticipate on expected future prices as a result of

‘staggered’ price setting. If it is assumed that all firms are identical and that there are enough firms so that the probability of a price change equals the fraction of firms that change their price, the expected future price change of a firm is equal to expected future inflation. A straightforward specification of the NPC that captures this element can be written as,

   

t  Et t1 yt  (1)

where, _t denotes quarterly inflation, E_t_t1 represents expectations formed in period t on

inflation in period t + 1, y is the output gap and ε an error term. The coefficient β denotes the _t

discount factor on expected inflation and λ is a strictly positive coefficient, which measures the sensitivity between inflation and the output gap.

3.2 Methodology: Expected Inflation

As mentioned in the introduction, Roberts (1995, 1997) assumes a unit discount factor for expected inflation. In addition, he states that this assumption is independent of the method used to measure expected inflation. This implies that an explicit measure of expected inflation is not required (i.e. the NPC as specified in equation (1) should work equally well for

different methodologies to measure expected inflation). This section further addresses the methodology employed to obtain various terms for expected inflation and why it is an appropriate method to do so.

Following GGL, I assume that agents in the market use only past information when they form expectations on future inflation, because not all new information of the current period might be available to the public at the time they form expectations. Therefore,

forecasters of inflation in period t are interested in the expected change of inflation in period t (i.e. E_t ), which is then used to compute the expected rate of inflation for period t + 1 (i.e. _t

1 

t t

E ). Effectively, I assume that inflation forecasters form expectations by forecasting the

change of inflation in period t. The sum of the forecasted change of inflation in period t and the actual rate of inflation in period t is defined as expected inflation for period t + 1.18

Since one of the goals of this research is to compare the magnitude of the discount factor on expected inflation across terms for expected inflation, a model is required that allows for the construction of different inflation expectations. Moreover, the model has to accommodate the use of different information sets (i.e. variables) on which expectations of future inflation are formed. An appealing model in this regard is that of a VAR model.

An important feature of VAR models is its flexibility and ease of generalisation of model specification. It should be noted that in the true spirit of VAR estimation, the model is as unrestricted as possible since then the model may be able to capture much of the structure of the data generating process. This, however, constitutes the a-theoretical nature of VAR models since little theoretical information about the relationships between the variables is used to construct the model. As a consequence VAR models are less amenable to theoretical analysis and therefore to policy prescriptions (Brooks (2008)). Since in this paper the VAR model is employed solely for its forecasts to obtain a measure of expected inflation and does not attempt to disentangle the dynamics thereof, the critique set forth by Brooks is not so

18_{This formulation implies that expectations of forecasters are adaptive, since each period the newly available}

much of a problem. Another argument for using a VAR model comes from a paper by De Mello (2009). She presents results for a first differenced VAR model for the purpose of one-step ahead forecasts. In fact, the research compares several (un)restricted, integrated and co integrated VAR models and compares the one-step (static) and multi-step (dynamic) forecast accuracy, where the unrestricted first differenced VAR model performs best in case of one-step ahead forecasting.19In order to exploit the findings of De Mello, the forecasting procedure will be of a static nature.

In general terms, the model can be written as a VAR(p) of the form,

        zt c0 c1 At1 ... cp Atp (2)

where  is a vector of first differenced variables included in the system, cz_t 0a vector of

constants, c1– cpare vectors of coefficients and ∆At-1– ∆At-pare vectors of first differenced

variables at the given lag-length denoted by p.

I derived the included variables from the VAR specifications of GGL, NN and Musy and Pommier (2007). None of these VAR specifications includes a futures commodities index and only the latter includes the money supply. Logically, the amount of money in circulation in junction with the amount of output is related to inflation and it is therefore appropriate to include the money supply. It is also likely that a futures commodities index has significant relevance for expectations of forecasting agents, more or less serving as an additional measure of price changes in traded goods (i.e. futures). Table A2.0 in Appendix A2 shows all 13 employed specifications. By specifying several VAR models to construct terms for expected inflation, I acquire a broad representation of different perspectives of forecasting agents in the market.

In order to arrive at the optimal VAR model for each specification, each specification was initially estimated as an unrestricted VAR(2) on the period 1995:Q1 – 2009:Q2 for Europe and 1976:Q1 – 2006:Q1 for the United States, Canada and Japan. This ensures that four observations are left out for Europe and 20 for the other countries which are used to assess the out-of-sample forecasting performance. Evidently, leaving four observations for out-of-sample forecast assessment is not a robust examination of the forecast accuracy,

19_{The difference between a ‘static’ and ‘dynamic’ forecast is that a static forecast is based upon all available}

however the Euro Area data set is limited.20When considering a maximum of VAR(6) and four out-of-sample observations, it is ensured that a minimum of 50 observations are preserved for the estimation of the forecast model.

The optimal lag-length was determined on the bases of the Akaike’s Information Criteria (AIC) and sequential modified likelihood ratio (LR). The AIC lag-length is selected in favor of the LR selection, however, after re-estimation with the AIC selected lag-length the stability (i.e. stationarity) of the system was assessed. If the system was found to be unstable, the system was again re-estimated with the LR selected lag-length and tested for stability. In all cases, when the AIC selected system was found to be unstable, the LR selected

specification was stable.

A further assessment of the VAR specifications was based on their static in- and out-of-sample forecast performance. Following Brooks and De Mello, I evaluate the VAR models on the basis of the Root Mean Squared Error (RMSE). The absolute value of RMSE does not mean a lot in and on itself and has to be compared across specifications. To this extent it is appropriate to compare the best performing model in terms of the RMSE with that of the individual RMSE’s.

I assume that the optimization of the VAR forecast models will produce representative inflation forecasts for the ‘true’ expectations of a broad range of agents in the market. On the basis of this assumption, the unrestricted discount factor estimates will be robust to different information sets to measure expected inflation.

3.3 Methodology: Output Gap

Recall from Section 2.3 that a variety of approaches exist to estimate the output gap. The main result of the study by Cayen and Van Norden (2005) is that it is difficult to conclude which method is ‘best’. On the basis of this result and provided that many authors employ the Hodrick-Prescott (HP) technique to estimate the output gap, this section describes in general terms the aspects of the HP ‘detrending’ technique. For example, GGL, NN and Musy and Pommier employ the HP-filter and this allows for a direct comparison of the results.

Hodrick and Prescott argue that any time series consists of two distinct components, namely a cyclical and trend (or growth) component. The cyclical component varies rapidly in response to a quickly changing environment, whereas the trend component adjusts more

20_{The limited data set of Europe is due to the availability of the Unit Labor Cost index. More data is available}

gradually and therefore represents the general growth rate over longer periods of time. In terms of the output gap, let GDP denote the log of the observed aggregate amount of output _t

at time t subject to, GDP_t c_t _t, where c and _t _tdenote the log of the cyclical and trend

components respectively. Assuming that in the long run aggregate output converges to trend output, the current deviation of GDP from _t _t corresponds to the output gap. Consequently

the measure of the output gap (eg. y ) is obtained by,t

t t t GDP

y   (3)

Hodrick and Prescott developed this method of smoothing in which by means of minimizing the variability of the trend component subject to a constant smoothing factor, both

components can be identified through a least squares fit of a linear time trend model. The higher the value of the smoothing factor, the smoother is the resulting time series of t. For

quarterly data, the appropriate constant smoothing factor equals 1600 and is the default setting in this paper.

It is well recognized that this method of smoothing the data may result in a significant loss of information inherent in the data as pointed out by for instance GGL. This is, however, the most straightforward and appealing method of measuring the output gap without having to specify the entire supply and demand side of the economy. Also, Roberts (1995) pointed out that a misspecification of any aspect of the model (i.e. the economy) renders inconsistent estimates of the coefficients in the NPC equation. Therefore, this relatively simple approach is easy to understand and requires fewer assumptions concerning the specifics of the economy.

3.4 Estimation and instruments

This section discusses the methods and reasons for estimating the NPC with Generalized Method of Moments (GMM) (as a special case of Instrumental Variables (IV)). An important step in this regard is the construction of an instrument set. This section concludes with a brief discussion of diagnostic tests that are employed to evaluate the validity of the estimated NPC’s.

Since it is likely that the regressors of equation (1) (eg. Ett1 and y ) are correlated t

homoscedastic. When this is not the case, the problem can be partially addressed by

calculating heteroscedasticity consistent errors., According to Baum, Schaffer and Stillman (2003), the conventional IV estimators are still consistent, but inefficient in the presence of heteroscedastic consistent errors. To address this issue, Hansen (1982) introduced the

Generalized Method of Moments (GMM), which produces consistent and efficient estimators in the presence of heteroscedastic errors of unknown form.21The orthogonality condition that needs to be satisfied for a valid GMM estimation of equation (1) is,









t y z

E    (4)

where z denotes a vector of variables, the instrument set, available at time t and all other t

terms are defined as in Section 3.1. Following GGL, I assume that the information set zt

consists of lagged instruments only, because not all new information of the current period might be available to the public at the time expectations are formed. In addition it is likely there is considerable measurement error in y , assuming this measurement error is _t

uncorrelated with past information, lagged instruments are appropriate.

The existing literature is unclear on what instruments are appropriate and what procedures should be followed to determine the most relevant instrument set. For instance, Dufour et al. (2005) and Nason and Smith (2008) argue that many GMM estimates of equation (1) have not properly addressed the issue of potential weak instruments, which may have led to biased inference. They propose a test for the validity of the over-identifying restrictions that is robust to weak instruments.22GGL note that the number of instruments should be kept as low as possible to avoid biased estimates when there are too many over-identifying restrictions, especially in small sample sizes.

I employ a test for Granger-Causality to select the instruments. The suitable instruments should be theory related and have high correlations with the instrumented variables. Given that the instrument set consists exclusively of lagged variables, the test for Granger-Causality suggests which variables and how many lags are likely to be relevant in explaining the current values of expected inflation and the output gap. That is, lagged values

21_{GMM estimation of the NPC has been applied in favour of the conventional IV by for instance GG, GGL,}

Sbordone, NN and Muto.

22_{In general the Hansen J-statistic is reviewed to infer whether the data supports the over-identifying}

of the included variables are hypothesized to Granger-cause expected inflation or the output gap. If the null of no Granger-Causality can be rejected for a specific lag of a variable, it is included in the initial instrument set.

After the initial instrument set has been formed, I estimate equation (1) with 2-Stage Least Squares (2SLS) and the initial instrument set. This allows for conducting White’s general test for heteroscedasticity, which is not supported under GMM. If heteroscedasticity is present, this justifies the use of GMM in favor of IV. The standard errors are calculated under a heteroscedasticity and autocorrelation consistent covariance (HAC).

Finally, the procedure for obtaining the optimal weighting matrix needs to be specified. Although no real increase in efficiency can be realized by iterating the optimal weighting matrix, Monte Carlo studies have shown that iteration may lead to slightly better estimates in small finite samples.23Given small sample sizes, I iterate the optimal weighting matrix to convergence.

3.5 Diagnostic tests

Following GGL, I perform a number of diagnostic tests to evaluate the GMM regressions. The diagnostic tests will indicate whether or not it is likely that the NPC estimates represent an empirically consistent and efficient short run trade-off between inflation and the output gap.

Since the choice of instruments is as difficult as it is important, a first evaluation will provide an indication of the potential weakness of the instruments. Stock and Yogo (2005) proposed the Cragg-Donald F-statistic. Although the formal test is not valid in case of GMM estimation, the magnitude of the statistic can be used to compare across instrument sets.24 Stock and Yogo argue that a Cragg-Donald statistic below 10 is reason for concern. Accordingly, an instrument set which reports a Cragg-Donald statistic < 10 is therefore dropped.25

The orthogonality condition in (4) is tested using the Hansen J-test. This test

effectively reveals whether the over-identifying restrictions are supported by the data. If the null hypothesis of the test is rejected, this indicates there are too many over-identifying

23

For a detailed discussion see, Hansen, Heaton and Yaron (1996).

24_{The Eviews manual mentions the invalidity of the test in case of GMM estimation, however notes that a}

comparative analysis can still be performed on the absolute magnitude of the statistic.

25_{If an instrument set reports an F-statistic < 10, the correlation matrix of all variables listed in Table 1.0}

restrictions and the specified NPC is invalid. A rejection of the null hypothesis of the Hansen test may point to a misspecification of the NPC (eg. forward-looking versus ‘hybrid’

specification) or may indicate that the model should be estimated with fewer instruments. An addition of this paper to existing literature is the formal test for endogenous regressors. In case of GMM, the Durbin-Wu Hausman test allows for an evaluation of endogenous regressors. The test hypothesizes that the specified relation should be estimated with OLS and tests for the null hypothesis of exogenous regressors. In effect, if the null cannot be rejected, this suggests that OLS will deliver consistent and efficient coefficient estimates. Moreover, the GMM estimates are still consistent, but may be inefficient. Given the theoretical basis of endogeneity inherent in the NPC and the dominant practice of employing GMM, OLS estimates will be presented for comparative purposes.

Finally the model’s errors are tested for a white noise process. For instance, GGL interpret the errors as random cost-push shocks to the economy and hence, the errors should not have a discernible structure. The Ljung-Box test of no autocorrelation is employed in this regard. Following Roberts (1995, 1997) and GGL, I include four lags of the potentially auto correlated residuals. If the null of no autocorrelation is rejected at any of the four lags, this rejects the models prediction of a white noise error. I recognize that the choice of the number of lags is relevant for the consequential inference on the errors process. No uniform best practice can be identified in the sense that the adopted test on four lags lacks a strong theoretical basis. I assume that including four lags of the potentially autocorrelated residuals is sufficient to conclude whether or not the errors can be interpreted as random cost-push shocks to the economy.

3.6 Data

The data set for each country consists of the variables listed in Table 1.0. All data are

Table 1.0 Overview of variables of each data set

Variable Measurement Symbol

Inflation Log-difference of Consumer Price Index _t

Output Gap Log-difference of real and trend Output (GDP) y_t

Consumption Log of aggregate consumption measured in

domestic currency Ct

Commodities Log of Futures Commodities Index FCIt

Money Supply Log of domestic currency MS_t

Unit Labor Cost Log of Unit Labor Cost Index ULCt

Nominal Interest Rate Log of 3-month nominal interest rate IR_t

Notes: All data are quarterly and first-differences represent the quarter-on-quarter change. Various

combinations of the listed variables are assessed to obtain forecasts of the inflation equation of the VAR system, which is then used as a proxy for expected future inflation in the ‘new’ Phillips-Curve.

Equation (1) consists of three variables, _t, E_t_t1 and y , where _t _t (eg. inflation) is

measured as the quarter-to-quarter log-difference of the country specific Consumer Price Index (CPI).26Furthermore, Ett1 (eg. expected inflation) is measured as the forecasted value of inflation as described in Section 3.2. y (eg. the output gap) is measured as the log-t

difference between the country-specific real GDP and the long term trend, described in Section 3.3.

3.7 Data plots of inflation and the output gap

In this paper, I hypothesize that co movements of inflation and the output gap have a positive relation. To this extent, Figures 1.0 – 1.3 depict the co movements of inflation and the output gap for the included countries, where the output gap is estimated by the HP-filter. The data plots in Figures 1.0 – 1.3 allow for an initial analysis of the hypothesized short run trade-off between inflation and the output gap. A summary of remarkable observations is provided after Figure 1.3.

26_{Various definitions of a Consumer Price Index are available. The one employed here consists of all items}

Figure 1.0 Co movements of inflation and the output gap: United States Figure 1.1 Co movements of inflation and the output gap: Europe

Notes: Data plot of quarterly inflation (base = 2005) and output gap as estimated Notes: Data plot of quarterly inflation (base = 2005) and output gap as estimated by the HP-filter. Correlation equals 28.90%. by the HP-filter. Correlation equals 28.53%.

Figure 1.2 Co movements of inflation and the output gap: Canada Figure 1.3 Co movements of inflation and the output gap: Japan

In all instances, observed in Figures 1.0 – 1.3, the hypothesized positive relation between inflation and the HP-filtered output gap is difficult to support. Given that correlations are positive but rather low (eg. between 19.49% and 28.90%) and all countries reveal

(possibly significant) deviations from the positive relation dictated by theory. Several remarkable observations include the observed negative output gaps from 2001 through 2003 for the United States in Figure 1.0. The negative output gaps do not appear to be strongly related with inflation, since prices consistently increased while the output gap took large negative values. For Europe in Figure 1.1, the negative output gaps in the late 1990’s appear to have led to a marginal decrease in price increases, however have not established a decrease in prices as predicted by theory. Figure 1.2 for Canada reveals several additional points in time in comparison to the United States where the output gap changed rapidly. In addition to the sharp decreases of output relative to its trend observed in the United States, Canada seems to have had more periods of considerable underproduction. More specifically, at the end of the 1970’s, during 1992 – 1993 and at the end of 2001 through 2002, the output gap took relatively large negative values for Canada. Finally, Figure 1.3 for Japan shows that from approximately 1995 until now, inflation in Japan has been quite steady around a 0% rate, with a notable increase in prices during 1996 - 1997. The corresponding values of the output gap appear to react heavily with large negative values at the end of the 1980’s, 2002 and also during the recent credit crunch in 2008. Although prices increase only marginally or even decrease during the last 10 years, the output gap reaches its sample maximum just before the crisis of 2008 and then drops to its sample minimum almost immediately after.

All-in-all there appears to be a positive co movement between inflation and the output gap in all economic regions. A positive correlation in all instances confirms this statement. However, all economic regions have experienced periods in which the co movements of inflation and the output gap deviate considerably from a positive relation. In this regard, the next section will present and discuss the regression results to provide further evidence on the hypothesized positive relation between inflation and the output gap in the NPC.

4. Results

results of equation (1) for each economic region separately. Finally, the test results for endogenous regressors are collectively discussed afterwards. Detailed OLS results are available in Appendix A3.

Table 1.1

Europe: GMM Estimates of the ‘new' Phillips-Curve, sample period: 1995Q1 - 2010Q2 Estimated Equation: t Ett1yt 

Instruments: Constant, lags 1 – 4 of _t, 2 lags of y and 2 lags of _t Comm_t

Specification Estimate of β Estimate of λ

for Ett₁ (St. Error) (St. Error)

VAR(5) t, y , t C , t ULCt 0.975 (0.052) -0.046 (0.043)

VAR(4) _t, y , _t C , _t IR_t 1.000 (0.049) -0.050 (0.030)

VAR(3) _t, y , _t ULC , _t IR , _t MS_t 1.078 (0.053) -0.049 (0.022)**

VAR(5) _t, y , _t C , _t FCI , _t ULC_t 1.031 (0.046) -0.033 (0.025)

VAR(3) _t, y , _t C , _t FCI , _t ULC , _t IR_t 1.024 (0.047) -0.047 (0.020)**

VAR(4) _t, y , _t C , _t ULC , _t IR_t 0.939 (0.050) -0.015 (0.028)

VAR(3) _t, y , _t C , _t ULC , _t MS_t 0.948 (0.043) -0.120 (0.032)***

VAR(3) _t, y , _t C , _t FCI , _t MS_t 1.079 (0.043)* -0.109 (0.024)***

VAR(3) t, y , t FCI , t ULC , t MSt 1.004 (0.043) -0.079 (0.027)***

VAR(5) t, y , t FCI , t ULC , t IR , t MSt 0.985 (0.051) -0.006 (0.026)

VAR(1) t, y , t C , t FCI , t IR , t MSt 1.016 (0.074) -0.064 (0.029)**

VAR(3) _t, y , _t C , _t FCI , _t ULC , _t MS_t 0.968 (0.044) -0.097 (0.025)***

VAR(4) _t, y , _t C , _t FCI , _t ULC , _t IR , _t MS_t 0.965 (0.043) 0.006 (0.019)

Notes: All data is quarterly where,_t is inflation, y_t is the output gap, C_t is consumption, FCI_t is a futures commodities index, ULC_t is a unit labor cost index, IR_t is the short term nominal interest rate and MS_t is the money supply. Standard errors are calculated under a heteroscedasticity and autocorrelation consistent

covariance (HAC). Testing for H0: β = 1; H1: β ≠ 1 and H0: λ < 0; H1: λ > 0. *, ** and *** indicate statistically

4.1 Comments and interpretation of results: Europe

The estimates of the discount factor β for Europe in Table 1.1 show a minimum of 0.939 and a maximum of 1.079. At the 10% level, the maximum of 1.079 has a statistically significant deviation from one. In the remainder of the estimated NPC’s, the null hypothesis of a discount factor of one cannot be rejected at any of the conventional significance levels. Concerning the output gap coefficient estimates λ, all except one are negative between (-)0.120 and (-)0.006. The positive estimate of λ equals 0.006 for the last specification of E_t_t1, however is not

statistically significant. Furthermore, it is easy to see that all of the negative estimates for the output gap coefficient do not allow for a rejection of the null hypothesis of λ < 0. Considering a two-sided test for λ, the third, fifth and 11th coefficient estimates are significantly negative at the 5% level. In addition, the seventh, eighth, ninth and 12th estimate for λ are significantly negative at the 1% level.

The instrument set is relevant on basis of Cragg-Donald F-statistics in range of 18.489 - 44.556. The Hansen test for too many over-identifying restrictions cannot be rejected in any case at the 10% level. Finally, the Ljung-Box test with four lags for a white noise error is rejected in eight out of 13 specifications. According to GGL, this may be due to the underlying Calvo specification, which may not capture all dynamics of the data generating process.

The results of Table 1.1 indicate that the specified model in equation (1) does not work for the data set of Europe. This is so because the estimates show a negative relation between inflation and the output gap. A negative coefficient on the output gap implies that an increase in unemployment would lead to an increase in output relative to long term trend output. It is easy to see this is simply not viable.

Table 1.2

United States: GMM Estimates of the ‘new' Phillips-Curve, sample period: 1976Q1 - 2010Q2 Estimated Equation: t Ett1yt 

Instruments: Constant, lags 1 – 5 of _t, lags 1 – 4 of y , 1 lag of _t MS and 2 lags of _t Comm_t

Specification Estimate of β Estimate of λ

for Ett1 (St. Error) (St. Error)

VAR(5) _t, y , _t C , _t ULC_t 0.987 (0.020) 0.011 (0.027)

VAR(4) t, y , t C , t IRt 0.980 (0.020) -0.060 (0.025)**

VAR(3) t, y , t ULC , t IR , t MSt 0.981 (0.019) -0.086 (0.025)***

VAR(5) t, y , t C , t FCI , t ULCt 0.989 (0.019) -0.024 (0.026)

VAR(3) t, y , t C , t FCI , t ULC , t IRt 0.983 (0.019) -0.068 (0.026)**

VAR(4) _t, y , _t C , _t ULC , _t IR_t 0.983 (0.016) -0.058 (0.026)**

VAR(3) _t, y , _t C , _t ULC , _t MS_t 0.981 (0.020) 0.001 (0.030)

VAR(3) _t, y , _t C , _t FCI , _t MS_t 0.972 (0.016)* -0.037 (0.028)

VAR(3) _t, y , _t FCI , _t ULC , _t MS_t 0.978 (0.016) -0.043 (0.025)*

VAR(5) _t, y , _t FCI , _t ULC , _t IR , _t MS_t 0.974 (0.018) -0.062 (0.029)**

VAR(1) _t, y , _t C , _t FCI , _t IR , _t MS_t 0.975 (0.021) -0.041 (0.034) VAR(3) t, y , t C , t FCI , t ULC , t MSt 0.980 (0.017) 0.000 (0.029)

VAR(4) t, y , t C , t FCI , t ULC , t IR , t MSt 0.960 (0.021)* -0.033 (0.034)

Notes: All data is quarterly where,_t is inflation, y_t is the output gap, C_t is consumption, FCI_t is a futures commodities index, ULC_t is a unit labor cost index, IR_t is the short term nominal interest rate and MS_t is the money supply. Standard errors are calculated under a heteroscedasticity and autocorrelation consistent

covariance (HAC). Testing for H0: β = 1; H1: β ≠ 1 and H0: λ < 0; H1: λ > 0. *, ** and *** indicate statistically

4.2 Comments and interpretation of results: United States

The results in Table 1.2 are similar to those of Europe in Table 1.1. For the United States, two out of 13 estimates for β reject the null hypothesis of a β equal to one at the 10% level. On the basis of this result, the discount factor for the United States is on average equal to one. Also similar to Europe, the output gap coefficient estimates in Table 1.2 are overall (significantly) negative. Three out of 13 estimates appear positive, however statistically insignificant. Basically, the model does not work for Europe and the United States.

In order to review the potential weakness of the instruments, the regressions of Table 1.2 report Cragg-Donald F-statistics between 22.035 and 33.347. In effect, the instruments are not extraordinary strong, however satisfy the condition of an F-statistic > 10. The J-test cannot reject the over-identifying restrictions, again in support of the specified model. Finally, in each instance the test for a white noise process is strongly rejected (i.e. at the 1% level in almost all cases).

On the basis of the results of Table 1.1 and 1.2, the null hypothesis of a discount factor equal to one cannot be rejected. Thus forward looking price setting by firms is highly relevant in the context of the NPC in the United States and Europe. Nevertheless, the model specified with the HP-filtered output gap does not work. The negative relation between inflation and the output gap is inconsistent with theory. My results are therefore in line with the findings of GGL and NN. A possible explanation for the negative output gap coefficients is that

‘detrending’ of actual output data results in a significant loss of information inherent in the data. An implication of this explanation is that it is impossible to conclude whether the HP-filter in particular or ‘detrending’ in general is the source of the significant information loss.

Table 1.3

Canada: GMM Estimates of the ‘new' Phillips-Curve, sample period: 1976Q1 - 2010Q2 Estimated Equation: _t E_t_t1y_t 

Instruments:

Constant, lags 1 – 5 of t, lags 1 – 3 of y , 1 lag of t MS , 2 lags of t Comm and 2 lags of t IRt

Specification Estimate of β Estimate of λ

for E_t_t1 (St. Error) (St. Error)

VAR(5) _t, y , _t C , _t ULC_t 0.951 (0.014)*** 0.002 (0.011)

VAR(4) _t, y , _t C , _t IR_t 0.965 (0.015)** 0.018 (0.010)**

VAR(3) _t, y , _t ULC , _t IR , _t MS_t 1.006 (0.017) 0.014 (0.011)*

VAR(5) t, y , t C , t FCI , t ULCt 0.983 (0.014) 0.022 (0.009)***

VAR(3) t, y , t C , t FCI , t ULC , t IRt 1.001 (0.017) 0.012 (0.010)

VAR(4) _t, y , _t C , _t ULC , _t IR_t 0.952 (0.016)*** 0.013 (0.010)*

VAR(3) _t, y , _t C , _t ULC , _t MS_t 0.978 (0.014) 0.005 (0.010)

VAR(3) _t, y , _t C , _t FCI , _t MS_t 0.980 (0.015) 0.019 (0.011)**

VAR(3) _t, y , _t FCI , _t ULC , _t MS_t 0.976 (0.016) 0.016 (0.010)*

VAR(5) _t, y , _t FCI , _t ULC , _t IR , _t MS_t 0.941 (0.016)*** 0.010 (0.011)

VAR(1) _t, y , _t C , _t FCI , _t IR , _t MS_t 0.999 (0.017) 0.004 (0.012)

VAR(3) t, y , t C , t FCI , t ULC , t MSt 0.980 (0.015) 0.027 (0.010)***

VAR(4) _t, y , _t C , _t FCI , _t ULC , _t IR , _t MS_t 0.997 (0.018) 0.016 (0.011)*

Notes: All data is quarterly where,_t is inflation, y_t is the output gap, C_t is consumption, FCI_t is a futures commodities index, ULC_t is a unit labor cost index, IR_t is the short term nominal interest rate and MS_t is the money supply. Standard errors are calculated under a heteroscedasticity and autocorrelation consistent

covariance (HAC). Testing for H0: β = 1; H1: β ≠ 1 and H0: λ < 0; H1: λ > 0. *, ** and *** indicate statistically

4.3 Comments and interpretation of results: Canada

In contrast to the results of Europe and the United States, the estimated relation between inflation and the output gap for Canada is significantly positive in eight out of 13

specifications. In four cases, the one-sided test of the null hypothesis of λ < 0 can be rejected at the 5% or 1% significance levels. The magnitude of the estimated output gap coefficients are very similar across specifications for Ett1 (i.e. in the range of 0.002 – 0.027) and the same applies to the discount factors (i.e. in the range of 0.941 – 1.006). Concerning the discount factors, four estimates allow a rejection of the null hypothesis of a β equal to one. This leads to the conclusion that on average, the discount factor does not differ significantly from one. Moreover, the two cases where the output gap coefficient is significantly positive at the 1% level, the discount factor is not significantly different from one.

The weak instrument statistics are in the range of 22.761 – 30.548, where the specifications with a significant output gap coefficient at the 1% level have statistics of 26.323 and 24.990. It thus appears that the instruments are marginally relevant and this may point to a weakness of the estimated results. The null hypothesis of too many over-identifying restrictions cannot be rejected in any case on basis of Hansen’s J-test (eg. p-values in the range of 0.110 – 0.392), however the hypothesis of a white noise error is strongly rejected in each case (i.e. uniformly at the 1% level).

Given the positive estimates of the output gap coefficient, the results of Canada allow for an interpretation of the magnitude of output gap coefficient. The interpretation follows from the model by Calvo described in Appendix A1. The relevant equation in this respect is given by,

 









where θ denotes the probability (or fraction) of firms receiving a signal, which enables a price change and φ is a measure of the firm’s supply elasticity. All remaining terms are defined as in equation (1) and β is assumed to equal one for simplicity. Note that equation (5) reduces to equation (1) when λ is defined as  

 









t

  

  2/1 _.