The Fisher’s hedge hypothesis: what about homogeneity and stability properties?

Abstract

The purpose of this study is to see if the Fisher’s hypothesis validation is robust in year or/and countries dimensions. We investigate whether nominal or real stock market returns are hedged against inflation rate, so as to determine the appropriate time and markets to invest in (from the 32 countries) over a period covering the 2008 global finance crisis (GFC) and the Covid 19 outbreak. Hedging property is found to be homogenous within countries and stable in time. Using either nominal or real return, based on cross-sectional data results, Fisher’s hypothesis is generally validated with a few exceptions, while the time-series based results show that the hedge property is robust only in some countries. Using time series data (cross section data), in terms of homogeneity (homogeneity and stability), there is no difference between hedge property between Euro and non-Euro countries (groups of countries or between sub-periods) for both periods covering either 2008 GFC or the Covid 19 outbreak. Robust results are also the outcome of panel data investigations with or without the interest rate role as macro control variable.

Citation

Neifar, M. and Harzallah, A. (2024), "The Fisher’s hedge hypothesis: what about homogeneity and stability properties?", Journal of Derivatives and Quantitative Studies: 선물연구, Vol. 32 No. 4, pp. 344-370. https://doi.org/10.1108/JDQS-07-2024-0028

1. Introduction

In the context of the relationship between stock market returns and the inflation rate, the generalized (Fisher, 1930) Fisher’s hypothesis implies a positive link between nominal stock returns and the inflation rate and the independence between real stock market returns and the expected inflation. In such a situation, the stocks offer a protection or hedge against inflation (Jaffe and Mandelker, 1976) [1]. According to Fisher (1930), assets that represent claims to physical or real assets, such as stocks, should offer a hedge against inflation.

The relationship between inflation rate and stock returns has been a subject of extensive empirical study by several financial economists around the world. The literature on Fisher hypothesis verification is conflicting. Some category of these studies showed a significant positive relationship between inflation rates and stock market (supports the hypothesis), some found a significant negative relationship and others found no significant relationship between the two variables.

In view of the wide range of conflicting empirical studies on how inflation rates affect the stock returns, one cannot draw conclusions about the reasons: is it because of the type of the data? (time series vs cross section or panel data), or because of the period of study? (pre-during and post-crisis as the 2008 GFC and the Covid 19 outbreak), or because of the used Fisher hypothesis definition? (in terms of nominal vs real return), or because of the considered countries? (Euro vs non Euro, or developed vs emerging, single vs group of countries, etc …).

Recently, a few authors have conducted investigations based on group of countries; panel data (Neifar and Hachicha, 2022; Salisu et al., 2019, 2020; Halit, 2016). But, each of these references does not consider comparative analysis by country groups.

This paper attempts not only to fill this gap but also to make a comparison of results based on three types of data: cross-sectional, time series and panel data. More precisely, this paper will verify the homogeneity and the stability of Fisher hedging hypothesis properties. Practically, this study will verify robustness results from Fisher’s (1930) hypothesis verification over time (covering two crisis: the 2008 global financial crisis (GFC) and the Covid 19 outbreak; pre, during and post the crisis), over countries or group of countries (European vs non-European countries, or emerging vs developed countries), as well as over year-country dimension.

The year dimension (with time series data) will concern the verification of the hedge robustness in two ways:

• (1)

  Results based on nominal returns (R) are compared with ones based on real returns (RR),

• (2)

  Results based on actual inflation (INF) are compared with ones based on expected (EINF) and unanticipated inflation (UEINF),

From the homogeneity side via two groups of countries (Euro and non-Euro countries).

The Fisher hypothesis will be tested using monthly data during 2000:01 − 2019:01 for the 2008 GFC case and during 2016:06–2024:04 for the Covid 19 outbreak. The verification of this hypothesis requires the decomposition of the current inflation rate into the unexpected and expected inflation rate using the so-called Hodrick–Prescott (HP) filter (Hodrick and Prescott, 1997).

Besides the hedging robustness verification, as done in the year dimension, the country dimension (with cross-section data) concerns in addition the homogeneity vs heterogeneity investigations of financial market behavior in the group of emerging vs developed countries. Moreover, for stability question, robustness verification concerns also three sub-periods pre-during and post-considered crisis (2008 GFC or Covid-19).

For the cross-sectional data creation, each time series will be reduced to a point which sum up the information in time by a point estimation; the time average for each country. With cross-sectional data, estimation will be made over three sub-periods; a first one for pre-crisis [(2000 − 2007) for the financial one and (2016:06–2019:11) for the Covid one], a second one for the crisis period [(2008 − 2009) for financial one and (2019:12–2022:06) for the Covid one] and a third one for post-crisis [(2010 − 2019) for financial one and (2022:07–2024:04) for the Covid 19 one].

Using time series or cross-section data, estimation will be carried out for two groups of countries: Group I (European vs non-European countries) and Group II (emerging vs developed countries). Static linear simple and multiple models will be estimated by Quasi-Generalized Least Squares (QGLS) or robust least squares.

The year-country dimension (with panel data) will particularly concern robustness check from previous robust results once macro-economic (namely interest rate) impact is taken into account.

Based on panel data, random effect model versus fixed effect model choice will be based on Hausman test. For more robustness investigation, some augmented models with the interest rate as control economic variable are considered.

Besides the robustness check of all results, we seek to determine which markets and at which point in time investors are mostly interested to invest in.

The text begins with an introduction. Section 2 provides a brief review of the literature. Section 3 presents the data analysis after description, creation and the sources. Section 4 gives several appropriate models for our study. Section 5 discusses in a first steps the finding for the baseline models in two dimensions (years and countries) for the period covering the 2008 GFC. Then, in a second steps, we propose three extensions for more robustness check: (1) to take into account more recent data covering the Covid 19 crisis, (2) the use of panel data for simultaneous check of year stability and country group homogeneity, (3) as well as the check the control variable (interest rate) impact. We conclude in Section 6.

2. Literature review

A vast literature on the link between equity returns (real or nominal) and the inflation rate (real, expected and/or unexpected) is available among the oldest economic and financial studies. The ability of financial assets returns to protect investors from inflation is not a new topic in both theoretical and empirical literature.

Numerous studies have empirically investigated the ability of stock returns to hedge inflation in both the developed and the emerging stock markets. Some studies found evidence of partial hedge for stock returns against inflation. Evidence of stock returns providing hedge against inflation at a given time periods and not in another periods also exist in the literature.

Most of the older studies are based on time-series data, with the exception of Gultekin (1983), which is based on both time-series and cross-section data.

From time series data side, three type of results can be summed up (see Table 1 for more details):

• (1)

  Some papers have found a positive relationship between stock returns and inflation (Boudoukh and Richardson, 1993; Alagidede and Panagiotidis, 2010; Owusu-Nantwi and Kuwornu, 2011; Oprea, 2014; Tiwari et al., 2015; Bampinas and Panagiotidis, 2016; Hau, 2017, …);

• (2)

  Others have found that stock prices and the inflation rate are negatively related, and then the Fisher hypothesis is not verified (Bodie, 1976; Fama and Schwert, 1977; Gultekin, 1983; Chatrath et al., 1997; Zhao, 1999; Eldomiaty and AboulSoud, 2020; Chiang, 2023);

• (3)

  Some others have found mixed results (Jaffe and Mandelker, 1976; Spyrou, 2004; Rushdi and al. 2012 …).

Gultekin (1983) examined the Fisher hypothesis in 26 countries from January 1947 to December 1979 and found that for most of the considered countries the relationship between these two variables is negative (Fisher hypothesis is not valid).

Some recent studies have used both time series and panel data (Salisu et al.,2019, 2020; Neifar and Hachicha, 2022). Salisu et al. (2019) found that both gold and palladium protect against inflation in OECD countries. The use of panel data in Salisu et al., (2019) is for the purposes of checking the robustness of their results. Salisu et al. (2020) in their investigations found that the equities become a good hedge against inflation only post the 2008 GFC. Subsequently, based on panel data Neifar and Hachicha (2022), showed that stock returns are hedged against inflation only during the 2008 GFC for the case of three developed countries (Canada, Suisse and the UK).

Studies on the relationship between stock market returns (nominal or real) and the inflation rate (real or anticipated) are numerous (see Table 1). However, to our knowledge, apart from Gultekin (1983) who tested Fisher’s hypothesis with cross-sectional data, all the literature considers (rarely) the time series (group of countries; panel) data in testing Fisher’s hypothesis.

For the Fisher’s hypothesis verification, this paper is the first which will consider both the years and countries dimensions as well as the year–country dimension to test homogeneity and/or stability of financial market behavior in European versus non-European countries, as well as in emerging versus developed countries covering two crisis: the 2008 GFC and the Covid 19 outbreak. Three types of data will be investigated: time series, cross section and panel data to see if hedging property is robust from different specifications.

3. Data analysis and processing

3.1 Data description and variable creation

The collected data are the consumer price indices (CPI), the stock market indices (SP) and the interest rate (IR). Sources of data are indicated in Table 2.

The considered sample of n=32 countries (for which data are available from January 2000 to January 2019 covering the 2008 GFC crisis and from June 2016 to April 2024 that cover the Covid 19 outbreak) will be gathered in two groups of countries (that as given in the 2021 IMF classification):

• (1)

  The first group includes 19 European countries and 13 non-European countries.

• (2)

  The second group involve 25 developed countries and 7 emerging countries.

The conventional considered variables; mainly the stock market returns and the inflation rate are defined and given in Table 2 (Panel A). Some other variables are useful in this paper: the real return, the expected inflation and the unanticipated inflation rates that are computed as shown in Table 2. Each variable computation and the sample of studied countries are given in Table 2 (Panel B).

To get rid of the time factor, and in order to obtain cross-sectional data, each variable Yit of country i, t=1, …, T, can be estimated by a point in the space; it is the mean of the T observations for the variable, which will be noted by Y̅i, for each country i=1,…,n=32.

3.2 Graphic analysis

Before starting modeling the return–inflation relationship, we give the graphical relationship between them in order to detect the direction of any association (positive or negative). We also seek to build an idea about the stability of these relationships. Several graphs are constructed in different spaces and different range of periods.

In the country dimension, we try to distinguish the behavior of individuals from different groups of countries (developed vs emerging or European vs non-European). In the time dimension, we are looking to see whether any association between variables remains stable from one sub-period to the next, particularly in relation to the 2008 GFC.

The graphical representation of the linear relationship between nominal yields and the inflation rate for Euro vs non-Euro countries and for emerging vs developed countries for the global period (2000 − 2019), the pre-financial crisis sub-period (2000 − 2007), the financial crisis sub-period (2008 − 2009) and the post-financial crisis sub-period (2010 − 2019) is illustrated in Figures 1–4, respectively.

We note a priori that for the overall period (2000 − 2019) and for the sub-period (2000 − 2007), the Fisher hypothesis is verified on average for both groups with a greater positive slope (β>0) for non-euro countries (group I) and for emerging countries (group II); see Figures 1 and 2. However, for the financial crisis sub-period (2008 − 2009), we find that Fisher’s hypothesis is on average not verified for both groups, since the slope is negative (β<0), see Figure 3. On the other hand, post the financial crisis (2010 − 2019), the slope becomes positive for both groups, and is higher for Euro countries (group I) and developed countries (group II). Then, on average the Fisher hypothesis is verified; see Figure 4.

4. Methodologies for the Fisher hypothesis testing

There are three possible definitions of inflation hedging (Bodie, 1976). The definition, which has been used in almost all empirical studies of inflation hedging, states that an asset is an inflation hedge if its real return (RRt) is independent of the rate of inflation (INFt), which imply a positive correlation between the nominal return (Rt) and inflation.

The traditional view that nominal rates of returns should move one-to-one with inflation is first attributed to Fisher (1930). A correlation of 1 is called a perfect hedge since price increases are perfectly compensated by the corresponding hedging asset returns. According to this hypothesis, rational individuals are concerned with real return on investment. If an asset does not provide a perfect hedge, a stable positive return–inflation relationship can still make the asset valuable.

In this paper, the most empirically applied Fisher hypothesis will be tested in a first step with two types of data. In the first subsection, we consider time series data based models, while in the second subsection we consider cross-section data based models (with either RRt or Rt and INFt or EINFt). Then, for further robustness check, we consider panel data based models.

4.1 Time series data based models

In this section, four specifications will be proposed. In each specification, we formulate the Fisher hypothesis which supposes some conditions (hedge property) on the appropriate marginal effect (slope coefficients).

The first baseline model is the following static linear relationship between nominal yield (Rt) and inflation (Gultekin, 1983):

(1)Rt=α+βINFt+ε1t,t=1,…,T,

where INFt is the inflation rate, α and β are real parameters, β is the marginal effect of the inflation rate and ε1t is the error term (ε1t ∼ (0, σ2)).

From Eq. (1), by definition, if the slope

• (1)

  β=1, the nominal returns (Rt) are fully hedged against inflation,

• (2)

  If 0<β<1, the nominal returns are partially hedged against inflation, and while

• (3)

  If β>1, the nominal returns are highly hedged against inflation.

The second baseline specification is the static linear relationship between real return (RRt) and inflation rate which can be presented as (Jaffe and Mandelker, 1976):

(2)RRt=α1+β1INFt+ε2t t=1,…,T,

where α1 and β1 are real parameters, β1 is the marginal effect of the inflation rate and ε2t is the error term (ε2t ∼ (0, σ2)).

Then, from Eq. (2), if the slope β1=0, the real returns (R Rt) are hedged against inflation.

According to Fisher (1930), the nominal interest rate is made up of the “real” rate plus the expected inflation rate. Thus, he argued that the expected real interest rate depends on real factors (such as the productivity of capital and the time preference of savers) and is independent of the expected inflation rate.

As previous empirical research on this topic, here we test the generalized Fisher hypothesis, which states that the market is efficient and that the expected real return on common stocks and the expected inflation rate vary independently so that, on average, investors are compensated for changes in purchasing power. The generalized Fisher hypothesis can be expressed within the following third baseline regression:

(3)RRt=α2+β2EINFt+ε3t,t=1,…,T

where EINFt is the expected inflation, α2 and β2 are real parameters, β2 represents the marginal effect of the expected inflation and ε3t is the error term.

Then, from Eq (3), if the slope β2=0, the real returns (R Rt) are hedged against expected inflation.

In the framework of the (Fama and Schwert, 1977), the Fisher hypothesis can be expressed in two joint hypotheses; stock market efficiency, and the independence of real returns from expected inflation (Chatrath and Song, 1997). By decomposing inflation into expected and unexpected inflation, Eq. (3) can be augmented as the following fourth model (Chatrath and Song, 1997):

(4)RRt=α3+β3EINFt+β4UNEINFt+ε4t,t=1,…,T,

where EINFt and UNEINFt are respectively the expected and the unexpected inflation rates, α3, β3 and β4 are real parameters, β3 and β4 represent the marginal effect of expected and unexpected inflation respectively, and ε4t is the error term (ε3t∼(0,σ2)).

From Eq. (4), by definition, if the slope

• (1)

  β3=0, the real returns (R Rt) are hedged against expected inflation, and

• (2)

  if β3=β4=0, the real returns are perfectly hedged against expected and unexpected inflation, which supports the Fisher hypothesis (Chatrath and Song, 1997; Adrangi et al., 2000).

Each equation will be estimated by ordinary least square (OLS) if the model is classical (in particular, if εjt ∼ i.i.d.(0,σ2),j=1,2,3,4). Otherwise, in the case of correlated errors [2], the quasi-generalized least squares (QGLS) method will be applied [3] (Neifar, 2011; Greene, 2018). The quality judgment of each regression will be based on the R² value. The significance of the parameters will be based on Student's t-test [4].

4.2 Cross-sectional data based model

Based on cross-sectional data, the relationship between nominal yields and the inflation rate (INF) is shown in Eq. (5), while the relationship between real yields and the expected inflation rate (EINF) is illustrated in Eq. (6). Eq. (7) shows the relationship between real yields and the expected and unexpected inflation rate as follow. We begin by

(5)Ri̅=α+βINF̅i+v1i,i=1,…,n

where

R̅i=∑tTRitT,INF̅i=∑tTINFitT,

which are respectively the average of nominal stock returns (Rit) and the average of inflation rate (INFit), α and β are real parameters and v1i is an error term (v1i ∼ i.i.d.(0, σ2)).

From Eq. (5), as in Eq. (1), the fully, partially or the highly hedge properties depend respectively on if β=1, 0<β<1 or if β>1.

We consider then

(6)RR̅i=α+β1 EINF̅i+v2i,i=1,…,n

where

RR̅i=∑tTRRitT,EINF̅i=∑tTEINFitT,

which are respectively the average of real stock returns (RRit) and the average of expected inflation rate (EINFit), α and β1 are real parameters and v2i is the error term (v2i ∼ i.i.d.(0,σ2)).

As in Eq. (2), from Eq. (6), if the slope β1=0, then the real returns are hedged against inflation.

And finally, we give

(7)RR̅i=α+β1EINF̅i+β2UNEINF̅i+v3i,i=1,…,n

where

UNEINF̅i=∑tTUNEINFitT,

which represents the average of unanticipated inflation rate, α,β1 and β2 are real parameters and v3i  is the error term (v3i ∼ i.i.d.(0,σ2)).

By definition, from Eq. (7), if the slope

• (1)

  β1=0, then real returns are hedged against expected inflation. And,

• (2)

  if β1=β2=0, then real returns are perfectly hedged against expected and unexpected inflation, which supports the Fisher hypothesis (Gultekin, 1983; Adrangi et al., 2000).

Again, each equation will be estimated by OLS for classical models (in particular, if vji ∼ i.i.d.(0,σ2),j=1,2,3). Otherwise, in the case of heteroscedasticity problems, the robust least squares method will be applied. Again, the quality question of each regression will be based on the value of the R² and the significance of the parameters will be based on Student's t-test (Neifar, 2011; Greene, 2018).

5. Empirical results

The Fisher hypothesis verification are carried out within two types of data and several models. In the first subsection, we consider time-series data case, while in the second we consider cross-sectional data investigations. A third sub-section is deserved to more robustness check. Basely, we use panel data covering the Covid 19 outbreak period for several countries (for which the hedge or no hedge properties are already found robust) within FE or RE linear static models to check the role of the interest rate as macroeconomic control variable.

5.1 Results based on time series data for the period covering the 2008 GFC

To check the validity of the Fisher hypothesis, Eqs. (1)–(4) are estimated for the case of Euro countries. When these equations are applied to non-Euro countries data, they are instead indicated by Eqs. (1)'–(4)'. To avoid autocorrelation error problem, all these models are estimated by the quasi-generalized least squares method [5].

Results for the period covering the 2008 GFC crisis are summed up in Table 3. For simplicity, Red color is used for the rejection case of Fisher hypothesis (indicated by NO), while green color is used for the non-rejection case of Fisher hypothesis (indicated by YES). Details of these results are available upon request.

Looking at Table 3’s results for the period covering the 2008 GFC crisis, we conclude that:

• (1)

  From columns (1) and (1)', there is no hedge against the inflation rate and the Fisher hypothesis is not verified except for the USA and Canada.

Based on Eq. (1) and (1)', the Fisher hypothesis (β>0) is verified only for the USA and Canada (with high coverage for both countries). These results are consistent with those of Boudoukh and Richardson (1993), Choudhry (1998), Spyrou (2004), Owusu-Nantw and Kuwornu (2011), Oprea (2014), Hau (2017). Findings therefore affirm the ability of the USA and Canada stock markets, as a long-term investment avenue, to protect investors against inflation in the long-run. As a result, investors are fully compensated for increases in the general price level through corresponding increases in nominal stock market returns and the real returns remain unaffected.

• (2)

  From columns (2) and (2)', there is a hedge against the inflation rate and the Fisher hypothesis is verified except for some countries.

Based on Eqs. (2) and (2)', Fisher’s hypothesis (β1=0) is supported except for some European countries (namely Estonia, Greece, Iceland, Ireland, Latvia and Luxemburg) and some non-European countries (namely Bangladesh, China, Chile, India, Malaysia and Turkey) whose inflation rate have a negative significant effect. These results are in line with those of (Bodie, 1976; Chatrath et al., 1997).

• (3)

  From columns (3) and (3)', there is a hedge against the expected inflation rate, and the Fisher hypothesis is verified except for some countries.

Based on Eqs (3) and (3)', real returns on European stock markets are hedged against the expected inflation rate (β2=0), with the exception of Finland, Hungary, Iceland, Latvia and Poland (associations are significantly negative). The majority of non-European real returns are hedged against the expected inflation rate, except for Chile and Turkey (where the relationship between real returns and the inflation rate is significant and negative), Japan and Malaysia (the relationship is positive but insignificant).

• (4)

  From columns (4) and (4)', there is no value perfectly hedged against expected and unexpected inflation with some exceptions.

Based on Eqs (4) and (4)', for European countries, the real yields of Austria, Germany, Belgium, Denmark, Spain, France, Italy, Portugal, the UK and Russia are perfectly hedged against expected and unexpected inflation (β3=β4=0). However, for non-European countries only the real returns of Brazil, Canada, Colombia, Indonesia, Mongolia and the USA which are perfectly hedged against expected and unexpected inflation.

There is a consistent lack of positive (insignificant) relation between nominal (real) stock returns and inflation (expected inflation) in some of the considered countries. For each of these countries, we recommend then to government to ensure a more realistic price level and to the monetary policies to find inflation stabilization that can be beneficial to the investors.

Moreover, robust time series data based results are found only in some cases:

• (1)

  The USA and Canada for which stock markets are hedged against inflation (Almeida et al., 2024), and

• (2)

  Iceland, Latvia, Chile, Malaysia and Turkey for which stock markets are not hedged against inflation (Geetha et al., 2011)

In the period covering 2008 GFC that is well known by deflation period.

Using time series data, in term of homogeneity, there is no difference between hedge property between Euro and Non Euro countries. This result may or not be true in the period covering the Covid 19 outbreak which is well-known by inflation crisis. This robustness check will be done in sub-section 5.3 sub-sub-section A.

5.2 Results based on cross-sectional data for the period covering the 2008 GFC

In this section we test the Fisher hypothesis within Eqs (5)–(7) using the robust least squares method to avoid hetescedasticity problem. The Fisher hypothesis verification will be discussed by group of countries and by period slice (pre − during and post − crises).

Results for the period covering the 2008 GFC are summed up in Table 4. Details of these results are also available upon request. Again for simplicity, Red color is used for the rejection case of Fisher hypothesis (indicated by NO), while green color is used for the non-rejection case of Fisher hypothesis (indicated by YES).

Looking at Table 4 results, we conclude that:

• (1)

  From Eq. (5), mixed results are derived about hedge properties against anticipated inflation, and then Fisher’s hypothesis can be verified.

From the cross-sectional regression Eq. (5), we can conclude that in the case of European countries, nominal stock market returns are highly hedged against inflation (β>1) Pre financial crisis, and that the Fisher hypothesis is verified only during this sub-period. In the case of emerging countries, the Fisher hypothesis is verified post-financial crisis, and nominal stock market returns are highly hedged against inflation (β>1), whereas in the case of developed countries, the Fisher hypothesis is verified pre- and post-financial crisis, and nominal returns are partially hedged against the inflation rate (β<1).

• (2)

  From Eq. (6), it can be seen that there is hedge against anticipated inflation, and then Fisher’s hypothesis is verified.

We can say that Fisher’s hypothesis is verified for both groups of countries (Group I: European vs non-European countries and Group II: emerging vs developed countries), and that the latter’s equities admit hedging against expected inflation (β1=0) pre-financial crisis (2000 − 2007), during the crisis (2008 − 2009) and post-financial crisis (2010 − 2019).

• (3)

  From Eq. (7), there is perfect hedge against expected and unexpected inflation with few exceptions.

The Fisher hypothesis is verified for both groups of countries (Group I and Group II); their equities are perfectly hedged against inflation (β1=β2=0) for all three sub-periods, except for equities in developed countries. The latter are only hedged against expected inflation during the financial crisis.

Moreover, cross-sectional data based results are found to be homogeneous and stable (robust) in almost all considered cases. Indeed, depending on the dependent variable type, we conclude that

• (2)

  From results based on nominal return, robust no hedge property against inflation is found in almost all cases (group I and group II). These results are in line with Geetha et al. (2011), Eita (2012), Naik and Padhi (2012), Khumalo (2013); Uwubanmwen and Eghosa (2015), Emeka and Aham (2016), Bin and Celis (2017), Chiang (2023b), Abdali and Alm (2024),

• (3)

  From results based on real return, robust hedge property against inflation is also found for almost all cases (group I and group II). These findings are consistent with studies of Adam (2010), Antonakakis et al. (2017), Chiang (2023), Chola (2024), Almeida et al. (2024).

Using cross-section data, in terms of homogeneity and stability, there is no difference about hedge properties between groups of countries or between sub-periods. This result may or not to be true in the period covering the Covid 19 outbreak which is well-known by inflation crisis. Based on time series, cross section and panel data, this robustness check will be done in the following sub-section.

5.3 More robustness check investigations for the period covering the covid 19 outbreak

For stability and homogeneity properties of hedging results verification, three others robustness check are the subject of this section:

• (1)

  Based on time series and cross section data, the first check is about comparison to the outcome from more recent period covering the Covid 19 outbreak,

• (2)

  The second is about the panel data based investigations,

• (3)

  The third is about the check of the interest rate impact as control variable.

5.3.1 Time series and cross-section data investigations

Homogeneity and stability of the hedging property with more recent data covering the Covid 19 crisis period that is well known by inflation period is the subject of this sub-section.

Results based on Eq(1)–Eq(4) (and on Eq(1’)–Eq(4’) for time series data covering the Covid 19 outbreak crisis are summed up at Table A1 (in Appendix). Hence, looking at Table A1, as for the case of the results of the 2008 GFC period (given at Table 3), we conclude that robust results are found only for some countries:

• (2)

  2 Euro countries: Hungary and Chile, and 2 Non Euro countries: Colombia and Malaysia for which stock markets are hedged against inflation.

• (3)

  3 Euro countries: Spain, Iceland, Russia and one Non Euro country: Canada for which stock markets are not hedged against inflation.

In addition, based on time series data, we tend to have the Fisher’s hypothesis verification more often in the period covering the Covid crisis than during the period covering the 2008 GFC (more green zone than red zone in Table A1 compared to Table 3).

As in the case of the 2008 GFC period, we also test the Fisher hypothesis within Eqs (5)–(7) using cross-section data covering the Covid 19 outbreak period. Results are presented at Table A2 (in Appendix).

Now, looking at Table A2 for the cross section data investigations, from the period covering the Covid 19 outbreak results, we conclude that:

• (1)

  As found in some works (Wongbampo, 2002; Gunasekarage, 2004; Sohail, 2009; Geetha et al., 2011; Eita, 2012; Naik and Padhi, 2012; Khumalo, 2013; Zoa, 2014; Uwubanmwen and Eghosa, 2015; Emeka and Aham, 2016; Bin and Celis, 2017; Chiang, 2023b; Abdali and Alm, 2024) once nominal return is used as dependent variable, no hedge property is found for all sub-periods and sub-groups (except for Euro-zone case post Covid period), and

• (2)

  In line with some (Adam, 2010; Tiwari et al., 2015; Antonakakis et al., 2017; Salisu et al., 2020; Almeida et al., 2024; Chiang, 2023; Chola, 2024)’ findings, from the use of real return as dependent variable, hedge property is not rejected for all groups as well as for sub-periods (with few exceptions).

The difference observed in the results when nominal versus real returns were used as the dependent variable can be explained by the fact that during the COVID-19 pandemic, the lack of hedging of stock market returns against inflation observed when nominal returns are used as the dependent variable, compared with existing hedging with real returns, can be explained by several factors. Nominal returns reflect gains in terms of current monetary value and are directly influenced by immediate fluctuations in inflationary expectations and central bank interventions. Massive expansionary monetary policies and economic uncertainties during COVID-19 disrupted this relationship, making nominal returns less predictable in response to inflation (Baker et al., 2020). By contrast, real returns, adjusted for inflation, better isolate the effect of rising prices on purchasing power, revealing a more stable and direct relationship between stock market returns and expected inflation (Fama, 1981; Mishkin, 1992). Investors, seeking to protect their real purchasing power in the face of anticipated inflation, adjusted their portfolios accordingly, which explains why hedging against inflation is more apparent with real returns.

We note also that the coefficients β during the covid 19 period (see Table A2 in the appendix) are less than those during the GFC 2008 crisis (see Table 4). This implies that hedging against the inflation rate is less important during the covid 19 period than during the GFC 2008 crisis (deflation period). This can be explained by the fact that the COVID-19 pandemic, as an exogenous public health shock, rapidly led to containment measures and global economic disruption. Investors initially reacted with a massive sell-off in risky assets, causing high volatility in the financial markets. However, rapid and massive policy responses, including direct transfers to households, increases in public spending and interest rate cuts by central banks, contributed to a relatively rapid recovery in investor confidence (Baker et al., 2020; Shiller, 2020). Inflationary expectations have risen as a result of expansionary policies, leading investors to adjust their portfolios in favor of assets likely to benefit from expected inflation, thereby increasing the hedging of equity returns against inflation (Andrade, 2021; Forbes, 2020).

Hence, based on cross-sectional data, the results are invariant with respect to both considered periods of the study: 2008 GFC and Covid 19 outbreak. Then, we can say that the results from the period covering the Covid 19 crisis are similar to those from the period covering the 2008 GFC.

5.3.2 Panel data models investigation

For the second robustness verification, we use panel data to see if Fisher hypothesis verification results are invariant simultaneously to both dimension year and country.

Eqs (1)–(4) can be rewritten as follow:

(8)Rit=αi+βINFit +ε1it,

(9)RRit=αi1+β1INFit+ε2it,

(10)RRit=αi2+β2EINFit+ε3it,

(11)RRit=αi3+β3EINFit+β4UNEINFit+ε4it,

where t=1, …, T,i=1,…,n, αi, αi1,αi2,αi3 are the individual effects (fixed or random), β, β1, β2, β3 and β4 are real parameters, and εjit∼(0,σj2).

As can be seen from Table A1 for the period covering the Covid-19 outbreak, group of the robust hedged stock returns includes Hungary, Chile, Colombia and Malaysia’ cases, while group of the robust no hedged stock returns includes Spain, Iceland, Russia and Canada’ cases. Hence, two panel data will be studied: one for the group of the countries with hedged stock returns and another for the group of the countries with no hedged stock returns.

Before estimation, correlation matrices for dependent variables are built to forestall multi-collinearity in linear model (11). This matrix is not reported here but are available upon request. Then, from both panel data, only countries for which model (11) does not suffer from collinearity problem are used.

Each of the considered models in Eqs (8)–(11) are estimated by fixed effect (FE) or random effect (RE) technic depending on the Hausman test result. Findings are presented at Table A3 (in Appendix). Now, looking at Table A3, from panel data based models, we conclude that generally hedge or no hedge property is a robust result for each of considered groups with only few exceptions.

5.3.3 Role of the interest rate with panel data

The channel of undesirable inflation impact on investments is explained by Asab and Al-Tarawneh (2018). They highlighted that […] “when inflation is high, monetary authorities respond by increasing real interest rates, thereby raising the marginal costs of business operations. As a consequence, investments fall. For financial intermediaries, they could just find out that the increase in real interest rates is just to the extent of compensating them for the reduction in the value of money due to the high inflation” (Adekoya et al., 2021).

Inflation is in general an increase of price level of goods and services in an economy resulting in a fall in purchasing power or value of money. The inflation hedge of any asset class is usually predicated on the (Fisher, 1930) hypothesis which renders the first attempt to formally state the hypothetical relation between asset returns and inflation. Under this hypothesis, the nominal interest rate is expressed as the sum of real returns and inflation rate.

For the third robustness verification about the role of interest rate, some augmented models will be considered to check if the interest rate can alternate hedge properties among considered groups and sub-periods. Eqs (8)–(11) can then be rewritten as follows:

(12)Rit=αi+βINFit+λ Iit+ε1it,

(13)RRit=αi1+β1INFit+λ Iit+ε2it,

(14)RRit=αi2+β2EINFit+λ Iit+ε3it,

(15)RRit=αi3+β3EINFit+β4UNEINFit+λ Iit+ε4it,

where t=1, …, T,i=1,…,n, αi, αi1,αi2,αi3 are the individual effects (fixed or random), β, β1, β2, β3, β4 and λ are real parameters and εjit∼(0,σj2).

Before estimation, correlation matrices for dependent variables are built to forestall multi-collinearity in linear models. These matrices are not reported here but are available upon request. Then, from the two panel data studied in the previous section, only countries for which considered models do not suffer from collinearity problem are used. Again, each of the considered models in Eqs (12)–(15) are estimated by FE or RE technic depending on the Hausman test result.

Estimation results from Eqs (12)–(15) are summed up at Table A4 (in Appendix). Now, looking at Table A4, from augmented models, we conclude that findings are similar to Table A3’s results. Hence taking into account the role of the interest rate in the augmented models does not impact the hedge or no hedge properties given by models in Eqs (8)–(11) in Table A3.

6. Conclusion

Theoretically, a strong association is supposed to exist between stock returns and domestic inflation rate. Indeed, if the value of money diminishes during high inflation rate, then inflation rate influences stock market risk (diminishes the level of stock returns). From Fisher’s hypothesis, it can be inferred that real assets returns should not move with expected inflation rates. Thus, there should be a positive relationship between inflation and nominal stock returns which provides investors a hedge against inflation.

In stock markets where the Fisher hypothesis holds true, nominal stock prices should fully reflect expected inflation and the relationship between stock returns and inflation should be positive. As a result, the real returns remain unaffected since investors are fully compensated for increases in the general price level through corresponding increases in nominal stock market returns.

The literature on the relationship between inflation and stock returns has been examined by numerous studies. The empirical findings are mixed. This paper contributes to the empirical discussion and has sought to find the relative impact of inflation rate on stock returns not only in time (for stability) but also in country dimension (for homogeneity). Besides the stability and the homogeneity question, Fisher hypothesis was investigated within two definitions (in term of nominal return or real return) for robustness question.

Practically, to test Fisher’s hypothesis, we proceed in two steps:

• (1)

  In the first step, we used two types of data: time series (19 European countries vs 13 non-European countries) and cross-sections (two groups; group I includes European vs non-European countries, and group II includes emerging vs developed countries). The study in this step is based on a sample of monthly data for the period 2000:01 − 2019:01 covering the 2008 GFC. Then, cross-sectional database was built by point estimation of each variable for each sub-period (pre, during and post 2008 GFC).

• (2)

  In a second step, we used 3 types of data for more robustness check. The study was based on a second sample of monthly data for a recent period from June 2016 to April 2024 that covers the Covid 19 outbreak. Then besides cross section data creation (as in the first step), we build panel data for only countries with robust hedge/no hedge properties based on time series data investigations. Then in this step, we used three types of data: time series, cross section and panel data. Moreover, a control macro variable (the interest rate) is used for additional robustness check.

From the first period investigations covering the 2008 GFC, we conclude that:

• (1)

  Using time series data, in term of homogeneity side, there is no difference between hedge property between Euro and Non Euro countries. Robust results are found only for some countries,

• (2)

  Using cross section data, in term of homogeneity and stability, there is no difference about hedge properties between groups of countries (group I of European vs non-European countries and group II of emerging vs developed countries) or between sub-periods (pre-, during and post- 2008 GFC)) in almost all considered cases; Particularly

  • •

    Robust no hedge property against inflation is found from results based on nominal return,

  • •

    Robust hedge property against inflation is found from results based on real return.

From the investigations on the second period covering the Covid 19 outbreak, we conclude that:

• (1)

  Using time series data, robust results are found only for some countries,

• (2)

  Using cross-section data,

  • •

    once nominal return is used as dependent variable, no hedge property is found for all sub-periods and sub-groups (with few exceptions),

  • •

    from the use of real return as dependent variable, hedge property is not rejected for all groups as well as for sub-periods (with few exceptions),

• (3)

  from panel data based models,

  • •

    Hedge or no hedge property is generally a robust result for each of considered groups (with only few exceptions).

  • •

    Taking into account of interest rate role as control variable, there is no impact on the hedge or no hedge properties.

Then, we can say that

• (1)

  Based on cross-sectional data, the results are invariant with respect to both considered periods of the study: 2008 GFC and Covid 19 outbreak.

• (2)

  Based on time series data, we tend to have the Fisher’s hypothesis verification more often in the period covering the Covid 19 outbreak than during the period covering the 2008 GFC.

• (3)

  Based on the nominal return as dependent variable, we tend to reject hedge property (with few exception).

Our study is among the first works to explore robustness hedging question in different dimensions and with three types of data. In light of the empirical findings our results can contribute to academic discourse. Hence, we propose our findings as an initial ground that needs further confirmation in future research.

In this paper, among the macroeconomic variables, inflation and interest rate are considered to be the most crucial factors affecting stock returns. However, our investigations may be subject to the critic that other variables can have significant impact on stock return since stocks may react to some micro-variables related to firm and customer behaviors, as well as to other macro-variables that react to the relationship of the company to the overall economy (Neifar and Harzallah, 2024; Neifar, 2021a, b; Neifar et al., 2021).

It is essential to acknowledge the limitations of our study by using only interest rate as control variable. But, we are confident that our framework will serve as an aide to future researchers in focusing their work on stock return robust hedging properties with larger dataset, longer data frame and additional models to explore other macroeconomic indicators impact like the money supply and the exchange rates (Chiang T. C., 2023b).

Also, besides academics and researchers, for investors [6], portfolio managers, policy-makers and economic decision-makers, our results have crucial policy and practical implications. It is essential for the potential investors within the robust hedged stock markets (namely in Chile, Colombia, Malaysia and Hungary for the period covering the Covid 19 outbreak) to note that the hedging potential of each stock varies across the sub-periods, which should be put into consideration in their investment designs. The portfolio managers in the robust hedged markets should care about deflationary periods (as the 2008 GFC period) since it leads to opposite results. It is necessary for policy makers to put time variation into consideration when modeling returns of stock market viz-a-viz inflation. For the economic decision-makers, when stock markets are hedged against inflation, they need to redirect their attention to other economic policy objectives, such as financial stability, risk management and economic growth stimulation.

Finally, we also have more suggestions for future researchers. Subsequent studies should endeavor to examine if some precious tradable metals (such as gold, silver, platinum, …) can hedge investors against inflation risks, against different macroeconomic risks including exchange rate, oil price and inflation across countries and against economic policy uncertainty.

Figures

Figure 1

Relationship between nominal returns (R) and inflation for different groups for the full period (2000 − 2019)

Figure 2

Relationship between nominal returns (R) and inflation for different groups for the pre-crisis period (2000 − 2007)

Figure 3

Relationship between nominal returns (R) and inflation for different groups during the crisis (2008 − 2009)

Figure 4

Relationship between nominal yields (R) and inflation for different groups over the post-crisis period (2010 − 2019)

Table A1

Sum up of the time series based fisher hypothesis verification results for the period covering the Covid-19 outbreak

Table A2

Sum up of cross-sectional regression results for the period covering the Covid-19 outbreak

Table A3

Robustness of the Inflation-hedging property: the case of panel data covering the Covid-19 outbreak

Table A4

Robustness of the Inflation-hedging property: the role of interest rate with panel data covering the Covid-19 outbreak

Notes

1. According to Fisher (1930) hypothesis, rational individuals are concerned with real return on investment. Hence, the nominal stock returns needed to sustain investors’ confidence in a stock market is the sum of the desired real return plus expected inflation. The real return on investment here implies rate of return that is earned after subtracting inflation rate.

2. The verification of non-auto-correlated errors of order one is based on Durbin–Watson (DW) statistics (Neifar, 2011; Greene, 2018).

3. With the cross-section (time series) data, we can have heteroscedasticity (autocorrelation) problem. Then, regressions are estimated with quasi-generalized lest square QGLS (Hildruth lue procedure or Cochrane − Orcutt procedure) so to get robust Student t-statistics (Neifar, 2011; Greene, 2018). The Cochrane-Orcutt procedure was used in this work.

4. Based on the quasi-generalized least squares (QGLS) results, Student’s t follows asymptotically the Normal distribution. At 5% (10%), significance will be based on the critical point 1.96 (1.64) (Neifar, 2011; Greene, 2018).

5. Student’s t asymptotically follows the Normal distribution. At 5% (10%), significance will be based on the critical point 1.96 (1.64).

6. […]“We identify four theories that explain the correlation between stock returns and subsequent investment. The first says that the stock market is a passive predictor of future activity that managers do not rely on to make investment decisions. The second theory says that, in making investment decisions, managers rely on the stock market as a source of information, which may or may not be correct about future fundamentals. The third theory, which is perhaps the most common view of the stock market’s influence, says that the stock market affects investment through its influence on the cost of funds and external financing. Finally, the fourth theory says that the stock market exerts pressure on investment quite aside from its informational and financing role, because managers have to cater to investors' opinions in order to protect their livelihood” (Morck et al., 1990). […]“According to Reilly and Brown (2006) investment is commitment of funds for a period of time in order to derive a rate of return that will compensate the investor for the period during which the funds are invested, for the expected rate of inflation during the investment horizon and for the uncertainty involved” (Mutua, 2015).

Competing interests: The authors declare that they have no competing interests.

Availability of data and materials: Data is available upon request

Appendix Robustness check results

Table 1

Sum up of a selected literature review

Reference	Period	Type of data	Model/Method	Hedge?
Gultekin (1983)	1947–1979	TS and CS	OLS	NO with nominal returns
Bodie (1976)	1953–1972	TS	OLS	NO with nominal returns
Fama and Schwert (1977)	1953–1971	TS	Simple regression	NO with nominal returns
Chatrath et al. (1997)	1984–1992	TS	OLS	NO with real returns
Zhao (1999)	1993–1998	TS	ARMA; GARCH	NO with nominal returns
Jaffe and Mandelker (1976)	1953–1971	TS	Multiple regression	NO with nominal returns
Choudhry (1998)	1981–1998	TS	Spectral regression method; Semi-parametric; Gaussian method	- YES with nominal returns - NO with real returns
Spyrou (2004)	1989–2000	TS	Simple regression	YES for some countries with nominal returns
Alagidede and Panagiotidis (2010)	Varies from country to country	TS	Johansen’s multivariate method	YES for 3 African countries with nominal returns
Rushdi et al. (2012)	1969–2008	TS	ARDL	YES for the case of expected inflation with real returns NO for the case of observed inflation with nominal returns
Tiwari et al. (2015)	1961–2012	TS	Causality and continuous Wavelet transform	YES with real returns
Bampinas and Panagiotidis (2016)	1993–2012	TS	OLS regression; Quantile regression	YES with nominal returns
Salisu et al. (2019)	2009–2019	TS and PD	Simple regression; Threshold-based regression models	YES
Salisu et al. (2020)	Not specified	TS and PD	Linear regression; Asymmetric model	YES after GFC with real returns
Eldomiaty and AboulSoud (2020)	1999–2016	TS	Johansen cointegration; Granger causality; VECM	NO with nominal returns
Neifar and Hachicha (2022)	1999–2018	TS and PD	ARDL model; MG; PMG	YES during 2008 GFC for the case of three developed countries with nominal returns
Chiang (2023b)	1990–2022	TS	GARCH; - APARCH-M	NO with real returns
Almeida et al. (2024)	1953–2023	TS	Fractional Cointegrating Vector Autoregressive (FCVAR)	YES with real returns
Abdali and Alm (2024)	2000–2023	TS	Linear regression models	NO with nominal returns
Farahani et al. (2023)	2020–2021	TS and PD	Mixed data sampling (MIDAS) regression model	YES with nominal returns
Chiang (2023)	1989–2023	TS	GED-APARCH	NO with both Nominal and Real return
Chola (2024)	2001–2021	TS	Johansen Tests	NO with nominal returns

Note(s): TS: time series and PD: panel data. CS: cross section. No: Fisher’s hypothesis not validated. Yes: the opposite of No. OLS: ordinary least square. ARDL: autoregressive dynamic linear. MG: mean group. PMG: pooled mean group. GARCH: generalized autoregressive conditional heteroscedastic

Source(s): Authors’ elaboration

Table 2

Data description

Panel A: variables
Variables	Descriptions	Sources
Stock market indices (SP)	Price of ordinary shares in companies listed on domestic or foreign stock exchanges	OECD and IMF
Consumer price index (CPI)	CPI is a measure of the overall change in the prices of goods and services consumed	IMF
Interest rate (IR)	Money Market Rate for Russia and Iceland and long run interest rate for other countries	OECD and IMF
	Time Series data	Definitions
Inflation (INF)	It is the difference between the log-transformed CPI at time (t) and time (t−1)	INFt=LCPIt−LCPIt−1
Expected inflation (EINF)	It is calculated using the (Hodrick and Prescott, 1997) filter	EINF = E(INFt|Ωt−1)
Unanticipated inflation (UNINF)	It is the difference between inflation and expected inflation	UNINF =INFt−EINF
Nominal return (R)	It is the difference between the log (LSP) share price at time (t) and time (t−1)	Rt=LSPt−LSPt−1
Real return (RR)	It is the difference between nominal return (R) and inflation (INF)	RRt=Rt−INFt
	Cross-sectional data creation
R̅i	It is the average of nominal yield over time, for i=1, …, 32	R̅i=∑tTRitT
RR̅i	It is the average of real yield over time, for i=1, …, 32	RR̅i=∑tTRRitT
INF̅i	It is the average of inflation over time, for i=1, …, 32	INF̅i=∑tTINFitT
EINF̅i	It is the average of expected inflation over time, for i=1, …, 32	EINF̅i=∑tTEINFitT
UNEINF̅i	It is the average of unexpected inflation over time, for i=1, …, 32	UNEINF̅i=∑tTUNEINFitT

Panel B: sample of countries
Group I: euro vs non-euro countries
European countries	Non-European countries
Austria, Germany, France, Belgium, Denmark, Estonia, Finland, Greece, Hungary, Island, Ireland, Italy, Latvia, Luxembourg, Poland, Portugal, Spain, United Kingdom, Russia	China, India, Indonesia, Japan, Turkey, Bangladesh, Mongolia, Malaysia, USA, Canada, Brazil, Colombia and Chile
Group II: developed vs emerging countries
Developed countries	Emerging countries
Austria, Germany, Belgium, Denmark, Spain, France, Italy, Portugal, United Kingdom, Russia, Estonia, Finland, Greece, Hungary, Iceland, Ireland, Latvia, Luxemburg, Poland, Canada, United States, Chile, Japan, Malaysia and Turkey	Brazil, Colombia, Indonesia, Mongolia, Bangladesh, China and India

Note(s): The (Hodrick and Prescott, 1997) HP filter requires the minimum of ∑t=0T(xt+xt̅)2+θ∑t=0T−1[(x̅t+1−x̅t)(x̅t−x̅t−1)], for θ>0. This method breaks down an x-series into its trend (the expected components) and cyclical components (the unexpected components). OECD: Organization for Economic Co-operation and Development. IMF: International Monetary Fund. The country classification is based on the 2021 IMF classification

Source(s): Authors’ elaboration

Table 3

Sum up of the time series based Fisher hypothesis verification results for the period covering the 2008 GFC

Table 4

Sum up of cross-sectional regression results for the period covering the 2008 GFC